Filtration and Transport of Colloids and Nanopaticles in Dense Emergent Vegetation

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Filtration and Transport of Colloids and Nanopaticles in Dense Emergent Vegetation Theory, Experiments, and Modeling
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Wu, Lei
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Doctorate ( Ph.D.)
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University of Florida
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Agricultural and Biological Engineering
Committee Chair:
Munoz-Carpena, Rafael
Committee Members:
Ziegler, Kirk Jeremy
Gao, Bin
Bonzongo, Jean-Claude J
Fox, Garey A

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attachment -- cnts -- colloid -- deposition -- filtration -- transport -- vegetation
Agricultural and Biological Engineering -- Dissertations, Academic -- UF
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Abstract:
A thorough understanding of filtration andtransport of colloidal contaminants in aquatic environment is of greatimportance to many environmental and biological processes. However, littleresearch has been conducted to investigate the transport of colloidal particlesthrough emergent vegetation in overland flow. In this work, a series ofsystemic laboratory experiments were conducted to measure the single-stemcontact efficiency (?0) and attachment efficiency (a) of colloid capture by asimulated plant stem in laminar lateral flow. The results showed that existingtheoretical and empirical models of colloid contact and attachment efficiencyfor porous media were found to fall short in describing the colloid filtrationby dense vegetation system in overland flow. New dimensionless equations ofsingle-stem efficiencies were developed to predict colloid filtration by densevegetation with reasonable accuracy. Except colloidal particles, theever-increasing use of engineered nanomaterials (e.g. carbon nanotubes (CNTs))will lead to heightened levels of these materials in the environment. CNTsaggregation and deposition behavior will dictate its transport potential andthus the environmental fate and potential ecotoxicological impacts of thesematerials. However, the unique properties of CNTs proposed challenges inexperimentally and theoretically quantifying its deposition and aggregation inthe environment. The surface element integration (SEI) technique was coupledwith the DLVO theory to determine the orientation-dependent interaction energybetween CNTs and an infinite isotropic planar surface. For the first time,analytical formula was developed to accurately describe the interaction betweennot only pristine but also surface charged CNTs and planar surfaces witharbitrary rotation angles. The new analytical expressions presented in thiswork can be used as a robust tool to describe the DLVO interaction between CNTsand planar surfaces under various conditions and thus to assist the design andapplication of CNT-based products. Normal 0 false false false EN-US ZH-CN X-NONE /* Style Definitions */ table.MsoNormalTable{mso-style-name:"Table Normal";mso-tstyle-rowband-size:0;mso-tstyle-colband-size:0;mso-style-noshow:yes;mso-style-priority:99;mso-style-parent:"";mso-padding-alt:0in 5.4pt 0in 5.4pt;mso-para-margin:0in;mso-para-margin-bottom:.0001pt;mso-pagination:widow-orphan;font-size:10.0pt;font-family:"Times New Roman","serif";}
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by Lei Wu.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: Munoz-Carpena, Rafael.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-11-30

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1 FILTRATION AND TRANSPORT OF COLLOIDS AND NANOPATICLES IN DENSE EMERGENT VEGETATION: THEORY, EXPERIMENT AND MODELING By LEI WU 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 201 3

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2 201 3 Lei Wu

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3 To my family

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4 ACKNOWLEDGMENTS I express my sincerest gratitude to my advisor, Dr. Rafael Mu oz Carpena and co Chair, Dr. Bin Gao, for their support, guidance, patience, and inspiration throughout my graduate study at the University of Florida. Their genuine interest and enthusiasm for this field are very contagious and I always left their office very optimistic and eager to tackle the next problem s at hand. I also thank the other members of my committee, Dr. Kirk J. Ziegler, Dr. Jean Claude Bonzongo, and Dr. Garey A. Fox for their valu able advice, help and support in the past three years. Special thanks go to Dr. L C S hen (University of Florida), Dr. Chongyang Shen (China Agricultural University), Dr. Xiqing Li (Peking Univeristy), Dr. Huilian Ma (University of Utah), Dr. Shihong Lin (Y ale University), Dr. Gregory V. Lowry (Carnegie Mellon University), Dr. Alexey N. Volkov (University of Virginia), Dr. Y. A. Pachepsky (UDSA), and Dr. David Kaplan (University of Florida), for their thoughtful suggestions and their insight to my research p roblems. My colleagu es also friends in the hydrologic modeling lab and nanotechnology lab deserve special acknowledgement. They not only ga ve me help and support to my research, but also helped me remember that there is more to life than research. I would also like to thank the entire office staff, technician s each of whom has helped me deal with at least crises. Above all others, I wish to thank my parents for their constant love, hout them.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURE S ................................ ................................ ................................ ........ 12 LIST OF ABBREVIATIONS ................................ ................................ ........................... 15 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 19 Scient ific Questions ................................ ................................ ................................ 19 Literature Review on Question 1 ................................ ................................ ............. 20 Colloidal Particles and Significance ................................ ................................ .. 20 Fate and Transport of Colloidal Particles in Subsurface Environment .............. 21 Fate and Transport of Colloidal Particles in Surface Environment ................... 22 Knowledge Gap on Quest ion 1 and Research Scope ................................ ............. 24 Research Objectives of Question 1 ................................ ................................ ........ 25 Literature Review on Question 2 ................................ ................................ ............. 27 Carbon Nanotubes and Significance ................................ ................................ 27 CNTs Releases to the Environment ................................ ................................ 28 Environmental Fate and transport of CNTs ................................ ...................... 29 Specific Knowledge Gap 2 and Research Scope ................................ ................... 31 Research Objectives 2 ................................ ................................ ............................ 32 Organization of the Dissertation ................................ ................................ .............. 32 2 EXPERIMENTAL ANALYSIS OF COLLOID CAPTURE BY A CYLINDRICAL COLLECTOR IN LAMINAR OVERLAND FLOW ................................ ................... 34 Introduct ory Remarks ................................ ................................ .............................. 34 Theory ................................ ................................ ................................ ..................... 37 Materials and Methods ................................ ................................ ............................ 39 Colloi ds and Collectors ................................ ................................ ..................... 39 Experimental Apparatus ................................ ................................ ................... 40 Experimental Methods ................................ ................................ ...................... 40 Results and Discussion ................................ ................................ ........................... 41 Effects of Flow Velocity and Colloid and Collector Sizes ................................ .. 41 Comparison of Experimental Data and Theoretical Predictions ....................... 43 A Regression Equation ................................ ................................ ..................... 45 Environmental Implication ................................ ................................ ....................... 46 3 SINGLE COLLECTOR ATTACHMENT EFFICIENCY OF COLLOID CAPTURE BY A CYLINDRICAL COLLECTOR ................................ ................................ ....... 54

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6 Introduct ory Remarks ................................ ................................ .............................. 54 Theory ................................ ................................ ................................ ..................... 57 Materials a nd M ethods ................................ ................................ ............................ 60 Materi als ................................ ................................ ................................ ........... 60 Experimental Methods ................................ ................................ ...................... 61 Results and D iscussion ................................ ................................ ........................... 63 Effect of Ionic Strength ................................ ................................ ..................... 63 Effect of Flow Velocity ................................ ................................ ...................... 64 Comparison of Experi mental Data and Theoretical Predictions ....................... 66 A Dimensionless Equation ................................ ................................ ................ 68 Environmental Implications ................................ ................................ ..................... 70 4 EXTENDED SINGLE STEM EFFICIENCY THEORY FOR COLLOID FILTRATION THROUGH SURFACE DENSE VEGETATION ................................ 80 Introduct ory Remarks ................................ ................................ .............................. 80 Theoretical Background and New Dimensionless Number (N STE ) .......................... 83 Materials and Methods ................................ ................................ ............................ 87 Materi als ................................ ................................ ................................ ........... 87 Vegetation Chamber Experiments ................................ ................................ .... 88 Characterize biopolymer brush layer (trichome) on vegetation stem ................ 89 Determine kinetic deposition rate ( k d ) ................................ ............................... 90 Results and Discussion ................................ ................................ ........................... 90 Effect of ionic strength on colloid filtration in dense vegetation ........................ 90 Coupled effect of flow velocity and stem density on colloid filtration in dense vegetation ................................ ................................ ................................ ...... 92 Extended single stem efficiency theory ................................ ............................ 94 Deposition Mechanisms and Other Potential Effects ................................ ........ 96 Environmental Implications ................................ ................................ ..................... 97 5 DLVO INTERACTIONS OF CARBON NANOTUBES WITH I SOTROPIC PLANAR SURFACE ................................ ................................ ............................. 108 Introduct ory Remarks ................................ ................................ ............................ 108 Theory ................................ ................................ ................................ ................... 113 Results and Discussion ................................ ................................ ......................... 116 DLVO Interactions between a Pristine SWNT and an Isotropic Planar Surface ................................ ................................ ................................ ........ 116 DLVO Interactions between a Surface Modified SWNT and a Charged Isotropic Planar Surface ................................ ................................ .............. 119 DLVO Forces and Torques of SWNTs with Planar Surfaces .......................... 123 DLVO Interactions of MWNTs and Planar Surfaces ................................ ....... 124 Environmental Implications ................................ ................................ ................... 125 6 CONCLUSIONS AND RECOMMENDATIONS ................................ ..................... 133 Conclusions ................................ ................................ ................................ .......... 133

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7 Recommendations for Further Study ................................ ................................ .... 136 Plant filtration theory in overland flow ................................ ............................. 136 Interactions between CNTs and interfaces ................................ ..................... 137 APPENDIX A SUPPORTING INFORMATION FOR CHAPTER 2 ................................ .............. 139 B SUPPORTING INFORMATION FOR CHAPTER 3 ................................ .............. 141 C SUPPORTING INFORMATION FOR CHAPTER 4 ................................ .............. 172 D SUPPORTING INFORMATION FOR CHAPTER 5 ................................ .............. 197 LIST OF REFERENCES ................................ ................................ ............................. 204 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 229

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8 LIST OF TABLES Table page 2 1. Summary of experimental conditions and results ................................ ............... 48 3 1. Summary of experimental conditions and results ................................ ............... 72 3 2. Calculated Maximum energy barriers ( ) and primary ( ) and secondary minimum ( ) under different experimental conditions .................. 73 3 3. Summary of dimensionless parameters governing attachment efficiency .......... 74 4 1. Summary of experimental conditions, biopolymer brush properties and best fit value of parameters in transport model ................................ .......................... 99 A 1. Experimental data of single stem contact efficiency ( 0 ) under different flow velocity conditions for a given colloid ( d p =1.05m) and collector ( d c =2cm) ..... 139 A 2. Experimental data of single stem contact efficiency ( 0 ) under different sizes of colloid and collector at a given flow velocity (u=0.02cm/s) ........................... 140 B 1. Definition of Dimensionless parameters ................................ ........................... 145 B 2. Summary of stepwise least square regression results ................................ ..... 146 B 3 conditions (IS=0.001M) for a given colloid (dp=1.05m) ................................ .. 150 B 4 conditions (IS=0.005M) for a given colloid (dp=1.05m) ................................ .. 151 B 5 conditions (IS=0.01M) for a given colloid (dp=1.05m ) ................................ .... 152 B 6. conditions (IS=0.05M) for a given colloid (dp=1.05m) ................................ .... 153 B 7. conditions (IS=0.1M) for a gi ven colloid (dp=1.05m) ................................ ...... 154 B 8. conditions (IS=0.00 1M) for a given colloid (dp=0.1m) ................................ .... 155 B 9. conditi ons (IS=0.005M) for a given colloid (dp=0.1m) ................................ .... 156 B 10. conditions (IS=0.01M) for a given colloid (dp=0.1m) ................................ ...... 157

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9 B 11. conditions (IS=0.05M) for a given colloid (dp=0.1m) ................................ ...... 158 B 12. conditions (IS=0.1M) for a given colloid (dp=0.1m ) ................................ ........ 159 B 13. and ionic strength conditions (u=0.0002cm/s and IS=0.01M) for a given colloid (dp=1.05m) ................................ ................................ .......................... 160 B 14. nt flow velocities and ionic strength conditions (u=0.002cm/s and IS=0.01M) for a given colloid (dp=1.05m) ................................ ................................ ................................ ..... 161 B 15. Experimental and ionic strength conditions (u=0.2cm/s and IS=0.01M) for a given colloid (dp=1.05m) ................................ ................................ ................................ ..... 162 B 16. and ionic strength conditions (u=0.0002cm/s and IS=0.1M) for a given colloid (dp=1.05m) ................................ ................................ ................................ ..... 163 B 17. and ionic strength conditions (u=0.002cm/s and IS=0.1M) for a given colloid (dp=1.05m) ................................ ................................ ................................ ..... 164 B 18. and ionic strength conditions (u=0.2cm/s and IS=0.1M) for a given colloid (dp=1.05m) ................................ ................................ ................................ ..... 165 B 19. ties and ionic strength conditions (u=0.0002cm/s and IS=0.01M) for a given colloid (dp=0.1m) ................................ ................................ ............................ 166 B 20. Experimental data of attac and ionic strength conditions (u=0.002cm/s and IS=0.01M) for a given colloid (dp=0.1m) ................................ ................................ ................................ ....... 167 B 21. and ionic strength conditions (u=0.2cm/s and IS=0.01M) for a given colloid (dp=0.1m) ................................ ................................ ................................ ....... 168 B 22. and ionic strength conditions (u=0.0002cm/s and IS=0.1M) for a given colloid (dp=0.1m) ................................ ................................ ................................ ....... 169

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10 B 23. and ionic strength conditions (u=0.002cm/s and IS=0.1M) for a given colloid (dp=0.1m) ................................ ................................ ................................ ....... 170 B 24. a nd ionic strength conditions (u=0.2cm/s and IS=0.1M) for a given colloid (dp=0.1m) ................................ ................................ ................................ ....... 171 C 1. Relevant parameters and constants for interaction between colloid and plant stem surface ................................ ................................ ................................ ..... 172 C 2. Experimental data of breakthrough curve under DI water conditions ............... 178 C 3. Experimental data of breakthrough curve under medium IS conditions (IS=0.01M) ................................ ................................ ................................ ........ 179 C 4. Experimental data of breakthrough curve under high IS conditions (IS=0.1M) 180 C 5. Experimental data of breakthrough curve under high IS conditions (IS=0.2M) 181 C 6. Experimental data of breakthrough curve under plant high density conditions (u=0.002 cm/s) ................................ ................................ ................................ 182 C 7. Experimental data of breakthrough curve under plant high density conditions (u=0.01cm/s) ................................ ................................ ................................ .... 183 C 8. Experimental data of breakthrough curve under plant high density conditions (u=0.05 cm/s) ................................ ................................ ................................ ... 184 C 9. Experimental data of breakthrou gh curve under plant high density conditions (u=0.1 cm/s) ................................ ................................ ................................ ..... 185 C 10. Experimental data of breakthrough curve under plant medium den sity conditions (u=0.002 cm/s) ................................ ................................ ................ 186 C 11. Experimental data of breakthrough curve under plant medium density conditions (u=0.01 cm /s) ................................ ................................ .................. 187 C 12. Experimental data of breakthrough curve under plant medium density conditions (u=0.05 cm/s) ................................ ................................ .................. 188 C 13. Experimental data of breakthrough curve under plant medium density conditions (u=0.1 cm/s) ................................ ................................ .................... 189 C 14. Experimental data of breakthrough curve under plant low density conditions (u=0.002 cm/s) ................................ ................................ ................................ 190 C 15. Experimental data of breakthrough curve under plant low density conditions (u=0.01 cm/s) ................................ ................................ ................................ ... 191

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11 C 16. Experimental data of breakthrough curve under plant low density conditions (u=0.05 cm/s) ................................ ................................ ................................ ... 192 C 17. Experimental data of breakthrough curve under plant low density conditions (u=0.1 cm/s) ................................ ................................ ................................ ..... 193 C 18. Experimental data of breakthrough curve under different sizes of colloid conditions (dp=0.1m) ................................ ................................ ...................... 194 C 19. Experimental data of breakthrough curve under different sizes of colloid conditions (dp=1.05m) ................................ ................................ .................... 195 C 20. Experimental data of breakthrough curve under different sizes of colloid conditions (dp=2.0m) ................................ ................................ ...................... 1 96

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12 LIST OF FIGURES Figure page 1 1. Fate and transport of colloids and nanoparticles in aquatic and terrestrial environments ................................ ................................ ................................ ...... 19 1 2. Illustration of colloid sizes and categories. ................................ ......................... 21 1 3. Illustration of vegetative filter strips (VFS). ................................ ......................... 24 1 4. Research scope of development of colloid filtration theory in dense vegetation in overland flow. ................................ ................................ ................ 25 1 5. Illustration of SWNT and MWNT. ................................ ................................ ........ 28 1 6. Illustration of CNTs releases to environment. ................................ ..................... 29 1 7. CNTs challenges to traditional DLVO theory. ................................ ..................... 30 1 8. Research scope of interactions between CNTs and planar surface. .................. 32 2 1. Graphical content of chapter 2 ................................ ................................ ............ 34 2 2. Schematic of experimental setup for measuring single collector contact efficiency. ................................ ................................ ................................ ........... 49 2 3. Effect of flow velocity on single collector contact efficiency. ............................... 50 2 4. Effect of colloid and collector sizes on single collector contact efficiency. .......... 51 2 5. Comparison of experimental data of single collector contact efficiency with predictions of (a) equations 1 3, (b) the YAO model, (c) the RT model, and (d) the TE mod el. ................................ ................................ ................................ 52 2 6. Comparison of experimental data of single collector contact efficiency with predictions of the new dimensionless equa tion (equation 2 6). .......................... 53 3 2. colloid capture by the cylinder in the flow chamber. ................................ ........... 75 3 3. DLVO interaction energy between the collector and the colloids: (a) 0.1 m colloids and (b) 1.05 m colloids. ................................ ................................ ....... 76 3 4. colloids captured by the cylinder in the flow chamber. ................................ ........ 77

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13 3 5. Maxwell, m odified Maxwell, and Bai Tien models for colloids captured by the cylinder in the flow chamber at ionic strength of 0.01M. ................................ ..... 78 3 6. new dimensionless equation. ................................ ................................ .............. 79 4 1. Graphical conte nt of chapter 4 ................................ ................................ ............ 80 4 2. (A) Schematic of adsorbed polymer layer; (B) Schematic of grafted polymer brush layer; (C) Schematic of a model for a spherical colloid with diameter of dp impinging upon a biopolymer brush in a solution. ................................ ........ 101 4 3. (A) Morphology of tri chomes on the plant stem and (B) air bubbles attached on the surface of trichomes under water. ................................ ......................... 102 4 4. Schematic and photos of experimental set up for vegetation chamber experiment. ................................ ................................ ................................ ....... 103 4 5. Effect of ionic strength on the colloid deposition onto the plant stem. .............. 104 4 6. Effect of coupled flow velocity and grass density on the colloid deposition onto the plant stem. ................................ ................................ .......................... 105 4 7. Goodness of fit evaluation of the extended model. ................................ .......... 106 4 8. Schematic illustration of three basic mechanisms of colloidal particles deposition on the plant stems. ................................ ................................ .......... 107 5 1. Graphical content of chapter 5 ................................ ................................ .......... 108 5 2. Schem atic illustration of interaction of a SWNT with an infinite isotropic planar surface. ................................ ................................ ................................ .. 127 5 3. The van der Waals interaction energy between a pristine SWNT and an isotropic planar surface. ................................ ................................ ................... 128 5 4. The electrostatic double layer interaction energy ( between a surface modified SWNT and a charged isotropic planar surface. ..................... 129 5 5. Total interaction energy between a surface modified SWNT and a charged isotropic planar surface. ................................ ................................ ... 130 5 6. DLVO force and torque acting on a surface modified SWNT a nd a charged isotropic planar surface. ................................ ................................ ................... 131 5 7. Normalized total interaction energy between a surface modified SWNT or MWNT and a charged isotropic planar surface. ................................ ................ 132

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14 B 1. Comparison of experimental attachment efficiency with predictions of the new dimensio nless equation for development dataset. ................................ .... 149 C 1. Breakthrough curves with and without plaster. ................................ ................. 174 C 2. Morris sensitivity analysis result chart ................................ .............................. 175 C 3. Sobol sensitivity analysis indices chart ................................ ............................. 176 C 4. Comparison of experimental deposition rate with predictions of the new dimensionless equation for development dataset ................................ ............. 177

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15 LIST OF ABBREVIATION S ADE A dvection dispersion equation CNTs Carbon nanotubes CFT Colloid filtration theory DA Derjaguin Approximation DI D eionized DLVO Derjaguin Landau Verwey Overbeek EDL E lectrical double layer E NPs Engineered nanoparticles EPM Electrophoretic mobility HA H umic acid IFBL I nteraction force boundary layer MWNTs M ulti walled nanotubes NOM Natural Organic M atter PV P ore volume PVC polyvinyl chloride SDBS Sodium dodecylbenzene sulfonate SEI Surface element integration SWNTs S ingle walled nanotubes VDW Van der Waals

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16 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 FILTRATION AND TRANSPORT OF COLLOIDS AND NANOPARTICLES IN DENSE EMERGENT VEGETATION: THEORY, EXPERIMENT, AND MODELING By LEI WU May 201 3 Chair: Rafael Mu oz Carpena Major: Agricultural and Biological Engineering A thorough understanding of filtration and transport of colloidal contaminants in the aquatic environment is of great importance to many environm ental and biological problems ( e.g., contaminant transport in flow, water quality, life cycles of microorganism s, and wetland geomorphology ). However, little research has been conducted to investigate the overland flow transport of colloidal particles through emergent vegetation. Understanding this process is critical since, compared to subsurface paths, overland f low constitutes a quick path for environmental transport of pollutants with immediate effects to surface water bodies that dense vegetation (natural or planted) could help to ameliorate In this work, first a series of laboratory experiments were conducted to measure the single stem contact efficiency ( 0 ) and attachment efficiency ( ) of colloid capture by a simulated plant stem in laminar lateral flow. The results showed that existing theoretical and empirical models of colloid contact and attachment efficiency developed originally for porous media fall short in describing the colloid filtration by dense vegetation system in overland flow. For the first time, n ew single stem efficiency theory (SSET) was developed to predict collo id filtration by dense vegetation with reasonable accuracy. In order to upscale SSET from clean single stem

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17 to real dense vegetation, a new dimensionless number was developed to account for the effect of plant stem surface properties on the colloid deposit ion in overland flow. Laboratory scale dense vegetation chamber experiments and model simulations were conducted to obtain the effective value s of colloid kinetic deposition rate ( k d ) in the vegetation system under different experimental conditions. The results showed that in addition to flow hydrodynamics (e.g. flow velocity) and solution chemistry (e.g. ionic strength), steric repulsion afforded by the biopolymer brush layer formed by trichomes on the p l ant stem surface also plays a significant role in the plant stem colloid deposition during overland flow. An extended model including the steric repulsion effect was developed that matched the experimental data well This extended model can be used to not only help construct and refine dynamic models o f colloid transport and filtration through dense vegetation in overland flow but also is applicable to prediction of colloid deposition on various polymer brush surfaces in natural, engineered and biomedical systems. In addition to colloidal particles, th e ever increasing use of engineered nanomaterials (e.g. carbon nanotubes CNTs) will likely lead to heightened levels of these materials in the environment. CNTs aggregation and deposition behavior will dictate their transport potential and thus the ir environmental fate and potential ecotoxicological impacts of these materials. However, the uni que properties of CNTs pose challenges to experimentally and theoretically quantifying their deposition and aggregation in the environment. The surface element i ntegration (SEI) technique was coupled with the Derjaguin Landau Verwey Overbeak ( DLVO ) theory to determine the orientation dependent interaction energy between CNTs and an infinite isotropic planar

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18 surface. For the first time, analytical formula s were dev eloped to accurately describe the interaction s between not only pristine but also surface charged CNTs and planar surfaces with arbitrary rotation angles. The new analytical expressions presented in this work can be used as a robust tool to describe the D LVO interaction between CNTs and planar surfaces under various conditions and thus to assist not only the development of effective strategies to reduce the environmental impact of CNTs but also the design and application of CNT based products.

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19 CHAPTER 1 INTRODUCTION Scientific Questions Over the last decade, colloids and nanoparticles have been used more frequently in agricultural, industrial applications and in consumer and medical products. [ 1 ] [ 2 3 ] And these applications will likely continue to increase. Concerns about their environmental fate, transport and ultimate environmental impacts have stimulated studi es to predict environmental concentrations in aquatic system s These particl es may enter the aquatic system either directly through wastewater treatment effluents or indirectly through surface run off through dense vegetation system s (Figure 1 1). Figure 1 1. F ate and transport of colloids and nanoparticles in aquatic and terrestrial environments The former has received a lot of attentions in the past few decades; however, much less effort has been dedicated to the fate and transport of colloidal particle s in surface runoff, especially through dense vegetation that might help to reduce transport through deposition in the soil vegetation system. In addition, unique properties of nanoparticles and their suspensions ( e.g., shape, size, structure, and chemical composition) challenge the ability of colloid science to understand nanoparticles

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20 aggregation behavior and the subsequent effects on environmental fate and transport of nanoparticles. Therefore, this dissertation focuses on two main questions: (1) a re th e current colloid filtration theories developed for porous media applicable to a vegetation system in surface water? And (2) a re the current approaches and models used in quantifying colloidal interactions and transport applicable to nanoparticles? Literature Review on Question 1 Colloidal P articles and Significance A thorough understanding of deposition of colloidal particles in surface water flow is of great importance to many environmental and biological processes [ 4 7 ] (e.g. transport and fate of contaminants, deterioration of water quality, life cycles of microorganisms and changes in wetland geomorphology). Colloidal particles with eff ective diameters of a 2 ) can be categorized into two categories: abiotic colloids ( e.g., amorphous iron, and manganese oxides, engineered nanomaterial) and biotic colloids ( e.g., viruses, bacteria, and protozoa) [ 8 9 ] Having relatively high specific surface areas and charge densities, colloids serve as efficient carrier of various pollutants and enhance their mobility along hydrologic pathways [ 10 12 ] Furthermore, ma ny of biotic colloids pose a risk to public health and are therefore contaminants of concern in surface water and drinking water supplies and on agricultural produce [ 13 16 ] Hence, effective treatmen t processes for many colloidal particles and contaminants rel y on the optimization of colloid transport and retention in surface water flow.

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21 Figure 1 2 I llustration of colloid sizes and categories Fate and Transport of Colloidal Particles in Subsurfac e Environment Considerable research ha s been devoted to study the fate and transport of colloidal particles in the subsurface environment (vadose zone and groundwater). Reviews have been given by Ryan and Elimelech, 1996 [ 17 ] ; Schij ven and Hassanizadeh, 2000 [ 18 ] ; Harvey and Harms, 2007 [ 19 ] ; Jin and Flury, 2002 [ 20 ] ; Ginn et al., 2002 [ 21 ] ; de Jonge et al., 2004 [ 22 ] ; DeNovio et al., 2004 [ 23 ] ; Rockhold et al., 2004 [ 24 ] ; Sen and Khilar, 2006 [ 25 ] ; Tufenkji et al., 2006 [ 26 ] Briefly, with the exception of a few field scale studies that examined the effect of infiltration on colloid mobilization [ 27 29 ] undisturbed soil columns have been used to mimic the subsurface in laboratory res earch [ 30 34 ] Bench scale soil packed column experiments have been conducted to examine the transport behaviors of different types of colloids including viruses, bacteria, clay particles, and synthetic microspheres, and engineered nanoparticles [ 35 38 ] Relationships between system physicochemical p roperties (e.g., flow velocity, solution chemistry, and surface properties) and colloid mobility in porous media were evaluated [ 39 42 ] In addition the influences of biological factors (e.g., cell size and shape, cell motility, and micro molecular length and composition) on bio colloid

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22 fate and transport in porous media have been assessed [ 43 44 ] Findings from these experimental investigations have revealed some of the fundamental transport mechanisms and enhanced current ability to predict colloid fate and transport in subsurface environments [ 25 45 47 ] Fate and Tra nsport of Colloidal Particles in Surface Environment Considerably less attention has been dedicated to the fate and transport of colloidal particles in surface flow par ticularly with respect to colloid transport through vegetation in overland flow [ 48 50 ] Dense submerged vegetation in aquatic systems have been shown to reduce the flow velocity in open channels to promote the deposition of sediments [ 51 53 ] suppress turbulence to favorably influence growth and distribution of aquatic organisms such as phytoplankton [ 54 56 ] and alter the resident time to affect water quality [ 57 59 ] Plant surrogates ( e.g., vertical cylinders) have often been used in the laboratory for exploring the key determinants of flow dynamics and governing mechanisms of contaminant transport through submerged vegetation [ 60 62 ] Findings from the laboratory experiments with simulated systems have greatly enhanced the understanding of flow and transport processes in rivers, estuaries, and natural and constructed wetlands [ 63 67 ] In addition to flow and sediment transport, the influence of submerged aquatic vegetation on the fate and transport of suspended fine particles ha s also been investigated in laboratory and field environments. Leonard et al [ 53 ] observed that the capture of suspended particles on the stems and leaves of Juncus roemerianus marsh contributed up to 10% of the total sediment deposition to a tidal marsh. Similarly, Pluntke and Kozerski [ 58 ] suggested that sedimentation onto plant structures should be considered when quantifying particle retention in submerged macrophyte stands.

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23 Particle retention in a sea grass meadow ( Posidonia oceanica ) was found to be up to 15 times greater than the equivalent non vegetated bed [ 68 ] In a wetland field site in the Florida Everglades Huang et al. [ 69 ] found that submerged aquatic vegetation could also remove colloidal particles from surface flow. These evidences strongly suggest that filtration by plant structures, such as stems, has a s ignificant effect on the fate and transport of colloidal particles in surface flow. Unfortunately, current understanding of the capture of colloidal particles by emergent terrestrial vegetation in overland flow is still very limited. Dense emergent vegeta tion in terrestrial systems has been proven to be effective in removal the non point source pollutants (including sediment, plant nutrients, and pesticides) from agricultural field and urban areas [ 70 72 ] Vegetative filter strips (VFS) (Figure 1 3) a common runoff pollution control practice, ha ve been promoted to help control the movement of pollutants from cropland and urban runoff. Many laboratory and field studies have been c onducted to determine the efficiency of VFS in protecting water resources from non point source pollution [ 73 75 ] For instance, it was reported that a well installed VFS can remove suspended sediments (up to 90%), phosphorus (75%), nitrogen ( up to 87%), and pesticides (40%) [ 76 79 ] Recently, a grow ing research effort is aimed at reducing the transport of biocolloids (particularly pathogens) in overland flow [ 80 83 ] Emergent terrestrial vegetation ( e.g., VFS) has been suggested as being effective in attenuating the loading of manure borne microorganisms from farms and other agricultural and urban lands to runoff [ 84 86 ] For ins tance, Fox, et al. [ 87 ] recently determined vegetative filter strips (VFS) effectiveness in removing E.coli from runoff relative to inflow rate, infiltration

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24 capacity, and flow concentration in a laboratory scale VFS soil box. Field experiments conducted by Ferguson et al. [ 88 ] also showed that colloid size played an im portant role in controlling the mobility of microorganisms (biocolloids) in dense vegetation. Results from those studies have informed the optimization of the design and maintenance of the emergent terrestrial vegetation filters to remove sediments and agr icultural chemicals [ 89 92 ] Figure 1 3. Illustration of v egetative filter strips (VFS). Knowledge Gap on Question 1 and Research Scope From the evidence presented above, it can be conclude d that although colloid and colloid facilitated transport in water flow is a well known contamination process, little research has been conducted to investigate the transport of colloidal particles through emergent vegetation in overland flow. There exists a knowledge gap regarding theories and mechanisms that govern colloid fate and transpor t in terrestrial dense vegetation in overland flow. Therefore, systemic studies to identify the fundamental

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25 processes of colloid transport through dense emergent terrestrial vegetation are needed. In overland flow, the depth of water is usually below the top of the sheath s of dense vegetation and thus plant stem s may control flow and transport processes [ 93 94 ] Un der these conditions plant stems can be modeled as rigid, cylindrical collectors for colloid deposition [ 95 ] Therefore, establishing a single stem efficiency theory of colloids filtration by dense emergent vegetation will advance the understanding the fate and transport of colloids in surface flow. The scope of this research is shown in Figure 1 4. Figure 1 4 R esearch scope of development of coll oid filtration theory in dense vegetation in overland flow Research Objectives of Question 1 The overall (long term) research goal i s to develop a singl e stem efficiency theory for plant filtration of colloids through dense vegetation in overland flow It is our central hypothesis that the stems of the surface vegetation can be modeled as rigid

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26 filtration collectors for colloids in shallow overland flow. S pecific hypothesis and objectives are as follows: Hypothesis 1: System physical factors will affect colloid capture by vegetation stem in shallow overland flow, and colloid filtration theory in porous media can be used to predict the colloid capture by dense vegetation in laminar overland flow Objective 1: develop a theory for predicting the single ste m contact efficiency ( 0 ) of colloid filtration by emergent dense vegetation in shallow overland flow. The specific objectives are to ( 1) determine how flow velocity, colloid size, and collector size affect the single stem efficiency of colloid capture by a cylindrical collector in laminar overland flow through flow chamber experiment, (2) test whether existing single collector contact efficiency models can be used to predict colloid capture by a cylinder in laminar overlan d flow, and (3) develop a dimensionless equation to describe the single stem efficiency of colloid transport through emergent vegetation in laminar overland flow. Hypothesis 2 : System physicochemical properties and flow velocity will affect colloids attach ment onto the surface of stem and DLVO theory coupled with torque balance approach can be used to interpret the colloid attachment onto the surface of collector in overland flow Objective 2 : develop a theory for predicting the single stem attachment effi ciency ( ) of colloid filtration by emergent vegetation in shallow overland flow. The specific objectives are to : ( 1) determine how ionic strength, colloid size and flow velocity affect the attachment efficiency of colloid capture by a cylindrical collector in laminar overland flow through flow chamber experiment ; (2) test whether existing

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27 attachment efficiency models can be used to predict colloidal particles attachment onto vege tation stems in laminar overland fl ow ; (3) if existing theories prove limited, develop a new equation to describe the attachment process of colloidal particles onto vegetation stems in laminar overland flow ; and (4) if it is necessary to develop a new equa tion, test the performance of attachment efficiency through column experiment s Hypothesis 3 : surface properties of plant stem will affect colloid kinetic deposition rate, and these can be used to improve the predict ion of colloid filtration by dense veget ation in shallow overland flow Objective 3 : to apply and modify the single stem efficiency theory to dense vegetation system in overland flow. The hypothesis and objectives will be tested and developed through the following experimental tasks: (1) determine the effect of flow velocity, vegetation density, colloid size and ionic strength on the colloid kinetic deposition rate; (2) determine whether existing single kinetic depositi on rate of colloid in dense vegetation system in overland flow; (3) develop a new theory (extended model) to predict the deposition of colloidal particles on plant stem in laminar overland flow. Literature Review on Question 2 Carbon N anotubes and Significance Nanoparticles (NPs), defined as particles with at least one dimension smaller than 10 0 nm have received much recent attention because of the ir potential toxic effects and the rapid development of n anotechnology [ 96 103 ] Carbon nanotubes (CN Ts) are among the top N Ps of concern in the environment [ 104 106 ] Entirely composed of carbon w ith a significantly large length to diameter ratio and unique physicochemical

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28 properties CNTs are rolled up graphene sheets with exceptional mechanical, electrical, optical, and thermal properties [ 107 110 ] There are mainly two types of CNTs: single and multi walled. Single walled carbon nanotubes (SWNTs) are one layered gra phitic cylinders having diameters on the order of a few nanometers, while multi walled carbon nanotubes (MWNTs) comprise of 2 to 30 concentric cylinders having outer d iameters o ften between 2 25 nm (Figure 1 5 ). They are largely used in many novel applicat ions in nanotechnology, electronics, optics, thermal conductors, and other fields in material science and engineering [ 111 114 ] Figure 1 5 I llustration of SWNT and MWNT CNTs R eleases to the Environment T he exponential growth in production of CNTs and their widespread application s in consumer products will inevitably result in their release into the environment ( e.g., air, water soil, and sediment, Figure 1 6 ) [ 115 116 ] Release m ay come from point sources (e.g., manufacturing and wastewater effluent) or from non point sources (e.g., attrition from CNTs products). Biochemical cycling of CNTs may involve photochemical reactions in the air; aggregatio n and filtration in the soil; sus pension, flocculation sedimentation, deposition and aggregation in the water and uptake, accumulation and

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29 degradation in organisms. Human exposure to CNTs is most likely during manufacturing, nut inhalation of CNTs released to the atmosphere and ingestion of drinking water or food. Dermal exposure from sunscreens and cosmetics is also likely. [ 1 ] Figure 1 6 I llustration of CNTs releases to environment Environmental Fate and transport of CNTs Once CNTs are released into the environment through any of the release pathways, their mobility and colloidal stability are expected to control their bioavailability and impact on the environment. While CNTs release occurs within all of these environments, this dissertation focuses on CNTs deposition behavior in aquatic systems. When released into aquatic environments, CNTs deposition is controlled by CNTs specific properties (e.g., shape, size, chemical composition, surface structure and coating), the surr ounding solution chemistry (e.g., pH, ionic strength, and natural organic matter), and hydrodynamic conditions. [ 117 ] In recent years several studies have been conducted to investigate CNTs deposition on solid surface in aquatic system

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30 either by experimental ap proach or by theoretical approach. [ 118 121 ] Traditionally, the interactions between CNTs and other solid surfaces have been investigated through column filtration experime nts. [ 118 123 ] However, the theoretical interpretation of results from such systems is still far from satisfaction because most of th e theoretical studies are based on colloid science principles Derjaguin Landau Verwey Overbeak (DLVO) [ 117 120 ] CNTs ch a llenge the limits of colloid science due to their small size, tub ular shape, struct ure, surface coating (Figure 1 7 ). Among these challenges, shape effect was reported to play an significant role in the DLVO framework. [ 2 ] Figur e 1 7 CNTs challenges to traditional DLVO theory In DLVO theory (modeling), one of the primary assumptions is that particles are spherical. The assumption is reasonable for ideal latex par ticles and some ideal colloidal contaminants. However, CNTs come in tube shapes, thereby, complicating traditional DL VO theory. Both van der Waals and electrostatic double layer forces are affected by the change s in shape. Several researchers have investigated these

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31 changes. [ 124 126 ] And these results imply that shape can theoretically control interactions between particles and different interface s. Several techniques have been developed to calculate the interaction force/energy between curved surfaces/bodies, including the Derjaguin Approximation (DA) and surface element integration (SEI). [ 124 ] The DA method estimates the interaction energy between two finite size bodies by relating it to that between two infinite parallel flat plates. [ 125 ] It can only be applied to surfaces that are separated by a small distance and to circumstances when the interaction rang e is substantially smaller than the radii of curvature of the surfaces. For very small non spherical particles, such as SWNTs, the DA method may lead to inaccuracies in calculating their interaction with planar surfaces. [ 127 ] The SEI technique takes into account curvature effects over the whole object, by integrating the interaction energy between a surfac e element of the object and the plane surface using the exact surface geometry of the object. It can precisely determine the interaction forces between a planar surface and a curved body with any defined shape, including CNTs. [ 117 ] Specific Knowledge Gap 2 and Research Scope From the evidence presented above, it can be conclude d that the traditional DLVO theory often failed to provide an accurate estimation of the interaction forces between CNTs and planar surfaces. Furthermore, the interaction of CNTs and planar surfaces is orientation dependent, which gives rise to a torque orienting the CNTs in an energetically favorable configuration to approach/depart the planar surfaces. Such a dynamic behavior cannot be explai ned merely on the basis of spherically symmetric inter action potentials of the traditional DLVO theory. A theory/model that can accurately

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32 describe the interaction between a CNT and a planar surface therefore is in critical need. The scope of this research is shown in the Figure 1 8. Figure 1 8 R esearch scope of interactions between CNTs and planar surface Research Objectives 2 The overarching objective of this work was to develop analytical formula s that can precisely describe the orientation dependent interaction energy/forces between a CNT and an isotropic planar surface It was hypothesized that the interaction of CNTs with planar surfaces is mainly controlled by the van der Waals and electrical double layer (EDL) forces, which are the same as the cl assic DLVO forces. S pecific objectives are as follows: Objective 1: develop an analytical formula of the orientation dependent interaction energy between a pristine SWNT and an isotropic planar surface. Objective 2 : develop an analytical formula of the orientation dependent interaction energy between a surface charged SWNT and an isotropic charged planar surface. Objective 3: evaluate DLVO f or ces and t orques of SWNTs with planar s urface s Organization of the Dissertation This Ph.D. dissertation has s ix chapters, including the present introductory chapter (Chapter 1). In C hapter 2, laboratory experiments were conducted to measure the single stem contact efficiency ( 0 ) of colloid capture by a cylindrical collector in

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33 laminar overland flow A dimensionless equation of 0 as a function of collector Reynolds number ( Re c ) and Peclet number ( N Pe ) was developed and matched t he experimental data very well In C hapter 3 the single stem attachment efficiency ( ) of colloid capture by a simulated plant stem (i.e. cylindrical collector) in laminar overland flow was measured directly in laboratory flow chamber experiments. A new dimensionless equation was proposed that predicts the of colloid capture by a cylindrical collector in laminar overland flow with reasonable accuracy. In addition, the equation was also effective in predicting the attachment efficiency of colloid deposition in porous media. In C hapter 4, i n order to upscale single stem efficiency theory to rea l dense vegetation, a new dimensionless number was developed to account for the effect of plant stem surface properties on the colloid deposition onto the plant stem in overland flow. An e xtended model including steric repulsion effect was developed that fit the experimental data with acceptable accuracy. This extended single stem efficiency theory can be used to help construct and refine mathematical models of colloid transport and filtration in laminar overland flow on vegetated surfaces. In C hapter 5, t he surface element integration (SEI) technique was coupled with the DLVO theory to determine the orientation dependent interaction energy between a single walled carbon nanotube (SWNT) and an infinite isotropic planar surface. For the first time, an analyt ical formula was developed to accurately describe the interaction between CNTs and planar surfaces with arbitrary rotation angles, which can be used to predict CNTs deposition on plant stem surface. Chapter 6 summarizes the results of all the previous chap ters and makes recommendations on future work. References are included at the end of this document.

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34 CHAPTER 2 EXPERIMENTAL ANALYSIS OF COLLOID CAPTURE BY A CYLINDR ICAL COLLECTOR IN LAMINAR OVERLAND FLOW 1 Figure 2 1. Graphical content of chapter 2 Introduct ory Remarks Transport of colloidal particles in water flow is an important contamination process that can deteriorate both surface and groundwater quality. Suspended colloids are capable of carrying a variety of contaminants and enhance their mob ility in aquatic systems [ 4 ] In addition, move ment of colloidal particles in soils may also affect their primary productivity, nutrient cycling, and species composition [ 128 ] A substantial research effort has been made to understand colloid and colloidal facilitated transport in porous media including the soil vadose zone and groundwater. Bench scale packed co lumn experiments have been conducted to examine the transport behaviors of different types of colloids including viruses, bac teria, clay 1 Reprinted with permission from Wu, L., B. Gao, and R. Munoz Carpena (2011), Experimental Analysis of Colloid Capture by a Cylindrical Collector in Laminar Overland Flow, Environmental Science & Technology 45 (18), 7777 7784. doi: 10.1021/es201578n

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35 particles, and synthetic microspheres and engineered nanoparticles [ 35 38 ] Relationships between system physicochemical p roperties (e.g., flow velocity, solution chemistry, and surface properties) and colloid mobility in porous media were evaluated [ 39 42 ] In addition the influences of biological factors (e.g., cell size and shape, cell motility, and micro molecular length and composition) on bio colloid fate and transport in porous media have been assessed [ 43 44 ] Finding from these investigations have enhanced current ability to predict and monitor the fate and transport of colloidal partic les in subsurface flow. Considerably less attention has been dedicated to the fate and transport of colloidal particles in surface flow par ticularly with respect to colloid transport through vegetation in overland flow [ 48 50 ] Several studies have shown that vegetation structures (submerged or emergent) can remove suspended particles including colloidal particles from surface flow [ 50 52 69 ] Leonard et al [ 53 ] observed that the capture of suspended particles on the stems and leaves of Juncus roemerianus marsh contributed up to 10% of the total sediment deposition to a tidal marsh. Similarly, Pluntke and Kozerski [ 58 ] suggested that sedimentation onto plant structures should be considered when quantifying particle retention in submerged macrophyte stands. Particle retention in a sea grass meadow ( Posidonia oceanica ) was found to be up to 15 times greater than the equivalent non vegetated bed [ 68 ] In a wetland field site in the Florida Everglades Huang et al. [ 69 ] found that aquatic vegetation could also remove colloidal particles from surface flow. These evidences strongly suggest that filtration by plant structures, such as stems, has a significant effect on the fate and transport of colloidal

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36 particles in surface flow. Unfortunately, current understanding of the capture of colloidal particles by pl ant structures in surface water is still very limited. In laminar overland flow, the depth of water is usually below the top of the sheath s of grassy vegetation and thus plan stem s may dominant flow and transport processes [ 93 94 ] Under many circumstances, plant stems can be m odeled as rigid, cylindrical collectors for colloid deposition [ 95 ] Therefore, establishing a single coll ector efficiency theory of colloids captured by a cylinder will advance the understanding the fate and transport of colloids in surface flow. The single collector concept has not only been widely used in colloid filtration in porous media [ 129 130 ] but also been successfully applied to sediment removal by aquatic plants [ 131 ] However, only few studies ha ve directly measured particle capture by a single collector, particularly with respect to a spherical o r cylindrical collector [ 131 132 ] Their measurements validated the mathematical models of single collector contact efficiency of colloids and suspended sediments Neverthel ess, it is unclear whether existing models can be used to de scribe the c apture of colloids by a cylindrical collector in overland flow In this study, laboratory experiments were conducted to measure the single collector contact efficiency of colloid capture by a cylindrical collector in laminar overland flow. A glass cylinder installed in a small size flow chamber was used as the collector. Silicone grease was appl ied to the collector surface to facilitate colloid deposition. Florescent microsphere suspension was used in the experiment as colloid flux. The amount of colloids deposition onto the cylinder surface was measured to determine the single collector contact efficiency under various experimental conditions. O ur objectives are to: 1) determine how perturbations in flow velocity, colloid size, and

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37 collector size affect the single collector efficiency of colloid capture by a c ylindrical collector in laminar overland flow 2) test whether existing single collector contact efficiency models can be used to predict colloid capture by a cylinder in laminar overland flow, and 3) develop a dimensionless equation to describe the single collector efficiency of colloid transport through emergent vegetation in laminar overla nd flow. Theory The contact efficiency of a single collector ( 0 ) is a ratio of the rate at which colloids strike the collector divided by the rate at which colloids flow toward the collector [ 129 ] The magnitude of 0 is assumed to be controlled by three transport mechanisms: interception, sedimentation, and diffusion. Interception takes place when a suspended colloid moving along flow streamlines come into contact with the collector by virtue of its size. Sedimentation occurs when a suspended colloid has a density greater than the fluid density, and the particle can then collide with a collector. Diffusion reflects the Brownian motion of the suspended colloid in fluid that leads to diffusive transport of the particle to the collector surface. The intercept ion and sedimentation processes contribute significantly to the single collector contact efficiency for colloids with diameters greater than 1 m ; while the diffusion mechanism becomes significant when colloids are smaller than 1 m [ 129 130 133 ] Because the sedimentation process is gravity driven, it a ffects the single collector contact efficiency only when the collectors are assembled along the gravity line. In case of laminar overland flow on a flat surface or a moderate slope, we can assume the contribution from the sedimentation processes to 0 is t rivial. If only interception and diffusion transport mechanisms are considered, the single collector contact efficiency of colloids in laminar overland flow captured by a cylinder can be expressed as:

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38 (2 1 ) where I and D are the contributions from interception and diffusion, respectively. Usually the contact efficiency of each mechanism is first determined separately and then the overall single collector contact efficiency can be obtained by summing the individual contri butions [ 129 130 133 ] Several models have been d eveloped to calculate the single collector contact efficiency of colloids, but almost all of them are for spherical collectors. For example, the Yao [ 129 ] RT [ 130 ] and TE [ 133 ] models have been widely used to determine the single collector contact efficiency of colloid filtration in porous media. It is unclear whether these models can be applied to describe the single collector contact efficiency of colloids to a cylinder collec tor. Recently, Palmer et al. [ 131 ] established a theory to calculate the single collector contact efficiency of suspended sediments for cylindrical collectors in aquatic systems. Based on their approach, each component of the single collector contact efficiency of the cylindrical collector can then be determined analytically as functions of the particle/collector size ratio ( R = d p / d c where d p is particle diameter and d c is cylinder diameter in this case), and the collector Reynolds number, R ec .= ud c where u is flo w velocity, and is the kinematic viscosity [ 131 ] For instance, under creeping flow conditions (i.e. Rec <1), particle contact efficiency due to direct interception ( I ) to a cylinder can be expressed as [ 134 ] : ( 2 2)

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39 The contact efficiency due to colloid diffusion ( D ) for creeping flow can be written as [ 135 ] : ( 2 3 ) where D is the particle diffusion coefficient, which can be obtained from the Einstein's diffusion equation [ 136 ] Equ ations 2 2 and 2 3 are based on the aerosol filtration theory of mass transfer to a cylinder, which could be very similar to transport and deposition of colloids in overland flow through emergent dense vegetation. Therefore, we conducted a range of experim ents to test whether the single collector contact efficiency equations can be used to describe the capture of colloids by the cylindrical collector in laminar overland flow. Materials and M ethods C olloids and Collectors Fluorescent, carboxylated, polystyre ne latex microspheres (Magsphere, Inc) of four different sizes (0.1, 1.05, 2.0 and 10.5 ) were used in the experiment as model colloids. The density of the colloids, as reported by the manufacture, was 1.05 g/cm 3 Colloid suspensions for testing were made by diluting the stock solution (1.05g/mL, corresponding to 1.010 15 8.610 11 1.210 11 and 8.610 8 no./mL for 0.1, 1.05, 2.0, and 10.5 colloids) to the target concentrations ( 10 mg/L, corresponding to 1.010 10 8.610 6 1.210 6 and 8.610 3 no./m L for 0.1, 1.05, 2.0, and 10.5 colloids ) with deionized (DI) water. The pH of the colloid suspensions were around 5.3 Two glass cylinders of diameters 1.0 and 2.0 cm were used in the experiment as the collectors to simulate plant stems. Clear s ilicone grease (Baysilone, GE Bayer) was applied to the collector to mark off a 1 cm test section at the bottom end (Figure 2 2 ) to

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40 collector area and make the attachment efficiency ( ) equal to one [ 131 ] T he final grease thickness ( 0.5 mm) was regarded not thick enough to significant change the diameter of the cylinder. This process was repeatable such that the grease layer thickness was constant every test. Experimental Apparatus The experimental appara tus used in this study was similar to that of Palmer et al. [ 131 ] but at a much smaller scale (Figure 2 2 ). The main component was an open flow channel flow chamber made of Plexiglas of 20 cm long, 10 cm wide, and 10 cm high. A recirculating peristaltic pump (Mas terflex L/S, Cole Parmer) was used to provide the desired system flow velocities. An aluminum screen (holes diameter 3.0 mm, 55% open area) was installed in the flow chamber to straighten the flow. A flat velocity profile could be obtained near the center part of the flow channel as long as low velocities (< 0. 3 cm/s) were used. Therefore, the longitudinal location 10 cm downstream of the inlet was chosen for the cylinder test position. The water depth in the flow chamber was controlled to be slightly above 1cm to ensure that the collector area was under water surface. Experimental Methods Before each test, the flow channel, the pipe and collector were cleaned thoroughly with DI water. The colloid suspension was then poured into the flume, and then stirred until the colloids spread out over the whole channel. A peristaltic pump was then used to circulate the flow in the chamber system for about 2 minutes. After the flow system properties (i.e., flow rate, water table, colloid distribution) stabilized, the cy linder collector was then carefully positioned into the chamber for durations ranging from 5 to

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41 120 minutes (i.e., 5, 10, 30, 60, 120 minutes). Nine different flow velocities (0.02 0.2 cm/s), two collector sizes (1 and 2 cm), and four colloid sizes (0.1 1 0.5 m) were tested in the experiment. At the end of each run, the flow was stopped and the collector was gently pulled out to measure the amount of colloids attached. Pre experimental tests showed that the colloids attached to the silicon grease on the co llector surface can be fully recovered with a 4% surfactant (sodium dodecylbenzene sulfonate, 10 mL) solution. A fluorescent spectrophotometer (PerkinElmer LS 45) was used to determine the amount of colloids recovered. Each experiment was repeated at least three times. The colloid capture rate ( r c ) by the collector was determined by measuring the increases in number of colloids on the collector over different time intervals. (2 4) where dN the number of colloids increased on the collector surface over a time interval ( dt ). Thus, the single collector contact efficiency ( 0 ) of colloids captured by a cylindrical collector in laminar flow can be written as: ( 2 5) where N 0 is the number of colloids in the suspension, u is the flow approach velocity, d c is the diameter of collector, and l c is the height of coated area of collector. Results and Discussion Effects of Flow Velocity and Colloid and Collector Sizes For all the experimental conditions tested, the number of colloids increased on the collector surface ( dN ) and the experimental time intervals ( dt ) showed good linear relationships with almost all R 2 larger than 0.9 except one (Table 1 1). Therefore, the

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42 slopes of the linear regre ssions were used as the colloid capture rates ( r c ) to determine the experimental single collector contact efficiencies ( 0 ). Standard errors were computed for three replicate trials to estimate the uncertainties. These results demonstrate the dependence of 0 on flow velocity ( u ), colloid particle diameter ( d p ), and collector diameter ( d c ). Increases in u reduced 0 when d p and d c were 1.05 m and 2 cm, respectively (Figure 2 3 ). For example, 0 decreased by 2 orders of magnitude when the u increased from 0.002 to 0.2 cm/s, indicating a negative correlation between the single collector contact efficiency and the colloid approach velocity. This trend is consistent with the findings of previous studies on colloid transport in porous media (spherical collector s). A number of experimental and modeling studies have demonstrated that the filtration/removal rate of colloids through a porous medium filter decreases when the flow rate increases [ 137 139 ] Compere et al. [ 139 ] observed that deposition rate of clay colloids decreases with flow velocit y whereas the collector efficiency increases by a factor of 5.1 as flow velocity decreases by a factor of 0.11. Similarly, Camesano and Logan [ 138 ] observed that, for passive colloids, the fractional r etention would increase by more than 800% as the pore velocity was decreased from 120 to 0.56 m/day. For suspended sediments captured by a cylinder collector, however, Palmer et al [ 131 ] observed an opposite result showing the increases of contact efficiency with hi gher flow rates. This divergence might be attributed to the high settling/sedimentation rate of suspended sediments in the overland flow, which is much higher than that of colloids (negligible in this study). In the study of Palmer et al [ 131 ] increasing in flow r ate could

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43 offset the sedimentation processes and increase the capture rate of sediments by the cylinder collector, and thus increase the single collector contact efficiency. For collectors of different sizes (i.e., d c = 1 or 2 cm), the single collector efficiency ( 0 ) varied with colloid diameters ( d p ), suggesting that a minimum value of 0 might exist at a critical colloid size (Figure 2 4 ). This is consistent with the classic single collector efficiency theory of col loid filtration in porous media [ 129 ] For colloid transport in porous media under unfavorable conditions, Elimelech [ 140 ] found that particles with diameter of 1.15 m had a lower collector efficiency than particles with diameters of 0.08, 0 .17, or 2.52 m. In a test of colloids with a wide range of particle sizes, Zhuang et al. [ 141 ] found that depend ence of colloid retention on particle size was nonlinear and there existed a fraction of colloids with greater mobility (i.e., minimum value of 0 ) than other fractions. On the other hand, however, several studies have also observed the independence of col lector efficiency on particle size [ 115 142 ] Further investigations are still needed to quantify the relationship between colloid size and collector efficiency. Comparisons of the 0 values between the two collectors of different sizes also revealed that, for the same u and d p the smaller collector (i.e., 1 cm) had higher values of 0 than the larger collector (i.e., 2 cm). Similar relationship between 0 and d c was observed for the removal of suspended sediments by cylinder collectors in overland flow [ 131 ] Comparison of Experimental Data and Theoretical Predictions The experimental data discussed above were compared with their corresponding values of 0 based on the theoretical predications of suspended particles captured by a cylinder collector in creeping flow (i.e., equations 2 1 2 3). Under shallow overland flow conditions, average overland flow velocity is often used to determine the single collector

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44 contact efficiency at a cross section of the cylinder (two dimensional) [ 131 ] As a result, mathematical formualtions of the single collector contact efficiency on colloid transport in porous media, such as the Yao model [ 129 ] RT model [ 130 ] and TE model [ 133 ] can be also used to calculate the theoretical predications of colloids captured by a cylinder collector. For all conditions tested, the experimental single collector contact efficiencies were larger than the corresponding theoretical values of equations 2 1 2 3 (Figure 2 5 a). This discrepancy is probably due to the relatively high collector Reynolds numbers in the experiments. Although the experimental flow velocities was controlled to be low, the Rec was between 0.4 and 40 (Laminar flow), still larger than the limitation o f the equations 2 1 2 3 (Rec<1, creeping flow). Under creeping flow conditions, it is reasonable to assume that only interception and diffusion processes are the main contributors to the single collector contact efficiency. When the R ec is higher, however, other processes, such as mechanical dispersion, could also alter the contact of colloids to the collector. A number of studies of the transport of colloids and other suspended particles in aquatic systems have emphasized the importance of longitudinal and vertical dispersion on their removal by vegetation [ 69 143 144 ] Similarly, the sphere collector models also underestimated the experimental 0 for all the experimental conditions (Figure 2 5 b d). The failures of the theoretical predictions suggested that none of the existing equations/models of single collator efficiency could be applied directly to determine the filtration rate of colloidal particles by dense, non submerged vegetation in lam inar overland flow.

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45 A Regression Equation Our experimental data indicated that the actual single collector contact efficiency should be a function of the flow velocity, collector size, and colloid size. Therefore, a dimensionless equation in the form of 0 a ( R ec ) b ( N Pe ) c can be formulated, where N pe is the Peclet number ( N pe = ud c / D ). Based on the experimental data obtained ( R ec =0.42 42 and N Pe =4.510 5 9.710 7 ), the best fit ( R 2 > 0.98, Figure 2 6 ) dimensionless equation can be written as: 0 = 0.0044 R ec 0.94 N Pe 0.03 ( 2 6) Although this dimensionless equation can be applied to a wide range of P eclet numbers (two orders), it is only valid for predicting the single collector contact efficiency of colloids approaching cylindrical collectors (plan stems) under laminar flow conditions. It was impossible to further validate the dimensionless equation for field conditions because only few/no studies have been conducted to measure the removal of colloidal particles by plant stems in laminar overland flow. In a recent study, Huang et al. [ 69 ] measured the filtration of 1 m latex microspheres within emergent vegetation at a wetland field site located in the Florida Everglades. They found that plant stems could effectively remove the colloids (microspheres) from flow and the average single collector removal efficiency was 0.0 02. Unfortunately, the Rec value of Huang et al. [ 69 ] was above the limitation of the dimensionless equation, as a result, the dimensionless equation could not be applied to their experiment. Additional investigations, particularly experimental studies, are in critical need to measure the r emoval of colloids by plant stems in laminar overland flow.

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46 Environmental Implication Laboratory experiments were conducted to measure the single collector contact efficiency of colloids by cylindrical collectors in a flow chamber under laminar flow condit ions. Our results indicated that 0 decreased with flow velocity ( u ) and collector diameter ( d c ), and a minimum value of 0 might exist at a critical colloid size ( d p ). We also found that existing single collector contact efficiency models underestimated the 0 of colloid capture by the cylinders in laminar overland flow. A new dimensionless equation was thus developed to determine 0 as a function of Reynolds and Pecl et numbers ( R ec and N Pe ) that matched the experimental data very well. Although additional investigations of its generality are still needed, this dimensionless equation can be used to determine the colloid filtration/deposition rate in dense, non submerge d vegetation (e.g., vegetative filter strips) in laminar overland flow, and to enhance current capacity to predict the fate and transport of colloidal contaminants in surface runoff. For colloid removal by emergent vegetation in laminar overland flow, th e single collector removal efficiency ( ) is often lower than the single collector contact efficiency ( 0 ) because the contacts between colloids and plant stems may not guarantee 100% removal. Therefore, single collector removal efficiency is often express ed as a product of an empirical attachment (collision) efficiency ( ) and the single collector contact efficiency [ 129 ] : = 0 ( 2 7) The attachment efficiency is defined as the fraction of contacts between colloids and the collector that result in attachment, which reflects the chemistry of the system [ 129 ] Several theoretically formulations have been established to calculate based on

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47 the Derjaguin Landau Verwey Overbeek (DLVO) interaction energy profiles between colloids and the collector upon close separation [ 145 147 ] For colloids in overland flow, convection dispersion equations coupled with deposition kinetics are commonly used to predict their fate and transport in dense vegetation [ 148 ] The kinetic deposition rate is often represented by the particle deposition rate coefficient, k d [ 129 133 ] For a vegetation system of a spacing density ( f ), which is defined as the ratio of the empty area among the plant stems divided by the total vegetated area, the relationship between k d and can be written as: ( 2 8 ) where the ratio of the approach velocity to the spacing density ( u / f ) is the interstitial fluid velocity commonly used in modeling colloid transport in filter media.

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48 Table 2 1 Summary of experimental conditions and results Test No. Particle diameter d p ( m) Flow velocity u (cm/s) Collector diameter d p (cm) Increased colloids no. as a function of time intervals (r c = dN/dt ) R 2 Single collector contact 0 ) Mean (%) I.1 1.05 0.002 2 13310821 0.991 6.4E 034.0E 04 I.2 1.05 0.004 2 13764969 0.994 3.3E 032.3E 04 I.3 1.05 0.008 2 14145899 0.990 1.7E 031.0E 04 I.4 1.05 0.01 2 14346600 0.986 1.4E 035.8E 05 I.5 1.05 0.02 2 14477629 0.991 6.9E 043.0E 05 I.6 1.05 0.04 2 14656629 0.989 3.5E 043.0E 05 I.7 1.05 0.08 2 14934629 0.981 1.8E 049.7E 06 I.8 1.05 0.10 2 14895617 0.992 1.4E 046.0E 06 I.9 1.05 0.20 2 15143566 0.983 7.3E 051.8E 06 II.1 0.1 0.02 2 169571145 0.942 8.2E 045.5E 05 II.2 0.1 0.02 1 14977876 0.954 1.5E 038.5E 05 II.3 1.05 0.02 1 12102481 0.981 1.1E 034.6E 05 II.4 2.0 0.02 2 154211194 0.985 7.4E 045.7E 05 II.5 2.0 0.02 1 12664718 0.978 1.2E 036.9E 05 II.6 10.5 0.02 2 23612589 0.886 1.1E 031.1E 05 II.7 10.5 0.02 1 22603402 0.972 2.1E 033.8E 05 *No. I.1 I.9 summarize the effect of flow velocity; No. II.1 II.7 summarize the effect of colloid size and collector size

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49 Figure 2 2 Schematic of experimental setup for measuring single collector contact efficiency

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50 0exp E Figure 2 3 Effect of flow velocity on single collector contact efficienc y

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51 0exp 0 experimental Figure 2 4 Effect of colloid and collector sizes on single collector contact efficienc y

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52 0exp 0 experimental; 0 equations1 3 0 equations 2 1 2 3; 0 Yao 0 Yao; 0 RT 0 RT; 0TE 0 TE Figure 2 5 Comparison of experimental data of single collector contact efficiency with predictions of (a) equations 1 3, (b) the YAO model, (c) the RT model, and (d) the TE model.

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53 0exp 0 0Eqn.6 0 Equation 6 Figure 2 6 Comparison of experimental data of single collector contact efficiency with predictions of the new dimensionless equation (equation 2 6).

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54 CHAPTER 3 SINGLE COLLECTOR ATT ACHMENT EFFICIENCY O F COLLOID CAPTURE BY A CYLINDRICAL COLLECTO R 1 Figure 3 1. Graphical content of chapter 3 Introduct ory Remarks Colloidal particles in surface runoff may have adverse effects on many environmental and biological processes, e.g., facilitating contaminant transport in flow, impairing water quality, disturbing life cycles of microorganisms, and altering wetland geomorp hology [ 39 42 149 150 ] It has been suggested that vegetation systems in surface flow can act as a filter or as storage to reduce contaminant loading into natural water bodies [ 72 151 152 ] Recent studies found both emergent and submergent plants were effective to remove colloids from wate r flow [ 153 154 ] However, only 1 Reprinted with permission from Wu, L. B. Gao, R. Munoz Carpena, and Y. A. Pachepsky (2012), Single collector attachment efficiency of colloid capture by a cylindrical collector in laminar overland flow, Environme ntal Science & Technology 46 (16), 8878 8886. doi: 10.1021/es301365f

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55 limited research has been conducted to explore the govern ing mechanisms of colloid filtration by plants in overland flow [ 155 ] Field deposition of colloids in overland flow through dense vegetation is comprised of subsurface soil infiltration and surface filtration by vegetation processes. The surface filtration of colloids by emergent plants can be considered to be governed by two sequential interactions between colloids and plants: physically controlled contact process and chemically controlled attachment process [ 129 145 146 156 ] The transport of colloidal particles from the bulk suspension to make a contact to a plant stem collector is mainly controlled by the interception and Brownian diffusion mechanisms [ 129 155 ] The single contact efficiency ( 0 ) theory of colloid filtration by vegetat ion has been established recently based on the two mechanisms [ 155 ] Under unfavorable chemical conditions (i.e., in the presence of repulsive electric double layer interactions), however, the contact with a collector surface per se does not ensure the capture of colloidal particles by the collector as a result of the repulsive interaction forces between the colloid and collector surfaces [ 129 145 ] Under most circumstances, the surface water environment is chemically unfavorable for the attachment of colloids on plant collectors because not only do most colloids and plants carry overall negative surface charges [ 157 158 ] but also the ionic strength of surface water is typically low ( 0.0001 0.01M) [ 149 159 ] To fully understand the fate and transport of co lloids in surface water, it is therefore critical to develop a theory to describe the attachment process of colloids on plant collectors. Based on the framework of the classic filtration theory (CFT) [ 129 ] the concept of single collector attachment effici ency ( ) was introduced together with the single

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56 contact efficiency ( 0 ) theory to predict the removal of colloids by vegetation in laminar overland flow under unfavorable conditions [ 155 ] : ( 3 1) id filtration by surface vegetation; however, a substantial research effort has been made to understand the attachment process of colloids in porous media under unfavorable conditions. According on chemical properties of the system, such as surface charge and solution chemistry, which exert significant effect on the interaction forces between the colloid and collector surfaces [ 129 145 156 ] Derjaguin Landau Verwey Overbeek (DLVO) theory is of ten used to predict the attachment efficiency of colloids in porous media; however, discrepancies were reported between the theoretical predictions and experimental observations [ 142 160 161 ] To acco unt for these discrepancies, various assumptions have been made to modify the DLVO [ 146 147 ] collector charge variabili ty [ 162 ] and surface charge heterogeneities [ 39 1 63 164 ] Researchers have also suggested that the discrepancies may also occur as a r esult of physical effects that are not included in filtration theory. For example, effect of especially when colloids are weakly deposited in the secondary minimum [ 165 166 ] In addition, other physical factors, such as surface roughness [ 41 167 ] pore geometry [ 168 169 ] and stagnation zones [ 170 171 ] wer colloids in porous media. Findings from investigations studying these effects could be

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57 very useful to the development of single collector attachment efficiency theory for plant filtration of colloids in overland flow. The overarching goal of this work was to establish the single collector attachment efficiency ( ) theory for colloid transport through dense emergent vegetation, such as vegetative filter strips, in laminar overland flow. Under such condition, the water de pth is usually below the top of the sheaths of vegetation and only plant stems are submerged to affect flow and transport processes [ 155 ] Laboratory experiments were conducted to measure the colloid attachment efficiency on a cylindrical collector in laminar overland flow under unfavorable chemical conditions. A glass cylinder was installed in a small size flow chamber to simulate plant stem collector under laminar flow conditions. Rigid cylinders, such as glass or plastic rods and nails have been demonstrated to be effective representations of plant stems in surface runoff under various flow conditions [ 131 155 172 ] Fluorescent microspheres were applied to the flow system as experimental colloids and the amount of colloids deposited onto the collector surface was measured to determine the attachment efficiency under various conditions. Specific objectives were as follows: (1) quantify the effect of ionic strength, colloid size, and flow velocity on the single collector attachment efficiency of colloid capture by the cylindrical collector in laminar overland flow, (2) determin e whether existing single collector attachment efficiency models of porous media can be applied directly to vegetation systems, (3) establish a theory to predict the deposition of colloidal particles on vegetation in laminar overland flow conditions. Theor y The single collector attachment efficiency ( ) is defined as the ratio of the rate at which particles successfully attach to the collector divided by the rate at which colloids

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58 strike the collector [ 129 ] Under unfavorable chemical conditions, the magnitude assumed to be mainly controlled by the interaction forces between the colloidal particle and the collector [ 129 ] The classic and extended DLVO theories, which describe the attractive (Van der Waals) and repulsive (electrostatic double layer) forces between colloid and collector surfaces, are often used to determine the inter surface interactions. colloid deposition in porous media under unfavorable conditions using either the interaction force boundary layer (IFBL) approximation or the Maxwell approach [ 142 145 146 ] Mod els based on the IFBL approximation or its extensions determine the of colloid deposition in porous media through analytically or numerically solving the convective diffusion equation with repulsive colloidal interactions. Previous studies, however, foun d the predictions from the IFBL models had rather poor agreement with measurements due to several factors [ 142 145 146 ] The most important one was that the IFBL models do not consider the contributions of the secondary minimum energy well, which may play an important role in colloid deposition under unfavorable chemical conditions [ 146 ] The Maxwell approach was then developed to determine colloid attachment efficiency through deposition in the secondary minimum [ 146 ] If both primary and secondary energy minimum depositions are considered, the single collector attachment efficiency ( ) of colloid deposition in porous media can then be expressed as [ 146 173 ] : ( 3 2)

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59 where pri and sec are the fractions of single collector attachment efficiency of colloid deposition in the primary energy minimum and secondary energy minimum, respectively. In the Maxwell model, the singl e collector attachment efficiency is considered to be only influenced by chemical factors, such as solution chemistry and surface properties [ 129 142 145 146 ] Predictions of the Maxwell model were more accurate for column experiments under unfavorable conditions than that of the IFBL models [ 146 173 ] Some deviations from experimental measurements, however, were also observed for the Maxwell model, particularly with respect to deposition of relatively large size particles in porous media [ 147 173 ] media may also be affected by hydrodynamic factors (e.g., flow velocity) [ 165 174 176 ] analysis [ 166 177 ] It is reported that the Maxwell model coupled with the hydrodynamic torques approach can provide an improve d prediction of of colloid deposition in porous media under unfavorable conditions [ 178 ] The attachment efficiency can be expressed as [ 178 ] : (3 3 ) where f pri and f sec are the fractions of single collector surface area over which the adhesive torques acting on the colloids retained in the primary and secondary minimum are greater than the fluid hydrodynamic drags, respectively. In addition to the theoretical approach, e mpirical expressions (e.g., Bai Tien model) have also been developed to predict the of colloid deposition in porous media in terms of dimensionless parameters [ 179 182 ] Detailed description of the theoretical

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60 APPENDIX Because of the differences in flow a nd transport processes between surface vegetation and subsurface porous media systems, however, it is unclear whether the processes of colloids on the plant collector under laminar flow conditions. Materials a nd M ethods Materials Fluorescent, carboxylated, polystyrene latex microspheres (Magsphere, Inc) of two sizes (0.1 and 1.05 m) were used as experimental colloids. As reported by the manufacturer, the density of the colloids is 1.05 g/cm 3 and the surface carboxyl group coverage is 8.2810 17 and 1.1910 18 /m 2 for 0.1 and 1.05 m colloids, respectively. Experimental solutions were made by diluting the stock colloid solution (1.05 g/mL, corresponding to 1.010 15 and 8.610 11 no./mL for 0.1 and 1.05 m colloids, respectively) to the target concentration (10.5 mg/L) with deionized (DI) water. A circular cylinder glass rod with diameter 0.5 cm was used as the collector to simulate plant stems. The glass rod was cleane d with acetone and then soaked in a 6 M HNO 3 solution for 5 h at 80 C to remove metal oxides and other impurities on its surface [ 141 142 166 ] For each measurement, a new glass rod without any coating was used to measure the under various experimental conditions. Analytical reagent grade KCl (Fisher Scientific) and DI water were u sed to prepare electrolyte solutions at desired ionic strengths. The pH for all the electrolyte solutions was adjusted to 7 with 1 mM KHCO 3 solution The experiments were conducted at five ionic strengths (0.001, 0.005, 0.01, 0.05, and 0.1 M) so that differ ent

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61 attachment efficiencies could be measured. While the first three were selected to represent the typically low ionic strengths of overland flow [ 149 159 ] the latter two (i.e., 0.05 and 0.1 M) were used to exam the effect of high ionic strength on attachment efficiency. Because carboxyl group is acidic and has a pKa value of 5 43 almost all (>99%) of the function groups on the colloids surface would deprotonate ( COO ) and because negative charged for all the tested experimental conditions. Thus, the (i.e., electrokinetic potential were 80.4, 70.3, 60.8, 48.1, 38.0 mV and 68.6, 63.9, 59.2, 41.2, 35.4mV, 57.8, 52.1, 50.5, 32.0, 18.8mV, which were determined with a ZetaPlus (Brookhaven of the glass rod were determined with colloidal glass suspensions (obtained from sonicating the glass rod) under various chemical conditions following the method developed by Johnson et al. [ 183 ] Experimental Methods The experimental apparatus and procedures used in this study were described in detail in our previous work [ 155 ] Briefly, the Plexiglass flow chamber was 20 cm long, 10 cm wide and 10 cm high. For each run, one clean glass cylinder was installed on the chamber bed as the single collector. A recirculating peristaltic pump (Masterflex L/S, Cole Parmer) was used to provide the desired system flow velocities. Once the colloid suspension was stabilized (i.e., flow rate, water table, colloid distribution), the collector was the n positioned into the chamber for durations of 5, 10, 30, 60, or 120 minutes to determine the colloid attachment rate on the collector surface over different time intervals [ 155 ] At the end of each run, the flow was stopped and the collector was pulled out to measure the amount of colloids attached. Pre experiments showed that

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62 colloids attached to the clean collector sur face under all experimental conditions could be fully recovered in DI water after 5 minutes of ultra sonication. A fluorescent spectrophotometer (PerkinElmer LS 45) was used to determine the colloid concentration. Each experiment was repeated at least thre e times. Two sets of experiments were conducted in the flow chamber system. The first set of experiments was designed to measure the under different ionic strength conditions (0.001, 0.005, 0.01, 0.05, and 0.1 M) at an approach flow velocity of 210 2 cm /s for both 0.1 and 1.05 m colloids. The second set of experiments was designed to measure the under different flow velocity conditions (210 4 210 3 210 2 210 1 and 1 cm/s) at 0.01 and 0.1M ionic strength with both 0.1 and 1.05 m colloids (Table 3 1). The colloid capture rate ( r c ) by the single collector was determined by measuring the increases in number of colloids on the collector over different time intervals. ( 3 4) where (no.) is the increment in number of colloids on the clean collector surface over a time interval (s).Thus, the single collector removal ef ficiency ( ) of colloids by a cylindrical collector in laminar flow can be written as follows: ( 3 5) where (no./m 3 ) is the number concentration of colloids in the suspension, (m/s) is the flow approach velocity, (m) is the diameter of collector, (m) is the height of test area of collector. T he values of the single collector contact efficiency ( 0 ) of the microspheres captured by the cylinder in the same flow chamber system had been

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63 determined previously [ 155 ] therefore, the single each experiment in this work was calculated using equation ( 3 1). Results and D iscussion Effect of Ionic Strength Measurements from the flow chamber experiments showed that the single collector attachment efficiencies ( ) varied by several orders of magnitude depending on experimental conditions (Table 1). When the flow ionic strength increased, increased for both 0.1 and 1.05 m colloids (Figure 1), which matched the trend from the DLVO calculations. The DLVO energy profiles between the colloids and the collector surface (Figure 3 3 ) confirmed the experimental conditions were unfavorable for attachment. When the ionic strength increased from 0.001 to 0.1M, the depth of the secondary minimum energy well ( sec ) increased from 0.03 to 0.41 kT and from 0.72 to 12.8 kT for 0.1 and 1.05 m colloids, respectively (Table 3 2). At the same time, however, the height of the energy barrier ( max ) decreased from 162.1 to 49.1 kT and from 1145.3 to 412.8 kT for particl e diameters 0.1 m and 1.05 m, respectively. Calculations from the Maxwell theory [ 146 173 ] showed tha t the pri was close to zero because the max was too high for all the tested experimental conditions, suggesting that deposition of colloids in the primary minimum energy well was insignificant. Instead, most the colloid attachment was in the secondary minimum and sec was notable when sec was larg er than 0.5 kT For any given ionic strength, the 1.05 m colloids had a larger than the 0.1 m ones, which is also consistent with both the DLVO and the Maxwell theory predictions that increase in colloid diameter would also increase the sec and thus e nhance the deposition in the secondary minimum.

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64 Although no previous investigations examined the effect of perturbations in solution chemistry or variations in particle size on the attachment processes of colloids on vegetation surfaces in overland flow, similar research has been well documented in the literatures of colloid deposition in porous media [ 17 141 142 184 187 ] A number of studies of colloid transport in porous media have demonstrated that a rise in ionic strength can increase the attachment of colloids on grain surfaces by reducing the thickness of the diffuse double layer betwee n colloid and collector surfaces and thus reducing the repulsive forces [ 17 188 ] which is consistent with the observations in this study. Several recent studies have also shown that colloid sizes can significantly affect colloid deposition in porous media [ 141 184 187 ] In a column transport study with different sizes latex microspheres, Hahn et al [ 161 ] observed that larger colloids had larger than the smaller ones and emphasized the dominance of secondary minimum deposition on colloid retention in porous media. In this work, observations from the ionic strength experiments indicated that the secondary minimum may play a dominant role in colloid deposition on the cylindrical collector under all the tested experimental conditions. Hence, the Ma xwell theory [ 146 173 ] could be used to determine the of colloid deposition on stem collectors if the attachment processes were only controlled by chemical factors. Effect of Flow Velocity Flow velocity also had a strong effect on the of the 0.1m and 1.05 m colloids in the flow chamber at ionic strength s of 0.01 or 0.1M (Figure 3 4 ), which suggested physical factors such as hydrodynamics also play an important role in controlling attachment efficiency of colloid deposition on cylindrical collectors in overland flow. Increases in flow velocity reduced th e attachment of colloids on the cylinder probably

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65 due to the hydrodynamic shear forces [ 166 ] Similar phenomena were observed in studies of colloid filtration and transport in emergent vegetation in wetland systems. In a field flume in the Everglades (FL), Harvey et al. [ 150 ] observed that particle attachment to vegetation stems decreased when flow velocity increased from 0.3 cm/s up to 6 cm/s. Similarly, Huan g et al. [ 154 ] found that high flow velocity in their field flume in the Everglades significantly reduced the removal of colloids by the emergent vegetation. These findings are consistent with current experimental observations; however, none of the previous studies has quantitatively examined the effect of hydrodynamic shear forces on the attachment efficiency of colloid capture by vegetation surfaces. Previous studies of colloid transport in porous media have emp hasized the hydrodynamic effect on colloid deposition and demonstrated that the decreased with increasing flow velocity under unfavorable chemical conditions [ 166 177 189 ] In a laboratory column experiment, Tong and Johnson [ 189 ] found a decrease of attachment efficiency when flow velocity increased from 2.310 5 to 9.210 5 m/s for colloids of 0.1 2. 0 m. They attributed the changes in the to the hydrodynamic drag force which could shear the colloids off the porous medium surfaces. It is suggested that hydrodynamic drag may indirectly prevent colloid deposition into the primary minimum to cross the energy barrier [ 190 ] Under certain conditions, the hydrodynamic drag may also reduce the retention of colloidal particles in the secondary minimum via following mechanisms: (1) reduce colloid retention capacity due to reduction of stagnant flow zone volumes [ 166 ] (2) enhance hydrodyn amic collisions between mobile and surface associated colloids [ 166 ]

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66 driven by increased colloid concentration gradients away from zones of accumulation (i.e., rear stagnation points) [ 146 ] Both current experimental results and findings f rom colloid transport studies in porous media suggest that the original assumption of chemical governing attachment processes in the CFT may need to be revisited. Physical factors such as flow velocity loid deposition in both vegetation and porous media systems. In the literature, a hydrodynamic torque approach was media [ 178 ] Surface vegetation, however, may have very different flow dynamics as compared with porous media. It is anticipated that the modified Maxwell theory may not be applicable to estimate the attachme nt efficiency of colloid capture by cylindrical collectors even under laminar flow conditions. Comparison of Experimental Data and Theoretical Predictions The Maxwell model [ 146 ] the modified Maxwell model (i.e., coupled by hydrodynamic torques) [ 178 ] and the empirical Bai Tien model [ 180 ] were used to estimate the of colloid capture by the cylindrical collecto r under all experimental conditions (Figure 3 5 ). Detailed information about the three models used in the study can be found in the APPENDIX Although it was reported that the Maxwell model works well for describing attachment processes of colloids (pa rticularly small size colloids) in porous media [ 146 191 ] it failed to match the experimental data obtained from the flow chamber (Figure 3 5 ). The model overestimated the of colloid deposition on the cylinder up to 1 2 orders of magnitude for almost all experimental measurements. This result further con firmed that physical factors, such as hydrodynamics, should be considered in the theory of

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67 colloid attachment efficiency. The modified Maxwell model, which considers the hydrodynamic factor, did show a better agreement with the experimental measurements th an the original Maxwell model (Figure 3 5 ). However, significant differences between model and experimental results were still observed, especially when flow velocity became high (e.g., 0.2 cm/s and 1cm/s), suggesting that, even after the consideration of hydrodynamic effect, the modified Maxwell model cannot be applied directly to describe the of colloid deposition on vegetation in laminar overland flow. Similarly, the Bai Tien model [ 180 ] overestimated up to 1 order of magnitude to the corresponding experimental results. In general, the experimental attachment efficiencies of colloid capture by the cylinder in laminar overland flow were much smaller than the predictions of the porous media models, which could be mainly attributed to the different flow dynamics in surface vegetation and porous media. Other possible causes of the discrepancy between the experimental data and theoretical predictions (i.e., Maxwell approaches) might include (1) the Maxwell models (original and modified) use the deepest point of the secondary minimum well to determine the but colloid may be initially attached in other positions with the secondary minimum; and (2) none DLVO forces, such as shor t range hydration and hydrophobic forces are not included in the Maxwell models [ 146 192 ] Although progresses have been made, current understanding of the governing factors of colloid attachment process in porous media is still quite limited. Because the surface property and geometry of the stem collector, as well as flow dynamics in vegetation systems could be much more complicated than that in porous media, it may not be feasible to establish a theoretical model to describe the of colloid

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68 capture by the cylinder under laminar flow conditions at this point. However, the direct experimental measurements obtained in this work provide an opportunity to develop an empirical model to predict the at tachment efficiency of colloid filtration by plants in overland flow. A D imensionless E quation Based on the Buckingham could be a function of 8 dimensionless parameters [ 179 180 ] (de fined in APPENDIX ). The experimental data was divided into a calibration/development subset with 58 data points randomly selected from the original 78 point dataset (Table 3 1) while ensuring a good distribution that covers the experimental conditions, and a verification subset consisting of the remaining 20 data points. The step wise least square method was used in this study to fit the development data subset. The coefficients of determination ( R 2 ) and Nash Sutc liffe efficiency ( E ) were used as quantitative descriptors of the predictive accuracy of the new dimensionless equation. The Nash Sutcliffe efficiency coefficient is defined as: ( 3 6) where m is the number of observations, exp pre and are the experimentally observed, model predicted, and mean single collector attachment efficiency, respectively. The R 2 value reflects strength of the linear relationship between observed and simulated data and the E value indicates how well observed and simulated data fit

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69 the 1:1 line. Predictions of the new dimensionless equati on are near perfect when both R 2 and E are close to one. The analysis of the development data subset showed that only three dimensionless parameters (defined in Table 3 3) were strongly correlated with the (p value smaller than 0.05). The results of step wise regression are pres ented in APPENDIX C 3. A dimensionless/regression eq uation ( R 2 =0.92 and E =0.85) can be written as follows: ( 3 7) ( N E1 ) than to other two dimensionless parameters, the N E1 term is a key component of the equation to reflect the effect of sur face potential ( APPENDIX ). To further validate the new dimensionless the rest of the 20 experimental data (Figure 5a, R 2 =0.89 and E =0.77), which indicate that the new dimensionless equation is effective in fitting t he experimental observations of colloid attachment efficiency on the cylinder with reasonable accuracy. Unfortunately, it was not possible to further validate the dimensionless equation for field conditions because only a few studies have been conducted to measure the removal of colloidal particles by plant stems in laminar overland flow. Hence, additional investigations, particularly experimental studies, are needed to measure the removal of colloids by plant stems in laminar overland flow. Although its ge nerality still needs to be tested, the dimensionless equation can be used to determine the attachment efficiency of colloid capture by a single vegetation stem in laminar overland flow with stem diameter based Reynolds number ( Re c =ud c /v )

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70 smaller than 50. If stems in a vegetation system have similar physicochemical properties, this equation, when coupled with spacing and geometry factors, can be applied to describe the attachment efficiency of colloid capture by stems in the vegetation syst em [ 129 155 ] Because most of the flow in porous media is creeping flow with Reynolds number smaller than 1 [ 17 ] the regression equation could also be applicable to describe the attachment processes of colloid deposition in porous media when the hydrodynamic effect is negligible (i.e., is dominated by chemical factors). To test this hypothesis, simulations of the new dimensionless equation were tested against measurements of obtained from well controlled column experiments of colloid transport in porous media under unfavorable conditions [ 156 160 178 189 193 ] (Figure 3 6 b). Results showed that the new dimensionless equation indeed can be used to predict the of colloid deposition in porous media with an R 2 value of 0.80 and E value of 0.69. Environmental Implications Only a very limited number of studies have been made to examine colloid transport in surface runoff, particularly with respect to develop theor ies or models to predict the filtration processes of colloid capture by vegetation collectors in overland flow. For the first time, laboratory experiments were conducted to directly measure attachment efficiency of colloid capture by a cylindrical collecto r in surface runoff under laminar flow conditions. Our results showed that both solution chemistry and hydrodynamic shear played important roles in colloid attachment processes. Because existing attachment efficiency models overestimated the of colloid a ttachment onto cylinders in laminar overland flow, a dimensionless equation was thus derived from experimental data and was found to predict the colloid attachment onto cylinders in

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71 surface water flow with reasonable accuracy. In addition, this new equatio n may also be used to predict colloid attachment efficiency in porous media if the attachment processes were mainly controlled by chemical factors. This attachment efficiency equation, when combined with the contact efficiency equation reported previously [ 133 155 194 ] can be used to determine kinetic deposition rate of colloidal particles in both vegetation and porous media systems. Based on this work, for the case of colloid filtration and transport in a vegetation system on a real field under laminar flow condition s, the kinetic deposition rate ( k d ) at the field scale can be written as: ( 3 8) where f is the ratio of the empty area among the plant s tems divided by the total vegetated area, d c is diameter of the vegetation stem, u is the approaching velocity, Re c and N pe are Reynolds number ( Re c .=ud c /v ) and Peclet number ( N pe =ud c /D ), respectively. Because equation ( 3 8) is completely untested, additional experiments would need to be designed to test the suitability of this equation to monitor and predict the fate and transport of colloids in vegetation in laminar surface flow, and perhaps inform the design of engineere d/natural surface runoff filtration systems, such as vegetative filter strips and constructed wetlands, to remove colloidal contaminants from surface runoff.

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72 Table 3 1 Summary of experimental conditions and results Test no. a ionic strength flow velocity particle diameter attachment efficiency IS (M) u (cm/s) d p (m) 0.001 0.02 1.05 8.010 3 6.210 4 0.005 0.02 1.05 9.110 3 8.010 4 0.010 0.02 1.05 1.910 2 3.610 3 0.050 0.02 1.05 8.310 2 1.110 2 0.100 0.02 1.05 1.210 1 1.510 2 0.001 0.02 0.1 2.610 3 3.510 4 0.005 0.02 0.1 4.410 3 6.310 4 0.010 0.02 0.1 1.510 2 2.110 3 0.050 0.02 0.1 5.010 2 4.310 3 0.100 0.02 0.1 1.010 1 9.010 3 0.01 0.0002 1.05 2.510 2 2.210 3 0.01 0.002 1.05 2.010 2 1.910 3 0.01 0.2 1.05 9.010 3 8.110 4 0.01 1 1.05 6.410 3 7.010 4 0.1 0.0002 1.05 2.210 1 2.310 2 0.1 0.002 1.05 1.510 1 2.110 2 0.1 0.2 1.05 5.010 2 6.310 3 0.1 1 1.05 3.410 2 4.110 3 0.01 0.0002 0.1 2.310 2 2.910 3 0.01 0.002 0.1 1.510 2 2.010 3 0.01 0.2 0.1 2.010 3 3.610 4 0.01 1 0.1 8.910 4 1.110 4 0.1 0.0002 0.1 1.310 1 2.110 2 0.1 0.002 0.1 8.910 2 8.010 3 0.1 0.2 0.1 8.110 3 1.110 3 0.1 1 0.1 6.010 3 7.310 4 a coupled effect of flow velocity and ionic strength.

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73 Table 3 2 Calculated Maximum energy barriers ( ) and primary ( ) and secondary minimum ( ) under different experimental conditions Ionic strength ( kT a ) Depth ( kT ) Distance (nm) Depth ( kT ) Distance (nm) I (M) 0.1m 1.05m 0.1m 1.05m 0.1m 1.05m 0.1m 1.05m 0.1m 1.05m 0.001 162.1 1145.3 NA b NA NA NA 0.03 0.72 103 93 0.005 148.9 1077.4 NA NA NA NA 0.07 1.86 43 36 0.01 143.4 1065.5 NA NA NA NA 0.11 2.83 30 25 0.05 85.8 708.8 NA NA NA NA 0.25 7.9 11 9 0.1 49.1 412.8 NA NA NA NA 0.41 12.8 7 6 a kT : k 23 C 2 J K 1 ), T is the absolute temperature, b NA: Not applicable, no primary minimum existing in the DLVO interaction energy profiles.

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74 Table 3 3 Summary of dimensionless parameters governing attachment efficiency Parameter Definition London number First electrokinetic parameter Double layer force parameter *A is the Hamaker constant, is the fluid viscosity, is the colloidal particle diameter, is the flow velocity, is the relative permittivity of the fluid (78.4 for water), is the permittivity in a vacuum (8.85410 12 C 2 N 1 m 2 ), and are the surface potential of the colloidal particles and collectors respectively, is the reciprocal of double layer thickness

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75 Figure 3 2 for colloid capture by the cylinder in the flow chamber

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76 Figure 3 3 DLVO interaction energy between the collector and the colloids: (a) 0.1 m c olloids and (b) 1.05 m colloids.

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77 Figure 3 4 u ) for colloids captured by the cylinder in the flow chamber

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78 Figure 3 5 predictions of the Maxwell, modified Maxwell, and Bai Tien models for colloids captured by the cylinder in the flow chamber at ionic strength of 0.01M.

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79 Figure 3 6 the new dimensionless equation

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80 CHAPTER 4 EXTENDED SINGLE STEM EFFICIENCY THEORY FO R COLLOID FILTRATION THROUGH SURFACE DENS E VEGETATION Figure 4 1. Graphical content of chapter 4 Introduct ory Remarks The presence of colloidal particles ( e.g., pathogen and engineered nanoparticles) in runoffs has raised increasing concerns regarding water quality and public health recently. [ 150 155 195 197 ] A number of previous inve stigations showed that both natural and engineered vegetation systems, such as vegetative filter strips and wetland, can act as a filter to remove colloidal contaminants in runoffs from agricultural and urban lands. [ 72 87 151 ] Lab and field scaled experiments and model simulations thus have been conducted to explore the movements of colloids with water flow in vegetation systems and advanced the understanding of theories and m echanisms that govern these processes. [ 78 79 198 200 ] Nevertheless, current ability to predict th e fate and

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81 transport of colloidal particles in surface runoff through emergent dense vegetation systems is still limited. To fill the knowledge gap, we recently developed a single stem efficiency theory to predict the filtration and transport of colloids in in surface runoff through emergent dense vegetation systems. [ 155 197 ] Laboratory flow chamber experiments demonstrated that the theory predicted both the contact efficiency ( 0 ) and the single stem attachment efficiency ( ) for colloid capture by a simulated plant stem in laminar overland flow very well. [ 155 197 ] Based on the single stem efficiency theory, a plant colloid filtration theory to quantify the kinetic deposition rate of colloid s on plan surfaces in vegetation systems under laminar flow conditions was developed. [ 197 ] It is anticipated that this plant colloid f iltration theory could be applied directly to determine the filtration rate of colloids by plant stems in vegetative filter strips or wetlands; however, none of the previous studies has experimentally tested the theory against data collected from real vege tation systems. One of the most important assumptions of the single stem efficiency theory was that the plant stems could be modeled as rigid rods with homogeneous surfaces. In reality, however, the surfaces of plant stems are commonly covered by non gl andular and glandular trichomes (hair or brush like protuberances) [ 201 ] which, if not considered, may introduce errors in the per ditions of the single stem efficiency theory. For example, previous studies showed that non glandular trichomes could reduce the attachment of particles, including bio colloids, on plant surfaces. [ 202 203 ] In addition, glandular trichomes on plant surfaces may secrete lipophilic substances to provide chemical or physicochemical protection against the adhesion of pathogens and other

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82 microorganisms. [ 201 204 ] More importantly, trichomes can also serve as a dense (biopolymer layer) around plant stem or other organs, [ 205 206 ] which may introduce steric repulsion forces to inhibit particle attachment. To better predict the filtrat ion and transport of colloids in real vegetation systems, it is thus important to consider the effect of steric repulsion on the single stem efficiency theory. In the literature of colloid transport in porous media, the importance of steric repulsion to c olloid attachment processes has been emphasized by many recent studies and several correlation equations have been successfully developed to include the steric repulsion effect in attachment efficiency of colloids on porous medium surfaces coated with poly mers. [ 207 212 ] The biopolymer brush layers on the surfaces of plant stems, however, are different from those studied in porous media which are adsorbed on medi um surfaces as clusters (Figure 4 2 A). [ 207 ] They are attached (grafted) by one end to the stem surfaces at relatively high coverage and stretch away along normal direction to avoid overlapping (Figure 4 2 B). These differences may cause the steric repulsion equations developed for porous media to be inappropriate in determinin g the attachment efficiency of colloids on plant stems. The steric repulsion between polymer brush layers and colloidal particles has been examined previously. [ 213 215 ] de Kerchove and Elimelech experimentally studied the effect of alginate conditioning film on the deposition kinetics of motile and non motile Pseudomonas aeruginosa and confirmed that steric interactions between the alginate brushes and the flagellated bacteria could hindered the deposition [ 213 ] Theoretical models have been also developed to predict the interaction of polymer brush layer with incoming particles. [ 212 216 217 ] A [ 216 ]

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83 an incoming particle may penetrate the brush and adsorb in the primary minimum at the substrate surface, or it may be trapped in the secondary minimum at the edge of the brush layer. Fi ndings from these investigations have advanced the knowledge of steric repulsion of polymer brushes and could potentially be applied to determine the interactions between colloidal particles and biopolymer brushes of plant stems. To our knowledge, however, none of the previous studies have attempted to us e the polymer brush theories to quantify the effect steric repulsion on attachment efficiency of colloids on plant stems. The overarching objective of this work was to modify the plant filtration theory so it can accurately predict colloid attachment rat e on plant stems for real vegetation systems. L aboratory flow chamber experiments and model simulation s were conducted to determine the deposition rate s of colloids on stems of real plants under various physicochemical conditions. The results were used to test and refine the single stem efficiency theory of colloid filtration in dense vegetation systems. Our specific objectives were as follows: (1) measure the filtration of colloids by dense vegetation in laminar flow over impermeable soil under different f low velocity, grass density, colloid size and ionic strength conditions, (2) determine the kinetic deposition rates of colloids on the plant stems in the vegetation systems, (3) test the current single stem efficiency theory against the obtained results, and (4) develop a n extended single stem efficiency theory with considerations of the effects of the steric repulsion of biopolymer brush es on natural plant stems. Theoretical Background and New Dimensionless Number (N STE ) Based on the colloid plant filtra tion theory, the deposition rate ( k d ) of colloids on plant stems in dense vegetation in laminar overland flow can be written as: [ 155 197 ]

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84 ( 4 1) where f is the ratio of the empty area among the plant stems divided by the total vegetated area, d c is diameter of the vegetation stem, u is the approaching flow velocity, and 0 and are the single stem contact efficiency and the single stem attachment efficiency, respectively. The current single stem efficiency theory assumes that the 0 is a function of the interception and diffusion pr ocesses (i.e. physical processes), while the is a function of the van der Waals attraction, electrostatic double layer repulsion, and hydrodynamic shear interactions (i.e. physical and chemical processes) and can be written as: [ 155 197 ] ( 4 2) ( 4 3) where Re c is the Reynolds number ( Re c = ud c / v ), N pe is the Peclet number ( N pe = ud c / D ), and N LO is the London number ( ), N E1 is the first electrokinetic parameter ( ), N DL is the double layer force parameter ), A is the Hamaker constant, is the fluid viscosity, is the colloidal particle diameter, is the relative permittivity of the fluid (78.4 for water), is the permittivity in a vacuum (8.85410 12 C 2 N 1 m 2 ), and are the surface potential of the colloidal particles and collectors respectively and is the reciprocal of double layer thickness. As discussed previously, the single stem efficiency theory (equations 4 2 4 3) was developed without considering the effects of biopolymer brushes (trichomes) on plant stem surfaces. Although the biop olymer brush layers might not affect the long

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85 distance transport/contact process much ( 0 ), they could significantly affect the short distance attachment process, particularly through introducing the steric repulsive forces to reduce the attachment efficie ncy ( ). [ 207 ] In addition, the presence of biopolymer brush could also reduce the friction forces and lead to a greater chance of colloid rolling and detachment by hydrodynamic shearing. [ 218 219 ] As a result, the single stem attachment efficiency mod el (i.e., equation 3), which only includes terms of the van der Waals attraction, electrostatic double layer repulsion, and hydrodynamic shear interactions, may overestimate the actual attachment rate of colloids on real plant stems with biopolymer layers. To improve the accuracy of the single stem efficiency theory, we propose to introduce a new term to single stem attachment efficiency model (i.e., equation 4 3) with a dimensionless number to reflect the biopolymer brush effects. Figure 1C shows a simple representation of a spherical particle with diameter of d p impinging upon a biopolymer brush, which has a uniform density of where n is the total number of polymer chains and A is the surface area of the substrate. Each polymer chain has a natur al e nd to end length of where a is the average segment length and N is the number of segments per chain, a dependent of the polymer molecular weight ( M W ) Based on the theory of grafted polymer brush, the equilibrium free energy ( ) of ea ch polymer chain can be written as: [ 220 221 ] ( 4 4) where for and for x >1. L 0 is the height (thickness) of the biopolymer brush and L is the separation distance (Figure 4 2 C). The

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86 repulsion energy exerted on a spherical particle by the polymer brush thus can be derived using equation (4 4 ) coupled with the Derjaguin Approximation: [ 217 ] ( 4 5) where for and for x >1. Although there is theoretical evidence that the Derjaguin Approximation may overestimate the steric repulsion energy, especially for small size particles, equations ( 4 5) still can provide a clear and fundamental estimation of steric inte raction. [ 217 ] As shown in equation ( 4 5), steric repulsion is a function of several parameters related to particle and biopolymer brush properties, including d p L 0 Mw In addition, the enhanced rolling and detachment caused by the biopolymer brush should a function of fluid viscosity and fluid velocity ( u ). [ 207 ] Based on the Buckingham methods, the new dimensionless number ( N STE ) that should be included as a term in the single collect attachment efficiency model to reflect the biopolymer brush effect can be expressed as: ( 4 6) where N STE can be found in the APPENDIX. Thus, in the extended single collector theory for real plants with biopolymer brushes, the single collector attachment ef ficiency equation can be written as: ( 4 7) where m and n are unknown constant that will be experimentally determined.

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87 Materials and Methods Materials Fluorescent, carboxylated, polystyrene latex microspheres (Magsphere, Inc) of three sizes (0.3, 1.05 and 2 m) were used as experimental colloids. As reported by the manufacturer, the density of the colloids is 1.05 g/cm 3 and the surface carboxyl grou p cov erage is 9 .2810 17 1.1910 18 and 1.68 10 18 /m 2 for 0.1, 1.05 and 2 m colloids, respectively. Experimental solutions were made by diluting the stock colloid solution (1.05 g/mL ) to the target concentration (10.5 mg/L corresponding to 3.7 10 10 8.610 8 and 1.1 10 8 no./mL for 0.1, 1.05 and 2 m colloids, respectively ) with deionized (DI) water. Analytical reagent grade KCl (Fisher Scientific) and DI water were used to prepare electrolyte solutions at desired ionic strengths. The pH for all the electrolyt e solutions was adjusted to 7 with 1 mM KHCO 3 solution The experiments were conducted at four ionic strengths ( DI water, 0.01, 0.1 and 0.2 M) so that different deposition kinetic rates could be tested T (i.e., electrokinetic potential ) of the 0.1, 1.05 and 2 m colloids were 80.4, 60.8, 38.0 29.4 mV, 68.6, 59.2, 35.4, 28.1mV and 56.7, 47.3, 32.1, 26.2 mV, respectively, grass stem were 57.8, 50.5, 18.8 and 16.9 mV, which were determined with a ZetaPlus (Brookhaven Instrument Co., grass stem were determined with colloidal grass suspensions (ob tained from sonicating the grass stem ) under various chemical conditions following the method used in the previous study. [ 197 ]

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88 Vegetation Chamber Experiments Small size vegetation chambers were used in the experiments (Figure 4 4 ). The chambers were made of Plexiglas of 20 cm long, 20 cm wide, and 10 cm high and equipped with a run off outlet Quartz sand ( Standard Sand & Silica Co. ) with a size range between 0.5 to 0.6 mm and a density of 1.54 g/cm 3 was added into the chambers (5 cm) as the growth soil. Brown Top Millet seedlings (1 2 weeks) were selected as experimental dense vegetation. The average diameter of fully grown stems ( d c ) was around 1.2 mm. High, medium and low stem densities with 2545, 4900 and 8184 stems/m 2 respectively, were used in the experiments. A peristaltic pump (Masterflex L/S, Cole Parmer) was used to apply the inflow, bromide and colloid solutions to the chambers at four different overland flow velocities (0.02 0.2 cm/s). Prior to the runo ff experiment, plaster (DAP Products. Inc) was used to seal the top sand surface to prevent infiltration and to eliminate the filtration of colloids by the soil (Figure 4 3A). Pre experiment with the flow chambers under the same treatments but without of d ense vegetation showed that more than 98% bromide and colloids were recovered from the system, indicating no deposition of colloids on the plaster layer Comparison of the breakthrough curves of colloid with and without plaster layer was shown in APPENDIX Colloid transport data were collected in duplicate from 17 vegetation surface flow chamber experiments with different combinations of four flow velocities, three plant stem densities, four ionic strengths, and three colloid sizes (Table 4 1). For each bre akthrough study, DI water was first applied to flush the vegetation system for about 60 min until the flow reach steady state. The breakthrough experiment was then initiated by switching from DI water to the colloid suspension for 30 min, and then the colu mn was flushed with DI water again for 90 min. Effluent samples were collected from the

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89 outlet with a fraction collector. B romide was also applied to the system as a conservative tracer. Colloid and b romide concentrations in the samples were determined wit h a fluorescent spectrophotometer (PerkinElmer LS 45) and an Ion Chromatograph (Dionex ICS 90 ), respectively. Characterize biopolymer brush layer (trichome) on vegetation stem The structure and composition of the trichome varies largely among plants, organs, and growth stages, but it is mainly composed by esterified fatty acids hydroxylated and epoxy hydroxylated with chain lengths mostly between 16 to 18 atoms of carbon. [ 222 ] The 9 or 10, 16 dihydroxyhexadecanoic acid, 16 hydroxyhexadecanoic acid, 18 hydroxy 9,10 epoxyoctadecaonoic acid and 9,10,18 trihydroxyoctadecan oic acid are the major components of C 16 and C 18 family. Their reported average of molecular weight (400 kg / mol) was thus used in this study. [ 223 224 ] Five plant stems were sampled from the vegetation chambers to determine the morphology, density, and length of the trichome on their surfaces. Images were obtained using self referencing system fitted with Navitar Precise Eye optics at 1.20x magnification. Average optical density of trichome was calculated using an image processing software (ImageJ 1.46, NIH). The results ( Table 4 1) were found in good agreement with reported data. [ 225 ] We assume that the biopolymer brush density is a constant for the vegetation stems used in the experiments. Previous studies reported that the height of the polymer brush layer extends with i ncreasing charge density or decreasing ionic strength ( I ) [ 226 227 ] Several models have been propo sed to relate the brush layer height to ionic strength and everyone has a general form of where m is a fractional exponent. [ 228 ] In this study, th e height of biopolymer brush layer was thus determined by assigning m as 2/3. 36 45

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90 Determine kinetic deposition rate ( k d ) Filtration and transpor t of colloids in dense vegetation system can be described by the advection d ispersion e quation coupled with a first order kinetic deposition. [ 197 ] The governing equations can be written as follows: ( 4 8) where C is the concentration of colloid suspension, z is the travel distance in the direction of flow, and D is the dispersion rate The eq uation w as run as an inverse problem to obtain the optimized parameters by fitting an analytical solution [ 229 ] to experimental breakthrough curves using a nonlinear least squares method. This inverse optimization method was first applied to bromide breakthrough data to estimate D and assumed that the dispersion coefficient ( D ) of colloid is the same as that of the bromide tracer in this study The best fit values of the kinetic deposition rate ( k d ) of colloids on stems of the vegetation systems were determined by fitting the colloid breakthrough curves. Predictions of the single s tem efficiency theory were tested against the best fit k d values. In addition, the best fit values were also used to refine the extended stem efficiency theory and to determine the m and n value of equation ( 4 7). Results and Discussion Effect of ionic str ength on colloid filtration in dense vegetation The effect of ionic strength on colloid deposition on plant stems is shown in breakthrough curves obtained from the vegetation chamber experiments (Figure. 4 5 A). The results showed that the removal of colloi ds in the dense vegetation system wa s a dependent of solution ionic strength. When the solution ionic strength increased from DI water to 0.2 M, the mass recovery of colloids from the vegetation system reduced from

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91 90.3% to 72.4%. This trend was also obser ved in many previous studies of colloid transport in porous media and was attributed to the fact that higher ionic strength could reduce the repulsive electric double layer forces to promote colloid deposition [ 188 197 ] Simulations of the advection dispersion model (equation 4 8) matched the colloid breakthrough curves very well for all the ionic strength conditions tested. The best fit k d values increased from 6.3 10 5 to 3.210 4 s 1 as the ionic strength increased, confirming the importance of ionic strength to colloid deposition on plant stems in the dense vegetation system. The classical DLVO theo ry, which includes the van der Waals and the elec trostatic double layer interactions, has been often used to describe the deposition of colloidal particles on surfaces under various conditions. [ 188 197 ] In this study, h owever, predict ions of the DLVO theor y were inconsistent with the experimental data, probably because this classical theory neglects the steric repulsion afforded by biopolymer brush layer. The DLVO energy profiles showed that deep secondary minimum energy wells (e.g., 1 2.5 and 58.7 kT ) exist under both medium and high ionic strength conditions (Figure. 4 5 C F, blue line in inset), corresponding to perfect attachment efficiency (i.e., =1) as predicted by the Maxwell theory T he observed removal of colloids by the dense vegetation under the two ionic strength conditions, however, was much smaller than the predictions by the theory. Extended DLVO theory that considers the steric repulsion (equation 4 5) thus was applied in this study to determine the effect of ionic stren gth on the interactions between colloids and plant stems. R ecent studies have shown that that ionic strength ( salt concentration) could also affect the steric repulsion forces by altering the

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92 dimensions of the biopolymer brush layer ( scaling relations, Fig ure 4 5 B). [ 226 ] To test this hypothesis, two scenarios of steric repulsion (with and without scaling rel ations) were evaluated. The results indicated that the steric repulsion without scaling relations (Figure. 4 5 C F, yellow dash line in inset) produced unrealistically high repulsive energy barriers and no secondary minimum existed under all tested conditio ns, indicating no deposition of colloids in the systems Removal of colloids by the dense vegetation, however, was observed for all the experimental conditions tested in this work. When the scaling relation was included, the extended DLVO e nergy profiles ( Figure. 4 4C F, red line in inset) were consistent with the experimental observations of colloid deposition on plant stem s (Figure. 4 5 A). Shallow secondary minimum wells were identified, indicating that colloid s may deposited on the edge of biopolymer bru sh layer in the se shallow secondary minimum wells. In addition, the high repulsive energy barriers at all ionic strength conditions indicated that removal of colloids by the plant stems through primary minimum deposition might not be feasible These resul ts showed that including steric repulsion with scaling relations into the total energy profile improved the accuracy of the predictions of the DLVO theory Additional investigations are still needed to refine the extended DLVO theory to accurately predict the interactions between colloids and vegetation surfaces. Coupled effect of flow velocity and stem density on colloid filtration in dense vegetation Both flow velocity and stem density affected the filtration and transport of colloids in the vegetation ch ambers ( Figure 4 6 ) For all the three stem densities, colloid removal decreased with increasing of flow velocity (Table 4 1), which is consistent with findings from our previo us flow chamber experiments with glass rods as simulated plant stems

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93 [ 155 197 ] For example, m as s balance calculations showed that at low density condition, the recovery of colloid was 76.7%, 80.8%, 86.5%, and 90.1% for flow velocity of 0.002, 0.01, 0.05 and 0.1 cm/s, respectively. Previou s studies have demonstrated that high flow velocity may introduce hydrodynamic drag force to reduce colloid deposition on surfaces [ 166 177 197 230 ] whi ch also applies in this work Similarly, the advection dispersion model (equation 4 8 ) described the colloid breakthrough curves very well for all the com binations of flow velocities and stem densities The best fit k d values are listed in Table 4 1. Although some theoretical studies [ 231 232 ] showed that weak she ar flow may have no effect on the polymer brush density profile and its height, others reported that hydrodynamic thickness (Figure 4 6 D) of surface attached polymer layers decreased with increasing of flow velocity. [ 233 ] This may reduce the net friction forces and thus reduce the deposition of colloids on the stem surfaces. It has also been reported that although low flow velocities may not affect the normal force between the polymer brush su rface and approa ching particles, a sharp o nset of additional repulsion force could appear when the velocity is above a certain threshold. [ 233 ] [ 234 ] [ 235 ] T he origin of the repulsion could be traced to the swelling of the polymer brushes with the increasing of flow velocity (Figure 4 6 E) These findings are also consi stent with the observed experimental data that colloid deposition decreased with increasing of flow velocity. P lant stem density also showed strong effect on the filtration and transport of colloid s in the vegetation chambers (Figure 4 6 and Table 4 1). F or a given flow velocity, m uch less colloids were recovered from the system wh en grass density increased Results of global sensitivity analysis ( APPENDIX) showed that vegetation

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94 density not only have a significant first order effect on the kinetic deposition rate of colloid in dense vegetation but also has a large effect by interactions with other parameters mainly through altering the flow field around the plant stem s to affect b oth the contact and attachment processes. Extended single stem efficiency theory The best fit k d data were divided into a calibration/development subset with 18 experimental data, which were randomly selected from the original 34 point dataset while ensur ing a good distribution that covers the experimental conditions, and a verification subset consisting of the remaining 16 experimental data. The least square method was used to fit the development data subset to determine the constant n in equation ( 4 7). A combination of graphical results, absolute value error statistics (e.g., root mean square error, RMSE ), and normalized goodness of fit statistics (e.g., Nash Sutcliffe efficiency, NSE ) was used as quantitative statistics of the predictive accuracy of t he new extended model. More details of the least square method and model goodness of fit assessment can be found in previous publications. [ 197 236 ] Based on this method, the best fit m and n was 1.74 and 0.45. Thus, the extended single stem model ( RMSE =0 and NSE =0.97, APPENDIX ) including effect of steric repulsion afforded by biopolymer brush layer on the plant stem surface can be written as follows: ( 4 9) The exponent coefficient for is negative ( 0.45), indicating that steric repulsion caused by biopolymer brush layer inhibit colloid deposition onto plant stem as expected.

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95 To further validate the extended single stem efficiency theory, equations ( 4 1), ( 4 2), and ( 4 9) were us ed to predict the kinetic deposition rate ( k d ) under different experimental conditions. The predicted k d was tested against the rest of the 16 experimental data (Figure 4 6A, RMSE =0 and NSE =0.88). The results indicated that the extended single stem effici ency theory is effective in filtration of colloids in real dense vegetation system with the fitted results span from acceptable to very good (Figure 4 7 A D). Furthermore, extended theory validity is significant since there is no probability of the fit bein g unsatisfactory ( NSE <0.65). The evolution of observed and predicted values (Figure 4 7 C) indicated that the predictions by extended theory are in good agreement with the experimental observations throughout the whole time series, although there is a slig ht discrepancy between the model predictions and experimental observations in the first period of the time series. Figure 4 6A also illustrated that the extended single stem efficiency theory significantly improves the predictions of over the origina l one (equations 4 1 4 3, RMSE =0 and NSE = 2.35) which overestimated the in most cases. In addition to neglecting of steric repulsion, the overpredictions of original equation may also be caused by actual velocities within the brush layer which follow the Darcy law and are larger than the approaching velocities. [ 233 ] Furthermore, it is worth noting that especially under either low flow velocity or high ionic strength conditions, the prediction of the original single stem efficiency theory deteriorated (Figure 4 7 A). This could be attributed to follows: (1) low flow velocity might make the effect of hydrodynamic shear force on the attachment process less important relative to interactions between colloid and stem surface especially in the presence of

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96 steric repulsion; (2) under high ionic strength conditions, compared to electrostatic double layer, steric repulsion is less sensitive to the Debye screening effect since the charge on the plant stem surface is governed by the Donnan potential inside the brush layer and not just the comp ression of EDL alone like bare surface. [ 207 ] And this may cause inaccuracy when previous equations were used to describe the ste ric repulsion dominated interaction. Therefore, the above results clearly illustrated the critical need to include this biopolymer brush effect into the previous equations since the extended theory indeed capture the fundamental mechanisms (including conta ct process and attachment process) which govern the colloidal particles deposition onto plant stem in overland flow, particularly in the presence of steric repulsions and decrease of the friction promoted by biopolymer brush layer. Deposition Mechanisms and Other Potential Effects Based on this new extended single stem efficiency theory and previous studies of adsorption of particles on grafted polymers, [ 237 ] we proposed the deposition of colloidal particles on plant stem in overland flow could through three ways (Figure 4 8 ): I II diameter of colloids is larger than the separation distance between the trichomes and III takes place when the diameter of colloids is larger than the separation distance between the trichomes and the particles are at tached on the tips of the trichome. Although these mechanisms are similar to those of colloid stabilized by grafted polymer, trichomes on plant surfaces in overland flow could behavior different from the grafted

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97 polymer, further theoretical and experimenta l studies are thus in critical need to better understand the mechanisms governing colloid deposition on stem surfaces. In this study, air bubbles were found attaching on the plant stem surfaces (Figure 4 3 B), which is consistent with recent findings that also that trichomes could keep a long lasting air (gas) film under water. [ 201 ] As a result, other well documented colloid deposition mechanisms for multiphase media, such as air water solid interface capture, [ 238 ] water film straining, [ 239 ] storage in immobile water zones, [ 240 ] and air water interface capture [ 241 ] could also potentially affect the filtration of colloids in dense vegetation systems. Additional investigations are still needed to incorporate these potential mechanisms to further refine the extended single stem efficiency theory to better predict the filtration and transport of colloids in dense vegetation in overland flow. Environmental Implications A n extended single stem efficiency theory of colloid deposition on p lant stem in overland flow was developed. The new theory represents an importan t step in advancing current understanding of fate and transport of colloidal particles in dense vegetation in overland flow. It also provides several insights into the fund amental mechanisms governing colloid filtration by plant stem s First, steric repuls ion afforded by the trichomes (brush li ke structure) on the plant surface plays an important role in affecting the interaction between colloid s and stem s Second, structure of the biopolymer brush layer on stem surface is depending on the solution chemistry and is one of the essential factors control ling colloid deposition process. Third, the extended single stem efficiency theory predicted the filtration of colloid s by real vegetation

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98 reasonable well and thus co uld be incorporate d in large scale models, such as VFSMOD [ 72 ] to predict the fate and transport of colloidal pa rticles in the field. Although the extended single stem efficiency theory is developed for colloid deposition on stems, its application is not just limited to vegetation systems. The theory should also be applicable to colloid deposition on various polym er brush surfaces in natural, engineered and biomedical systems.

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99 Table 4 1 Summary of experimental conditions, biopolymer brush properties and best fit value of parameters in transport model Test no a u (cm/s) f IS (M) d p (m) b ( no./cm 2 ) (Mean+SD) L 0 c (cm) (Mean+SD) D d (cm 2 /s) k d (S 1 ) (Mean) MR (%) (Mean) R 2 (Mean) 1 0.002 (Re c =0.2) 0.62 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 9.710 4 3.210 5 84.8 0.95 2 0.01 (Re c =1.0) 0.62 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 4.110 3 1.910 5 90.9 0.97 3 0.05 (Re c =5.0) 0.62 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 6.410 3 8.910 6 94.1 0.97 4 0.1 (Re c =10) 0.62 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 9.610 3 7.010 6 95.0 0.97 1 0.002 (Re c =0.2) 0.33 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 1.210 3 1.310 4 79.2 0.94 2 0.01 (Re c =1.0) 0.33 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 4.810 3 6.310 5 90.3 0.97 3 0.05 (Re c =1.0) 0.33 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 7.010 3 1.210 5 92.1 0.97 4 0.1 (Re c =10) 0.33 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 9.910 3 9.210 6 93.6 0.97 1 0.002 (Re c =0.2) 0.15 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 1.610 3 1.910 4 76.7 0.94 2 0.01 (Re c =1.0) 0.15 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 5.010 3 1.810 4 80.9 0.96 3 0.05 (Re c =5.0) 0.15 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 8.910 3 8.310 5 86.5 0.97 4 0.1 (Re c =10) 0.15 DI water 1.05 4.510 3 3.210 2 3.310 3 8.110 4 1.210 2 2.110 5 90.1 0.96 0.01 (Re c =1.0) 0.33 0.01 1.05 4.510 3 3.210 2 1.4 10 4 3.7 10 5 4.810 3 8.410 5 87.3 0.95 0.01 (Re c =1.0) 0.33 0.1 1.05 4.510 3 3.210 2 1.2 10 5 3.2 10 6 4.810 3 1.410 4 79.7 0.95

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100 Table 4 1. Continued Test no a u (cm/s) f IS (M) d p (m) b ( no./cm 2 ) (Mean+SD) L 0 c (cm) (Mean+SD) D d (cm 2 /s) k d (S 1 ) (Mean) MR (%) (Mean) R 2 (Mean) 0.01 (Re c =1.0) 0.33 0.2 1.05 4.510 3 3.210 2 8.7 10 6 2.4 10 6 4.810 3 3.210 4 72.4 0.94 0.01 (Re c =1.0) 0.33 0.05 2.0 4.510 3 3.210 2 4.8 10 5 1.3 10 5 4.810 3 7.410 5 93.2 0.97 0.01 (Re c =1.0) 0.33 0.05 0.3 4.510 3 3.210 2 4.8 10 5 1.3 10 5 4.810 3 2.010 5 90.7 0.96 a 1 [ 225 ] ; c for low IS condition data obtained from literature [ 225 ] for medium and high ionic IS cond itions data estimated from equation (7); d determined fro m the bromide breakthrough curv

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101 Figure 4 2 (A) Schematic of adsorbed polymer layer; (B) Schematic of grafted polymer brush layer; (C) Schematic of a model for a spherical colloid with diameter of dp impinging upon a biopolymer brush in a solution.

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102 Figure 4 3 (A) Morphology of trichomes on the plant stem and (B) air bubbles attached on the surface of trichomes under water.

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103 Figure 4 4 Schematic and photos of experimental set up for vegetation chamber experiment

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104 Figure 4 5 Effect of ionic strength on the colloid deposition onto the plant stem

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105 Figure 4 6 Effect of coupled flow velocity and grass density on the colloid deposition onto the plant stem

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106 Fig ure 4 7 Goodness of fit evaluation of the extended model

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107 Figure 4 8 Schematic illustration of three basic mechanisms of colloidal particles deposition on the plant stems.

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108 CHAPTER 5 5 DLVO INTERACTIONS OF CARBON NANOTUBES WITH ISOTROPIC PLANAR SURFACE 1 Figure 5 1. Graphical content of chapter 5 Introduct ory Remarks Carbon nanotubes (CNTs) are cylinder shaped nanoparticles with an extremely high length to diameter ratio [ 107 ] Single walled carbon nanotubes (SWNTs) possess the simplest geometry among the CNTs, and have diameters ranging from 0.4 to 3 nm. Multi walled carbon nanotubes (MWNTs) are composed of a concentric arrangement of many SWNTs, which can reach diameters up to 100 nm. Their novel properties, such as exceptional mechanical strength, and superior electrical and the rmal conductivity, prompt their applications in quantum wires [ 242 ] high resolution scanning probes [ 243 ] transistors [ 244 ] electron field emission sources [ 245 ] chemical and biological 1 Reprinted with permission from Wu, L. B. Gao, Y, Tian, R. Munoz Carpena, and Kirk J. Zigler (2013) DLVO i nteraction s between a carbon nanotube and an isotropic planar surface. Langmuir 2013, 29 (12):3976 3988. doi: 10.1021/la3048328

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109 sensors [ 246 247 ] reinforced composite materials [ 248 ] nanomedicine [ 249 ] and in many other areas. Some of these applications require assembling or depositing individual CNTs on surfaces of bulk materials with desired parameters, including location, orientation, geometry and density. [ 250 252 ] As a result, good understanding of the interaction forces between CNTs and the host surfaces is essential to creation and optimization of CNT based products. Furthermore, this knowledge may be also used to inform the development of effective strategies to reduc e the environmental impacts of the CNTs because surface interaction is one of most important factors governing the fate and transport of manufactured materials in soil and aquatic systems [ 117 119 123 ] A theory/model that can accurately describe the interaction between a CNT and a planar surface therefore is in critical need. Pristine CNTs are crystalline graphitic rods and are often considered to have no surface charge. Their interaction with a surface therefore is mainly controlled by van der Waals forces. [ 253 ] In the literature, the van der Waals interaction between CNTs and substrate surfaces is determined either by the continuum Lennard Jones (LJ) model (nanoscopic) with considerations of all pairs of interacting atoms [ 254 258 ] or by the Lifshitz theory (microscopic) in terms of the Hamaker coefficients. [ 253 259 ] The shape and range of the attractive van der Waals interaction (potential) varies with different dimensions (nanoscopic and microscopic). Th e LJ model is successful in describing the short range van der Waals interaction potential in CNT systems. The latter has recently been adapted in a form to describe long range van der Waals interaction between pristine SWNTs and anisotropic surfaces with good accuracy. [ 253 ] However, application

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110 of this method requires ab initio optical properties, which are not only barely documented in the literature but also difficult to measure. Because of strong inter tube attractions, pristine CNTs tend to aggregate and conducted to dis perse CNTs in aqueous or organic media. [ 260 262 ] Surface modification methods, such as chemical functionalization ( e.g., acid oxidization) and polymeric coating ( e.g., surfactant, dissolved organic matter, and ligand) ar e often used to improve the stability of CNT suspensions by introducing repulsive electrostatics forces. [ 262 264 ] For example, Mamedov et al., [ 265 ] used nitric aci d oxidization to introduce negative charges on SWNTs, which enable the assemble of a stable SWNT composite film. Thus, for highly charged CNTs, dispersion of CNTs is mainly controlled by electrostatic forces. Unfortunately, there are only limited amount of studies that have been attempted to theoretically determine the interaction, particularly electrostatic interaction, of surface modified CNTs with charged surfaces. For example, Chapot et al ., [ 266 ] and Lowen [ 267 ] proposed frameworks allowing one to compute the interactions between charged rodlike colloidal particles, respectively. However, it is still unknown whether these frameworks can be directly applied to calculate the electrostatic double layer repulsion between charged CNTs and planar interface. The surface charge form an electric double layer at the CNTs surfaces, which is similar to the phenomenon observed with colloidal particles. [ 268 ] In addition, previous studies have indicated that, although CNTs are molecular objects with two dimensions in the nanometric range, their dispersion, deposition and aggregation behaviors follow the principles of the classical colloidal system (especially for MWNTs). [ 269 ] 11 13 [ 270 ] The Derjaguin Landau Verwey

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111 Overbeek (DLVO) theory, which was originally developed for spherical colloids, thus has been used to semi quantitatively describe the stability of surface modified CNTs and their interaction with planar surfaces. [ 117 119 122 123 271 272 ] Because an used, the DLVO theory often failed to provide an accurate estimation of the interaction forces under various conditions. [ 117 ] Furthermore, the interaction of CNTs and planar surfaces is orientation dependent, which gives rise to a torque orienting the CNTs in an energetically favorable configuration to approach/depart the planar surfaces. Such a dynamic behavior cannot be explained merely on the basis of spherically symmetric interaction potentials of the classic DLVO theory. Several techniques have been developed to calculate the interaction force/energy between curved surfaces/bodies, including the Derjaguin Approximation (DA) and surface element integration (SEI) [ 124 ] The DA method estimates the interaction energy between two finite size bodies b y relating it to that between two infinite parallel flat plates. It can only be applied to surfaces that are separated by a small distance and to circumstances when the interaction range is substantially smaller than the radii of curvature of the surfaces For very small non spherical particles, such as SWNTs, the DA method may lead to inaccuracies in calculating their interaction with planar surfaces. [ 127 ] The SEI technique takes into account curvature effects over the whole object, by integrating the interaction energy between a surface element of the object and the plane surface using the exact surface geometry of the object. It can precisely determine the interaction forces between a planar surface and a curved body with any defined shape, including CNTs. [ 117 ] For instance, Stolarczyk et al ., [ 127 ]

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112 s uccessfully applied the SEI method to numerically evaluate the interaction forces between functionalized MWNTs and ligand stabilized gold nanoparticles (modeled as cylinder sphere system). However, the calculation is quite complex and time consuming. Thus, although the SEI has made a remarkable breakthrough in the accurate calculation of interaction energy between a curved body and planar surface or two curved bodie s, the difficulty in numerical implement restrict its wide application in describing the interface interactions. Nevertheless, little research effort has been made to apply the SEI to quantify the interaction of CNTs with planar surfaces, particularly with respect to obtaining analytical expressions. Therefore, accurate and efficient analytical calculation of interaction energy between CNTs and planar surface is of great scientific and practical significance. The overarching objective of this work was to develop analytical formula s that can precisely describe the orientation dependent interaction energy/forces between a CNT and an isotropic planar surface. It was hypothesized that the interaction of C NTs with planar surfaces is mainly controlled by the van der Waals and electrical double layer (EDL) forces, which are the same as the classic DLVO forces. The SEI method was thus integrated into the DLVO theory framework to obtain the analytical expressions of the orientation dependent interaction energy b etween a n SWNT and an isotropic planar surface. The interaction energy was evaluated for two different situations: 1) a pristine SWNT and an isotropic planar surface and 2) a surface charged SWNT and an isotropic planar surface. After validations, the anal ytical expression s were also extended to determine the interaction between a MWNT and the planar surface.

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113 Theory The general expression representing the interaction between an SWNT and planar surface are presented in Fig. 5 2 For convenience, two coordina te systems were used: a set of body fixed coordinates ( x,y,z ) to account for the internal geometric properties of the tubular SWNT and a space fixed coordinate system ( X,Y,Z ) to account for the orientation of the tubular SWNT relative to the planar surface The SWNT was modeled as a hollow cylinder ( e.g., SWNTs can be converted from nearly endless, highly tangled ropes into short, open ended pipes after acid oxidation). [ 273 ] The top, side, and bottom surfaces of the SWNT were defined as S 1 S 2 and S 3 respectively. It is worth noting that the presence of hemispherical caps at the ends of the SWMTs is also important and is currently a topic of ongoing investigation but is beyond the scope of this study. The SEI method was used in this work to determine the total inte raction energy between an SWNT of finite length and an infinite planar surface. [ 124 126 ] The governing equation of the SEI method can be written as: ( 5 1) where ( D, ) is the total interaction energy when the center of the SWNT with arbitrary angle ( ) is located at a distance from the planar surface, is the outer unit normal vector to the SWNT surface element dS ( i =1,2,3, and represents top, side, and bottom surfaces of the SWNT, respectively), is a unit vector normal to the planar surface, E ( h i ) represents the unit interaction energy between the surface element and the planar surface at a distance of h i

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114 It is worth noting that the sign of governs the magnitude of the total interaction energy. When the unit normal to the surface at a given point makes an obtuse angle with the sign of is negative, and hence, the energy contribution of the corresponding surface element is also negative. Thus, we need to subtract the The differential area of the surface element dS of the SWNT then can be written as: for outer S 2 (5 2 1 ) for inner S 2 (5 2 2 ) for S 1 and S 3 (5 3 ) where a is semi axes of the SWNT directed along x and y axes, and are radial and angular coordinate in a cylindrical coordinate system, respectively, and R is diameter of carbon atom ( m). The expressions of in the equation ( 5 1) can be written as: for S 2 (5 4 ) for S 1 (5 5 ) for S 3 (5 5 ) where is orientation angle of the tubular SWNT. The distance between the surface element and the planar surface, h i can be written as: for S 1 (5 6 ) for outer S 2 (5 7 1 ) for inner S 2 (5 7 2 ) for S 3 (5 8 )

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115 where D is separation distance of infinite planar surface from the center of the SWNT and L is semi axe of an SWNT directed along z axis. The interaction between the surface element and the planar surface was assumed to be mainly controlled by the DLVO forces, [ 268 274 275 ] thus the E ( h i ) can be written as: (5 9 ) where are the van der Waals and EDL interaction energy per unit area between two infinite flat plates, respectively. For van der Waals interaction, instead of the Lifshitz approach which is obtained from complex optical properties, the Hamaker approach was used in this study and can be written as: [ 276 ] (5 10 ) where A is the effective Hamaker constant. For EDL interaction linear superposition approach is regarded as the most accurate physical description of the EDL interaction for CNTs since it gives intermediate values between those for the constant potential (mobile charges that keep the potential between the two surfac es constant) and constant charge (assuming immobile charges) cases, and is given by Gregory: [ 277 ] (5 11 ) where is the permittivity of vacuum, is the relative permittivity of the solution, is the surface potential, is the Debye Huckel parameter, k is the T is the temperature, is the valence of electrolyte, e is the electron charge. Ideally, the surface pot ential must be used in Eqn. (5 11 ). In recent studies, is often approximated by the zeta potential ( ), the potential located at the

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116 electrokinetic plane of shear. [ 184 278 282 ] It is worth noting that there is debate of charges distribution on the surface of CNTs: Paillet et al., [ 283 ] provided the experimental confirmation that charges are distributed uniformly along the nanotubes while open ends and defect sites were reported to be preferentially charged areas. [ 270 284 285 ] By using theoretically an atomic charge dipole model and experimentally electrostatic force microscopy, Wang et al., [ 286 ] demonstrate that the charge enhancement at the end already becomes insignificant for an SWNT with length around 30nm, and will become negligible for micrometer long CNTs. Hence, in this study, we assume that charge on the SWNTs surface is uniformly distributed on the surfa ce. Results and Dis cussion DLVO Interactions between a Pristine SWNT and an Isotropic Planar Surface The interaction between a pristine SWNT (without charge) and an isotropic planar surface is mainly controlled by the attractive van der Waals forces. For this case, mathematical ana lysis of the interaction energy between a pristine SWNT and an isotropic planar surface with arbitrary angle position yields the analytical solution:

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117 (5 12 ) where Detailed math derivations of Equation (5 12 ) can be found in the Appendix. Equation (5 12 ) was used to determine the interaction energy profiles (scaled to kT ) between a pristine SWNT and a planar quartz surface in water for arbitrary angle approaching patterns. The interaction energy between the pristine SWNT and the flat surface was attractive, suggesting pristine SWNTs intend to attach to surfaces due to the attractive van der Waals forces. Overall, the magnitude of van der Waals interaction energy depends on the orientation of the SWNT with respect to the planar surface (Figure 5 3 A). The attractive interaction energy increases when the arbitrary angle incre ases from tube side). The attractive energy of the side on pattern was much higher (at least two orders of magnitude) than that of the end on pattern, indicating that it is more favorabl e

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118 for pristine SWNTs to attach on planar surfaces through the side on approaching pattern (Figure 5 3 B). For a randomly positioned pristine SWNT in the system, the difference in attractive energy between the two approaching patterns may generate a torque to drive the SWNT to attach on the planar surface through the side on pattern. Because of their nanosized diameter, SWNTs are often modeled as solid cylinders (without open ends) instead of tubes for convenience. [ 127 ] This practice, however, may generate errors when used to determine the interaction energy between SWNTs and flat surfaces using the SEI met hod. When a pristine SWNT of a relatively small diameter ( e.g., 0.2 nm) was modeled as a cylinder, the energy profile matched that of the tubular SWNT closely (Figure 5 2C). When a pristine SWNT of a relatively large diameter ( e.g., 1.5nm) was modeled as a cylinder, however, the results showed large deviations between the two energy profiles. These results suggest that SWNTs, especially with large diameters, should be modeled as tubes instead of cylinders to accurately describe their interaction with planar surfaces. Recently, Rajter et al., [ 253 ] developed a model to calculate the van der Waals interaction energy between optically anisotropic SWNTs and planar surfaces based on the optical properties of the SWNTs and the Lifshitz theory. This model was applied in this work to determine the e nergy profile between the pristine SWNT and the planar surface for the side on pattern. The interaction energy profile obtained from the Rajter model was compared with that of this work and of the widely used DA method (Figure 5 3 D). The results demonstrat ed excellent agreements between the new model and the Rajter model, indicating the SEI approach can be integrated into the DLVO theory to accurately describe the interaction between SWNTs and planar surfaces. In contrast,

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119 the DA method overestimated the i nteraction of between a SWNT and planar surface up to three order of magnitude, confirming the method is not suitable for nanosized non spherical particles [ 127 ] DLVO Interaction s between a Surface M odified SWNT and a Charged Isotropic Planar S urface Both van der Waals and EDL forces are important to the interaction between a surface modified SWNT and a ch arged isotropic planar surface. While the attractive van der Waals interaction energy is the same as discussed previously (Eqn.(5 12 )), analytical expression of the EDL interaction energy between the surface charged SWNT and the planar surface with arbitra ry angle can be written as: (5 13 ) where ; ; ; I 0 (x) I 0 (x), and I 2 (x) are modified Bessel function of order zero, first, and second, respectively; and L 1 (x) is modified Struve function of order 1. Deta iled math derivation s of Eqn. (5 13 ) can be found in the appendix. Here, the electrostatic interaction was considered as the only force introduced by the surface modification, which is reasonable for surface charged CNTs. Other interaction forces, such as steric repulsion and hydrophobic interaction, however, may be trigged by other surface modification methods ( e.g., coating with surfactants or polymers) [ 256 258 ] which are beyond the scope of this study. Thus,

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120 further investigations are necessary to include these non DLVO interaction forces to descr ibe the extended DLVO interaction of surfactant or polymer modified SWNTs and planar surfaces in the future. Eqn. ( 5 13 ) was used to determine the scaled EDL energy profiles between a charged SWNT ( i.e., humic acid coated SWNT [ 271 ] ) and a planar quartz surface in water for arbitrary angle approaching patterns (Figure 5 3A). The potential values of the SW NT and the glass beads under two ionic strength conditions were obtained from reported values in the literature. [ 122 271 ] It is worth noting that by measuring the electrophoretic mobility and then relating to th e reported potential of SWNTs was calculated from the Smoluchowski approximation for spherical particles, which may overestimate the actual potential of the tubular SWNTs by up to 20%. [ 264 ] Several research efforts have been made to modify the Smoluchowski equation to apply to higher aspect ratio structures ( i.e., cylindrical particles). [ 287 ] [ 288 289 ] Additional investigations, however, are still needed to determine whether the modi fied Smoluchowski equations can provide accurate solution to the potential of tubular SWNTs. The EDL interaction energy between the charged SWNTs and the quartz surface was repulsive and the repulsive energy was four orders of magnitude stronger for the side on pattern than for the end on pattern under both ionic strength conditions (Figure 5 4 B). Furthermore, the predicted EDL energy was very sensitive to ionic strength and decreased dramatically for arbitrary angle pattern when solution ionic strength increased from 0.001M to 0.1M. This is consistent with the DLVO theory that increases

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121 in ionic strength can compress the double layer to reduce the repulsive electrostatic forces between two charged surfaces in electrolyte solutions. [ 268 275 ] Based on the equations ( 5 12) and (5 13 ), the total interaction energy between the surface modified SWNT and the charged planar surface for arbitrary angle attachment thus can be written as: (5 14 ) Eqn. (9) was used to determine the scaled DLVO energy profiles between a surface modified SWNT and a planar quartz surface (using the physicochemical and surface properties mentioned above) in water for arbitrary angl e approaching patterns (Figure 5 A). The coupled effects of orientation and ionic str ength on the total interaction energy profile of SWNTs and planar surfaces were also investigated (Figure 5 5 B E). Both the height of the energy barrier and the depth of the secondary minimum increased with the increase of the approaching angle for the tw o tested ionic strength conditions. The total interaction energy of the side on pattern was several orders of magnitude higher than that of end on pattern over the entire range of separation distances. The energy barrier of the side on pattern was (0.001M: 173.52 kT and 0.1M: 87.64 kT ) more than three hundred times higher than that of the end on pattern (0.001M: 0.55 kT and 0.1M: 0.47 kT ). These results indicated that it might be much easier for the SWNT to overcome the energy barrier to attach to the plan ar surface through the end on pattern than through the side on pattern when repulsive EDL forces presence. For all arbitrary angle patterns, the EDL interaction became short ranged with the presence of electrolyte ( i.e., 0.1M). As a result, shallow seconda ry minimum energy

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122 wells were identified in the energy profiles for all approaching patterns. For example, the depth of secondary minimum for side on pattern is 0.87 kT indicating that the SWNT could also attach to the planar surface through side on patter n in the secondary minima under high energy barrier conditions although the deposition in secondary minimum is temporary since the depth is close to the average kinetic energy of a particle (1.5 kT ). No secondary minimum wells were found in the energy prof iles for the low ionic strength conditions over the entire range of separation distance shown (0 100nm) for both patterns. Figure 5 5 F I show the coupled effects of radii and orientation (approaching angle) on the total inte raction energy profile of SWNTs and planar surface s F or easy comparison, zeta po tential and length of SWNTs of di fferent radii were assumed to the same under the tested conditions. Overall, both of the interaction energy barriers and depths of secondary minimum increase when the radii of the SWNTs increase for all the approaching patterns. W hen the SWNTs radii are fixed heights of the i nteraction energy barrier and depths of secondary minimum increase d as the rotation angles increase from 0 to /2 which is consistent with the discussion above. W hen the orientation angle is fixed t he sepa ration distance at the secondary minimum well is independent of SWNTs radius For example, the secondary minimum locations ( separation distances ) of three tested SWNTs wit h different radii were at 10.1, 7.4, 7.3 and 9.5 nm when the approaching in insets). These results are in agreem ent with findings from previous studies using the LJ potential approach [ 255 ] Based on the results shown in Figure 5 5 F I there may exist a critical rotation angle at which the secondary minimum well separation distance

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123 reaches the minimum value Volkov and Zhigilei [ 257 ] also reported that the equilibrium distance betwe en two CNTs of finite length depends on their rotat ion angle s DLV O Forces and T orques of SWNTs with Planar S urface s The analytical expressions of interaction energy of SWNTs with planar surfaces enable a straightforward analysis of force and torque which can be written as the derivatives of the energy potential: (5 15 ) (5 16 ) Thus, the DLVO force and torque between a charged SWNT and a planar quartz surface (of the same properties as mentioned above) in water for arbitrary angle approaching patterns can be determined using the two equations (Figure 5 6 A and Figure 5 6 B, respect ively). Figures 5 6 C shows the DLVO force profiles of SWNTs of different radii interacting with the planar surface under side on approaching pattern. The results indicated that the distance where the attractive force reaches its maximum is a constant (9.7 nm) for all the tested conditions, which is in consistent with the fact that the corresponding secondary energy minimum wells are located at around 9.5 nm (Figure 5 6 D) as discussed above. The inset in Figure 5 6 C indicates that the surfaces jumped to a pr imary minimum from H=0.87 nm. Figure 5 6 D shows the dependences of torque on the rotation angle of the SWNT approaching to the surface. The results suggested that that torque direction is depended on the separation distance. For close separation distance (e.g., 2 nm) between the SWNT and the surface, where repulsion (perpendicular) to the planar surface. For intermediate separation distance, in vicinity of

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124 the equilibrium position, the equilibrium angle (when the torque is zero) may not relatively long separation distance, where attraction may dominate, and the torque (positive) acts to align the CNTs t DLVO Interactions of MWNTs and Planar Surfaces The analytical expressions can be extended to describe the interaction of MWNTs and planar surfaces. Because MWNTs are made of several layers of SWNTs, it is r easonable to assume that each layer interacts with the planar surface as one SWNT. Thus, the total van der Waals interaction energy between a MWNT and a planar surface can be obtained by summation over all layers with an assumption that the interlayer spac ing is a constant (i.e., 3.39 nm). [ 290 ] For the EDL interaction of modified MWNTs, previous studies have shown that all the surface charge may uniformly distributed on their out layer surfaces [ 291 ] and thus the EDL expression (i.e., Eqn. ( 5 13 )) can be applied directly. Figure 5 7 compares normalized interaction energy profiles ( vs. where is the secondary minimum and is the corresponding se paration distance) of a modified SWNT and a modified MWNT (same surface charge density and length) interact with the charged planar surface at three different rotation angles. When the rotation angle is fixed, the normalized interaction energy profiles of SWNTs and MWNTs were almost identical regardless of the differences in their radii. This result is consistent with the findings from previous studies based on the continuum LJ potential model [ 254 255 292 293 ] and confirmed the analytical expressions are applicable to the MWNT systems. Because the normalized energy profile is independent of the

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125 Hamaker constant, surface charge density, and CNT properties, further research is needed to explore the mathematical c onnections among the maximum energy barrier, secondary energy minimum, and their corresponding separation distances (equilibrium distance). This will allow the development of new mathematical tools to determine the interaction of CNTs and planar surfaces. Environmental Implications Interaction between a tubular SWNT and an isotropic planar surface with arbitrary orientation angles was quantified by integrating the SEI method into the DLVO theory. For the first time, exact analytical solutions of the DLVO interaction energy were developed not only for pristine SWNTs and planar surfaces, but also for surface modified SWNTs and charged planar surfaces with arbitrary orientation angles Simplified formulas were also given for t patterns. Compared to the results of other methods, the new solutions were either convenient or more accurate than existing approach to describe the interaction of SWNTs with isotropic surfaces. The analytical formulas derived for SWNTs can also be applied to MWNTs with minor modifications. The analysis of DLVO force and torque showed that in the region close to the planar surface, the repulsive interaction creates preferential alignment of CNTs perpendicular to the planar surface; without this interaction, parallel alignment is favored. The new model presented in this work provides a clear picture of the interaction energy/forces/torques between CNTs and planar surfaces with arbitrary orientation and sheds light on understanding the approaching patterns of CNTs to the planar surfaces under various conditions. It can be used as an effective tool by end users to predict and optimize the interaction between CNTs and planar surfaces for a wide variety of fields of in terests ( e.g., bio devices,

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126 biomedicine, etc .). Although in this work the interaction energy is developed for CNTs, it is not limited to CNTs and can be readily applied for various types of nano and micro tubular structures for analysis of their interacti on with planar surfaces.

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127 Figure 5 2 Schematic illustration of interaction of a SWNT with an infinite isotropic planar surface

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128 Figure 5 3 The van der Waals interaction energy between a pristine SWNT and an isotropic planar surface

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129 Figure 5 4 T he electrostatic double layer interaction energy ( between a surface modified SWNT and a charged isotropic planar surface

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130 Figure 5 5 T otal interaction energy between a surface modified SWNT and a charg ed isotropic planar surface

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131 Figure 5 6 DLVO force and torque acting on a surface modified SWNT and a charged isotropic planar surface A B C D

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132 Figure 5 7 Normalized total interaction energy between a surface modified SWNT or MWNT and a charged isotropic planar surface

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133 CHAPTER 6 6 CONCLUSIONS AND RECOMMENDATIONS Conclusions T h is PhD dissertation systematically addresses fundamental research on colloid filtration and transport in overland flow through dense emergent vegetation syste m s and interfacial interactions between nanoparticles (CNTs) and planar surface. In the first part of the study, a series of laboratory experiments were successfully conducted to measure the single collector contact efficien cy ( 0 ) and attachment efficiency ( ) of colloid capture by a simulated plant stem in laminar overland flow. Florescent microspheres of various sizes were used as experimental colloids. The colloid suspensions were applied to a glass cylinder installed in a small size flow chamber at different flow rates. S ilicone grease was applied to the cylinder surface to determine the single collector contact efficiency ( 0 ) under favorable conditions Different solution ionic strengths ( IS ) were used in the experiments to simulate unfavorable attachment c onditions. Our results showed that increases in flow rate and collector size reduced the value of 0 and a minimum value of 0 might exist at a critical colloid size. increased with IS and decreased with flow velocity. The experimental observations of 0 and were compared to theoretical predictions of different single colle ctor contact efficiency models and attachment models, respectively. The results indicated that both of existing single collector contact efficiency models and attachment models fall short in matching the 0 and of colloid capture by the simulated plant in laminar overland flow. For the first time single stem efficiency theory including dimensionless equation s of 0 and was developed that matched the experimental data with r easonable accuracy In addition, for the case of colloid filtration and transport in a

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134 vegetation system under shallow laminar flow conditions, a new equation was also proposed to calculate the colloid kinetic deposition rate at the field scale. In order to upscale s ingle stem efficiency theory to real dense vegetation, a new dimensionless number was developed to account for the effect of plant stem surface properties on the colloid deposition in overland flow. Laboratory scale dense vegetation ch amber experiments and model simulations were conducted to obtain the effective value of colloid kinetic deposition rate in vegetation system under different experimental conditions. The results showed that in addition to flow hydrodynamics ( e.g., flow velo city) and solution c hemistry ( e.g., ionic strength), steric repulsion afforded by biopolymer brush player on the pant stem also plays a significant role in colloid deposition onto the plant in overland flow. For the first time, extended single stem efficie ncy theory including steric repulsion effect was developed that fit the experimental data with acceptable accuracy. F indings from this work filled the existing knowledge gap regarding the fundamental mechanisms that govern overland flow colloid transport t hrough emergent vegetation and a re regarded as first steps for accurate quantitative prediction of colloid fate and transport in surface flow through these vegetation systems. They can also inform guidelines for the design, establishment, and maintenance o f surface vegetation as filters for colloidal contaminants, such as pathogens. Because establishment of dense vegetated areas in the form of grass filters is a low cost and potentially effective pollution control practice, optimization of the design and im plementation of these filters can produce significant societal and environmental benefits. In addition, although t he extended single stem efficiency theory is developed for prediction of colloid kinetic deposition onto plant stem, its application is not li mited to

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135 plant stem and can be also used for various polymer brush surfaces in natural, engineered and biomedical systems. In the second part of the study, the surface element integration (SEI) technique was coupled with the DLVO theory to determine the o rientation dependent interaction energy between a single walled carbon nanotube (SWNT) and an infinite isotropic planar surface. For the first time, an analytical formula was successfully developed to accurately describe the interaction between not only pristine but also surface charged CNTs and planar surfaces with arbitrary rotation angles. Simplified formulas were also Compared to other methods, the new analytical formulas were either m ore convenient or more accurate to describe the interaction between CNTs and planar surface especially with respect to arbitrary angles. The results revealed complex dependences of both force and torque between SWNTs and planar surfaces on the separation d istances and rotation angles. With minor modifications, the analytical formulas derived for SWNTs can also be applied to multi walled carbon nanotubes (MWNTs). The findings from this part of my work provide a clear picture of the interaction energy/forces/ torques between CNTs and planar surfaces with arbitrary orientation and sheds light on understanding the approaching patterns of CNTs to the planar surfaces under various conditions. They can be used as effective tool s by end users to predict and optimize the interaction between CNTs and planar surfaces for a wide variety of fields of interests (e.g., bio devices, biomedicine, etc.). Although in this work the interaction energy is developed for CNTs, it is not limited to CNTs and can be readily applied for various types of nano and micro tubular structures for analysis of their interaction with planar surfaces.

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136 Recommendations for Further Study During the course of this study, several unresolved issues that need to be addressed in more detail have been ide ntified A brief discussion of these issues is presented below. In addition to these unresolved issues, future work resulting from interesting preliminary results is also suggested. Plant filtration theory in overland flow One of the primary trends of curr ent and future research is moving from ideal and well controlled systems to real and more complex systems. For example, investigating natural colloidal contaminants (e.g., bacteria and virus) deposition on plant stem surface under real field conditions is a necessary next step The extended single stem efficiency theory is based on a limited set of data including shallow laminar flow conditions, biopolymer brush layer properties and colloid kinetic deposition rate s fitted from breakthrough curves. Addition al characterizations of broader range of flow velocit ies plant stem surface properties and measurements of deposition kinetics of colloids under different environmental conditions are needed to further evaluate the robustness of extended model. Other po ssible effects controlling the colloid deposition on plant stem further investigations. For example, long lasting air film (bubble s ) due to hair like structure s on the stem surface may also play an important role in c olloid deposition. To address this, s ome of the mechanisms that control the colloid deposition in unsaturated conditions (e.g., air water solid interface capture, water film straining, storage in immobile water zones, and air water interface capture) are worth further investigation. O n the ba sis of findings from this study, classic colloid filtration theory in porous media and theoretical studies of adsorption of particles on grafted polymers, we

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137 proposed three different mechanisms of colloidal particles deposition on plant stem in overland fl ow : Type primary deposition T ype secondary deposition and T ype ternary deposition Although these mechanisms are similar to those of colloid s stabilized by grafted polymer, plant stem surface properties and dynamic overland flow conditions make the se poorly understood mechanisms, so in depth theoretical and experimental studies are in critical need. Interactions between CNTs and interfaces As the capability of CNTs to be modified with various surface functional groups is utilized in consumer product s, a related research effort should be to provide models to predict how certain surface functionalization and coatings influence their environmental fate and transport in aquatic system. For example, CNTs are commonly manufactured with surface coatings (e. g., polymers and surfactants and polyelectrolytes) to enhance dispersion stability in solution. In this case, i n addition to traditional DLVO interaction (van der Waals attraction and el ectrostatic double layer repulsion), steric repulsion afforded by an a dsorbed polymeric layer also needs to be considered. Therefore, theoretical models to calculate accurately steric interactions between CNTs and different interfaces are in critical need. Similarly to the future work stated above, another important trend for future research in th is second topic is moving from well defined systems to more complex and environmentally relevant systems. For example, investigating deposition of CNTs on the more complex surface s such as collectors with different mineralogical compositions and different types of soils is more challenging but critical for a full understanding of their environmental transport.

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138 For the framework of interactions between CNTs and diff erent interfaces, in addition to modeling interactions between CNTs and planar surface, develop ments of the theories to quantify the interactions between two CNTs (tube tube system), as well as CNTs and other nanoparticles (tube sphere system ) are also of great importance and in critical need to fully understand the interactions between CNTs and different interfaces.

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139 APPENDIX A SUPPORTING INFORMATI ON FOR CHAPTER 2 Table A 1 Experimental data of single stem contact efficiency ( 0 ) under different flow velocity conditions for a given colloid ( d p =1.05m) and collector ( d c =2cm) Replicate 1 Replicate 2 Replicate 3 u (cm/s) r c = R 2 0 r c = R 2 0 r c = R 2 0 0.002 1.35E+04 0.96 6.53E 03 1.24E+04 0.95 5.98E 03 1.40E+04 0.97 6.75E 03 0.004 1.44E+04 0.98 3.48E 03 1.27E+04 0.97 3.05E 03 1.42E+04 0.97 3.43E 03 0.008 1.47E+04 0.98 1.77E 03 1.31E+04 0.98 1.58E 03 1.47E+04 0.96 1.77E 03 0.01 1.45E+04 0.98 1.40E 03 1.37E+04 0.96 1.32E 03 1.49E+04 0.96 1.43E 03 0.02 1.47E+04 0.98 7.10E 04 1.38E+04 0.98 6.64E 04 1.49E+04 0.97 7.21E 04 0.04 1.50E+04 0.98 3.62E 04 1.39E+04 0.97 3.36E 04 1.50E+04 0.97 3.62E 04 0.08 1.54E+04 0.97 1.86E 04 1.42E+04 0.96 1.72E 04 1.52E+04 0.97 1.83E 04 0.1 1.54E+04 0.98 1.49E 04 1.42E+04 0.97 1.37E 04 1.51E+04 0.98 1.45E 04 0.2 1.55E+04 0.98 7.47E 05 1.48E+04 0.95 7.12E 05 1.52E+04 0.98 7.33E 05

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140 Table A 2. Experimental data of single stem contact efficiency ( 0 ) under different sizes of colloid and collector at a given flow velocity (u=0.02cm/s) Replicate 1 Replicate 2 Replicate 3 d p ( m) d c (cm) r c R 2 0 r c R 2 0 r c R 2 0 0.1 2 1.69E+04 0.91 4.30E 06 1.58E+04 0.92 4.02E 06 1.81E+04 0.95 4.60E 06 1.05 2 1.47E+04 0.98 3.74E 06 1.38E+04 0.98 3.49E 06 1.49E+04 0.97 3.79E 06 2 2 1.53E+04 0.95 3.88E 06 1.43E+04 0.97 3.63E 06 1.67E+04 0.93 4.23E 06 10.5 2 2.34E+04 0.90 5.95E 06 2.36E+04 0.92 5.99E 06 2.38E+04 0.86 6.04E 06 0.1 1 1.48E+04 0.91 7.54E 06 1.42E+04 0.92 7.20E 06 1.59E+04 0.87 8.08E 06 1.05 1 1.21E+04 0.95 6.15E 06 1.16E+04 0.97 5.90E 06 1.26E+04 0.95 6.38E 06 2 1 1.27E+04 0.91 6.43E 06 1.19E+04 0.93 6.07E 06 1.34E+04 0.92 6.80E 06 10.5 1 2.23E+04 0.92 1.13E 05 2.24E+04 0.92 1.14E 05 2.31E+04 0.93 1.17E 05

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141 APPENDIX B SU PPORTING INFORMATION FOR CHAP TER 3 DLVO Interaction Energy Profiles. DLVO theory [ 294 295 ] was used to calculate the total interaction energy (sum of London van der Waals attraction and electrostatic double layer repulsion) b etween colloid and glass cylinder surfaces under different conditions. The Lifshitz van der Waals attraction energy ( ) for a sphere plate system can be written as [ 296 ] : ( B 1) where A (110 20 J) is the Hamaker constant for the polystyrene water glass system [ 142 145 ] h is the separation distance, and r is the radius of the particle. The EDL repulsion energy ( )for a sphere plate system can be written as [ 296 ] : ( B 2 ) where is the dielectric constant of the medium (78.4 for water), 0 is the vacuum permittivity (8.85410 12 C 2 N 1 m 2 ), k 23 C 2 J K 1 ), T is the temperature, z is the valence of electrolyte, e is the electron charge (1.60210 19 C), and are the surface potential of the colloid and the collector surface, and is the reciprocal of the Debye length. The surface potential of colloids and collector can be de termined following van Oss et al. [ 297 ] : ( B 3) where d is the distance between the surf ace of the charged particle and the slipping plane and usually taken as 5 angstroms (10 10 m). Existing models of estimating attachment efficiency

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142 I. Maxwell model Detailed description of the Maxwell model can be found in the literature [ 146 191 ] The model assumes that velocity distribution of a colloid in the secondary minimum follows the Maxwell function: ( B 4 ) ( B 5 ) where is mass of the colloid and is the velocity. The fraction of successful collision resulting in the colloid deposition in the secondary minimum, can be written as: ( B 6 ) ( B 7 ) Similarly, the fraction of successful collision resulting in the colloid deposition in the primary minimu m, can be written as: ( B 8 ) The single collector attachment efficiency, can be written as: ( B 9 ) II. Modified Maxwell model To account for the influence of fluid hydrodynamic drag on the attachment efficiency, two hydrodynamic factors, and are introduced to modify the Maxwell model [ 298 ] : ( B 10 )

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143 where and are the fractions of single collector surface area over which the adhesive torques acting on the colloids retained in the primary and secondary minimum are greater than the fluid hydrodynamic drags, respectively. The values of adhesive force ( ) are estimated as the sum of and for colloids retained in the energy wells [ 298 ] : ( B 11 ) where and are primary and secondary minimum, respectively; and is separation distance. The value of lever arm ( ) of the adhesive torque can be estimated with: [ 298 ] ( B 12 ) where is colloid radius, and 9 Nm 2 ). The adhesive torque can then be expressed as: ( B 13 ) The drag force ( F H ) that acts on a colloid attached on the collector interface at a separat ion distance ( h ) can be written as [ 299 300 ] : ( B 14 ) where is fluid viscosity and is hy drodynamic shear, is a constant defined as [ 299 300 ] : (B 15 ) On a smooth surface the value of the applied hydrodynamic torque that acts on the colloid at is given as [ 299 300 ] :

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144 ( B 16 ) In this case, is a second dimensionless function that depends on is given as [ 299 300 ] : ( B 17 ) Where is much greater than the value of is more simply given as [ 177 298 ] : ( B 18 ) III. Bai Tien model. An empirical correlation equation developed by Bai and Tien [ 180 ] was also used in this work. The equation of the Bai Tien model can be written as: ( B 19 ) Definitions of the dimensionless parameters are listed in Table B 1.

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145 Table B 1. Definition of Dimensionless parameters Parameter Definition London number First electrokinetic parameter Second electrokinetic parameter Third electrokinetic parameter Double layer force parameter Reynolds number Aspect ratio Peclet number A is the Hamaker constant, is the fluid viscosity, is the colloidal particle diameter, is the flow velocity, is the relative permittivity of the fluid, is the permittivity in a vacuum, and are the surface potential of the colloidal particles and collectors respectively, is the reciprocal of double layer thickness, is the ionic strength, is kinetic viscosity, is the bulk diffusion coefficient (described by Stokes Einstein equation).

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146 Table B 2. Summary of stepwise least square regression results Step 1 Entering variable: log(NE2) Summary measures Multiple R 0.7046 R Square 0.4964 Adj R Square 0.4850 StErr of Est 0.4167 ANOVA Table Source df SS MS F p value Explained 1 7.5325 7.5325 43.3739 0.0000 Unexplained 44 7.6412 0.1737 Regression coefficients Coefficient Std Err t value p value Lower limit Upper limit Constant 2.0256 0.0875 23.1621 0.0000 2.2019 1.8494 log(NE2) 9.2662 1.4070 6.5859 0.0000 12.1018 6.4306 Step 2 Entering variable: log(NE1) Summary measures Change % Change Multiple R 0.7683 0.0638 9.0% R Square 0.5903 0.0939 18.9% Adj R Square 0.5713 0.0863 17.8% StErr of Est 0.3802 0.0365 8.8% ANOVA Table Source df SS MS F p value Explained 2 8.9575 4.4787 30.9813 0.0000 Unexplained 43 6.2162 0.1446 Regression coefficients Coefficient Std Err t value p value Lower limit Upper limit Constant 2.2197 0.1009 21.9934 0.0000 2.4232 2.0161 log(NE2) 10.0772 1.3094 7.6959 0.0000 12.7179 7.4365 log(NE1) 0.1588 0.0506 3.1396 0.0031 0.0568 0.2608

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147 Table B 2. Continued Step 3 Entering variable: log(NDL) Summary measures Change % Change Multiple R 0.8894 0.1211 15.8% R Square 0.7911 0.2007 34.0% Adj R Square 0.7761 0.2049 35.9% StErr of Est 0.2747 0.1055 27.7% ANOVA Table Source df SS MS F p value Explained 3 12.0034 4.0011 53.0086 0.0000 Unexplained 42 3.1702 0.0755 Regression coefficients Coefficient Std Err t value p value Lower limit Upper limit Constant 3.3878 0.1978 17.1261 0.0000 3.7870 2.9886 log(NE2) 6.4480 1.1053 5.8339 0.0000 8.6786 4.2175 log(NE1) 0.2592 0.0398 6.5100 0.0000 0.1788 0.3396 log(NDL) 0.5649 0.0889 6.3525 0.0000 0.3855 0.7444 Step 4 Entering variable: log(NLO) Summary measures Change % Change Multiple R 0.9233 0.0338 3.8% R Square 0.8524 0.0613 7.8% Adj R Square 0.8380 0.0619 8.0% StErr of Est 0.2337 0.0410 14.9% ANOVA Table Source df SS MS F p value Explained 4 12.9342 3.2335 59.2001 0.0000 Unexplained 41 2.2394 0.0546

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148 Table B 2. Continued Regression coefficients Coefficient Std Err t value p value Lower limit Upper limit Constant 3.2475 0.1717 18.9171 0.0000 3.5942 2.9008 log(NE2) 1.1545 1.5901 0.7260 0.4719 4.3658 2.0568 log(NE1) 0.2072 0.1179 1.7564 0.0865 0.4453 0.0310 log(NDL) 0.9858 0.1270 7.7647 0.0000 0.7294 1.2422 log(NLO) 0.4453 0.1079 4.1280 0.0002 0.2274 0.6631 Step 5 Leaving variable: log(NE2) Summary measures Change % Change Multiple R 0.9222 0.0010 0.1% R Square 0.8505 0.0019 0.2% Adj R Square 0.8398 0.0018 0.2% StErr of Est 0.2324 0.0013 0.6% ANOVA Table Source df SS MS F p value Explained 3 12.9054 4.3018 79.6545 0.0000 Unexplained 42 2.2682 0.0540 Regression coefficients Coefficient Std Err t value p value Lower limit Upper limit Constant 3.2470 0.1707 19.0216 0.0000 3.5915 2.9025 log(NE1) 0.2726 0.0757 3.6019 0.0008 0.4253 0.1199 log(NDL) 1.0623 0.0705 15.0774 0.0000 0.9201 1.2045 log(NLO) 0.5084 0.0634 8.0168 0.0000 0.3804 0.6364

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149 Evaluation of new dimensionless equation on development dataset Figure B 1 Comparison of experimental attachment efficiency with predictions of the new dimensionless equation for development dataset

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150 Table B 3 conditions (IS=0.001M) for a given colloid (dp=1.05m) IS=0.001M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 1.00E 03 1.00E 03 1.00E 03 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+08 1.04E+08 1.04E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 8.00E 02 8.00E 02 8.00E 02 Z eta potentials of collector C 5.70E 02 5.70E 02 5.70E 02 S urface potentials of particles P 0.084284 0.084284 0.084284 S urface potentials of collector C 0.060038 0.060038 0.060038 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+06 2.41E+06 2.41E+06 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 6.42E 03 6.42E 03 6.42E 03 First electrokinetic parameter N E1 3.39E+00 3.39E+00 3.39E+00 Second electrokinetic parameter N E2 9.45E 01 9.45E 01 9.45E 01 Third electrokinetic parameter N E3 7.53E+13 7.53E+13 7.53E+13 Double layer force parameter N DL 1.09E+02 1.09E+02 1.09E+02 Attachment efficiency 0.0076 0.0079 0.0088

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1 51 Table B 4 strength conditions (IS=0.005M) for a given colloid (dp=1.05m) IS=0.005M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 5.00E 03 5.00E 03 5.00E 03 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 2.32E+08 2.32E+08 2.32E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 7.00E 02 7.00E 02 7.00E 02 Z eta potentials of collector C 5.20E 02 5.20E 02 5.20E 02 S urface potentials of particles P 0.078638 0.078638 0.078638 S urface potentials of collector C 0.058403 0.058403 0.058403 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+06 2.41E+06 2.41E+06 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 6.42E 03 6.42E 03 6.42E 03 First electrokinetic parameter N E1 2.67E+00 2.67E+00 2.67E+00 Second electrokinetic parameter N E2 9.57E 01 9.57E 01 9.57E 01 Third electrokinetic parameter N E3 3.76E+14 3.76E+14 3.76E+14 Double layer force parameter N DL 2.44E+02 2.44E+02 2.44E+02 Attachment efficiency 0.015 0.009 0.0033

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152 Table B 5 conditions (IS=0.01M) for a given colloid (dp=1.05m) IS=0.01M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 1.00E 02 1.00E 02 1.00E 02 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 3.28E+08 3.28E+08 3.28E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 6.00E 02 6.00E 02 6.00E 02 Z eta potentials of collector C 5.00E 02 5.00E 02 5.00E 02 S urface potentials of particles P 0.070725 0.070725 0.070725 S urface potentials of collector C 0.058924 0.058924 0.058924 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+06 2.41E+06 2.41E+06 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 6.42E 03 6.42E 03 6.42E 03 First electrokinetic parameter N E1 2.14E+00 2.14E+00 2.14E+00 Second electrokinetic parameter N E2 9.84E 01 9.84E 01 9.84E 01 Third electrokinetic parameter N E3 7.53E+14 7.53E+14 7.53E+14 Double layer force parameter N DL 3.45E+02 3.45E+02 3.45E+02 Attachment efficiency 0.006 0.019 0.023

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153 Table B 6 strength conditions (IS=0.05M) for a given colloid (dp=1.05m) IS=0.05M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 5.00E 02 5.00E 02 5.00E 02 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 7.34E+08 7.34E+08 7.34E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 4.80E 02 4.80E 02 4.80E 02 Z eta potentials of collector C 3.20E 02 3.20E 02 3.20E 02 S urface potentials of particles P 0.069314 0.069314 0.069314 S urface potentials of collector C 0.046198 0.046198 0.046198 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+06 2.41E+06 2.41E+06 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 6.42E 03 6.42E 03 6.42E 03 First electrokinetic parameter N E1 1.17E+00 1.17E+00 1.17E+00 Second electrokinetic parameter N E2 9.23E 01 9.23E 01 9.23E 01 Third electrokinetic parameter N E3 3.76E+15 3.76E+15 3.76E+15 Double layer force parameter N DL 7.71E+02 7.71E+02 7.71E+02 Attachment efficiency 0.091 0.061 0.076

PAGE 154

154 Table B 7 strength conditions (IS=0.1M) for a given colloid (dp=1.05m) IS=0.1M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 1.00E 01 1.00E 01 1.00E 01 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+09 1.04E+09 1.04E+09 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 3.80E 02 3.80E 02 3.80E 02 Z eta potentials of collector C 1.80E 02 1.80E 02 1.80E 02 S urface potentials of particles P 0.063888 0.063888 0.063888 S urface potentials of collector C 0.030255 0.030255 0.030255 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+06 2.41E+06 2.41E+06 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 6.42E 03 6.42E 03 6.42E 03 First electrokinetic parameter N E1 6.21E 01 6.21E 01 6.21E 01 Second electrokinetic parameter N E2 7.74E 01 7.74E 01 7.74E 01 Third electrokinetic parameter N E3 7.53E+15 7.53E+15 7.53E+15 Double layer force parameter N DL 1.09E+03 1.09E+03 1.09E+03 Attachment efficiency 0.129 0.09 0.12

PAGE 155

155 Table B 8 strength conditions (IS=0.001M) for a given colloid (dp=0.1m) IS=0. 00 1M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 1.00E 03 1.00E 03 1.00E 03 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+08 1.04E+08 1.04E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 6.80E 02 6.80E 02 6.80E 02 Z eta potentials of collector C 5.70E 02 5.70E 02 5.70E 02 S urface potentials of particles P 0.071804 0.071804 0.071804 S urface potentials of collector C 0.060038 0.060038 0.060038 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+05 2.29E+05 2.29E+05 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 7.08E 01 7.08E 01 7.08E 01 First electrokinetic parameter N E1 2.90E+01 2.90E+01 2.90E+01 Second electrokinetic parameter N E2 9.85E 01 9.85E 01 9.85E 01 Third electrokinetic parameter N E3 7.53E+13 7.53E+13 7.53E+13 Double layer force parameter N DL 1.04E+01 1.04E+01 1.04E+01 Attachment efficiency 0.002 0.0026 0.0033

PAGE 156

156 Table B 9 ic strength conditions (IS=0.005 M) for a given colloid (dp=0.1m) IS=0. 00 5 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 5.00E 03 5.00E 03 5.00E 03 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 2.32E+08 2.32E+08 2.32E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 6.30E 02 6.30E 02 6.30E 02 Z eta potentials of collector C 5.20E 02 5.20E 02 5.20E 02 S urface potentials of particles P 0.070934 0.070934 0.070934 S urface potentials of collector C 0.058403 0.058403 0.058403 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+05 2.29E+05 2.29E+05 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 7.08E 01 7.08E 01 7.08E 01 First electrokinetic parameter N E1 2.46E+01 2.46E+01 2.46E+01 Second electrokinetic parameter N E2 9.82E 01 9.82E 01 9.82E 01 Third electrokinetic parameter N E3 3.76E+14 3.76E+14 3.76E+14 Double layer force parameter N DL 2.32E+01 2.32E+01 2.32E+01 Attachment efficiency 0.0048 0.004 0.0035

PAGE 157

157 Table B 10 strength conditions (IS=0.01M) for a given colloid (dp=0.1m) IS=0. 01 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 1.00E 02 1.00E 02 1.00E 02 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 3.28E+08 3.28E+08 3.28E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 5.90E 02 5.90E 02 5.90E 02 Z eta potentials of collector C 5.00E 02 5.00E 02 5.00E 02 S urface potentials of particles P 0.069704 0.069704 0.069704 S urface potentials of collector C 0.058924 0.058924 0.058924 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+05 2.29E+05 2.29E+05 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 7.08E 01 7.08E 01 7.08E 01 First electrokinetic parameter N E1 2.21E+01 2.21E+01 2.21E+01 Second electrokinetic parameter N E2 9.86E 01 9.86E 01 9.86E 01 Third electrokinetic parameter N E3 7.53E+14 7.53E+14 7.53E+14 Double layer force parameter N DL 3.28E+01 3.28E+01 3.28E+01 Attachment efficiency 0.0092 0.01 0.017

PAGE 158

158 Table B 11. strength conditions (IS=0.05M) for a given colloid (dp=0.1m) IS=0. 05 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 5.00E 02 5.00E 02 5.00E 02 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 7.34E+08 7.34E+08 7.34E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 4.00E 02 4.00E 02 4.00E 02 Z eta potentials of collector C 3.20E 02 3.20E 02 3.20E 02 S urface potentials of particles P 0.057892 0.057892 0.057892 S urface potentials of collector C 0.046198 0.046198 0.046198 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+05 2.29E+05 2.29E+05 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 7.08E 01 7.08E 01 7.08E 01 First electrokinetic parameter N E1 9.68E+00 9.68E+00 9.68E+00 Second electrokinetic parameter N E2 9.76E 01 9.76E 01 9.76E 01 Third electrokinetic parameter N E3 3.76E+15 3.76E+15 3.76E+15 Double layer force parameter N DL 7.34E+01 7.34E+01 7.34E+01 Attachment efficiency 0.05 0.043 0.059

PAGE 159

159 Table B 12 Experimental strength conditions (IS=0.1M) for a given colloid (dp=0.1m) IS=0. 1 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 04 2.00E 04 2.00E 04 Ionic strength IS 1.00E 01 1.00E 01 1.00E 01 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+09 1.04E+09 1.04E+09 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 3.50E 02 3.50E 02 3.50E 02 Z eta potentials of collector C 1.80E 02 1.80E 02 1.80E 02 S urface potentials of particles P 0.058977 0.058977 0.058977 S urface potentials of collector C 0.030255 0.030255 0.030255 Reynolds number N Re 1.05E+00 1.05E+00 1.05E+00 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+05 2.29E+05 2.29E+05 Froude number N Fr 8.15E 07 8.15E 07 8.15E 07 London number N LO 7.08E 01 7.08E 01 7.08E 01 First electrokinetic parameter N E1 5.71E+00 5.71E+00 5.71E+00 Second electrokinetic parameter N E2 8.13E 01 8.13E 01 8.13E 01 Third electrokinetic parameter N E3 7.53E+15 7.53E+15 7.53E+15 Double layer force parameter N DL 1.04E+02 1.04E+02 1.04E+02 Attachment efficiency 0.096 0.136 0.14

PAGE 160

160 Table B 13 velocities and ionic strength conditions (u=0.0002cm/s and IS=0.01M) for a given colloid (dp=1.05m) U=0.00 0 2cm/s & IS=0. 0 1 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 06 2.00E 06 2.00E 06 Ionic strength IS 1.00E 02 1.00E 02 1.00E 02 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 3.28E+08 3.28E+08 3.28E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 6.00E 02 6.00E 02 6.00E 02 Z eta potentials of collector C 5.00E 02 5.00E 02 5.00E 02 S urface potentials of particles P 0.070725 0.070725 0.070725 S urface potentials of collector C 0.058924 0.058924 0.058924 Reynolds number N Re 1.05E 02 1.05E 02 1.05E 02 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+04 2.41E+04 2.41E+04 Froude number N Fr 8.15E 11 8.15E 11 8.15E 11 London number N LO 6.42E 01 6.42E 01 6.42E 01 First electrokinetic parameter N E1 2.14E+02 2.14E+02 2.14E+02 Second electrokinetic parameter N E2 9.84E 01 9.84E 01 9.84E 01 Third electrokinetic parameter N E3 7.53E+14 7.53E+14 7.53E+14 Double layer force parameter N DL 3.45E+02 3.45E+02 3.45E+02 Attachment efficiency 0.025 0.018 0.033

PAGE 161

161 Table B 14 velocities and ionic strength conditions (u=0.002cm/s and IS=0.01M) for a given colloid (dp=1.05m) U=0.002cm/s & IS=0. 01 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 05 2.00E 05 2.00E 05 Ionic strength IS 1.00E 02 1.00E 02 1.00E 02 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 3.28E+08 3.28E+08 3.28E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 6.00E 02 6.00E 02 6.00E 02 Z eta potentials of collector C 5.00E 02 5.00E 02 5.00E 02 S urface potentials of particles P 0.070725 0.070725 0.070725 S urface potentials of collector C 0.058924 0.058924 0.058924 Reynolds number N Re 1.05E 01 1.05E 01 1.05E 01 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+05 2.41E+05 2.41E+05 Froude number N Fr 8.15E 09 8.15E 09 8.15E 09 London number N LO 6.42E 02 6.42E 02 6.42E 02 First electrokinetic parameter N E1 2.14E+01 2.14E+01 2.14E+01 Second electrokinetic parameter N E2 9.84E 01 9.84E 01 9.84E 01 Third electrokinetic parameter N E3 7.53E+14 7.53E+14 7.53E+14 Double layer force parameter N DL 3.45E+02 3.45E+02 3.45E+02 Attachment efficiency 0.02 0.012 0.024

PAGE 162

162 Table B 15 velocities and ionic strength conditions (u=0.2cm/s and IS=0.01M) for a given colloid (dp=1.05m) U=0.2cm/s & IS=0. 01 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 03 2.00E 03 2.00E 03 Ionic strength IS 1.00E 02 1.00E 02 1.00E 02 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 3.28E+08 3.28E+08 3.28E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 6.00E 02 6.00E 02 6.00E 02 Z eta potentials of collector C 5.00E 02 5.00E 02 5.00E 02 S urface potentials of particles P 0.070725 0.070725 0.070725 S urface potentials of collector C 0.058924 0.058924 0.058924 Reynolds number N Re 1.05E+01 1.05E+01 1.05E+01 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+07 2.41E+07 2.41E+07 Froude number N Fr 8.15E 05 8.15E 05 8.15E 05 London number N LO 6.42E 04 6.42E 04 6.42E 04 First electrokinetic parameter N E1 2.14E 01 2.14E 01 2.14E 01 Second electrokinetic parameter N E2 9.84E 01 9.84E 01 9.84E 01 Third electrokinetic parameter N E3 7.53E+14 7.53E+14 7.53E+14 Double layer force parameter N DL 3.45E+02 3.45E+02 3.45E+02 Attachment efficiency 0.008 0.009 0.0099

PAGE 163

163 Table B 16 velocities and ionic strength conditions (u=0.0002cm/s and IS=0.1M) for a given colloid (dp=1.05m) U=0.0002cm/s & IS=0. 1 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 06 2.00E 06 2.00E 06 Ionic strength IS 1.00E 01 1.00E 01 1.00E 01 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+09 1.04E+09 1.04E+09 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 3.80E 02 3.80E 02 3.80E 02 Z eta potentials of collector C 1.80E 02 1.80E 02 1.80E 02 S urface potentials of particles P 0.063888 0.063888 0.063888 S urface potentials of collector C 0.030255 0.030255 0.030255 Reynolds number N Re 1.05E 02 1.05E 02 1.05E 02 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+04 2.41E+04 2.41E+04 Froude number N Fr 8.15E 11 8.15E 11 8.15E 11 London number N LO 6.42E 01 6.42E 01 6.42E 01 First electrokinetic parameter N E1 6.21E+01 6.21E+01 6.21E+01 Second electrokinetic parameter N E2 7.74E 01 7.74E 01 7.74E 01 Third electrokinetic parameter N E3 7.53E+15 7.53E+15 7.53E+15 Double layer force parameter N DL 1.09E+03 1.09E+03 1.09E+03 Attachment efficiency 0.23 0.1889 0.237

PAGE 164

164 Table B 17 velocities and ionic strength conditions (u=0.002cm/s and IS=0.1M) for a given colloid (dp=1.05m) U=0.0 02cm/s & IS=0. 1 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 05 2.00E 05 2.00E 05 Ionic strength IS 1.00E 01 1.00E 01 1.00E 01 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+09 1.04E+09 1.04E+09 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 3.80E 02 3.80E 02 3.80E 02 Z eta potentials of collector C 1.80E 02 1.80E 02 1.80E 02 S urface potentials of particles P 0.063888 0.063888 0.063888 S urface potentials of collector C 0.030255 0.030255 0.030255 Reynolds number N Re 1.05E 01 1.05E 01 1.05E 01 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+05 2.41E+05 2.41E+05 Froude number N Fr 8.15E 09 8.15E 09 8.15E 09 London number N LO 6.42E 02 6.42E 02 6.42E 02 First electrokinetic parameter N E1 6.21E+00 6.21E+00 6.21E+00 Second electrokinetic parameter N E2 7.74E 01 7.74E 01 7.74E 01 Third electrokinetic parameter N E3 7.53E+15 7.53E+15 7.53E+15 Double layer force parameter N DL 1.09E+03 1.09E+03 1.09E+03 Attachment efficiency 0.164 0.144 0.16

PAGE 165

165 Table B 18 velocities and ionic strength conditions (u=0.2cm/s and IS=0.1M) for a given colloid (dp=1.05m) U=0.2cm/s & IS=0. 1 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 03 2.00E 03 2.00E 03 Ionic strength IS 1.00E 01 1.00E 01 1.00E 01 P article diameter d p 1.05E 06 1.05E 06 1.05E 06 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+09 1.04E+09 1.04E+09 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.16E 13 4.16E 13 4.16E 13 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 3.80E 02 3.80E 02 3.80E 02 Z eta potentials of collector C 1.80E 02 1.80E 02 1.80E 02 S urface potentials of particles P 0.063888 0.063888 0.063888 S urface potentials of collector C 0.030255 0.030255 0.030255 Reynolds number N Re 1.05E+01 1.05E+01 1.05E+01 A spect ratio N R 2.10E 04 2.10E 04 2.10E 04 Peclet number N pe 2.41E+07 2.41E+07 2.41E+07 Froude number N Fr 8.15E 05 8.15E 05 8.15E 05 London number N LO 6.42E 04 6.42E 04 6.42E 04 First electrokinetic parameter N E1 6.21E 02 6.21E 02 6.21E 02 Second electrokinetic parameter N E2 7.74E 01 7.74E 01 7.74E 01 Third electrokinetic parameter N E3 7.53E+15 7.53E+15 7.53E+15 Double layer force parameter N DL 1.09E+03 1.09E+03 1.09E+03 Attachment efficiency 0.05 0.043 0.059

PAGE 166

166 Table B 19 Experimental data of attachment efficiency velocities and ionic strength conditions (u=0.0002cm/s and IS=0.01M) for a given colloid (dp=0.1m) U=0.0002cm/s & IS=0. 01 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 06 2.00E 06 2.00E 06 Ionic strength IS 1.00E 02 1.00E 02 1.00E 02 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 3.28E+08 3.28E+08 3.28E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 5.90E 02 5.90E 02 5.90E 02 Z eta potentials of collector C 5.00E 02 5.00E 02 5.00E 02 S urface potentials of particles P 0.069704 0.069704 0.069704 S urface potentials of collector C 0.058924 0.058924 0.058924 Reynolds number N Re 1.05E 02 1.05E 02 1.05E 02 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+03 2.29E+03 2.29E+03 Froude number N Fr 8.15E 11 8.15E 11 8.15E 11 London number N LO 7.08E+01 7.08E+01 7.08E+01 First electrokinetic parameter N E1 2.21E+03 2.21E+03 2.21E+03 Second electrokinetic parameter N E2 9.86E 01 9.86E 01 9.86E 01 Third electrokinetic parameter N E3 7.53E+14 7.53E+14 7.53E+14 Double layer force parameter N DL 3.28E+01 3.28E+01 3.28E+01 Attachment efficiency 0.03 0.023 0.018

PAGE 167

167 Table B 20 velocities and ionic strength conditions (u=0.002cm/s and IS=0.01M) for a given colloid (dp=0.1m) U=0.002cm/s & IS=0. 01 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 05 2.00E 05 2.00E 05 Ionic strength IS 1.00E 02 1.00E 02 1.00E 02 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 3.28E+08 3.28E+08 3.28E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 5.90E 02 5.90E 02 5.90E 02 Z eta potentials of collector C 5.00E 02 5.00E 02 5.00E 02 S urface potentials of particles P 0.069704 0.069704 0.069704 S urface potentials of collector C 0.058924 0.058924 0.058924 Reynolds number N Re 1.05E 01 1.05E 01 1.05E 01 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+04 2.29E+04 2.29E+04 Froude number N Fr 8.15E 09 8.15E 09 8.15E 09 London number N LO 7.08E+00 7.08E+00 7.08E+00 First electrokinetic parameter N E1 2.21E+02 2.21E+02 2.21E+02 Second electrokinetic parameter N E2 9.86E 01 9.86E 01 9.86E 01 Third electrokinetic parameter N E3 7.53E+14 7.53E+14 7.53E+14 Double layer force parameter N DL 3.28E+01 3.28E+01 3.28E+01 Attachment efficiency 0.009 0.015 0.019

PAGE 168

168 Table B 21. velocities and ionic strength conditions (u=0.2cm/s and IS=0.01M) for a given colloid (dp=0.1m) U=0.2cm/s & IS=0. 01 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 03 2.00E 03 2.00E 03 Ionic strength IS 1.00E 02 1.00E 02 1.00E 02 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 3.28E+08 3.28E+08 3.28E+08 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 5.90E 02 5.90E 02 5.90E 02 Z eta potentials of collector C 5.00E 02 5.00E 02 5.00E 02 S urface potentials of particles P 0.069704 0.069704 0.069704 S urface potentials of collector C 0.058924 0.058924 0.058924 Reynolds number N Re 1.05E+01 1.05E+01 1.05E+01 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+06 2.29E+06 2.29E+06 Froude number N Fr 8.15E 05 8.15E 05 8.15E 05 London number N LO 7.08E 02 7.08E 02 7.08E 02 First electrokinetic parameter N E1 2.21E+00 2.21E+00 2.21E+00 Second electrokinetic parameter N E2 9.86E 01 9.86E 01 9.86E 01 Third electrokinetic parameter N E3 7.53E+14 7.53E+14 7.53E+14 Double layer force parameter N DL 3.28E+01 3.28E+01 3.28E+01 Attachment efficiency 0.0032 0.002 0.0018

PAGE 169

169 Table B 22. velocities and ionic strength conditions (u=0. 0002cm/s and IS=0. 1M) for a given colloid (dp=0.1m) U=0.0002cm/s & IS=0. 1 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 06 2.00E 06 2.00E 06 Ionic strength IS 1.00E 01 1.00E 01 1.00E 01 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+09 1.04E+09 1.04E+09 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 3.50E 02 3.50E 02 3.50E 02 Z eta potentials of collector C 1.80E 02 1.80E 02 1.80E 02 S urface potentials of particles P 0.058977 0.058977 0.058977 S urface potentials of collector C 0.030255 0.030255 0.030255 Reynolds number N Re 1.05E 02 1.05E 02 1.05E 02 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+03 2.29E+03 2.29E+03 Froude number N Fr 8.15E 11 8.15E 11 8.15E 11 London number N LO 7.08E+01 7.08E+01 7.08E+01 First electrokinetic parameter N E1 5.71E+02 5.71E+02 5.71E+02 Second electrokinetic parameter N E2 8.13E 01 8.13E 01 8.13E 01 Third electrokinetic parameter N E3 7.53E+15 7.53E+15 7.53E+15 Double layer force parameter N DL 1.04E+02 1.04E+02 1.04E+02 Attachment efficiency 0.13 0.11 0.18

PAGE 170

170 Table B 23. velocities and ionic strength conditions (u=0.002cm/s and IS=0. 1M) for a given colloid (dp=0.1m) U=0.0 02cm/s & IS=0. 1 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 05 2.00E 05 2.00E 05 Ionic strength IS 1.00E 01 1.00E 01 1.00E 01 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+09 1.04E+09 1.04E+09 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 3.50E 02 3.50E 02 3.50E 02 Z eta potentials of collector C 1.80E 02 1.80E 02 1.80E 02 S urface potentials of particles P 0.058977 0.058977 0.058977 S urface potentials of collector C 0.030255 0.030255 0.030255 Reynolds number N Re 1.05E 01 1.05E 01 1.05E 01 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+04 2.29E+04 2.29E+04 Froude number N Fr 8.15E 09 8.15E 09 8.15E 09 London number N LO 7.08E+00 7.08E+00 7.08E+00 First electrokinetic parameter N E1 5.71E+01 5.71E+01 5.71E+01 Second electrokinetic parameter N E2 8.13E 01 8.13E 01 8.13E 01 Third electrokinetic parameter N E3 7.53E+15 7.53E+15 7.53E+15 Double layer force parameter N DL 1.04E+02 1.04E+02 1.04E+02 Attachment efficiency 0.08 0.089 0.096

PAGE 171

171 Table B 24. velocities and ionic strength conditions (u=0.2cm/s and IS=0. 1M) for a given colloid (dp=0.1m) U=0.2cm/s & IS=0. 1 M Parameters Replicate 1 Replicate 2 Replicate 3 Approaching velocity u 2.00E 03 2.00E 03 2.00E 03 Ionic strength IS 1.00E 01 1.00E 01 1.00E 01 P article diameter d p 1.00E 07 1.00E 07 1.00E 07 C ollector diameter d c 5.00E 03 5.00E 03 5.00E 03 Hamaker constant A 1.00E 20 1.00E 20 1.00E 20 Boltzman's constant K B 1.38E 23 1.38E 23 1.38E 23 A bsolute temperature T 2.98E+02 2.98E+02 2.98E+02 R eciprocal of double layer thickness 1.04E+09 1.04E+09 1.04E+09 Avogadro's number N A 6.02E+23 6.02E+23 6.02E+23 Elementary charge e 1.60E 19 1.60E 19 1.60E 19 D ensity of suspension 1.05E+03 1.05E+03 1.05E+03 F luid viscosity 1.00E 03 1.00E 03 1.00E 03 Brownian diffusivity D 4.37E 12 4.37E 12 4.37E 12 R elative permittivity of fluid media 7.85E+01 7.85E+01 7.85E+01 P ermittivity in vacuum 0 8.85E 12 8.85E 12 8.85E 12 Z eta potentials of particles P 3.50E 02 3.50E 02 3.50E 02 Z eta potentials of collector C 1.80E 02 1.80E 02 1.80E 02 S urface potentials of particles P 0.058977 0.058977 0.058977 S urface potentials of collector C 0.030255 0.030255 0.030255 Reynolds number N Re 1.05E+01 1.05E+01 1.05E+01 A spect ratio N R 2.00E 05 2.00E 05 2.00E 05 Peclet number N pe 2.29E+06 2.29E+06 2.29E+06 Froude number N Fr 8.15E 05 8.15E 05 8.15E 05 London number N LO 7.08E 02 7.08E 02 7.08E 02 First electrokinetic parameter N E1 5.71E 01 5.71E 01 5.71E 01 Second electrokinetic parameter N E2 8.13E 01 8.13E 01 8.13E 01 Third electrokinetic parameter N E3 7.53E+15 7.53E+15 7.53E+15 Double layer force parameter N DL 1.04E+02 1.04E+02 1.04E+02 Attachment efficiency 0.013 0.0099 0.0087

PAGE 172

172 APPENDIX SU PPORTING INFORMATION FOR CHAP TER 4 Buckingham approach to develop ( N STE ) The first step of dimensional analysis using the Buckingham theorem is listing the relevant material parameters, process related parameters, and universal physical constants. For the case of the interaction between colloid and plant stem surface, re levant parameters are summarized in Table S1. Table C 1. Relevant parameters and constants for interaction between colloid and plant stem surface No. Symbol Description Dimension Type Phenomena 1 H Hamaker constant ML 2 T 2 Material parameter Van der Waals attraction 2 Debye parameter L 1 Material parameter Electrostatic double layer repulsion 3 Permittivity of vacuum T 4 I 2 M 1 L 3 Material parameter Electrostatic double layer repulsion 4 Dielectric constant Material parameter Electrostatic double layer repulsion 5 Surface potential of colloid L 2 MT 3 I Material parameter Electrostatic double layer repulsion 6 Surface potential of collector L 2 MT 3 I Material parameter Electrostatic double layer repulsion 7 d p Colloid diameter L Material parameter Multiple phenomena 8 u/f Porewater velocity LT 1 Process parameter Transport, shear, and detachment 9 Viscosity ML 1 T 1 Constant Multiple phenomena 10 M W Molecular weight of polymer M/mole Material parameter Steric repulsion 11 N A number Mole 1 Constant Steric repulsion 12 L 0 Height of brush layer L Material parameter Steric repulsion 13 Brush density L 2 Material parameter Steric repulsion Based on the traditional DLVO theory, van der Waals attraction and electrostatic double layer repulsion are controlled by parameters 1 to 9. In the previous studies, four

PAGE 173

173 dimensionless numbers such as N LO N E1 N E2 and N DL have been developed to represent these two DLVO interactions. According to grafted polymer brush layer theory, steric repulsion afforded by biopolymer brush layer and decrease of friction due to biopolymer layer are mainly governed by parameters 7 to 13, which are used to develop a new dimensionless number: N STE The derivation of this dimensionless number can be found as follows: L 0 M w u /f Na d p L 1 0 1 0 1 1 2 M 0 1 0 0 1 0 0 T 0 0 1 0 1 0 0 Mole 0 1 0 1 0 0 0 After several matrix operations to transform the core matrix to unity matrix, the final matrices are: L 0 M w u /f Na d p Z 1 =L+T 1 0 0 0 2 1 2 Z 2 =M 0 1 0 0 1 0 0 Z 3 = T 0 0 1 0 1 0 0 Z 4 =M+Mole 0 0 0 1 1 0 0 Therefore, Core matrix Residual matrix Unity matrix Residual matrix

PAGE 174

174 Breakthrough curves of both colloid and Br transport with and without plaster Prior to the runoff experiment, plaster was used to seal the top sand surface to prevent infiltration and to eliminate the filtration of colloids by the soil. Pre experiment with the flow chambers under the same treatments but without of dense vegetation showed that more than 98% bromide and colloids were r ecovered from the system, indicating no deposition of colloids on the plaster layer and contribution of infiltration and exchange are insignificant under tested conditions. Figure C 1 Breakthrough curves with and without plaster. Global sensitivity analysis results

PAGE 175

175 To get a qualitative sensitivity analysis o f model output Morris methods was used. According to this method, only parameters separated from the origin of the plane ar e considered important. Figure C 2 showed the g raphical representation of the Morris results for model output. The qualitative ranking of factors is as follows: (1) Flow velocity ( u ), (2) vegetation density (1 f ), (3 ) ionic strength ( IS ), (4 ) diameter of collector ( d c ), ( 5 ) diameter of colloid ( d p ), ( 6) zeta potential of colloid ( ) and collector ( ). Figure C 2. Morris sensitivity analysis result chart Flow velocity ( u ), vegetation density (1 f ) and ionic strength ( IS ) are shown away rough first order effects but also interaction component. The diameter of collector size ( d c ) and particle size ( d p ) are two input factors are mostly through first order effect with small in teraction components. The rest input factors ( and ) are very close to origin which indicated they are insensitive to the model output.

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176 To get the quantitative sensitivity analysis of model output, Sobol method was also tested. Base on the qualitative ranking of t he input factors, the first five input factors were chosen to test. The first order and total sensitivity indices of the Sobo l analysis calculated with 24576 samples were shown in Figure C 3. Figure C 3. Sobol sensitivity analysis indices chart The results of first order indices indicated that flow velocity (the first significant factor) has a larger first order index than the vegetation density (1 f ) which is the second significant factor. However, the total sensitivity indices of flow velocity and vegetation density are pretty close, which means vegetation density has a large effect by

PAGE 177

177 interactions with other parameters. For example, vegetation density has an important effect on the flow filed around the plant stem which controls on not only physical contact process but also physicochemical attachment process. It is also noted that the sum of all first order indices (0.6983) is less than 1, which means the model is non additive, as could be expected. The results showed that all tot al sensitivity indices are higher than the first order sensitivity indices. This is of course theoretically nece ssary, since first order indice is a part of the total order indices. For example, the total order indices of the most influential factor flow velocity (u) ( 0.4061 ) included the first order effect (0. 2265 ) and interactions with other factors. Comparison of experimental deposition rate with predictions of the new dimensionless equation for development dataset Figure C 4. Comparison of experimen tal deposition rate with predictions of the new dimensionless equation for development dataset

PAGE 178

178 Table C 2 E xperimental data of breakthrough curve under DI water conditions DI water Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.0277 0.00 0.0163 2 0.01 0.01 0.0092 4 0.1789 0.12 0.1519 6 0.29 0.23 0.2621 8 0.4 0.35 0.3769 10 0.55 0.50 0.5251 12 0.69 0.64 0.6649 14 0.79 0.74 0.7666 16 0.868 0.79 0.8309 18 0.88 0.82 0.8488 20 0.92 0.84 0.8801 22 0.948 0.87 0.9114 24 0.95 0.88 0.9134 26 0.945 0.89 0.916 28 0.947 0.89 0.9197 30 0.944 0.89 0.9156 32 0.947 0.89 0.9161 34 0.942 0.90 0.9187 36 0.862 0.80 0.8292 38 0.68 0.63 0.6552 40 0.42 0.35 0.3836 42 0.263 0.23 0.248 44 0.19 0.12 0.1565 46 0.15 0.09 0.1221 48 0.1 0.06 0.0813 50 0.072 0.05 0.0599 52 0.06 0.03 0.0437 54 0.068 0.02 0.0417 56 0.039 0.02 0.0274 58 0.05 0.00 0.026 60 0.032 0.01 0.0197

PAGE 179

179 Table C 3 E xperimental data of breakthrough curve under medium IS conditions (IS=0.01M) IS=0.01M Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.0287 0.00 0.0137 2 0.0321 0.00 0.0116 4 0.078 0.03 0.0537 6 0.254 0.16 0.2058 8 0.43 0.38 0.4048 10 0.66 0.56 0.6123 12 0.74 0.65 0.6947 14 0.77 0.74 0.7547 16 0.82 0.80 0.8084 18 0.86 0.82 0.8421 20 0.9 0.84 0.8713 22 0.899 0.85 0.8744 24 0.91 0.87 0.8918 26 0.91 0.87 0.8915 28 0.92 0.86 0.8913 30 0.92 0.86 0.8923 32 0.92 0.88 0.901 34 0.9 0.88 0.8908 36 0.82 0.78 0.8023 38 0.68 0.64 0.6591 40 0.54 0.45 0.4943 42 0.34 0.27 0.3035 44 0.23 0.17 0.1993 46 0.13 0.10 0.1166 48 0.09 0.06 0.0771 50 0.08 0.05 0.0642 52 0.06 0.02 0.0419 54 0.0487 0.01 0.0297 56 0.045 0.01 0.0253 58 0.042 0.01 0.0261 60 0.032 0.01 0.0195

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180 Table C 4 E xperimental data of breakthrough curve under high IS conditions (IS=0.1M) IS=0.1M Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.009 0.01 0.0074 2 0.02 0.00 0.0109 4 0.04 0.01 0.0244 6 0.21 0.15 0.1813 8 0.39 0.37 0.3788 10 0.6 0.55 0.5757 12 0.69 0.66 0.676 14 0.76 0.74 0.7478 16 0.8 0.75 0.7766 18 0.84 0.79 0.8137 20 0.83 0.80 0.8147 22 0.85 0.80 0.8272 24 0.87 0.79 0.8322 26 0.87 0.81 0.8415 28 0.87 0.83 0.852 30 0.9 0.82 0.8597 32 0.89 0.83 0.8591 34 0.87 0.83 0.8491 36 0.84 0.78 0.809 38 0.67 0.64 0.6541 40 0.5 0.46 0.4815 42 0.34 0.29 0.3131 44 0.21 0.17 0.1906 46 0.14 0.10 0.1186 48 0.09 0.07 0.0797 50 0.08 0.03 0.054 52 0.06 0.00 0.0312 54 0.04 0.01 0.0251 56 0.05 0.00 0.0209 58 0.04 0.00 0.0199 60 0.05 0.00 0.0199

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181 Table C 5 E xperimental data of breakthrough curve under high IS conditions (IS=0.2M) IS=0.2M Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.019 0.00 0.0116 2 0.022 0.01 0.0177 4 0.026 0.01 0.0199 6 0.2 0.13 0.1633 8 0.36 0.30 0.3323 10 0.51 0.47 0.4915 12 0.66 0.60 0.6302 14 0.78 0.73 0.7558 16 0.79 0.75 0.768 18 0.82 0.73 0.7763 20 0.83 0.74 0.7862 22 0.83 0.76 0.7966 24 0.83 0.79 0.8093 26 0.85 0.79 0.8193 28 0.84 0.79 0.8154 30 0.83 0.80 0.8154 32 0.84 0.79 0.8154 34 0.83 0.78 0.8054 36 0.79 0.75 0.77 38 0.62 0.59 0.6032 40 0.46 0.42 0.4388 42 0.28 0.28 0.2782 44 0.24 0.17 0.204 46 0.17 0.13 0.1515 48 0.14 0.08 0.1106 50 0.09 0.05 0.0708 52 0.08 0.05 0.0641 54 0.08 0.03 0.0536 56 0.07 0.02 0.047 58 0.06 0.01 0.0348 60 0.06 0.00 0.0243

PAGE 182

182 Table C 6 E xperimental data of breakthrough curve under plant high density conditions (u=0.002 cm/s) u=0.002cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.0004 0.00 0.0002 2 0.005 0.00 0.003 4 0.0088 0.00 0.0056 6 0.0097 0.00 0.0058 8 0.025 0.00 0.0133 10 0.032 0.00 0.0172 12 0.036 0.00 0.0181 14 0.08 0.00 0.0416 16 0.14 0.05 0.0947 18 0.24 0.13 0.1856 20 0.33 0.24 0.284 22 0.43 0.32 0.3736 24 0.5 0.41 0.4566 26 0.54 0.46 0.499 28 0.57 0.51 0.5394 30 0.64 0.53 0.5873 32 0.64 0.56 0.5999 34 0.65 0.58 0.6139 36 0.68 0.56 0.6185 38 0.66 0.58 0.6208 40 0.67 0.58 0.6232 42 0.68 0.57 0.6255 44 0.67 0.58 0.6229 46 0.66 0.54 0.5999 48 0.57 0.51 0.5385 50 0.51 0.43 0.4694 52 0.42 0.36 0.3899 54 0.36 0.26 0.3089 56 0.31 0.21 0.2615 58 0.26 0.17 0.2135 60 0.23 0.14 0.1873 62 0.2 0.12 0.1582 64 0.18 0.08 0.1319 66 0.14 0.09 0.1161 68 0.15 0.07 0.1112 70 0.12 0.07 0.0928 72 0.11 0.05 0.0805 74 0.09 0.04 0.0649 76 0.09 0.03 0.0598 78 0.08 0.03 0.0547 80 0.08 0.00 0.0491

PAGE 183

183 Table C 7 E xperimental data of breakthrough curve under plant high density conditions (u=0.01cm/s) u=0.01cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.0098 0.00 0.0073 2 0.01 0.01 0.0083 4 0.02 0.02 0.0179 6 0.02 0.02 0.0184 8 0.08 0.03 0.0537 10 0.2 0.17 0.1842 12 0.38 0.33 0.3542 14 0.49 0.45 0.4708 16 0.59 0.55 0.5691 18 0.69 0.61 0.6509 20 0.7 0.66 0.6805 22 0.73 0.68 0.7028 24 0.74 0.70 0.7217 26 0.74 0.72 0.7298 28 0.77 0.72 0.7467 30 0.78 0.73 0.7543 32 0.76 0.75 0.7543 34 0.78 0.73 0.7543 36 0.77 0.74 0.7572 38 0.77 0.71 0.7398 40 0.65 0.59 0.6186 42 0.49 0.43 0.4615 44 0.38 0.30 0.3388 46 0.29 0.22 0.2543 48 0.22 0.14 0.1803 50 0.16 0.11 0.1365 52 0.11 0.08 0.0958 54 0.1 0.05 0.0732 56 0.08 0.03 0.0566 58 0.06 0.03 0.0453 60 0.06 0.02 0.038 62 0.05 0.02 0.0331 64 0.04 0.03 0.0328 66 0.03 0.03 0.0277 68 0.028 0.02 0.025 70 0.03 0.01 0.0218

PAGE 184

184 Table C 8 E xperimental data of breakthrough curve under plant high density conditions (u=0.05 cm/s) u=0.05cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.009 0.01 0.0074 2 0.012 0.01 0.0109 4 0.03 0.02 0.0244 6 0.209 0.15 0.1813 8 0.42 0.34 0.3788 10 0.61 0.54 0.5757 12 0.69 0.66 0.676 14 0.77 0.73 0.7478 16 0.79 0.76 0.7766 18 0.83 0.80 0.8137 20 0.84 0.79 0.8147 22 0.84 0.81 0.8272 24 0.85 0.81 0.8322 26 0.86 0.82 0.8415 28 0.87 0.83 0.852 30 0.88 0.84 0.8597 32 0.87 0.83 0.8491 34 0.84 0.80 0.819 36 0.69 0.62 0.6541 38 0.52 0.44 0.4815 40 0.33 0.30 0.3131 42 0.22 0.16 0.1906 44 0.13 0.11 0.1186 46 0.11 0.05 0.0797 48 0.06 0.05 0.054 50 0.05 0.01 0.0312 52 0.04 0.01 0.0251 54 0.04 0.00 0.0209 56 0.03 0.01 0.0199 58 0.03 0.01 0.0177 60 0.02 0.01 0.013

PAGE 185

185 Table C 9 E xperimental data of breakthrough curve under plant high density conditions (u=0.1 cm/s) u=0.1cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.03 0.02 0.0231 2 0.04 0.01 0.0263 4 0.16 0.10 0.1284 6 0.45 0.36 0.405 8 0.69 0.61 0.6499 10 0.8 0.72 0.7624 12 0.85 0.82 0.8365 14 0.87 0.82 0.8446 16 0.88 0.83 0.857 18 0.9 0.83 0.864 20 0.89 0.83 0.8609 22 0.89 0.84 0.8659 24 0.9 0.85 0.8763 26 0.9 0.86 0.8781 28 0.9 0.85 0.876 30 0.88 0.88 0.8809 32 0.9 0.86 0.8815 34 0.81 0.75 0.7796 36 0.56 0.48 0.5191 38 0.29 0.26 0.2771 40 0.15 0.12 0.1357 42 0.09 0.06 0.0738 44 0.08 0.02 0.0489 46 0.06 0.02 0.038 48 0.05 0.00 0.0265 50 0.04 0.00 0.0216 52 0.03 0.01 0.0198 54 0.03 0.01 0.0179 56 0.02 0.01 0.0169 58 0.02 0.01 0.0169 60 0.0163 0.02 0.0161

PAGE 186

186 Table C 10 E xperimental data of breakthrough curve under plant medium density conditions (u=0.002 cm/s) u=0.002cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.005 0.00 0.0033 2 0.005 0.00 0.004 4 0.009 0.01 0.0081 6 0.01 0.01 0.0086 8 0.02 0.00 0.01 10 0.039 0.01 0.0264 12 0.11 0.08 0.0932 14 0.28 0.24 0.2579 16 0.49 0.44 0.4673 18 0.6 0.56 0.5778 20 0.69 0.63 0.6602 22 0.74 0.68 0.7116 24 0.77 0.70 0.736 26 0.79 0.74 0.766 28 0.8 0.75 0.7738 30 0.82 0.78 0.7995 32 0.83 0.77 0.802 34 0.84 0.78 0.8081 36 0.84 0.76 0.8013 38 0.83 0.77 0.8015 40 0.82 0.77 0.7947 42 0.57 0.50 0.535 44 0.44 0.38 0.4098 46 0.34 0.28 0.3103 48 0.26 0.20 0.23 50 0.183 0.18 0.179 52 0.179 0.13 0.1567 54 0.156 0.12 0.1368 56 0.156 0.08 0.12 58 0.132 0.08 0.1065 60 0.099 0.08 0.0877 62 0.0879 0.07 0.0788 64 0.0865 0.06 0.0711 66 0.089 0.05 0.07 68 0.078 0.06 0.0699 70 0.0773 0.06 0.0678 72 0.0821 0.05 0.0666 74 0.0798 0.05 0.0658 76 0.0787 0.05 0.0643 78 0.0754 0.05 0.0621 80 0.0754 0.04 0.06

PAGE 187

187 Table C 11. E xperimental data of breakthrough curve under plant medium density conditions (u=0.01 cm/s) u=0.01cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.002 0.00 0.0012 2 0.021 0.00 0.0117 4 0.03 0.01 0.0219 6 0.08 0.05 0.0635 8 0.139 0.10 0.1177 10 0.37 0.31 0.3422 12 0.57 0.50 0.5346 14 0.69 0.63 0.6618 16 0.79 0.75 0.7717 18 0.81 0.77 0.7898 20 0.84 0.80 0.8178 22 0.84 0.80 0.8195 24 0.85 0.80 0.8238 26 0.86 0.81 0.8366 28 0.86 0.83 0.8449 30 0.87 0.83 0.8476 32 0.88 0.82 0.8478 34 0.87 0.83 0.8476 36 0.79 0.74 0.765 38 0.55 0.46 0.5049 40 0.328 0.29 0.3075 42 0.19 0.16 0.175 44 0.15 0.07 0.1106 46 0.11 0.05 0.0811 48 0.09 0.03 0.0597 50 0.08 0.03 0.0537 52 0.07 0.01 0.0383 54 0.06 0.01 0.0364 56 0.05 0.02 0.0331 58 0.043 0.00 0.0216 60 0.03 0.01 0.0202 62 0.02 0.00 0.0107 64 0.01 0.00 0.0059 66 0.01 0.00 0.0052 68 0.01 0.00 0.005 70 0.006 0.00 0.0036

PAGE 188

188 Table C 12 E xperimental data of breakthrough curve under plant medium density conditions (u=0.05 cm/s) u=0.05cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.017 0.01 0.0133 2 0.059 0.05 0.056 4 0.17 0.10 0.1374 6 0.51 0.46 0.4869 8 0.72 0.65 0.6861 10 0.84 0.78 0.8121 12 0.89 0.79 0.8419 14 0.9 0.86 0.8822 16 0.93 0.86 0.8943 18 0.94 0.85 0.8971 20 0.94 0.86 0.8994 22 0.95 0.85 0.9022 24 0.95 0.86 0.9061 26 0.95 0.87 0.91 28 0.94 0.88 0.9092 30 0.94 0.88 0.9112 32 0.95 0.85 0.9018 34 0.82 0.74 0.7781 36 0.48 0.42 0.4489 38 0.3 0.21 0.2564 40 0.17 0.08 0.1264 42 0.09 0.04 0.0626 44 0.007 0.06 0.0317 46 0.04 0.01 0.0243 48 0.027 0.00 0.0137 50 0.01 0.01 0.0086 52 0.007 0.00 0.0043 54 0.007 0.00 0.0059 56 0.008 0.00 0.0047 58 0.004 0.00 0.002 60 0.001 0.00 0.0008

PAGE 189

189 Table C 13 E xperimental data of breakthrough curve under plant medium density conditions (u=0.1 cm/s) u=0.1cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.004 0.00 0.0025 2 0.02 0.00 0.0112 4 0.36 0.27 0.3165 6 0.73 0.69 0.7096 8 0.86 0.81 0.8372 10 0.92 0.87 0.8973 12 0.93 0.89 0.9087 14 0.93 0.90 0.9161 16 0.95 0.90 0.927 18 0.96 0.90 0.9277 20 0.94 0.92 0.9295 22 0.95 0.90 0.9267 24 0.96 0.89 0.927 26 0.95 0.90 0.924 28 0.96 0.90 0.9297 30 0.94 0.87 0.9052 32 0.69 0.66 0.6749 34 0.2 0.16 0.178 36 0.09 0.06 0.0774 38 0.07 0.01 0.0411 40 0.08 0.02 0.032 42 0.04 0.02 0.0294 44 0.03 0.00 0.0155 46 0.04 0.01 0.0137 48 0.009 0.01 0.0079 50 0.005 0.00 0.0043 52 0.005 0.00 0.0038 54 0.004 0.00 0.0038 56 0.005 0.00 0.0041 58 0.004 0.00 0.0033 60 0.004 0.00 0.0033

PAGE 190

190 Table C 14 E xperimental data of breakthrough curve under plant low density conditions (u=0.002 cm/s) u=0.002cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.005 0.00 0.0034 2 0.009 0.00 0.0062 4 0.016 0.01 0.0122 6 0.019 0.01 0.0156 8 0.019 0.01 0.0159 10 0.0199 0.01 0.0162 12 0.058 0.05 0.0519 14 0.15 0.09 0.1213 16 0.26 0.22 0.2412 18 0.38 0.33 0.3546 20 0.48 0.42 0.4493 22 0.59 0.53 0.5604 24 0.66 0.58 0.6207 26 0.72 0.67 0.6964 28 0.76 0.72 0.7406 30 0.77 0.72 0.7463 32 0.78 0.73 0.7531 34 0.79 0.73 0.7591 36 0.79 0.74 0.7656 38 0.81 0.75 0.7806 40 0.78 0.69 0.7347 42 0.65 0.61 0.6281 44 0.56 0.50 0.5292 46 0.499 0.45 0.4728 48 0.42 0.36 0.3903 50 0.38 0.30 0.3384 52 0.3 0.26 0.28 54 0.25 0.21 0.2316 56 0.21 0.17 0.1908 58 0.2 0.15 0.1755 60 0.17 0.11 0.1414 62 0.15 0.11 0.1295 64 0.14 0.09 0.1165 66 0.13 0.08 0.1037 68 0.1 0.07 0.0856 70 0.09 0.05 0.0697 72 0.09 0.04 0.0663 74 0.08 0.02 0.0507 76 0.06 0.04 0.0508 78 0.07 0.03 0.0502 80 0.05 0.03 0.0414

PAGE 191

191 Table C 15 E xperimental data of breakthrough curve under plant low density conditions (u=0.01 cm/s) u=0.01cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.0098 0.01 0.0083 2 0.019 0.01 0.0169 4 0.058 0.05 0.0534 6 0.083 0.08 0.0794 8 0.15 0.13 0.139 10 0.23 0.17 0.2015 12 0.33 0.28 0.3039 14 0.45 0.39 0.4222 16 0.53 0.50 0.5163 18 0.63 0.59 0.61 20 0.74 0.67 0.7069 22 0.79 0.75 0.7704 24 0.79 0.75 0.7718 26 0.81 0.76 0.7863 28 0.81 0.78 0.7972 30 0.83 0.77 0.8003 32 0.84 0.79 0.8146 34 0.84 0.81 0.8237 36 0.85 0.80 0.826 38 0.83 0.78 0.8029 40 0.73 0.70 0.715 42 0.59 0.53 0.5607 44 0.5 0.46 0.4803 46 0.38 0.31 0.347 48 0.332 0.25 0.293 50 0.25 0.18 0.2165 52 0.21 0.14 0.1745 54 0.17 0.12 0.1452 56 0.14 0.09 0.1149 58 0.12 0.06 0.0902 60 0.08 0.02 0.0485 62 0.06 0.01 0.0371 64 0.05 0.02 0.034 66 0.04 0.02 0.0288 68 0.05 0.00 0.0254 70 0.03 0.01 0.0223

PAGE 192

192 Table C 16 E xperimental data of breakthrough curve under plan t low density conditions (u=0.05 cm/s) u=0.05cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.0176 0.01 0.0141 2 0.026 0.02 0.022 4 0.047 0.04 0.0448 6 0.16 0.15 0.1552 8 0.43 0.37 0.3986 10 0.6 0.55 0.5725 12 0.72 0.70 0.7093 14 0.83 0.79 0.8117 16 0.86 0.82 0.8422 18 0.9 0.87 0.8829 20 0.91 0.87 0.8906 22 0.91 0.87 0.8906 24 0.91 0.90 0.9057 26 0.92 0.90 0.9075 28 0.93 0.89 0.9106 30 0.93 0.91 0.9183 32 0.93 0.91 0.9191 34 0.92 0.89 0.9034 36 0.78 0.73 0.7528 38 0.54 0.46 0.5005 40 0.37 0.30 0.3345 42 0.26 0.20 0.2316 44 0.2 0.15 0.1747 46 0.14 0.09 0.116 48 0.1 0.07 0.0873 50 0.09 0.04 0.0674 52 0.08 0.02 0.0517 54 0.06 0.03 0.0443 56 0.06 0.01 0.0341 58 0.05 0.01 0.031 60 0.04 0.01 0.0272

PAGE 193

193 Table C 17 E xperimental data of breakthrough curve under plant low density conditions (u=0.1 cm/s) u=0.1cm/s Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.04 0.02 0.0303 2 0.059 0.05 0.0563 4 0.197 0.17 0.1851 6 0.49 0.43 0.4621 8 0.68 0.59 0.6333 10 0.76 0.71 0.7343 12 0.85 0.77 0.8081 14 0.9 0.82 0.8621 16 0.9 0.85 0.8727 18 0.92 0.88 0.8992 20 0.93 0.89 0.9119 22 0.94 0.89 0.9157 24 0.95 0.89 0.9194 26 0.97 0.87 0.9222 28 0.96 0.89 0.927 30 0.97 0.89 0.9313 32 0.96 0.91 0.9343 34 0.89 0.81 0.8475 36 0.57 0.47 0.5199 38 0.35 0.29 0.3207 40 0.23 0.15 0.1924 42 0.15 0.13 0.1394 44 0.11 0.07 0.0907 46 0.06 0.05 0.0525 48 0.05 0.02 0.0341 50 0.03 0.01 0.0189 52 0.02 0.00 0.0111 54 0.02 0.00 0.0101 56 0.016 0.00 0.0088 58 0.01 0.01 0.0083 60 0.02 0.00 0.0109

PAGE 194

194 Table C 18 E xperimental data of breakthrough curve under different siz es of colloid conditions (d p =0.1 m) d p =0.1m Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.02 0.003 0.0116 2 0.023 0.012 0.0177 4 0.024 0.016 0.0199 6 0.172 0.155 0.1633 8 0.374 0.291 0.3323 10 0.521 0.462 0.4915 12 0.663 0.597 0.6302 14 0.767 0.745 0.7558 16 0.788 0.748 0.768 18 0.799 0.754 0.7763 20 0.802 0.770 0.7862 22 0.832 0.761 0.7966 24 0.834 0.785 0.8093 26 0.844 0.795 0.8193 28 0.846 0.785 0.8154 30 0.849 0.782 0.8154 32 0.84 0.771 0.8054 34 0.78 0.760 0.77 36 0.621 0.585 0.6032 38 0.453 0.425 0.4388 40 0.287 0.269 0.2782 42 0.243 0.165 0.204 44 0.165 0.138 0.1515 46 0.122 0.099 0.1106 48 0.0876 0.054 0.0708 50 0.076 0.052 0.0641 52 0.067 0.040 0.0536 54 0.054 0.040 0.047 56 0.043 0.027 0.0348 58 0.034 0.015 0.0243 60 0.021 0.017 0.0188

PAGE 195

195 Table C 19 E xperimental data of breakthrough curve under different sizes of colloid conditions (d p =1.05 m) d p =1.05m Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.0023 0.002 0.0019 2 0.0044 0.004 0.0042 4 0.0087 0.007 0.0076 6 0.19 0.190 0.1898 8 0.43 0.401 0.4155 10 0.576 0.562 0.5692 12 0.721 0.695 0.7079 14 0.779 0.760 0.7697 16 0.832 0.817 0.8244 18 0.842 0.836 0.8392 20 0.863 0.852 0.8577 22 0.877 0.860 0.8683 24 0.878 0.859 0.8685 26 0.888 0.848 0.8681 28 0.889 0.854 0.8717 30 0.892 0.858 0.8749 32 0.909 0.853 0.8808 34 0.887 0.845 0.866 36 0.679 0.656 0.6677 38 0.467 0.449 0.4579 40 0.338 0.303 0.3205 42 0.221 0.183 0.2021 44 0.134 0.113 0.1236 46 0.098 0.053 0.0757 48 0.065 0.042 0.0537 50 0.045 0.021 0.0331 52 0.034 0.019 0.0265 54 0.033 0.007 0.0202 56 0.021 0.008 0.0146 58 0.017 0.010 0.0134 60 0.013 0.009 0.0112

PAGE 196

196 Table C 20 E xperimental data of breakthrough curve under different sizes of colloid conditions (d p =2.0 m) d p =2.0m Time C/C 0 (Minute) Replicate 1 Replicate 2 Mean 0 0.0098 0.005 0.0074 2 0.0112 0.011 0.0109 4 0.0276 0.021 0.0244 6 0.199 0.164 0.1813 8 0.387 0.371 0.3788 10 0.589 0.562 0.5757 12 0.687 0.665 0.676 14 0.761 0.735 0.7478 16 0.789 0.764 0.7766 18 0.823 0.804 0.8137 20 0.829 0.800 0.8147 22 0.834 0.820 0.8272 24 0.856 0.808 0.8322 26 0.867 0.816 0.8415 28 0.88 0.824 0.852 30 0.887 0.832 0.8597 32 0.888 0.810 0.8491 34 0.88 0.758 0.819 36 0.699 0.609 0.6541 38 0.501 0.462 0.4815 40 0.32 0.306 0.3131 42 0.199 0.182 0.1906 44 0.132 0.105 0.1186 46 0.082 0.077 0.0797 48 0.06 0.048 0.054 50 0.04 0.022 0.0312 52 0.03 0.020 0.0251 54 0.028 0.014 0.0209 56 0.022 0.018 0.0199 58 0.0189 0.017 0.0177 60 0.015 0.011 0.013

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197 APPENDIX D B SU PPORTING INFORMATION FOR CHAP TER 5 D 1 Derivation of Van der Waal Interaction Energy (Eqn.7). ( D 1 ) Using Eqn. (2.1), (3.1), (4.2) and (6.1), for outer S 2 we obtain ( D 2 ) Similarly, from Eqn. (2.2), (3.2), (4.1), (6.1) and Eqn. (2.2), (3.3), (4.3), (6.1), we obtain Eqn. ( D 3) and ( D 4 ), respectively, ( D 3 ) ( D 4 ) For E qn. ( D 2 ), to simplify the notation, we let and Then Eqn. ( D 2 ) can be written as ( D 5 ) To evaluate Eqn. ( D 5 ), we appeal to the following formulas ( D 6 ) ( D 7 )

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198 Together with the following integral identity ( D 8 ) The right hand side of Eqn. ( D 5 ) becomes ( D 9 ) Now, let and Then and the Eqn. ( D 9 ) becomes ( D 10 ) By differentiating Eqn. (20) with respect to and appealing to Eqn. ( D 7 ), we obtain ( D 11 )

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199 Similarly, for inner S 2 we obtain ( D 12 ) For Eqn. ( D 3 ), to simplify the notation, we let and Then Eqn. ( D 3 ) can be written as ( D 13 ) We recall a well known integral identity ( D 14 ) Eqn. ( D 12 ) can be obta ined by differentiating Eqn. ( D 14 ) with respect to 1 and integrating with respect to after routine algebraic operation, we obtain ( D 15 ) Similarly, for Eqn. ( D 15 ), we obtain

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200 ( D 16 ) D 2 Derivation of Electrostatic Double Layer Interaction Energy (Eqn.8 ) ( D 17 ) Using Eqn. (2.1), (3.1), (4.2) and (6.2), we obtain ( D 18 ) Similarly, from Eqn. (2.2), (3.2), (4.1), (6.2) and Eqn. (2.2), (3.3), (4.3), (6.2), we obtain Eqn. ( D 19 ) and ( D 20 ), respectively, ( D 19 ) ( D 20 ) After letting and integrating with respect to z Eqn. ( D 20 ) can be written as ( D 21 ) Let and Eqn. ( D 21 ) can be obtained by differentiation with respect to Now we recall the integral representation of modified Struve function of order zero L 0 (z):

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201 ( D 22 ) And rewrite in term of L 0 (z): ( D 23 ) Differentiating with respect to we obtain ( D 24 ) Based on the recurrence relations of modified Struve functions, we know that ( D 25 ) We obtain ( D 26 ) For Eqn. ( D 19 ) and Eqn. ( D 20 ), letting we obtain ( D 27 ) We now rewr ite the integral in the Eqn. ( D 27 ) in term of Bessel function. To do this, first we convert the integral from Polar coordinate to Cartesian coordinate. Let and we obtain ( D 28 )

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202 Let Eqn. ( D 28 ) can be also written as ( D 29 ) We recall the integral representations o f Bessel function of order zero ( J 0 (x) ) and modified Bessel function of order zero ( I 0 (x) ): ( D 30 ) (C 31 ) Let in the Eqn. ( D 30 ), the first integral on the right hand in the Eqn. ( D 29 ) can be written as ( D 32 ) Differentiating Eqn. ( D 32 ) with respect to i a the second integral o n the right hand in the Eqn. ( D 29 ) can be written as ( D 33 ) Based on the recurrence relations of Bessel function, we obtain ( D 34 ) Inserting Eqn. ( D 32 ) and ( D 33 ) into Eqn. ( D 29 ), we obtain ( D 35 ) Similarly, we obtain ( D 36 )

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203 Inserting Eqn. ( D 35 ) and ( D 36 ) into Eqn. ( D 37 ), we obtain ( D 37 )

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229 BIOGRAPHICAL SKETCH Lei Wu was born in Shandong province China She received her degree in e nvironmental e ngineering from School of Environmental S cience and Engineering, Qingdao University, China in 2004 After three year study and research in Peking University graduate school, s he got a m aster s degree in environmental science in 200 7 Sh e then started her doctoral research in Department of Agricultural and Biological Engineering, University of Florida, Gainesville, FL under the supervisory of Dr. Ra fael Mu oz Carpena and Dr. Bin Gao.