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An Investigation of the nodular microstructure of selected silsesquioxane and epoxy thermosetting resins

University of Florida Institutional Repository
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PAGE 1

INVESTIGATION OF THE NODULAR MICROSTRUCTURE OF SELECTED SILSESQUIOXANE AND EPO XY THERMOSETTING RESINS By KYLE KATHAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Kyle Kathan

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This work is dedicated to my grandparent s William and Shirley Gray of Paulsboro, New Jersey.

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iv ACKNOWLEDGMENTS I would first and foremost like to tha nk my family, my parents Chris and Russ Kathan, my sister Kendra Kathan, my uncle Steven Gray, and my wife Dana for their support and love while pursuing my graduate studies. I would like to additionally thank my a dvisor, Dr. Ronald Baney, who has made graduate school a worthwhile endeavor for me and has helped me through all of the academic challenges I have faced over the past four years. As far as my research goes, I must firs t acknowledge the University of Florida Alumni Foundation for supporting me with a ge nerous fellowship over the past four years, which has allowed me to pursue the research that is of the most interest to me. Additionally I would like to acknowledge the University of Florida McKnight Brain Institute (MBI) for allowing me to use its phenomenal facilities I would like to acknowledge Jim Rocca and Tim Vaught of the MBI for their help with my research. I would like to thank Jim Rocca for all his help with NMR theory and practice and thank Tim Vaught for helping me with my optical microscopy work. Additionally I would like to acknowledge the University of Florida Particle Engineering Research Center (ERC) for its help with my research. Specifically, I would like to thank Dr. Kevin Powers of the ERC fo r guiding my early work when I first started at UF.

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v I would like to thank Dr. Jack Mecholsky of the UF Materials Science and Engineering Department for helping me to better understand mechan ical properties and fractals, topics which consumed a large portion of this research. Lastly I would like to thank the entire Bane y group. It is my feeling that the Baney group is far and away the most academically diverse group in the MSE department with people studying everything from nu clear fuel to bioactive materi als. Four years of the Baney group have proved to be invaluable.

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vi TABLE OF CONTENTS ACKNOWLEDGMENTS.................................................................................................iv ABSTRACT....................................................................................................................... xv CHAPTER 1 INTRODUCTION AND OVERVIEW OF CHAPTERS.............................................1 1.1 Research Introduction.............................................................................................1 1.2 Fractal Basics..........................................................................................................2 1.3 Materials Overview................................................................................................4 1.4 Chapter Outline.......................................................................................................6 2 FRACTALS AND FRACTURE..................................................................................7 2.1 Introduction to Fractals...........................................................................................7 2.2 Fractals in Nature....................................................................................................8 2.3 Fractal Dimension...................................................................................................9 2.4 Mechanical Properties..........................................................................................15 2.5 Conclusions...........................................................................................................18 3 EPOXIES, SILSESQUIOXANES, AND NODULAR STRUCTURE REVIEW......20 3.1 Introduction...........................................................................................................20 3.2 Review of Nodular Epoxies..................................................................................20 3.2.1 Introduction to Epoxies..............................................................................20 3.2.2 Processing and Synthesis Fa ctors Affecting Nodule Size..........................21 3.2.3 Observations of Nodule Size......................................................................24 3.2.4 Mechanical Properties of Nodular Epoxy Resins.......................................28 3.3 Review of Silsesquioxanes...................................................................................31 3.3.1 Introduction to Silsesquioxanes..................................................................31 3.3.2 Applications of Silsesquioxanes.................................................................34 3.3.3 Characterization of Silsesquioxanes...........................................................36 3.4 Conclusions...........................................................................................................41 4 FRACTURE PROPERTI ES OF EPOXY RESIN......................................................42 4.1 Introduction...........................................................................................................42 4.2 Epoxy Synthesis and Processing..........................................................................44 4.3 Methodology and Experimental...........................................................................45

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vii 4.3.1 Characterization of Mechani cal Properties of Epoxy Resins.....................46 4.3.1.1 Introduction......................................................................................46 4.3.1.2 Modulus............................................................................................46 4.3.1.3 Failure stress..................................................................................49 4.3.1.4 Flaw size...........................................................................................49 4.3.1.5 Fracture mechanics and toughness...................................................52 4.3.2 Studies of Nodular Micros tructure and Nodule Size..................................54 4.3.2.1 Scanning electron microscopy.........................................................54 4.3.2.2 Solvent extraction and particle size..................................................56 4.3.2.3 Iodine staining of surfaces................................................................58 4.3.3 Investigation of Fractal Dimensional Increment........................................60 4.3.3.1 Introduction......................................................................................60 4.3.3.2 Flaw to mirror size...........................................................................61 4.3.3.3 Non-destructive slit-island method..................................................64 4.3.3.4 Hand calculations of fractal dimensional increment........................70 4.3.3.5 Calculation of fractal dime nsional increment by Image-Pro............73 4.3.3.6 Comparison of methods of measuring fractal imension...................74 4.4 Structural Parameter a0.........................................................................................75 4.4.1 Calculating a0..............................................................................................75 4.4.2 Relationship of Fractal Dimensional Increment to a0.................................76 4.4.3 Error Analysis of a0....................................................................................78 4.5 Results and Discussion.........................................................................................78 4.6 Conclusions...........................................................................................................80 5 SILSESQUIOXANES: GROWTH, STRU CTURE, AND CHARACTERISTICS...82 5.1 Introduction...........................................................................................................82 5.2 Polysilsesquioxane Synthesis and Processing......................................................84 5.2.1 Polymer Synthesis......................................................................................84 5.2.2 Polysilsesquioxane Monolith Synthesis.....................................................86 5.3 Experimental.........................................................................................................87 5.3.1 Characterization of Polymer.......................................................................87 5.3.1.1 Viscosity...........................................................................................87 5.3.1.2 Matrix assisted lase r desorption ionization......................................90 5.3.1.3 Nuclear magnetic resonance.............................................................92 5.3.1.4 Fourier transform infrared spectroscopy..........................................99 5.3.2 Condensation of Polysilsesquioxanes.......................................................100 5.3.3 Characterization of N odular Microstructure............................................101 5.4 Polymethylsilsesquioxane...................................................................................106 5.6 Conclusions.........................................................................................................107 6 RESULTS, DISCUSSIONS, AND FUTURE WORK.............................................109 6.1 Results and Discussion.......................................................................................109 6.1.1 Calculating Fractal Dimension with Optical Microscopy........................109 6.1.2 Nodular Microstructure of Epoxie s and Fractal Analysis of Failure.......109 6.1.3 Synthesis and Characteri zation of Silsesquioxanes..................................110

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viii 6.2 Future Work........................................................................................................111 APPENDIX A ERROR ANALYSIS OF THE WEST MECHOLSKY PASSOJA THEORY........114 A.1 Introduction........................................................................................................114 A.2 Sources of Error.................................................................................................115 A.3 Error Analysis Equations...................................................................................115 A.4 Error Analysis Graphs.......................................................................................116 B SILICON 29 NMR STUDY OF SILSESQUIOXANES..........................................119 B.1 Introduction and Methodology...........................................................................119 B.2 Si-29 NMR Spectra............................................................................................119 C STUDY OF POLYSILSEQUIOXANES.................................................................126 D MEASUREMENTS OF EPOXY SAMPLES..........................................................128 LIST OF REFERENCES.................................................................................................137 BIOGRAPHICAL SKETCH...........................................................................................145

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ix LIST OF TABLES Table page 2-1. Fractal Dimensional Increment for Common Families of Materials.........................17 3-1. GPC and MALDI Molecular Weights of Silsesquioxanes from Tecklenburg..........37 4-1. Modulus and Percent Error of Epoxy Resins at Different Strain Rates....................46 4-2. Student’s t-test Results of Mo dulus of Different Strain Rates..................................48 4-3. Failure Stress of Epoxy Resi n at Different Strain Rates...........................................49 4-4. Student’s t-test of Fa ilure Stress of Epoxy Resins....................................................49 4-5. Toughness of Epon 825 with 8ph DETA (MPa*m1/2)...............................................52 4-6. Toughness of Epon 825 with 10phr DETA (MPa*m1/2)...........................................52 4-7. Average Critical Crack Size for Epoxy Resins ( m)................................................54 4-8. Fractal Dimensional Increment by Flaw to Mirror Size for Epoxy Resins...............62 4-9. Student’s T-test of Fractal Di mensional Increment of Epoxy Resins.......................63 4-10. Hand Measurements of Fractal Dimension.............................................................71 4-11. Hand Calculations of Fract al Dimension of Epoxy Resins.....................................72 4-12. D* of Epoxy Resins Ca lculated With Image Pro....................................................73 4-13. Comparison of Fractal Dimensional Increment D* Values for Different Methods....................................................................................................................74 4-14. Calculated a0 values ( m) for Epoxy Resins...........................................................75 4-15. a0 Values ( m) calculated from the Slope of the Fractal Dimension vs. Toughness Plot.........................................................................................................77 4-16. Cumulative Error Values for a0 Calculations for Epoxy Resins.............................78 4-17. Calculated a0 Values and Associated Error.............................................................79 5-1. Usage of Common Polysilsesquioxanes....................................................................83

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x 5-2. Synthesized Polysilsesquioxanes and Structures.......................................................85 5-3. Total Hydroxide Cont ent of Silsesquioxanes............................................................95 5-4. Hydroxide Content of Trim ethylsilylated Silsesquioxanes.......................................97 5-5. Ratio of Network to Cage St ructure of Silsesquioxanes from FTIR.......................100 6-1. a0 and Error Values for Epoxy Resins by Strain Rate.............................................110 C-1. Crosslinked Silsesquioxane Compositions.............................................................126 D-1. Modulus Values for Epoxy Samples......................................................................128 D-2. Failure Stress Values for Epoxy Samples...............................................................129 D-3. Flaw Size and Toughness Measurements for Epon 825 with 8phr DETA Strained at 0.1 mm/min........................................................................................................129 D-4. Flaw Size and Toughness Measurements for Epon 825 with 8phr DETA Strained at 10 mm/min.........................................................................................................129 D-5. Flaw Size and Toughness Measurements for Epon 825 with 8phr DETA Strained at 100 mm/min.......................................................................................................130 D-6. Flaw Size and Toughness Measur ements for Epon 825 with 10phr DETA Strained at 0.1 mm/min..........................................................................................130 D-7. Flaw Size and Toughness Measur ements for Epon 825 with 10phr DETA Strained at 10 mm/min...........................................................................................130 D-8. Flaw Size and Toughness Measur ements for Epon 825 with 10phr DETA Strained at 100 mm/min.........................................................................................131 D-9. Mirror Measurements for Epon 825 w ith 8phr DETA Strained at 0.1mm/min.....131 D-10. Mirror Measurements for Epon 825 w ith 8phr DETA Strained at 10mm/min....131 D-11. Mirror Measurements for Epon 825 w ith 8phr DETA Strained at 100mm/min..132 D-12. Mirror Measurements for Epon 825 with 10phr DETA Strained at 0.1mm/min.132 D-13. Mirror Measurements for Epon 825 w ith 10phr DETA Strained at 10mm/min..132 D-14. Mirror Measurements for Epon 825 w ith 8phr DETA Strained at 100mm/min..133 D-15. Image Pro Measurements for Fract al Dimensional Increment by Sample...........133

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xi LIST OF FIGURES Figure page 1-1. Mandelbrot Set, Generated with Fractal Explorer 2.02...............................................3 1-2. Chemical Structure of Epoxy Re sin Monomer and Crosslinking Agent.....................5 2-1. Koch Curve, Five Iterations. Generated with Fractal Explorer 2.02...........................8 2-2. IFS Fractal of Fern Leaf. Ge nerated with Fractal Explorer 2.02.................................9 2-3. Perimeters of Nations by Richardson........................................................................11 2-4. Relationship of D* to Mechanical Toughness...........................................................17 3-1. Polymerization Reactions between Epoxides and Amines........................................21 3-2. Schematic of an AFM Tip on Epoxy Surface............................................................27 3-3. AFM of Fracture Surface...........................................................................................27 3-4. TEM of Epoxy Fracture Surface...............................................................................27 3-5. Fractal Structure of Fracture Surface of Epoxy.........................................................30 3-6. MALDI Molecular Weights of Silsesquioxane Fractions.........................................37 4-1. Stress-Strain Curves of Epon 825 with 8 phr DETA)...............................................47 4-2. Stress-Strain Curves of Epon 825 with 10 phr DETA...............................................47 4-3. Example Critical Crack Size Produced through Slow Crack Growth.......................50 4-4. Flaw Size of Epon 825 with 8phr DETA at 100mm/min Strain Rate.......................50 4-5. Flaw Size of Epon 825 with 10 phr DETA strained at 10mm/min...........................51 4-6. Ln-Ln plot of Strain Rate vs. KIc for Epon 825 Resins.............................................53 4-7. SEM Micrograph of E pon 825 with 8 phr DETA.....................................................56 4-8. SEM Micrograph of E pon 825 with 10 phr DETA...................................................56 4-9. EDS Spectra of Epoxy Staine d with Iodine for 5 minutes........................................59 4-10. EDS SEM Image of Iodine (Red) Stained Epoxy Surface.......................................59 4-11. Flaw Size and Mirror for Epon 825 with 8 phr DETA strained at 0.1 mm/min......62 4-12. Fractal Dimension vs Toughness for Epoxy Resins................................................64

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xii 4-13. Slit-Island Method from Hill...................................................................................65 4-14. 3-D Composite Image of Specimen.........................................................................67 4-15. 2-D Image of Specimen...........................................................................................67 4-16. 0-50 Elevation Image Section..................................................................................68 4-17. 0-100 Elevation Image Section................................................................................68 4-18. 0-150 Elevation Image Section................................................................................69 4-19. 50-255 Elevation Image Section..............................................................................69 4-20. Magnification of Fracture Surface...........................................................................70 4-21. Richardson Plot of Pe rimeter of Fracture Surface...................................................71 4-22. Square Root of Fractal Dimension vs. Toughness Calculated by Hand from Slit Island Contours...........................................................................................................72 4-23. Square Root of Fractal Dimension vs. Toughness Calculated by Image Pro from Slit Island Contours....................................................................................................74 4-24. Comparison of Fractal Dimension In crement to Toughness for Three Different Techniques of Measuring...........................................................................................77 5-1. Comparison of Silicone and Polysilsesquioxane Structures......................................82 5-2. Common Polysilsesquioxane Structures...................................................................83 5-3. Substrate Configurations...........................................................................................86 5-4. Schematic of Ubbelhode Viscometer........................................................................87 5-5. Viscosity of Polymethylsilsesquioxane.....................................................................89 5-6. Bruker Avance 600 Mhz Vertical Bore Spectrometer at MBI.................................94 5-7. NMR Spectra of Synthesized Silsesquioxanes..........................................................95 5-8. Bis(Trimethylsilyl)Acetamide Structure (BTMSAA)...............................................96 5-9. NMR Spectra of Trimethylsilylated Silsesquioxanes................................................97 5-10. Total, Reactive, a nd Non-reactive Hydroxide Content of Silsesquioxanes............98 5-11. FTIR Spectra of Synthesized Silsesquioxanes......................................................100

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xiii 5-12. Polymethylsilsesquioxane Microstructure.............................................................103 5-13. Polyethylsilsesquioxane Microstructure................................................................103 5-14. Polypropylsilsesqui oxane Microstructure.............................................................104 5-15. Polybutylsilsesquioxane Microstructure................................................................104 5-16. Polymethylsilsesquioxane Cluster Structure.........................................................105 5-17. Highlighted Two-Phase Imag e of Polypropylsilsesquioxane................................105 5-18. Cao-Baney Route Polymethylsilsesquioxane........................................................106 A-1. Effect of Variables on a0.........................................................................................117 A-2. Effect of Variables on Error of a0...........................................................................118 B-1. Polymethylsilsesquioxane.......................................................................................120 B-2. Derivitized Polymethylsilsesquioxane....................................................................120 B-3. Polyethylsilsesquioxane..........................................................................................121 B-4. Derivitized Polyethylsilsesquioxane.......................................................................121 B-5. Polypropylsilsesquioxane.......................................................................................122 B-6. Derivitized Polypropylsilsesquioxane....................................................................122 B-7. Polybutylsilsesquioxane..........................................................................................123 B-8. Derivitized Polybutylsilsesquioxane.......................................................................123 B-9. Polyvinylsilsesquioxane..........................................................................................124 B-10. Derivitized Polyvinylsilsesquioxane.....................................................................124 B-11. Polychloropropylsilsesquioxane...........................................................................125 B-12. Derivitized Polych loropropylsilsesquioxane........................................................125 D-1. Flaw Size of Epon 825 with 8phr DETA Strained at 0.1 mm/min.........................133 D-2. Flaw Size of Epon 825 with 8phr DETA Strained at 10 mm/min..........................134 D-3. Flaw Size of Epon 825 with 8phr DETA Strained at 100 mm/min........................134 D-4. Flaw Size of Epon 825 with 8phr DETA Strained at 100 mm/min........................134

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xiv D-5. Flaw Size of Epon 825 with 10phr DETA Strained at 10 mm/min........................135 D-6. Flaw Size of Epon 825 with 10phr DETA Strained at 100 mm/min......................135

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xv Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN INVESTIGATION OF THE NODULAR MICROSTRUCTURE OF SELECTED SILSESQUIOXANE AND EPO XY THERMOSETTING RESINS By Kyle Kathan December, 2005 Chair: Ronald H. Baney Major Department: Materials Science and Engineering The role of the nodular microstructure in the failure of thermosetting resins is unknown. The goal of the research presented he re is to determine the mechanisms by which the nodular microstructure aff ects failure of thermosetting resins. Epoxy samples were fractured in tension a nd characterized for fractal dimension, nodular size, and toughness. Samp les of epoxy resins have b een shown in this work to have a relationship between nodular size and toughness. Two compositions were chosen to study the effect of strain rate and nodule size on toughness and a structural parameter, a0. Additionally fractal dimensional increm ent was measured using two separate techniques, Flaw to Mirror ratio (F-M), and Slit-Island contours. The fractal dimensional increment for slit-island contours was calcula ted using two similar algorithms; by hand, and with the aid of software. It was shown through the course of this work that nodule size does not directly relate to toughness. Additionally, it was al so shown that the epoxy resins characterized

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xvi for this research are not structurally homoge nous. It is believed that the inhomogeneities in the structure can be related to a0. Silsesquioxanes were synt hesized using a novel two pha se reaction. Samples were characterized for molecular weight and struct ure using matrix assi sted laser desorption and ionization (MALDI) and silicon-29 nucle ar magnetic resonance (Si-29 NMR). Additionally samples were char acterized during growth usin g an Ubbelohde viscometer to characterize the growth processes of the two phase reaction. It was found that as the temperature of the growth phase synthesi s increases, the viscos ity of the polymer decreases; this is contrary to what one w ould normally expect in polymer condensation reactions. This result is due to immiscible two phase nature of the reaction process. Additional experiments were performed to inquire about the ro le of the organic group on nodular size. It was hypothesized that nodule size is dependent on the difference in surface energy between the nodule and the free energy of the so lution. Organically different silsesquioxanes and different solvents were used to investig ate this hypothesis.

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1 CHAPTER 1 INTRODUCTION AND OVERVIEW OF CHAPTERS 1.1 Research Introduction Thermosetting resins are polymeric materials that react or cure with the addition of heat or energy. Thermosetting resins are ge nerally very brittle and do not melt after curing. Research has indicated that some thermosetting resins such as epoxies and silsesquioxanes do not have homogenous microstructures, as once thought. Epoxies and other thermosetting resins have been shown to have a nodular microstructure. (Duchet et al 2003) A nodular microstructure can be loosely described as one containing spherical feat ures of uniform size in a rand omly packed order. Nodules can range from tens of nanometers to a few micrometers, depending on the system investigated and the processing used in synthe sis. It is of interest to understand how nodules influence the toughness and mechanic al properties of epoxies and other thermosetting resins. Previous research has in dicated that there is a possible correlation between nodular size and ultimat e mechanical properties. (Baney et al. 1999) Using conventional thought, it is not entirely obvious as to what role nodular size plays in the toughness of thermoset resins. By applyi ng a new philosophy of failure in brittle materials, this dissertation research ha s sought to find a better understanding of how nodules affect toughness. Epoxies and other thermosets are unique in that they can be considered a hybrid between ceramics and plastics. They are po lymeric in nature and have a low modulus compared to ceramics, yet they fail in a brittle manner, similar to glasses and crystalline

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2 materials. Recently, researchers have shown that the toughness of brittle materials can be linked to a structural parameter, a0, by relating the fracture surface of a specimen to a fractal model. The WMP theory, de veloped by West, Mecholsky and Passoja, (Mecholsky et al. 2002), indicates that toughness of a brittle material can be determined from the following equation: (1) 2 1 0D a E KIc wherein KIc is the toughness, E is Young’s Modulus, a0 is a structural parameter, and D* is the fractal dimensional increment. This theorem will be re viewed in depth in Chapter 2. It was my hypothesis that this work could show a clear relationshi p between the nodular size and the structural parameter a0. Ideally this work would be used as a guide to help better understa nd what microstructural factors alter the toughness of thermosetting resins. By understanding exactly how nodule size affects the toughness of an epoxy, it should be possible to engineer a system with increased toughness. Additionally this work would serve to increase the body of knowledge to which the WMP theory has been applied. 1.2 Fractal Basics Fractals are a class of comple x geometric structures that are similar on all length scales. In other words, fractals show essentia lly the same structural features, regardless of magnification. While often very comp lex, fractals can often be described very simply. On the most fundamental level all fractals can be describe d using three terms, the initiator, the generator, and the rule. Basi cally the initiator is the structure that starts off a fractal, the generator is the change in th e initiator with each su ccessive iteration, and the rule is how the generator is applied to th e initiator. Additionally, fractals can also be characterized by their non integer dimension, D. Fractals are not like normal geometric

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3 structures. A simple geometric structure woul d have a topological dimension of 2 if it was a flat surface or 1 if it was a line. Fr actals however, have dimensions greater than their topological dimension. An example woul d be the surface of an orange. Ideally one would say the surface area is equivalent to 4 r2 where the exponent, 2, is the topological dimension for a surface. However, upon closer inspection one would see that the surface of an orange is full of pits and valleys and e xhibits a very irregular surface. The surface of an orange is in fact fractal and has a surface area equal to 4 rD, where D is the fractal dimension and is between 2 and 3. The fractal dimensional increment, D*, as used in the WMP theory, is the decimal portion of the fractal dimension. The fractal dimension can be described as a quan tification of the complexity of a given fractal st ructure and has been used to quantify everything from basic fractals to coastlines of nations. (Richardson 1961) Figure 1-1 Mandelbrot Set, Genera ted with Fractal Explorer 2.02 Fractals have been applied to everything from art to science. Figure 1-1 is an example of a common fractal known as th e Mandelbrot Set and was generated with Fractal Explorer 2.02. (Arthur 2005) Recent res earch has used fractals as a method of

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4 quantifying the nature of brittle fracture in materials. It has long been established that the fracture surface of a material is self similar, wh ich is much like a fractal. A great deal of research has been conducted defining the fract ure surface in fractal terms. Researchers have applied fractal basics to fracture to determine the relationship of microstructure on toughness. (Borodich 1997; Borodich 1999) Current research into application of fractals to fracture toughness has focused on glasses and cr ystalline materials. In previous work it has been demonstrated that fractal dimensi on and toughness of a material are related by the structural parameter, a0. (West et al. 1999; Mecholsky et al. 2002) Values for a0 are typically on the order of lattice parameters. This present work e ndeavors to expand on current knowledge by seeking out materials which should have a larger value of a0. For the purpose of this work, nodular materials have been investigated under the assumption that a0 will be a function of the nodular size. Two separate materials have been investigated, epoxy and silsesquioxanes. 1.3 Materials Overview Epoxy materials were originally thought to be homogenous. However work in the late 1970’s and early 1980’s, primarily by K outsky and associates, demonstrated that epoxies are not homogenous. It was reported that epoxies have a nodular microstructure, in which the nodular size is very uniform a nd closely packed together. (Racich et al. 1976) Additionally, it was found that the si ze of nodules depends on the amount of crosslinking catalyst used and the curing sc hedule employed. (Mijovic et al. 1979) Also of note, it was found that the mechanical prop erties of said systems are dependent on nodular size. (Mijovic et al. 1979 ) Figure 1-2 is the chemical structure of the monomers employed in this research.

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5 H2N N H NH2 OO O O Figure 1-2. Chemical Structure of Epoxy Resin Monomer and Crosslinking Agent. Top: Diethylenetriamine, Crosslinking Agent Bottom: Diglycidyl Ethers Bisphe nol-A (DGEBA), Epoxy Resin Monomer Silsesquioxanes are a technologically importa nt inorganic polymers. (Baney et al. 1995) Most work with silsesquioxanes has fo cused on developing materials to be applied to integrated circuits as a lo w-k dielectric interconnect laye r. Silsesquioxanes are also used as fillers and thin films in other a pplications. Traditiona lly, silsesquioxanes are synthesized through a direct condensation reaction from chlorosilanes and water and generally have the chemical formula RSiO3/2 where R is virtually any organic group. Depending on the nature of silsesquioxane mo nomer, many possible structures can be formed. Most commonly investigated silses quioxanes have been reported to be a large network polymer, although ladde rs and polyhedral structures can also be found. These structures will be covered in Chapter 3. (Ban ey et al. 1995) Work presented here will focus on a novel process for synthesizing sils esquioxane polymers using two immiscible phases. (Kondo et al. 2000) It is postulat ed that hydrogen bonding organic solvents can be used to guide and control the structure by hydrogen bonding to th e silanol groups of the growing polymer. In effect, this drives the polymer into the organic phase where

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6 condensation reactions are slowed greatly, an d the formation of small cage structures such as the cubical octomer is retarded. 1.4 Chapter Outline Chapter 2 begins a critical review of the applications of fractals to fracture mechanics. An overall view of fractals and th eir applications in science is presented to illustrate the relevance to this work. This chapter includes a detailed review of the topical literature. Chapter 3 is a re view of the materials used in this researc h. This incorporates topics ranging from char acterizing nodularity in epox ies to the synthesis and characterization of silsesquioxanes. Chap ter 4 highlights the relationship between nodular microstructure and toughness of epoxy resi ns. High purity diglycidyl ethers of bisphenol-A, DEGBA, (Epon 825) were cross-li nked using diethylene triamine (DETA). After samples were cured, tensile testing was preformed at a variety of strain rates and test conditions. Chapter 5 focuses on the two-phase process for synthesizing silsesquioxanes. Viscosity studies were c onducted to comprehend the nature of growth of the silsesquioxane polymers. Additio nally, NMR was conducted to better understand the effects of the organic group on structure of the polymer. Finally, samples were crosslinked using tin octoate, a co mmon catalyst in this technolo gy, and different solvents to investigate the effects of solvent/polymer in teraction on nodular growth. Additionally an appendix was prepared to discuss er ror analysis of the WMP theory.

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7 CHAPTER 2 FRACTALS AND FRACTURE 2.1 Introduction to Fractals The term fractal was coined by Mandelb rot to describe a set of objects with seemingly infinite detail. (Mandelbrot 1977) Fractals are complex geometric structures that can be related to many scientific and natural phenomena. Fractals are self-similar or self affine and scale invariant. Self-similar structures are the same in all directions in all magnifications. Self-affine structures very si milar to self-similar, but differ in one scale in one direction. Scale invariant means th at no matter what magnification or scale a fractal is being viewed at, the structure is fo r all intents and purposes the same. Visually fractals can look extremely complex; however they are often very simply described. Fractals are often said to possess an infinite level of detail, but ar e often generated by a simple iterative process. All mathematical fractals can be describe d by three terms; initiator, generator, and rule. The initiator is the starting structur e from which a fractal will be formed. The generator is how the structure changes with each successive step. The rule describes how the generator is applied to the structure. Fi gure 2-1 is an example of the Koch curve. The initiator in this case is a straight line; the generator is formed by two lines of equal length that connect at a 60 degr ee angle. The rule for the Ko ch curve states that each straight length of the curve is broken into three separate segments of equal length and the middle segment is replaced with the generator. The generator is scaled to the length of the segment it is replacing.

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8 Figure 2-1. Koch Curve, Five Iterati ons. Generated with Fractal Explorer 2.02 Self-similar objects are ones that have been said to be invariant of scale. Selfaffine objects are very similar to self-similar fractals, but are scaled differently in one dimension, x, y, or z. Fracture surfaces are considered self-affine because the perturbations which form the features of the fracture surface are the same from one region to another but are scaled differently. (Mechol sky et al. 2002) 2.2 Fractals in Nature Fractal structures are commonl y found in nature. Everything from trees and plant life to clouds and weather patterns to the dist ribution of stars and gasses in the cosmos can be described using fractals. (Bailey et al. 2004; Falgarone et al. 2004) Commonly, geological structures ar e defined using fractals. (Volland et al. 2004) Properties such as coastlines and erosion have been related to fractals. (Turcotte 1992 ; Carpinteri et al. 2004) Figure 2-2 is an example of an Iterat ive Functional System (IFS) fractal generated with Fractal Explorer Software. Figure 2-2 is auspiciously similar to a fern leaf and is an example of how natural structures ca n often be described with fractals. Over the past two decades or so, fractals have been used extensively to characterize natural phenomenon. Researchers have looked ex tensively at applying fractal theories to failure, diffusion, and spread of disease and ma ny more fields of research. Additionally, chaos theory has used fractals to mathematica lly describe the apparent random nature of the universe. (Gleick 1998)

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9 Figure 2-2 IFS Fractal of Fern Leaf. Generated with Fractal Explorer 2.02 2.3 Fractal Dimension In Mandelbrot’s famous treatise the question was asked, how long is the coast of England? (Mandelbrot 1967) How long is the perimeter of any country? How long is the Koch Curve for that matter? A simple question on the surface, but depending on what scale is employed, the length is essentially infi nite. The Koch curve begins as a straight line of a defined length from which a fractal is formed. The problem becomes what size ruler does one use to measure th e length of the Koch curve? A smaller ruler will give a larger length because as one measures along the curve, the smaller ruler will see the smaller features that a larger ruler would not. An answer to Mandelbrot’s question is that the length of the coast of England is relati ve, being determined by the person measuring it and the measuring stick they are using. Euclid ean structures such as lines, disks, and spheres have dimensions of 1, 2, and 3 respec tively; however fractals are not so simple. Effectively, fractals are non-Euclidean, as such the dimension must be different than in Euclidean geometry. For Euclidean geometry, the dimension of a line is 1, but in fractals, the dimension is a non-integer number.

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10 The Hausdorff or fractal dimension is often used to characterize a fractal. The Hausdorff dimension is a measure of the number of repeated features in a structure with a decrease in scale or an applic ation of an iteration of the generator to the initiator. The Hausdorff dimension can be de rived from equation 2-1: (2-1) D r C N Where N is the number of objects, r is the level of reduction and D is the Hausdorff or fractal dimension. A relevant example would be the dimensi on of the Koch Curve. Upon inspection one would see there are 3 pieces to the Koch Curve, after applying an iteration, the middle piece is replaced with 2 new pieces ma king 4 pieces total. The fractal dimension of the Koch curve is 1.26. Th e fractal dimension can be tho ught of as a quantification of the tortuosity of a fractal. The fractal dimensi on is the unifying value for which structures or phenomena are characterized and related to a desired property. Lewis Fry Richardson established that ther e is a linear relationship between the measured perimeter of a country and the length of the rule used to measure. (Richardson 1961) Figure 2-3 is a representation of Ri chardson’s work. Richardson’s work was originally intended to explain why nations of Europe went to war by comparing the differences reported by two nations in the leng th of their shared border. Richardson’s work was later adapted for use in materials science and mathematics. The Richardson method measures the peri meter of a complex object or fractal using different ruler lengths. As ruler le ngth decreases, overall perimeter increases. This is because smaller rulers identify small features on the perimeter that a larger ruler

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11 would not see. Richardson s howed that the log of the rule r length versus the log of the perimeter gives a linear relationship. The Richardson equation is given by equation 2-2 (2-2) L(s) = sD+c Where s is the length, D is the fractal dimension and c is a constant Figure 2-3 Perimeters of Nations by Richardson (Richardson 1961) Later Mandelbrot would identif y the slope of the Richards on work to be equivalent to 1-D where D is the fractal dimension. It s hould be noted that in this work the fractal dimension is between 1 and 2. For a line the fractal dimension is between 1 and 2, for a surface, the dimension is between 2 and 3. (Borodich 1999) Rule rs were used in the case of Richardson’s work; to calculate the fractal dimension of surfaces and bodies, a dimension appropriate measuring device must be used such as squares for surfaces. The fractal dimension is a measure of the to rtuosity of a fractal. (Chen et al. 1993) Commonly the fractal dimensional increment is reported. The fractal dimensional increment or D* is the decimal portion of the fractal dimension. Fractal dimensional increment is reported to show correlation betw een different techniques. As previously

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12 mentioned a line fractal has a dimension betwee n 1 and 2, and a surface between 2 and 3. Fractal dimensional increment can allow one to compare line and surface fractals. In recent years the fractal dimensional of a fracture surface has been related to the mechanical properties of a material. (Mande lbrot et al. 1984) As such it has become important to determine methods to measure the fractal dimension of a fracture surface. There are many different techniques for measuring the fractal dimension. Mandelbrot pioneered the use of fractal dimension as a descriptive tool in analyzing fracture surfaces. It has been shown extensively that fracture surfaces for brittle and ductile materials ar e fractal. Many research groups have attempted to relate fractal dimension to mechanical properties (Mecholsky et al. 1991; Chen et al. 1997; Charkaluk et al. 1998) There is some disagr eement as to the rela tionship between fractal dimension and mechanical toughness. Bigerell e has attributed this disagreement to the different methods used to m easure fractal dimension. (Big erelle et al. 2004) Many different techniques have been used to meas ure fractal dimension; slit-island, vertical section, box counting, and projec ted area. These are just a few of the techniques that have been reported in the litera ture, each of these methods can result in a different fractal dimension for the same fracture surface. Hill has reported a protocol for measuring fractal dimension using coast lines of polished fracture surfaces set in epoxy. (Hill et al. 2001) Further work of Hill and Della Bona using the same technique indicated th at fractal dimension is dependent on the contour angle of the sample. (D ella Bona et al. 2001). The protocol described by Hill can be called a slit-island method, which was firs t described by Mandelbrot. (Mandelbrot et al. 1984) Samples are set into epoxy and polished down until islands appear. The islands

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13 are formed by removing the top of a rough s ection on the fracture surface. The perimeter of the islands is measured using several differe nt ruler lengths in a fashion in accordance with the Richardson method. The slit-island technique has been used exte nsively in the litera ture as a method of measuring fractal dimension for comparison to fracture properties. Bigerelle has laid out certain criteria that the con cept of slit island was deduced from: (Bigerelle et al. 2004) (i) When islands are derived from initial self-affine fractal surface of dimension Ds by sectioning with a plane, their coastlines are self-similar fractals with dimension D = Ds 1. (ii) The relation between perimeter and area is given by the equation: (2-2) ) ( ) ( ) (/ 1 A P RD where R() is a constant which depends on th e choice of the yardstick length, h, used to measure the length along the walki ng path. This equation is only true for self-similar whose perimeter and ar ea are measured in the same way. (iii) When the graph of log( P ) versus log( A ) is rectilinear, the fractal dimension is deduced from the slope. Bigerelle further finds that the slit-island method can be a statistically significant measure of the fractal dimension with prop er choice of ordinates and abscissa. Additionally it was also s hown that the slit-island met hod can produce artifacts when correlating the fractal dimension of a surface to a particular physical process or testing parameter. It was additionally reported by Hill that fractal di mension was dependent on the technique employed. It was found that the slit -island technique gave fractal dimensional increment values between 0.08 and 0.28. The ve rtical profile techni que and indentation technique also reported resulti ng in much lower values.

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14 The slit-island techniques such as th at reported by Hill, produce dimensions between 1 and 2. This method gives an appr oximation to the true fractal surface. Although generally accepted, these values are not true fractal di mensions for the surface. A surface fractal should have a dimension betw een two and three. Joseph and associates used atomic force microscopy to generate a three dimensional image of a fracture surface of epoxy samples. (Joseph et al. 1998) Th e values found by AFM are between 2 and 3 and can be considered to be the true fractal dimension of a surface. The drawback of AFM is that only very small regions are measured, generally less then 100 m2. Conversely, the work of Xie used laser prof ilometery to generate a 3-D map over very large samples, several millimeters on a side. (Xie et al. 1998) The algorithm used by the AFM in the work of Joseph uses a technique similar to the Richardson method for calcula ting dimension of a line. The AFM uses progressively smaller cubes to measure the volume. Conversely the Richardson method uses progressively smaller lines to measure a peri meter. The technique reported by Xie uses a projective covering method where the fractur e surface is broken up into progressively smaller squares. The surface area of the fracture surface is found by measuring the surface area of the portion of th e fracture surface projected in the square. The projective covering method has found much more use in th e literature than AFM. (Stach et al. 2001; Stach et al. 2003; Stach et al. 2003; Stem p et al. 2003). It should be noted that samples characterized using surface projectio n techniques are ofte n considered to be multifractal. A surface is considered multifract al if there are multiple fractal dimensions depending on scale length.

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15 Dubac reported a technique for measuring fr actal dimension usi ng vertical profiles of fracture surfaces. The profile is laid against several grids of squares of different sizes. The number of squares for each size used to cove r the entire trace of the profile is used to calculate the fractal dimension. (D ubuc et al. 1989) Hill ha s reported that using vertical profiles gives vastly different results than the slit-island technique. (Hill et al. 2001) Stach has argued that part of the discrepancy of profile tech niques is that the amount of surface characterized is very low and can no t be descriptive of the entire surface, especially for surfaces that are multifractal or self-affine. (Stach et al. 2003) There is currently no standard method for m easuring fractal dimension of a surface. In general the technique used to measure fr actal dimension is dependent on the author’s preference. Due to the disp arity in techniques and thus di mension of measured surfaces, there is some debate in regards to the relationship of fractal dimension to mechanical properties of material. 2.4 Mechanical Properties Fractal analysis has been used extensively in materials science to relate the geometry of the fracture surface to mechanical properties. (Mecholsky et al. 1991; Lyu et al. 1994; Balankin 1995; Charkaluk et al. 1998; Su et al. 2000; Issa et al. 2003) Many different properties of materials have been related to fractal dimension, including impact energy, fracture toughness, and crack propagation. (Big erelle et al. 2004) The wide range of fractal dimension values, as measured using different techniques, has led some researchers to believe that fractal dimension is merely a universal value or parameter that has no correlation to mechanical properties. Some further argue that fractal dimension is a measure of the amount of plastic deformati on associated with failure of a specimen.

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16 Experimental data has indicated that th ere is a relationship between mechanical properties and fractal struct ures. (Borodich 1999) Re searchers have related the parameter fractal dimension to the toughness and fracture surface of brittle and ductile materials. (Mandelbrot et al. 1984; West et al. 1999; Stach et al. 2001). Mandelbrot’s early work pioneered the study of fractals on fracture surfaces West et al. have shown that surface energy of formation of a fractur e surface for brittle materials is directly related to the fractal dimensional increm ent of a surface. (West et al. 1999) This relationship is give n by equation 2-3. (2-3) 02 1D Ea Where is the surface energy of formation, E is Young’s modulus, ao is a structural parameter and D* is the fractal dimensional increment. Often fracture surfaces and fractals are quantified for the fractal dimension. Mecholsky has shown there to be a roughly linear relationshi p between fractal dimension and toughness for materials of the same family (glasses, ceramics, and crystals). (Hill et al. 2000) Figure 2-4 is a graph pl otting the results of Mecholsky. Table 2-1 is a table of fractal dimensi onal increment for common materials from West. (West et al. 1999) It should be noted that on small scales, fracture surfaces are purely fractal. However it has been reported that the disorder of long range structures gives rise to multifractal surfaces. (Xie et al 1998; Xie et al. 1999; Babadagli et al. 2001; Stach et al. 2001; Carpinteri et al. 2002; Carpinteri et al. 2003; Stach et al. 2003; Stach et al. 2003) Multifractal surfaces are ones where there are more then one fractal dimensions for the same surface.

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17 Figure 2-4 Relationship of D* to Mechanical Toughness(Hill et al. 2000) Table 2-1 Fractal Dimensional Increm ent for Common Families of Materials Class D* Single Crystals 0.07-0.12 Glasses 0.007-0.1 Glass-ceramics 0.06-0.3 Polycrystaline ceramics 0.06-0.35 Polymers 0.2-0.29 (West et al. 1999) The fracture surface of a material can be viewed as being composed of perturbations that are deviati ons from a flat plane. These perturbations are formed by constructive and destructive addition of break ing atomic bonds along the crack plane. It has been shown by several res earch groups that fractal di mensional increment can be related to surface energy of a fracture su rface and toughness. (Mandelbrot et al. 1984;

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18 Issa et al. 2003) The toughness of a materi al can be related to fractal dimensional increment through the following equation: (2-4) 2 1 0 *a D E KIc Where E is the Young’s Modulus, D* is the fractal dimensional increment, KIc is the plane strain mechanical toughness, and a0 is a structural parameter. The parameter a0 has been extensively investig ated by West, Mecholsky, and Passoja. (Mecholsky et al. 2002) They report the use of molecular dynamic calculations to calculate the fractal dimension of silica gla sses and silicon crystals. They report that a0 is a measure of the free volume that is formed by breaking bonds at the crack tip. Therefore a0 can be described as the initiator of the fractal to the D* generator. Using the West Mecholsky Passoja Theory one can relate the fractal dimensional increment directly to the toughness of a material. However a0 can not be directly measured and must be calculated from other meas urements. It is of interest to understand the processing parameters that affect a0; if one were able to control a0, one should be able to control toughness. 2.5 Conclusions Fractals are complex geometric structures which have been shown to have a major role in nature. Many research groups have used fractals to describe the fracture surface of a material. Additionally there is signifi cant evidence that the fractal dimensional increment is directly related to toughness. West, Mecholsky, a nd Passoja have shown that toughness can be related to fractal dimension by a structural parameter a0. The fractal dimensional increment is a meas ure of the fractal nature of a fracture surface. There currently exist several different techniques for measuring fractal

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19 dimensional increment such as slit-island, and projected area. Commonly the slit-island technique is the accepted method of measuri ng fractal dimensional increment of fracture surfaces.

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20 CHAPTER 3 EPOXIES, SILSESQUIOXANES, AND NODULAR STRUCTURE REVIEW 3.1 Introduction Two materials of commercial interest were investigated in this dissertation, epoxy resins and silsesquioxane resins. These materials were chosen because of their microstructure and extensive supportive literature. In th is chapter a review of the relevant literature in regards to bot h materials and nodular microstructure. 3.2 Review of Nodular Epoxies 3.2.1 Introduction to Epoxies Epoxies are organic polymers containing epoxide rings. An epoxide ring is a three member ether ring containing an oxygen and is under strain and thus very reactive. Epoxies are commonly used in composites and st ructural materials. Additionally epoxies are used as adhesives and electrical insulators Epoxies are two-part thermosetting resins that require a catalyst to initiate polymerizati on. The addition of energy and catalyst cure an epoxide resin into a ri gid three dimensional struct ure that does not melt upon reheating. Frequently, amine or anhydrid e catalysts are used to cr osslink epoxy resins. Figure 3-1 is a figure of common crosslinking re actions between epoxide and amine groups. Due to their complexity the microstructure of epoxy resins is not very well understood. Some groups have claimed that epoxies have a nodular micros tructure, while others claim that any asserted proof is purely circumstantial.

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21 Figure 3-1. Polymerization Reactio ns between Epoxides and Amines 3.2.2 Processing and Synthesis Factors Affecting Nodule Size The discovery of an inhomogeneous microstr ucture in epoxy resi ns dates back to the late 1950’s. (Errath et al. 1959) Since that time few research groups have investigated epoxy resins for nodularity. Researchers ha ve generally focuse d on three processing parameters when investigating nodule size; subs trate, cure time/temperature, and resin to cross linker ratio. (Rac ich et al. 1976; Mijovic et al. 1979; Takahama et al. 1982) It has been shown that the nodules are connected by an interstitial material forming a sort of micro gel, although the nature of this inters titial material is un known. (Vanlandingham et al. 1999; Lopez et al. 2002; Kozlov et al. 2004) Conversely, not all research groups are convinced that epoxy resins are nodular. (Dus ek et al. 1978; Duchet et al. 2003) Most notably, the early work of Koutsky et. al. from the University of Wisconsin focused heavily on determining the parameters that influence nodular size. (Racich et al. 1976) This work seemed to indicate a correla tion between the synthesis conditions and nodular size. Koutsky and Racich found that free energy of the surface on which the epoxy was cured had an effect on final nodule si ze. At first, one would assume this would be a result of heterogeneous nuclea tion of the nodule at the interface. However

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22 this does not explain why the bulk has the same nodule size as the surface. Their work used completely different substrates ranging fr om copper to various plastics to change the free energy of the surface. Samples were prepared from Epon 825 resi n and Diethylenetriamine. Nodule size was determined using Transmission Electron Microscopy. Kouts ky and Racich drew eight distinct conclusion s initially quoted belo w: (Racich et al. 1976) 1. There is a definite nodular formation th rough the bulk of the epoxy resins studied 2. The nodules have sizes ranging from 10 to 60 nm 3. At times, the nodules appear to align and fo rm networks either at interfaces or in the bulk 4. Nodules are observed to agglomer ate into larger supernodules 5. Teflon and silicone rubber contact with epoxy inhibits the ap pearance of nodules while copper and glass indu ce large nodules or increase packing density of small nodules 6. No consistent relation has been found among composition or cure, nodule size, and nodule density 7. Slightly cured, soft epoxy resins show what may be interpreted as nodule precursors 8. Some individual nodules surprisingly show a very subtle ridgelike fine structure on both fracture and free surfaces What is not clear from that work, however, is if it is truly the surface energy of the mold that affects nodule formation, or if another surface phenomenon such as crystal structure or difference in physic al properties of the various s ubstrates such as porosity is responsible. A better study would have been to use sili cone molds. The silicone could be easily modified with any one of numerous agents to change the surface energy. Doing this would result in molds or substrates that are similar to one another in terms of structure, porosity and other physical properties, but have different surface energies. To date no one has investigated the eff ects of surface energy modified silicone molds on nodule size.

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23 Additional experiments by K outsky and other researchers focused on the impact of chemical composition on nodule size. (Vanlandi ngham et al. 1999) Results indicate that an increase in cross-linking catalyst results in smaller nodule sizes. This would seem to suggest that nodules form from nucleation events similar to crystals precipitating from a melt. A higher content of cross-linker woul d produce more nucleation events in a given volume of resin. The difference between nodules and crystallization is that the nodules seem to all be the same size and shape, and th ey are connected with a similar material as the nodule itself. However crystallization of a melt can result in a large distribution of grains and irregular shapes. Additionally, the grains im pinge upon one another and there is no additional material connecting them togeth er as in the epoxies It is unclear from this analogy as to why nodules are so perfectly spherical and of constant size. Many of the detractors of the nodule microstructure of epoxies cite the fact that no conclusive proof of a difference between the interstitial material and the nodule, which they believe illustrates that the microscopic evidence for nodules are artifacts of either the imaging system or the manner in which the sample was prepared. (Oberlin et al. 1982; Duchet et al. 2003) Although Pollard confirms this obs ervation by relating the gelation of epoxy the Avrami theory of phase ch ange.(Pollard et al. 1987) Th e Avrami theory appears to hold true until at least the gel point of the resin. Another variable investigated for its in fluence on nodule size of epoxy resins is the processing parameter. Mijovic and Wu ha ve independently shown that mixing and processing conditions has an e ffect on nodular size of the fi nal product. (Mijovic et al. 1985; Wu 1991) Epoxy resins are thermosetti ng and generally need heat to initiate curing. The temperature and length of curing can greatly shape the final properties of a

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24 resin, including the nodule size. Mijovic f ound that by adjusting th e curing schedule, one was able to adjust the presen ce of nodules. Proper contro l of curing resulted in a resin with no nodular structure, but a rougher fracture surface. Wu, on the other hand, studied the conse quences of stirring on nodule size. His work found that greater care in stirring resu lted in smaller nodule sizes, but the mechanical toughness did not change. This is an interesting result. One could infer from this work that a higher rate of stirring mi xes the resin and catalyst better, forming a more homogenous starting material. Using the cr ystallization analogy from earlier, a more homogenous mixture would result in more nuc leation events and t hus more nodules. The reasoning as to why the toughness remains the same is that while the microstructure changes the total cross-link density remains the same. From this, one could infer that nodule growth and size is related to the solu bility of the nodule in the resin/catalyst solution. (Wu et al. 1985) As nodules grow th eir solubility decrease s, which determines the future growth capacity of nodules. The exact nature of nodule growth is unknown however. Researchers have related the growth of an epoxy network to a fractal structure. (Kozlov et al. 2004) Fractals are complex geometric structures that are findi ng tremendous use in characterization of polymerization reactions and growth kinetics. (Schaef er et al. 1986) 3.2.3 Observations of Nodule Size Many researchers believe that epoxy resi ns are inhomogeneous and have a nodular microstructure; conversely numerous others beli eve that the observed structure is merely an artifact. Duchet lists thr ee reasons as to why nodules are not real: (Duchet et al. 2003) 1. Epoxy networks cured with various am ine curing agents, having stoichiometric and nonstoichiometric compositions, have been studied. From electron microscopy observations, no correlation between the nodular structure and the crosslinking

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25 density has been obtained. Moreover, etched uncross-linked polymers such as poly(methylmethacrylate) can present a nodul ar structure similar to that of epoxy networks. Oberlin et al. analyzed more deeply epoxy resin morphology by transmission electron microscopy and f ound that cured epoxy networks based on DGEBA and diamine were homogeneous and remained stable while under study in a clean vacuum. (Oberlin et al. 1982) However, in a poorer vacuum, electron irradiation etches the sample. Progressive ly, nodules about 100 nm in size appear. These investigations show that this nodular structure is not linked to inhomogeneous cross-linking, and the author s have ascribed the nodular structure to the interactions of etchi ng agents with the sample su rface (linear or crosslinked). 2. Fluctuations in cross-linking densities should be put into evidence by scattering methods. Dusek and coworkers (Dusek et al. 1978) observed no difference between scattering curves realized on linear and amorphous thermoplastics and curves realized on cured epoxy networks. Epoxy–amine networks exhibit only one sharp and well-defined relaxation peak related to the glass-transition temperature (Tg). Therefore, there is no physical proof of structural inhomogeneity. 3. Moreover, if inhomogeneities are formed during the curing process, the kinetics, the evolution of the distribu tion of i-mers, the gel point conversion, and so forth should be affected. However, these paramete rs for a nodular structure are quite well described by equations determined by st atistical calculation if a single reaction mechanism and quite homogeneous curing are assumed. This perfect agreement is proof of the epoxy network homogeneity. Many researchers have tried to prove the existence of nodules and investigate the nature of the interstitial or connective ma terial. Most have focused on two separate methods; microscopy such as scanning elec tron microscopy (SEM) and transmission electron microscopy (TEM), and scattering me thods such as Light Scattering and Smallangle X-Ray Scattering (SAXS). Electron microscopy techniques have ofte n confirmed the existence of nodules. Works of Koutsky and Mijovic have all used SEM and TEM to show conclusively that epoxies have a nodular microstr ucture. (Mijovic et al. 1977; Mijovic et al. 1979; Mijovic et al. 1979) However, work of Koutsky and Ulhmann using SAXS did not show any structure at the length scale wh ich nodules have been observed. (Matyi et al. 1980) This has led many researchers to believe they are nonexistent and are surface artifacts of

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26 sample processing. Conversely, the work of Gupta argues that the el ectron beam can etch the surface of epoxy samples and is thus the origin of the nodular structure observed by other groups. (Gupta et al. 1985). Stevens used Light Scattering to show th at as epoxy cures, two phases principally form. Stevens studied two epoxy system s and found different primary phase sizes depending on the system. Because the exact in dexes of refraction of the solid and solvent phases were unknown, true size and distribution could not be calculated. It was, however, estimated that the difference in indices in re fraction was as low as 1%. One could infer that cross-link density and thus modulus is pr oportional to index of refraction. Using this, one could further infer that the difference in modulus between nodule and connective material is very low. (Stevens et al. 1982) Dusek provided an in-depth review of the formation of epoxy networks and potential sources of inhomogeneities. (Dusek 1986) He cited evidence claiming results found in light scattering are due to inhom ogeneities of the epoxy resin without any crosslinking catalyst added and are often larger th en nodules observed with electron microscopy. Dusek goes on further to state that epoxies are hom ogenous and not nodular due to the closeness of gel point of a resi n to the predicted value, as determined kinetically. Many researchers have used at omic force microscopy (AFM) to study the fracture surfaces of epoxies. AFM is a proj ection of the surface and does not give exact nuances on the underlying structure. Figure 3-2 is a sc hematic of an AFM tip on the surface of a nodular material. (Duchet et al. 20 03) Figure 3-3 is an AFM image of an epoxy resin as reported by Akari. (Araki et al. 2002). Conversely Figure 3-4 is a TEM image generated by Koutsky of a fract ure surface. (Mij ovic et al. 1979)

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27 Figure 3-2. Schematic of an AFM Tip on Epoxy Surface. (Duchet et al. 2003) Figure 3-3. AFM of Fracture Surface. (Araki et al. 2002) Figure 3-4. TEM of Epoxy Fractur e Surface. (Mijovic et al. 1979) From Figure 3-2 one can see how an AFM can “miss” data as the tip traverses across the surface. By comparing Figure 3-4 to Figure 3-3 one can clearly see nodules in

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28 the surface; however Akari has reported that those are not no dules, but merely a complex surface structure. (Araki et al 2002) AFM has often been used to disprove the existence of nodular microstructure. With proper pr eparation of a sample surface and test parameters, it has been shown that AFM can be used to map compositional differences on the surface of a material. Researchers have used these techniques to investigate the surface of epoxy fracture surf aces for differences in modulus between nodule and connective material. Duchet used AFM to explore the nature of the interstitial material and to measure differences in modulus between the two pha ses. He reports that the AFM gives a homogenous surface and therefore there are no differences between nodules and interstitial material; this nodul es are artifacts of other imag ing techniques. (Duchet et al. 2003) Additionally, work by Ba i using Small Angle Neut ron Scattering (SANS) to investigate cross-link density indicates th at the cross-link density is homogenous throughout the structure. (Bai 1985). The arguments presented by Duchet above ar e invalid because they assume that the nodule and the connective or interstitial material s are inherently dissim ilar. Currently no one has proven conclusively that the nodule and connective material are significantly different from one another. Many of the tec hniques which can be used to investigate the differences in density or modulus of the tw o phases may not be sensitive enough to see the differences, if any exist. 3.2.4 Mechanical Properties of Nodular Epoxy Resins In general, the toughness of epoxy re sins is between 0.4 and 1.8 MPa*m1/2. (Plangsangmas et al. 1999; Araki et al. 2002) Epoxies are generally thought to be brittle in nature, although at temperatures above the glass transition temperature, there can be

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29 significant plastic deformation at the crack ti p. Studies have shown that epoxies age slightly after initial curing a nd post curing. (Jo et al. 1991) The consequences of aging can result in changes to the mechanical and th ermal properties. The nature of the change depends on the how close the formulati on is to the stoichiometric ratio. Many works have observed that a failure crack in an epoxy resin propagates through the connective material as opposed to through the nodule. (M ijovic et al. 1981; Woo et al. 1991; Wu 1991; Vanl andingham et al. 1999). It is unclear, however, how nodules affect toughness. It has been shown that samples with the same glass transition temperature (Tg) but different nodule sizes have the same toughness. This indicates that it is the cross-link densit y, (which is proportional to Tg) th at is the determining factor of toughness not the nodule size. (Mijovic et al. 19 85) However it has also been shown that nodule size does affect toughness, but in simila r manner to how cross-link density shapes toughness. (Mijovic et al. 1979; Mijovic et al. 1981) Properties are generally at a maximum wh en the ratio of amine to epoxide is stoichiometric. However as noted by Vanla ndingham, amine molecules can aggregate at the surface and result in a localized region that is not at the stoichiometric point and thus has different properties than the bulk. (Van landingham et al. 1999) Mijovic has shown a correlation between toughness and dynamic mechanical properties of nodular epoxy resins. (Mijovic et al. 1979; M ijovic et al. 1981) However it should be noted that mechanical properties do not follow a linea r relationship to nodular size. Instead properties follow a parabolic relationship, first increasing with nodular size and amine content and then decreasing. Peak values ar e found to be around the stoichiometric point.

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30 Like brittle ceramics, when epoxy sample s are loaded until failure, a fractal structure forms on the fracture surface. This fractal structure is characterized by three distinct regions on the fracture surface; mirror, mist, and hackle. Figure 3-5, adapted from Plangsangmas, is a schematic of an epoxy fracture surface. (Plangsangmas et al. 1999) Figure 3-5. Fractal Structure of Fracture Surface of Epoxy. (Plangsangmas et al. 1999) Fractal structures are often characterized for fractal dimension. Fractal dimension is a measure of the tortuosity of a fracture surface. Joseph et al. has shown that AFM can be used to calculate the fract al dimension of epoxy fracture surfaces. (Joseph et al. 1998) Their work has shown that epoxies have a fr actal dimension of approximately 0.26 for all fracture regions (mirror, mist, and hackle); th is indicates that frac ture surfaces are selfsimilar. It is unclear at th is point how fractal dimension relates to nodule size and epoxy resins, but it has been shown that fractal dime nsion can be related to toughness. An indepth review of fract al dimension can be found in Chapter 2.

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31 Additional materials have been shown to have nodular microstructures. Examples of other nodular materials ar e silsesquioxanes and certain po lymer blends. (Lopez et al. 2002; Auad et al. 2003) Nodular microstructures in polymer bl ends are attributed to the insolubility of one phase in another. Sils esquioxanes, however, are similar to epoxies in structure and are also consider ed to be thermosetting polymers; it is unknown as to why they form nodular structures. 3.3 Review of Silsesquioxanes 3.3.1 Introduction to Silsesquioxanes Silsesquioxane materials are hybrid inor ganic-organic materials combining key properties of both ceramic materials and polymer ic materials. Silsesquioxanes are silicon based materials defined as having three bri dging oxygens and a fourth organic group. The classification and ultimate properties of silsesquioxane are dependent on the fourth group attached to the ce ntral silicon. There are virtually limitless possible organic units that can be attached to the central silicon atom This research will focus on the properties of novel polymethylsislesquioxanes synthesi zed from methyltrimethoxysilane and the relationship between microstruc ture and fractal failure. Polymethylsilsesquioxane (PMSQ) has a nodular micros tructure. The microstructure is similar to th at of epoxy resins. (Racich et al 1976) It will be postulated by Dr. Baney that the scale of the nodular microstructure determines the ultimate mechanical properties of the material. Silsesquioxanes are synthesized by sol-gel like reactions. Traditionally, silica solgel reactions use acids or bases to promot e hydrolysis and condensation reactions in precursors. (Brinker et al. 1989) Monomers condense to form very small colloidal particles which then connect together to fo rm the gel network. The two chemical

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32 reactions that typify most sol-gel syst ems are an alkoxy hydrolysis reaction and a hydrolytic condensation reacti on. The typical reactions for silica sol-gel are: O H SiOSi HOSi SiOH and ROH SiOH O H SiOR2 2 Studies have shown the pure silica sol-ge l, derived from tertramethoxysilane or similar compounds can have an inherently fract al nature on the colloidal scale. (Schaefer et al. 1986) The clusters have a mass fract al; essentially the ma ss of the cluster is proportional to rdm where dm is the mass fractal constant. Mass fractal values are easily measured using Small Angle X-ray Scattering (S AXS). (Orcel et al. 1986) This value is not to be confused with D*, a brittle fracture fractal constant used in the WMP Theory. Traditionally sol-gel reactions have very larg e volumes of solvents that must be removed from the final body. The result is an extremely porous network. The combined effects of Ostwald ripening and si ntering to remove the porosity el iminates virtually all of the microstructural evidence of a cluster microstr ucture. The techniques used to synthesize the polymethylsislesquioxane that will be studi ed in this research result in a dense body with zero porosity. As a result the samples do not have to be heat-t reated and the desired structure is not lost. It has been previously es tablished that silsesquioxa ne resins have a nodular microstructure. (Baney et al 1999) The structure of the resin is a result of both intramolecular condensation, where the molecule folds back on itself creating a closed cluster, and intermolecular condensation, wher e two large silsesquioxane clusters connect to form a larger network. The degree of interversus intramolecular condensation is dependent on the size and charge distributi on of the organic group and the processing parameters. A larger more steric organi c group will lead to more intramolecular

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33 condensation and a smaller group will result in more intermolecular condensation. Unique to methylsilsesquioxanes, the very small organic group (methyl) can lead to an extremely high degree of intramolecular conde nsation under certain reaction conditions. This structure is commonly called the T8 cubical octomer, which c onsists of a silica cube with methyl groups on each corner. Several novel techniques have been developed to prevent the formation of cubical octomer and force the structure to form an open network. Many of these techniques focus on complex solvent interactions that serve to control the organic/inorganic nature of the resin, or un ique hydrolysis condensation routes. (Bourget et al. 1999; Takamura et al. 1999; Kondo et al. 2000; Crouzet et al. 2003) The research in this work focuse s on a non-catalytic route, foregoing the traditional use of acid or base in sol-gel chemistry. In addition to the cubical octomer and the random network, a third possible structure can be found in silsesquioxane ma terials. First id entified in phenyl silsesquioxanes, a ladder structure can be fo rmed upon hydrolysis. (Baney et al. 1995) The presence of a ladder structure is charac terized via x-ray diffrac tion (XRD). When a silsesquioxane resin is examined with XRD, two broad halos are formed. One halo corresponds to the size of the silica tetrahedron. The second halo occurs at smaller 2 values, corresponding to the spacing across th e width of the ladder or the X-Si-O-Si-X length. For polymethylsilsesquioxane, this spacing corresponds to approximately 8.14. Computer modeling has long been used in sol-gel reactions to model how clusters form and approximate sizes of the clusters. It should be noted that clusters are not nodules. Generally clusters are less than 1 nm in diameter, nodules, conversely, are generally 10 nm to 1 m in diameter. It is convenient to think of clusters as the building

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34 blocks of nodules. The model most commo nly used is called the Eden model and assumes that monomers will add randomly to available sites on the cluster. A modification of the Eden model, called the Poisoned Eden model, uses model monomers that have been poisoned, or not completely hydrolyzed, blocking gr owth along particular sites in the cluster. This is very similar to the monomers used to synthesize silsesquioxanes in that they are permanently poisoned with only three available reaction sites. The results of this research indicate that high levels of poi soning increase cluster size for the same number of monomers. (Sch aefer et al. 1986) Th e clusters in these studies and in most sol-gel reactions are not to be confused with the nodules which are the focus of this research. Early work on polymethylsislesquioxane has shown nodules to have a radius around 200 nm. It is unknown however, what the substructure of the nodules is at this time, whether it is com posed of small spherica l clusters creating a hierarchy of structure, or if polymethylsi slesquioxane nodules are homogenous and grow continuously from solution to their observed size. It is the belief of the author that when the inherently poisoned methyltrimethoxysilane is reacted in non-catalytic environments, it will result in different structures, which are not necessarily cluster-like, on the nanoscale. This fact will ultimately affect the mesoscale nodules. 3.3.2 Applications of Silsesquioxanes Silsesquioxane materials have nearly as wi de a variety of uses as possible organic groups that can be attached to the silica networ k. In recent years, the drive for smaller, faster electronic devices has helped pushed re search into silsesquioxanes. (Wang et al. 1999; Yang et al. 2001) From a materials scie nce aspect, the most crucial element to making faster, smaller electronics is the interconnect material. This material must have a low dielectric constant, which is necessary to prevent cr oss talk betw een neighboring

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35 wires and increase signal propagation speed. Most of the work in low-k materials revolves around creating controlled porosity f ilms. Using the rule of mixtures, the dielectric constant of the film is a function of the dielectric constant of the material and the porosity. Higher porosity materials have lower dielectric constants because air has a constant slightly above unity. The silses quioxane organic groups help increase porosity and have an inherently lo wer dielectric constant. In addition to having a low dielectric constant, silsesquioxanes are also very thermally resistant in comp arison to traditional carbon based polymers. Depending on the organic group, silsesquioxanes can have de composition temperatures in excess of 550 C. (Fan et al. 2001) This prope rty is desired in a variety of applicatio ns; in particular it is extremely important in the case of elect ronic devices, where the entire package is subjected to multiple high heat treatments during processing. Silsesquioxane resins can also be used as ceramic precursors. Upon pyrolyzation in a non-oxidizing atmosphere, si lsesquioxanes convert into si licon oxycarbide (SiOC). (Babonneau et al. 1994; Bujalski et al. 1998; Eguchi et al. 1 998) Silicon oxycarbide is a high temperature ceramic material that when derived from silsesquioxanes has properties in excess of traditional ceramic processi ng synthesis routes Silicon oxycarbide synthesized from silsesquioxane is non-stoich iometric, having a variab le carbon to silicon ratio. The properties can be engineered by c ontrolling the amount of carbon present in the resin, which is a function of the organic gr oup present in the silsesquioxane. Preceramic precursors such as polymethylsislesquioxane ha ve found a niche in the synthesis of high temperature and toughness materials, and are a novel route to better ceramics through chemistry.

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36 3.3.3 Characterization of Silsesquioxanes The techniques used to characterize si lsesquioxanes most commonly relate structure to a particular parameter. Generally silsesquioxanes are characterized for molecular structure or molecular weight and related to some physical property, whether it be dielectric constant or toughness and so forth. Many different techniques can be used to measure molecular weight of polymers such as Gel Permutation Chromatography (G PC) and Matrix Assisted Laser Desorption Ionization Mass Spectroscopy (MALDI). Howeve r, it has been shown in the literature that GPC is not a viable technique for measuri ng the molecular weight of silsesquioxanes. Tecklenburg has directly compared GPC and MALDI for measurements of a silsesquioxane based polymer. (Tecklenburg et al. 2001) The polymer synthesized by Tecklenburg was not a pure silsesquioxane, bu t contained some siloxane linkages; however the polymer was fundamentally a si lsesquioxane. The synthesized polymer was fractionated using super-critical fluids into 21 distinct fractions. This was done to make measuring molecular weight easier by concentrating similar molecular weight species together. The results of Tecklenbur g are shown in Table 3-1 a nd show that as molecular weight increases, the validity of GPC decreases. He further cites the reason for this is that the disparity in molecular weight is due to the fact that the molecular radius of silsesquioxanes does not follow a normal curve used for calibration of GPC. This is further evidenced by Figure 3-6. Figure 3-6 plots the molecular weight of several fractions as found by MALDI.

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37 Table 3-1. GPC and MALDI Molecular Wei ghts of Silsesquioxanes from Tecklenburg Fraction GPC Mn MALDI Mn 8 3415 g/mol 3778 g/mol 13 9070 12,087 18 24,130 45,434 20 43,770 100,995 21 56,160 124,847 Figure 3-6. MALDI Molecular Weights of S ilsesquioxane Fractions (Tecklenburg et al. 2001) One can see in Figure 3-6 that the dist ribution of molecular weights from one fraction to another overlap one another. Th is is a byproduct of the fractionation method. As previously mentioned super-critical fluids were used to fractiona te this particular polymer. Super-critical fract ionation works by changing the solubility parameters of a super-critical fluid by adjusting the temperatur e or pressure. The changes in temperature or pressure allow one to fractionate a po lymer roughly by molecular weight, which in general is the biggest f actor determining the solu bility of a polymer.

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38 Silsesquioxanes differ in that there are nearly an infinite number of complex three dimensional structures. In many cases there are multiple isomeric conformations of the same molecular weight. Different structur es of the same isomer have different solubilities, as evidenced by the molecular weig ht distributions in Figure 3-6. This can be further related to structure and the inaccuracies of GPC. Gel Permeation Chromatography assumes that every molecule of the same molecular weight will have the same hydrodynamic radius. However with some imagination one can visual a high molecular silsesquioxane molecule that is very condensed forming a small tight cluster or one that is very large forming a giant networ k. Although they have the same molecular weight, the GPC retention times would be vastly different. Not all researchers have used fractionati on when studying molecular weight. Mori et al have use MALDI to characterize synthe sized silsesquioxane pol ymers. (Mori et al. 2004) However it should be noted that the pol ymer studied by Mori was shown to have a much lower molecular weight than that examined by Tecklenburg. Additional researchers have shown similar results when using MALDI to characterize low molecular weight silsesquioxanes. (Wallace et al. 1999 ) As such, fractionation was likely not required to increase sensitivity. (McEwen et al. 1997) Matrix Assisted Laser Desorption Ionization is a very powerful tool for characterizing large molecular weight silses quioxanes. However most of the research reported in the literature using MALDI for characteriza tion of silsesquioxanes has focused on low molecular weight polyhedra l oligomeric silsesquioxane structures (POSS). (Falkenhagen et al. 2003; dell'Erba et al. 2004; Anderson et al. 2005)

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39 MALDI can also be used as a method of investigating structure. For low molecular weight polymers, ther e are generally one or very fe w isomers of an oligomer. Many research groups have developed softwa re for calculating the structure of an oligomer from molecular weight. Wall ace has used MALDI data for several low molecular weight silsesquioxanes to inve stigate the role the organic group has on intramolecular condensation. (Wallace et al. 2000) Overall it was s hown that the nature of the R-group and the nature of the reacti on process greatly affected the degree of intramolecular condensation. Wallace contrasted the ability of NMR a nd FTIR to determine the structure of silsesquioxanes against that of MALDI. Us ing MALDI he was able to determine exact structures, and classified t echniques commonly used to characterize the structure of silsesquioxanes, NMR and FTIR, as semi-quantitative at best. Fourier Transform Infrared Spectroscopy (FTI R), while not able to interrogate the exact structure of a silsesquioxane polymer, ca n be used to investigate the nature of intramolecular binding. There are two IR ac tive Si-O-Si peaks in silsesquioxanes. Lee has identified the tw o peaks as 1120 cm-1 and 1030 cm-1. (Lee et al. 2002) Lee further states that these peaks are the cage and networ k structures respectivel y. Cage structures are generally molecules with high levels of intramolecular condens ation on a short range order. An example would be the polymethylsi lsesquioxane cubical octomer, which is an entirely cage structure. It should be noted however, th at not all research groups are convinced that the 1120 cm-1 and 1030 cm-1 peaks in a silsesquioxane FTIR spectra are separate structural entities. Oh states that absorb ance peaks near 1133 cm-1 are from a Si-O stretch and absorbance

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40 peaks near 1031 cm-1 are due to a Si-O-Si asymmetr ic stretch. (Oh et al. 2002) Conversely the work of both Oh and Lee indicat es that the ratio of the two peaks changes with thermal curing of the investigated material s. This would lead one to believe that at elevated temperatures there is a structural rearrangement of the polymer which results in a change in peak absorbance of the aforementioned peaks. Within limitations MALDI can be used to determine isomeric structure oligomers, and FTIR can be used to semi-quantitatively investigate physical st ructure; conversely, Nuclear Magnetic Resonance (NMR) can be used to determine chemical structure of silsesquioxane polymers. Nuclear Magnetic Resonance is a powerful tool for quantifying the chemical nature of a desired atom in a sample. Silicon29 NMR is commonly employed to investigate the silsesquioxanes. Arkles and Larson re port in a detailed revi ew Silicon-29 NMR peaks of many silicon compounds found in the Gelest Annual Catalog. (Arkles et al. 2004) As previously mentioned, a silicon atom in a silsesquioxane molecule has one organic group and three oxygens bonded to it. Some of these oxygens are bridging oxygens in that they connect two silicon atoms together. Conversely some oxygen atoms are bound to hydrogen atoms forming hydroxide gr oups or silanols. The chemical shift of a silicon atom in a silsesquioxane is dependent on the number of silanols and bridging oxygens. Kondo has used Silicon-29 NMR to investigate the structure of polymethylsilsesquioxane polymers. (Kondo et al. 2000) The polymet hylsilsesquioxane synthesized by Kondo was characterized for the amount of hydroxide present in the

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41 structure. Because silsesquioxanes are polymer s, the peaks of the NMR spectra are broad peaks, not the sharp, narrow peaks commo nly found in smaller structures. Kondo reported a range of peaks for each type of silicon in a silsesquioxane. (Kondo et al. 2000) Nuclear Magnetic Resonance is a very power ful tool for determining the chemical structure of silsesquioxanes and silicon based materials. It has been used extensively to measure the hydroxide content of silsesquioxanes. 3.4 Conclusions While there are dissenting views on the existence of nodules in epoxies, it is the author’s opinion that they are in fact real. Two explanations exist as to why researchers come up with different results re garding nodular microstructures. It is very plausible that not all compositions result in a nodular microstructure. As reported above, chemical composition and processing parameters can gr eatly influence final nodule size of an epoxy. Conversely, it is possi ble that techniques used to investigate stru cture such as SAXS or AFM are not sensitive enough to pick up differences in structure. It is unknown at this time how nodules form or what their role is on mechanical properties. Silsesquioxanes, on the other hand, are complex hybrid inorganic-organic polymers which have also been shown to form nodular structures. The structure of silsesquioxanes is very complex, however different silsesqui oxanes are easy to synt hesize. This should allow one to investigate the na ture of the R-group and steric effects on nodule formation. There are many different tec hniques for characterizing s ilsesquioxanes, although some such as FTIR are not without controversy.

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CHAPTER 4 FRACTURE PROPERTIES OF EPOXY RESIN 4.1 Introduction Analyzing the effects of a nodular microstr ucture on the mechan ical properties of epoxy resins is a novel applic ation of the West Mechols ky Passoja theory (WMP). Should a relationship between nodular size a nd toughness be found, it would be possible to endeavor to engineer an epoxy with a diff erent nodular size and thus different toughness. Mechanical properties have been previ ously related to fractal relationships. (Mandelbrot et al. 1984) The fracture surface of many brittle materials is a self-similar pattern consisting of a mirror, mist, and hackle region. The WMP theory takes the fractal structures found in fracture surfaces one step further by relating the toughness of a sample to the fractal dimension of the fracture surface. The West Mecholsky Passoja theory a sserts that toughness is related to a fundamental structural parameter a0. (Mecholsky et al. 2002) The relationship between toughness and a0 is given by the equation: (4-1) 2 / 1 0) ( D a E KIc Where KIc is the toughness, E is the modulus, a0 is the structural parameter, and D* is the fractal dimensional increment. The fractal dimensional increment is the decimal portion of the fractal dimension. The fract al dimension and dimensional increment are reviewed in detail in Chapter 2.

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43 We have postulated for this work that a0 is proportional to the nodular size and might be akin to the size of the intersti tial volume formed by the packing of nodules together. Scanning electron micrographs of nodular epoxies have revealed a randomlypacked structure. Assuming a narrow dist ribution of nodule size, a randomly packed structure should be 63% nodules by volume, and 36% interstitial materi al. This brings up the question; What is the nature of the interstitial material in relationship to the nodule, and how does it affect the toughness of an epoxy resin? There has been significant research on es tablishing a link between nodular size of epoxy systems and mechanical properties. (M ijovic et al. 1979; Mi jovic et al. 1979) Additionally many researchers ha ve investigated the nature of the nodular microstructure. (Errath et al. 1959) Not all researchers, how ever, are convinced that nodules do exist. (Duchet et al. 2003) It is possible that not all epoxy compositions studied are nodular. This would be one explanation of the discre pancies between multiple works of research. The epoxy composition investigated in this wo rk has been confirmed to have a nodular microstructure and is derived from a met hod of preparation found in the literature. (Racich et al. 1976) One possible explanation for nodule formation is that as the curing reaction begins minor inhomogeneities begin to form in the bul k of the epoxy resin-curing agent solution. This results in a cluster that has a slightly di fferent chemical potential or solubility than the rest of the bulk. Initially this difference in chemical potential or solubility could be a function of the surface energy in the cluster in relation to the free energy of the uncured bulk. The difference in energies would drive the curing reaction at a faster rate at the

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44 surface of the cluster than in the bulk of th e solution. Eventually the nodule would reach a size where the surface energy has to decrease to a point where it no longer dominates the reaction process and nodule gr owth slows greatly, but does not stop altogether. This possible explanation would answer the question of why nodules are so uniform in size as reported by researchers in the literature. Epoxy samples were characterized for thei r toughness, nodular size, modulus, and fractal dimension with the intent of fitting the data to the West Mecholsky Passoja theory. It was expected that the toughness of the epoxy samples will be proportional to the size of the nodules of the sample. It is the author’s hypothesis that nodule si ze can be related to the size of the interstitial volume found be tween nodules. Additional experiments were performed to better understand the nature of the system being inve stigated and inquire about nodules and nodular formation. 4.2 Epoxy Synthesis and Processing Samples were prepared from Epon 825 epoxy resin and cross-linked with Diethylenetriamine (DETA). Epon 825 is a hi gh purity diglycidylether of bisphenol-A manufactured by Shell Chemical, and it has an equivalent mass of 176 grams. An equivalent is defined as the mass of a polym er corresponding to one mole of reactant group, in this case the epoxide group. Di ethylenetriamine is a penta-functional amine curing agent with three amine groups; two prim ary and one secondary amine. DETA has an equivalent mass of 20.6 grams. The st oichiometric ratio of DETA to Epon 825 is 11.7 grams per hundred grams resin (phr). Samples were cured in silicone molds with 8 or 10 phr of DETA, or a substoichiometric ratio. These compositions were chosen to closely follow reported

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45 literature. (Racich et al. 1976) The two co mpositions were chosen because they have been previously shown to be nodular and have different nodule sizes. A sample of Epon 825 resin was weighed and manually mixed with DETA. The samples were then degassed for 30 minutes using a vacuum pump. After removing bubbles, the resin was pour ed into the molds. Samples were cured at 50C for 24 hrs a nd removed from the molds. The molds were fabricated from Dow Corning Silastic T-2 silicone. Samples were produced in a dog-bone shape that conforms to ASTM standard D638. This standard is primarily used for calculating the modulus of polymer resins. Samples were strained at three different strain rates, 0.1 mm/min 10 mm/min, and 100 mm/min. Ten samples were measured for each composition and strain rate. Samples were loaded until there was failure in tension. A laser extensometer was used to calculate displacement and modulus. Modulus was calcu lated as the slope of the stress-strain curve. Optical microscopy was used to calcula te the flaw size of broken samples so that toughness could be calculated. Nodular size was determined using a scanning electron microscope. Additionally fractal dimensiona l increment was calculated using an optical microscopy technique and a derivation of the Richardson method. Additional techniques for measuring fractal dimension were used to corroborate data gathered by optical microscopy. 4.3 Methodology and Experimental Methodology used for investigation can be broken into three distinct tasks; characterization of mechanical propertie s, characterization of nodule size, and characterization of fractal dimension. The following section will detail the methods used for each of these tasks and present a brief di scussion on the results gathered in each task.

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46 4.3.1 Characterization of Mechanic al Properties of Epoxy Resins 4.3.1.1 Introduction Tensile testing was used to break all samples investigated in this research. Tensile testing was selected because of ease of sample preparation and reproducibility of data. As previously mentioned, samples were pr epared in accordance with ASTM standard D638. This standard describes a process of measuring the modul us of a polymeric resin. Samples were testing on an Instron 1122 frame with a 1000 lb load cell. Mechanical vice grips were used to secure the sample. An attached laser extensometer was used to gauge displacement. The nominal gauge lengt h for samples investigated was 12.5 mm. Load was recorded for a given displacement. Failure stress and modulus was recorded from these experiments. Optical microscopy wa s used to determine flaw size. The flaw size combined with the failure stre ss was used to calculate toughness. 4.3.1.2 Modulus As previously mentioned, two compositions and three strain rates were investigated. Modulus was calculated as the sl ope of the stress-strain curve. Ten samples were broken for each data point. Table 4-1 represents the average of all samples for each data point and the associated error of one standard deviation from the mean. Table 4-1. Modulus and Percent Error of Epoxy Resins at Different Strain Rates 8phr 10phr Strain Rate E (GPa) %Err E (GPa) %Err 0.1 1.41 4.9 1.41 3.9 10 1.45 4.8 1.42 3.3 100 1.51 9.2 1.44 1.5 The following figures are examples of th e stress-strain curve generated by the tensile test. Figure 4-1 indicates three s uperimposed stress-strain curves for 8 phr DETA. Figure 4-2 indicates three superimposed stress strain curves for 10 phr DETA.

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47 Stress Strain Curves of Epon 825 with 8phr DETA 0 10 20 30 40 50 60 70 80 90 100 00.010.020.030.040.050.060.070.080.09 StrainStress (MPa) 0.1 mm/min 10mm/min 100 mm/min Figure 4-1 Stress-Strain Curves of Epon 825 with 8 phr DETA) Stress Strain Curves of Epon 825 with 10phr DETA0 10 20 30 40 50 60 70 80 90 100 00.010.020.030.040.050.060.070.08 StrainStress (MPa) 0.1 10 100 Figure 4-2 Stress-Strain Curves of Epon 825 with 10 phr DETA

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48 The curves in the figures are for samples that were close to the mean of all the samples for that strain rate and composition. The values for each specimen measured can be found in Appendix D in Table D-1. The sa mples in Figure 4-1 are #8 for 0.1 mm/min, #6 for 10 mm/min and #3 for 100 mm/min. Th e samples in Figure 4-2 are #3, #6, and #9 for 0.1, 10, and 100 mm/min respectively. A two-tailed Student’s t-test was used to gauge if the moduli for each strain rate were different from one another. The results of this analysis can be found in Table 4-2. Columns 1 and 2 are the moduli being compared The percentages listed in Column P of Table 4-2 indicate the probabilities that the modulus of one strain rate is equivalent to the modulus of another strain rate. It is cl ear from the data in Table 4-2 that for each composition, the moduli for each strain rate are sufficiently different fr om one another. It should be noted that a t-test compari ng the modulus of 0.1 mm/min 8 phr to 0.1 mm/min 10 phr returns a very high probability th at the moduli are the same. However, as strain rate increases the moduli change s is greater for the phr 8 then the phr 10 samples. This indicates that the compositions are different. Table 4-2. Student’s ttest Results of Modulus of Different Strain Rates 8 phr DETA 10 phr DETA 1 2 P 1 2 P 0.1 100 3% 0.1 100 7% 0.1 10 15% 0.1 10 34% 10 100 11% 10 100 12% It appears from the data in table 4-1 that th e modulus, decreases with increasing strain rate for both compositions, which wa s to be expected. Epoxies are brittle polymers; while they fail in a manner similar to ceramics, they still have time and

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49 temperature dependent properties. The change in modulus with strain rate is due to the ability of polymer chains to rearrange and accommodate strain. 4.3.1.3 Failure stress Failure stress ( f) was recorded as part of the tens ile test for each sample. Failure stress was defined as the lo ad under which a sample woul d fail. Table 4-3 lists the average failure stress for all compositions and strain rates tested. Additionally Table D-2 in Appendix D lists the failure stress for each sample tested Although the data listed in Table 4-3 are similar, a t-test implemented co mparing each strain rate with a composition to one another indicates that the failure stresse s are different. The results of the t-tests can be found in Table 4-4. Table 4-3. Failure Stress of E poxy Resin at Different Strain Rates 8phr 10phr Strain Rate f (MPa) %Err f (MPa) %Err 0.1 60 3.9 69 4.3 10 74 10.5 75 17.4 100 74 27.7 64 24.4 Table 4-4 Student’s t-test of Failure Stress of Epoxy Resins 8 phr DETA 10 phr DETA 1 2 P 1 2 P 0.1 100 3% 0.1 100 20% 0.1 10 0% 0.1 10 9% 10 100 47% 10 100 6% 4.3.1.4 Flaw size The flaw size for each sample was measur ed using optical microscopy. For the purpose of this work, a Zeiss Axioplan 2 Micr oscope retrofitted with a custom built LEI XYZ stage capable of 0.1 m was used to image samples. Images were taken at 5x magnification. Images were processed usin g the bundled MCID Elite 6.0 Morphometric

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50 Software package. The following figures ar e examples of flaws found in the samples tested. Figure 4-3 Example Critical Crack Size Produced through Slow Crack Growth Figure 4-4 Flaw Size of Epon 825 with 8phr DETA at 100mm/min Strain Rate

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51 Figure 4-5 Flaw Size of Epon 825 with 10 phr DETA strained at 10mm/min It was often found that that the flaw was produced by slow crack growth. What this means is that the flaw starts out small from some initial defect in the structure and grows outward to a large size with an applied load. When the crack grows to a large enough size, fast brittle failure occurs, producing a mirro r and hackle region. The region of slow crack growth can be seen close up in Figure 43 above. Note the size of the initial defect, 40 by 33 micrometers, and the growth of the flaw to a much larger size. The region of slow crack growth is visually similar to othe r polymers reported in the literature. (Kurtz et al. 1998) It was found that the flaws were asymme tric. As such, an approximation was used to model the flaw as roughly circular. The size of the flaw can be found from the following equation: (4-2) 2 1) ( ab c

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52 Where a and b are one half the dimensions of the flaw and c is the effective radius of the flaw. Tables D-3 through D-8 in Ap pendix D list the flaw sizes of all samples measured in this research and the modeled radius. 4.3.1.5 Fracture mechanics and toughness Toughness was calculated using the modele d flaw size (c) and the failure stress f using the following equation: (4-3) c Y Kf Ic Where KIc is the toughness, f is the failure stress, c is the flaw size and Y is the stress intensity factor. The value of Y depends on the shape of the flaw and location. A surface flaw has a greater Y value than a body flaw. As mentio ned in the previous section, all flaws were modeled to be circular, as such, two Y valu es were used, 1.13 and 1.26. Surface flaws require an approximate 12% correction factor over body flaws. The location and Y value for each flaw for every sample can be found in the last columns of Tables D-3 through D8 in Appendix D. Tables 4-5 and 4-6 are th e toughness for all three strain rates for 8 phr DETA and 10 phr DETA respectively. Table 4-5 Toughness of Epon 825 with 8ph DETA (MPa*m1/2) 0.1mm10mm100mm Average1.34 1.09 1.06 Std Dev 0.28 0.28 0.23 Error 21% 26% 25% Table 4-6 Toughness of Epon 825 with 10phr DETA (MPa*m1/2) 0.1mm10mm100mm Average 1.56 1.28 0.77 Std Dev 0.23 0.35 0.23 Error 15% 27% 29%

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53 The accuracy of the data reported in the above tables can be checked by an Ln-Ln plot. An Ln-Ln plot is a comp onent of the WLF theory. The WLF theory states that for a given temperature, the toughness of a material va ries linearly with stra in rate on a plot of the natural log of KIc vs. the natural log of strain ra te.(Green 1998) Figure 4-6 shows the Ln-Ln plot of the strain rate and temper ature for both compositi ons studied in this research. Ln Ln Plot of Rate vs KIc R2 = 0.94 R2 = 0.99 6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4 -4-20246 Ln Strain RateLn KIc Ln 8 Ln10 Linear (Ln 8) Linear (Ln10) Figure 4-6 Ln-Ln plot of Strain Rate vs. KIc for Epon 825 Resins The high R2 values indicate that the toughnes s values calculated in this study closely match what would be predicted by the WLF theory. It was mentioned in the previous section that the critical crack size is a actually the by product of slow crack growth. The con cept of slow crack growth presents an interesting problem. For both compositions, all samples characterized were made at the same time. As such it is safe to assume that the population of flaws from sample to sample for a particular composition is the sa me. Slow crack growth however, can result

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54 in a change in critical crack size. The follo wing table lists the aver age critical crack size weighted by stress intensity factor Y. Addi tionally, measurements of the initial flaw size revealed an average flaw radius of 20 m. This can be seen in Figure 4-3. Table 4-7 Average Critical Crack Size for Epoxy Resins ( m) 8 phr DETA 10 phr DETA Strain Rate 0.1mm 10mm 100mm 0.1mm 10mm 100mm Average 524 225 233 522 309 191 Std. Dev 191 100 118 141 126 89 Error 36% 45% 50% 27% 41% 47% Utilization of Students t-test reveals that stat istically, the critical flaw sizes for all three strain rates for 10 phr DETA are different from one another. However, for 8phr DETA 10 and 100mm/min were shown to be statistically similar. At higher strain rates, c is a smaller value; conversely, low strain rates result in large critical crack tip sizes. This would indicate that sub-critical crack growth is a strain rate dependent phenomenon. 4.3.2 Studies of Nodular Microstructure and Nodule Size Epoxy samples were characterized for size and nature of nodules. Nodule size was measured with an SEM and the nature of the nodules was investigated using two different techniques. Three separate studies were performed; measurement of nodule size with SEM, extraction of nodules from resins with solvents, and mapping of iodine stain in relationship to nodular structure. The results of the studies presented here can be used to give insight into the nodular mi crostructure of epoxy resins. 4.3.2.1 Scanning electron microscopy The nodular size of our epoxy samples was found with the aid of scanning electron microscopy (SEM). Samples were coated wi th a gold palladium alloy and images were taken at 10 KeV on a JEOL 6400 SEM. Se veral magnifications were used. In the

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55 literature, Transmission Electron Microscopy (TEM) has been extensively used; SEM was selected for this research because of the ease of sample preparation. Additionally, images were taken on both the fracture surface and exterior surfaces of the epoxy samples. It should be not ed that initially a diamond blade saw was employed to investigate the nature of the n odule in the bulk of the sample. Cutting the epoxy with a diamond blade resulted in no ticeable deformation of the nodules and was therefore not found to be a valid method of preparing surfaces for imaging. Nodule size was determined with Image Pro Software. Multiple images from each composition were taken to ensure statistical significance. Samples were prepared by first washing the surface with acetone to remove excess organic matter. Previous work by Koutsky et. al. found that acetone can be used to etch the surface of epoxy and preferentially remove the interstitial material (Mijovic et al. 1979) Care was taken to ensure that samples were not over-etched. Furthermore samp les were exposed to iodine to stain the surface. This was done to increase the average atomic number of the surface. Higher atomic number coatings on SEM samples help to increase the signal-to-noise ratio of an image. Figures 4-7 and 4-8 are typi cal SEM micrographs of Epon 825 with 8 and 10 phr of DETA respectively. Image Pro was utilized to determine an average value for nodule size. The images were loaded into the soft ware and an algorithm was used to identify the nodules. The software allows the user to se t threshold limits for upper and lower size. By setting the lower limit sufficiently high it is possible to remove the embedded nodules from the count produced by the soft ware. Judging visually, the nodule size in the

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56 picture below was estimated to be on average around 100 nm for both compositions. As such a lower threshold for nodule size wa s chosen to be half or 50 nm. Figure 4-7 SEM Micrograph of Epon 825 with 8 phr DETA Figure 4-8 SEM Micrograph of Epon 825 with 10 phr DETA The nodule sizes for the investigated resins appear to be approximately 75nm +/10nm and 100nm +/20nm for 8 and 10 phr respec tively. These values are slightly larger than those found in the literature for similar composition. However, the differences can be explained by taking into account the roles different processing methods, cure schedules, and substrates. 4.3.2.2 Solvent extraction and particle size As previously mentioned, acetone can be us ed to etch the surface of epoxies. Koutsky et al have shown that by exposing epoxy samples to acetone for long periods of

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57 time nodules are actually etched. Additiona lly, evidence indicates that the acetone does not have a significant effect on the nodule, indicating that the acetone is more reactive with the interstitial material than with the n odule. This could indicate that the nodule is more highly cross-linked than the interstitial material. Epoxy samples were treated with acetone for extended periods of time to separate the discontinuous material. It has been reported in the literature by Koutsky that acetone washing does not influence nodule size, but can instead, with extended exposure, wash nodules out from the gel network. (Mijovic et al. 1979) Samples were exposed to acetone for extended periods of time, 24 hrs, 48 hrs, and 2 weeks. The bulk epoxy was removed, leaving a mixture of acetone and particles. The resulting solution was concentrated by evaporating the acetone out at room temperature. This resulted in a white, viscous liquid with no visible nodules. This liquid is a mixture of nodules and unreacted resin monomers. Samples of the acetone-nodule solution were characterized using optical microscopy. No nodules were found using optical microscopy; instead there is featureless residue on the surface of the slide. Recalling that the composition studied here is sub-stoichiometric, it appears that not only are nodules washed out, but so is a small portion of the resin. Further experiments using Soxhlet extraction confir med the presence of unreacted monomer and low molecular weight oligomers. Soxhlet extraction was performed using a standard setup and acetone as the solvent. (Kim 2004 ) After running the extraction for 2 hours, the acetone was evaporated, leavin g behind a viscous, white liquid. The fact that nodules can be washed out of epoxies with a suitable solvent presents some interesting questions. Why do only some of the nodules wash out but not all? As

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58 previously mentioned in the SEM study, nodules have a narrow distribu tion of sizes. It would appear that as nodules grow, the overa ll surface functionality density decreases. This reduction in functionality greatly slows down nodule growth. The ultimate size is a function of the ratio of epoxi de to catalyst group (phr). This nodule size to catalyst relationship has been demonstrated previously in the literature. (Racich et al. 1976) Additionally it has been reporte d in the literature that failure occurs around nodules and not through nodules. This would indicate that nodules are only loosely bound to the surrounding structure. 4.3.2.3 Iodine staining of surfaces One of the arguments against the existence of nodules is that some researchers have postulated that the nodule and interstitial material should have different densities. Small Angle X-Ray Scattering (SAXS) and Atomic Force Microscopy (AFM) have both failed to show any difference in density. In this study epoxy surfaces were stained with iodine in an attempt to elucidate the nature of the nodule and interstitial mate rial. Three samples were investigated in this st udy, no staining, 5 minutes stai ning and 12 hours staining. Samples introduced to the confined iodi ne chamber quickly changed color, from transparent to a brownish colo r. The longer sample s were exposed to iodine, the darker the sample would become. Washing the sample s in acetone after staining would result in slight discoloration of the acetone from excess iodine washing off. It is postulated that the iodine forms a charged transfer comp lex with the carbonoxygen-carbon bonds found in the diglycidyl ethers bisphenol-A (DGEBA) chain. This effectively forms a bond that is stronger than hydrogen bonds but not as strong as covalent bonds.

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59 Figure 4-9. EDS Spectra of Epoxy Stained with Iodine for 5 minutes Figure 4-9 is a representative EDS spect rum for an epoxy sample stained for 5 minutes. From the EDS spectra it is clear th at there is, in fact, iodine present on the surface of the sample. EDS is not sensitive enough though, to give information about the local chemical structure of the iodine that would indicate whether or not the charged transfer complex hypothesis is correct. A dditionally, it was noted that under normal secondary electron imaging, the epoxy sample stained for 12 hours appeared to degrade in real time from electron beam damage. Th is indicates that the sample may have been over-stained and the iodine may have in fact cleaved the C-O-C bonds. Figure 4-10. EDS SEM Image of Io dine (Red) Stained Epoxy Surface

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60 The last figure in this section, Figure 4-10, is an EDS mapping of iodine on the surface of the specimen. The red areas in the image indicate the presence of iodine. There appears to be no evidence of preferen tial segregation of io dine in the nodular epoxy specimen. The EDS spectra and mapping do indicate the presence of iodine in the epoxy sample. However there is no indication that th e iodine prefers to re side in the nodule or interstitial material. It is po ssible that the iodine is too small of a molecule to observe any differences in density as it diffuses through the epoxy. Additionally it also plausible that the difference in density between the nodule and the interstitial material is very small and therefore the difference in concentration of iodine is below the resolution limit of the microscope. 4.3.3 Investigation of Fracta l Dimensional Increment The fractal dimensional increment, D*, is very important in the West Mecholsky Passoja theory. As previously stated the fract al dimension can be related to the tortuosity of the fracture surface. In Chapter 2 methods for calculating fractal dimension and D* were described. These methods are time consuming and in some cases destroy the surfaces being characterized. This research has also focused on developing a technique that is comparable to other methods em ployed for calculating fractal dimension of fracture surfaces that is quick, simple, and non-destructive. 4.3.3.1 Introduction The ractal dimensional increment was calcul ated using three sepa rate techniques. Primarily a novel non-destructive slit-island method was used to generate images from which the fractal dimensional increment could be derived. The images were processed using Image-Pro software. The fractal dime nsion was calculated by hand using a small

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61 selection of images and the Ri chardson method to confirm the validity of the Image-Pro software algorithm. Additionally the ratio of the mirror to the size of the flaw was used as a method to compare the values derived from the non-destructive slit-island method. 4.3.3.2 Flaw to mirror size There are many different techniques for calcu lating fractal dimensional increment. Many of them are time consuming or require a gr eat deal of sample preparation. Fractal dimensional increment can be quickly estimate d by comparing the size of a flaw to the size of the mirror. When a sample fails in a brittle manner, three distinct regions are formed on the fracture surface; mirror, mist, an d hackle. The ratio of the radii for each of the regions is constant rega rdless of size for any material The size of these regions depends on the failure stress. For a given material, a sample that fails at a higher stress will have a smaller mirror than one that fails at a lower stre ss. Additionally the ratio of the mirror size to the size of the flaw can be used to quickly estimate fractal dimensional increment. Fractal dimensional increment can be found using the following equation: (4-3) mr c D Where D* is fractal dimensional increment, c is flaw size and rm is the size of the mirror. The mirror size was measured using optical microscopy. The size of the mirror was compared to the size of the flaw for a given sample as found in Section 4.3.2. Additionally, because the mirrors were often asymmetrical, the radius of the mirror was calculated as the average of two measurements using the same formula to calculate the size of the asymmetric flaws. Figure 4-11 is an example of the mirror and flaw for an epoxy sample where the flaw has the dimens ions 239 by 166 micrometers and the mirror

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62 has a radius from 719 to 664 micrometers. Additionally, example figures for each strain rate and composition can be found in Appendix D Figure 4-11 Flaw Size and Mirror for Epon 825 with 8 phr DETA strained at 0.1 mm/min For each of the strain rates and compositions tested, three samples were measured for mirror size. Table 4-8 lists the average fr actal dimensional increment for each of the compositions and strain rates investigated. Ad ditionally Table 4-9 list the t-test results for comparing fractal dimensional increment distributions. Table 4-8 Fractal Dimensional Increment by Flaw to Mirror Size for Epoxy Resins 8 phr DETA 10 phr DETA 0.1mm10mm100mm0.1mm10mm 100mm Avereage 0.23 0.15 0.15 0.24 0.22 0.11 Std Dev 0.06 0.06 0.04 0.07 0.05 0.05 Error 26% 40% 24% 28% 25% 45%

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63 Table 4-9 Student’s T-test of Fractal Dimensional Increment of Epoxy Resins 8 phr DETA 10 phr DETA 1 2 P 1 2 P 0.1 100 0% 0.1 100 0% 0.1 10 1% 0.1 10 24% 10 100 97% 10 100 0% The fractal dimensional increment seems to follow established trends in the literature. Tougher resins have slightly greater D* values. However for the 8 phr DETA, the fractal dimensional increment appears to be the same for 10 and 100 mm/min strain rate. The Student’s T-test comparing 10 an d 100mm/min strain rate for 8phr DETA returns a 97% probability that the distributio ns are the same. The measurements of the mirror radii, flaw size, and examples of fl aws and mirrors for each sample can be found in Appendix D in Tables D-9 through D-14 and in Figures D-1 through D-6. It should be noted th at this method is merely an estimate and cannot be used to solely describe the fractal dimension of a surf ace. The ratio of th e flaw to mirror size can be shaped by residual stresses in the sa mple. It is unknown at this time if there are any residual stresses in the sample, however it is unlikely. Additionally, operator error can play a large role in calculated values. Where the operator defines the start of the hackle region and sample geometry can affect final values. Additional experiments using different techniques should be conducted to confirm the results of Table 4-7. Figure 4-12 plots the square root of th e fractal dimensional increment versus the toughness of epoxy resins. This type of plot has been commonly used in the literature to show a relationship between fractal dimens ional increment and toughness. The R2 value is rather low, which could indicate that the fl aw to mirror size ratio is not necessarily the best method for calculating fractal dimension.

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64 Figure 4-12. Fractal Dimension vs Toughness for Epoxy Resins 4.3.3.3 Non-destructive slit-island method The non-destructive slit-island method is an adaptation of the commonly used slitisland method. The slit-island method calls for setting a specimen in an epoxy resin and polishing the surface to a fine grit to produ ce islands formed by the removal of the top of the fracture surface. This is shown in Figure 4-13. The fractal dimension is then calculated by measuring the perimeter of the islands with different length rulers (Richardson method). This technique can be very time consuming, although it is acknowledged to be the best method fo r calculating fractal dimension and gives most accurate representation of the surface.

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65 Figure 4-13. Slit-Island Method from Hill (Hill et al. 2001) One of the drawbacks of the slit-island met hod is that it requires the sample, or in some cases a replica of the sample, to be po lished, destroying the specimen or replica surface. One study of this research has focused on developing a suitable method of producing slit-island contours for a specimen without destroying the fracture surface. This was done by producing three dimensional images of a fracture surface using optical microscopy. This preserves the fracture su rface should additional measurements become necessary. The same microscope described in Section 4.3.2 to measure flaw size was used for this study. However, instead of focusing on the region surrounding the flaw, images were taken of the hackle region. The M3D package of the MCID E lite 6.0 Morphometric Software package was utilized for this study. This software package was used to generate 3D images from fifteen 2D “slices ” taken at different focal planes with equivalent spacing. The 3D images were then manipulated using the software to produce digitally polished surfaces from which the fractal dimension can be calculated. The digitally polished images could then be characterized for fractal dimension using either hand calculations or software measurements.

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66 This process is ideal for determining the fr actal dimension of the hackle region of a fracture surface. As mentioned in Chapter 2, the hackle region is often characterized by the very large features on the fracture surface. This technique does not work well for the small features found in mirror region. This is because the perturbations found in the mirror region are often sub-micron in scal e and cannot be properly imaged with an optical microscope. Figure 4-13 is an example of a 3-D composite image generated using the MCID software. This image is taken down the z-axis of the composite. This is done to make image polishing easier and more compatible with other software. The accompanying figure, Figure 4-15, is a 2-D im age of the same area of a sample as Figure 4-14. Figure 4-15 was generated using z-axis imaging. This is similar to the process used to generate the 3-D images, but does not result in a pictur e that can be digitally manipulated like the 3-D pictures. The 2-D pictures were used as a referenc e to ensure that the 3-D images were generated properly. The 2-D images are in color, whereas the 3-D images are in black and white. Because each of the sections used to gene rate a 3-D image are often only a couple of microns apart, and the cross-sectional ar eas of each picture are often several hundred microns, 3-D data sets are very diffi cult to view at a perspective angle. The process assigns a gray scale value to each pixel, from 0 to 255, where 0 is black and 255 is white. The higher the number, th e higher the pixel is in the 3-D picture. From each specimen, several sections were di gitally cut by specifying a range of gray scale values to display. Doing this, generates pi ctures similar to those used by Hill et al to calculate fractal dimension using a slit-island technique.(Hill et al. 2001) In most cases,

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67 the higher elevations were removed; however, in some instances the lower sections were removed. Figures 4-16 through 4-19 are all sections taken from Figure 4-14. For each specimen at least 3 sections were generated al ong with a full height composite spectrum (Figure 4-14) and a 2-D real color composite image (Figure 4-15) Figure 4-14 3-D Composite Image of Specimen Figure 4-15 2-D Image of Specimen

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68 Figure 4-16 0-50 Elevation Image Section Figure 4-17 0-100 Elevation Image Section

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69 Figure 4-18 0-150 Elevation Image Section Figure 4-19 50-255 Elevation Image Section

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70 4.3.3.4 Hand calculations of fractal dimensional increment In order to confirm the data gathered using the Image-Pro software, a sample was characterized by hand using the Richardson Method. Figure 4-20 is a subsection of Figure 4-19. The contrast was enhanced and the section magnified to make hand measurements easier. Additionally the interi or parts were removed to produce a more simple perimeter. The pixilation of the figure below is due to the fact that it is a very large magnification of a small section of Figure 4-19. The image in Figure 4-19 was characterized for fractal dimension by hand. The image was magnified to a sufficiently large size and then printed. The perimeter of the white portion of the image was measured from four separate starting points, approximately 90 degrees apart. Table 4-10 is the data gathered in this process. Additionally Figure 4-21 is the plot of the natural log of the ruler length versus the natural log of the perimeter. The fractal di mension of this image can be determined from equation 4-4. Figure 4-20 Magnification of Fracture Surface

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71 (4-4) Ln (P) = (1-D)Ln (R) +C Where R is the ruler length, P is the perimeter, D is the fractal dimension, and C is a constant. Table 4-10 Hand Measurements of Fractal Dimension Number of Steps Ruler 1 2 3 4 Av e Per Ln R Ln P 10 6 5 4 4 4.8 47.5 2.30 3.861 9 6 5 4 6 5.3 47.3 2.20 3.855 8 6 5 6 7 6.0 48.0 2.08 3.871 7 8 7 8 7 7.5 52.5 1.95 3.961 6 9 8 9 9 8.8 52.5 1.79 3.961 5 11 11 11 10 10.8 53.8 1.61 3.984 4 14 14 14 14 14.0 56.0 1.39 4.025 3 21 21 19 18 19.8 59.3 1.10 4.082 2 34 37 35 36 35.5 71.0 0.69 4.263 1 76 74 77 78 76.3 76.3 0 4.334 Figure 4-21. Richardson Plot of Perimeter of Fracture Surface Three sub-sections for each of the six para meters tested in this research were measured for fractal dimensional increment using hand calculations on images generated Fractal Dimension of Fracture Surface Ln(P) = -0.221Ln(R) + 4.3535 R2 = 0.973.8 3.9 4.0 4.1 4.2 4.3 4.4 0.0 0.5 1.0 1.5 2.0 2.5 Ln Ruler Slope = 1-D D = 1.22 D*= 0.22 L n P e r i m e t e r

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72 by the non-destructive slit-island method. These D* values for each subsection, and the average and error for each composition and stra in rate can be found in the Table 4-11. Table 4-11. Hand Calculations of Fractal Dimension of Epoxy Resins 8 phr DETA 10 phr DETA Section 0.1mm10mm100mm0.1mm10mm 100mm 1 0.23 0.21 0.15 0.20 0.18 0.15 2 0.21 0.18 0.19 0.21 0.15 0.12 3 0.22 0.15 0.16 0.25 0.16 0.12 Average 0.22 0.18 0.17 0.22 0.16 0.13 Error 5% 17% 12% 5% 9% 17% The values in the above table are similar to those found previously when comparing the flaw to mirror size in Section 4.3.3.2. Fractal dimensional increment for all samples appears to be around 0.2. A plot of KIc vs. D*1/2 for the fractal dimensions found by hand calculations can be found in Fi gure 4-22. Note the higher R2 value for a linear regression in this plot versus Figure 4-12 Figure 4-22 Square Root of Fractal Dimensio n vs. Toughness Calculated by Hand from Slit Island Contours KIc vs D*1/2R2 = 0.8973 800 900 1000 1100 1200 1300 1400 1500 0.30 0.35 0.40 0.45 0.50 D*1/2 KIc (MPa*m 1/2 )

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73 4.3.3.5 Calculation of fractal dimensional increment by Image-Pro The hand calculations of fractal dimens ion can be extremely time-consuming and difficult. For that reason, software was employ ed to gather fractal dimension across large volumes of data. The fractal dimensional increment was calculated from images generated using the non-destruc tive slit-island method. “Image-Pro” software was used to calculate fractal dimension for all specimens for all samples. Image-Pro uses a derivation of the Richardson method to calculate fractal dimensional. Image-Pro first analyzes the image for light and dark regions to identify islands. Multiple ruler lengths are used to measure the perimeter in accord ance with the Richardson method. Fractal dimension is calculated for regions with a perimeter larger than 30 pixels because the specific algorithm used by the software breaks down for very small features. The data wasis then tabulated and averaged together across all regions for a sample to produce a fractal dimension for a surface. For each composition and strain rate, the im ages were used to obtain an average fractal dimension. Table 4-12 lists the average fractal dimension and error of each composition and strain rate for epoxy resins. Table 4-12. D* of Epoxy Resi ns Calculated With Image Pro Strain 8 phr DETA 10 phr DETA Rate D* %Error D* %Error 0.1 0.2212% 0.2313% 10 0.1913% 0.1717% 100 0.189% 0.1611% Finally, the square root of the fractal di mensional increment calculated with Image pro for images generated with optical micr oscopy was plotted versus the toughness of epoxy resins. This result can be found in Figure 4-23.

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74 Figure 4-23 Square Root of Fractal Dimens ion vs. Toughness Calculated by Image Pro from Slit Island Contours 4.3.3.6 Comparison of methods of measuring fractal imension Three separate methods were used to calculate fractal dimension for samples in this research; flaw to mirror ratio (F-M), hand calcu lations of slit island contours (Hand), and computer calculations of slit island contou rs (Computer. The results of these three methods are tabulated in table 4-13. Table 4-13. Comparison of Fractal Dimens ional Increment D* Values for Different Methods 8 phr DETA 10 phr DETA 0.1mm10mm100mm0.1mm10mm 100mm F-M 0.23 0.15 0.15 0.24 0.22 0.11 Hand 0.22 0.18 0.17 0.22 0.16 0.13 Computer 0.22 0.19 0.18 0.23 0.17 0.15

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75 It can be seen in the above table that al l three methods return similar values for each composition and strain rate. 4.4 Structural Parameter a0 The goal of this research was to determin e the nature of the relationship of nodular size and toughness. It was proposed that nodular size could be related to toughness through the West Mecholsky Passoja theory Using the data presented above, it is possible to calculate the WMP structural parameter, a0, and the relationship to nodule size for epoxy resins investig ated in this research. 4.4.1 Calculating a0 The structural parameter of the West Mecholsky Passoja theory, a0, can be calculated from the modulus, fractal dime nsional increment, and toughness using the following equation: (4-5) 2 01 D E K aIc Where KIc is the toughness, E is the modulus, and D* is the fractal dimensional increment. The structure parameter for all three strain rates and each composition can be found in the following table: Table 4-14. Calculated a0 values ( m) for Epoxy Resins Strain Rate (mm/min)8phr DETA 10 phr DETA 0.1 4 5 10 4 4 100 3 3 The data used to calculate these results are the average values for a particular composition and strain rate. The fractal dimensional increment values used in this calculation come from calculations with Image Pro. These values were chosen because of

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76 the low error for the measurements made. Because the error associated with a0 is a function of the individual va riables and not a direct summ ation, Appendix A has been prepared describing how the error was calcula ted. Additionally Section 4.4.3 describes statistical significance of these results. 4.4.2 Relationship of Fractal Dimensional Increment to a0 Through the course of measuring fractal di mensional increment, several plots were derived that plotted the square root of D* versus the toughness of epoxy resins. The figures are 4-12, 4-22, and 4-23. These t ypes of graphs have been reported in the literature to show that for families of material s, such as glasses, or crystalline materials, the fractal dimensional increment follows a tren d that as toughness increases so does D*. Additionally, it has been hypothesized that these graphs woul d extrapolate through zero. The data in Figures 4-12, 4-22, and 423 were extrapolated back through zero; these results can be found in Figure 4-24. Th e data in Figure 4-24 s eems to indicate that the flaw to mirror size ratio is the best method of calcula ting D*. Even though the error is very high compared to other methods, th e flaw to mirror ratio best fits predicted models for fractal dimension to toughness. This argument is made based on the fact that when extrapolates the data for a ll three techniques through zero, the R2 value for flaw to mirror ratios is the highest (0.7311) as evidenced on the next page. The data in Figure 4-24 serves a second purpose, not only can it help to better understand where problems in measuring fracta l dimension occur, but also the slope of the linear regression line can be used to calculate a0. The slope of a plot of D*1/2 versus KIc should be equivalent to Ea0 1/2. Table 4-15 lists the a0 values calculated by comparing the slope of the regression line to the modulus of epoxy resins.

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77 Comparision of Fractal Dimensional Incrementsy = 2564.1x R2 = 0.5824 y = 2638.6x R2 = 0.6718 y = 2622.4x R2 = 0.7311 800 900 1000 1100 1200 1300 1400 1500 0.300.350.400.450.50 D*1/2KIc (KPa M1/2) Flaw/Mirror Hand Image Pro Linear (Image Pro) Linear (Hand) Linear (Flaw/Mirror) Flaw/Mirror Hand Image Pro Figure 4-24 Comparison of Fractal Dime nsion Increment to Toughness for Three Different Techniques of Measuring Table 4-15 a0 Values ( m) calculated from the Slope of the Fractal Dimension vs. Toughness Plot Strain Rate(mm/min) 8phr DETA 10 phr DETA 0.1 4 4 10 3 3 100 3 3 The values of a0 are similar to those calculated from the data gathered in this research using equation 4-5. This would seem to further i ndicate that there is no strain rate component to a0. However, the calculated values of a0 are still significantly higher than nodule size. As previously mentione d the nodule size for the investigated compositions ranges between 75 and 100 nm, the calculated values of a0 are well over one order of magnitude larger. Addi tionally it should be noted that a0 values are identical for both compositions, ever though the nodule sizes are different.

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78 4.4.3 Error Analysis of a0 The West Mecholsky Passoja theory is a complex equation that requires several measurements to be made. Each measuremen t has its own error, which can be the result of many factors such as machine compliance a nd user specific error. As a result, the cumulative error for calculation of a0 can not be calculated from directly summing the error of the individual measurements. The indi vidual errors for each variable in the WMP theory must be weighed for their magnitude and power in the equation. As previously mentioned, Appendix A has been prepared to describe how error was calculated and also describes the individual contribution of each variable on the final value of toughness or a0. Table 4-16 list the error for each composition and strain rate tested in this research. Table 4-16 Cumulative Error Values for a0 Calculations for Epoxy Resins Strain Rate 8phr DETA 10 phr DETA 0.1 53% 44% 10 68% 61% 100 44% 74% The cumulative errors asso ciated with calculated a0 from the data presented are very large. As such, one can only make the j udgment that there is no statistical difference between the a0 values calculated for one strain rate to another. Additionally, it is also apparent that the error in the strain rate values for the two DETA compositions are the same. Each composition has a different stra in rate, however this data would seem to indicate that a0 is not a function of nodule size. 4.5 Results and Discussion The data found in the various studies in this re search is similar to what can be expected from the literature (KIc 1.5 MPa*m1/2, E 1.5 GPa, D* 0.2, Nodule Size 100 nm). The value of a0 was found to be around 4 m, which is significantly larger than

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79 nodule size. While the cumulative error of a0 as listed in Table 4-16 may be very large, it should be noted that calculated values of a0 are almost two orders of magnitude greater than nodule size, and would be almost thr ee orders of magnitude larger than the interstitial formed by nodule packing. As such it can be concluded that there is no direct relation between nodule size, nodule packing, and a0, which is contrary to our initial hypothesis. Table 4-17. Calculated a0 Values and Associated Error Strain 8 phr DETA 10 phr DETA Rate a0( m) %Error a0( m) %Error 0.1 3.9 53% 5.2 44% 10 3.8 68% 3.8 61% 100 3.3 61% 2.6 74% There are a few possible explanations for this result. First and foremost it could be that nodule size simply does not relate to tough ness in any meaningful way. Literature has shown that as the nodule size changes cr oss-link density also changes, which has been shown extensively to re late to toughness for other polymeric systems.(Racich et al. 1976) It could be that toughne ss is only controlled by cross-li nk density. As previously mentioned, previous research has reported th at nodules and interstitial material have the same density and modulus. Therefore, it could m ean that nodules are merely an artifact of the polymerization reaction and do not positivel y or adversely affect toughness. Another possible explanation can be found from th e results of the nodule extraction study. As mentioned above and in the literature, ace tone and other solvents can be used to swell the epoxy resin and wash nodules out of the structure. However not all nodules wash out of the structure, only a very small percentage. The structural parameter, a0 can be linked to the weakest link th eory of failure. When a body is stressed, failure occurs at

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80 the weakest link; in the case of the WMP theory, the fracture tip propagates through the weakest links. One possible explanation for why a0 is so large could stem from the fact that not all nodules are tightly bound to the structure. As the crack tip propagates through the sample, it paths along between weakly or unbound nodules in the bulk. Additionally it should be noted that the sub critical crack growth found as part of the flaw size measurements has not been previously shown to occur in homogenous epoxy resins. This phenomenon is usually fo und in heterogeneous materials such as composites where R-curve behavior can occur. It is possible th at individual nodules serve as points for crack tip deflection, the chief mechanism for toughening in composites.(Watchman 1996) This lends further credence to the beli ef that there are structural inhomogeneities such as free or loosely bound nodules that might contribute to the a0 parameter. 4.6 Conclusions The original hypothesis of this work was that a0 can be related to nodule size, however the results indicate that there is no relation between these two values. While the hypothesis was shown not to be valid, this work has opened up some possible avenues to investigate further. It could be further postulated that a0 may not be directly related to nodule size, but is a function of concentration of loosel y bound nodules. Researchers have shown that there is a very low concen tration of nodules that are not tightly bound to the resin and can be washed out of the structur e with extended exposure to solvents. It is possible that a0 is a function of the average dist ance between weakly or unbound nodules. Additionally part of this research focuse d on developed a new ad aptation of the slit island method for producing contours for calcu lating fractal dimension. Contours were produced using optical microscopy and ch aracterized using hand and computer

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81 calculations. The values calculated fr om the optical microscopy contours were compared to fractal dimensions derived from th e ratio of mirror to flaw size of fracture surfaces. The values for both techniques were found to be similar to one another. This means that optical microscopy can be used to calculate fractal dimension of fracture surfaces.

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82 CHAPTER 5 SILSESQUIOXANES: GROWTH, STRU CTURE, AND CHARACTERISTICS 5.1 Introduction Silsesquioxanes are technol ogically important hybrid in organic-organic polymers. Polysilsesquioxanes have been used in ever ything from microelect ronics to scratchresistant coatings. Chemically they are similar to polysiloxanes or silicone, with a backbone of silicon and oxygen and an organic side groups; they differ in that polysilsesquioxanes have one organic group while polysiloxanes have two. The general structures of a polysilsesquioxane and pol ysiloxane are compared in Figure 5-1. O Si O Si Si O Si O O Si Si O O OH OH HO HOPolysilsesquioxaneHO Si O Si O Si OHPolysiloxane Figure 5-1. Comparison of Silicone and Polysilsesquioxane Structures There are many different types of polysil sesquioxanes. Polysilsesquioxanes are classified by their organic gr oup. A methyl polysilsesquioxane is one that contains a CH3 group, a phenyl polysilses quioxane contains a C6H5 group and so forth. The properties, uses, and structure of a polys ilsesquioxane are dependent on the organic group. Table 5-

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83 1 lists some of the more common polysil sesquioxanes and gives the common use and structure. Table 5-1. Usage of Co mmon Polysilsesquioxanes Silsesquioxane Usage Hydrido Interlayer Dielectrics Methyl Additive, Binder, Precursor Phenyl Coatings, Precursor Silsesquioxanes are synthesized from trifunctional monomers. This results in a complex non-linear polymer. Many possible structures can be formed from polysilsesquioxanes. There are generally thr ee classes of polysilsesquioxane structures: random network, cage, and ladder, as found in Figure 5-2. It shoul d be noted, however, that the ladder structure has almost exclus ively been shown to be only found in the phenyl polysilsesquioxane polymer. O O Si Si O O Si O Si O Si Si O Si O Si OO OCagedO Si O Si Si O Si O O Si Si O O OH OH HO HOLadderO Si O O Si O Si Si O Si O Si O Si O Si O Si O Si O Si O O O Si O Si O Si OH O Si HO OH OH OH HO OH OH OH OH OHRandom Network Figure 5-2. Common Polysi lsesquioxane Structures

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84 An independent study performed by Cao and Baney indicated that bulk polysilsesquioxanes have a nodular microstructure similar to what is found in epoxies. These results have not be report ed or duplicated elsewhere, th is is mostly due to the fact that polysilsesquioxanes are very rarely ever studied as bulk resins. Traditionally, polysilsesquioxanes are used as thin films or as fillers in other polymers. This research will build on the work of Cao and Baney by investigating what parameters influence the formation of nodules in polysilsesquioxanes. This research was broken down into two di stinct portions, characterization of the polymerization process, and characteri zation of cross-linked polysilsesquioxane monoliths. The polymerization process was investigated for effects on physical properties such as viscosity and molecular weight, and ch emical properties such as structure. The cross-linked monoliths were investigated for nodular microstructure as seen in the epoxies and methods of formation. 5.2 Polysilsesquioxane Synthesis and Processing 5.2.1 Polymer Synthesis Silsesquioxanes are synt hesized using many differe nt techniques. All polysilsesquioxanes studied in th is research were synthesized in the same manner using a novel two phase reaction technique that has b een reported in the literature.(Kondo et al. 2000) The Itoh method for synthesizing poly silsesquioxanes uses a hydrogen bonding solvent, such as a ketone, to direct the condens ation reaction. The ketone used in the Itoh method hydrogen bonds to the hydroxide groups of the hydrolyzed monomer. Silsesquioxanes were prepared using alky ltrichlorosilanes purchased from Gelest Inc. The ratio of silane to water was held constant for each polysilsesquioxane, 1.5 moles silane to 600 ml of water. A solution of 1. 5 moles of silane and 75 ml of methyl isobutyl

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85 ketone (MiBK) was dripped into a vigorously stirred mixture of 600ml of water and 450 ml of MiBK in a 2-liter, 3-necked round-bo ttomed flask. The water-MiBK mixture was held at 10 C while the alkyltrichlorosilane-MiBK mi xture was added over the course of one hour. After the monomer was added, the solution was he ated to a desired growth temperature, usually 50 C and stirred for 3 additional hours. Table 5-2 Synthesized Polysils esquioxanes and Structures R-Group Name Polymer Structure -CH3 Methyl PMSQ Si Cl Cl Cl -C2H5 Ethyl PESQ Si Cl Cl Cl -C3H7 Propyl PPSQ Si Cl Cl Cl -C4H9 Butyl PBSQ Si Cl Cl Cl -C2H3 Vinyl PVSQ Si Cl Cl Cl -C3H6Cl Chloropropyl PCPSQ Si Cl Cl Cl Cl

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86 Upon completion of the growth phase, the ketone-water mixture was removed from the flask and separated. The ketone phase containing the polysilsesquioxane polymer, was then washed repeatedly to neutral usin g deionized water, which was done to remove the large excess of hydrochloric acid that is a byproduct of the reaction process. Excess MiBK was removed using vacuum distillation. The purified polymer was then removed for characterization and furthe r use. Six polysilsesquioxanes were synthesized in this research. Table 5-2 is a list of the polysilsesquioxanes synthe sized in this research and the structure of each monomer used. 5.2.2 Polysilsesquioxane Monolith Synthesis The polysilsesquioxanes synthesized in this research were cross linked to form freestanding monoliths. Tin octoate was used as a catalyst to cross-link the polysilsesquioxanes through a condensation reaction. Before cross-linking, the polysilsesquioxanes were diluted with a solvent, and the tin octoate was dissolved in a matching solvent. Solvents used in this portion of the research include chloroform, acetone, carbon tetrachloride, and hexane. The ratio of polysilsesquioxane to solvent and the concentr ation of tin octoate were varied. Additionally the substrate upon wh ich the polysilsesquioxane was cured was also investigated. Two substrates were i nvestigated in two geometries, glass and Teflon in single and double sided configurations. Thes e configurations are found in figure 5-3. Figure 5-3 Substrate Configurations. Substrate Substrate Substrate Spacer Pol y silses q uioxane + Catal y st

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87 5.3 Experimental 5.3.1 Characterization of Polymer 5.3.1.1 Viscosity Viscosity of polysilsesquioxane polymer in ketone was measured as the polymer was grown at elevated temperatures as part of the synthesis process. An Ubbelhode kinematic viscometer was used to collect viscosit y data. Figure 5-4 is a representation of an Ubbelhode viscometer. The viscometer is loaded by filling th e lower chamber A, and then charged by applying a vacuum to tube D. The viscometer is fully charged when the liquid level is above line F. The vacuum is removed and the liquid is allowed to flow back down through chamber B and back into chamber A. Viscosity is calculated by taking the amount of time it takes for the liquid level to pass from Line F through Line G and multiplying the time by a constant inherent to the viscometer used. A type 1 viscometer was used; the constant for this model is 0.1 cSt/s. Figure 5-4 Schematic of Ubbelhode Viscometer

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88 Samples were removed from the ketone-wat er-silsesquioxane ma terial and cooled to room temperature. The water was remove d from the sample and the viscosity of the organic phase was measured. Samples were taken every 30 minutes from the start of the drip of the silane into the water and ketone and a total of 9 data points were taken for each polysilsesquioxane synthesized. Each sa mple was measured three times to average out an error. Polymethylsilsesquioxane samples were characterized for viscosity. Viscosity has long been used to study polymerization reactions of linear polymers as a way to indirectly measure molecular weight. In this study, viscosity was used to gauge the effects of aging temperature on polysilsesquioxanes grown usi ng the Itoh method. Figure 5-5 is the viscosity of polymethylsilsesquioxa ne. Three separate batches of polymethylsilsesquioxanes were synthesized with different aging temperatures, 40 C, 50 C and 60 C. The right portion of the graph is the initial addition phase, where the chlorosilane-ketone solution is dripped into th e flask. All three samples have very similar viscosity measurements through the additi on phase, which follows a linear trend. However, upon heating the sample to the gr owth temperature, the viscosity greatly changes from batch to batch. All three samples follow a logarithmic increase in viscosity with time. Normally one would expect the condensation rate of a sol-gel reaction like this in the Itoh method to increase with increasing temperatures. The results in Figure 5-5 are somewhat surp rising. It is apparent from the graph that increased growth temperat ure decreases viscosity, which is contrary to the popularly held belief that increasing temperature incr eases reaction rates. There are two possible explanations for this result.

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89 Figure 5-5. Viscosity of Polymethylsilsesquioxane Because polysilsesquioxanes are networ k polymers, two different condensation reactions can occur; intermolecular conde nsation, where two separate molecules condense to become one, or intramolecular condensation, where tw o functional groups on one molecule condense. It is possible that at higher temperatures the polysilsesquioxane molecules have more mobility and the abili ty to bend back on themselves more easily, resulting in an increase in intramolecula r condensation. With an increase in intramolecular condensation, one would expect a lower molecular weight. Additionally intramolecular condensation consumes hydroxi de groups in the polymer, which should result in the polymer mol ecule becoming more hydrophobic and more soluble in the organic ketone phase. It has been postulated that the ketone phase mediates condensation reactions, removing the polymer from the wa ter phase when the polymer becomes more Viscosity of Polymethylsilsesquioxane0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 060120180240 Time (min)Viscosity (cSt) 40C Add 40C Growth 60C Add 60C Growth 50C Add 50C Growth A ddition Phase Growth Phase

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90 hydrophobic. In the ketone phase one would ex pect condensation reac tions to be reduced by virtue of ketone hydrogen bonding to the hydroxides blocking reaction sites. There is another possible explanation for the lower viscosity of polysilsesquioxanes with increasing temperature. At room temp erature water is only slightly soluble in MiBK, however as temperature increases, the so lubility of water in MiBK increases. Figure 5-6 is the solubility of water in MiBK with temperature. It serves that as water becomes more soluble in MiBK with higher temperatures, po lysilsesquioxanes with high amounts of hydroxide would also become more soluble in the ketone. One could assume that the higher temperatures result in lower molecular weight hydroxide rich polysilsesquioxanes dissolving into the ketone phase more quickly, where the reaction slows down. The result of this would be that the viscosity of the polysilsesquioxane polymer would be decreased. It is unclear from the viscosity measuremen ts alone which of these concepts are the proper explanation of the unique results found in Figure 5-6. It is possible that both of these phenomena are occurring simultaneously. An investigation of the molecular weight and structure could give a better i ndication of what is happening for future research. 5.3.1.2 Matrix assisted laser desorption ionization Matrix assisted laser desorption ionization (MALDI) is a relatively new technique for accurately measuring the molecular weight of high mass molecules. Traditionally this technique has been reserved for proteins and other bio-molecules. Recent research has used MALDI for various polymers, including polysilsesquioxanes. MALDI functions by precipitating the material that is to be analyzed in a matrix. The sample is then irradiated with a laser, which is preferentially absorbed by the matrix

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91 rather than the analyte. The energy absorbed from the laser causes the matrix to vaporize taking some of the analyte with it. Additionally at this point the matrix serves to ionize the analyte, usually by transferring a sodium ion to the material being studied. After desorption and ionization the an alyte is then subjected to an electric field which accelerates the ion towards the detector. Th e amount of time it takes for the ion to reach the detector is proportional to the mass of the ion. The data from MALDI can be matched up exactly to various possible structures of the analyte, which makes it an extremely powerful tool for protein synthesis and polysilsesquioxanes. The most difficult part of MALDI is proper selection of the matrix in which to disperse the sample. The matrix selected for this research was sinapic acid. Samples were prepared in a manner similar to that found in the literature. (Falkenhagen et al. 2003) A solution of sinapic acid in tetrahydrofuran (THF), 10 g/ml, was added to an equivalent volume of a solution of polysilsesquioxane in THF, 1 g/ml. 100 l of the sample was placed drop-wise on the surface of the MALDI sample plate and allowed to dry. The samples were irradi ated using a nitrogen laser ( = 337 nm) with a 20 kV acceleration voltage, 100 pulses we re acquired for each spectrum. Matrix Assisted Laser Desorption Ioniza tion Time of Flight Mass Spectroscopy (MALDI-TOF MS) was used as a method to ca lculate molecular weight of synthesized polysilsesquioxanes. This technique was chosen over gel permeation chromatography due to its higher accuracy in calculating molecular weight for complex polymers and polysilsesquioxanes, as described in Chapter 3. The matrix employed in this research is identical to the one described in the literature. (Falkenhagen et al. 2003) It wa s found that MALDI is not an applicable

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92 technique for the polysilsesquioxanes synt hesized here. The polysilsesquioxanes synthesized as part of this research have a vary large molecular weight distribution, similar to those studied by Tecklenberg. (Tecklenburg et al. 2001) However unlike Tecklenberg’s work, the polys ilsesquioxanes studied here we re not fractionated. As such, the amount of signal from any one mo lecular weight spec ies was too low in comparison to the background. For this research, the molecu lar weight was not possible to determine. It should be noted, however, that most of the polysilsesq uioxanes synthesized in this work were visually similar. In other words, they were transparent and free of particles. Polymethylsilsesquioxane was opaque, addition ally after sitting, small white particles would settle on the bottom of the flask. Furthermore, distillation of the other polysilsesquioxanes resulted in hi gh viscosity liquids, whereas the polymethylsilsesquioxane would form a solid mass upon cooling down. 5.3.1.3 Nuclear magnetic resonance Nuclear magnetic resonan ce (NMR) was employed to determine the relative amounts of hydroxide functionalized silicon gr oups in each silsesquioxane. NMR has been extensively used in all fields of chem istry to characterize th e nature of compounds and reactions. NMR is a chemically sensitive technique in that it can be used to determine the local chemical environment of a partic ular group or functionality of a sample. NMR functions by aligning the nuclear spins of all the atoms in a sample using a very powerful magnet. With all the spins poin ting in the same direction, a radio pulse is used to knock the spins over. As the nuclei return towards alignment with the magnetic field, a signal is given off that is indicative of the chemical nature of the particular atom.

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93 The amount of shielding or de-shielding as a result of an atom being bound to another changes the chemical shift in an NMR Spectra. Silicon 29 NMR was used to calculate the ratio of the T2 group to the sum of the T2 and T3 peak in this research. For the purposes of this research and as a tradition in the field a T group is a s ilicon atom with 3 oxygens and one organic group bound to it. A T2 group is one that has two bridgi ng oxygens and one hydroxide, a T3 group is one that has three bridging oxygens. Additionally a T1 group and T0 group are silicon atoms that have one or no bridging oxygen groups, respectively. The NMR spectra were collected on a Bruker Avance 600 MHz Spectrometer with a 14.1 T/51mm Magnet and a 5 mm BBO Probe non-spun at 27 C. The spectral width was 100ppm and ranged from -30ppm to -130ppm. A five microsecond pulse at 3dB was used to collect data at a frequency of re sonant frequency 119.2 MHz. The acquisition time was 0.8585 seconds and the number of scans taken was 2048. A pulse acquire program was used with no decoupling on the hydrogen atom. The samples were prepared using deuterated acetone as a solvent. Additionally some samples of silsesquioxanes were tr eated with bis-(Trimethylsilyl)acetamide (BTMSAA) to derivitize a po rtion of the hydroxide groups of the silsesquioxanes. Samples were treated with a ratio of 41 by volume of BTMSAA to silsesquioxanes. All of the polysilsesquioxane s studied had two peaks, T2 and T3. Because polysilsesquioxanes are polymeric materials, th e NMR spectra are not simple neat spectra with sharp narrow lines. The NMR peaks in polysilsesquioxanes and other polymeric materials are often broad humps or coll ections of peaks. For the alkyl polysilsesquioxanes the T2 peak was defined as the range of peaks between -52ppm and -

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94 62ppm, and the T3 peak was defined as the range of peaks between -62ppm and -72ppm. The T2 and T3 peaks for polyvinylsilsesquioxane were defined as -67ppm to -77ppm and -77ppm to -87ppm respectively. Figure 5-6 Bruker Avance 600 Mhz Vert ical Bore Spectrometer at MBI Figure 5-7 is a composite image of all 6 s ilsesquioxanes characterized in this study. All spectra were gathered using the same cond itions as described above. Additionally Table 5-3 is an accumulation of th e integrated values for the T2 and T3 peaks. For all spectra the integral of the T3 peak was set to one. The hydroxide content was found from the following formula: (5-1) 100 %3 2 2T T T OH Assuming an inherent error of 5% in the processing of NMR spectra, the total hydroxide for ethyl through chloropropyl silses quioxane can be taken to be nominally equivalent. Conversely, the hydroxide content for methyl silsesquiox ane is significantly lower than the other five synthesized silsesquiox anes. This would indicate that there is a stearic component to the polymerization pr ocess. The methyl group is significantly smaller than the other five groups investigated. It is likely that the methyl silsesquioxane can condense and cross-link to a much higher degree than any of the other

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95 silsesquioxanes due to the ease with which the smaller structure can form various intramolecular conformations. Figure 5-7. NMR Spectra of Synthesized Silsesquioxanes Table 5-3. Total Hydroxide Content of Silsesquioxanes PSQ T2 T3 %OH Methyl 0.271 1 21.32% Ethyl 0.671 1 40.16% Propyl 0.608 1 37.81% Butyl 0.537 1 34.94% Vinyl 0.614 1 38.04% Chloropropyl0.689 1 40.79% A second set of NMR experiments were pe rformed to investig ate the structural nature of the hydroxide groups of the various silsesquioxanes studi ed. Silsesquioxanes were trimethylsilylated with bis(trimethyl silyl)acetamide (BTMSAA). Derivitization

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96 attaches a trimethylsilyl group to a hydroxide group on a target molecule. Figure 5-8 is the structure of BTMSAA. O N Si Si Figure 5-8. Bis(Trimethylsilyl)Ac etamide Structure (BTMSAA) Samples in this research were trimethylsilylated to determine the amount of reactive hydroxide of the synthesized sils esquioxanes. As previously mentioned silsesquioxanes are very complex, large molecu les. It would be unreasonable to assume that all of the hydroxides in a silsesqui oxane molecule are capable of further condensation reactions, such as those used to cross-link the silsesquioxane polymer into a monolith. Derivitization allows one to char acterize the amount of hydroxides that are reactive, those that can react further, and non -reactive hydroxides, those that are not able to react further. N on-reactive hydroxides are ones that are likely trapped in a large, complex molecule and are unable to maneuver or configure in such a fashion that would easily allow condensation. Samples were treated with a large exce ss of BTMSAA to derivitize the reactive hydroxides. Figure 5-9 is a com posite image of the 6 derivitzed silsesquioxanes. In this figure the BTMSAA consumes a por tion of the hydroxides in the T2 peak. This portion consumed is the reactive hydroxide. The remaining portion is the non-reactive hydroxide. For the purpose of this research the non-reactive hydroxides are ones that are defined as being locked within the structure of the polymer molecule, or unavailable to the rather large BTMSAA molecule

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97 Figure 5-9. NMR Spectra of Tr imethylsilylated Silsesquioxanes The trimethylsilylated silses quioxanes do not appear to follow any trends regarding reactive hydroxide content and R-group nature Table 5-4 lists the reactive hydroxide content of each silsesquioxane. Table 5-4. Hydroxide Content of Trimethylsilylated Silsesquioxanes PSQ T2 T3 %OH Methyl 0.118 1 10.55% Ethyl 0.244 1 19.61% Propyl 0.363 1 26.63% Butyl 0.347 1 25.76% Vinyl 0.003 1 0.30% Chloropropyl0.129 1 11.43% From these results it is very difficult to gauge what role the R group has on structure formation and guiding the ability of hydroxides to condense. Additionally

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98 Figure 5-10 shows the percentage the total, reactive, and non-reactive silanol of each silsesquioxane. Hydroxide Content of Silsesquioxanes0% 5% 10% 15% 20% 25% 30% 35% 40% 45% PMSQPESQPPSQPBSQPVSQPCPSQ Silsesquioxanes% Hydroxide Total OH Reactive OH Nonreactive OH Figure 5-10. Total, Reactive, and Non-re active Hydroxide Conten t of Silsesquioxanes The figure above reveals no consistent tre nd in hydroxide content or nature among the synthesized silsesquioxanes. It is unknown at this time as to why that is the case. The most notable result of this investiga tion is the nature of the hydroxide of the polyvinylsilsesquioxane. It appears from the results that the hydroxide content of the vinyl silsesquioxane is entirely reactive, me aning the structure is very open. One possible explanation for this is that some form of long range order is present in the polymer, such as the previously described ladder st ructure found in polyphenylsilsesquioxane. The ladder structure does not tr ap hydroxides within the polymer molecule in the same fashion as a hi ghly branched network structure would.

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99 Appendix B is a compilation of the NMR study done for this research. In Appendix B, one can find individual spectra of all the samples run for this research and can compare the integral regions of each sample. 5.3.1.4 Fourier transform infrared spectroscopy Fourier transform infrared spectroscopy (FT-IR) was used to gather information in regards to the structure of the silsesquioxanes. Literature has shown that there are two SiO-Si bands found in FT-IR spec tra of silsesquioxanes.(Lee et al. 2002) These bands have been correlated to the ca ge and network struct ure respectively. Transmission mode FT-IR was used to collect spectra. One drop of sample was placed between two salt windows. Spectra were compared for ratio of the cage and network structure to the organic group, which was used as a reference peak. Fourier Transform Infrared Spectroscopy was employed in this research to quantify the structural nature of the synthesized sils esquioxanes. As reported in Chapter 3, there are two IR active peaks in silsesquioxanes which can be used to characterize the structure, 1120 cm-1 and 1030 cm-1. These peaks represent the cage and network structures respectively. Figure 5-11 is a figure of the FTIR Spectra of the six silsesquioxanes studied in this research. Tabl e 5-5 is the measured intensities of FTIR spectra of the silsesquioxanes studied in this research. One can see that polymethylsilsesquioxane and polyethylsilsesqu ioxane have similar ratios of network to cage. Additionally one can see th at the rest of the silsesquiox anes also have similar ratios of network to cage. This would seem to indi cate that R group has a limited influence on structure. Smaller R groups resulted in hi gher ratios of the two peaks, whereas larger groups resulted in smaller ratios. Pol yvinylsilsesquioxane has an R group roughly the same size as polyethylsilsesquioxane but ha d a ratio lower than ethyl or methyl and

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100 higher than the other three silsesquioxanes. It is possible that some sort of hydrogen bonding occurred between hydroxides and the vinyl groups resulting in the intermediate ratio of network to cage structure. Figure 5-11 FTIR Spectra of Synthesized Silsesquioxanes Table 5-5 Ratio of Network to Cage St ructure of Silsesquioxanes from FTIR PSQ Cage NetworkRatio N/C Methyl 0.86 0.91 0.95 Ethyl 0.92 0.95 0.97 Propyl 0.48 0.62 0.77 Butyl 0.53 0.70 0.76 Vinyl 0.64 0.75 0.85 Chloropropyl0.53 0.66 0.80 5.3.2 Condensation of Polysilsesquioxanes Curing the synthesized silsesquioxanes presented a very difficult problem. Samples were cross-linked using tin octoate as a catalyst. Two separate solvents were

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101 used, chloroform and carbon tetrachloride. These solvents were chosen because chloroform is a hydrogen bonding solvent and carbon tetrachloride is not. Additionally several different geometries and materials we re employed to make the samples. Unlike the samples of epoxies, the silsesquioxanes c ould not be cast in a silicone mold because the solvent would swell the si licone and deform the mold. Initially glass plates were used to cro ss-link the silsesquioxane. Two geometries were used, a single glass pane with the to p surface exposed to air, and a double glass pane separated by a thin spacer. The double glass pane in all situations resulted in a resin that had very fine porosity which was a re sult of the evaporation of the solvent. The single glass pane resulted in a ma terial that was dense and non-porous, however, because one side was exposed to air and the other not, it resulted in an uneven curing process. After removal from the plat e, the sample would slowly curl tighter. Experiments were performed to try and minimi ze this effect, such as allowing the sample to cure longer and at higher temperatures. This proved to be futile and either reduced the amount of curling, qualitatively measured, or did nothing at all. Appendix C has been prepared as a list of tables of compositions of various silsesquioxanes studied in this work. Table C-1 in Appendix C is a list of the various compositions and configurations stud ied as part of this research. 5.3.3 Characterization of Nodular Microstructure As previously mentioned, it has been re ported that silsesquioxanes can form a nodular microstructure, as found in epoxies. Th e goal of this research was to investigate the nature of nodule formation by varying the properties of the polymer and polymer environment during curing. It was my hope that one coul d find a relationship between the R-group of the silsesquioxane and the natu re of the solvent and nodule size. It was

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102 my hypothesis that the nodules nucleated out of the solvent-polymer mixture, forming small nodules with high surface energy. Th e high surface energy would preferentially drive condensation reactions at the surface of the nodule. As the nodule grew, the difference in surface energy with the solutio n would begin to approach zero and slow down additional condensation at the surface. Or iginally it was intended that different Rgroups on the silsesquioxanes and different solvents such as chloroform and carbon tetrachloride could be used to investigate this hypothesis. It was found, however, that not a ll silsesquioxanes form truly nodular microstructures. In fact, it was found that only polymethylsilsesquioxane formed a nodular microstructure. The following pictur es are SEM micrographs of methyl, ethyl, propyl, and butyl silsesquioxane There does appear to be some structure in other silsesquioxanes, but not a nodular structure. Instead the st ructure appears to be two separate phases, an interconnected light and dark phase. This can be best seen in Figure 5-14 of polypropylsilsesquioxane. Figure 5-17 highlights the differences in phases. From the pictures above some sim ilarity may be vi ewed, but only the polymethylsilsesquioxane has a truly nodular microstructure Figure 5-16 is a high magnification SEM micrograph of polymethylsi lsesquioxane. One can see what appear to be very small nodules or clusters n this figure. One can estimate the size of these clusters to be approximately 20 nm in diameter It is my assumption that these clusters are the primary building blocks of the nodules. The similar scale features can be found in the pictures of the other silsesquioxanes above. It is unknown at this time as to why only polymethylsilsesquioxane is nodular.

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103 \ Figure 5-12 Polymethylsilses quioxane Microstructure Figure 5-13 Polyethylsilses quioxane Microstructure

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104 Figure 5-14 Polypropylsilses quioxane Microstructure Figure 5-15 Polybutylsilses quioxane Microstructure

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105 Figure 5-16 Polymethylsilsesqu ioxane Cluster Structure Figure 5-17 Highlighted Two-Phase Image of Polypropylsilsesquioxane A possible explanation would be the small steric size of the methyl group allows for more possible configurations of a molecule and thus more intramolecular condensation. As a structure grows in th e polymethylsilsesquioxane the polymer comes back around on itself, in other silsesquiox anes the larger groups would prevent the

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106 molecule from bending back into the growing nodule and conti nue growth into a larger network. 5.4 Polymethylsilsesquioxane It should be noted that prior to undertaking the ma in method of synthesis of polymethylsilsesquioxane descri bed above, the author of this work experimented greatly with another route for synthesizing polym ethylsilsesquioxane. The method used previously was originally described by Cao and Baney. (Baney et al. 1999) This technique used a mixture of methyltrimethoxys ilane, methanol and a sub-stoichiometric amount of water which was then cast onto a polystyrene Petri dish. The solution was allowed to cure to a dense resin. Figure 5-18 is an SEM micrograph of a sample of polymethylsilsesquioxane generated with this te chnique. One can see in this picture that the Cao-Baney method of polymethylsilsesquioxan e preparation also results in a nodular microstructure. This indicat es that the formation of a nodular microstructure in polymethylsilsesquioxane is not dependent on a process. Figure 5-18 Cao-Baney Rout e Polymethylsilsesquioxane

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107 5.6 Conclusions The results found in this research have helped to better understand two-phase synthesis techniques for making silsesquio xanes. It was found that increasing temperature decreases viscosity. The decrea se in viscosity is likely due to a lower molecular weight of the polymer. It is likely that at higher temperatures the silsesquioxanes experience a hi gher level on intramolecular c ondensation. It was not shown however that the increase in intramol ecular condensation occurs on a short range order. That is, FTIR experiments do not i ndicate any increase in a short range cage structure versus a long ra nge network structure. Si-29 NMR was used to characterize the s ynthesized and derivitized silsesquioxane polymers. The larger R-group silses quioxanes, ethyl, pr opyl, butyl, vinyl, and chloropropyl, appeared to have the same relative amounts of hydroxide, approximately 40%. Conversely, it was shown that the polymethylsilsesquioxane had approximately 20% hydroxide. Derivitizatio n was used to determine th e amount of reactive hydroxide present in the structure. It was shown that there is no trend between group size and reactive hydroxide content. Of note, it was found that the hydroxide content of polyvinylsilsesquioxane was totally reactive, an d not locked within the structure of the molecule. Originally it was hypothesized that silsesquioxane based re sins could be used as a model for investigating the relationship between nodule size and toughness. It was shown by Cao and Baney that polymethyl silsesquioxane can form a nodular microstructure. In this research, five a dditional silsesqu ioxanes were synthesized, however, it was shown that onl y the polymethylsilsesquioxane forms a nodular structure. It is my hypothesis that this is due to the much lower content of hydroxide in the

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108 methylsilsesquioxanes than the other silsesqui oxanes. As the polymethylsilsesquioxane grows, the number of sites for additional condensation reactions greatly reduces. Additionally the tin-octoate acts as a cata lyst for both intraand intermolecular condensation. Since polymethylsilsesquioxane already has a low c ontent of hydroxide, it is possible that the catalyst enhanced intr amolecular condensation greatly reduces the hydroxide sites for intermo lecular condensation.

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109 CHAPTER 6 RESULTS, DISCUSSIONS, AND FUTURE WORK 6.1 Results and Discussion 6.1.1 Calculating Fractal Dimens ion with Opti cal Microscopy A new technique for calculating fractal di mension was developed for this work. This technique used optical microscopy to generate a three dimensional image of a fracture surface. The digital image was then sectioned using the software bundled with the microscope. The sectioned images were us ed to calculate fractal dimension. Fractal dimension was determined by using a Slit -Island technique popularized by Mandelbrot. (Mandelbrot et al. 1984) It was shown through the course of this wo rk that this technique returns fractal dimension values which are to be expected for the material studied. The results were compared to fractal dimension calculated for the same material using mirror to flaw size ratios. The algorithm used by the software was shown be true by measuring the fractal dimension by hand using the compass-walk method. 6.1.2 Nodular Microstructure of Epoxies and Fractal Analysis of Failure It was not shown conclusively that the size of the nodule in epoxy resins is directly related to the toughness through the West Mechol sky Passoja Theory. Table 6-1 lists the structural parameter a0 values for the compositions and strain rates tested in this research. There seems to be some effect of strain rate on calculated a0 values. At low strain rates a0 was shown to be much higher.

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110 Table 6-1 a0 and Error Values for Epoxy Resins by Strain Rate Strain 8 phr DETA 10 phr DETA Rate a0( m) %Error a0( m) %Error 0.1 4.1 53% 4.4 45% 10 3.0 68% 2.6 54% 100 2.1 68% 2.0 67% The data in table 6-1 would seem to indicate that a0 is not a static property but is affected by strain conditions. These result s have lead to another possible hypothesis for future work. It has been noted in the litera ture that a small percentage of the nodules in the microstructure are not bound to the micr ostructure. It can by hypothesized that a0 is the distance between unbound nodules. At low strain rates, the structure can accommodate more strain before failure a nd thus the average distance between unbound nodule sites increases greatly. At high strain rates, the failure load and strain are smaller, therefore a0 or the distance between unbound nodules remains low. 6.1.3 Synthesis and Characterization of Silsesquioxanes Six different silsesquioxanes were synthe sized for this research; methyl, ethyl, butyl, propyl, vinyl and ch loropropyl. Originally, it was proposed that the silsesquioxanes could be used as a model for th ermosetting resins. It had been previously shown that polymethylsilsesquioxane forms a nodular resin. Unfortunately for the purpose of this research it was found that only the polymethylsilsesquioxane forms a nodular microstructure. Additionally it was f ound that the techniques used to synthesis the silsesquioxane resins di d not produce samples from wh ich mechanical properties could be determined or that were statistically similar in structure. Silsesquioxanes were extensively charact erized for structure using Silicon 29-NMR and FTIR. It was found that the polymet hylsilsesquioxane contai ned significantly less

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111 hydroxide then the other silsesquioxanes. It was also found that methyl, ethyl and vinyl silsesquioxane had similar ratios of cage to network structure which were found to be lower then the other th ree silsesquioxanes studied. Also it was found empiri cally that the molecular weight of the polymethylsilsesquioxa ne was significantly higher then any of the other polymers. From the Silicon 29-NMR, FTIR, and molecular weight data, one could theorize the reason as to why the polymethylsilsesqui oxane forms nodules but none of the other silsesquioxanes do. It is possible the reas on PMSQ forms a nodular microstructure is because of the reduced functiona lity due to high molecular wei ght and shown with NMR. As the synthesized polymer molecules begin to condense in the presen ce of catalyst, the amount of functionality decrease s further resulting in fewer sites for more condensation. Eventually the nodule grows to a size where th e surface functionality is effectively zero, which greatly hampers additi onal nodule growth. This doe s not occur in the other silsesquioxanes because there is a much great er initial concentration of hydroxides, due in part to the lower molecular weight. 6.2 Future Work While the work presented here does seem to indicate a relationship between nodule sizes, nodule packing, and mech anical properties, more work should be done to further back this conclusion. To date, it is not entirely known as to why nodules form, or what parameter control nodule size. Additional work should be done to control nodule size. As mentioned in chapter 2, Koutsky has s hown that surface energy may be a deciding factor in nodule size. It would be of intere st to modify the silicone molds with various agents to change the surface energy. By using the same composition and different surface energy overall the bulk properties should remain the same such as modulus and

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112 cross link density. These experiments woul d go a long way to showing if surface energy of a mold or substrate has a significant effect on nodule size. It is still unknown how nodules differ from interstitial material. Techniques such as SAXS and AFM have not shown any difference between nodul e and interstitial material, this has lead some groups to believe that nodules are artifacts of the imaging process and are not real. As mentioned in chapter 4, soaking a nodular epoxy resin in acetone for long periods of time results in some of the nodules diffusing out of the bulk. This would seem to indicate that the nodules are not well bound with the connective phase and would indicate that the functionality of the surface of a nodule is very low. It would be of interest to qua ntify the surfaces of nodules for functionality. Also it was postulated from the results that a0 could be a function of th e distance between nodules that are loosely connected to the matrix. Additional experiments could be used to investigate what portion of the nodules in the resin can be extracted from the body and then modeled to find average distance between nodul es. It would be of interest to see if the distance between unbound nodul es was proportional to a0. Additional experiments should also be pr eformed to understand if the size of a0 is specific to nodular epoxies or all epoxies. Further experi ments could include calculating a0 for epoxy resins that do not have any nodular micros tructure. If similar a0 values to what were found in this research were found for a non nodular e poxy resin that would seem to indicate that there is no direct correlation between n odule size, unbound nodules, and toughness. Additionally it was postulated in chapter 4 that the nodules could server a mechanism for crack tip deflection tougheni ng. Experiments could be preformed measuring the toughness of two resins one nodular, and one non-nodular that have the

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113 same crosslink density. Theses experiment s would serve to not onl y show if nodules are a toughening mechanism but could also be used to compare a0 values. Lastly further experiments should be pr eformed with silsesquioxanes to understand the formation of nodules in silsesquioxanes. Through the course of th is research only the polymethylsilsesquioxane was shown to form a nodular microstructure. Larger R-Groups did not form nodular structures. It is not know why the larger R groups did not form nodular microstructures. It would be of in terest to look at co-polymers of various silsesquioxanes with different R-Groups. By synthesizing silsesqui oxane polymers with various ratios of precursors one should be able to empirically control molecular weight, hydroxide content, and structure. From ther e one could form a phase diagram of sort showing what regions form nodular structures. There is still a large am ount of work to be done for understanding how nodules form in both epoxies and silsesquioxanes. Assuming the distance between loosely or unbound nodules are the origin of a0, a better understanding of how nodules form and why they are more akin to inhomogeneities c ould lead to engineering a tougher thermoset resin.

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114 APPENDIX A ERROR ANALYSIS OF THE WEST MECHOLSKY PASSOJA THEORY A.1 Introduction Error analysis of an equati on is a crucial step in unde rstanding the effects of each variable on the output of an equation and how the asso ciated error in measuring each variable affects the overall re sult. This appendix will se rve as a thought exercise in understanding error in measurem ents and results and the ef fect on the West Mecholsky Passoja Theory. The WMP Theory identifies three key variab les that must be measured using any of a variety of techniques. Thes e variables are the toughness, KIc, the fractal dimension D*, and the modulus, E. Each of these variable s introduces error into the final product. Because the equations investigated in the research are not simple linear addition equations, the cumulative error in a0 is not a simple summation of the error of the variables. The cumulative erro r of an equation is defined as the square root of the sum of the squares of the partial deri vative of the equation with re spect to a variable times the error of the variable The cumulative for any equation can be found from equation A-1 (A-1) j i i i T TE E E E1 2 Where ET is the magnitude of the cumulative error for equation ET, ET/ Ei is the partial derivative of ET with respect to variable i, and Ei is the magnitude of the error of variable i.

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115 A.2 Sources of Error With any characterization process there ar e two sources of error, intrinsic and extrinsic. Intrinsic errors are generally those defined as independent of the test conditions. These are errors that are inherent to the te sting process and are often extremely difficult to remove. Extrinsic errors are often errors intr oduced into the system or measurement by the user or an outside s ource. Extrinsic errors can be minimized by repeating the same experiment multipl e times and averaging the results. A.3 Error Analysis Equations The WMP equation is defined in equation A-2. Equation A-3 is the WMP equation rearranged to solve for the structural parameter a0. Equation A-8 is the cumulative error associated with calculating a0 and measuring the parameters KIc, D*, and E. Equations A-4 through A-7 are the equations us ed to derive equation A-8. (A-2) 2 1 0D a E KIc (A-3) 2 2 0D E K aIc (A-4) 2 0 2 0 2 0 0 E E a K K a D D a aIc Ic (A-5) 2 2 2 0D E K D aIc (A-6) 2 02D E K K aIc Ic (A-7) 3 2 02 D E K E aIc

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116 (A-8) 2 3 2 2 2 2 2 2 2 02 2 E D E K K D E K D D E K aIc Ic Ic Ic Where KIc is the Toughness, KIc is the magnitude of the error of KIc, E is the Modulus, E is the magnitude of the error of E, D* is fractal dimensional increment and D* is the magnitude of the error of D*. A.4 Error Analysis Graphs Using equations A-3, A-8 were used to de termine the effects of variables and error on the WMP theory. For each graph there are 3 separate sets of data plotted. Each data set represents the effect of a particular vari able on the WMP theory or on the cumulative error. For each data set only the variable investigated was changed while the others remained constant, the values of which can be found in the figures. The values chosen are proportional to value measured in this research.

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117 Effect of Variable on a00 100 200 300 400 500 600 700 800 900 1000 50%60%70%80%90%100%110%120%130%140%150% % of Nominal Valuea0 (nm) E KIc D* This graph plots the effect of each variable in the West Mecholsky Passoja Equation with respect to each other. From this graph it is possible to see that modulus has the largest effect on a0 over a given range (111-1000nm). Conversely D* has the lowest net effect (167-500nm) Nominal Values E: 1000 MPa KIc: 250 KPa*m1/2D*: 0.25 Figure A-1 Effect of Variables on a0

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118 Effect of Variable on Error10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 0%2%4%6%8%10%12%14%16%18%20% % Error of Variables% Error of a0 D* KIc E This graph shows the effect of one variable on the overal error associated with a0 by holding the other two variables constant. In this case, the error for D* was held to 10%, E to 5% and Kic to 15%. It is apparent from this graph that D* under these constrictions has the least effect on error of a0. Conversely KIchas the greatest effect. Figure A-2 Effect of Variables on Error of a0

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119 APPENDIX B SILICON 29 NMR STUDY OF SILSESQUIOXANES B.1 Introduction and Methodology Silicon 29 NMR was used to characterize the amount of hydroxide present in synthesized silsesquioxane polymers. A dditionally the silsesquioxane polymers were derivitized to determine the amount of reactiv e hydroxide present in the structure. Samples were run on an Bruker Avan ce 600 MHz spectrometer tuned to the silicon 29 resonant frequency, for this par ticular machine the resonant frequency is approximately 119.23 MHz. Samples were dissol ved in deuterated acetone to obtain a proper lock. Spectra were acquired for four hours or 5120 scans. Sp ectra were collected between zero and -130 ppm, with tetramethylsilane as the reference. Samples were run in high purity glass tubes. This resulted in a large SiO4 peak which was removed through careful integration of peaks. The releva nt peaks are found in chapter 5 and in the literature.(Kondo et al. 2000) B.2 Si-29 NMR Spectra The following figures are the NMR spectra ga thered for this research. Present in each figure are the integrals used to find the concentration of each peak. The total hydroxide and reactive hydroxide conten t is tabulated in chapter 5.

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120 120 Figure B-1 Polymethylsilsesquioxane Figure B-2. Derivitized Po lymethylsilsesquioxane

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121 121 Figure B-3 Polyethylsilsesquioxane Figure B-4 Derivitized Polyethylsilsesquioxane

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122 122 Figure B-5 Polypropylsilsesquioxane Figure B-6 Derivitized Polypropylsilsesquioxane

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123 123 Figure B-7 Polybut ylsilsesquioxane Figure B-8. Derivitized Polybutylsilsesquioxane

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124 Figure B-9 Polyvi nylsilsesquioxane Figure B-10 Derivitized Polyvinylsilsesquioxane

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125 125 Figure B-11 Polychloropropylsilsesquioxane Figure B-12 Derivitized Poly chloropropylsilsesquioxane

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126 APPENDIX C STUDY OF POLYSILSEQUIOXANES Table C-1 lists the compositions studied in this research. The samples above are the polysilsesquioxanes synthesized to inves tigate the nature of the cross linked polysilsesquioxane. The codes for Table C1 can be found at the end of the table. Table C-1. Crosslinked S ilsesquioxane Compositions # PSQ Sol PSQ Con Tin Con Ratio (l/ml) Substrate 1 M 1 80 10 50/10 1 2 M 1 80 10 50/10 2 3 M 1 80 10 50/10 3 4 M 1 80 10 50/10 4 5 M 1 50 10 50/10 1 6 M 1 50 10 50/10 2 7 M 2 50 10 50/10 3 8 M 2 50 10 50/10 4 9 M 2 50 10 50/10 1 10 M 2 50 10 50/10 1 11 E 1 80 10 50/10 1 12 E 1 80 10 50/10 2 13 E 1 80 10 50/10 3 14 E 1 80 10 50/10 4 15 E 1 80 10 100/10 1 16 E 1 80 10 100/10 2 17 E 1 80 10 100/10 3 18 E 1 80 10 100/10 4 19 E 2 60 10 50/10 1 20 E 2 60 10 50/10 2 21 E 2 60 10 50/10 3 22 E 2 60 10 50/10 4 23 E 1 60 5 50/10 1 24 E 1 60 5 150/10 1 25 E 1 60 5 250/10 1 26 M 1 60 5 100/10 1 27 M 2 60 5 100/10 2 28 M 3 60 5 100/10 3 29 M 4 60 5 100/10 4 30 E 1 60 5 100/10 1

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127 127 31 E 1 60 5 100/10 2 32 E 1 60 5 100/10 3 33 E 1 60 5 100/10 4 34 P 1 60 5 100/10 1 35 P 1 60 5 100/10 2 36 P 1 60 5 100/10 3 37 P 1 60 5 100/10 4 38 B 1 60 5 100/10 1 39 B 1 60 5 100/10 2 40 B 1 60 5 100/10 3 41 B 1 60 5 100/10 4 42 V 1 60 5 100/10 1 43 V 1 60 5 100/10 2 44 V 1 60 5 100/10 3 45 V 1 60 5 100/10 4 46 CP 1 60 5 100/10 1 47 CP 1 60 5 100/10 2 48 CP 1 60 5 100/10 3 49 CP 1 60 5 100/10 4 50 M 1 20 5 50/10 1 The # is the sample number tested PSQ is the polysilsesquioxane investigated in that sample, M = Methyl, E = Ethyl, P = Propyl, B = Butyl, V = Vinyl, CP = Chloropropyl Sol is the solvent investigated, 1 =Chl oroform, 2 = Carbon Tetrachloride, 3 = Acetone, 4 = Decane PSQ Con is the concentra tion of the polysilsesquioxa ne in the solvent by mass Tin Con is the concentration of Tin Octoate in the solvent Ratio is the ratio of Tin Octoate solution to polysilsesquioxane solution in microliters to milliliters. Substrate is the configurati on of substrate investigated. 1 = Single Glass Pane, 2 = Double Glass Pane, 3 = Single Teflon Co ated Pane, 4 = Double Teflon Coated Pane

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128 APPENDIX D MEASUREMENTS OF EPOXY SAMPLES This appendix has been prepared to list the individual measurements made upon samples in chapter four. The following tables are broken up by composition, strain rate and sample number. Error was calculated as the width of one standard deviation. Table D-1 Modulus Values for Epoxy Samples Modulus (MPa) 8 phr DETA 10 phr DETA Sample 0.1mm10mm100mm0.1mm10mm 100mm 1 1466 1421 1602 1380 1358 1406 2 1298 1427 1602 1284 1487 1441 3 1443 1437 1482 1392 1437 1462 4 1453 1339 1626 1439 1463 1471 5 1311 1341 1783 1368 1356 1424 6 1500 1453 1391 1463 1428 1435 7 1455 1492 1363 1469 1366 1405 8 1383 1463 1465 1429 1451 1443 9 1366 1551 1374 1422 1430 1436 10 1444 1523 1398 1424 1388 1435 Average 1412 1445 1509 1407 1416 1436 Std Dev 69 69 139 54 46 21 Error 4.9% 4.8% 9.2% 3.9% 3.3% 1.5%

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129 Table D-2 Failure Stress Values for Epoxy Samples Failure Stress (f) 8 phr DETA 10 phr DETA Sample 0.1mm10mm100mm0.1mm10mm 100mm 1 60 53 69 68 83 68 2 60 76 67 75 69 53 3 57 74 89 68 56 47 4 56 73 98 64 89 64 5 62 78 48 68 68 70 6 64 78 91 70 88 67 7 58 78 91 72 86 94 8 59 80 87 67 72 75 9 62 71 37 68 54 38 10 62 76 67 70 85 69 Average 60 74 74 69 75 64 Std Dev 2 8 21 3 13 16 Error 3.9% 10.5%27.7% 4.3% 17.4% 24.4% Table D-3 Flaw Size and Toughness Measurements for Epon 825 with 8phr DETA Strained at 0.1 mm/min Sample 2a 2b a b c LocationY fKIc 1 416 333 208 167 186 Surface 1.26 60 1039 2 1255 968 628 484 551 Body 1.13 60 1598 3 690 976 345 488 410 Surface 1.26 57 1468 4 893 908 447 454 450 Surface 1.26 56 1513 5 724 618 362 309 334 Surface 1.26 62 1426 6 239 166 120 83 100 Body 1.13 64 721 7 691 614 346 307 326 Surface 1.26 58 1334 8 694 414 347 207 268 Surface 1.26 59 1219 9 827 792 414 396 405 Surface 1.26 62 1565 10 566 1036 283 518 383 Surface 1.26 62 1531 Table D-4 Flaw Size and Toughness Measurements for Epon 825 with 8phr DETA Strained at 10 mm/min Sample 2a 2b a b c LocationY f KIc 1 219 176 110 88 98 Surface 1.26 53 666 2 477 377 239 189 212 Body 1.13 76 1252 3 134 156 67 78 72 Surface 1.26 74 790 4 623 760 312 380 344 Body 1.13 73 1531 5 633 258 317 129 202 Surface 1.26 78 1406 6 290 228 145 114 129 Surface 1.26 78 1123 7 480 513 240 115 166 Surface 1.26 78 1272 8 155 169 78 116 95 Surface 1.26 80 981 9 120 70 60 117 84 Surface 1.26 71 820 10 382 297 191 118 150 Body 1.13 76 1054

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130 Table D-5 Flaw Size and Toughness Measurements for Epon 825 with 8phr DETA Strained at 100 mm/min Sample 2a 2b a b c LocationY fKIc 1 331 119 165.559.5 99.2 Surface 1.264 69 863 2 221 240 110.5120 115.2Surface 1.264 67 904 3 246 206 123 103 112.6Surface 1.264 89 1069 4 202 426 101 213 146.7Surface 1.264 98 1340 5 611 598 305.5299 302.2Body 1.128 48 932 6 401 262 200.5131 162.1Body 1.128 91 1313 7 176 179 88 89.5 88.7 Body 1.128 91 966 8 No Flaw X X X X X X NF 9 303 272 151.5136 201.5Body 1.128 87 1393 10 998 536 499 268 365.7Body 1.128 37 798 Table D-6 Flaw Size and Toughness Measurements for Epon 825 with 10phr DETA Strained at 0.1 mm/min Sample 2a 2b a b c LocationY fKIc 1 772 563 386 282 330 Surface 1.264 68 1566 2 953 916 477 458 467 Surface 1.264 75 1832 3 1176 1000 588 500 542 Surface 1.264 68 1798 4 617 550 309 275 291 Surface 1.264 64 1378 5 557 452 279 226 251 Surface 1.264 68 1360 6 600 424 300 212 252 Surface 1.264 70 1396 7 1128 593 564 297 409 Surface 1.264 72 1632 8 250 600 125 300 194 Surface 1.264 67 1177 9 977 980 489 490 489 Surface 1.264 68 1889 10 772 564 386 282 330 Surface 1.264 70 1603 Table D-7 Flaw Size and Toughness Measurements for Epon 825 with 10phr DETA Strained at 10 mm/min Sample 2a 2b a b c LocationY fKIc 1 598 376 299 188 237.1Surface 1.264 83 1445 2 540 860 270 430 340.7Body 1.128 69 1440 3 248 188 124 94 108.0Surface 1.264 56 732 4 350 428 175 214 193.5Surface 1.264 89 1560 5 237 228 118.5114 116.2Surface 1.264 68 933 6 111 121 55.5 60.5 57.9 Surface 1.264 88 845 7 612 450 306 225 262.4Surface 1.264 86 1566 8 504 770 252 385 311.5Surface 1.264 72 1597 9 524 478 262 239 250.2Surface 1.264 54 1081 10 524 398 262 199 228.3Surface 1.264 85 1629

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131 Table D-8 Flaw Size and Toughness Measurements for Epon 825 with 10phr DETA Strained at 100 mm/min Sample 2a 2b a b c LocationY fKIc 1 179 378 89.5 189 130.1Surface 1.264 68 980 2 235 200 117.5 100 108.4Surface 1.264 53 694 3 212 180 106 90 97.7 Surface 1.264 47 587 4 No Flaw VisibleX X X X X NF 5 400 400 200 200 200.0Surface 1.264 70 1259 6 150 150 75 75 75.0 Surface 1.264 67 732 7 714 302 357 151 232.2Body 1.128 54 652 8 90 99 45 49.5 47.2 Surface 1.264 75 652 9 400 300 200 150 173.2Surface 1.264 38 627 10 106 140 53 70 60.9 Surface 1.264 69 682 Table D-9 Mirror Measurements for Epon 825 with 8phr DETA Strained at 0.1mm/min Sample c M1 M2 MA D* 1 186.1 1306 1320 1313 0.14 2 551.1 1856 1695 1776 0.31 3 410.3 4 450.2 5 334.5 1501 1397 1449 0.23 6 99.6 718 644 681 0.15 7 325.7 1269 1453 1361 0.24 8 268.0 1224 1194 1209 0.22 9 404.7 1559 1370 1465 0.28 10 382.9 1348 1515 1432 0.27 Table D-10 Mirror Measurements for Epon 825 with 8phr DETA Strained at 10mm/min Sample c M1 M2 MA D* 1 98 1100 1241 1171 0.08 2 212 1029 1014 1022 0.21 3 72 622 791 707 0.10 4 344 1368 1211 1290 0.27 5 202 1111 812 962 0.21 6 129 1364 1234 1299 0.10 7 166 868 1026 947 0.18 8 95 670 778 724 0.13 9 84 969 606 788 0.11 10 150 1110 1241 1176 0.13

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132 Table D-11 Mirror Measurements for Epon 825 with 8phr DETA Strained at 100mm/min Sample c M1 M2 MA D* 1 99.2 695 885 784 0.13 2 115.2 945 1239 1082 0.11 3 112.6 676 724 700 0.16 4 146.7 561 859 894 0.16 5 302.2 6 162.1 907 982 944 0.17 7 88.7 964 842 901 0.10 8 9 201.5 1089 1095 1092 0.18 10 365.7 1890 1890 1890 0.19 Table D-12 Mirror Measurements for E pon 825 with 10phr DETA Strained at 0.1mm/min Sample c M1 M2 MA D* 1 330 1046 1005 1026 0.32 2 467 1827 1827 1827 0.26 3 542 1554 1544 1549 0.35 4 291 1430 1328 1379 0.21 5 251 1300 952 1126 0.22 6 252 1343 978 1161 0.22 7 409 1855 2060 1958 0.21 8 194 1333 1500 1417 0.14 9 489 10 330 1667 1767 1717 0.19 Table D-13 Mirror Measurements for Epon 825 with 10phr DETA Strained at 10mm/min Sample c M1 M2 MA D* 1 237.1 727 1083 905 0.261979 2 340.7 1820 1657 1738.5 0.195993 3 108.0 412 374 393 0.274715 4 193.5 1036 1036 1036 0.186795 5 116.2 1183 1183 1183 0.098249 6 57.9 356 277 316.5 0.183084 7 262.4 1120 1120 1120 0.234279 8 311.5 1355 1321 1338 0.232795 9 250.2 1157 1198 1177.5 0.212515 10 228.3 817 874 845.5 0.270062

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133 Table D-14 Mirror Measurements for Epon 825 with 8phr DETA Strained at 100mm/min Sample c M1 M2 MA D* 1 130.1 1206 1190 1198 0.11 2 108.4 1355 1360 1357.5 0.08 3 97.7 918 1234 1076 0.09 4 5 200.0 907 980 943.5 0.21 6 75.0 1135 979 1057 0.07 7 232.2 1641 1328 1484.5 0.16 8 47.2 329 320 324.5 0.15 9 173.2 1752 1412 1582 0.11 10 60.9 1195 1410 1302.5 0.05 Table D-15 Image Pro Measurements for Fr actal Dimensional Increment by Sample 8 phr DETA 10 phr DETA Sample # 0.1mm 10mm 100mm 0.1mm 10mm 100mm 1 0.25 0.19 0.18 0.26 0.13 0.19 2 0.20 0.22 0.19 0.21 0.15 0.14 3 0.22 0.21 0.15 0.23 0.19 0.16 4 0.24 0.20 0.19 0.17 0.14 0.15 5 0.26 0.18 0.21 0.22 0.16 0.17 6 0.19 0.15 0.16 0.26 0.19 0.14 7 0.22 0.19 0.17 0.23 0.21 0.18 8 0.21 0.14 0.19 0.27 0.23 0.18 9 0.18 0.19 0.19 0.23 0.16 0.19 10 0.23 0.18 0.18 0.25 0.17 0.15 Figure D-1 Flaw Size of Epon 825 with 8phr DETA Strained at 0.1 mm/min

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134 Figure D-2 Flaw Size of Epon 825 with 8phr DETA Strained at 10 mm/min Figure D-3 Flaw Size of Epon 825 with 8phr DETA Strained at 100 mm/min Figure D-4 Flaw Size of Epon 825 with 8phr DETA Strained at 100 mm/min

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135 Figure D-5 Flaw Size of Epon 825 with 10phr DETA Strained at 10 mm/min Figure D-6 Flaw Size of Epon 825 with 10phr DETA Strained at 100 mm/min

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136

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137 LIST OF REFERENCES Anderson, S. E., Baker, E. S., Mitchell, C., Haddad, T. S. and Bowers, M. T. (2005). "Structure of hybrid polyhe dral oligomeric silsesquioxane propyl methacrylate oligomers using ion mobility mass spectrometry and molecular mechanics." Chemistry of Materials 17(10): 2537-2545. Araki, W., Adachi, T., Yamaji, A. and Gamou, M. (2002). "Fracture toughness of bisphenol A-type epoxy resin." Jour nal of Applied Polymer Science 86(9): 22662271. Arkles, B. and Larson, G. (2004). S ilicon Compounds: Silanes and Silicones Morrisville, PA, Gelest. Arthur, S. (2005). Fractal Explorer 2.02. Auad, M. L., Borrajo, J. and Aranguren, M. I. (2003). "Morphology of rubber-modified vinyl ester resins cured at different temperatures." Journal of Applied Polymer Science 89(1): 274-283. Babadagli, T. and Develi, K. (2001). "On the ap plication of methods used to calculate the fractal dimension of fracture surfaces." Fractals-Complex Geometry Patterns and Scaling in Nature and Society 9(1): 105-128. Babonneau, F., Bois, L., Yang, C. Y. and Interra nte, L. V. (1994). "Sol-Gel synthesis of a siloxypolycarbosilane gel and Its pyrolyt ic conversion to silicon oxycarbide." Chemistry of Materials 6(1): 51-57. Bai, S. J. (1985). "Crosslink distribution of epoxy networks studied by small-angle neutron-scattering." Polymer 26(7): 1053-1057. Bailey, J. K., Bangert, R. K., Schweitzer, J. A., Trotter, R. T., Shuster, S. M. and Whitham, T. G. (2004). "Fractal geometry is heritable in trees." Evolution 58(9): 2100-2102. Balankin, A. S. (1995). "Mechanics of self -affine cracks." Revista Mexicana De Fisica 41(4): 473-479. Baney, R. and Cao (1999). "Unpublished results presented to the University of Florida and Dow Corning."

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145 BIOGRAPHICAL SKETCH Kyle Kathan was born in Clarksboro, New Jersey, where he lived with his parents Chris and Russ Kathan, and sister, Kendra. Kyle graduated high school in 1997 from Kingsway Regional High School with a strong in terest in math and science. After graduating from high school, he attended Ru tgers University in New Brunswick, New Jersey, where he studied ceramic engineeri ng and materials scien ce. Dr. Kathan first developed a love for silicon chemistry while working for Jeff Brinker at Sandia National Labs on a NSF research experience for unde rgraduates program. After graduating from Rutgers University, Kyle enrolled in the University of Florida to pursue a Ph.D. in Materials Science and Engineering in the fall of 2001. Kyle was married to Dana Cole of Ocala, Florida on August 7th, 2004.


Permanent Link: http://ufdc.ufl.edu/UFE0012142/00001

Material Information

Title: An Investigation of the nodular microstructure of selected silsesquioxane and epoxy thermosetting resins
Physical Description: Mixed Material
Language: English
Creator: Kathan, Kyle R. ( Dissertant )
Baney, Ronald H. ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Materials Science and Engineering thesis, Ph.D
Dissertations, Academic -- UF -- Materials Science and Engineering

Notes

Abstract: The role of the nodular microstructure in the failure of thermosetting resins is unknown. The goal of the research presented here is to determine the mechanisms by which the nodular microstructure affects failure of thermosetting resins. Epoxy samples were fractured in tension and characterized for fractal dimension, nodular size, and toughness. Samples of epoxy resins have been shown in this work to have a relationship between nodular size and toughness. Two compositions were chosen to study the effect of strain rate and nodule size on toughness and a structural parameter, a0. Additionally fractal dimensional increment was measured using two separate techniques, Flaw to Mirror ratio (F-M), and Slit-Island contours. The fractal dimensional increment for slit-island contours was calculated using two similar algorithms; by hand, and with the aid of software. It was shown through the course of this work that nodule size does not directly relate to toughness. it was also shown that the epoxy resins characterized for this research are not structurally homogenous. It is believed that the inhomogeneities in the structure can be related to a0. Silsesquioxanes were synthesized using a novel two phase reaction. Samples were characterized for molecular weight and structure using matrix assisted laser desorption and ionization (MALDI) and silicon-29 nuclear magnetic resonance (Si-29 NMR). Additionally samples were characterized during growth using an Ubbelohde viscometer to characterize the growth processes of the two phase reaction. It was found that as the temperature of the growth phase synthesis increases, the viscosity of the polymer decreases; this is contrary to what one would normally expect in polymer condensation reactions. This result is due to immiscible two phase nature of the reaction process. Additional experiments were performed to inquire about the role of the organic group on nodular size. It was hypothesized that nodule size is dependent on the difference in surface energy between the nodule and the free energy of the solution. Organically different silsesquioxanes and different solvents were used to investigate this hypothesis.
Subject: dimension, epoxy, fractal, microscopy, nmr, nodule, optical, silsesquioxane, thermosetting, toughness
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 161 pages.
General Note: Includes vita.
Thesis: Thesis (Ph.D.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 003504015
System ID: UFE0012142:00001

Permanent Link: http://ufdc.ufl.edu/UFE0012142/00001

Material Information

Title: An Investigation of the nodular microstructure of selected silsesquioxane and epoxy thermosetting resins
Physical Description: Mixed Material
Language: English
Creator: Kathan, Kyle R. ( Dissertant )
Baney, Ronald H. ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Materials Science and Engineering thesis, Ph.D
Dissertations, Academic -- UF -- Materials Science and Engineering

Notes

Abstract: The role of the nodular microstructure in the failure of thermosetting resins is unknown. The goal of the research presented here is to determine the mechanisms by which the nodular microstructure affects failure of thermosetting resins. Epoxy samples were fractured in tension and characterized for fractal dimension, nodular size, and toughness. Samples of epoxy resins have been shown in this work to have a relationship between nodular size and toughness. Two compositions were chosen to study the effect of strain rate and nodule size on toughness and a structural parameter, a0. Additionally fractal dimensional increment was measured using two separate techniques, Flaw to Mirror ratio (F-M), and Slit-Island contours. The fractal dimensional increment for slit-island contours was calculated using two similar algorithms; by hand, and with the aid of software. It was shown through the course of this work that nodule size does not directly relate to toughness. it was also shown that the epoxy resins characterized for this research are not structurally homogenous. It is believed that the inhomogeneities in the structure can be related to a0. Silsesquioxanes were synthesized using a novel two phase reaction. Samples were characterized for molecular weight and structure using matrix assisted laser desorption and ionization (MALDI) and silicon-29 nuclear magnetic resonance (Si-29 NMR). Additionally samples were characterized during growth using an Ubbelohde viscometer to characterize the growth processes of the two phase reaction. It was found that as the temperature of the growth phase synthesis increases, the viscosity of the polymer decreases; this is contrary to what one would normally expect in polymer condensation reactions. This result is due to immiscible two phase nature of the reaction process. Additional experiments were performed to inquire about the role of the organic group on nodular size. It was hypothesized that nodule size is dependent on the difference in surface energy between the nodule and the free energy of the solution. Organically different silsesquioxanes and different solvents were used to investigate this hypothesis.
Subject: dimension, epoxy, fractal, microscopy, nmr, nodule, optical, silsesquioxane, thermosetting, toughness
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 161 pages.
General Note: Includes vita.
Thesis: Thesis (Ph.D.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 003504015
System ID: UFE0012142:00001


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INVESTIGATION OF THE NODULAR MICROSTRUCTURE OF SELECTED
SILSESQUIOXANE AND EPOXY THERMO SETTING RESINS













By

KYLE KATHAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Kyle Kathan


































This work is dedicated to my grandparents William and Shirley Gray of Paulsboro, New
Jersey.















ACKNOWLEDGMENTS

I would first and foremost like to thank my family, my parents Chris and Russ

Kathan, my sister Kendra Kathan, my uncle Steven Gray, and my wife Dana for their

support and love while pursuing my graduate studies.

I would like to additionally thank my advisor, Dr. Ronald Baney, who has made

graduate school a worthwhile endeavor for me and has helped me through all of the

academic challenges I have faced over the past four years.

As far as my research goes, I must first acknowledge the University of Florida

Alumni Foundation for supporting me with a generous fellowship over the past four

years, which has allowed me to pursue the research that is of the most interest to me.

Additionally I would like to acknowledge the University of Florida McKnight Brain

Institute (MBI) for allowing me to use its phenomenal facilities. I would like to

acknowledge Jim Rocca and Tim Vaught of the MBI for their help with my research. I

would like to thank Jim Rocca for all his help with NMR theory and practice and thank

Tim Vaught for helping me with my optical microscopy work.

Additionally I would like to acknowledge the University of Florida Particle

Engineering Research Center (ERC) for its help with my research. Specifically, I would

like to thank Dr. Kevin Powers of the ERC for guiding my early work when I first started

at UF.









I would like to thank Dr. Jack Mecholsky of the UF Materials Science and

Engineering Department for helping me to better understand mechanical properties and

fractals, topics which consumed a large portion of this research.

Lastly I would like to thank the entire Baney group. It is my feeling that the Baney

group is far and away the most academically diverse group in the MSE department with

people studying everything from nuclear fuel to bioactive materials. Four years of the

Baney group have proved to be invaluable.















TABLE OF CONTENTS


A C K N O W L E D G M E N T S ................................................................................................. iv

A B STR A C T .................................................................. .... ........ ...... .. xv

CHAPTER

1 INTRODUCTION AND OVERVIEW OF CHAPTERS ............................................1

1.1 R research Intro du action .................................................................. ..................... 1
1.2 F ractal B asics .................................................. ................. .. 2
1.3 M materials Overview ...................................... ........................ ... .
1.4 C chapter O utline...................................................... 6

2 FRACTALS AND FRACTURE ............................................ .......................... 7

2.1 Introduction to Fractals......................................... ................................. 7
2.2 Fractals in Nature .................. ........................ ............... .... .............
2 .3 F ractal D im en sion .......... ..... ............................................................ ....... ....... ... 9
2.4 M echanical Properties ................................................ .............................. 15
2 .5 C o n c lu sio n s ..................................................................................................... 1 8

3 EPOXIES, SILSESQUIOXANES, AND NODULAR STRUCTURE REVIEW......20

3 .1 In tro d u ctio n ................................ ................. .................................................. 2 0
3.2 Review of Nodular Epoxies ............. .. ......... ..................... 20
3.2.1 Introduction to Epoxies ......................................................20
3.2.2 Processing and Synthesis Factors Affecting Nodule Size........................21
3.2.3 Observations of Nodule Size..................................... ................. 24
3.2.4 Mechanical Properties of Nodular Epoxy Resins.................. ......... 28
3.3 R eview of Silsesquioxanes ...................................................................................3 1
3.3.1 Introduction to Silsesquioxanes.................... .... ....................... 31
3.3.2 A applications of Silsesquioxanes............................................................. 34
3.3.3 Characterization of Silsesquioxanes................................ .....................36
3 .4 C o n clu sio n s.................................................. ................ 4 1

4 FRACTURE PROPERTIES OF EPOXY RESIN.....................................................42

4 .1 Introduction ................... ................... ........................... ................. 42
4.2 Epoxy Synthesis and Processing ........................................ ....... ............... 44
4.3 M ethodology and Experim ental ........................................ ........ ............... 45









4.3.1 Characterization of Mechanical Properties of Epoxy Resins...................46
4 .3.1.1 Introduction ............................ ............ .................. ............... 46
4 .3.1.2 M odulu s .............................. ........................ .... ........ 46
4 .3 .1.3 F ailu re stress............. .................................... ........ ... ........... 4 9
4 .3.1.4 F law size................ ............................. ................. 49
4.3.1.5 Fracture m echanics and toughness ......... ................................... 52
4.3.2 Studies of Nodular Microstructure and Nodule Size ..............................54
4.3.2.1 Scanning electron m icroscopy ............................... ............... .54
4.3.2.2 Solvent extraction and particle size............................................56
4.3.2.3 Iodine staining of surfaces.... ...................................58
4.3.3 Investigation of Fractal Dimensional Increment .....................................60
4 .3.3.1 Introduction ............................ ............ .................. ............... 60
4.3.3.2 Flaw to m irror size ..................... ............................ 61
4.3.3.3 Non-destructive slit-island method ..........................................64
4.3.3.4 Hand calculations of fractal dimensional increment........................70
4.3.3.5 Calculation of fractal dimensional increment by Image-Pro............73
4.3.3.6 Comparison of methods of measuring fractal imension.............. 74
4 .4 Structural P aram eter ao ........................................ ...................... .....................75
4.4.1 Calculating ao................................. .. .... ................... 75
4.4.2 Relationship of Fractal Dimensional Increment to ao.............................76
4.4.3 Error A analysis of ao ............................................................................. 78
4 .5 R results and D iscu ssion ........................................ ...................... .....................78
4 .6 C o n c lu sio n s ..................................................................................................... 8 0

5 SILSESQUIOXANES: GROWTH, STRUCTURE, AND CHARACTERISTICS ...82

5 .1 In tro d u ctio n ................................ ............................ ................ 8 2
5.2 Polysilsesquioxane Synthesis and Processing ............................................... 84
5.2.1 Polym er Synthesis ................................ .... ........ ............ .. .......... 84
5.2.2 Polysilsesquioxane Monolith Synthesis ............................................. 86
5.3 E xperim mental ........................................................ ...................... 87
5.3.1 Characterization of Polymer ............. ............................................. 87
5.3.1.1 V iscosity ............... ..................................... ..... .........87
5.3.1.2 Matrix assisted laser desorption ionization ....................................90
5.3.1.3 N nuclear m agnetic resonance.................................. ............... 92
5.3.1.4 Fourier transform infrared spectroscopy .......................................99
5.3.2 Condensation of Polysilsesquioxanes............. ...... ..............100
5.3.3 Characterization of Nodular Microstructure ................. ................... 101
5.4 Polym ethylsilsesquioxane ...................................................................... 106
5 .6 C on clu sion s................................................. ................ 10 7

6 RESULTS, DISCUSSIONS, AND FUTURE WORK............................................ 109

6.1 R results and D iscu ssion ..................................................................................... 109
6.1.1 Calculating Fractal Dimension with Optical Microscopy ........................109
6.1.2 Nodular Microstructure of Epoxies and Fractal Analysis of Failure .......109
6.1.3 Synthesis and Characterization of Silsesquioxanes...............................110









6 .2 F u tu re W ork ...................................... ............................. ................ 1 1 1

APPENDIX

A ERROR ANALYSIS OF THE WEST MECHOLSKY PASSOJA THEORY ........114

A 1 In tro d u ctio n .................................................... .. ...................... ................ .. 1 14
A.2 Sources of Error .......... .................................. .......... ... .. ............ 115
A.3 Error Analysis Equations ......... ........................................ ......................... 115
A .4 Error Analysis Graphs ............. ................................ ...... ...............116

B SILICON 29 NMR STUDY OF SILSESQUIOXANES............... .....................119

B 1 Introduction and M methodology ......... ............ ....... ........... ..................... 119
B .2 Si-29 N M R Spectra ............................................................................. ... ... 119

C STUDY OF POLYSILSEQUIOXANES ...................................... ............... 126

D MEASUREMENTS OF EPOXY SAMPLES .................................. ...............128

L IST O F R E F E R E N C E S ...................................................................... ..................... 137

BIOGRAPHICAL SKETCH ............................................................. ............... 145































viii















LIST OF TABLES
Table pge

2-1. Fractal Dimensional Increment for Common Families of Materials.........................17

3-1. GPC and MALDI Molecular Weights of Silsesquioxanes from Tecklenburg..........37

4-1. Modulus and Percent Error of Epoxy Resins at Different Strain Rates ..................46

4-2. Student's t-test Results of Modulus of Different Strain Rates .............................48

4-3. Failure Stress of Epoxy Resin at Different Strain Rates ..........................................49

4-4. Student's t-test of Failure Stress of Epoxy Resins ......................................... 49

4-5. Toughness of Epon 825 with 8ph DETA (MPa*m2)........................ ...............52

4-6. Toughness of Epon 825 with 10phr DETA (MPa*ml/2) ............ ......................52

4-7. Average Critical Crack Size for Epoxy Resins ([tm) .............. ................. 54

4-8. Fractal Dimensional Increment by Flaw to Mirror Size for Epoxy Resins ..............62

4-9. Student's T-test of Fractal Dimensional Increment of Epoxy Resins .....................63

4-10. Hand M easurements of Fractal Dimension .................................. .................71

4-11. Hand Calculations of Fractal Dimension of Epoxy Resins................................72

4-12. D* of Epoxy Resins Calculated With Image Pro ......................................... 73

4-13. Comparison of Fractal Dimensional Increment D* Values for Different
M methods ............................................................... ..... ..... ......... 74

4-14. Calculated ao values ([tm) for Epoxy Resins ............. ........................................ 75

4-15. ao Values ([tm) calculated from the Slope of the Fractal Dimension vs.
Toughness Plot .................................... ............................... .........77

4-16. Cumulative Error Values for ao Calculations for Epoxy Resins ...........................78

4-17. Calculated ao Values and Associated Error .......................................................79

5-1. Usage of Common Polysilsesquioxanes........................................... .................. 83









5-2. Synthesized Polysilsesquioxanes and Structures................... ..................................85

5-3. Total Hydroxide Content of Silsesquioxanes............. ...............................................95

5-4. Hydroxide Content of Trimethylsilylated Silsesquioxanes.................................97

5-5. Ratio of Network to Cage Structure of Silsesquioxanes from FTIR.....................100

6-1. ao and Error Values for Epoxy Resins by Strain Rate ...........................................110

C-1. Crosslinked Silsesquioxane Com positions ............................................................126

D-1. M odulus Values for Epoxy Samples ........................................... ............... 128

D -2. Failure Stress V alues for Epoxy Sam ples.................................... .....................129

D-3. Flaw Size and Toughness Measurements for Epon 825 with 8phr DETA Strained
at 0 .1 m m /m in .................................................................... 12 9

D-4. Flaw Size and Toughness Measurements for Epon 825 with 8phr DETA Strained
at 10 m m /m in ....................................................................... 129

D-5. Flaw Size and Toughness Measurements for Epon 825 with 8phr DETA Strained
at 100 m m /m in ......................................................................130

D-6. Flaw Size and Toughness Measurements for Epon 825 with 10phr DETA
Strained at 0.1 m m /m in ............................................................................ ... 130

D-7. Flaw Size and Toughness Measurements for Epon 825 with 10phr DETA
Strained at 10 m m /m in ........................................................................ 130

D-8. Flaw Size and Toughness Measurements for Epon 825 with 10phr DETA
Strained at 100 m m /m in ............................................................................ ... 13 1

D-9. Mirror Measurements for Epon 825 with 8phr DETA Strained at 0. 1mm/min .....131

D-10. Mirror Measurements for Epon 825 with 8phr DETA Strained at 10mm/min ....131

D-11. Mirror Measurements for Epon 825 with 8phr DETA Strained at 100mm/min ..132

D-12. Mirror Measurements for Epon 825 with 10phr DETA Strained at 0. mm/min .132

D-13. Mirror Measurements for Epon 825 with 10phr DETA Strained at 10mm/min ..132

D-14. Mirror Measurements for Epon 825 with 8phr DETA Strained at 100mm/min ..133

D-15. Image Pro Measurements for Fractal Dimensional Increment by Sample ...........133









LIST OF FIGURES
Figure page

1-1. Mandelbrot Set, Generated with Fractal Explorer 2.02.........................................3

1-2. Chemical Structure of Epoxy Resin Monomer and Crosslinking Agent..................5

2-1. Koch Curve, Five Iterations. Generated with Fractal Explorer 2.02...........................

2-2. IFS Fractal of Fern Leaf. Generated with Fractal Explorer 2.02..............................9

2-3. Perimeters of Nations by Richardson ............. ................................. .............. 11

2-4. Relationship of D* to M echanical Toughness.............................................. 17

3-1. Polymerization Reactions between Epoxides and Amines.....................................21

3-2. Schematic of an AFM Tip on Epoxy Surface..............................27

3-3. A FM of Fracture Surface ................................................................... ...........27

3-4. TEM of Epoxy Fracture Surface. ........................................ .......................... 27

3-5. Fractal Structure of Fracture Surface of Epoxy.................................................... 30

3-6. MALDI Molecular Weights of Silsesquioxane Fractions .............. ...............37

4-1. Stress-Strain Curves of Epon 825 with 8 phr DETA) ............................................47

4-2. Stress-Strain Curves of Epon 825 with 10 phr DETA.............................................47

4-3. Example Critical Crack Size Produced through Slow Crack Growth.....................50

4-4. Flaw Size of Epon 825 with 8phr DETA at 100mm/min Strain Rate.....................50

4-5. Flaw Size of Epon 825 with 10 phr DETA strained at 10mm/min .........................51

4-6. Ln-Ln plot of Strain Rate vs. KI, for Epon 825 Resins ....................................... 53

4-7. SEM Micrograph of Epon 825 with 8 phr DETA ............................................. 56

4-8. SEM Micrograph of Epon 825 with 10 phr DETA .................. ............... 56

4-9. EDS Spectra of Epoxy Stained with Iodine for 5 minutes .....................................59

4-10. EDS SEM Image of Iodine (Red) Stained Epoxy Surface................. ................59

4-11. Flaw Size and Mirror for Epon 825 with 8 phr DETA strained at 0.1 mm/min......62

4-12. Fractal Dimension vs Toughness for Epoxy Resins..............................................64









4-13. Slit-Island M ethod from H ill ............................................................................65

4-14. 3-D Composite Im age of Specim en................................... .......................... 67

4-15. 2-D Im age of Specim en ....................... ........ ................................. ............... 67

4-16. 0-50 Elevation Im age Section........................................... ........................... 68

4-17. 0-100 Elevation Im age Section.................... ......... ........................ ............... 68

4-18. 0-150 Elevation Im age Section.................... ......... ........................ ............... 69

4-19. 50-255 Elevation Im age Section........................................ .......................... 69

4-20. Magnification of Fracture Surface..................................................70

4-21. Richardson Plot of Perimeter of Fracture Surface....................... ............... 71

4-22. Square Root of Fractal Dimension vs. Toughness Calculated by Hand from Slit
Island Contours ............. ...... .............. ....................... 72

4-23. Square Root of Fractal Dimension vs. Toughness Calculated by Image Pro from
Slit Island C ontours .......................... ........................... .. .. .. .. ..... ..... .. 74

4-24. Comparison of Fractal Dimension Increment to Toughness for Three Different
T echniqu es of M measuring ........................................ .............................................77

5-1. Comparison of Silicone and Polysilsesquioxane Structures .................................... 82

5-2. Common Polysilsesquioxane Structures ....................................... ............... 83

5-3. Substrate C configurations. ........................................ ............................................86

5-4. Schematic of Ubbelhode Viscometer ............................................87

5-5. Viscosity of Polymethylsilsesquioxane................................. ...............89

5-6. Bruker Avance 600 Mhz Vertical Bore Spectrometer at MBI.................................94

5-7. NMR Spectra of Synthesized Silsesquioxanes ......................................................95

5-8. Bis(Trimethylsilyl)Acetamide Structure (BTM SAA) ...............................................96

5-9. NMR Spectra of Trimethylsilylated Silsesquioxanes..............................................97

5-10. Total, Reactive, and Non-reactive Hydroxide Content of Silsesquioxanes ...........98

5-11. FTIR Spectra of Synthesized Silsesquioxanes ............................................... 100









5-12. Polymethylsilsesquioxane M icrostructure..........................................................103

5-13. Polyethylsilsesquioxane M icrostructure............... .............................................103

5-14. Polypropylsilsesquioxane M icrostructure .................................. ............... 104

5-15. Polybutylsilsesquioxane Microstructure.......................................................104

5-16. Polymethylsilsesquioxane Cluster Structure ............................... ............... .105

5-17. Highlighted Two-Phase Image of Polypropylsilsesquioxane............................105

5-18. Cao-Baney Route Polym ethylsilsesquioxane....................................................... 106

A E effect of V ariables on ao............................................... .................................... 117

A -2. Effect of V ariables on Error of ao ................................ ......................... .. .......... 118

B Polym ethylsilsesquioxane......... ................. .................................. ............... 120

B-2. Derivitized Polymethylsilsesquioxane......................... ........................120

B -3. Polyethylsilsesquioxane......... ......... ................ .. ...................... ............... 121

B-4. Derivitized Polyethylsilsesquioxane.............................. ...............121

B-5. Polypropylsilsesquioxane ............... ............ ........... ............... ...............122

B-6. D erivitized Polypropylsilsesquioxane ............. ............ ................... .............. 122

B -7. Polybutylsilsesquioxane......... ................. ..................................... ............... 123

B-8. Derivitized Polybutylsilsesquioxane ................................... .................123

B -9. Polyvinylsilsesquioxane......... ................. ..................................... ............... 24

B-10. Derivitized Polyvinylsilsesquioxane..................................... ...............124

B-11. Polychloropropylsilsesquioxane ................................ ..................125

B-12. Derivitized Polychloropropylsilsesquioxane ............................. ..................1..25

D-1. Flaw Size of Epon 825 with 8phr DETA Strained at 0.1 mm/min......................133

D-2. Flaw Size of Epon 825 with 8phr DETA Strained at 10 mm/min......................... 134

D-3. Flaw Size of Epon 825 with 8phr DETA Strained at 100 mm/min....................... 134

D-4. Flaw Size of Epon 825 with 8phr DETA Strained at 100 mm/min....................... 134









D-5. Flaw Size of Epon 825 with 10phr DETA Strained at 10 mm/min.....................135

D-6. Flaw Size of Epon 825 with 10phr DETA Strained at 100 mm/min..................135














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

AN INVESTIGATION OF THE NODULAR MICROSTRUCTURE OF SELECTED
SILSESQUIOXANE AND EPOXY THERMO SETTING RESINS
By

Kyle Kathan

December, 2005

Chair: Ronald H. Baney
Major Department: Materials Science and Engineering

The role of the nodular microstructure in the failure of thermosetting resins is

unknown. The goal of the research presented here is to determine the mechanisms by

which the nodular microstructure affects failure of thermosetting resins.

Epoxy samples were fractured in tension and characterized for fractal dimension,

nodular size, and toughness. Samples of epoxy resins have been shown in this work to

have a relationship between nodular size and toughness. Two compositions were chosen

to study the effect of strain rate and nodule size on toughness and a structural parameter,

ao. Additionally fractal dimensional increment was measured using two separate

techniques, Flaw to Mirror ratio (F-M), and Slit-Island contours. The fractal dimensional

increment for slit-island contours was calculated using two similar algorithms; by hand,

and with the aid of software.

It was shown through the course of this work that nodule size does not directly

relate to toughness. Additionally, it was also shown that the epoxy resins characterized









for this research are not structurally homogenous. It is believed that the inhomogeneities

in the structure can be related to ao.

Silsesquioxanes were synthesized using a novel two phase reaction. Samples were

characterized for molecular weight and structure using matrix assisted laser desorption

and ionization (MALDI) and silicon-29 nuclear magnetic resonance (Si-29 NMR).

Additionally samples were characterized during growth using an Ubbelohde viscometer

to characterize the growth processes of the two phase reaction. It was found that as the

temperature of the growth phase synthesis increases, the viscosity of the polymer

decreases; this is contrary to what one would normally expect in polymer condensation

reactions. This result is due to immiscible two phase nature of the reaction process.

Additional experiments were performed to inquire about the role of the organic

group on nodular size. It was hypothesized that nodule size is dependent on the difference

in surface energy between the nodule and the free energy of the solution. Organically

different silsesquioxanes and different solvents were used to investigate this hypothesis.














CHAPTER 1
INTRODUCTION AND OVERVIEW OF CHAPTERS

1.1 Research Introduction

Thermosetting resins are polymeric materials that react or cure with the addition of

heat or energy. Thermosetting resins are generally very brittle and do not melt after

curing. Research has indicated that some thermosetting resins such as epoxies and

silsesquioxanes do not have homogenous microstructures, as once thought.

Epoxies and other thermosetting resins have been shown to have a nodular

microstructure. (Duchet et al. 2003) A nodular microstructure can be loosely described

as one containing spherical features of uniform size in a randomly packed order. Nodules

can range from tens of nanometers to a few micrometers, depending on the system

investigated and the processing used in synthesis. It is of interest to understand how

nodules influence the toughness and mechanical properties of epoxies and other

thermosetting resins. Previous research has indicated that there is a possible correlation

between nodular size and ultimate mechanical properties. (Baney et al. 1999) Using

conventional thought, it is not entirely obvious as to what role nodular size plays in the

toughness of thermoset resins. By applying a new philosophy of failure in brittle

materials, this dissertation research has sought to find a better understanding of how

nodules affect toughness.

Epoxies and other thermosets are unique in that they can be considered a hybrid

between ceramics and plastics. They are polymeric in nature and have a low modulus

compared to ceramics, yet they fail in a brittle manner, similar to glasses and crystalline









materials. Recently, researchers have shown that the toughness of brittle materials can be

linked to a structural parameter, ao, by relating the fracture surface of a specimen to a

fractal model. The WMP theory, developed by West, Mecholsky and Passoja,

(Mecholsky et al. 2002), indicates that toughness of a brittle material can be determined

from the following equation:

(1) K = E(aoD )Y

wherein Kic is the toughness, E is Young's Modulus, ao is a structural parameter, and D*
is the fractal dimensional increment.

This theorem will be reviewed in depth in Chapter 2. It was my hypothesis that this work

could show a clear relationship between the nodular size and the structural parameter ao.

Ideally this work would be used as a guide to help better understand what microstructural

factors alter the toughness of thermosetting resins. By understanding exactly how nodule

size affects the toughness of an epoxy, it should be possible to engineer a system with

increased toughness. Additionally this work would serve to increase the body of

knowledge to which the WMP theory has been applied.

1.2 Fractal Basics

Fractals are a class of complex geometric structures that are similar on all length

scales. In other words, fractals show essentially the same structural features, regardless

of magnification. While often very complex, fractals can often be described very

simply. On the most fundamental level all fractals can be described using three terms,

the initiator, the generator, and the rule. Basically the initiator is the structure that starts

off a fractal, the generator is the change in the initiator with each successive iteration, and

the rule is how the generator is applied to the initiator. Additionally, fractals can also be

characterized by their non integer dimension, D. Fractals are not like normal geometric









structures. A simple geometric structure would have a topological dimension of 2 if it

was a flat surface or 1 if it was a line. Fractals however, have dimensions greater than

their topological dimension. An example would be the surface of an orange. Ideally one

would say the surface area is equivalent to 4x7r2 where the exponent, 2, is the topological

dimension for a surface. However, upon closer inspection one would see that the surface

of an orange is full of pits and valleys and exhibits a very irregular surface. The surface

of an orange is in fact fractal and has a surface area equal to 47xrD, where D is the fractal

dimension and is between 2 and 3. The fractal dimensional increment, D*, as used in the

WMP theory, is the decimal portion of the fractal dimension. The fractal dimension can

be described as a quantification of the complexity of a given fractal structure and has

been used to quantify everything from basic fractals to coastlines of nations. (Richardson

1961)

















Figure 1-1 Mandelbrot Set, Generated with Fractal Explorer 2.02


Fractals have been applied to everything from art to science. Figure 1-1 is an

example of a common fractal known as the Mandelbrot Set and was generated with

Fractal Explorer 2.02. (Arthur 2005) Recent research has used fractals as a method of









quantifying the nature of brittle fracture in materials. It has long been established that the

fracture surface of a material is self similar, which is much like a fractal. A great deal of

research has been conducted defining the fracture surface in fractal terms. Researchers

have applied fractal basics to fracture to determine the relationship of microstructure on

toughness. (Borodich 1997; Borodich 1999) Current research into application of fractals

to fracture toughness has focused on glasses and crystalline materials. In previous work

it has been demonstrated that fractal dimension and toughness of a material are related by

the structural parameter, ao. (West et al. 1999; Mecholsky et al. 2002) Values for ao are

typically on the order of lattice parameters. This present work endeavors to expand on

current knowledge by seeking out materials which should have a larger value of ao. For

the purpose of this work, nodular materials have been investigated under the assumption

that ao will be a function of the nodular size. Two separate materials have been

investigated, epoxy and silsesquioxanes.

1.3 Materials Overview

Epoxy materials were originally thought to be homogenous. However work in the

late 1970's and early 1980's, primarily by Koutsky and associates, demonstrated that

epoxies are not homogenous. It was reported that epoxies have a nodular microstructure,

in which the nodular size is very uniform and closely packed together. (Racich et al.

1976) Additionally, it was found that the size of nodules depends on the amount of

crosslinking catalyst used and the curing schedule employed. (Mijovic et al. 1979) Also

of note, it was found that the mechanical properties of said systems are dependent on

nodular size. (Mijovic et al. 1979) Figure 1-2 is the chemical structure of the monomers

employed in this research.













H2N ~N NH2
H






Figure 1-2. Chemical Structure of Epoxy Resin Monomer and Crosslinking Agent.

Top: Diethylenetriamine, Crosslinking Agent
Bottom: Diglycidyl Ethers Bisphenol-A (DGEBA), Epoxy Resin Monomer

Silsesquioxanes are a technologically important inorganic polymers. (Baney et al.

1995) Most work with silsesquioxanes has focused on developing materials to be applied

to integrated circuits as a low-k dielectric interconnect layer. Silsesquioxanes are also

used as fillers and thin films in other applications. Traditionally, silsesquioxanes are

synthesized through a direct condensation reaction from chlorosilanes and water and

generally have the chemical formula RSiO3/2 where R is virtually any organic group.

Depending on the nature of silsesquioxane monomer, many possible structures can be

formed. Most commonly investigated silsesquioxanes have been reported to be a large

network polymer, although ladders and polyhedral structures can also be found. These

structures will be covered in Chapter 3. (Baney et al. 1995) Work presented here will

focus on a novel process for synthesizing silsesquioxane polymers using two immiscible

phases. (Kondo et al. 2000) It is postulated that hydrogen bonding organic solvents can

be used to guide and control the structure by hydrogen bonding to the silanol groups of

the growing polymer. In effect, this drives the polymer into the organic phase where









condensation reactions are slowed greatly, and the formation of small cage structures

such as the cubical octomer is retarded.

1.4 Chapter Outline

Chapter 2 begins a critical review of the applications of fractals to fracture

mechanics. An overall view of fractals and their applications in science is presented to

illustrate the relevance to this work. This chapter includes a detailed review of the topical

literature. Chapter 3 is a review of the materials used in this research. This incorporates

topics ranging from characterizing nodularity in epoxies to the synthesis and

characterization of silsesquioxanes. Chapter 4 highlights the relationship between

nodular microstructure and toughness of epoxy resins. High purity diglycidyl ethers of

bisphenol-A, DEGBA, (Epon 825) were cross-linked using diethylenetriamine (DETA).

After samples were cured, tensile testing was preformed at a variety of strain rates and

test conditions. Chapter 5 focuses on the two-phase process for synthesizing

silsesquioxanes. Viscosity studies were conducted to comprehend the nature of growth

of the silsesquioxane polymers. Additionally, NMR was conducted to better understand

the effects of the organic group on structure of the polymer. Finally, samples were cross-

linked using tin octoate, a common catalyst in this technology, and different solvents to

investigate the effects of solvent/polymer interaction on nodular growth. Additionally an

appendix was prepared to discuss error analysis of the WMP theory.














CHAPTER 2
FRACTALS AND FRACTURE

2.1 Introduction to Fractals

The term fractal was coined by Mandelbrot to describe a set of objects with

seemingly infinite detail. (Mandelbrot 1977) Fractals are complex geometric structures

that can be related to many scientific and natural phenomena. Fractals are self-similar or

self affine and scale invariant. Self-similar structures are the same in all directions in all

magnifications. Self-affine structures very similar to self-similar, but differ in one scale

in one direction. Scale invariant means that no matter what magnification or scale a

fractal is being viewed at, the structure is for all intents and purposes the same. Visually

fractals can look extremely complex; however they are often very simply described.

Fractals are often said to possess an infinite level of detail, but are often generated by a

simple iterative process.

All mathematical fractals can be described by three terms; initiator, generator, and

rule. The initiator is the starting structure from which a fractal will be formed. The

generator is how the structure changes with each successive step. The rule describes how

the generator is applied to the structure. Figure 2-1 is an example of the Koch curve.

The initiator in this case is a straight line; the generator is formed by two lines of equal

length that connect at a 60 degree angle. The rule for the Koch curve states that each

straight length of the curve is broken into three separate segments of equal length and the

middle segment is replaced with the generator. The generator is scaled to the length of

the segment it is replacing.














Figure 2-1. Koch Curve, Five Iterations. Generated with Fractal Explorer 2.02



Self-similar objects are ones that have been said to be invariant of scale. Self-

affine objects are very similar to self-similar fractals, but are scaled differently in one

dimension, x, y, or z. Fracture surfaces are considered self-affine because the

perturbations which form the features of the fracture surface are the same from one

region to another but are scaled differently. (Mecholsky et al. 2002)

2.2 Fractals in Nature

Fractal structures are commonly found in nature. Everything from trees and plant

life to clouds and weather patterns to the distribution of stars and gasses in the cosmos

can be described using fractals. (Bailey et al. 2004; Falgarone et al. 2004) Commonly,

geological structures are defined using fractals. (Volland et al. 2004) Properties such as

coastlines and erosion have been related to fractals. (Turcotte 1992; Carpinteri et al.

2004) Figure 2-2 is an example of an Iterative Functional System (IFS) fractal generated

with Fractal Explorer Software. Figure 2-2 is auspiciously similar to a fern leaf and is an

example of how natural structures can often be described with fractals.

Over the past two decades or so, fractals have been used extensively to characterize

natural phenomenon. Researchers have looked extensively at applying fractal theories to

failure, diffusion, and spread of disease and many more fields of research. Additionally,

chaos theory has used fractals to mathematically describe the apparent random nature of

the universe. (Gleick 1998)
























Figure 2-2 IFS Fractal of Fern Leaf. Generated with Fractal Explorer 2.02

2.3 Fractal Dimension

In Mandelbrot's famous treatise the question was asked, how long is the coast of

England? (Mandelbrot 1967) How long is the perimeter of any country? How long is the

Koch Curve for that matter? A simple question on the surface, but depending on what

scale is employed, the length is essentially infinite. The Koch curve begins as a straight

line of a defined length from which a fractal is formed. The problem becomes what size

ruler does one use to measure the length of the Koch curve? A smaller ruler will give a

larger length because as one measures along the curve, the smaller ruler will see the

smaller features that a larger ruler would not. An answer to Mandelbrot's question is that

the length of the coast of England is relative, being determined by the person measuring it

and the measuring stick they are using. Euclidean structures such as lines, disks, and

spheres have dimensions of 1, 2, and 3 respectively; however fractals are not so simple.

Effectively, fractals are non-Euclidean, as such the dimension must be different than in

Euclidean geometry. For Euclidean geometry, the dimension of a line is 1, but in

fractals, the dimension is a non-integer number.









The Hausdorff or fractal dimension is often used to characterize a fractal. The

Hausdorff dimension is a measure of the number of repeated features in a structure with a

decrease in scale or an application of an iteration of the generator to the initiator. The

Hausdorff dimension can be derived from equation 2-1:

C
(2-1) N
rD
Where N is the number of objects, r is the level of reduction and D is the Hausdorff

or fractal dimension.



A relevant example would be the dimension of the Koch Curve. Upon inspection

one would see there are 3 pieces to the Koch Curve, after applying an iteration, the

middle piece is replaced with 2 new pieces making 4 pieces total. The fractal dimension

of the Koch curve is 1.26. The fractal dimension can be thought of as a quantification of

the tortuosity of a fractal. The fractal dimension is the unifying value for which structures

or phenomena are characterized and related to a desired property.

Lewis Fry Richardson established that there is a linear relationship between the

measured perimeter of a country and the length of the rule used to measure. (Richardson

1961) Figure 2-3 is a representation of Richardson's work. Richardson's work was

originally intended to explain why nations of Europe went to war by comparing the

differences reported by two nations in the length of their shared border. Richardson's

work was later adapted for use in materials science and mathematics.

The Richardson method measures the perimeter of a complex object or fractal

using different ruler lengths. As ruler length decreases, overall perimeter increases.

This is because smaller rulers identify small features on the perimeter that a larger ruler







11


would not see. Richardson showed that the log of the ruler length versus the log of the

perimeter gives a linear relationship. The Richardson equation is given by equation 2-2

(2-2) L(s) = sD+c
Where s is the length, D is the fractal dimension and c is a constant

4.5 1 1 1 1
SAustralia
----- LogL(s))=-,13Log(s)+4,4
4 ---------- ----- - .
South Africa
Log(L(s))=-.04Log(s)+3.8



2.5 I- I
SLog(L(s))=-.12Log(s)+3.7
Great Britain
Log(L(s))=-.24Log(s)+3.7


y Log(L(s))=-.l12Log(s)+3.1

Log(s)


Figure 2-3 Perimeters of Nations by Richardson (Richardson 1961)

Later Mandelbrot would identify the slope of the Richardson work to be equivalent

to 1-D where D is the fractal dimension. It should be noted that in this work the fractal

dimension is between 1 and 2. For a line the fractal dimension is between 1 and 2, for a

surface, the dimension is between 2 and 3. (Borodich 1999) Rulers were used in the case

of Richardson's work; to calculate the fractal dimension of surfaces and bodies, a

dimension appropriate measuring device must be used such as squares for surfaces.

The fractal dimension is a measure of the tortuosity of a fractal. (Chen et al. 1993)

Commonly the fractal dimensional increment is reported. The fractal dimensional

increment or D* is the decimal portion of the fractal dimension. Fractal dimensional

increment is reported to show correlation between different techniques. As previously









mentioned a line fractal has a dimension between 1 and 2, and a surface between 2 and 3.

Fractal dimensional increment can allow one to compare line and surface fractals.

In recent years the fractal dimensional of a fracture surface has been related to the

mechanical properties of a material. (Mandelbrot et al. 1984) As such it has become

important to determine methods to measure the fractal dimension of a fracture surface.

There are many different techniques for measuring the fractal dimension.

Mandelbrot pioneered the use of fractal dimension as a descriptive tool in

analyzing fracture surfaces. It has been shown extensively that fracture surfaces for

brittle and ductile materials are fractal. Many research groups have attempted to relate

fractal dimension to mechanical properties. (Mecholsky et al. 1991; Chen et al. 1997;

Charkaluk et al. 1998) There is some disagreement as to the relationship between fractal

dimension and mechanical toughness. Bigerelle has attributed this disagreement to the

different methods used to measure fractal dimension. (Bigerelle et al. 2004) Many

different techniques have been used to measure fractal dimension; slit-island, vertical

section, box counting, and projected area. These are just a few of the techniques that

have been reported in the literature, each of these methods can result in a different fractal

dimension for the same fracture surface.

Hill has reported a protocol for measuring fractal dimension using coast lines of

polished fracture surfaces set in epoxy. (Hill et al. 2001) Further work of Hill and Della

Bona using the same technique indicated that fractal dimension is dependent on the

contour angle of the sample. (Della Bona et al. 2001). The protocol described by Hill can

be called a slit-island method, which was first described by Mandelbrot. (Mandelbrot et

al. 1984) Samples are set into epoxy and polished down until islands appear. The islands









are formed by removing the top of a rough section on the fracture surface. The perimeter

of the islands is measured using several different ruler lengths in a fashion in accordance

with the Richardson method.

The slit-island technique has been used extensively in the literature as a method of

measuring fractal dimension for comparison to fracture properties. Bigerelle has laid out

certain criteria that the concept of slit island was deduced from: (Bigerelle et al. 2004)

(i) When islands are derived from initial self-affine fractal surface of dimension Ds
by sectioning with a plane, their coastlines are self-similar fractals with dimension
D = Ds- 1.

(ii) The relation between perimeter and area is given by the equation:

[P(y)]1/D
(2-2) R() = [P
VA(17)

where R(r) is a constant which depends on the choice of the yardstick length, h,
used to measure the length along the walking path. This equation is only true for
self-similar whose perimeter and area are measured in the same way.

(iii) When the graph of log(P) versus log(A) is rectilinear, the fractal dimension is
deduced from the slope.

Bigerelle further finds that the slit-island method can be a statistically significant

measure of the fractal dimension with proper choice of ordinates and abscissa.

Additionally it was also shown that the slit-island method can produce artifacts when

correlating the fractal dimension of a surface to a particular physical process or testing

parameter.

It was additionally reported by Hill that fractal dimension was dependent on the

technique employed. It was found that the slit-island technique gave fractal dimensional

increment values between 0.08 and 0.28. The vertical profile technique and indentation

technique also reported resulting in much lower values.









The slit-island techniques such as that reported by Hill, produce dimensions

between 1 and 2. This method gives an approximation to the true fractal surface.

Although generally accepted, these values are not true fractal dimensions for the surface.

A surface fractal should have a dimension between two and three. Joseph and associates

used atomic force microscopy to generate a three dimensional image of a fracture surface

of epoxy samples. (Joseph et al. 1998) The values found by AFM are between 2 and 3

and can be considered to be the true fractal dimension of a surface. The drawback of

AFM is that only very small regions are measured, generally less then 100 trm2.

Conversely, the work of Xie used laser profilometery to generate a 3-D map over very

large samples, several millimeters on a side. (Xie et al. 1998)

The algorithm used by the AFM in the work of Joseph uses a technique similar to

the Richardson method for calculating dimension of a line. The AFM uses progressively

smaller cubes to measure the volume. Conversely the Richardson method uses

progressively smaller lines to measure a perimeter. The technique reported by Xie uses a

projective covering method where the fracture surface is broken up into progressively

smaller squares. The surface area of the fracture surface is found by measuring the

surface area of the portion of the fracture surface projected in the square. The projective

covering method has found much more use in the literature than AFM. (Stach et al.

2001; Stach et al. 2003; Stach et al. 2003; Stemp et al. 2003). It should be noted that

samples characterized using surface projection techniques are often considered to be

multifractal. A surface is considered multifractal if there are multiple fractal dimensions

depending on scale length.









Dubac reported a technique for measuring fractal dimension using vertical profiles

of fracture surfaces. The profile is laid against several grids of squares of different sizes.

The number of squares for each size used to cover the entire trace of the profile is used to

calculate the fractal dimension. (Dubuc et al. 1989) Hill has reported that using vertical

profiles gives vastly different results than the slit-island technique. (Hill et al. 2001)

Stach has argued that part of the discrepancy of profile techniques is that the amount of

surface characterized is very low and can not be descriptive of the entire surface,

especially for surfaces that are multifractal or self-affine. (Stach et al. 2003)

There is currently no standard method for measuring fractal dimension of a surface.

In general the technique used to measure fractal dimension is dependent on the author's

preference. Due to the disparity in techniques and thus dimension of measured surfaces,

there is some debate in regards to the relationship of fractal dimension to mechanical

properties of material.



2.4 Mechanical Properties

Fractal analysis has been used extensively in materials science to relate the

geometry of the fracture surface to mechanical properties. (Mecholsky et al. 1991; Lyu et

al. 1994; Balankin 1995; Charkaluk et al. 1998; Su et al. 2000; Issa et al. 2003) Many

different properties of materials have been related to fractal dimension, including impact

energy, fracture toughness, and crack propagation. (Bigerelle et al. 2004) The wide range

of fractal dimension values, as measured using different techniques, has led some

researchers to believe that fractal dimension is merely a universal value or parameter that

has no correlation to mechanical properties. Some further argue that fractal dimension is

a measure of the amount of plastic deformation associated with failure of a specimen.









Experimental data has indicated that there is a relationship between mechanical

properties and fractal structures. (Borodich 1999) Researchers have related the

parameter fractal dimension to the toughness and fracture surface of brittle and ductile

materials. (Mandelbrot et al. 1984; West et al. 1999; Stach et al. 2001). Mandelbrot's

early work pioneered the study of fractals on fracture surfaces. West et al. have shown

that surface energy of formation of a fracture surface for brittle materials is directly

related to the fractal dimensional increment of a surface. (West et al. 1999) This

relationship is given by equation 2-3.


(2-3)y = EaoD
2
Where y is the surface energy of formation, E is Young's modulus, ao is a structural

parameter and D* is the fractal dimensional increment.



Often fracture surfaces and fractals are quantified for the fractal dimension.

Mecholsky has shown there to be a roughly linear relationship between fractal dimension

and toughness for materials of the same family (glasses, ceramics, and crystals). (Hill et

al. 2000) Figure 2-4 is a graph plotting the results of Mecholsky.

Table 2-1 is a table of fractal dimensional increment for common materials from

West. (West et al. 1999) It should be noted that on small scales, fracture surfaces are

purely fractal. However it has been reported that the disorder of long range structures

gives rise to multifractal surfaces. (Xie et al. 1998; Xie et al. 1999; Babadagli et al. 2001;

Stach et al. 2001; Carpinteri et al. 2002; Carpinteri et al. 2003; Stach et al. 2003; Stach et

al. 2003) Multifractal surfaces are ones where there are more then one fractal dimensions

for the same surface.



















K 3 ,. ,
K Ic 3
(MPa-mi2i)





Cows Graiin CeoiariN

0.1 0.2 03 0,4 0.5 OA


(D*)1"


Figure 2-4 Relationship of D* to Mechanical Toughness(Hill et al. 2000)

Table 2-1 Fractal Dimensional Increment for Common Families of Materials
Class D*
Single Crystals 0.07-0.12
Glasses 0.007-0.1
Glass-ceramics 0.06-0.3
Polycrystaline ceramics 0.06-0.35
Polymers 0.2-0.29
(West et al. 1999)

The fracture surface of a material can be viewed as being composed of

perturbations that are deviations from a flat plane. These perturbations are formed by

constructive and destructive addition of breaking atomic bonds along the crack plane. It

has been shown by several research groups that fractal dimensional increment can be

related to surface energy of a fracture surface and toughness. (Mandelbrot et al. 1984;









Issa et al. 2003) The toughness of a material can be related to fractal dimensional

increment through the following equation:

(2-4) Kc = E(D ao)12
Where E is the Young's Modulus, D* is the fractal dimensional increment, KI, is

the plane strain mechanical toughness, and ao is a structural parameter.



The parameter ao has been extensively investigated by West, Mecholsky, and

Passoja. (Mecholsky et al. 2002) They report the use of molecular dynamic calculations

to calculate the fractal dimension of silica glasses and silicon crystals. They report that

ao is a measure of the free volume that is formed by breaking bonds at the crack tip.

Therefore ao can be described as the initiator of the fractal to the D* generator.

Using the West Mecholsky Passoja Theory one can relate the fractal dimensional

increment directly to the toughness of a material. However ao can not be directly

measured and must be calculated from other measurements. It is of interest to understand

the processing parameters that affect ao; if one were able to control ao, one should be able

to control toughness.

2.5 Conclusions

Fractals are complex geometric structures which have been shown to have a major

role in nature. Many research groups have used fractals to describe the fracture surface

of a material. Additionally there is significant evidence that the fractal dimensional

increment is directly related to toughness. West, Mecholsky, and Passoja have shown

that toughness can be related to fractal dimension by a structural parameter ao.

The fractal dimensional increment is a measure of the fractal nature of a fracture

surface. There currently exist several different techniques for measuring fractal







19


dimensional increment such as slit-island, and projected area. Commonly the slit-island

technique is the accepted method of measuring fractal dimensional increment of fracture

surfaces.














CHAPTER 3
EPOXIES, SILSESQUIOXANES, AND NODULAR STRUCTURE REVIEW

3.1 Introduction

Two materials of commercial interest were investigated in this dissertation, epoxy

resins and silsesquioxane resins. These materials were chosen because of their

microstructure and extensive supportive literature. In this chapter a review of the

relevant literature in regards to both materials and nodular microstructure.

3.2 Review of Nodular Epoxies

3.2.1 Introduction to Epoxies

Epoxies are organic polymers containing epoxide rings. An epoxide ring is a three

member ether ring containing an oxygen and is under strain and thus very reactive.

Epoxies are commonly used in composites and structural materials. Additionally epoxies

are used as adhesives and electrical insulators. Epoxies are two-part thermosetting resins

that require a catalyst to initiate polymerization. The addition of energy and catalyst cure

an epoxide resin into a rigid three dimensional structure that does not melt upon

reheating.

Frequently, amine or anhydride catalysts are used to crosslink epoxy resins. Figure

3-1 is a figure of common crosslinking reactions between epoxide and amine groups.

Due to their complexity the microstructure of epoxy resins is not very well understood.

Some groups have claimed that epoxies have a nodular microstructure, while others claim

that any asserted proof is purely circumstantial.










--CH-CH2 + --I PH. -CH-CH2,N-
0 COH H



CH-H-CHz-N- -- -+CHCH-CH-N-
0 OH H OH



-CH-CH2 + -CH- -- -CH-
O OH C-CH2-CH-
I
OH

Figure 3-1. Polymerization Reactions between Epoxides and Amines

3.2.2 Processing and Synthesis Factors Affecting Nodule Size

The discovery of an inhomogeneous microstructure in epoxy resins dates back to

the late 1950's. (Errath et al. 1959) Since that time few research groups have investigated

epoxy resins for nodularity. Researchers have generally focused on three processing

parameters when investigating nodule size; substrate, cure time/temperature, and resin to

cross linker ratio. (Racich et al. 1976; Mijovic et al. 1979; Takahama et al. 1982) It has

been shown that the nodules are connected by an interstitial material forming a sort of

micro gel, although the nature of this interstitial material is unknown. (Vanlandingham et

al. 1999; Lopez et al. 2002; Kozlov et al. 2004) Conversely, not all research groups are

convinced that epoxy resins are nodular. (Dusek et al. 1978; Duchet et al. 2003) Most

notably, the early work of Koutsky et. al. from the University of Wisconsin focused

heavily on determining the parameters that influence nodular size. (Racich et al. 1976)

This work seemed to indicate a correlation between the synthesis conditions and

nodular size. Koutsky and Racich found that free energy of the surface on which the

epoxy was cured had an effect on final nodule size. At first, one would assume this

would be a result of heterogeneous nucleation of the nodule at the interface. However









this does not explain why the bulk has the same nodule size as the surface. Their work

used completely different substrates ranging from copper to various plastics to change the

free energy of the surface.

Samples were prepared from Epon 825 resin and Diethylenetriamine. Nodule size

was determined using Transmission Electron Microscopy. Koutsky and Racich drew

eight distinct conclusions initially quoted below: (Racich et al. 1976)

1. There is a definite nodular formation through the bulk of the epoxy resins studied
2. The nodules have sizes ranging from 10 to 60 nm
3. At times, the nodules appear to align and form networks either at interfaces or in
the bulk
4. Nodules are observed to agglomerate into larger supemodules
5. Teflon and silicone rubber contact with epoxy inhibits the appearance of nodules
while copper and glass induce large nodules or increase packing density of small
nodules
6. No consistent relation has been found among composition or cure, nodule size, and
nodule density
7. Slightly cured, soft epoxy resins show what may be interpreted as nodule
precursors
8. Some individual nodules surprisingly show a very subtle ridge-like fine structure
on both fracture and free surfaces
What is not clear from that work, however, is if it is truly the surface energy of the

mold that affects nodule formation, or if another surface phenomenon such as crystal

structure or difference in physical properties of the various substrates such as porosity is

responsible.

A better study would have been to use silicone molds. The silicone could be easily

modified with any one of numerous agents to change the surface energy. Doing this

would result in molds or substrates that are similar to one another in terms of structure,

porosity and other physical properties, but have different surface energies. To date no

one has investigated the effects of surface energy modified silicone molds on nodule size.









Additional experiments by Koutsky and other researchers focused on the impact of

chemical composition on nodule size. (Vanlandingham et al. 1999) Results indicate that

an increase in cross-linking catalyst results in smaller nodule sizes. This would seem to

suggest that nodules form from nucleation events, similar to crystals precipitating from a

melt. A higher content of cross-linker would produce more nucleation events in a given

volume of resin. The difference between nodules and crystallization is that the nodules

seem to all be the same size and shape, and they are connected with a similar material as

the nodule itself. However crystallization of a melt can result in a large distribution of

grains and irregular shapes. Additionally, the grains impinge upon one another and there

is no additional material connecting them together as in the epoxies It is unclear from

this analogy as to why nodules are so perfectly spherical and of constant size. Many of

the detractors of the nodule microstructure of epoxies cite the fact that no conclusive

proof of a difference between the interstitial material and the nodule, which they believe

illustrates that the microscopic evidence for nodules are artifacts of either the imaging

system or the manner in which the sample was prepared. (Oberlin et al. 1982; Duchet et

al. 2003) Although Pollard confirms this observation by relating the gelation of epoxy

the Avrami theory of phase change.(Pollard et al. 1987) The Avrami theory appears to

hold true until at least the gel point of the resin.

Another variable investigated for its influence on nodule size of epoxy resins is the

processing parameter. Mijovic and Wu have independently shown that mixing and

processing conditions has an effect on nodular size of the final product. (Mijovic et al.

1985; Wu 1991) Epoxy resins are thermosetting and generally need heat to initiate

curing. The temperature and length of curing can greatly shape the final properties of a









resin, including the nodule size. Mijovic found that by adjusting the curing schedule, one

was able to adjust the presence of nodules. Proper control of curing resulted in a resin

with no nodular structure, but a rougher fracture surface.

Wu, on the other hand, studied the consequences of stirring on nodule size. His

work found that greater care in stirring resulted in smaller nodule sizes, but the

mechanical toughness did not change. This is an interesting result. One could infer from

this work that a higher rate of stirring mixes the resin and catalyst better, forming a more

homogenous starting material. Using the crystallization analogy from earlier, a more

homogenous mixture would result in more nucleation events and thus more nodules. The

reasoning as to why the toughness remains the same is that while the microstructure

changes the total cross-link density remains the same. From this, one could infer that

nodule growth and size is related to the solubility of the nodule in the resin/catalyst

solution. (Wu et al. 1985) As nodules grow their solubility decreases, which determines

the future growth capacity of nodules.

The exact nature of nodule growth is unknown however. Researchers have related

the growth of an epoxy network to a fractal structure. (Kozlov et al. 2004) Fractals are

complex geometric structures that are finding tremendous use in characterization of

polymerization reactions and growth kinetics. (Schaefer et al. 1986)

3.2.3 Observations of Nodule Size

Many researchers believe that epoxy resins are inhomogeneous and have a nodular

microstructure; conversely numerous others believe that the observed structure is merely

an artifact. Duchet lists three reasons as to why nodules are not real: (Duchet et al. 2003)

1. Epoxy networks cured with various amine curing agents, having stoichiometric
and nonstoichiometric compositions, have been studied. From electron microscopy
observations, no correlation between the nodular structure and the crosslinking









density has been obtained. Moreover, etched uncross-linked polymers such as
poly(methylmethacrylate) can present a nodular structure similar to that of epoxy
networks. Oberlin et al. analyzed more deeply epoxy resin morphology by
transmission electron microscopy and found that cured epoxy networks based on
DGEBA and diamine were homogeneous and remained stable while under study in
a clean vacuum. (Oberlin et al. 1982) However, in a poorer vacuum, electron
irradiation etches the sample. Progressively, nodules about 100 nm in size appear.
These investigations show that this nodular structure is not linked to
inhomogeneous cross-linking, and the authors have ascribed the nodular structure
to the interactions of etching agents with the sample surface (linear or crosslinked).

2. Fluctuations in cross-linking densities should be put into evidence by scattering
methods. Dusek and coworkers (Dusek et al. 1978) observed no difference
between scattering curves realized on linear and amorphous thermoplastics and
curves realized on cured epoxy networks. Epoxy-amine networks exhibit only one
sharp and well-defined relaxation peak related to the glass-transition temperature
(Tg). Therefore, there is no physical proof of structural inhomogeneity.

3. Moreover, if inhomogeneities are formed during the curing process, the kinetics,
the evolution of the distribution of i-mers, the gel point conversion, and so forth
should be affected. However, these parameters for a nodular structure are quite well
described by equations determined by statistical calculation if a single reaction
mechanism and quite homogeneous curing are assumed. This perfect agreement is
proof of the epoxy network homogeneity.

Many researchers have tried to prove the existence of nodules and investigate the

nature of the interstitial or connective material. Most have focused on two separate

methods; microscopy such as scanning electron microscopy (SEM) and transmission

electron microscopy (TEM), and scattering methods such as Light Scattering and Small-

angle X-Ray Scattering (SAXS).

Electron microscopy techniques have often confirmed the existence of nodules.

Works of Koutsky and Mijovic have all used SEM and TEM to show conclusively that

epoxies have a nodular microstructure. (Mijovic et al. 1977; Mijovic et al. 1979; Mijovic

et al. 1979) However, work of Koutsky and Ulhmann using SAXS did not show any

structure at the length scale which nodules have been observed. (Matyi et al. 1980) This

has led many researchers to believe they are nonexistent and are surface artifacts of









sample processing. Conversely, the work of Gupta argues that the electron beam can etch

the surface of epoxy samples and is thus the origin of the nodular structure observed by

other groups. (Gupta et al. 1985).

Stevens used Light Scattering to show that as epoxy cures, two phases principally

form. Stevens studied two epoxy systems and found different primary phase sizes

depending on the system. Because the exact indexes of refraction of the solid and solvent

phases were unknown, true size and distribution could not be calculated. It was, however,

estimated that the difference in indices in refraction was as low as 1%. One could infer

that cross-link density and thus modulus is proportional to index of refraction. Using this,

one could further infer that the difference in modulus between nodule and connective

material is very low. (Stevens et al. 1982)

Dusek provided an in-depth review of the formation of epoxy networks and

potential sources of inhomogeneities. (Dusek 1986) He cited evidence claiming results

found in light scattering are due to inhomogeneities of the epoxy resin without any cross-

linking catalyst added and are often larger then nodules observed with electron

microscopy. Dusek goes on further to state that epoxies are homogenous and not nodular

due to the closeness of gel point of a resin to the predicted value, as determined

kinetically. Many researchers have used atomic force microscopy (AFM) to study the

fracture surfaces of epoxies. AFM is a projection of the surface and does not give exact

nuances on the underlying structure. Figure 3-2 is a schematic of an AFM tip on the

surface of a nodular material. (Duchet et al. 2003) Figure 3-3 is an AFM image of an

epoxy resin as reported by Akari. (Araki et al. 2002). Conversely Figure 3-4 is a TEM

image generated by Koutsky of a fracture surface. (Mijovic et al. 1979)















- 4


AF*
"N \


SSurac c


Figure 3-2. Schematic of an AFM Tip on Epoxy Surface. (Duchet et al. 2003)


Figure 3-3. AFM of Fracture Surface. (Araki et al. 2002)


















Figure 3-4. TEM ofEpoxy Fracture Surface. (Mijovic et al. 1979)

From Figure 3-2 one can see how an AFM can "miss" data as the tip traverses

across the surface. By comparing Figure 3-4 to Figure 3-3 one can clearly see nodules in


ip"~









the surface; however Akari has reported that those are not nodules, but merely a complex

surface structure. (Araki et al. 2002) AFM has often been used to disprove the existence

of nodular microstructure. With proper preparation of a sample surface and test

parameters, it has been shown that AFM can be used to map compositional differences on

the surface of a material. Researchers have used these techniques to investigate the

surface of epoxy fracture surfaces for differences in modulus between nodule and

connective material.

Duchet used AFM to explore the nature of the interstitial material and to measure

differences in modulus between the two phases. He reports that the AFM gives a

homogenous surface and therefore there are no differences between nodules and

interstitial material; this nodules are artifacts of other imaging techniques. (Duchet et al.

2003) Additionally, work by Bai using Small Angle Neutron Scattering (SANS) to

investigate cross-link density indicates that the cross-link density is homogenous

throughout the structure. (Bai 1985).

The arguments presented by Duchet above are invalid because they assume that the

nodule and the connective or interstitial materials are inherently dissimilar. Currently no

one has proven conclusively that the nodule and connective material are significantly

different from one another. Many of the techniques which can be used to investigate the

differences in density or modulus of the two phases may not be sensitive enough to see

the differences, if any exist.

3.2.4 Mechanical Properties of Nodular Epoxy Resins

In general, the toughness of epoxy resins is between 0.4 and 1.8 MPa*ml/2

(Plangsangmas et al. 1999; Araki et al. 2002) Epoxies are generally thought to be brittle

in nature, although at temperatures above the glass transition temperature, there can be









significant plastic deformation at the crack tip. Studies have shown that epoxies age

slightly after initial curing and post curing. (Jo et al. 1991) The consequences of aging

can result in changes to the mechanical and thermal properties. The nature of the change

depends on the how close the formulation is to the stoichiometric ratio.

Many works have observed that a failure crack in an epoxy resin propagates

through the connective material as opposed to through the nodule. (Mijovic et al. 1981;

Woo et al. 1991; Wu 1991; Vanlandingham et al. 1999). It is unclear, however, how

nodules affect toughness. It has been shown that samples with the same glass transition

temperature (Tg) but different nodule sizes have the same toughness. This indicates that

it is the cross-link density, (which is proportional to Tg) that is the determining factor of

toughness not the nodule size. (Mijovic et al. 1985) However it has also been shown that

nodule size does affect toughness, but in similar manner to how cross-link density shapes

toughness. (Mijovic et al. 1979; Mijovic et al. 1981)

Properties are generally at a maximum when the ratio of amine to epoxide is

stoichiometric. However as noted by Vanlandingham, amine molecules can aggregate at

the surface and result in a localized region that is not at the stoichiometric point and thus

has different properties than the bulk. (Vanlandingham et al. 1999) Mijovic has shown a

correlation between toughness and dynamic mechanical properties of nodular epoxy

resins. (Mijovic et al. 1979; Mijovic et al. 1981) However it should be noted that

mechanical properties do not follow a linear relationship to nodular size. Instead

properties follow a parabolic relationship, first increasing with nodular size and amine

content and then decreasing. Peak values are found to be around the stoichiometric point.









Like brittle ceramics, when epoxy samples are loaded until failure, a fractal

structure forms on the fracture surface. This fractal structure is characterized by three

distinct regions on the fracture surface; mirror, mist, and hackle. Figure 3-5, adapted

from Plangsangmas, is a schematic of an epoxy fracture surface. (Plangsangmas et al.

1999)








Flaw Hackle

Region



Mirror Region Mist Region


Figure 3-5. Fractal Structure of Fracture Surface of Epoxy. (Plangsangmas et al. 1999)

Fractal structures are often characterized for fractal dimension. Fractal dimension

is a measure of the tortuosity of a fracture surface. Joseph et al. has shown that AFM can

be used to calculate the fractal dimension of epoxy fracture surfaces. (Joseph et al. 1998)

Their work has shown that epoxies have a fractal dimension of approximately 0.26 for all

fracture regions (mirror, mist, and hackle); this indicates that fracture surfaces are self-

similar. It is unclear at this point how fractal dimension relates to nodule size and epoxy

resins, but it has been shown that fractal dimension can be related to toughness. An in-

depth review of fractal dimension can be found in Chapter 2.









Additional materials have been shown to have nodular microstructures. Examples

of other nodular materials are silsesquioxanes and certain polymer blends. (Lopez et al.

2002; Auad et al. 2003) Nodular microstructures in polymer blends are attributed to the

insolubility of one phase in another. Silsesquioxanes, however, are similar to epoxies in

structure and are also considered to be thermosetting polymers; it is unknown as to why

they form nodular structures.

3.3 Review of Silsesquioxanes

3.3.1 Introduction to Silsesquioxanes

Silsesquioxane materials are hybrid inorganic-organic materials combining key

properties of both ceramic materials and polymeric materials. Silsesquioxanes are silicon

based materials defined as having three bridging oxygens and a fourth organic group.

The classification and ultimate properties of silsesquioxane are dependent on the fourth

group attached to the central silicon. There are virtually limitless possible organic units

that can be attached to the central silicon atom. This research will focus on the properties

of novel polymethylsislesquioxanes synthesized from methyltrimethoxysilane and the

relationship between microstructure and fractal failure.

Polymethylsilsesquioxane (PMSQ) has a nodular microstructure. The

microstructure is similar to that of epoxy resins. (Racich et al. 1976) It will be postulated

by Dr. Baney that the scale of the nodular microstructure determines the ultimate

mechanical properties of the material.

Silsesquioxanes are synthesized by sol-gel like reactions. Traditionally, silica sol-

gel reactions use acids or bases to promote hydrolysis and condensation reactions in

precursors. (Brinker et al. 1989) Monomers condense to form very small colloidal

particles which then connect together to form the gel network. The two chemical









reactions that typify most sol-gel systems are an alkoxy hydrolysis reaction and a

hydrolytic condensation reaction. The typical reactions for silica sol-gel are:

SiOR + H20 <-> SiOH + ROH
and
SiOH + HOSi <--- SiOSi +H20
Studies have shown the pure silica sol-gel, derived from tertramethoxysilane or

similar compounds can have an inherently fractal nature on the colloidal scale. (Schaefer

et al. 1986) The clusters have a mass fractal; essentially the mass of the cluster is

proportional to rdm where dm is the mass fractal constant. Mass fractal values are easily

measured using Small Angle X-ray Scattering (SAXS). (Orcel et al. 1986) This value is

not to be confused with D*, a brittle fracture fractal constant used in the WMP Theory.

Traditionally sol-gel reactions have very large volumes of solvents that must be removed

from the final body. The result is an extremely porous network. The combined effects of

Ostwald ripening and sintering to remove the porosity eliminates virtually all of the

microstructural evidence of a cluster microstructure. The techniques used to synthesize

the polymethylsislesquioxane that will be studied in this research result in a dense body

with zero porosity. As a result the samples do not have to be heat-treated and the desired

structure is not lost.

It has been previously established that silsesquioxane resins have a nodular

microstructure. (Baney et al. 1999) The structure of the resin is a result of both

intramolecular condensation, where the molecule folds back on itself creating a closed

cluster, and intermolecular condensation, where two large silsesquioxane clusters connect

to form a larger network. The degree of inter- versus intramolecular condensation is

dependent on the size and charge distribution of the organic group and the processing

parameters. A larger more steric organic group will lead to more intramolecular









condensation and a smaller group will result in more intermolecular condensation.

Unique to methylsilsesquioxanes, the very small organic group (methyl) can lead to an

extremely high degree of intramolecular condensation under certain reaction conditions.

This structure is commonly called the T8 cubical octomer, which consists of a silica cube

with methyl groups on each corer. Several novel techniques have been developed to

prevent the formation of cubical octomer and force the structure to form an open

network. Many of these techniques focus on complex solvent interactions that serve to

control the organic/inorganic nature of the resin, or unique hydrolysis condensation

routes. (Bourget et al. 1999; Takamura et al. 1999; Kondo et al. 2000; Crouzet et al.

2003) The research in this work focuses on a non-catalytic route, foregoing the

traditional use of acid or base in sol-gel chemistry.

In addition to the cubical octomer and the random network, a third possible

structure can be found in silsesquioxane materials. First identified in phenyl

silsesquioxanes, a ladder structure can be formed upon hydrolysis. (Baney et al. 1995)

The presence of a ladder structure is characterized via x-ray diffraction (XRD). When a

silsesquioxane resin is examined with XRD, two broad halos are formed. One halo

corresponds to the size of the silica tetrahedron. The second halo occurs at smaller 20

values, corresponding to the spacing across the width of the ladder or the X-Si-O-Si-X

length. For polymethylsilsesquioxane, this spacing corresponds to approximately 8.14A.

Computer modeling has long been used in sol-gel reactions to model how clusters

form and approximate sizes of the clusters. It should be noted that clusters are not

nodules. Generally clusters are less than 1 nm in diameter, nodules, conversely, are

generally 10 nm to 1 |tm in diameter. It is convenient to think of clusters as the building









blocks of nodules. The model most commonly used is called the Eden model and

assumes that monomers will add randomly to available sites on the cluster. A

modification of the Eden model, called the Poisoned Eden model, uses model monomers

that have been poisoned, or not completely hydrolyzed, blocking growth along particular

sites in the cluster. This is very similar to the monomers used to synthesize

silsesquioxanes in that they are permanently poisoned with only three available reaction

sites. The results of this research indicate that high levels of poisoning increase cluster

size for the same number of monomers. (Schaefer et al. 1986) The clusters in these

studies and in most sol-gel reactions are not to be confused with the nodules which are

the focus of this research. Early work on polymethylsislesquioxane has shown nodules to

have a radius around 200 nm. It is unknown however, what the substructure of the

nodules is at this time, whether it is composed of small spherical clusters creating a

hierarchy of structure, or if polymethylsislesquioxane nodules are homogenous and grow

continuously from solution to their observed size. It is the belief of the author that when

the inherently poisoned methyltrimethoxysilane is reacted in non-catalytic environments,

it will result in different structures, which are not necessarily cluster-like, on the

nanoscale. This fact will ultimately affect the mesoscale nodules.

3.3.2 Applications of Silsesquioxanes

Silsesquioxane materials have nearly as wide a variety of uses as possible organic

groups that can be attached to the silica network. In recent years, the drive for smaller,

faster electronic devices has helped pushed research into silsesquioxanes. (Wang et al.

1999; Yang et al. 2001) From a materials science aspect, the most crucial element to

making faster, smaller electronics is the interconnect material. This material must have a

low dielectric constant, which is necessary to prevent cross talk between neighboring









wires and increase signal propagation speed. Most of the work in low-k materials

revolves around creating controlled porosity films. Using the rule of mixtures, the

dielectric constant of the film is a function of the dielectric constant of the material and

the porosity. Higher porosity materials have lower dielectric constants because air has a

constant slightly above unity. The silsesquioxane organic groups help increase porosity

and have an inherently lower dielectric constant.

In addition to having a low dielectric constant, silsesquioxanes are also very

thermally resistant in comparison to traditional carbon based polymers. Depending on

the organic group, silsesquioxanes can have decomposition temperatures in excess of 550

C. (Fan et al. 2001) This property is desired in a variety of applications; in particular it

is extremely important in the case of electronic devices, where the entire package is

subjected to multiple high heat treatments during processing.

Silsesquioxane resins can also be used as ceramic precursors. Upon pyrolyzation in

a non-oxidizing atmosphere, silsesquioxanes convert into silicon oxycarbide (SiOC).

(Babonneau et al. 1994; Bujalski et al. 1998; Eguchi et al. 1998) Silicon oxycarbide is a

high temperature ceramic material that when derived from silsesquioxanes has properties

in excess of traditional ceramic processing synthesis routes. Silicon oxycarbide

synthesized from silsesquioxane is non-stoichiometric, having a variable carbon to silicon

ratio. The properties can be engineered by controlling the amount of carbon present in the

resin, which is a function of the organic group present in the silsesquioxane. Preceramic

precursors such as polymethylsislesquioxane have found a niche in the synthesis of high

temperature and toughness materials, and are a novel route to better ceramics through

chemistry.









3.3.3 Characterization of Silsesquioxanes

The techniques used to characterize silsesquioxanes most commonly relate

structure to a particular parameter. Generally silsesquioxanes are characterized for

molecular structure or molecular weight and related to some physical property, whether it

be dielectric constant or toughness and so forth.

Many different techniques can be used to measure molecular weight of polymers

such as Gel Permutation Chromatography (GPC) and Matrix Assisted Laser Desorption

Ionization Mass Spectroscopy (MALDI). However, it has been shown in the literature

that GPC is not a viable technique for measuring the molecular weight of silsesquioxanes.

Tecklenburg has directly compared GPC and MALDI for measurements of a

silsesquioxane based polymer. (Tecklenburg et al. 2001) The polymer synthesized by

Tecklenburg was not a pure silsesquioxane, but contained some siloxane linkages;

however the polymer was fundamentally a silsesquioxane. The synthesized polymer was

fractionated using super-critical fluids into 21 distinct fractions. This was done to make

measuring molecular weight easier by concentrating similar molecular weight species

together.

The results of Tecklenburg are shown in Table 3-1 and show that as molecular

weight increases, the validity of GPC decreases. He further cites the reason for this is

that the disparity in molecular weight is due to the fact that the molecular radius of

silsesquioxanes does not follow a normal curve used for calibration of GPC. This is

further evidenced by Figure 3-6. Figure 3-6 plots the molecular weight of several

fractions as found by MALDI.










Table 3-1. GPC and MALDI Molecular Weights of Silsesquioxanes from Tecklenburg
Fraction GPC Mn MALDI Mn
8 3415 g/mol 3778 g/mol
13 9070 12,087
18 24,130 45,434
20 43,770 100,995
21 56,160 124,847























0 50000 o0oooo t5OO00 200000 250ooo
mass (u)
Figure 3-6. MALDI Molecular Weights of Silsesquioxane Fractions (Tecklenburg et al.

i2001) 2



fraction to another overlap one another. This is a byproduct of the fractionation method.
D -
5 0 Mr 1000 MON 200000 2500CO
mass (u)


Figure 3-6. MALDI Molecular Weights of Silsesquioxane Fractions (Tecklenburg et al.
2001)

One can see in Figure 3-6 that the distribution of molecular weights from one

fraction to another overlap one another. This is a byproduct of the fractionation method.

As previously mentioned super-critical fluids were used to fractionate this particular

polymer. Super-critical fractionation works by changing the solubility parameters of a

super-critical fluid by adjusting the temperature or pressure. The changes in temperature

or pressure allow one to fractionate a polymer roughly by molecular weight, which in

general is the biggest factor determining the solubility of a polymer.









Silsesquioxanes differ in that there are nearly an infinite number of complex three

dimensional structures. In many cases there are multiple isomeric conformations of the

same molecular weight. Different structures of the same isomer have different

solubilities, as evidenced by the molecular weight distributions in Figure 3-6. This can

be further related to structure and the inaccuracies of GPC. Gel Permeation

Chromatography assumes that every molecule of the same molecular weight will have the

same hydrodynamic radius. However with some imagination one can visual a high

molecular silsesquioxane molecule that is very condensed forming a small tight cluster or

one that is very large forming a giant network. Although they have the same molecular

weight, the GPC retention times would be vastly different.

Not all researchers have used fractionation when studying molecular weight. Mori

et al have use MALDI to characterize synthesized silsesquioxane polymers. (Mori et al.

2004) However it should be noted that the polymer studied by Mori was shown to have a

much lower molecular weight than that examined by Tecklenburg. Additional

researchers have shown similar results when using MALDI to characterize low molecular

weight silsesquioxanes. (Wallace et al. 1999) As such, fractionation was likely not

required to increase sensitivity. (McEwen et al. 1997)

Matrix Assisted Laser Desorption Ionization is a very powerful tool for

characterizing large molecular weight silsesquioxanes. However most of the research

reported in the literature using MALDI for characterization of silsesquioxanes has

focused on low molecular weight polyhedral oligomeric silsesquioxane structures

(POSS). (Falkenhagen et al. 2003; dell'Erba et al. 2004; Anderson et al. 2005)









MALDI can also be used as a method of investigating structure. For low

molecular weight polymers, there are generally one or very few isomers of an oligomer.

Many research groups have developed software for calculating the structure of an

oligomer from molecular weight. Wallace has used MALDI data for several low

molecular weight silsesquioxanes to investigate the role the organic group has on

intramolecular condensation. (Wallace et al. 2000) Overall it was shown that the nature

of the R-group and the nature of the reaction process greatly affected the degree of

intramolecular condensation.

Wallace contrasted the ability of NMR and FTIR to determine the structure of

silsesquioxanes against that of MALDI. Using MALDI he was able to determine exact

structures, and classified techniques commonly used to characterize the structure of

silsesquioxanes, NMR and FTIR, as semi-quantitative at best.

Fourier Transform Infrared Spectroscopy (FTIR), while not able to interrogate the

exact structure of a silsesquioxane polymer, can be used to investigate the nature of

intramolecular binding. There are two IR active Si-O-Si peaks in silsesquioxanes. Lee

has identified the two peaks as 1120 cm-1 and 1030 cm-1. (Lee et al. 2002) Lee further

states that these peaks are the cage and network structures respectively. Cage structures

are generally molecules with high levels of intramolecular condensation on a short range

order. An example would be the polymethylsilsesquioxane cubical octomer, which is an

entirely cage structure.

It should be noted however, that not all research groups are convinced that the 1120

cm-1 and 1030 cm-1 peaks in a silsesquioxane FTIR spectra are separate structural entities.

Oh states that absorbance peaks near 1133 cm-1 are from a Si-O stretch and absorbance









peaks near 1031 cm-1 are due to a Si-O-Si asymmetric stretch. (Oh et al. 2002)

Conversely the work of both Oh and Lee indicates that the ratio of the two peaks changes

with thermal curing of the investigated materials. This would lead one to believe that at

elevated temperatures there is a structural rearrangement of the polymer which results in

a change in peak absorbance of the aforementioned peaks.

Within limitations MALDI can be used to determine isomeric structure oligomers,

and FTIR can be used to semi-quantitatively investigate physical structure; conversely,

Nuclear Magnetic Resonance (NMR) can be used to determine chemical structure of

silsesquioxane polymers.

Nuclear Magnetic Resonance is a powerful tool for quantifying the chemical nature

of a desired atom in a sample. Silicon-29 NMR is commonly employed to investigate

the silsesquioxanes. Arkles and Larson report in a detailed review Silicon-29 NMR

peaks of many silicon compounds found in the Gelest Annual Catalog. (Arkles et al.

2004)

As previously mentioned, a silicon atom in a silsesquioxane molecule has one

organic group and three oxygens bonded to it. Some of these oxygens are bridging

oxygens in that they connect two silicon atoms together. Conversely some oxygen atoms

are bound to hydrogen atoms forming hydroxide groups or silanols. The chemical shift

of a silicon atom in a silsesquioxane is dependent on the number of silanols and bridging

oxygens.

Kondo has used Silicon-29 NMR to investigate the structure of

polymethylsilsesquioxane polymers. (Kondo et al. 2000) The polymethylsilsesquioxane

synthesized by Kondo was characterized for the amount of hydroxide present in the









structure. Because silsesquioxanes are polymers, the peaks of the NMR spectra are broad

peaks, not the sharp, narrow peaks commonly found in smaller structures. Kondo

reported a range of peaks for each type of silicon in a silsesquioxane. (Kondo et al. 2000)

Nuclear Magnetic Resonance is a very powerful tool for determining the chemical

structure of silsesquioxanes and silicon based materials. It has been used extensively to

measure the hydroxide content of silsesquioxanes.

3.4 Conclusions

While there are dissenting views on the existence of nodules in epoxies, it is the

author's opinion that they are in fact real. Two explanations exist as to why researchers

come up with different results regarding nodular microstructures. It is very plausible that

not all compositions result in a nodular microstructure. As reported above, chemical

composition and processing parameters can greatly influence final nodule size of an

epoxy. Conversely, it is possible that techniques used to investigate structure such as

SAXS or AFM are not sensitive enough to pick up differences in structure. It is unknown

at this time how nodules form or what their role is on mechanical properties.

Silsesquioxanes, on the other hand, are complex hybrid inorganic-organic polymers

which have also been shown to form nodular structures. The structure of silsesquioxanes

is very complex, however different silsesquioxanes are easy to synthesize. This should

allow one to investigate the nature of the R-group and steric effects on nodule formation.

There are many different techniques for characterizing silsesquioxanes, although some

such as FTIR are not without controversy.














CHAPTER 4
FRACTURE PROPERTIES OF EPOXY RESIN

4.1 Introduction

Analyzing the effects of a nodular microstructure on the mechanical properties of

epoxy resins is a novel application of the West Mecholsky Passoja theory (WMP).

Should a relationship between nodular size and toughness be found, it would be possible

to endeavor to engineer an epoxy with a different nodular size and thus different

toughness.

Mechanical properties have been previously related to fractal relationships.

(Mandelbrot et al. 1984) The fracture surface of many brittle materials is a self-similar

pattern consisting of a mirror, mist, and hackle region. The WMP theory takes the fractal

structures found in fracture surfaces one step further by relating the toughness of a

sample to the fractal dimension of the fracture surface.

The West Mecholsky Passoja theory asserts that toughness is related to a

fundamental structural parameter ao. (Mecholsky et al. 2002) The relationship between

toughness and ao is given by the equation:

(4-1) K1E = E(aoD*)1/2

Where Kic is the toughness, E is the modulus, ao is the structural parameter, and D*

is the fractal dimensional increment. The fractal dimensional increment is the decimal

portion of the fractal dimension. The fractal dimension and dimensional increment are

reviewed in detail in Chapter 2.











We have postulated for this work that ao is proportional to the nodular size and

might be akin to the size of the interstitial volume formed by the packing of nodules

together. Scanning electron micrographs of nodular epoxies have revealed a randomly-

packed structure. Assuming a narrow distribution of nodule size, a randomly packed

structure should be 63% nodules by volume, and 36% interstitial material. This brings up

the question; What is the nature of the interstitial material in relationship to the nodule,

and how does it affect the toughness of an epoxy resin?

There has been significant research on establishing a link between nodular size of

epoxy systems and mechanical properties. (Mijovic et al. 1979; Mijovic et al. 1979)

Additionally many researchers have investigated the nature of the nodular microstructure.

(Errath et al. 1959) Not all researchers, however, are convinced that nodules do exist.

(Duchet et al. 2003) It is possible that not all epoxy compositions studied are nodular.

This would be one explanation of the discrepancies between multiple works of research.

The epoxy composition investigated in this work has been confirmed to have a nodular

microstructure and is derived from a method of preparation found in the literature.

(Racich et al. 1976)

One possible explanation for nodule formation is that as the curing reaction begins

minor inhomogeneities begin to form in the bulk of the epoxy resin-curing agent solution.

This results in a cluster that has a slightly different chemical potential or solubility than

the rest of the bulk. Initially this difference in chemical potential or solubility could be a

function of the surface energy in the cluster in relation to the free energy of the uncured

bulk. The difference in energies would drive the curing reaction at a faster rate at the









surface of the cluster than in the bulk of the solution. Eventually the nodule would reach

a size where the surface energy has to decrease to a point where it no longer dominates

the reaction process and nodule growth slows greatly, but does not stop altogether. This

possible explanation would answer the question of why nodules are so uniform in size as

reported by researchers in the literature.

Epoxy samples were characterized for their toughness, nodular size, modulus, and

fractal dimension with the intent of fitting the data to the West Mecholsky Passoja theory.

It was expected that the toughness of the epoxy samples will be proportional to the size of

the nodules of the sample. It is the author's hypothesis that nodule size can be related to

the size of the interstitial volume found between nodules. Additional experiments were

performed to better understand the nature of the system being investigated and inquire

about nodules and nodular formation.

4.2 Epoxy Synthesis and Processing

Samples were prepared from Epon 825 epoxy resin and cross-linked with

Diethylenetriamine (DETA). Epon 825 is a high purity diglycidylether of bisphenol-A

manufactured by Shell Chemical, and it has an equivalent mass of 176 grams. An

equivalent is defined as the mass of a polymer corresponding to one mole of reactant

group, in this case the epoxide group. Diethylenetriamine is a penta-functional amine

curing agent with three amine groups; two primary and one secondary amine. DETA has

an equivalent mass of 20.6 grams. The stoichiometric ratio of DETA to Epon 825 is

11.7 grams per hundred grams resin (phr).

Samples were cured in silicone molds with 8 or 10 phr of DETA, or a sub-

stoichiometric ratio. These compositions were chosen to closely follow reported









literature. (Racich et al. 1976) The two compositions were chosen because they have

been previously shown to be nodular and have different nodule sizes.

A sample of Epon 825 resin was weighed and manually mixed with DETA. The

samples were then degassed for 30 minutes using a vacuum pump. After removing

bubbles, the resin was poured into the molds.

Samples were cured at 500C for 24 hrs and removed from the molds. The molds

were fabricated from Dow Coming Silastic T-2 silicone. Samples were produced in a

dog-bone shape that conforms to ASTM standard D638. This standard is primarily used

for calculating the modulus of polymer resins. Samples were strained at three different

strain rates, 0.1 mm/min, 10 mm/min, and 100 mm/min.

Ten samples were measured for each composition and strain rate. Samples were

loaded until there was failure in tension. A laser extensometer was used to calculate

displacement and modulus. Modulus was calculated as the slope of the stress-strain

curve. Optical microscopy was used to calculate the flaw size of broken samples so that

toughness could be calculated. Nodular size was determined using a scanning electron

microscope. Additionally fractal dimensional increment was calculated using an optical

microscopy technique and a derivation of the Richardson method. Additional techniques

for measuring fractal dimension were used to corroborate data gathered by optical

microscopy.

4.3 Methodology and Experimental

Methodology used for investigation can be broken into three distinct tasks;

characterization of mechanical properties, characterization of nodule size, and

characterization of fractal dimension. The following section will detail the methods used

for each of these tasks and present a brief discussion on the results gathered in each task.









4.3.1 Characterization of Mechanical Properties of Epoxy Resins

4.3.1.1 Introduction

Tensile testing was used to break all samples investigated in this research. Tensile

testing was selected because of ease of sample preparation and reproducibility of data.

As previously mentioned, samples were prepared in accordance with ASTM standard

D638. This standard describes a process of measuring the modulus of a polymeric resin.

Samples were testing on an Instron 1122 frame with a 1000 lb load cell. Mechanical

vice grips were used to secure the sample. An attached laser extensometer was used to

gauge displacement. The nominal gauge length for samples investigated was 12.5 mm.

Load was recorded for a given displacement. Failure stress and modulus was recorded

from these experiments. Optical microscopy was used to determine flaw size. The flaw

size combined with the failure stress was used to calculate toughness.

4.3.1.2 Modulus

As previously mentioned, two compositions and three strain rates were

investigated. Modulus was calculated as the slope of the stress-strain curve. Ten samples

were broken for each data point. Table 4-1 represents the average of all samples for each

data point and the associated error of one standard deviation from the mean.

Table 4-1. Modulus and Percent Error of Epoxy Resins at Different Strain Rates
8phr 10phr
Strain Rate 8phr 1 Ophr
E (GPa) %Err E (GPa) %Err
0.1 1.41 4.9 1.41 3.9
10 1.45 4.8 1.42 3.3
100 1.51 9.2 1.44 1.5

The following figures are examples of the stress-strain curve generated by the

tensile test. Figure 4-1 indicates three superimposed stress-strain curves for 8 phr

DETA. Figure 4-2 indicates three superimposed stress strain curves for 10 phr DETA.









47





Stress Strain Curves of Epon 825 with 8phr DETA


90


80


70


60


50


40


30


20


10


0
0 001 002 003 004 005 006 007 008 009
Strain


Figure 4-1 Stress-Strain Curves of Epon 825 with 8 phr DETA)


Stress Strain Curves of Epon 825 with 10phr DETA


IL
50


40


30


20


10


0 001 002 003 004 005 006 007 008
Strain


Figure 4-2 Stress-Strain Curves of Epon 825 with 10 phr DETA


-0 1 mm/min
10mm/min
-100 mm/min


0 1

-100









The curves in the figures are for samples that were close to the mean of all the

samples for that strain rate and composition. The values for each specimen measured can

be found in Appendix D in Table D-1. The samples in Figure 4-1 are #8 for 0.1 mm/min,

#6 for 10 mm/min and #3 for 100 mm/min. The samples in Figure 4-2 are #3, #6, and #9

for 0.1, 10, and 100 mm/min respectively.

A two-tailed Student's t-test was used to gauge if the moduli for each strain rate

were different from one another. The results of this analysis can be found in Table 4-2.

Columns 1 and 2 are the moduli being compared. The percentages listed in Column P of

Table 4-2 indicate the probabilities that the modulus of one strain rate is equivalent to the

modulus of another strain rate. It is clear from the data in Table 4-2 that for each

composition, the moduli for each strain rate are sufficiently different from one another.

It should be noted that a t-test comparing the modulus of 0.1 mm/min 8 phr to 0.1

mm/min 10 phr returns a very high probability that the moduli are the same. However,

as strain rate increases the moduli changes is greater for the phr 8 then the phr 10

samples. This indicates that the compositions are different.

Table 4-2. Student's t-test Results of Modulus of Different Strain Rates
8 phr DETA 10 phr DETA
1 2 P 1 2 P
0.1 100 3% 0.1 100 7%
0.1 10 15% 0.1 10 34%
10 100 11% 10 100 12%


It appears from the data in table 4-1 that the modulus, decreases with increasing

strain rate for both compositions, which was to be expected. Epoxies are brittle

polymers; while they fail in a manner similar to ceramics, they still have time and









temperature dependent properties. The change in modulus with strain rate is due to the

ability of polymer chains to rearrange and accommodate strain.

4.3.1.3 Failure stress

Failure stress (of) was recorded as part of the tensile test for each sample. Failure

stress was defined as the load under which a sample would fail. Table 4-3 lists the

average failure stress for all compositions and strain rates tested. Additionally Table D-2

in Appendix D lists the failure stress for each sample tested. Although the data listed in

Table 4-3 are similar, a t-test implemented comparing each strain rate with a composition

to one another indicates that the failure stresses are different. The results of the t-tests

can be found in Table 4-4.

Table 4-3. Failure Stress of Epoxy Resin at Different Strain Rates
Strain Rate 8phr 1 Ophr
cyf (MPa) %Err of (MPa) %Err
0.1 60 3.9 69 4.3
10 74 10.5 75 17.4
100 74 27.7 64 24.4

Table 4-4 Student's t-test of Failure Stress of Epoxy Resins
8 phr DETA 10 phr DETA
1 2 P 1 2 P
0.1 100 3% 0.1 100 20%
0.1 10 0% 0.1 10 9%
10 100 47% 10 100 6%

4.3.1.4 Flaw size

The flaw size for each sample was measured using optical microscopy. For the

purpose of this work, a Zeiss Axioplan 2 Microscope retrofitted with a custom built LEI

XYZ stage capable of 0.1 |tm was used to image samples. Images were taken at 5x

magnification. Images were processed using the bundled MCID Elite 6.0 Morphometric









Software package. The following figures are examples of flaws found in the samples

tested.


Figure 4-3 Example Critical Crack Size Produced through Slow Crack Growth


Figure 4-4 Flaw Size of Epon 825 with 8phr DETA at 100mm/min Strain Rate




























Figure 4-5 Flaw Size of Epon 825 with 10 phr DETA strained at 10mm/min

It was often found that that the flaw was produced by slow crack growth. What this

means is that the flaw starts out small from some initial defect in the structure and grows

outward to a large size with an applied load. When the crack grows to a large enough

size, fast brittle failure occurs, producing a mirror and hackle region. The region of slow

crack growth can be seen close up in Figure 4-3 above. Note the size of the initial defect,

40 by 33 micrometers, and the growth of the flaw to a much larger size. The region of

slow crack growth is visually similar to other polymers reported in the literature. (Kurtz

et al. 1998)

It was found that the flaws were asymmetric. As such, an approximation was used

to model the flaw as roughly circular. The size of the flaw can be found from the

following equation:


(4-2) c = (ab)2









Where a and b are one half the dimensions of the flaw and c is the effective radius

of the flaw. Tables D-3 through D-8 in Appendix D list the flaw sizes of all samples

measured in this research and the modeled radius.

4.3.1.5 Fracture mechanics and toughness

Toughness was calculated using the modeled flaw size (c) and the failure stress of

using the following equation:

(4-3) Kic = Yf c

Where Kic is the toughness, of is the failure stress, c is the flaw size and Y is the stress

intensity factor.

The value of Y depends on the shape of the flaw and location. A surface flaw has

a greater Y value than a body flaw. As mentioned in the previous section, all flaws were

modeled to be circular, as such, two Y values were used, 1.13 and 1.26. Surface flaws

require an approximate 12% correction factor over body flaws. The location and Y value

for each flaw for every sample can be found in the last columns of Tables D-3 through D-

8 in Appendix D. Tables 4-5 and 4-6 are the toughness for all three strain rates for 8 phr

DETA and 10 phr DETA respectively.

Table 4-5 Toughness of Epon 825 with 8ph DETA (MPa*ml/2)
0.1mm 10mm 100mm
Average 1.34 1.09 1.06
Std Dev 0.28 0.28 0.23
Error 21% 26% 25%


Table 4-6 Toughness of Epon 825 with 10phr DETA (MPa*ml2)
0.1mm 10mm 100mm
Average 1.56 1.28 0.77
Std Dev 0.23 0.35 0.23
Error 15% 27% 29%










The accuracy of the data reported in the above tables can be checked by an Ln-Ln

plot. An Ln-Ln plot is a component of the WLF theory. The WLF theory states that for a

given temperature, the toughness of a material varies linearly with strain rate on a plot of

the natural log of Kic vs. the natural log of strain rate.(Green 1998) Figure 4-6 shows the

Ln-Ln plot of the strain rate and temperature for both compositions studied in this

research.


Ln Ln Plot of Rate vs Kic

7.4




7T.1 Ln 8
R2 = 0.94
S* LnlO
C -- Linear (Ln 8)
6. Linear (Ln10)


6.7 R2 = 0.99 -


-4 -2 0 2 4 6
Ln Strain Rate

Figure 4-6 Ln-Ln plot of Strain Rate vs. Kic for Epon 825 Resins

The high R2 values indicate that the toughness values calculated in this study

closely match what would be predicted by the WLF theory.

It was mentioned in the previous section that the critical crack size is a actually the

by product of slow crack growth. The concept of slow crack growth presents an

interesting problem. For both compositions, all samples characterized were made at the

same time. As such it is safe to assume that the population of flaws from sample to

sample for a particular composition is the same. Slow crack growth however, can result









in a change in critical crack size. The following table lists the average critical crack size

weighted by stress intensity factor Y. Additionally, measurements of the initial flaw size

revealed an average flaw radius of 20 |tm. This can be seen in Figure 4-3.

Table 4-7 Average Critical Crack Size for Epoxy Resins ([tm)
8 phr DETA 10 phr DETA
Strain Rate 0.1mm 10mm 100mm 0.1mm 10mm 100mm
Average 524 225 233 522 309 191
Std. Dev 191 100 118 141 126 89
Error 36% 45% 50% 27% 41% 47%


Utilization of Students t-test reveals that statistically, the critical flaw sizes for all three

strain rates for 10 phr DETA are different from one another. However, for 8phr DETA

10 and 100mm/min were shown to be statistically similar. At higher strain rates, c is a

smaller value; conversely, low strain rates result in large critical crack tip sizes. This

would indicate that sub-critical crack growth is a strain rate dependent phenomenon.

4.3.2 Studies of Nodular Microstructure and Nodule Size

Epoxy samples were characterized for size and nature of nodules. Nodule size was

measured with an SEM and the nature of the nodules was investigated using two different

techniques. Three separate studies were performed; measurement of nodule size with

SEM, extraction of nodules from resins with solvents, and mapping of iodine stain in

relationship to nodular structure. The results of the studies presented here can be used to

give insight into the nodular microstructure of epoxy resins.

4.3.2.1 Scanning electron microscopy

The nodular size of our epoxy samples was found with the aid of scanning electron

microscopy (SEM). Samples were coated with a gold palladium alloy and images were

taken at 10 KeV on a JEOL 6400 SEM. Several magnifications were used. In the









literature, Transmission Electron Microscopy (TEM) has been extensively used; SEM

was selected for this research because of the ease of sample preparation.

Additionally, images were taken on both the fracture surface and exterior surfaces

of the epoxy samples. It should be noted that initially a diamond blade saw was

employed to investigate the nature of the nodule in the bulk of the sample. Cutting the

epoxy with a diamond blade resulted in noticeable deformation of the nodules and was

therefore not found to be a valid method of preparing surfaces for imaging.

Nodule size was determined with Image Pro Software. Multiple images from each

composition were taken to ensure statistical significance. Samples were prepared by first

washing the surface with acetone to remove excess organic matter. Previous work by

Koutsky et. al. found that acetone can be used to etch the surface of epoxy and

preferentially remove the interstitial material. (Mijovic et al. 1979) Care was taken to

ensure that samples were not over-etched. Furthermore samples were exposed to iodine

to stain the surface. This was done to increase the average atomic number of the surface.

Higher atomic number coatings on SEM samples help to increase the signal-to-noise ratio

of an image.

Figures 4-7 and 4-8 are typical SEM micrographs of Epon 825 with 8 and 10 phr of

DETA respectively. Image Pro was utilized to determine an average value for nodule

size. The images were loaded into the software and an algorithm was used to identify the

nodules. The software allows the user to set threshold limits for upper and lower size.

By setting the lower limit sufficiently high, it is possible to remove the embedded

nodules from the count produced by the software. Judging visually, the nodule size in the









picture below was estimated to be on average around 100 nm for both compositions. As

such a lower threshold for nodule size was chosen to be half or 50 nm.














Figure 4-7 SEM Micrograph of Epon 825 with 8 phr DETA














Figure 4-8 SEM Micrograph of Epon 825 with 10 phr DETA

The nodule sizes for the investigated resins appear to be approximately 75nm +/-

10nm and 100nm +/- 20nm for 8 and 10 phr respectively. These values are slightly larger

than those found in the literature for similar composition. However, the differences can

be explained by taking into account the roles different processing methods, cure

schedules, and substrates.

4.3.2.2 Solvent extraction and particle size

As previously mentioned, acetone can be used to etch the surface of epoxies.

Koutsky et al have shown that by exposing epoxy samples to acetone for long periods of









time nodules are actually etched. Additionally, evidence indicates that the acetone does

not have a significant effect on the nodule, indicating that the acetone is more reactive

with the interstitial material than with the nodule. This could indicate that the nodule is

more highly cross-linked than the interstitial material.

Epoxy samples were treated with acetone for extended periods of time to separate

the discontinuous material. It has been reported in the literature by Koutsky that acetone

washing does not influence nodule size, but can instead, with extended exposure, wash

nodules out from the gel network. (Mijovic et al. 1979) Samples were exposed to

acetone for extended periods of time, 24 hrs, 48 hrs, and 2 weeks. The bulk epoxy was

removed, leaving a mixture of acetone and particles. The resulting solution was

concentrated by evaporating the acetone out at room temperature. This resulted in a

white, viscous liquid with no visible nodules. This liquid is a mixture of nodules and

unreacted resin monomers. Samples of the acetone-nodule solution were characterized

using optical microscopy. No nodules were found using optical microscopy; instead

there is featureless residue on the surface of the slide.

Recalling that the composition studied here is sub-stoichiometric, it appears that

not only are nodules washed out, but so is a small portion of the resin. Further

experiments using Soxhlet extraction confirmed the presence of unreacted monomer and

low molecular weight oligomers. Soxhlet extraction was performed using a standard

setup and acetone as the solvent. (Kim 2004) After running the extraction for 2 hours,

the acetone was evaporated, leaving behind a viscous, white liquid.

The fact that nodules can be washed out of epoxies with a suitable solvent presents

some interesting questions. Why do only some of the nodules wash out but not all? As









previously mentioned in the SEM study, nodules have a narrow distribution of sizes. It

would appear that as nodules grow, the overall surface functionality density decreases.

This reduction in functionality greatly slows down nodule growth. The ultimate size is a

function of the ratio of epoxide to catalyst group (phr). This nodule size to catalyst

relationship has been demonstrated previously in the literature. (Racich et al. 1976)

Additionally it has been reported in the literature that failure occurs around nodules and

not through nodules. This would indicate that nodules are only loosely bound to the

surrounding structure.

4.3.2.3 Iodine staining of surfaces

One of the arguments against the existence of nodules is that some researchers have

postulated that the nodule and interstitial material should have different densities. Small

Angle X-Ray Scattering (SAXS) and Atomic Force Microscopy (AFM) have both failed

to show any difference in density. In this study epoxy surfaces were stained with iodine

in an attempt to elucidate the nature of the nodule and interstitial material. Three samples

were investigated in this study, no staining, 5 minutes staining and 12 hours staining.

Samples introduced to the confined iodine chamber quickly changed color, from

transparent to a brownish color. The longer samples were exposed to iodine, the darker

the sample would become. Washing the samples in acetone after staining would result in

slight discoloration of the acetone from excess iodine washing off. It is postulated that

the iodine forms a charged transfer complex with the carbon-oxygen-carbon bonds found

in the diglycidyl ethers bisphenol-A (DGEBA) chain. This effectively forms a tx-xt bond

that is stronger than hydrogen bonds but not as strong as covalent bonds.










Full scale = 388 cps
Epon 825 with Iodine







II

0 2 4 6 8 10 12 14 16 18 20
keV

Figure 4-9. EDS Spectra of Epoxy Stained with Iodine for 5 minutes

Figure 4-9 is a representative EDS spectrum for an epoxy sample stained for 5

minutes. From the EDS spectra it is clear that there is, in fact, iodine present on the

surface of the sample. EDS is not sensitive enough though, to give information about the

local chemical structure of the iodine that would indicate whether or not the charged

transfer complex hypothesis is correct. Additionally, it was noted that under normal

secondary electron imaging, the epoxy sample stained for 12 hours appeared to degrade

in real time from electron beam damage. This indicates that the sample may have been

over-stained and the iodine may have in fact cleaved the C-O-C bonds.


Figure 4-10. EDS SEM Image of Iodine (Red) Stained Epoxy Surface









The last figure in this section, Figure 4-10, is an EDS mapping of iodine on the

surface of the specimen. The red areas in the image indicate the presence of iodine.

There appears to be no evidence of preferential segregation of iodine in the nodular

epoxy specimen.

The EDS spectra and mapping do indicate the presence of iodine in the epoxy

sample. However there is no indication that the iodine prefers to reside in the nodule or

interstitial material. It is possible that the iodine is too small of a molecule to observe any

differences in density as it diffuses through the epoxy. Additionally it also plausible that

the difference in density between the nodule and the interstitial material is very small and

therefore the difference in concentration of iodine is below the resolution limit of the

microscope.

4.3.3 Investigation of Fractal Dimensional Increment

The fractal dimensional increment, D*, is very important in the West Mecholsky

Passoja theory. As previously stated the fractal dimension can be related to the tortuosity

of the fracture surface. In Chapter 2 methods for calculating fractal dimension and D*

were described. These methods are time consuming and in some cases destroy the

surfaces being characterized. This research has also focused on developing a technique

that is comparable to other methods employed for calculating fractal dimension of

fracture surfaces that is quick, simple, and non-destructive.

4.3.3.1 Introduction

The ractal dimensional increment was calculated using three separate techniques.

Primarily a novel non-destructive slit-island method was used to generate images from

which the fractal dimensional increment could be derived. The images were processed

using Image-Pro software. The fractal dimension was calculated by hand using a small









selection of images and the Richardson method to confirm the validity of the Image-Pro

software algorithm. Additionally the ratio of the mirror to the size of the flaw was used

as a method to compare the values derived from the non-destructive slit-island method.

4.3.3.2 Flaw to mirror size

There are many different techniques for calculating fractal dimensional increment.

Many of them are time consuming or require a great deal of sample preparation. Fractal

dimensional increment can be quickly estimated by comparing the size of a flaw to the

size of the mirror. When a sample fails in a brittle manner, three distinct regions are

formed on the fracture surface; mirror, mist, and hackle. The ratio of the radii for each

of the regions is constant regardless of size for any material. The size of these regions

depends on the failure stress. For a given material, a sample that fails at a higher stress

will have a smaller mirror than one that fails at a lower stress. Additionally the ratio of

the mirror size to the size of the flaw can be used to quickly estimate fractal dimensional

increment. Fractal dimensional increment can be found using the following equation:


(4-3) D* c


Where D* is fractal dimensional increment, c is flaw size and rm is the size of the

mirror.

The mirror size was measured using optical microscopy. The size of the mirror

was compared to the size of the flaw for a given sample as found in Section 4.3.2.

Additionally, because the mirrors were often asymmetrical, the radius of the mirror was

calculated as the average of two measurements using the same formula to calculate the

size of the asymmetric flaws. Figure 4-11 is an example of the mirror and flaw for an

epoxy sample where the flaw has the dimensions 239 by 166 micrometers and the mirror









has a radius from 719 to 664 micrometers. Additionally, example figures for each strain

rate and composition can be found in Appendix D





















Figure 4-11 Flaw Size and Mirror for Epon 825 with 8 phr DETA strained at 0.1 mm/min



For each of the strain rates and compositions tested, three samples were measured

for mirror size. Table 4-8 lists the average fractal dimensional increment for each of the

compositions and strain rates investigated. Additionally Table 4-9 list the t-test results

for comparing fractal dimensional increment distributions.



Table 4-8 Fractal Dimensional Increment by Flaw to Mirror Size for Epoxy Resins
8 phr DETA 10 phr DETA
0.1mm 10mm 100mm 0.1mm 10mm 100mm
Average 0.23 0.15 0.15 0.24 0.22 0.11
Std Dev 0.06 0.06 0.04 0.07 0.05 0.05
Error 26% 40% 24% 28% 25% 45%









Table 4-9 Student's T-test of Fractal Dimensional Increment of Epoxy Resins
8 phr DETA 10 phr DETA
1 2 P 1 2 P
0.1 100 0% 0.1 100 0%
0.1 10 1% 0.1 10 24%
10 100 97% 10 100 0%


The fractal dimensional increment seems to follow established trends in the

literature. Tougher resins have slightly greater D* values. However for the 8 phr DETA,

the fractal dimensional increment appears to be the same for 10 and 100 mm/min strain

rate. The Student's T-test comparing 10 and 100mm/min strain rate for 8phr DETA

returns a 97% probability that the distributions are the same. The measurements of the

mirror radii, flaw size, and examples of flaws and mirrors for each sample can be found

in Appendix D in Tables D-9 through D-14 and in Figures D-1 through D-6.

It should be noted that this method is merely an estimate and cannot be used to

solely describe the fractal dimension of a surface. The ratio of the flaw to mirror size

can be shaped by residual stresses in the sample. It is unknown at this time if there are

any residual stresses in the sample, however, it is unlikely. Additionally, operator error

can play a large role in calculated values. Where the operator defines the start of the

hackle region and sample geometry can affect final values. Additional experiments using

different techniques should be conducted to confirm the results of Table 4-7.

Figure 4-12 plots the square root of the fractal dimensional increment versus the

toughness of epoxy resins. This type of plot has been commonly used in the literature to

show a relationship between fractal dimensional increment and toughness. The R2 value

is rather low, which could indicate that the flaw to mirror size ratio is not necessarily the

best method for calculating fractal dimension.














SVS [*1"2


1500

1400

1300

1200

1100

10O0

900

800


0.35 0.40 0.45
D*f T2


Figure 4-12. Fractal Dimension vs Toughness for Epoxy Resins

4.3.3.3 Non-destructive slit-island method

The non-destructive slit-island method is an adaptation of the commonly used slit-

island method. The slit-island method calls for setting a specimen in an epoxy resin and

polishing the surface to a fine grit to produce islands formed by the removal of the top of

the fracture surface. This is shown in Figure 4-13.

The fractal dimension is then calculated by measuring the perimeter of the islands

with different length rulers (Richardson method). This technique can be very time

consuming, although it is acknowledged to be the best method for calculating fractal

dimension and gives most accurate representation of the surface.

















Figure 4-13. Slit-Island Method from Hill (Hill et al. 2001)
One of the drawbacks of the slit-island method is that it requires the sample, or in
some cases a replica of the sample, to be polished, destroying the specimen or replica
surface. One study of this research has focused on developing a suitable method of
producing slit-island contours for a specimen without destroying the fracture surface.
This was done by producing three dimensional images of a fracture surface using optical
microscopy. This preserves the fracture surface should additional measurements become
necessary.
The same microscope described in Section 4.3.2 to measure flaw size was used for
this study. However, instead of focusing on the region surrounding the flaw, images
were taken of the hackle region. The M3D package of the MCID Elite 6.0 Morphometric
Software package was utilized for this study. This software package was used to
generate 3D images from fifteen 2D "slices" taken at different focal planes with
equivalent spacing. The 3D images were then manipulated using the software to produce
digitally polished surfaces from which the fractal dimension can be calculated. The
digitally polished images could then be characterized for fractal dimension using either
hand calculations or software measurements.


CM fl









This process is ideal for determining the fractal dimension of the hackle region of a

fracture surface. As mentioned in Chapter 2, the hackle region is often characterized by

the very large features on the fracture surface. This technique does not work well for the

small features found in mirror region. This is because the perturbations found in the

mirror region are often sub-micron in scale and cannot be properly imaged with an

optical microscope.

Figure 4-13 is an example of a 3-D composite image generated using the MCID

software. This image is taken down the z-axis of the composite. This is done to make

image polishing easier and more compatible with other software. The accompanying

figure, Figure 4-15, is a 2-D image of the same area of a sample as Figure 4-14. Figure

4-15 was generated using z-axis imaging. This is similar to the process used to generate

the 3-D images, but does not result in a picture that can be digitally manipulated like the

3-D pictures. The 2-D pictures were used as a reference to ensure that the 3-D images

were generated properly. The 2-D images are in color, whereas the 3-D images are in

black and white.

Because each of the sections used to generate a 3-D image are often only a couple

of microns apart, and the cross-sectional areas of each picture are often several hundred

microns, 3-D data sets are very difficult to view at a perspective angle.

The process assigns a gray scale value to each pixel, from 0 to 255, where 0 is

black and 255 is white. The higher the number, the higher the pixel is in the 3-D picture.

From each specimen, several sections were digitally cut by specifying a range of gray

scale values to display. Doing this, generates pictures similar to those used by Hill et al to

calculate fractal dimension using a slit-island technique.(Hill et al. 2001) In most cases,









the higher elevations were removed; however, in some instances the lower sections were

removed. Figures 4-16 through 4-19 are all sections taken from Figure 4-14. For each

specimen at least 3 sections were generated along with a full height composite spectrum

(Figure 4-14) and a 2-D real color composite image (Figure 4-15)


Figure 4-14 3-D Composite Image of Specimen


Figure 4-15 2-D Image of Specimen


















































Figure 4-16 0-50 Elevation Image Section


Figure 4-17 0-100 Elevation Image Section


----~
~ .c.-


I
$I
































Figure 4-18 0-150 Elevation Image Section


Figure 4-19 50-255 Elevation Image Section









4.3.3.4 Hand calculations of fractal dimensional increment

In order to confirm the data gathered using the Image-Pro software, a sample was

characterized by hand using the Richardson Method. Figure 4-20 is a subsection of

Figure 4-19. The contrast was enhanced and the section magnified to make hand

measurements easier. Additionally the interior parts were removed to produce a more

simple perimeter. The pixilation of the figure below is due to the fact that it is a very

large magnification of a small section of Figure 4-19.

The image in Figure 4-19 was characterized for fractal dimension by hand. The

image was magnified to a sufficiently large size and then printed. The perimeter of the

white portion of the image was measured from four separate starting points,

approximately 90 degrees apart. Table 4-10 is the data gathered in this process.

Additionally Figure 4-21 is the plot of the natural log of the ruler length versus the

natural log of the perimeter. The fractal dimension of this image can be determined from

equation 4-4.


Figure 4-20 Magnification of Fracture Surface
Figure 4-20 Magnification of Fracture Surface






71


(4-4) Ln (P)= (1-D)Ln (R) +C
Where R is the ruler length, P is the perimeter, D is the fractal dimension, and C is a
constant.

Table 4-10 Hand Measurements of Fractal Dimension


Number of Steps
1 2 3 4


Av e Per LnR LnP


4.8
5.3
6.0
7.5
8.8
10.8
14.0
19.8
35.5
76.3


47.5
47.3
48.0
52.5
52.5
53.8
56.0
59.3
71.0
76.3


2.30
2.20
2.08
1.95
1.79
1.61
1.39
1.10
0.69
0


3.861
3.855
3.871
3.961
3.961
3.984
4.025
4.082
4.263
4.334


Fractal Dimension of Fracture Surface


Slope = 1-D
D = 1.22
D*= 0.22


0.0 0.5 1.0 1.5 2.0 2.5
Ln Ruler


Figure 4-21. Richardson Plot of Perimeter of Fracture Surface

Three sub-sections for each of the six parameters tested in this research were

measured for fractal dimensional increment using hand calculations on images generated


Ruler

10
9
8
7
6
5
4
3
2
1










by the non-destructive slit-island method. These D* values for each subsection, and the

average and error for each composition and strain rate can be found in the Table 4-11.

Table 4-11. Hand Calculations of Fractal Dimension of Epoxy Resins
8 phr DETA 10 phr DETA
Section 0.1mm 10mm 100mm 0.1mm 10mm 100mm
1 0.23 0.21 0.15 0.20 0.18 0.15
2 0.21 0.18 0.19 0.21 0.15 0.12
3 0.22 0.15 0.16 0.25 0.16 0.12
Average 0.22 0.18 0.17 0.22 0.16 0.13
Error 5% 17% 12% 5% 9% 17%

The values in the above table are similar to those found previously when comparing

the flaw to mirror size in Section 4.3.3.2. Fractal dimensional increment for all samples

appears to be around 0.2. A plot of Ki vs. D*1/2 for the fractal dimensions found by hand

calculations can be found in Figure 4-22. Note the higher R2 value for a linear regression

in this plot versus Figure 4-12


Kc vs D

1500

1400

1300
R1300 = 0.8973

1200

E 1100

1000

900

800 -
0.30 0.35 0.40 0.45 0.50
D* 1/2



Figure 4-22 Square Root of Fractal Dimension vs. Toughness Calculated by Hand from
Slit Island Contours









4.3.3.5 Calculation of fractal dimensional increment by Image-Pro

The hand calculations of fractal dimension can be extremely time-consuming and

difficult. For that reason, software was employed to gather fractal dimension across large

volumes of data. The fractal dimensional increment was calculated from images

generated using the non-destructive slit-island method. "Image-Pro" software was used

to calculate fractal dimension for all specimens for all samples. Image-Pro uses a

derivation of the Richardson method to calculate fractal dimensional. Image-Pro first

analyzes the image for light and dark regions to identify islands. Multiple ruler lengths

are used to measure the perimeter in accordance with the Richardson method. Fractal

dimension is calculated for regions with a perimeter larger than 30 pixels because the

specific algorithm used by the software breaks down for very small features. The data

wasis then tabulated and averaged together across all regions for a sample to produce a

fractal dimension for a surface.

For each composition and strain rate, the images were used to obtain an average

fractal dimension. Table 4-12 lists the average fractal dimension and error of each

composition and strain rate for epoxy resins.

Table 4-12. D* of Epoxy Resins Calculated With Image Pro
Strain 8 phr DETA 10 phr DETA
Rate D* %Error D* %Error
0.1 0.22 12% 0.23 13%
10 0.19 13% 0.17 17%
100 0.18 9% 0.16 11%


Finally, the square root of the fractal dimensional increment calculated with Image

pro for images generated with optical microscopy was plotted versus the toughness of

epoxy resins. This result can be found in Figure 4-23.











K, vs [712


SCO



aoo


.30


025


0.40
D*1-2


0.45


Figure 4-23 Square Root of Fractal Dimension vs. Toughness Calculated by Image Pro
from Slit Island Contours

4.3.3.6 Comparison of methods of measuring fractal imension

Three separate methods were used to calculate fractal dimension for samples in this

research; flaw to mirror ratio (F-M), hand calculations of slit island contours (Hand), and

computer calculations of slit island contours (Computer. The results of these three

methods are tabulated in table 4-13.

Table 4-13. Comparison of Fractal Dimensional Increment D* Values for Different
Methods
8 phr DETA 10 phr DETA
0.1mm 10mm 100mm 0.1mm 10mm 100mm
F-M 0.23 0.15 0.15 0.24 0.22 0.11
Hand 0.22 0.18 0.17 0.22 0.16 0.13
Computer 0.22 0.19 0.18 0.23 0.17 0.15


Fe =0.111






iz









It can be seen in the above table that all three methods return similar values for

each composition and strain rate.


4.4 Structural Parameter ao

The goal of this research was to determine the nature of the relationship of nodular

size and toughness. It was proposed that nodular size could be related to toughness

through the West Mecholsky Passoja theory. Using the data presented above, it is

possible to calculate the WMP structural parameter, ao, and the relationship to nodule size

for epoxy resins investigated in this research.

4.4.1 Calculating ao

The structural parameter of the West Mecholsky Passoja theory, ao, can be

calculated from the modulus, fractal dimensional increment, and toughness using the

following equation:


(4-5) a=
E D
Where Kic is the toughness, E is the modulus, and D* is the fractal dimensional

increment. The structure parameter for all three strain rates and each composition can be

found in the following table:

Table 4-14. Calculated ao values ([tm) for Epoxy Resins
Strain Rate (mm/min) 8phr DETA 10 phr DETA
0.1 4 5
10 4 4
100 3 3


The data used to calculate these results are the average values for a particular

composition and strain rate. The fractal dimensional increment values used in this

calculation come from calculations with Image Pro. These values were chosen because of









the low error for the measurements made. Because the error associated with ao is a

function of the individual variables and not a direct summation, Appendix A has been

prepared describing how the error was calculated. Additionally Section 4.4.3 describes

statistical significance of these results.

4.4.2 Relationship of Fractal Dimensional Increment to ao

Through the course of measuring fractal dimensional increment, several plots were

derived that plotted the square root of D* versus the toughness of epoxy resins. The

figures are 4-12, 4-22, and 4-23. These types of graphs have been reported in the

literature to show that for families of materials, such as glasses, or crystalline materials,

the fractal dimensional increment follows a trend that as toughness increases so does D*.

Additionally, it has been hypothesized that these graphs would extrapolate through zero.

The data in Figures 4-12, 4-22, and 4-23 were extrapolated back through zero;

these results can be found in Figure 4-24. The data in Figure 4-24 seems to indicate that

the flaw to mirror size ratio is the best method of calculating D*. Even though the error

is very high compared to other methods, the flaw to mirror ratio best fits predicted

models for fractal dimension to toughness. This argument is made based on the fact that

when extrapolates the data for all three techniques through zero, the R2 value for flaw to

mirror ratios is the highest (0.7311), as evidenced on the next page.

The data in Figure 4-24 serves a second purpose, not only can it help to better

understand where problems in measuring fractal dimension occur, but also the slope of

the linear regression line can be used to calculate ao. The slope of a plot of D*1/2 versus

KIc should be equivalent to Ea01/2. Table 4-15 lists the ao values calculated by comparing

the slope of the regression line to the modulus of epoxy resins.











Comparision of Fractal Dimensional Increments

1500
Flaw/Mirror

1400 A* a Hand
A Image Pro

1300 -Linear (Image Pro)
Linear (Hand)
S1200- Linear (Flaw/Mirror)
1 1200

1100- y = 2622.4x
100 R2 = 0.7311 Flaw/Mirror
1000
y= 2638.6x
R2 = 0.6718 Hand
900
y= 2564.1x Image Pro
800 A- R2 = 0.5824
0.30 0.35 0.40 0.45 0.50
D*112


Figure 4-24 Comparison of Fractal Dimension Increment to Toughness for Three
Different Techniques of Measuring

Table 4-15 ao Values ([tm) calculated from the Slope of the Fractal Dimension vs.
Toughness Plot
Strain Rate(mm/min) 8phr DETA 10 phr DETA
0.1 4 4
10 3 3
100 3 3


The values of ao are similar to those calculated from the data gathered in this

research using equation 4-5. This would seem to further indicate that there is no strain

rate component to ao. However, the calculated values of ao are still significantly higher

than nodule size. As previously mentioned the nodule size for the investigated

compositions ranges between 75 and 100 nm, the calculated values of ao are well over

one order of magnitude larger. Additionally it should be noted that ao values are identical

for both compositions, ever though the nodule sizes are different.









4.4.3 Error Analysis of ao

The West Mecholsky Passoja theory is a complex equation that requires several

measurements to be made. Each measurement has its own error, which can be the result

of many factors such as machine compliance and user specific error. As a result, the

cumulative error for calculation of ao can not be calculated from directly summing the

error of the individual measurements. The individual errors for each variable in the WMP

theory must be weighed for their magnitude and power in the equation. As previously

mentioned, Appendix A has been prepared to describe how error was calculated and also

describes the individual contribution of each variable on the final value of toughness or

ao. Table 4-16 list the error for each composition and strain rate tested in this research.

Table 4-16 Cumulative Error Values for ao Calculations for Epoxy Resins
Strain Rate 8phr DETA 10 phr DETA
0.1 53% 44%
10 68% 61%
100 44% 74%

The cumulative errors associated with calculated ao from the data presented are

very large. As such, one can only make the judgment that there is no statistical difference

between the ao values calculated for one strain rate to another. Additionally, it is also

apparent that the error in the strain rate values for the two DETA compositions are the

same. Each composition has a different strain rate, however this data would seem to

indicate that ao is not a function of nodule size.

4.5 Results and Discussion

The data found in the various studies in this research is similar to what can be

expected from the literature (KIc z 1.5 MPa*ml/2, E z 1.5 GPa, D*z0.2, Nodule Size

100 nm). The value of ao was found to be around 4 itm, which is significantly larger than









nodule size. While the cumulative error of ao as listed in Table 4-16 may be very large,

it should be noted that calculated values of ao are almost two orders of magnitude greater

than nodule size, and would be almost three orders of magnitude larger than the

interstitial formed by nodule packing. As such, it can be concluded that there is no direct

relation between nodule size, nodule packing, and ao, which is contrary to our initial

hypothesis.

Table 4-17. Calculated ao Values and Associated Error
Strain 8 phr DETA 10 phr DETA
Rate ao([tm) %Error ao([tm) %Error
0.1 3.9 53% 5.2 44%
10 3.8 68% 3.8 61%
100 3.3 61% 2.6 74%


There are a few possible explanations for this result. First and foremost it could be

that nodule size simply does not relate to toughness in any meaningful way. Literature

has shown that as the nodule size changes cross-link density also changes, which has

been shown extensively to relate to toughness for other polymeric systems.(Racich et al.

1976) It could be that toughness is only controlled by cross-link density. As previously

mentioned, previous research has reported that nodules and interstitial material have the

same density and modulus. Therefore, it could mean that nodules are merely an artifact of

the polymerization reaction and do not positively or adversely affect toughness. Another

possible explanation can be found from the results of the nodule extraction study.

As mentioned above and in the literature, acetone and other solvents can be used to

swell the epoxy resin and wash nodules out of the structure. However not all nodules

wash out of the structure, only a very small percentage. The structural parameter, ao can

be linked to the weakest link theory of failure. When a body is stressed, failure occurs at









the weakest link; in the case of the WMP theory, the fracture tip propagates through the

weakest links. One possible explanation for why ao is so large could stem from the fact

that not all nodules are tightly bound to the structure. As the crack tip propagates

through the sample, it paths along between weakly or unbound nodules in the bulk.

Additionally it should be noted that the sub critical crack growth found as part of

the flaw size measurements has not been previously shown to occur in homogenous

epoxy resins. This phenomenon is usually found in heterogeneous materials such as

composites where R-curve behavior can occur. It is possible that individual nodules

serve as points for crack tip deflection, the chief mechanism for toughening in

composites.(Watchman 1996) This lends further credence to the belief that there are

structural inhomogeneities such as free or loosely bound nodules that might contribute to

the ao parameter.

4.6 Conclusions

The original hypothesis of this work was that ao can be related to nodule size,

however the results indicate that there is no relation between these two values. While the

hypothesis was shown not to be valid, this work has opened up some possible avenues to

investigate further. It could be further postulated that ao may not be directly related to

nodule size, but is a function of concentration of loosely bound nodules. Researchers

have shown that there is a very low concentration of nodules that are not tightly bound to

the resin and can be washed out of the structure with extended exposure to solvents. It is

possible that ao is a function of the average distance between weakly or unbound nodules.

Additionally part of this research focused on developed a new adaptation of the slit

island method for producing contours for calculating fractal dimension. Contours were

produced using optical microscopy and characterized using hand and computer






81


calculations. The values calculated from the optical microscopy contours were

compared to fractal dimensions derived from the ratio of mirror to flaw size of fracture

surfaces. The values for both techniques were found to be similar to one another. This

means that optical microscopy can be used to calculate fractal dimension of fracture

surfaces.














CHAPTER 5
SILSESQUIOXANES: GROWTH, STRUCTURE, AND CHARACTERISTICS

5.1 Introduction

Silsesquioxanes are technologically important hybrid inorganic-organic polymers.

Polysilsesquioxanes have been used in everything from microelectronics to scratch-

resistant coatings. Chemically they are similar to polysiloxanes or silicone, with a

backbone of silicon and oxygen and an organic side groups; they differ in that

polysilsesquioxanes have one organic group while polysiloxanes have two. The general

structures of a polysilsesquioxane and polysiloxane are compared in Figure 5-1.


HO /

S-/ /
HO / 0 Si- O \

\-s \ /, 0


Si OHO--Si--OH


Polysiloxane Polysilsesquioxane

Figure 5-1. Comparison of Silicone and Polysilsesquioxane Structures

There are many different types of polysilsesquioxanes. Polysilsesquioxanes are

classified by their organic group. A methyl polysilsesquioxane is one that contains a CH3

group, a phenyl polysilsesquioxane contains a C6H5 group and so forth. The properties,

uses, and structure of a polysilsesquioxane are dependent on the organic group. Table 5-









1 lists some of the more common polysilsesquioxanes and gives the common use and

structure.

Table 5-1. Usage of Common Polysilsesquioxanes
Silsesquioxane Usage
Hydrido Interlayer Dielectrics
Additive, Binder,
Methyl
l Precursor
Phenyl Coatings, Precursor


Silsesquioxanes are synthesized from trifunctional monomers. This results in a

complex non-linear polymer. Many possible structures can be formed from

polysilsesquioxanes. There are generally three classes of polysilsesquioxane structures:

random network, cage, and ladder, as found in Figure 5-2. It should be noted, however,

that the ladder structure has almost exclusively been shown to be only found in the

phenyl polysilsesquioxane polymer.

\SI HO /
s, S'- O I0
/ 0 /HO S
o / ",oI s-. _O

CO-SI--OH

Caged Ladder


Figure 5-2. Common Polysilsesquioxane Structures









An independent study performed by Cao and Baney indicated that bulk

polysilsesquioxanes have a nodular microstructure similar to what is found in epoxies.

These results have not be reported or duplicated elsewhere, this is mostly due to the fact

that polysilsesquioxanes are very rarely ever studied as bulk resins. Traditionally,

polysilsesquioxanes are used as thin films or as fillers in other polymers. This research

will build on the work of Cao and Baney by investigating what parameters influence the

formation of nodules in polysilsesquioxanes.

This research was broken down into two distinct portions, characterization of the

polymerization process, and characterization of cross-linked polysilsesquioxane

monoliths. The polymerization process was investigated for effects on physical

properties such as viscosity and molecular weight, and chemical properties such as

structure. The cross-linked monoliths were investigated for nodular microstructure as

seen in the epoxies and methods of formation.

5.2 Polysilsesquioxane Synthesis and Processing

5.2.1 Polymer Synthesis

Silsesquioxanes are synthesized using many different techniques. All

polysilsesquioxanes studied in this research were synthesized in the same manner using a

novel two phase reaction technique that has been reported in the literature.(Kondo et al.

2000) The Itoh method for synthesizing polysilsesquioxanes uses a hydrogen bonding

solvent, such as a ketone, to direct the condensation reaction. The ketone used in the Itoh

method hydrogen bonds to the hydroxide groups of the hydrolyzed monomer.

Silsesquioxanes were prepared using alkyltrichlorosilanes purchased from Gelest

Inc. The ratio of silane to water was held constant for each polysilsesquioxane, 1.5 moles

silane to 600 ml of water. A solution of 1.5 moles of silane and 75 ml of methyl isobutyl