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POLYMERINDUCED FORCES AT INTERFACES By MURALI RANGARAJAN 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 2006 Copyright 2006 by Murali Rangarajan This dissertation is dedicated with all my love, prayers and humble pranams to AMMA, the source and soul of my life. ACKNOWLEDGMENTS I offer my prayers and pranams to the Almighty Parabrahman and Parashakti; it is entirely due to Divine Grace that I have been able to accomplish this meager work. My deepest respect, gratitude and pranams are also due to my Guru Amma whose unerring and eternal guidance and blessings have shaped me in every way, shape and form. I would like to offer my sincere thanks to Dr. Anthony J. C. Ladd for his whole hearted support, guidance and patience. The years of my doctoral work have been some of the most challenging years and I have grown as a person in v~ I never even imag ined. Dr. Ladd was very patient with me, was ahvli available for help and guidance in academic and personal matters, and motivated me when some personal problems tried to overwhelm me. In times where being an academician has increasingly become a job of seeking funds and managing a small company consisting of graduate students, when true teacherstudent interactions are hard to come by, Dr. Ladd has showed me how one can still find time and v,v to mentor students and educate them in issues concerning their research and beyond. I shall remember my interactions with him in shaping my future interactions with colleagues and students in my career. I would also like to thank Dr. Richard Dickinson and Dr. Daniel Purich for their support and guidance. While working with them, I learned the value of collaboration and free exchange of ideas. Both of them were very supportive during times of personal crisis, for which they have my sincere gratitude. I would like to thank Dr. Raj R i, I palan, Dr. Jorge Jimenez, Dr. Jason de Joannis, Ms. Tiffany Cloud, Ms. Patricia Socias, Mr. Shelton Wright and Mr. Michael Smith for interactions and collaborations on the work on polymerinduced forces examined using meanfield theories. Dr. Raj also paid for my conference presentations and I would like to acknowledge his support. My colleagues in the lab of Dr. Ladd made my stay one of pleasure. They sup ported me in all my work and we enjol, 1 discussing everything under the sun and beyond! I would like to thank my colleagues and friends Dr. Jonghoon Lee, Dr. Piotr Symczak, Dr. NhanQuyen Nguyen, Mr. Byoungjin Chun, Mr. Berk Usta, Mr. Gaurav Misra and Dr. Dazhi Yu. I would particularly like to thank Byoungjin who was been helpful in more v,i< than I can count. Every time I ever had a question, particularly on C programming or LaTeX, he unhesitatingly helped me in any way he could. I would like to thank the Department of C'!I. i,, il Engineering and the University of Florida Alumni Graduate Fellowship for funds. I had a great blessing of finding amazing friends, my family away from home, during my stay in Gainesville. They are simply far too many to list. I sincerely hope I have told each and every one of them enough times that I love them deeply and that all my prayers and best wishes go for them. The Maitri family, as we have called ourselves, has been the source of my emotional and spiritual strength. Growing together with them has been such a blessing and pleasure; I pray to God that this love and atmosphere of growth will continue to exist for the rest of our lives. Last but definitely not the least, I would like to thank my parents for their love and support. Their love has been a constant source of strength. They taught me to never give up and alv i strive to better myself and succeed. For everything they have ever done for me, mere words will not do any justice. My prv .i are for them to find happiness and peace in every moment of their life. TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................... ......... iv LIST OF TABLES ...................... ............ ix LIST OF FIGURES ........... ................. ...... x ABSTRACT . . . . . . ... . xiii CHAPTER 1 INTRODUCTION ................... ............. 1 1.1 Flexible and Semiflexible Polymers .. ............... 1 1.1.1 Freely Jointed ('! Ch, A Model of Flexible Polymers ..... 1 1.1.2 Freely Rotating C'!i ,i ..................... 3 1.1.3 Wormlike ('!i ,ii A Model of Semiflexible Polymers ..... 4 1.1.4 Physical Interpretation of Persistence Length . . .. 5 1.2 Scope of the Present Work ............... ... 6 1.3 Organization of the Present Work ............... . 9 2 ADSORPTION OF FLEXIBLE POLYMERS AT INTERFACES: A LIT ERATURE REVIEW ..... ........... ... ........ 10 2.1 Introduction ............. . . ... 10 2.2 Theories of Polymer Adsorption: Overview . . 12 2.2.1 DensityFunctional Theories: Cahnde Gennes Approach . 12 2.2.2 FreeEnergyFunctional (FEF)/Scaling Approach . ... 15 2.2.3 Analytical MeanField Theories ..... . . ..... 16 2.2.4 Numerical MeanField Theory .................. .. 19 2.3 Computational Studies of Polymer Adsorption . . 19 2.4 Experimental Studies of Polymer Adsorption . . .. 20 2.5 Estimation of PolymerInduced Forces: Validity of the Theoretical and Computational Results .................. ... 21 3 POLYMERINDUCED FORCES FROM NUMERICAL MEANFIELD THE ORIES ................... ............ ...... 25 3.1 Introduction .................. ............. .. 25 3.1.1 Background Information ................. . .. 25 3.1.2 Organization of the C!i lpter ........... . .. 27 3.2 System and Surroundings .................. ..... 28 3.3 Thermodynamics of Inhomogeneous Systems . . .... 30 3.4 Free Energy from Lattice Numerical MeanField Theory . ... 32 3.4.1 System, Preliminaries, and Partition Function . ... 32 3.4.2 Configurational Entropy ................. . .. 35 3.4.3 Equilibrium Distribution ............... . .. 37 3.4.4 Free Energy ..... . . ..... ........... 40 3.5 Inhomogeneous Continuum Description of the Film . ... 42 3.5.1 Continuum Formulation of the Fundamental Equation . 42 3.5.2 Estimation of Di, i iiii.; Pressure, Film Tension and Interfa cial Tension ............ . . .... 44 3.6 Film Tension, Interfacial Tension and Force of Compression from Lat tice M odels .................. ............. .. 46 3.6.1 Full Equilibrium .................. ..... .. 46 3.6.2 Restricted Equilibrium ................ .... .. 49 3.7 Validity of the Approach .................. ..... .. 50 3.8 Results and Discussion .................. ..... .. 50 3.8.1 Full Equilibrium .................. ..... .. 51 3.8.2 Restricted Equilibrium .................. ... .. 52 3.8.3 Implications on the MeanField Predictions . .... 56 3.9 Concluding Remarks .................. ......... .. 58 4 EFFECTS OF POLYMERLAYER ANISOTROPY ON THE INTERAC TION BETWEEN ADSORBED LAYERS ................ ..61 4.1 Introduction ..... ...... ... ......... ... .... 61 4.2 SelfConsistent Anisotropic MeanField Theory (SCAFT) ...... ..63 4.2.1 Preliminaries and Notations .............. .. 63 4.2.2 Anisotropic Mean Field ............ ... . .. 66 4.2.3 Statistical Weights and Composition Rule . . ... 67 4.2.4 Structure of the Adsorbed L iv.r ................ .. 69 4.2.5 Estimation of Interaction Forces ................ .. 73 4.3 Results and discussion .................. ...... .. .. 73 4.3.1 Structure of the Adsorbed L rs . . . 74 4.3.2 Interaction Forces .................. ..... .. 76 4.4 Concluding Remarks .................. ......... 83 5 BENDING OF SEMIFLEXIBLE POLYMERS ................ .. 86 5.1 Introduction ............ . . . .... 86 5.2 Wormlike C('! ,ii as Slender Elastic Rods . . ..... 86 5.3 Bending of Slender Elastic Rods ............... . .. 87 5.3.1 Special Cases .................. ........ .. 88 5.3.2 Boundary Conditions .... . . ..... 89 5.3.3 Equation of Pure Bending of a Rod of Circular Cross Section 89 5.4 Results and discussion .... ........... ...... .. 90 5.4.1 DoubleHinged Case ................ .. .. 91 5.4.2 ClampedHinged Case ............... . .. 92 6 FACTIN AS A SEMIFLEXIBLE ELASTIC ROD: MOTILITY OF Liste ria I,,,.:., ',.,/, ,:, PROPELLED BY ACTIN FILAMENTS . ... 98 6.1 Introduction .................. ............. .. 98 6.2 The ListeriaActin System . . ..... ... 98 6.3 Biophysical Model of the Motility of Listeria: Actoclampin ...... 102 6.4 Results and Discussion ..... ........ . .. 105 6.4.1 ActinBased Motility of Listeria . . 105 6.4.2 Long LengthScale Rotation of Listeria . . .. 107 vii 7 CONCLUSIONS AND FUTURE WORK ................... .112 7.1 Problems Addressed ........... . . ... 112 7.2 Conclusions .. .. .. .. .. .. .. ... .. .. ... .. .. 112 7.2.1 Interacting Polymer Layers . . . .. 112 7.2.2 Bending of Semiflexible Polymers: FActin Propelled Motility of Listeria ....... ........ ....... ....... 113 7.3 Future Work .................. ...... ....... 114 7.3.1 Interacting Polymer Layers . . . .. 114 7.3.2 Modeling of Semiflexible Polymers: FActin Propelled Motil ity of Listeria .................. ......... 116 APPENDIX A STRUCTURE OF THE ADSORBED POLYMER LAYERS USING SELF CONSISTENT ANISOTROPIC MEANFIELD THEORY (SCAFT): AD DITIONAL RESULTS .................. ........... .. 119 B SEGMENT DENSITY DISTRIBUTIONS: OVERALL, TAIL, BRIDGE, AND LOOP DENSITIES .................. .......... 126 REFERENCES ................... ..... .... ........ 130 BIOGRAPHICAL SKETCH ................... ......... 136 LIST OF TABLES Table Page B1 Segment Density Distribution for F = 0.75, H/a = 5.0, r = 200, = 1.0 kT . . . . .. . . . . . 127 B2 Segment Density Distribution for F = 0.25, H/a = 5.0, r = 200, = 1.0 kT . . . . .. . . .... .. . 128 B3 Segment Density Distribution for F = 1.25, H/a = 6.0, r = 200, X, 1.0 kT . . . . .. . . .... .. . 129 LIST OF FIGURES Figure Page 11 The freely jointed chain model of a polymer on a 2D lattice . . 2 12 The wormlike chain model of a polymer ................. 4 13 Two interacting physisorbed linear, flexible, homopolymer 1 irS con fined by flat, parallel, adsorbing surfaces ..... . . .. .. 7 14 An illustration of the actoclampin model: An ensemble of elongating clamped filaments under compression or tension propelling the motile surface ............................... ........ 8 31 Two interacting adsorbed lr ,i s separated by a distance H: Definition of the system . . . . . .. . . 29 32 Film tension 7 and interfacial tension 2a as a function of surface separa tion (H/a) under full equilibrium conditions. The results are shown for an adsorption energy X, 0.5 kT, chain length r100, bulk concentra tion (b 0.05, and good solvent conditions, X 0.0 . . ... 52 33 Average density of polymer segments (8t/H) in the interface as a func tion of surface separation (H/a) under full equilibrium conditions. X, 0.5 kT, r100, ob 0.05, and X 0.0 ................ 53 34 Force per unit area f as a function of surface separation (H/a) under full equilibrium conditions for r=100 and X = 0.0. The negative deriva tive of excess grand canonical free energy is also shown for comparison. (a), 0.5 kT, Ob 0.05; (b), 1.0 kT, b 0.005 . .... 54 35 Interaction potential W between the surfaces in full equilibrium with so lutions of varying bulk concentrations. X, = 0.5 kT, r 100, and X = 0.0 55 36 Correct calculation of excess semigrand free energy under restricted equi librium conditions. The results shown here are for a surface coverage 7 0.75, X, 1.0 kT, r 200, and X = 0.0 . . ...... 56 37 Force per unit area as a function of surface separation (H/a) under re stricted equilibrium conditions. 7 = 0.5, X, = 1.0 kT, r=200, and X = 0.0 57 38 The variation of deviatoric stress per segment upon compression, for two surface coverages 7 = 0.5 and 7 = 0.75. The results are shown for X, 1.0 kT, r 200, and X = 0.0, under restricted equilibrium . ... 58 39 Interaction potential W between surfaces in restricted equilibrium for coverages 7 = 0.5 and 7 = 0.75. The results shown here are for X, = 1.0 kT, r 200, and X = 0.0 .................. ......... .. 59 310 Force per unit area f as a function of surface coverage 7 under restricted equilibrium conditions for a fixed separation of (H/a) 4.5 . ... 60 41 Bond orientations and anisotropic mean field ............... ..65 42 Overall segment density distribution: Comparison of lattice meanfield results (SF1, SF2, SCAFT) with lattice Monte Carlo simulations . 74 43 Number and size distribution of bridges, as a function of H/a, for a con stant adsorption energy, X, = 1.0 kT and F = 0.75 ............ ..76 44 Force per unit area between the surfaces f as a function of surface sepa ration H/a ................... ... ........ 79 45 Force per unit area f at a fixed separation of H/a = 4.5 as a function of surface coverage F .................. ...... ..... 80 46 Number and size distribution of bridges, as a function of rescaled sur face coverage F/Fo, for a constant rescaled adsorption energy, X, Xc = 0.74, and H /a = 5.0 .................. ........... .. 81 47 Force per bridge fbr as a function of F for a fixed wall separation H/a 4.5 . . . . . . . .. . .. 82 51 Shapes of a doublehinged rod under compression for forces greater than the critical force .. ... .. .. .. .. .. .. .. .. .. .... .. 93 52 Force on the doublehinged rod as a function of end position, correspond ing to the shapes shown in Figure 51 ................ 94 53 Possible filament end positions for a filament clamped at the origin and oriented along the zaxis for various axial and compressive forces. Note the existence of multiple solutions for each force: Fy > 0 . ... 96 54 Possible filament end positions for a filament clamped at the origin and oriented along the zaxis for various axial and compressive forces: Fy < 0 97 55 Shape of a filament clamped at the origin and oriented at an angle 45 degrees from the zaxis under compression ................. ..97 61 Shape of a filament under tension. F, = 19.2 pN . ..... 106 62 Motility of the surface due to a single filament: Surface position and Fil ament end position ............... .......... 107 63 Motility of the surface due to a single filament: Force . . ... 108 64 Trajectory of the motile surface propelled by five filaments . ... 109 A1 Average number of bridges, nbr, as a function of surface coverage F for a constant adsorption energy, X, = 1.0 kT and H/a = 5.0 . ... 119 A2 Average size of bridges, lbr, as a function of surface coverage F for a con stant adsorption energy, X, = 1.0 kT and H/a = 5.0 . . ... 120 A3 Average number of loops, nlo, as a function of H/a for a constant ad sorption energy, X, 1.0 kT and F = 0.75 ................. ..120 A4 Average size of loops, lio, as a function of H/a for a constant adsorption energy, X, 1.0 kT and F = 0.75 ................... ...... 121 A5 Average number of loops, nlo, as a function of surface coverage F for a constant adsorption energy, X, = 1.0 kT and H/a = 5.0 . ... 121 A6 Average size of loops, lio, as a function of surface coverage F for a con stant adsorption energy, X, = 1.0 kT and H/a = 5.0 . . .... 122 A7 Average number of tails, nta, as a function of H/a for a constant adsorp tion energy, X, = 1.0 kT and F = 0.75 ................. ..122 A8 Average size of tails, Ita, as a function of H/a for a constant adsorption energy, X, 1.0 kT and F = 0.75 ................... ...... 123 A9 Average number of tails, nta, as a function of surface coverage F for a constant adsorption energy, X, = 1.0 kT and H/a = 5.0 . ... 123 A10 Average size of tails, Ita, as a function of surface coverage F for a con stant adsorption energy, X, = 1.0 kT and H/a = 5.0 . . .... 124 A11 Average number of loops, nlo, as a function of rescaled surface coverage F/F0, for a constant rescaled adsorption energy, X, Xc = 0.74, and H/a = 5.0 ................. ......... ....... 124 A12 Average size of loops, lio, as a function of rescaled surface coverage F/F0, for a constant rescaled adsorption energy, X, Xc = 0.74, and H/a = 5.0 125 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 POLYMERINDUCED FORCES AT INTERFACES By Murali Rangarajan August 2006 Chair: Anthony J. C. Ladd Major Department: Chemical Engineering This dissertation concerns studies of forces generated by confined and physisorbed flexible polymers using lattice meanfield theories, and those generated by confined and clamped semiflexible polymers modeled as slender elastic rods. Lattice meanfield theories have been used in understanding and predicting the behavior of polymeric interfacial systems. In order to efficiently tailor such systems for various applications of interest, one has to understand the forces generated in the interface due to the polymer molecules. The present work examines the abilities and limitations of lattice meanfield theories in predicting the structure of physisorbed poly mer lV, i and the resultant forces. Within the lattice meanfield theory, a definition of normal force of compression as the negative derivative of the partitionfunctionbased excess free energy with surface separation gives misleading results because the theory does not explicitly account for the normal stresses involved in the system. Correct expressions for normal and tangential forces are obtained from a continuummechanics based formulation. Preliminary comparisons with lattice Monte Carlo simulations show that meanfield theories fail to predict significant attractive forces when the surfaces are undersaturated, as one would expect. The corrections to the excluded volume (nonreversal chains) and the meanfield (anisotropic field) approximations improve the predictions of lv. r structure, but not the forces. Bending of semiflexible polymer chains (elastic rods) is considered for two bound ary conditions where the chain is hinged on both ends and where the chain is clamped on one end and hinged on the other. For the former case, the compressive forces and chain shapes obtained are consistent with the inflexional elastica pub lished by Love. For the latter, multiple and higherorder solutions are observed for the hingedend position for a given force. Preliminary studies are conducted on actinbased motility of Listeria morn.'. ;//. ',,i ,' i by treating actin filaments as elastic rods, using the actoclampin model. The results show qualitative agreement with calculations where the filaments are modeled as Hookean springs. The feasibility of the actoclampin model to address long lengthscale rotation of Listeria during actinbased motility is addressed. CHAPTER 1 INTRODUCTION This dissertation concerns studies of forces generated by confined and physisorbed flexible polymers using lattice meanfield theories, and those generated by confined and clamped semiflexible polymers modeled as slender elastic rods. In this chapter, we first define and outline models of flexible and semiflexible polymers and identify the contributions of the present work. Finally, we discuss the organization of the rest of the dissertation. 1.1 Flexible and Semiflexible Polymers A polymer is a large molecule consisting of many small, simple chemical units called monomers, joined together by a chemical reaction. They can be classified as flexible or semiflexible polymers on the basis of a microscopic property (persistence length) or a macroscopic one (flexural rigidity) that determines polymer flexibility. In this section, we derive microscopic models of flexible and semiflexible polymers and relate persistence length and flexural rigidity. These microscopic models consider a single polymer chain under the action of thermal forces and define various quantitative properties of the chain. It is important to note that the size of a polymer molecule is such that it is impossible to neglect the effect of thermal forces on the molecule, as one often does for macroscopic objects. For a detailed description of polymer models and the concepts involved, the interested reader is referred to classic texts (e.g., Doi [1], Yamakawa [2]). 1.1.1 Freely Jointed Chains: A Model of Flexible Polymers A polymer molecule has many internal degrees of freedom. For example, one may imagine the rotational, vibrational, and torsional freedoms about C C bonds in polyethylene molecules. It is because of this high degree of flexibility that one often models a polymer rod as a long, flexible piece of string. The simplest mathematical model of a polymer chain assumes the chain to follow a regular lattice. The portions of Figure 11: The freely jointed chain model of the polymer on a 2D lattice. The filled circles are the segments and the thick lines are the bonds. the polymer lying on the lattice points are called , jir, ',1 and the rods connecting the segments are called bonds. Let b be the length of each bond. Figure 11 shows a twodimensional lattice. Due to random thermal motion of the surrounding medium, the polymer chain has the flexibility to take on different configurations (all on the 2D lattice). The simplest possible model then would be when there is no correlation between the directions that different bonds take and that all directions have the same probability. This is the socalled freelyjointedchain model (also called randomcoil model or randomflightchain model). In this case, the configuration of a polymer will be the same as a random walk on the lattice. The endtoend vector R joining one end of the polymer to the other is a measure of the size of the polymer. If the polymer is made of N bonds, with r, the vector of the nth bond, we have N R r,. (11) n=l The average value of R, (R) = 0, since the probability of the endtoend vector being R is the same as it being R. Therefore, one calculates R2, and expresses the size of the polymer by the root mean square (rms) value of R. This is given by M N (R2 E E(r. rm) Nb2 + 2 ( E rm). (12) nl rl 1 nmrn Since for a freely jointed chain there is no correlation between different bond vectors, the second term on the right hand side of Equation (12) is zero. Therefore, for a freely jointed chain, the endtoend distance of the chain is given by (R2)= Nb2 (13) i.e., the size of the polymer is proportional to N1/2. Further, it can be shown that if the initial segment of a freely jointed chain is fixed at origin of a Cartesian coordinate system, and if P(R) is the probability distribution of finding the Nth segment at a position R, for large N and for R > b, the probability distribution function is Gaussian. P(R, N) ( 3 3/Vb 2 (_32. (14) 27Nb2 2Nb2 This probability distribution function is a Green's function which satisfies the diffusion equation associated with the random process (position) r(s) of a Brownian particle with L = Nb regarded as time. 1.1.2 Freely Rotating Chains In this model, the bonds have rotational freedom but the bond angles are fixed at r 0, where r 0 is the angle between any nth and (n + 1)th bonds (0 < 0 < r/2). Therefore (r, rn+) = b2cos0. It can easily be shown that for any n and m (n < m), (r, r,) b2cosm"O. From the right hand side of Equation (12), one then gets (R2) Nb2 1 + cos 2b2cosO ( cos .O) (15) 1 cosO (1 cosO)2 For large N, this result approaches (R2) Nb21 + (16) 1 cosO u(s) Figure 12: The wormlike chain model of a polymer. The figure shows an instantaneous configuration of the continuous chain of length L with an endtoend radius R. From these results, it is possible to derive an expression for the average projection of the endtoend vector R on the initial tangent of the chain ul = rl/b. N 1 cosNO (R ui) b b1 (ri r) cos (17) n=l As N tends to infinity, cosNO tends to zero for 0 < 0 < r/2, and therefore this result reduces to b LtUN, (R ul) = L. (18) 1 cosO The quantity on the lefthand side of Equation (18) provides an operational definition of an important property of a polymer chain, namely, the persistence length. It is important to note that this definition of persistence length is valid only for large N. Persistence length, as defined above, measures the distance from origin till the chain 'remembers' the initial direction ul. (Only until then will the dot product in the summation of Equation (17) be nonzero.) For a freely jointed chain, there are no restrictions in 0, i.e., (ri r,) = 0 (if n / 1). Therefore it is easy to see that in Equation (17), only the first term of the summation will survive, and the persistence length is simply b. On the other hand, for a freely rotating chain, the persistence length is ahiv greater than b. 1.1.3 Wormlike Chains: A Model of Semiflexible Polymers As discussed above, a freely rotating chain restricts the flexibility of the polymer chain by restricting the bond angles, and hence is the simplest model of a semiflexible polymer. From Equation (18), one can write b L cosO = L (19) L, N L, where L = Nb is the total length of the chain. The wormlike chain, illustrated in Figure 12 is defined as a limiting continuous chain formed from this discrete chain by letting N oo, b 0, and 0 0 under the restriction that L remain constant. Therefore, if we note that LtN ,oo,ocosNO = LtND 1 = exp (110) Equation (17) gives LtNo(R U) =L, p( exp  (111) which, for large L, reduces to Lp. Further, from Equation (15), one can write (R2) 2LLp 1 (1 exp (112) defining the (R2) for wormlike chains in terms of the persistence length Lp. For large L (L > Lp), (R2) 2LLp, and for large Lp (Lp > L), (R2) L2. In the intermediate regime, where L > Lp, the formula (R2) = 2LLp (113) can be used with less than 5'. error when L > 3Lp. 1.1.4 Physical Interpretation of Persistence Length In the discrete freely rotating chain model, we saw that (rn rn+1) = b2cos0. However, for a continuous wormlike chain, /L R r(L) r(0) u(s)ds, (114) where u(s) = dr(s)/ds is the tangent vector at s, the arc length. Further, u. uj (. 0 ., where Oij is the angle between the points i and j along the chain. The extent to which the chain is flexible is determined by the correlation (ui uj) =((. .). In general, (cosO(s)) denotes the mean cosine of the angle between chain segments separated by the contour length s. This function possesses the property of socalled multiplicativity: if the chain has two neighboring sections with lengths s and s', then (cos0(s + s')) = (cose(s))(cose(s')). (115) The function having this property is an exponential, i.e., (cosO(s)) = exp  (116) where the preexponential factor is unity, because cos0(s = 0) = 1, and Lp is a constant for each given polymer. This constant is the basic characteristic of polymer flexibility and Equation (116) is the exact definition of persistence length. The physical interpretation of persistence length is readily drawn from Equa tion (116). When s < L,, Equation (116) gives (cos0(s)) t 1. Hence the angle O(s) fluctuates around zero. This simply means that chain segments that are closer than Lp have nearly the same direction. For the opposite case s > Lp, (cosO(s)) } 0. This means, O(s) can be anything from 0 to 3600 with equal probability. So the chain direction gets completely 'forgotten' at lengths much greater than Lp. 1.2 Scope of the Present Work The present work focuses on two problems of interest: Forces generated by confined, adsorbed flexible polymers: Mean field studies: Adsorption of flexible polymers is of great importance in various processes of technological and biological relevance (e.g., cell .r'regation, inhibition of viral replication, bacterial adhesion to surfaces, colloidal/nanoparticle interactions, adhesion, lubrication, and bilayer membranes). A number of experimental stud ies [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] have appeared in the literature that have indicated the importance of understanding polymerinduced forces in tailoring polymer 1lV. r for specific applications. In addition to substantial experimental studies on polymer adsorption at interfaces, a number of theories of polymer adsorption have been devel oped over the past four decades. A model case that has been investigated widely is the adsorption of linear, flexible polymers onto one or two uniform flat surfaces [14]. The structure of the adsorbed polymer 1v.r is described in terms of trains, loops, and tails, Tall \ j r  ; :  I ail Adsorbed chains Free chain \ "1 .............. Bridge I s ^ ..... \ \ IH . Z.. .......... Loop Train z=O z=H Figure 13: Two interacting physisorbed linear, flexible, homopolymer 1lr, i confined by flat, parallel, adsorbing surfaces separated by a distance H. The adsorbed chain conformations are described in terms of loops, trains, tails, and bridges. and additionally in terms of bridges in the case of two interacting 1,is, as illustrated in Figure 13. The primary purpose of theoretical attempts in this area is to develop predictive equations for the structure of the l.si~ and to relate the structure to the forces. The present work develops the appropriate thermodynamics of interacting polymer 1i.. r and a correct formulation for estimating polymerinduced forces in the model case described above. While the method of estimating polymerinduced forces has been discussed in some previous works [15, 16], there are three contributions from the present work: A complete, thorough, and general discussion of estimating interfacial tension and normal force of compression of interacting physisorbed polymer l. ri confined by two flat, parallel, solid surfaces, The first correct results of forces between two planar physisorbed polymer l~i. rs in good solvents using lattice meanfield theories, and, Figure 14: An illustration of the actoclampin model: An ensemble of elongating clamped filaments under compression or tension propelling the motile surface. Re produced from Dickinson and Purich [18]. An examination of the effects of polymer i .r anisotropy (bond correlations) on the structure of the polymer 1. r and the interaction forces under restricted equilibrium conditions [17]. Biophysical modeling of the motility of Listeria monocytogenes pro pelled by actin filaments: Actinbased motility is seen in cell crawling using filipodial/lamellipodial extensions, intracellular propulsion of organelles, and the motil ity of microbial pathogens such as Listeria and S'i,. //.I The pushpull forces involved in the propulsion are generated by actin filament elongation. In order to understand the underlying '1.i,1l1, ,! i. mechanism that generates the necessary forces from the biochemical process of filament elongation, a number of models have been proposed. One of the models, proposed by Dickinson and Purich [18], treats the motile system as an affinitymodulated, processive motor complex which, while simultaneously clamped to the elongating end of the actin filament attached to the motile surface, facilitates filament elongation and uses the resulting forces to propel the motile surface (see Figure 14 for an illustration). The present work models FActin as a slender elastic rod and estimates the forces generated in the propulsion of Listeria by actoclampin motors [18, 19]. The stepwise motility observed experimentally by Kuo and McGrath [20] and predicted using the actoclampin mechanism by Dickinson and Purich [18] is reproduced. An estimate of the torques generated in the system is made to show how the filament elongation and clamp translocation results in longlength scale, righthanded helical trajectories of the motile surface and the attached actin tails. 1.3 Organization of the Present Work The rest of the dissertation is organized as follows: C'!i lpter 2 presents a brief sum mary of the theoretical and computational studies concerning the adsorption of flexible polymers at interfaces. In ('! Ilpter 3, we discuss the thermodynamics of inhomogeneous systems, develop the formalism to evaluate tangential and normal forces in interacting physisorbed polymer 1lV.i and estimate these forces using the lattice, numerical mean field theory of Schecl i,' . and Fleer. The effects of incorporating bond correlations and polymer!i, r anisotropy into the meanfield on the structure of the polymer 1v.. ri and the interaction forces are examined for good solvents under restricted equilibrium conditions in ('! Ilpter 4. An analysis of the bending of semiflexible polymers modeled as elastic rods is presented in Chapter 5, where two cases (a rod hinged at both ends and a rod clamped at one end and hinged at the other end) are considered. Modeling FActin as a slender elastic rod and employing the actoclampin model of force gener ation, C'! Ilpter 6 examines the motility of Listeria mon'... '. u', propelled by actin filaments. Finally, C'! Ilpter 7 summarizes the key results and identifies problems of future interest that evolve out of the present work. CHAPTER 2 ADSORPTION OF FLEXIBLE POLYMERS AT INTERFACES: A LITERATURE REVIEW 2.1 Introduction Flexible polymers physisorbed at interfaces are interesting from a fundamental point of view and in terms of their numerous applications of technological relevance (e.g., lubrication, adhesion, selfassembled drugdelivery vehicles, stabilization or flocculation of colloidal dispersions, bil I r membranes, and prevention of protein adsorption) [12]. A thorough understanding of the structure of these physisorbed polymer lIirs along with the forces generated by these rlV i~ is essential for tailoring polymer interfaces for specific applications. Experimental studies in this discipline have focused on: Characterizing the surfaces (e.g., active sites on surfaces) Probe the structural details of the l1 r (e.g., segment density distribution and average thickness), and, Measure the net polymerinduced forces arising from steric, electrostatic, hydro dynamic, and other interactions. They have contributed significantly to the understanding of polymer adsorption. How ever, experiments are still limited in their ability to discern the finer details of the l1v.r structure (e.g., the arrangements of the polymer segments on the surface as loops, tails, trains, see Figure 13), and the contributions of the various interactions towards the net polymerinduced force. As a result, the need for theoretical guidelines that relate the conformations of adsorbed polymers with the resulting forces is indispensable. In this context, a model case that has been widely investigated is the adsorption of linear, flexible, monodisperse homopolymers present in a monomeric solvent onto one or two uniform flat surfaces [14]. In this case, the polymer liv r structure is described in terms of loops, tails and trains (and bridges in case of two surfaces not very far apart). The conformational details of the adsorbed chains are crucial in determining whether the interaction between the surfaces is attractive or repulsive. The interactions between polymer segments and the solvent molecules can lead to the classification of the solvents as good solvents in which the segmentsolvent interaction is close to the same magnitude or more favored than the segmentsegment and solventsolvent interactions (the polymer chain spreads out), or poor solvents in which the segmentsolvent interactions are significantly less favored than the segment segment and solventsolvent interactions (the polymer chain tends to collapse). These interactions are characterized by the FloryHIl.ii; parameter X. For the case X = 0, the segments and solvent molecules have no specific preference over one another. In the present work, all calculations are considered for the case X = 0. The present work considers forces arising due to the confinement of polymers by two flat surfaces. Depending on the characteristic time of compression of the physisorbed 1ri.;i, one can distinguish three possibilities in such a system. When the characteristic time of compression of the li.r is larger than that of desorption, the chemical potentials of the polymer chains and solvent molecules are the same as the corresponding ones in the 'surrounding' bulk solution. This situation is referred to as full equilibrium situation. In order to maintain the constant chemical potential, the total amount of polymer between the surfaces varies as the system is compressed. The solvent is in equilibrium with the bulk solution whereas the total amount of polymer in the system is fixed. This would correspond to a case when the charac teristic time for desorption of polymer chains is larger than the characteristic time of compression. If the characteristic time for rearrangement of the conformations is significantly smaller than the time of compression, the solvent molecules 'drain' out while the polymer chains (whose total mass is fixed) between the surfaces rearrange themselves into new 'equilibrium' conformations. This situation is referred to as restricted equilibrium. The third possibility would correspond to the case in which the polymer chains take longer time to rearrange themselves than the time taken to compress the system. Then, the chains would not have reached equilibrium with the bulk solution. This is the nonequilibrium case. 2.2 Theories of Polymer Adsorption: Overview The theories of polymer adsorption are broadly based on a densityfunctional approach or a meanfield approach. In a densityfunctional approach, the free energy of the system is written as a density functional1 and it is required that the equilibrium segment density distribution corresponds to the free energy minimum. In the mean field approach, the equilibrium segment density distribution is determined by the conformation of a single polymer chain in the presence of an external mean field, appropriately defined to account for the rn i, ,v body interactions in the system. This aside, using the scaling results proposed by de Gennes [14] based on the socalled magnet analogy (analogy with highly fluctuating magnetic systems near the critical temperature), the results of the densityfunctional approach have been modified, giving rise to the socalled scaling/freeenergyfunctional (FEF) theory. Here we provide the salient features of the guidelines on the structure of adsorbed polymer l. ?i and polymerinduced forces provided by these approaches. 2.2.1 DensityFunctional Theories: Cahnde Gennes Approach The first successful attempt at describing polymer adsorption may be termed the Cahnde Gennes density functional approach [21, 22]. In this approach, a free energy functional (FEF) is derived for polymer 1.. i~ using Cahn and Hilliard's [23] ideas to obtain the free energy of a nonuniform system. The free energy is expressed as a sum of a surface interaction term accounting for the attractive/repulsive interactions between the surface and the polymer segments and a functional accounting for the nonuniform concentration that develops between the surface and the bulk. The typical surface interaction parameter is expressed as a linear function of the concentration of the segments at the surface, a boundary condition, and the binding energy between the segments and the surface. As long as the binding 1 In simple terms, a functional is a function of a variable which itself is a function of position and/or time. In this case, the free energy of the system is expressed as a func tion of the segment density, which is a function of position. Thus a functional is derived for free energy. energy for a chain is significantly higher than the thermal energy but weak when considered per segment (true for long chains), the linear approximation suffices. The functional term contains a 'local' free energy contribution given by the virial expansion of the FloryH l.iii' free energy of interaction between the segments and the solvent, and a nonlocall' term accounting for the entropic constraints in placing the chains due to the presence of other chains and other segments (excluded volume) as well as the energetic terms accounting for mechanical and chemical equilibrium with the bulk, depending on the type of equilibrium that exists between the adsorbed chains and the bulk. The entropic contribution is a function of not only the segment density but also of its ,j,,,l. l,/' in the system, with the density and its gradients treated as independent variables. The number of virial coefficients (and their values) considered in the FloryflrI'ii_  virial expansion depends on the quality of the solvent. For instance, for good solvents it is sufficient to keep only the second virial coefficient v[21].2 For theta conditions v = 0 and hence only the third virial coefficient remains [24]. For poor solvents both the second and third virial coefficients are used (in this case v has a negative value)[25, 26]. The details of the derivation of the free energy are presented in Fleer et al. [12], and are similarities between this approach and the meanfield approach are discussed therein. The segment density profile as a function of distance from the surface by minimiz ing the FEF with respect to position. The free energy functional is derived in terms 2 The condition where only the second virial coefficient is important (and a mean field approximation is appropriate) is also referred to as i,,irj.:,,l solvent condition. The assumption that only the second virial coefficient is important implies that only pair correlations are significant and that threebody and higherorder correlations are neg ligible. On the other hand, meanfield approximation is appropriate when the spatial fluctuations in the segment concentration are small. Therefore, this definition implicitly assumes that the segment concentration is large enough so that its spatial fluctuations are small. So, the marginal solvent regime is a region between the semidilute and the concentrated regimes, often encountered in experimental systems and most of the re sults in the present work. of a single order parameter b(z), which is related to the segment volume fraction as O(z) = (z), where z is the normal distance from the surface. Before proceeding to summarize some useful results, the following comments are in order. De Gennes [21, 22] refers to the above formulation as a meanfield theory. The label 'mean field' in this context differs from the meanfield approximation used in the numerical as well as analytical meanfield theories we discuss later. There is clearly a meanfield assumption in the virial expansion for the 'local' contribution of the free energy functional. Nevertheless, fundamentally the Cahnde Gennes formulation is a densityfunctional approach with the functional being obtained using some meanfield approximations. Further, it has be shown by Jimenez [27] that under marginal solvent conditions, the Cahnde Gennes theory is equivalent to the socalled Ground State Dominance Approximation (GSDA), a simplified solution of a meanfield formulation (see Section 2.2.3) in which the adsorption is described by the conformations of a single polymer chain in the presence of an external field [28]. This formulation leads to a Schr6dingerlike diffusion equation, which can be solved in terms of an eigenfunction expansion. The leading term of this solution corresponds to the chains in the lowest energy state (the 'ground state'), i.e. the adsorbed chains. The following are the important predictions by the Cahnde Gennes approach: This approach ignores the chain end effects and considers chains of infinite molecular weight (N  oc). Hence, it does not account for the presence of tails and their effects in the structure and forces. The existence of three regions in space, namely, the proximal, central and the distal regions, in the concentration profiles is observed. The proximal region is dominated by shortrange surface interaction parameter. The concentration profile in the central region is universal. The adsorption livr here has the local structure of a semidilute solution and the correlation length of concentration fluctuations corresponds to the thickness of the adsorbed lir. In the distal region, the concentration profile decays exponentially to the bulk value. In the central regime, the segment concentration follows the power law decay as a function of distance from the surface and is independent of the bulk concentra tion. Under marginal solvent conditions, the concentration profile has an inverse square dependence on the position from the surface. This result will be contrasted with the scaling results in Section 2.2.2. The force between the two walls is alv,v attractive for an undersaturated l1. r (i.e., (F/Fo) < 1.0, where, F is the surface coverage, and Fo is the saturation surface coverage, in equivalents of rni_..,1l i, rs) under full equilibrium conditions, with a power law dependence on the surface separation, h, f ~ h3. These conditions are hard to reproduce in experiments because the chains are usually so strongly adsorbed that they cannot desorb or diffuse much within the timescale of compression. The experimental results, therefore, mostly indicate repulsive forces. Restricted equilibrium conditions, where the total amount of polymers between the surfaces is fixed, are probably closer to experiments. However, using the Cahnde Gennes approach, de Gennes[22] determined that the interaction force (and the free energy) for the case of restricted equilibrium vanished for all separation distances between the two surfaces. As a means to examining forces under restricted equilibrium conditions, de Gennes employ, .1 the selfsimilarity arguments of scaling in bulk polymer solutions (based on the 'magnet analogy') in modifying the FEF. This study resulted in a set of works which are presented under the next subsection. 2.2.2 FreeEnergyFunctional (FEF)/Scaling Approach In this approach, de Gennes [22] modified the FEF using renormalized scaling arguments, wherein the local and nonlocal terms of the FEF are modified based on the scaling results of the correlation length and the osmotic pressure in semidilute solutions. Once again, the theory only considers chains with infinite molecular weight (N  o0). In the central region, the power law has a coefficient of 4/3, a decay less pronounced than the one predicted from meanfield arguments (2). Under full equilibrium conditions, both scaling/FEF and the Cahnde Gennes approach, along with the GSDA (discussed in Section 2.2.3) show the same attraction between the surfaces. Under restricted equilibrium conditions, a repulsive force is observed at all distances for saturated surfaces. Further attempts in this approach include generalization to various solvent conditions [26, 29, 30] and study of interactions between undersaturated 1.. s [31]. The FEFScaling approach has since been extended by Rossi and Pincus [31, 32] to the case of undersaturated polymer 1., r. Their results show that, for moderate undersaturations, i.e., 0.5 < F/Fo < 1.0, the forces become less repulsive (as compared to the saturated case). For large undersaturations, i.e., F/Fo < 0.4, an attractive force occurs. The next subsection summarizes the key aspects of analytical meanfield ap proaches to examine the structure of the polymer lI.r. This approach is based on the Edwards equation, which exploits the analogy between the conformation of a polymer chain and the diffusion equation. A numerical analog of the same approach is the lat tice, meanfield theory of Schecl i' .' and Fleer [33, 34]. An analysis of this theory is the primary focus of this work and is presented in ('! Ilpters 3 and 4. 2.2.3 Analytical MeanField Theories The Edwards Equation The adsorption of polymers at interfaces can be described, in a meanfield treat ment, by the conformations of a single polymer chain in the presence of an appropri ately defined external field U(r) (r being the spatial position vector), which in turn depends on the polymer conformations or segment density distribution. In the case of adsorption on a flat surface, with the assumption of lateral homogeneity in the system, the external field U(r) is only a function of the normal distance z from the surface. Such a type of problem requires a selfconsistent approach in which the external field U(r) is dependent on the local concentration along the z direction, while the con centration is itself a function of the external field. The mathematical description of the polymer conformation is based on the statistical weight, G(z, n), of a chain that contains n segments, with the endsegment being located at a distance z from the surface. As first shown by Edwards [28], G(z, n) satisfies the following Schr6dingerlike equation3 known as the Edwards equation: G(z, n) a2 G(z, n) U (z S 6 9z2 U(z)G(z, n). (21) On 6 OZ2 The interactions between the segments and the wall can be introduced in two different viv. One possibility is to include an additive delta function (at z = 0) in the external field U(z). The other option is to account for the interaction by using an 3 A lattice version of Equation (21), sometimes referred to as the propagation equa tion, is discussed in ('!i lpter 3. The numerical solution of the lattice version is for chains of arbitrary length, whereas the analytical solutions to the Edwards equation shown in this and the following section address the limit of infinite chain length. effective boundary condition: G(z, n) G(O,n) (22) On 0 b where b is a characteristic length, known as the extrapolation length, associated with the strength of the adsorption energy. If the latter method is used, the external field U(z) contains only the meanfield excludedvolume potential acting on a segment. In the case of a polymer lv,,r under marginalsolvent conditions, U(z) is given by U(N) =v(), (23) where v is the second virial or excludedvolume coefficient and Q(z) the segment concentration at a distance z from the surface. Selfconsistency requires U(z) to be related to the statistical weight G(z, n). This relation is given by the socalled composition rule , () = C j G(z, n m)G(z, m)dm (24) The evaluation of the constant C depends on whether one is considering the case of a polymer 1v r in equilibrium with a bulk solution, i.e., full equilibrium [35], or the case of restricted equilibrium [36]. GroundState Dominance Approximation Analytical solutions of Equation (21) may be obtained as an eigenfunction expansion. The leading term in the eigenfunction expansion corresponds to chains in the lowest energy state, the i,..;;,./ state, which corresponds to adsorbed chains. The approximation where all but the leading term in the eigenfunction expansion are neglected is the socalled p.i.'ni,,ltate dominance approximation. This approximation, which in essence neglects the effect of tails in the adsorbed l~V,r, is equivalent to the freeenergy functional approach presented earlier (Section 2.2.1), as shown by 4 At this time we will not attempt to describe the physical meaning of the propaga tion equation or the composition rule. This can be done more easily when one considers the corresponding lattice equations discussed in Chapter 3. Jimenez [27]. As with the case of the Cahnde Gennes approach, the forces between two saturated lv. i~ under full equilibrium conditions are .li. , attractive and those under restricted equilibrium conditions are close to zero. Beyond GroundState Dominance: TwoOrderParameter Theory Going beyond the groundstate dominance approximation facilitates the char acterization of the contribution of tails to the segment density profile. Semenov and coworkers [35, 36, 37, 38] consider the tail contributions by distinguishing between the adsorbed and free chains in defining statistical weights and defining separate segment densities for loops and tails. The overall segment density profile is obtained simply adding the loop density and the tail density. In order to distinguish loop and tail contributions, they define a second order parameter and another partial differential equation for that order parameter. Details on the derivations for different conditions can be found in references [35, 36, 37, 38]. We shall focus here on the three characteris tic lengths that result from the above analysis. The first length is the extrapolation length which is related to the strength of the adsorptive potential and defines the extension of the proximal region. The second is the characteristic length z*, which represents the crossover of the loop and tail density profiles. Finally, the third length of importance is related to the cutoff distance that defines where the segment density starts to decay exponentially, and therefore is also referred to as the I7.,;. thickness. The interval (b, A) corresponds to the central region defined by de Gennes (see Sec tion 2.2.1). Within this region, z* delineates the loopdominated region from the taildominated portion of the 1lV.r. Here we make use of these characteristic lengths to present the results obtained for adsorption from dilute and semidilute conditions. The loop density profile for these cases is given by 2 d forb < (25) and S~ z8 for z* < z < A, (26) whereas the tail density profile is given by t ~ 3 n ) for b < z < z* (27) and 20 t ~ for z* < z < A. (28) As can be seen from the above equations, the overall density profile arising from the twoorderparameter theory leads to the same scaling relation obtained from the Cahnde Gennes approach, i.e., Total 2. (29) The strength of the approach summarized here is that it allows for the determination of the taildominant and loopdominant regions and for the determination of imptotic laws for the density of loops, tails and free chains. The twoorderparameter theory provides analytical insights into the nature of tailinduced effects such as tailinduced repulsion between 1. ri [36]. This theory predicts a decrease in repulsion for very small undersaturation, and attraction for any (F/Fo) < 0.98. 2.2.4 Numerical MeanField Theory The primary work in the numerical meanfield theory has been by the group of Fleer (cite Fleer's book). Fleer and coworkers have developed a latticebased numerical meanfield theory and have improved upon some of its limitations [33, 34, 39, 40, 41]. The present work will provide a preliminary examination of the capabilities and limitations of the numerical meanfield theory in providing quantitative guidelines on the structure of the adsorbed lv ris and the polymerinduced forces. A detailed description of the theory itself is provided in the next chapter. The equivalence of this theory with the analytical selfconsistent meanfield theory of Semenov and coworkers is discussed in [37, 42]. 2.3 Computational Studies of Polymer Adsorption Computer simulations provide the most convenient way to test these theories and the various approximations that go into them almost individually. They provide rigor ous, exact solutions of the problem within a given set of assumptions. By judiciously manipulating the simulation system, the accuracy of the meanfield predictions and the validity of the approximations in the meanfield theories can be tested in a targeted fashion. An efficient Monte Carlo algorithm can provide all the structural features of the polymer 11V,ir between the surfaces. De Joannis [43] has taken the first step in this direction by a careful examination of the structure of physisorbed 1.v.i~ for a wide range of adsorption energies and molecular weights. Jimenez [27] has taken the first step in correlating the structure of the polymer iv.r with the polymerinduced forces using lattice Monte Carlo simulations and the socalled contactdistribution method (CDM, [44]) to evaluate forces between finite/semiinfinite objects in the presence of polymer chains [45]. However, as discussed in Section 2.5 and C! Ilpter 3, there is a caveat as to whether this change in Helmholtz free energy calculated as above is indeed the normal force. Even under good solvent conditions, where some concerns regarding solvent equilibrium are no longer crucial, the veracity of the results [27] obtained using CDM is still an issue. There are no easy v~i of testing the validity of these calculations within a latticebased formalism unless one can calculate the osmotic pressures in the film. However, comparisons with Monte Carlo results of similar but continuous systems would shed some light in this regard. These calculations are beyond the scope of the present work but are merely sir.. I. here as a necessary exercise before one can venture to further use CDM as a method of estimating forces in interfacial lattice systems. 2.4 Experimental Studies of Polymer Adsorption In the previous subsections we have summarized some of the theoretical predictions for the density profile and the force of compression of physisorbed polymer 1i,r. A number of experimental techniques (e.g., hydrodynamic measurements, ellipsometry, evanescentwaveinduced fluorescence, smallangle neutron scattering, surface force apparatus, and atomic force microscopy) have been used to study the structure of physisorbed and grafted 1lv. ir and the resulting polymerinduced forces. The experimental efforts made so far can be classified broadly into three categories, namely, those which focus on the amount of adsorbed polymers (,i/. ., rl features), those which give information on the microstructure of the lr, and those focusing on force measurements. We present a brief discussion of some of these here. Hydrodynamic measurements, ellipsometry [46], and evanescentwaveinduced fluorescence [47] are some of the techniques that have been used to measure the l1vr thickness or the total amount of adsorbed polymers. These experiments have confirmed that the thickness of the adsorbed l1vr is of the order of the coil size and have been used extensively in studies of adsorption kinetics or competitive adsorption. Neutron scattering techniques, based on either smallangle scattering or reflec tivity experiments, have been used to probe the density profile of adsorbed and endgrafted lv, ri [48, 49]. The results obtained with these techniques are in qualitative agreement with theoretical predictions. The third group of experiments have studied the interaction force between physisorbed polymer lir. Most of the force experiments have made use of the SFA and have measured the forces under good, theta and belowtheta conditions [13, 50]. The analysis of the experimental data shows qualitative agreement with theories, but again, no definite comparisons with the theories are possible due to the large number of uncontrolled factors that typically exist in experiments. For example, it has been observed experimentally that polymer chains do not adsorb uniformly on surfaces but form 'islands' [51, 52]. Therefore, it is difficult to use 'average' measurements such as the ones obtained from SFA and compare those forces with theoretical predictions. Atomic force microscopy (AFM) could be used as a 'local' probe in this respect, but one must 1p .i attention to the influence of the shape and size of the AFM tip and its affinity (or lack thereof) to the polymer chains in order to interpret the measurements meaningfully. Moreover, current theories mostly ignore lateral surface inhomogeneities and are inadequate for predicting the force of compression by finitesized objects. 2.5 Estimation of PolymerInduced Forces: Validity of the Theoretical and Computational Results It is important to note that in all the theoretical and computational results discussed in this chapter pertaining to the calculation of the force of compression of two physisorbed polymer lI,i, the force is defined as the negative of the derivative of the free energy with respect to surface separation, under constant surface areas, temperature, and other appropriate 'extensive' variables. The present work contests the validity of such a definition within the lattice meanfield formalism, especially because the forcesurface separation pair of 'intensive/extensive' pair of variables are not positive homogeneous functions of the zeroth/first degree [53], as required of the terms in the integrated equation of state of a system. Other reasons for this argument, specific to the definition of partition function within the lattice meanfield formalism, are considered in Chapter 3. A homogeneous function is a function with multiplicative scaling behavior. If the argument is multiplied by some factor, the result is multiplied by the power of this factor. Let f : V W be a function between two vector spaces over a field F. We i that f is homogeneous of degree k if the equation f(av) = kf(v) (210) holds for all a c F and v c V. A function f(x) = f(xx, ..., Xn) that is homogeneous of degree k has partial derivatives of degree k 1. Furthermore, it satisfies Euler's homogeneous function theorem, which states that x'ixf (x) kf(x) (211) i=1 In classical thermodynamics, we write differential equations of state5 of a macro scopic system as, for instance, dF = SdT PdV + IdN. (212) In this equation, the extensive variables such as entropy (S), volume (V) and amount of substance (N) are homogeneous functions of first degree, whereas the intensive variables such as temperature (T), pressure (P) and chemical potential (p) are ho mogeneous functions of zeroth degree. Therefore, using Euler's homogeneous function theorem, one can write the integrated equation of state as F PV + pN, (213) 5 Note that Equation (212) is analogous to Equation (211), when k = 1. This is true for the free energy as it is an extensive variable. with the GibbsDuhem equation being SdT + VdP Ndp = 0. (214) Classical thermodynamics requires that the appropriate set of macroscopic vari ables to describe the energy of a system (using an equation of state) be a set compris ing of at least one extensive variable and the rest being either extensive variables or their corresponding intensive variable (such as ST, VP, and N/i). For physisorbed polymers confined between two surfaces of area A each, separated by a distance h and in equilibrium with a bulk solution, one cannot arbitrarily define the excess free energy F" at constant temperature, bulk osmotic pressure, and bulk chemical potentials as dF" = adA fAdh, (215) where a denotes the total surface tension, and f denotes the force. This is because, while A and a are homogeneous functions of degree one and zero respectively, h and fA are not. In other words, while doubling the surface area doubles the free energy without changing the surface tension, free energy and surface separation do not have such a linear relationship. This results in the fact that one cannot integrate a 'differential equation of state' such as Equation (215). Further, one cannot write a GibbsDuhem equation for the system consistent with Equation (215). However, when F" is defined appropriately, a modified version of Equation (215) is valid, with certain restrictions. This is the fundamental problem concerning the thermodynamics of nonuniform systems and is addressed in C'! plter 3. It appears that all the works reviewed in this chapter that concern the estimation of polymerinduced normal forces of compression, be it the densityfunctional approach or meanfield approach or the computer simulations of Jimenez, may not have consid ered the above argument in defining their forces in the interface. Based on the present study and the works on thermodynamics of nonuniform systems (see discussion in Section 3.3), it appears that a definition of normal force of compression as 1 ,ex f t M ) (216) A, OH / ,Vpi,As is fundamentally incorrect unless the partition function 'il'/. ./il accounts for the reversible workmode involved in the normal compression of the 1 i,r. In the above equation, A8 is the surface area of each of the adsorbing surfaces, H is the surface separation, and Q"x is the excess grand canonical free energy derived from the partition function. Moreover, there are additional restrictions as to the appropriate chemical potentials that are conserved in this equilibrium (see Section 3.6 for further discussion). These issues bring to question the validity of the simulation results of Jimenez [27] and the computational meanfield results of Scheul i' I,' and Fleer [34], and Semenov and coworkers [36, 38]. Whether the results of the scaling theories (which do not obtain forces from the partition function of the system) are valid is a question that needs further exploration. In the next chapter, we discuss the thermodynamics of interacting polymer 1 .i. i and develop expressions for evaluating forces of compression of two physisorbed polymer 1. rs confined by flat, parallel surfaces under full and restricted equilibrium conditions using the lattice, numerical meanfield theory of Sd!, n i' 11P' and Fleer. We present the first correct results of forces of compression using the meanfield theory, and then examine the capabilities and limitations of this theory and its various improvements in providing quantitative guidelines for relating the l ,rv structure to the interaction forces. CHAPTER 3 POLYMERINDUCED FORCES FROM NUMERICAL MEANFIELD THEORIES 3.1 Introduction In this chapter, we develop the thermodynamic formalism for estimating tangential and normal stresses in interacting polymer lv. rs within a 1D lattice, numerical mean field theory [34]. We show that, within a onedimensional meanfield approximation, the normal force of compression is not equal to the negative derivative of the free energy with respect to the normal distance H between the two adsorbing surfaces under good solvent conditions. This is because the partition function fails to explicitly account for the mechanical work in maintaining the distance H between the surfaces. Therefore, in order to obtain expressions for tangential and normal stresses in the interface, we consider a continuum analysis of the interface and then adopt the results for a lattice system. Using the above formalism, we examine the lattice meanfield predictions of interaction forces, interfacial tension, and interaction potentials of polymer 1. rlS in good solvents under full and restricted equilibrium conditions over a range of surface coverages. 3.1.1 Background Information Consider the problem of two interacting polymer li r physisorbed on flat, par allel, impenetrable surfaces placed at a fixed but arbitrary separation distance H (i.e., the dividing surfaces are not Gibbs' surfaces, as is usually defined in thermodynamics of thin films). In the context of interacting electrical double 1., ri, it has been shown by D. i ii 1ii Langmuir, Verwey and Overbeek [54] that the force acting between the two plates can be obtained either from the free energy or from the osmotic stresses in the system. Employing the former approach in obtaining the forces between interacting polymer l~V. rs, both the lattice and the continuumformulation analytical theories derive expressions for excess free energy of the solution between the surfaces, which is the difference between the free energies of the system in the presence and absence of the surfaces. Note that in the absence of the surfaces, the system is homogeneous with 25 the same composition as that of the bulk solution with which it is in (full) equilibrium. Therefore, the excess grand canonical free energy is defined as ` (H) = Q(H) (31) The free energy of interaction between the lz. i~ is defined as the change in free energy when the surfaces are moved reversibly from an 'infinite' separation distance to a given separation distance H, nt Qex(H) Qe(oo) (32) The force between the two surfaces is then defined as the derivative of the interaction free energy with respect to separation distance, 1 81 (aQe' f t (33) + T,Vjpi,As U / T,V1pi,As The above relation implies that the work performed for the compression of the polymer lv. is, namely, dW = fdH, is contained in the thermodynamic potential derived using the appropriate statistical mechanical formulation for the conformations of the chains in the interface, and that the appropriate chemical potentials of the polymer chains and the solvent molecules are invariant upon compression (see Section 3.6 for further discussion). However, this analysis breaks down in the context of lattice and analytical meanfield theories because of the inability of the partition function to explicitly build in the work of compression arising from the attractive or repulsive forces that might exist between the confining surfaces. In particular, the free energy formulated using the lattice meanfield theories only accounts for the tangential stresses in the interface relative to the bulk stress, as shown in this chapter. In a lattice meanfield description of polymer lv.i s (for details, see Section 3.4 and Scheul i, ' and Fleer [33, 34]), one writes the grand canonical partition function and obtains the equilibrium conformation of the polymer chains and solvent molecules by minimizing the corresponding grand canonical potential. The equilibrium conformation is then related to the density distribution in the system. This density distribution depends on the composition of the bulk solution with which the polymer 1 irs are in equilibrium. From the density distribution, one can obtain the excess free energy of the 1.ris relative to that of the bulk solution, at a given separation distance and area of the surfaces. In Sections 3.4 and 3.6, we show that this excess free energy is equal to the socalled film tension, which is the energy required to create a thin polymer film (consisting of two interacting polymer lI.i ) of a given area 2A, and maintain it at a thickness H. In order to obtain the correct estimate of the interaction forces between the l1.1rs, we turn to a continuum analysis of the polymer l.ri~ along the lines of Evans [15] and Ploehn [16, 55]. The force of compression, the film tension and the interfacial tension of the 1.iri are then obtained by relating the resulting normal and tangential stresses (respectively) to the thermodynamic potential given by the work function based on continuum mechanics. The polymerinduced force of compression per unit area is shown to be equal to the dii. ,ii:iii; pressure as defined in the theory of thin liquid film (see for instance, Babak [56]), i.e., the Volterra derivative of the excess grand canonical potential of the polymer film with respect to the surface separation. The dii. ,ili.i; ; pressure is the difference between the osmotic pressure in the midpoint and that in the bulk, when the polymer 1..ri are in full equilibrium with the bulk solution, analogous to the result in electrical double l.r1i [54]. The case of the polymer 1.. ir being in restricted equilibrium (total amount of polymer between the surfaces is fixed) is also discussed. The formulation presented here is analogous to the ones presented by Evans [15] and Ploehn [16, 55], but is simpler and more direct. Moreover, whereas Evans [15] and Ploehn [16] have examined the predictions of analytical meanfield theories based on Edwards' equation, there has been no analysis of the correct predictions of force of interaction by the lattice, numerical meanfield theory of Scheul i' ,' and Fleer [34], to the best of our knowledge. This forms the focus of the rest of the study. We present the first correct results of polymerinduced forces between two planar surfaces using lattice numerical meanfield theory. 3.1.2 Organization of the Chapter The remaining part of this chapter is organized as follows: In Section 3.2, the sys tem under consideration is defined. Section 3.3 discusses the general thermodynamics of inhomogeneous systems, and defines the appropriate partition functions, thermo dynamic potentials, and the criterion for equilibrium. Using the lattice meanfield theory of Schel i,' and Fleer [34], the expression for grand canonical potential and semigrand potential are derived under full equilibrium and restricted equilibrium, respectively, in Section 3.4. Expressions for the force of compression and interfacial ten sion are obtained in Section 3.5, using the principles of continuum mechanics. Following this, in Section 3.6, we develop the expressions for normal and tangential stresses for the lattice model and show that the excess grand canonical potential derived in Section 3.4 is indeed equal to the film tension. In Section 3.8, we illustrate the predictions of the lattice meanfield theory for a few sample cases. The discrepancies observed be tween the results based on the correct formulation derived in this work and some of the previous results published in the recent literature, based on the incorrect formulation, (Jimenez et al. [57]) are discussed. 3.2 System and Surroundings The system under consideration (see Figure 31) is a monodisperse, homopolymer solution in a monomeric good solvent that is confined between two parallel, solid, impenetrable, adsorbing surfaces. The polymer molecules physisorb on the surfaces to form two polymer l1v. rs, with the distance between the adsorbing surfaces denoted by H. We denote this region between the two surfaces as a thin or a thick film. The solution in the film is exposed to and is in equilibrium with an infinite reservoir of a homogeneous bulk solution whose properties are denoted with a superscript or subscript b. When the two lI. ir are noninteracting, i.e., the force between the two surfaces is zero, the film consists of two physisorbed polymer interfaces and a homogeneous bulk in the middle with the same composition as that of the phase b and is considered a thick film. However, when the two solid surfaces are close enough so that there is no uniform 'bulk' phase in the middle, the polymer solution between the surfaces are said to form a thin film. The 'filmexcess' properties are defined with reference to the homogeneous bulk phase b. The interfacial tension is then defined with respect to the normal stress in the film. In order to confine the thin film of polymer solution between the two surfaces, mechanical work has to be performed for maintaining the area of contact between the Solid Surfaces, Intel La ex,int.. ijex,int = Cjex~b r b l Reservoir; Phase b Reservoir; Phase b Figure 31: Two interacting adsorbed 'l. r separated by a distance H. Note that the dividing surfaces are not Gibbs' surfaces. Here, the superscript or phase b denotes bulk conditions, the superscript int denotes the interface, the subscript m denotes the conditions at the midpoint of the interface (z = H/2 in a 1D system), P refers to pres sure, II denotes the osmotic pressure, p / is the exchange chemical potential (defined in Section 3.6), and 7 is the film tension. solid surfaces and the polymer solution, and for maintaining the normal distance of separation H between the two surfaces. The former work is usually reported in terms of the interfacial tension a, while the latter work is typically represented in terms of the disjoining pressure of the thin film TTd or the normal force of compression of the polymer 1. is wT AswhereIn modeling the film, the dividing surfaces are defined to be at the z planes of contact between the surfaces and the solution. Therefore the dividing surfaces are not Gibbs' surfaces and the surface separation H is an independent variable. With this picture, one can understand why the disjoining pressure cannot be written as the negative derivative of the excess grand canonical potential derived from a partition function. Due to the presence of the parallel confining surfaces, the ensemble average properties in the film are inhomogeneous only in one dimension. (This is denoted as the zdirection.) Mechanical stability demands that the normal stress be uniform everywhere in the film while the tangential stress is uniform in each /;,plane. The normal force required to maintain the film is simply the excess normal stress in the film relative to the reservoir while the interfacial tension is simply the integral of the tangential stress relative to the normal stress in the film. However, this normal stress in the film is not known apriori when the film is thin, i.e., when the normal force is nonzero. What one does know is the pressure of the bulk fluid in the reservoir. Therefore, one writes the partition function of the thin film (within the meanfield approximation in this study) with reference to the bulk reservoir phase. In other words, the mechanical work required to maintain the surface separation of a thin film by compressing the polymer l V.i~ is not explicitly accounted for in the partition function. As will be shown in Sections 3.4 and 3.6, the excess grand canonical potential in such a formulation is indeed equal to the film tension 7. While the two reversible mechanical work modes are present in the film tension, the normal stress in the film is not explicitly present. This is the reason that one cannot define the normal force to be the negative derivative of a partitionfunctionbased thermodynamic potential with respect to the separation distance. 3.3 Thermodynamics of Inhomogeneous Systems The appropriate statistical mechanical ensemble for the problem of interacting polymer l1iv,rs under full equilibrium conditions is the grand canonical ensemble. However, the corresponding grand canonical potential is not equal to PV because the system under consideration has a nonuniform spatial distribution of species, and hence nonuniform pressure and chemical potential distributions. In this section, we review the thermodynamics of a general inhomogeneous system and identify some key results therein that are relevant to our discussion. For further details, the interested reader is referred to an elegant review by Wajnryb et al. [58] The grand canonical partition function of a general inhomogeneous system within the assumption of local equilibrium is given by S[V; dFN p[ drf(r)(r) + pp (r)Zd,(r)], (34) 2ref N. a= where ( denotes the terms to ensure indistinguishability of identical system configura tions, Na denotes the set of all possible particle numbers, FN, denotes the phase space variables, (r) and h,(r) denote the energy density field, and the number densities of components (a = 1, 2, ..., c), and P3(r) and po(r) denote the corresponding Lagrange multiplier fields. The partition function is a functional of these Lagrange multiplier fields. It follows from the definition of the grand canonical partition function that [Sln V;,p]  F(r) (35) and ^ ef I  n (r). (36) Here E(r) = ((r)) and no(r) = (r(r) are the ensemble averages of the energy density field and the number densities respectively. The grand canonical partition function leads to the definition of the appropriate grand canonical potential of the system  In ([V; ) P dr (r). (37) kT eef V Here b(r) is the 'density of l n and is a thermodynamic field. If one were to write the fundamental equation of the entropy of this inhomogeneous system, one would have S= j dra(r), (38) where c u(r) k [3(r)(r) + (r (r)na(r) + (r)]. (39) a=1 The variations of entropy and its density would turn out to be 6 = dr6s(r), (310) where c 6s k[O3(r) (r) + Pa(ri%, ,(r)] (311) a=l1 with the corresponding GibbsDuhem equation as cr C Sdr[ (r)6/3(r) + n(r)P(r) (r) (r)] 0 (312) Sa=1 where the integrand is not equal to zero at each point in r. This is because the thermo dynamic field Q(r) is a nonlocal functional of the Lagrange multiplier fields P(r) and pa(r). In the context of interaction between polymer 1_. i~, it is this nonlocal dependence of the 'grand canonical potential density' that accounts for the work of compression arising from the attractive or repulsive forces between the two surfaces, as well as the work required in maintaining the area of the surfaces exposed to the solution. However, in the problem of interacting polymer 1.. ir confined by two parallel solid surfaces, the integral in Equation (312) onedimensional, i.e., in zdirection alone. We shall use this in deriving the criterion for thermodynamic equilibrium in the film in Section 3.6.1. 3.4 Free Energy from Lattice Numerical MeanField Theory 3.4.1 System, Preliminaries, and Partition Function Consider a lattice system confined between two flat, parallel, impenetrable surfaces and filled with polymer segments (linear, monodisperse chains of finite length r) and solvent molecules (good solvent). We only consider fluctuations in a single direction normal to the surface, which we define as the zdirection. Each segment or solvent molecule occupies one lattice site. The lattice adjoins the two hard/adsorbing surfaces and is divided into M 1. rs of sites parallel to the adsorbing surfaces, in the z direction, i.e., H = 1,M, where lz, the length of each lattice site in the zdirection is taken to be unity. Each lIvr contains L lattice sites. The surfaces therefore correspond to z = 0 and z = M + 1, and adsorption takes place in z = 1 and z = M. If D is the coordination number of the lattice, then, a lattice site in any 1. r z has D nearest neighbors, of which a fraction A/,z is in lvr z'. The fraction of nearest neighbors that lie in the same l1 .r, then, is A0, that in the .,.i i:ent l1v.r is A1 and so on. Therefore, A can be viewed as the fraction of the nearest neighbors of a lattice site. In the case of a firstorder Markov chain statistics, A's are the stepweights of the random walk. Furthermore, in a simple cubic lattice, there are six nearest neighbors to a lattice site in a 1, r z, of which four are in 1. r z and one each in the lIri z 1. Therefore, for a cubic lattice, A1 = 1/6, Ao = 4/6, and, A = 0,  j I> 1. If there are n polymer chains and no solvent molecules in the film (defined as the region enclosed by the two surfaces), then it follows that, nr + no = ML. (313) We define the volume fractions of polymer and solvent at any given l vr as, ()= ); 0(z) n(Z (314) L L such that, n(z) + nO(z) = L; (z) = 1 (z) (315) and, n(J) n r; nO(z) = n. (316) z z When the solution in the film is open to a bulk solution with respect to numbers, the numbers of polymer chains and solvent molecules between the surface vary. The appropriate grand canonical partition function E is given by the sum of the appropri ately weighted canonical partition functions (the canonical partition function Q being defined for a fixed number of solvent molecules and a fixed number of polymer chains in a defined set of conformations corresponding to a given energy of the system) for different values of possible number of polymer chains (and hence different numbers of solvent molecules). a uns ({nc} V, A, T) exp ( exp (Us so1 = (317) exp ( n (z) exp (zc) (317 ex kT ex kT where, {nc} is a set of permissible conformations of the polymer chains in the film such that, n' = n, (318) c f is the configurational entropy of arranging the polymer segments and the solvent molecules in one of the n~ conformations within mean field, V is the volume of the system, A is the surface area, T is the temperature, p0 is the chemical potential of the solvent molecules corresponding to the bulk solvent concentration, and p is the chemical potential of the polymer segment corresponding to the bulk polymer concentration. The interaction energies are given by Usrf = kTxs(n(1) + n(M)) (319) and Use_ = kTx n(z) ((z)). (320) The equilibrium distribution of the polymer segments in the lattice is determined by maximizing the grand canonical partition function in Equation (317). A detailed derivation of the interaction energies, configurational entropy and the equilibrium distribution are given in the original Schecl i. i,Fleer papersxx and is not repeated here. Instead, we merely present the important equations in the theory that pertain to the estimation of the grand canonical potential. It is to be noted that the chain statistics is defined in terms of concentrations of chain conformations and not in terms of concentrations of individual segments. A chain is then treated as connected segments. A chain conformation is characterized by defining the l1 vr numbers in which each of the successive segments are present. A conformation can then be denoted as Conformation c : (1, i)(2,j)(3, k)...(r 1,)(r, m), (321) indicating that the first segment is in lIVr i, the second segment is in liv1r j and so on. This implies that many different actual arrangements of the segments (in the film) can correspond to a defined conformation. If a segment s is placed in l rv.r z i and segment s + 1 is placed in 1l .r z j, then the number of different allowed placements of s + 1 relative to s is DAo if j = i, and DA1I if j = i 1. It then follows that a dimer with conformation (1, i) (2,j) can assume LDAj_ different positions. A trimer with the conformation (1, i) (2,j) (3, k) can assume LD2 Xjj kj positions, if backfolding of the chain is allowed. A partial correction to the backfolding of segments will be applied later on. Using similar logic, the number of v iv of arranging r segments of a chain in conformation c given in equation (321), in an empty lattice is LDr1Aj_iAkj...A l, which can be written as L~cDr1, where Lc = (As,+1)c (322) where, s,s+ A0o, if s=s+ (323) Ai, otherwise Further, if one considers only the number of arrangements of a part of the polymer chain, then, the notation wc(s, t) is used, where the part of the chain considered is between segments s and t, including both of them. Similarly, the summation C:(,t) would account only for all the possible conformations of that part of the chain. It is evident that c = wc(1, r), and that c() = Zc(1,r)(). The definition of statistics of the chain in terms of conformations and in terms of individual segments are interchangeable and are related by the following relation: n(z) Y rzn^ Y r^ r, (324) c z where, r,,c is the number of segments of a chain in conformation c in 1lv. r z. The nota tion used to indicate the l r number corresponding to the segment s of conformation c is the subscript k(s, c), k being the l,vr number. Finally, we define the following reference system: nr amorphous polymer segments forming a pure polymer liquid (n chains each with r segments), occupying nr lattice sites Two flat, parallel, impenetrable surfaces, each with L sites on them, with no mL, sites between them, enclosing no solvent molecules Every site is filled with either a polymer segment or a solvent molecule. 3.4.2 Configurational Entropy The configurational entropy term f ({n} V, A, T) is the number of v,v of placing the n chains of r segments each and n solvent molecules between the two surfaces, corresponding to the energy given by the two exponential terms, i.e., the degeneracy of the system. In calculating the degeneracy, the BraggWilliams approximation of random mixing within each 1'. r is used, i.e., the polymer segments in each 1'i.r are considered to be randomly distributed over the L lattice sites in that l, r. As discussed in Section 3.4.1, the number of v~ i of placing a chain in a conformation c in an empty lattice is LcD'r1. If the lattice is partially occupied, then a chain can only be placed if the appropriate l., r (as defined in the conformation, e.g., equation (321)) have vacant sites. Therefore, we have to apply r correction factors to the combinatorics in order to partially account for the excluded volume, one factor for each segment of the chain. This correction factor, in the simplest case, is the probability that a given 1, r has at least one free site (usually called the vacancy probability). If the number of occupied sites in l1v r z at any given instant is v(z), then the vacancy probability is given by (1 v(z)/L). Therefore, the number of possibilities of placing a chain in conformation c is given by LcDr'1 [ 1 (1 vk(s,c)/L) = uc(D/L)r1 H 1(L k(s,c)), where Vk(s,c) is the number of previously occupied segments in the 1l.,r k where the segment s of the chain in conformation c is placed. Therefore, the number of arrangements u of placing the first chain of conformation c (of the n, chains) in an empty lattice is given by M rzc1 = c(D/L)r1 (L v)). (325) z=1 v(z)=0 Placing nc chains in conformation c would give rise to the factor wc"(D/L)(r1)"0, while the multiplication extends to v(z) = ncrz,c 1. Placing all n = e n~ chains would lead to the number of arrangements ( ) M n(z)1 w(n) (D/L)(r1)n ( ( nf J (L v)). (326) c z=l v(z) 0 Now the remaining L n(z) = no(z) sites in each of the M 1li.i~s have to be filled with no solvent molecules. The number of possibilities of arranging no(z) number of solvent molecules in lr z is given simply vby Y z)=n(z)(L~ v(z)). Upon simplification, solvent molecules in lv z is given simply by 11,(,) ( < vz)). Upon simplification, the expression for configurational entropy is found to be 1c, 1 (1(7!) nKI (171 nj)!) (327) The factorials nc! and no(z)! correct for indistinguishability of the rc chains in each conformation c and of the solvent molecules in each 1.r z. The configurational entropy of the reference system f+ can be derived using similar arguments. In the bulk, the distinction between 1v. rs is irrelevant. Since the number of (equivalent) lattice sites in the bulk polymer is nr, the factor (L!)M in equation (327) is replaced by (nr)!, and the factor L(r1)" by (nr)(r1)". Further, in the bulk, all conformations are equally probable. Hence, nc = n, and Uw 1. Also since there are no solvent molecules, no(z) = 0. Therefore, equation (327) reduces to Q (ur) (D/nr) 1)n (328) n! This expression is also derived by Flory [1]. Therefore, this formulation is consistent with earlier theories. 3.4.3 Equilibrium Distribution The grand canonical partition function of the system is given by Equation (317)). At equilibrium, the appropriate free energy of the system takes its minimum value, i.e., the partition function E is at its maximum value. This situation corresponds to the most probable set of conformations (with a corresponding number of chains nfq. and number of solvent molecules n%,), which will be obtained in this section. If we neglect fluctuations once the system has attained equilibrium, then the number of chains and solvent molecules are fixed. This means that the sum over all n's in equation (317) is replaced by the maximum term. To obtain the equilibrium distribution, i.e., the number of chains nd in conformation d in the equilibrium situation, the terms within the sum in equation (317) are differentiated with respect to nd and set to zero. Realizing that Ez no(z) = ML r E n~, we have nQ I+ (p ,) /kT= 0. (329) 8nd VA,T,{ncnd} This differentiation corresponds to adding one chain in conformation d from bulk and placing r,,d segments in each 1lv. r z, and removing r solvent molecules from the appropriate l i~s in order to maintain constant volume (i.e., all sites are occupied). This conservation of total volume can be expressed as O(noa(z)) (az) (330) rza =^ (330) 8Od ) V,A,T,{n nd} ( Ond ) V,A,T,{n nd}* This maximization is subject to an additional constraint that each liv.r in the lattice has to be completely filled with either polymer segments or solvent molecules. This constraint is implicitly included, because in performing the differentiation, no(z) = L c rzcc = L n(z) is used. However, for multicomponent systems, this constraint should be explicitly included with an appropriate Lagrange multiplier. Performing simple algebraic manipulations on equations (327) and (328), and using Stirling's approximation of the logarithm of a factorial, one obtains In = MLInL ncn ( c) no(z) nno(z) nr (r 1)nlnL. (331) Upon differentiation and simplification, the equilibrium set of conformations are then obtained as In lnd+r 1+lnr+ ( + L kT Sr,,d{ Xs (61,, + 6MK) + X KK(z)) ) + ln0(z)} (332) Defining In C = r 1 + Inr + k (333) kT we write, In = n wld + In C + r,d n G, L z M or, C dfJ G, (334) z=1 where, In G, xs (l,z + 6M,z) +X ((n z)) (0())) + n0(z). (335) Equation (335) is a discretized version of the Edwards equation, discussed in C'! plter 2. The number of chains in a particular equilibriumconformation d is proportional to WUd, a product of (r 1) stepweights, i.e., A's, which determine the conformation as is evident from Equation (321). Further, nd is also proportional to r weighting factors (G,'s), as shown in Equation (334). Each segment in the chain contributes a weighting factor Gz, which is a Boltzmanntype factor accounting for the change in free energy when a solvent molecule in liv, r z is replaced by a segment. It consists of a contribution for the exchange adsorption energy Xy kT, when the livT z is .,.li i,:ent to a surface (z = 1 or z = M); a factor for the exchange interaction energy between segments and solvent molecules X ((O(z)) (o(z))), for replacing a solvent molecule by a segment; and a factor for the local entropy k ln (z) of the solvent molecule. For the simplest case r = 1 (monomer), we have, from Equation (334), ((z) = n(z)/L = KG, exp(p po)/kT, where K is a constant. Since the volume fraction varies linearly with the weighting factor Gz for a monomer, Gz is also called monomer weighting factor. Another way of viewing Gz is as follows: The meanfield experienced by a segment in the presence of polymer chains and solvent molecules is given by u(z) = Usrf(z) + Useg,so(z) + u'(z), (336) where, Usurf(z) = kTX, (61, + ,z) (337) useso 5(z) kTX (((z)) (())) and, (338) u'(z) = kTInfo(z), (339) such that, G() exp ( (340) Here, uW(z) is the hardcore or excluded volume interaction in the I'iV'T. The unnormalized probability of finding the sth polymer segment of a chain in the 1,.r z is given by the function G(z, s) G(z, s) = G(z) A,, G(z', s 1), (341) where A is the fraction of nearest neighbor sites in the 1Ivr z' of the lattice. This equation is analogous to the propagation equation discussed in Section 2.2.3 in C'! Ilpter 2. It states that the sth segment of a chain can be in a given site in l1vr z if and only if the (s 1)th segment can be in one of its nearest neighbors. In enforcing this constraint of connectivity, a meanfield assumption that does not distinguish the sites within each 1ivr is used. In other words, connectivity is enforced 1l b.v1vr only. This constraint partially accounts for connectivity and excluded volume. Further, analogous to the composition rule discussed in Section 2.2.3 in C'! Ilpter 2, the segment volume fraction Q(z) is given by the composition rule (z) = 0 G(zs)G(zr s + 1), (342) where C is a normalization constant. For a polymer liv.r in equilibrium with a bulk solution, C = b/N, and for restricted equilibrium, C = F/G(N), where G(N) is the unnormalized probability of finding the end segment of the chain anywhere in the system. 3.4.4 Free Energy The free energy of interaction between the two l., ri is obtained from the grand canonical partition function as S= kTIn ). (343) ref The logarithm of the grand canonical partition function is given by ( Usurf Usegsol Y0 Z no () P Y, nc4 '( f In + + I (344) Yre kT kT kT kT The bulk chemical potentials are estimated using FloryHtIIi_ ; theory Si 1 + ',. + x,2 (345) kT r and = b 1 )+ In o + x (346) Substituting the various terms in (344), one obtains the excess grand canonical free energy to be k L ( )(() b + )) (347) KU z ( Obo ()/ which is the same as T L (1(z) fIb) (348) z where II is the osmotic pressure. The above relation of the excess grand canonical free energy is analogous to the wellknown KirkwoodBuff equation [16]. As mentioned in Section 3.2, by considering the excess osmotic pressure with reference to the bulk (reservoir) osmotic pressure, this potential accounts for the work done in creating the thin film interfaciall tension, 2Lcr) and maintaining it at the given thickness M (d i. Piii.; pressure TTdA. 11). In other words, this potential is indeed the film tension 7. This is consistent with the argument presented at the end of Section 3.3. When the system is in restricted equilibrium, the film is open with respect to solvent but not with respect to polymer. In this case, the semigrand potential is simply + (349) kT kT Therefore we get the expression for semigrand potential as Fe (H) n t lnO(z) + X () ((z)) (H ) (350) where the total amount of polymer in the gap Ot = Y, O(z), the solvent density distribution O(z) = 1 (z), and G(r) is the unnormalized probability of finding a chain of r monomeric (persistence) units in the gap region. We shall further discuss this potential and its interpretation in Section 3.8. 3.5 Inhomogeneous Continuum Description of the Film Based on the arguments in Sections 3.2 and 3.3, one concludes that there are two reversible mechanical workmodes involved in the problem of interacting polymer lV1.is, namely the interfacial tension a and the dii. iiii;.: pressure TTd. However, to obtain useful predictive equations for the disjoining pressure, one has to examine the stresses in the system. For this purpose, we employ a quasithermodynamic framework based on the principles of continuum mechanics. Within this framework, in each microscopic volume, the system is considered as an inhomogeneous continuous medium where the fundamental thermodynamic relations presented in Section 3.3 are valid. This hypothesis holds when the correlation length in the system is greater than the range of intermolecular forces. 3.5.1 Continuum Formulation of the Fundamental Equation We begin by writing the internal energy and entropy balance equations [59] for the polymer lV. ir and the reservoir of homogeneous bulk solution the 1v. r s are in equilibrium with. Internal energy balance: The rate of change of internal energy per unit mass of a material element of a multicomponent fluid is given by p dU (V.q) + tr(T.Vv) V. ,bi (351) P dt where denotes the material derivative, q is the heat flux vector, tr denotes trace of the tensor, T is the stress tensor, ,m'b is the chemical potential of species i defined on a mass basis, and ji is the corresponding mass flux. The first term on the right corresponds to energy conduction, the second term corresponds to the mechanical workmodes, while the last term corresponds to energy change due to material diffusion. The stress tensor T consists of the isotropic bulk pressure pb and the osmotic stress tensor T and can be written as T pbl + T. (352) Note that here the osmotic stresses are defined with reference to the reservoir bulk pressure rather than the .. vetundetermined normal stress in the film. Entropy inequality: For slow and reversible compression, the second law of thermodynamics becomes an equality. At constant temperature, it reads p + V.q 0, (353) dt T where S is the entropy per unit mass. Subtracting Equation (353) from (351) gives us the total Helmholtz free energy balance. Helmholtz free energy balance: dA t (354) St V.V t Vv). (354) Upon further simplifications, conversion into molar basis and integration over the volume of the system, equation (354) becomes dA t dV + b dni + tr(Vv)dV. (355) at dt 't .I For homogeneous systems, the osmotic stress tensor vanishes uniformly, i.e., r = 0. Therefore Equation (355) reduces to 6hom _pb + _/t6, (356) which is the wellknown fundamental equation at constant temperature. It is important to note that the above fundamental equation of the system, Equation (355), is for both the film as well as for the reservoir. The last term accounts for the net mechanical work performed in maintaining the area, 2A,, of contact between the surfaces and the solution as well as maintaining the two surfaces at a given distance H. This term can be broken down into two contributions one from the film and the other from the reservoir arising from the exchange of polymers and solvent molecules between the interface and the reservoir upon equilibration. Therefore it is evident that the variation of the excess grand canonical potential of the film at constant temperature is given by 69 e= tr (T 6v)dV, (357) JvI where V' denotes the volume of the film. This variation corresponds to the mechanical work contribution due to axisymmetric compression or expansion of the lV. r? S at constant temperature. Equation (357) is analogous the variation of the grand canonical potential defined in Equation (37) in Section 3.3. Pure compression is irrotational, so Vv is simply the rate of deformation tensor D with components Dii i = y, z. (358) Ai where the dilations Ai are the stretched lengths of line segments in the principal directions i = x, y, z with initial lengths of unity (in the undeformed state). 3.5.2 Estimation of Disjoining Pressure, Film Tension and Interfacial Tension For axisymmetric compression or expansion of two flat, parallel l. is, the osmotic stress tensor assumes the following form 7= TH (exe, + eyey) + Te ee, (359) where Tr and Tr are the tangential and normal stresses respectively. (The unit vector in each direction is denoted by e.) Substitution of the above equation in Equation (357) then leads tol 6ex= 6 T dV rdV ( T) A dV. (360) It follows from Equation (360) that ( Jqex\ jH  TH = r dz, (361) 6A T,H JO which defines the excess tangential stress (with respect to the reservoir bulk pressure) as the film tension. In evaluating the above derivative at constant temperature, it is to be noted that the corresponding GibbsDuhem equation given by Equation (312) must also be satisfied. It is not necessary that the chemical potentials of each species 1 Since the volume V1 is not constant during deformation, the evaluation of integral in Equation (357) is performed by mapping the volume V1 to an invariant reference volume VI'R using the transformation V' lV6VIR, where lv = Ilyl,. The above transformation also enables one to evaluate the variations of functionals according to the equality fv,, 6(lvf)dV'R 6 (fV,,, flvdV',) 6 (fs fdV). be constant in the film. We shall discuss the necessary criteria for equilibrium in the next section and make further observations in this regard. Moreover, from Equation (360) one also has = TdA8s As,,m (362) 6H /T,A, which shows that the disjoining pressure is simply the negative of the centerline normal osmotic stress T ,m. Here and elsewhere we denote the midpoint (H/2) between the two solid surfaces by the subscript m, i.e., Hm = H/2. The above expressions for 7 and 7rd can also be expressed in terms of the isotropic and deviatoric components of the tensor T. Since the osmotic stress tensor can be written in terms of the isotropic and deviatoric stresses in the system as 7 = iso + Tdev ( xe ee ee + ee, (363) with the isotropic component given by 1 2 1 Tso trT T 2 + T (364) and the deviatoric component given by 2 Tdev = r(T 7), (365) 3 one has, for the film tension, j7= H iso )dz, (366) and, for the disjoining pressure, TTd = [Tiso + Tdev]Hm (367) The interfacial tension a can then be obtained from the definition of film tension, 7 2o TdH. 3.6 Film Tension, Interfacial Tension and Force of Compression from Lattice Models We now reduce the general expressions presented above for the film tension and the disjoining pressure in Equations (366) and (367) for the case of lattice meanfield approximation. The purpose of this section is develop the expressions necessary to obtain the osmotic stress tensor (and hence the isotropic and deviatoric stresses) within the lattice formalism so that the force of interaction, the film tension and the interfacial tension can be obtained from lattice models. Although the results presented here can be generalized to any variations of the lattice theory, we shall primarily consider them in the context of the Scheui i' jFleer theory as it has been used extensively in the literature. 3.6.1 Full Equilibrium We first consider the case when the polymer 1i r~ are in full equilibrium with the bulk phase b, on a simple cubic lattice of size L x L x H, with each lattice element having unit length in each direction (i.e., the area A8 = L2, and z increases in unit steps). The total Helmholtz energy of the system (film + the reservoir) is then given by A PbV + jPir + As (p() ez)) W(z) Pz) (368) i z I Here the last term is consistent with the fact that the inhomogeneity of thermodynamic properties in the film is onedimensional (see earlier discussions on the thermodynamics of inhomogeneous systems). P(z) = pb + Hb H(), consistent with the definition of the stress tensor in Equation (352), pi(z) are the chemical potentials of species i evaluated for a homogeneous solution at local composition Qi(z), and ci(z) are positiondependent nonlocal fields which act on individual components. The fields ci are chosen such that (i(z) is an equilibrium distribution consistent with the configurational constraints imposed by the connectivity of polymer chains. The total amount of each species in the system is given by ni = ni + A i(z). (369) The criterion for equilibrium is determined by minimizing the total Helmholtz energy of the film and the reservoir at constant T, ni, As, and H. Therefore we have 6A = Pb6Vb + pn\ + A 6 ((z) ) )C (z ) P(z) (370) i z I since the terms Vb6Pb and n6pfi are zero. Note that the first two terms in Equa tion (370) are nonzero because they account for the change in the Helmholtz energy in the reservoir due to the exchange of material and energy with the film. Since the total amount of each species in the system is constant, we have 6bn As6Z{(z). (371) Using the above result and using partial molar volumes vi the first two terms of Equation (370) can be rewritten as pbVb + P 6n = A (pb Pb) i(z). (372) i z i Therefore Equation (370) becomes 6A A, Y, C 6pi(z) 6e(z) u6P(z) O(z) (373) 6A = < (373) S+ E ~i(z) C(z) v P(z) 1p + VPb 6bi(z) We identify the leading term on the right hand side of Equation (373) to be the GibbsDuhem equation for the film, analogous to Equation (312) and hence this term vanishes, leaving 6A A Pi(z) C(Z) P (P(z) Pb) i(Z) (374) For equilibrium, we have 6A = 0 for all variations at constant T, ni, As and H. This criterion gives us the appropriate nonlocal selfconsistent fields ci(z) as C(z) P(z) + 7, (n(z) Pb) (375) where we have incorporated the osmotic stresses. It is evident from the above expres sion that for the solvent molecules, c0(z) = 0, since they have no configurational degrees of freedom. These fields also vanish in the homogeneous bulk phase. Further, we can rewrite Equation (375) as (z) Ce(z) + vn(z) = p + vb. (376) This is analogous to a generalized 'membrane' equilibrium as discussed by Lyklema [60]. Alternatively, the criterion for equilibrium can be represented as the constancy of the exchange chemical potentials Pii ii ci (377) in the film and the bulk. Whilst one obtains polymerinduced forces from free energies in a grand canonical ensemble, it is essential that these transfer chemical chemicals are held constant. Substituting Equation (375) into (368) and simplifying, one obtains the excess grand canonical potential of the film as I 7A, =A, [f b n(z)], (378) equivalent to the result, Equation (348), obtained in Section 3.4 and analogous to the KirkwoodBuff formula. For compression under full equilibrium deviatoric stresses are absent. Therefore, Equations (364) and (366) imply that riso(z) = b (z). (379) It then follows that the normal stress in the film is simply the osmotic pressure UnI at the midpoint Hm and that the ldi .iiiii.; pressure or the force of compression per unit area under full equilibrium conditions is TTd = lm nb. (380) This result is in accordance with that for electrostatic interactions between charged surfaces. Whereas in the electrostatic interactions between two parallel plates one has an electric potential gradient balanced by the osmotic pressure, in the case of interacting polymer liv.i, one has a concentration gradient arising from entropic and energetic interactions of the polymer segments and the solvent, which is balanced by the osmotic pressure. For a symmetric system (where the adsorption energy per segment X, is the same for both surfaces and the surfaces have same area A8 exposed to the film and are otherwise identical), symmetry demands that the concentration gradient vanishes at the midpoint, leaving the normal stress everywhere in the film to be the midpoint osmotic pressure. Thus, under full equilibrium conditions, the normal force of compression of the film is simply the difference between the normal stress in the film and that in the reservoir. In our calculations, we only consider symmetric systems. Consistent with the definition of film tension, the interfacial tension is given by 2aj [In II(z)] (381) z 3.6.2 Restricted Equilibrium The case of more practical interest, commonly referred to as restricted equilibrium, arises when the total amount of polymer is fixed in the system during compression. This constraint on the total amount of polymer is imposed by introducing a Lagrange parameter g(z) that accounts for the additional potential (per segment in z) needed to maintain a fixed number of polymer chains in the system. Through a similar analysis as in the previous section, the criterion for equilibrium in the film can be shown to be [61] i(z) Ce(z) + Van(z) + g4(z) = P + Vab. (382) The film tension for the case of restricted equilibrium therefore becomes r' 7A A= > [nLb n(z) g(z) )] (383) Comparison of Equation (366) with Equation (383) shows that the expression for the deviatoric stress in the film is Tdev(z) = 2g(z)O(z). (384) As the isotropic stress is still given by Equation (379), the dlii 1111:; pressure for the case of restricted equilibrium is given by 7d Il, IIb 2 ., (385) The normal stress everywhere in the film is therefore 'ln 2.), . The interfacial tension under restricted equilibrium conditions does not follow from the definition of film tension because the deviatoric corrections differ for the film tension and the di]i 1ii11ii; pressure, as was shown in Section 3.5. The interfacial tension is simply defined with reference to the normal stress in the film and is given by 2a = [II I() 2 g(z)(z). (386) 3.7 Validity of the Approach The approach needed to calculate the force of axisymmetric normal compression of a binary system of interacting polymer livir at constant interfacial area and the work needed in maintaining the area of the surfaces exposed to the polymer 1., ris at a constant surface separation in lattice systems interfaciall tension) were described above. The traditional meaning of the interfacial tension as the work needed in lateral stretching or compression of the interface at constant surface separation does not apply to the lattice system. All the results derived in these sections follow from the thermodynamics of inhomogeneous systems, principles of continuum mechanics and the assumptions involved therein. Further all these relations are independent of the statisticalmechanical model one uses to calculate the composition of components in the film, osmotic pressure and appropriate chemical potentials. In what follows, we shall focus on the numerical meanfield theory of Sd!. il i' 1' and Fleer, a summary of which was presented in Section 3.4, as the statisticalmechanical model to estimate the forces of compression of polymer I iT. 3.8 Results and Discussion We begin here with the some results for the force of compression between two physisorbed polymer 1 i,ri using a lattice (numerical) meanfield theory. All calcu lations reported here are performed for firstorder Markovian chains in a monomeric good solvents (X = 0.0), on a simplecubic lattice under full or restricted equilibrium conditions. Since our calculations are performed on a lattice, the results are normalized per lattice site in a l ivr rather than per unit volume, as shown earlier. The deviatoric stress is therefore calculated on a per segment basis. Forces and segment densities are reported per unit area (i.e., per the number of lattice sites in a liv). 3.8.1 Full Equilibrium Figure 32 shows the variation of film tension and interfacial tension as a function of surface separation under full equilibrium conditions. The decrease in the film tension upon compression indicates that the polymer molecules in the film prefer to adsorb on the surfaces even for an adsorption energy2 of X = 0.5 kT. This is also evident from the fact that upon compression, polymer chains do not leave the film, resulting in an increase in the average density of the polymer segments (Ot/H) in the film (Figure 33). This makes it harder to compress the l'r iS, which is evident from the strong osmotic repulsive forces obtained in Figures 34a and 34b as well as in the sharp increase in the interfacial tension in Figure 32. The trend is very similar for stronger adsorption energies. The force profiles shown in Figures 34a and 34b reveal qualitative discrepancies between our results and the negative derivative of the excess grand canonical potential (as obtained from the meanfield theory), which has been interpreted in the literature as the force of interaction [34, 57]. The forces are predominantly repulsive, as opposed to the monotonically attractive behavior predicted in the literature for interactions under full equilibrium. Our results agree qualitatively with the analytical meanfield predictions of Ploehn [16]. Finally, in Figure 35 we show the interaction potentials W for different bulk concentrations for an adsorption energy of X, = 0.5 kT per segment for a chain of 100 segments. The interaction potential is obtained simply by integrating the force profile rH W(H)= f d( h)dh, (387) infty where h is used as a dummy variable for the distance between the two surfaces. This interaction potential is clearly not equal to the excess grand canonical potential interfaciall tension) shown in Figure 32. 2 The critical adsorption energy Xc for the lattice model is estimated to be 0.18 kT. The critical adsorption energy represents the magnitude of X, at which the entropic repulsion between the walls and the polymer chains cancel any wallsegment affinity. Adsorption occurs only for X, > Xc. A.04 0 C 0.05 Film Tension, y  Interfacial Tension, 2a C 0.06 =0.05 Sa. =0.5 k T S 0.07 v r =100 0.08 E 0.09 ir. 0. .1.1.1.1.1.1.1.1.. 6 10 14 18 22 26 30 Surface Separation, (H/a) Figure 32: Film tension 7 and interfacial tension 2a as a function of surface separa tion (H/a) under full equilibrium conditions. The results are shown for an adsorption energy X, 0.5 kT, chain length r100, bulk concentration Qb 0.05, and good solvent conditions, X = 0.0. 3.8.2 Restricted Equilibrium Under conditions of restricted equilibrium, the gap region is open with respect to the solvent but closed with respect to the polymer. In this case, the usual practice is to approximate the restricted equilibrium problem to an equivalent full equilibrium problem with an additional constraint that the total amount of polymer is invariant upon compression. By appropriately minimizing the partition function for a mixture of polymer chains and solvent molecules in the gap region, the segment density distri bution between the gap is obtained [34]. Once the segment densities are known, one can evaluate the semigrand canonical free energy relative to an 'effective' bulk solution with which the l~V. rs are in full equilibrium. Again, the negative derivative of this free energy has been interpreted in the literature as the force of compression. In this section, we shall first develop the equations for the deviatoric stresses and the force of compression within the lattice meanfield formulation by correctly incorporating the results of the generalized membrane equilibrium described earlier. 0.25 .. ... . .... .... 4 +Average number of segments 02 per unit volume 0.20 S0.15 E b = 0.05 ; X =0.5 Sr = 100 0.10  0.05 , 0 5 10 15 20 25 30 35 Surface Separation, (H/a) Figure 33: Average density of polymer segments (8t/H) in the interface as a function of surface separation (H/a) under full equilibrium conditions. X, = 0.5 kT, r 100, Qb 0.05, and x 0.0. As obtained in Section 3.4, the semigrand free energy is given by (also see Equation 24 in [34]), F ) n ( )ln0 (z) + x (z) ((z)) (O) (388) The second term in Equation (388) is an entropic mixing term while the third term contains contributions from enthalpic and entropic interactions. The first and the fourth terms partially account for the deviatoric stresses that arise in the system due to the confinement of polymer in the gap. Schel i,' and Fleer [34], in page 1886 of their manuscript, denote the last term as "a small attractive term accounting for the osmotic pressure of the solution outside the pl! Ii and neglect the term in their calculations. We interpret this term as an osmotic pressure contribution arising due to the confinement of polymer either outside or inside the surfaces depending upon the amount of polymer in the gap. Therefore, there could be conditions where this term is significant. This is demonstrated in Figure 36, which shows the negative derivative of the semigrand free energy with and without the last term. In what follows, we have recalculated the meanfield results in that work by including the above term. i(z) eC(z) + ,nl(z) + g (z) = + v_,b. (389) 0.0002 0.015 S0.0001 0.010 Force &y/H o.oooo : " S  ... : *.. ' 0.005   \ S0.0001 m 0.000 SForce 0 0.0002 .05 By/H 0.005 Sb =0.05 X =1 0.0003 5= 0 0.010 ,r =100 L.. r =100 0.0004 ....1 .. .... I .. .. i 0.015 I I 0 5 10 15 20 25 30 0 5 10 15 20 H/a Surface Separation, (H/a) Figure 34: Force per unit area f as a function of surface separation (H/a) under full equilibrium conditions for r 100 and X = 0.0. The negative derivative of excess grand canonical free energy is also shown for comparison. (a)X, = 0.5 kT, b 0.05; (b), 1.0 kT, Qb 0.005. A consequence of the approximation of restricted equilibrium as an equivalent full equilibrium problem is that, for a polymer segment, Equation (389) becomes p9(z) (z) + vfI(z) = ,bff + VIlff =b + vfb g, (390) where t fbf and II f are the chemical potential of the polymer chain and the os motic pressure corresponding to the 'effective' bulk solution. Thus, the effectivefull equilibrium approximation leads to g, an effective additional potential or stress needed per . iir that is a function of surface separation but independent of z g ( b b (pb ) I II f) (391) from which it follows that the interfacial tension under restricted equilibrium conditions is M 2a = [fIb () g(z)] (392) z=1 Correspondingly, for the d(i .iPii: pressure, Equation (385) is replaced by f = In nIb 2,/. (393) The chemical potentials and the osmotic pressures in Equations (391) and (393) are evaluated within the Floryfli,ii' : approximation for each lir using Equations 0.0020 i ^ 0.0015 \ 0.00005 = 0.00005 b = 0.005 0.0010 ... b 0.05 X 0.5 Sr =100 0.0005  ciP S0.0000 0.0005 * 5 10 15 20 25 Surface Separation, (H/a) Figure 35: Interaction potential W between the surfaces in full equilibrium with solu tions of varying bulk concentrations. X, = 0.5 kT, r 100, and X = 0.0. (345) and (346). (It follows from Equation (391) that for full equilibrium, for which b =b and I1b i b, g 0.) Figure 37 shows that, under certain conditions, the forces of compression predicted by Equation (385) can be qualitatively different from the derivative of semigrand free energy. In general, we observe that this difference is quantitative for extremely undersaturated 1lvri and for 1< irs close to saturation coverage and beyond. In the intermediate range, qualitative differences (i.e., repulsion instead of attraction) could result. To indicate the effects of the deviatoric stresses in the system, we show the variation of the effective deviatoric stress per segment for two surface coverages F = 0.5 and 0.75 (Figure 38). The former case corresponds to a starved 1 vr wherein the surface coverage is less than the saturation coverage F03 Here the polymer chains are constrained from entering the gap region, resulting in an attraction. On the other hand, the latter case corresponds to a slightly oversaturated regime (F > Fo = 0.7). 3 The saturation coverage oF is defined as the limiting value of F corresponding to b 0. For short chains, Fo is obtained from the adsorption isotherm (a plot of F as a function of Qb) from an extrapolation of the equilibrium coverage F beyond the initial rise to b = 0. 0.012 **Without solvent exchange term 0.010 * With solvent exchange term 0.008 I = 0.75 r = 200 0.006 0.004 0.002 0.002  3 4 5 6 7 8 9 10 11 Surface Separation, (H/a) Figure 36: Correct calculation of excess semigrand free energy under restricted equilib rium conditions. The results shown here are for a surface coverage 7 = 0.75, X, = 1.0 kT, r 200, and X = 0.0. Here the polymer chains are constrained from leaving the gap, resulting in an increased repulsion. The appropriate interaction potentials corresponding to the above cases are shown in Figure 39. 3.8.3 Implications on the MeanField Predictions We now turn to the implications of the new force calculations on the capabilities and limitations of the lattice meanfield theories. Here, we reinterpret the results [57] already published in the literature based on the incorrect thermodynamic formulation of defining the force to be the negative derivative of excess semigrand free energy. A plot of the force per unit area as a function of surface coverage F, for a surface separation of H/a = 4.5, where a is the lattice Ip 'i shown in Figure 310a reinforces the above observations. From our results, it is seen that SF2 predicts higher repulsive forces than SF1, probably because the elimination of backfolding through SF2 serves to decrease the entropy of the chains, making them more 'rigid' and harder to compress. The deviation of SFi from SF2 is particularly pronounced at higher coverages. Further, upon enlarging the lowcoverage region of Figure 310a, one observes that SF2 predicts slightly more attractive forces than SF1. This attraction, observed at low coverages, should correspond to that of ideal chains (SF1) and nonreversal chains (SF2). It is to 0.015 i U" 0.010 0.005 0.000 , ^ ... ... 0.005 S r0o.5 S 0.010 r = 200 I 0.015 88F/18H o0 I " Force L 0.020  0.025 I I I I 2 4 6 8 10 Surface Separation, H/a Figure 37: Force per unit area as a function of surface separation (H/a) under re stricted equilibrium conditions. 7 0.5, X, 1.0 kT, r=200, and X = 0.0. be noted that despite the rather significantly attractive effective deviatoric stress per segment at low coverages (as shown in Figure 38), the total force is only very weakly attractive. It has been observed [57] that, compared to Monte Carlo simulations with self avoidingwalk chains, SF1 and SF2 underestimate the surface saturation coverage, because the chain statistics permit overlapping of the polymer segments. For this reason, we examine the force as a function of reduced surface coverage (F/Fo); see Figure 310b. The rescaling preserves the observations already made in the context of Figure 310a. In addition, the corrected force calculations show that the forces at very low coverages are too small to provide any meaningful comparison with the linearity of force versus coverage predicted by scaling arguments, which are based on an analysis of the system as a collection of isolated bridges [62, 63]. The earlier results based on the SF2 formulation [57] appear to show an agreement with the scaling arguments for coverages up to about F ~ 0.4; but those results do not correspond to the force of interaction, as already noted. 0.020 0.010 tS g: o.ooo  *ee  S 0.000 S0.010 a. 4 0.050 0 .040 r=5 0040 r=200 .S X = 1.0 4 4.050 0.060 0 5 10 15 20 25 Surface Separation, (H/a) Figure 38: The variation of deviatoric stress per segment upon compression, for two surface coverages 7 = 0.5 and 7 = 0.75. The results are shown for sX = 1.0 kT, r 200, and x = 0.0, under restricted equilibrium. The crossover coverages4 obtained from the corrected calculations are significantly different from those obtained from the previous calculations. The corrected reduced crossover coverages are close to the prediction of the scaling/free energy functional theory (Rossi and Pincus [32], Fc,r ~ 0.4). In this context, it is interesting to contrast the numerical meanfield results with the twoorderparameter analytical meanfield theory of Semenov et al. [36], which predicts attraction for I,':u F/Fo < 0.98, for all values of H/a. Due to reasons as yet unclear, the analytical meanfield results [36] show surprisingly strong attraction compared to the numerical meanfield theory. 3.9 Concluding Remarks We have developed the thermodynamics of interacting polymer l1i. ri and have derived the appropriate expressions for polymerinduced force of compression between two surfaces, within the framework of quasithermodynamics. The main conclusion here is that onedimensional meanfield approximations do not i, /'/., ./li account for the mechanical workmode involved in maintaining the separation distance between the 4 Crossover coverage corresponds to the surface coverage at which the interaction between the surfaces changes from attractive to repulsive. 0.05 r              0.05 0.04  0.75  0.03 r II200 0.01 0 Surface Separation, (H/a) Figure 39: Interaction potential W between surfaces in restricted equilibrium for cov erages 7 = 0.5 and 7 = 0.75. The results shown here are for X, = 1.0 kT, r=200, and X 0.0. adsorbing surfaces. Therefore, one has to be careful in defining the interfacial tension and the normal force of compression. Further, we have presented the first correct results for the interaction forces between adsorbed polymer lIV'Trs based on the latticebased numerical meanfield theory. It is shown that qualitative differences between the correct results and the earlier incorrect formulation could occur under both full and restricted equilibrium conditions. The force of compression of the adsorbed lv.ri~ in full equilibrium with a homogeneous bulk solution is neither monotonically attractive as seen from the results of Schecl, i,' 1 and Fleer [34], nor is it monotonically repulsive, as claimed by Ploehn [16]. Under restricted equilibrium conditions, additional deviatoric stresses develop in the system. Depending on whether the polymer molecules are confined in the gap or outside the gap, these stresses cause additional repulsion or attraction respectively. It appears from these preliminary results and further results [17] shown in C'!h ipter 4 that meanfield theories highly underestimate the attractive forces at conditions when bridging effects are expected to dominate. The elimination of backfolding in the chain statistics hardly improves the predictions of the force of compression. The results presented here are significantly different compared to the 60 0.04 ....0.04 M 0.03 Force, SF / 0.03 Force, SF  0.02 Force, SF / 0.02 Fce,SF / / 1Force, / S 0.01 / I 0.01 0.01 8F/H, S 0. 01  S0.00 0.00 t. SF H/a =4.5 0.02 H/a = 4.5 0.02 8F/8H, SF r r=200 L SF/SH, SFi r = 200 8F/8H, SF2 0.03 I I 0.03 1 i 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 Surface Coverage, r Reduced Surface Coverage, F/FT Figure 310: Force per unit area f as a function of surface coverage 7 under restricted equilibrium conditions for a fixed separation of (H/a) 4.5. (a)Xs = 1.0 kT, r 200, and x = 0.0; (b) Effects of rescaling the surface coverage 7 by normalizing with the saturation coverage o7 on the force for (H/a) 4.5, X, = 1.0 kT, r 200, and X = 0.0. predictions of the twoorderparameter theory, even though the assumptions involved in the theories are similar in nature. CHAPTER 4 EFFECTS OF POLYMERLAYER ANISOTROPY ON THE INTERACTION BETWEEN ADSORBED LAYERS 4.1 Introduction In C!i ipter 2, we reviewed various theoretical (scaling, analytical and numerical meanfield theories) and computational approaches to the problem of physisorption of linear, flexible homopolymers onto one or two uniform, flat, parallel, impenetrable solid surfaces. We then developed a thermodynamic formalism for examining the forces of interaction of these lv. r and applied it using the lattice numerical mean field theory developed by Sd<, ul i, 1. and Fleer [34] with and without backfolding in Chapter 3. Both these Markov approximations consider an isotropic mean field. Further improvement in the theory arose due to the introduction of anisotropy in the mean field by partially accounting for the effects of bond orientations on the equilibrium properties of the system. This selfconsistent anisotropic meanfield theory (SCAFT) was first proposed by Leermakers and Scheul i' 1' [40] to study phase transitions in lipid 'ilii r membranes, in order to account for the anisotropic orientational interactions between the lipidlike molecules in a membrane. The theory was able to successfully predict the critical phase behavior of the membrane observed in experiments. This work also provided an elegant derivation of the theory from the basic principles of statistical thermodynamics. More recently, van der Linden et al. [41] extended the Scheul i' I' and Fleer theory to semiflexible polymers, in which bond correlations were incorporated. In both these works, the authors studied only the overall segment densities and some of the broad structural features. The focus in the former work was to develop a theory for membranes based on statistical thermodynamics and that of the latter work was to formulate a lattice meanfield theory for semiflexible polymers (e.g., 'wormlike' chains). Fleer et al. [42] have attempted to relate the numerical meanfield formalism and the twoorderparameter theory in an attempt to obtain closedform solutions that reproduce the numerical meanfield results. The numerical meanfield theory of Scheul i' I' and Fleer is, presently, the only theory from which one can obtain quantitative guidelines on the adsorption of short chains (of finite molecular weight). It is therefore important to ascertain the reliability of this theory in predicting the structural details of the adsorbed 1~ .r, and the forces of compression of two polymer l1 ir. Comparisons have been made with experiments in the literature (e.g., Fleer et al. [12]). However, many of the intricate structural details of the adsorbed l vr are either inaccessible or not easily accessible through experiments, such as specific information on loops, tails, and bridges. Moreover, experiments involve many parameters that cannot be precisely controlled or do not factor into most theories, such as surface roughness and polydispersity of the polymer. Another very important issue in comparisons with experiments is the ambiguity in relating a theoretical state of the system (as defined by the number of chains, number of Kuhn units, a theoretical adsorption energy, and a theoretical segment solvent interaction (X) parameter) with appropriate experimental conditions. These considerations limit the extent of any such comparison to one of a qualitative nature. In view of these, while it may be instructive and important to assess the validity of a theory against experiments, a simple, direct comparison would not only be limited but also, under most conditions, be misleading. On the other hand, computer simulations are 'exact' within their approximations (e.g., lattice approximation in lattice MC simulations), and comparisons against latticebased simulations would enable one to assess the limitations of the meanfield approximation in the context of lattice meanfield theories. We, therefore, use Monte Carlo results [27] of realistic chains (self avoiding walks) within the lattice formalism as a reference, to examine the predictions of the meanfield theories. In this chapter, we consider the effects of anisotropy in the 1v.r on the resulting meanfield predictions of the structure of the lIvr as well as the forces arising from the interactions between two l i r. Specifically, we introduce anisotropy in the SF2 formulation (S(!. u lP i ' and Fleer numerical meanfield theory with secondorder Markov chains). We also provide preliminary comparisons of the anisotropic meanfield predictions of the structure of the adsorbed lVir and the forces of compression of the Ii. i against results from rigorous computer simulations [57]. A summary of the anisotropic meanfield formulation is provided in the next section. (Further details are available elsewhere; see Fleer et al. [12]; Leermakers and Sdt, l i. 11,' [40].) We then discuss the improvements in the predictions due to the introduction of anisotropy and comment on the limitations of meanfield theories as seen from these comparisons. One of our objectives in the next section is to present a clear and easily understandable anisotropic meanfield formulation and to provide expressions for the various structural features of the adsorbed polymer l1.r. 4.2 SelfConsistent Anisotropic MeanField Theory (SCAFT) 4.2.1 Preliminaries and Notations Consider a lattice system confined between two flat, parallel, impenetrable surfaces and filled with polymer segments (chains of finite length, r) and solvent molecules. We only consider fluctuations in a single direction normal to the surface, which we define as the zdirection. Each segment or solvent molecule occupies one lattice site. The lattice adjoins one or two hard/adsorbing surfaces and is divided into M 1lIvi~s of sites parallel to the adsorbing surfaces, in the zdirection. Each lIr contains L lattice sites. The surfaces therefore correspond to z = 0 and z = M + 1, and adsorption takes place in z = 1 and z = M. If D is the coordination number of the lattice, then, a lattice site in any 1i.r z has D nearest neighbors, of which a fraction A/,z is in 1V. z'. The fraction of nearest neighbors that lie in the same 1.iVr, then, is A0, that in the .,li ,:ent 1iv.r is A1 and so on. Therefore, A can be viewed as the fraction of the nearest neighbors of a lattice site. In the case of a firstorder Markov chain statistics, A's are the stepweights of the random walk. Furthermore, in a simple cubic lattice, there are six nearest neighbors to a lattice site in a l. r z, of which four are in 1 r z and one each in the lI'. rs z 1. Therefore, for a cubic lattice, A i = 1/6, Ao = 4/6, and, Aj = 0,  j I> 1. If there are n polymer chains and no solvent molecules between the surfaces, then it follows that, nr + n = ML. (41) We define the volume fractions of polymer and solvent at any given 1.,r as, ~) z) ;() O) (42) L L such that, n(z) + nO(z) = L; (z) = 1 0(z) (43) and, n(z) nr; n(z) = n. (44) z z We now present the relevant equations for the case of restricted equilibrium. The polymer chains are modeled as stepweighted random walks in a simple cubic lattice. We consider a secondorderMarkov chain statistics in which immediate stepreversals (backfolding of segments) are disallowed. The dimensionless meanfield potential u(z) that a polymer segment experiences in 1 ir z is given as u(z) = Xs1 61z XsM 6Mz + uint(Z) + u'(z), (45) where Xsi is the Silberberg adsorption energy parameter for the polymer/ solvent pair on the surface at li .r z and %ij is the standard Kronecker delta function. For a symmetric system, Xs X=sM. The term uint(z) accounts for the energetic interactions between the polymer segments and solvent molecules within the BraggWilliams random mixing approximation. int(z) =x [(z()) o(z))]. (46) Here, X is the FloryHu'Iii: segmentsolvent interaction parameter that decides the solvent quality. In the present work, we examine only good solvent conditions, for which X = 0. The nearneighbor average of volume fraction of the segments in 1iv.r z is given by, M (A)) A (), (47) S1 which, for a cubic lattice with only the nearestneighbor interactions, becomes (O(z)) = A10(z 1) + AoQ(z) + A i(z + 1). The Lagrange parameter u'(z) in Equation (45) partially accounts for the excluded volume of the segment and solvent in a given 1 r, v. This is usually based on the BraggWilliams randommixing approximation. k=4 I I Layer z I I Layer z+1 k=4 1 2 3 4 5 0 U Empty Sites Filled Sites New segment Segments whose bond orientation will not 'block' the placing of the new segment ofthe chain in layer 2. Figure 41: Bond orientations and anisotropic mean field. (a) The notion of bond ori entations. (i) A typical polymer chain in a lattice. (ii) Segments and bonds. (iii) Bond Orientations. In the illustration above, the orientation of the bond between segment (s 1) and segment s (in (ii)) is k 1 (see (iii)). This way of representing bond orien tation thus indirectly specifies the position of the previous, e.g., (s )th, segment. In a cubic lattice there are six possible orientations. These six orientations can be thought of as three pairs of 'opposite' orientations. For instance, k = 1 and k = 3 are 'opposite' orientations. Consecutive segments with 'opposite' orientations will cause a backfolded conformation; (b) Conformations of consecutive bonds. (i) Straight (ii) Perpendicular (iii) Backfolded; (c) Schematic representation of the anisotropic mean field in a square lattice. Isotropic Mean Field: Probability of placing a new segment in 1.vr 2 = 3/6. Anisotropic Mean Field: Probability of placing a new segment in 1.vr 2 = 3/5. 4.2.2 Anisotropic Mean Field Anisotropic meanfield theory attempts to improve upon the BraggWilliams ap proximation in order to better account for the excluded volume. To better understand k=4 k=2 (iii) 7 I the idea of anisotropic meanfield, let us consider a polymer chain as a set of segments connected by bonds of defined orientations. The notion of bond orientations is illus trated in Figure 4la. Between any three consecutive segments, three conformations can be identified namely, a straight conformation, a perpendicular conformation and a backfolded conformation. These conformations are illustrated in Figure 4lb. A second order Markov chain statistics does not allow the physically unrealistic backfolded conformation. In the case of isotropic mean field, the excluded volume is accounted for by just requiring that the probability of having a segment in a 1v,vr is the fraction of empty sites in that 1~ , (This is the standard BraggWilliams random mixing approximation.) Therefore, all the bond orientations are equally likely (within the re strictions imposed by the chain statistics). In an anisotropic mean field, the orientation of a polymer segment in a given 1lvr depends on the orientations of its neighboring segments (the neighboring bonds), which apriori limit the probability of having the segment in that 1 I r. This is illustrated in Figure 41c. This therefore means that each orientation has to be weighted appropriately. This introduces a bias in the stepweights of the random walk, making the mean field anisotropic 1 .This idea is explained further below. In an isotropic mean field, the Lagrange parameter u'(z) is given by the fraction of 'empty' lattice sites in the 1 i,v z, which is equivalent to the volume fraction of solvents (for 'good' solvents). u'(z) = n(l (z)) (48) In case of anisotropic mean field, there are apriori more 'empty' sites available in the 1l.r z once we account for the fact that those segments in lir z with the same bond orientation and those in 1l,r z 1 with a complementary orientation will not 'block' the given bond orientation. Let Q(z, k) be the fraction of bonds with those orientations 1 Notice that the correction to the mean field is independent of the chain statistics used (in this case, a secondorder Markov statistics.) that will not block a bond with an orientation k. Q(z, k) = j (z, s, k) + (z', s, k')} (49) s Here, k and k' are 'opposite' bond orientations (see Figure 41(c)), and z' = z or z 1, as the case may be. Q(z, s, k) is the volume fraction of sth segments (in rmers) with a bond orientation k, in lv r z. Therefore, 1 Q(z, k) is the maximum available fraction of 'empty' sites in 1v.r z for a segment to occupy with a bond orientation k. This correction factor is given as, 1 g(z, k) (410) 1 Q(z, k) where Q(z, k) is the fraction of bonds with those orientations that will not block a bond with an orientation k. Note that in case of isotropic mean field, the correction factor is unity. 4.2.3 Statistical Weights and Composition Rule The Boltzmann factor GCzm(z) exp (uz (411) \ kT / defines the socalled free/monomer segment distribution function. It is evidently the unnormalized probability of a 'monomer' segment in liv.r z. We now define the unnormalized probability2 G(z, s, k), of a smer, with the endsegment located in l1vr z, with a bond orientation k (as per the definition of bond orientations in figure 41(a)). Since we model the chain using Markov statistics, the probability of a smer can be obtained by summing the probabilities of adding a monomer to the different possible orientations of a (s 1)mer that will give rise to the desired conformation (defined by G(z, s, k)), thereby resulting in a recursive relation for evaluating the statistical weights. To be consistent with the notation, we need to define the monomer statistical weights as G(z, 1, k) even though bond orientations do not have a physical meaning in this case. 2 We call this the statistical weight for obvious reasons. For a cubic lattice, we therefore write, G(z, k)= (z) (412) 6 The corrections for the anisotropy due to bond orientations should be introduced from dimers on. We now write the statistical weights for dimers. 6 G(z, 2, k) = Gm(z) g(z, k) G(z', 1,1) (413) = 1 Using secondorder Markov statistics, we can further write the statistical weights of a smer as G(z, s, k) = Gm(z) g(z, k) G(z', 1,1) (414) l1k' To introduce a shorthand notation, we write the above equations as G(z, s, k) = Gm(z) g(z, k) (G(z', s 1,1)) (415) Note that this way of evaluating the statistical weights automatically ensures chain connectivity as shown by Sd. i, l i: and Fleer [33]. We now proceed to evaluate the segment volume fractions (which we call segment densities) using the socalled composition rule G(z, s, k) ,,, G(z, r s + 1,l) (4, s, k) C if s / 1 or r (416) (5/6) Gm(z) G(z,s,k)y 6=1G(z,r +1,1) (z, s, k) = C if s = 1 or r (417) Gm, (z) r 6 i(z) (z s, k) (418) s=1 k=1 Here, C is a normalization constant to account for the fact that the statistical weights are not normalized. For the case of restricted equilibrium, it is defined as C = (419) r G(r) L G(r) where Ot is the total amount of polymer between the plates and G(r) is the socalled endsegment distribution, i.e., the statistical weight of finding a rmer anywhere between the two surfaces. The former is defined as Ot= z=n (420) z and, the latter is defined as M 6 G(r) = G(z, r, k) (421) z=1 k 1 A selfconsistent solution is obtained by assuming a density profile and isotropic meanfield (as initial guess) and using an iterative procedure to evaluate the volume fractions. 4.2.4 Structure of the Adsorbed Layer Once a selfconsistent solution is obtained for the segment densities, one can easily obtain details of the structure of the adsorbed lir and the free energy of the system. To do this, we see that the polymer chains between the two surfaces (0t) are in one of the five following groups 3 The chains are free, 0f. The chains are adsorbed only to surface '1', 0f. The chains are adsorbed only to surface '2', 0'. The chains form bridges, with the last chain end leaving from surface '1', 0 . The chains form bridges, with the last chain end leaving from surface '2', 20. Therefore we have, Ot = f + Oa + O + O + O (422) Since t CrG(r), we define, G(r)) Cf(r) () + GI(r) + G0(r) + G (r) (423) 3 We use the approach of Fleer [34] here. and further, M 6 G*(r) Z=G*(z,r,k) (424) z=1 k 1 for each of the group (represented by *). Now we proceed to define recursive relation ships for the statistical weights of free, adsorbed and bridged chains similar to the way we defined the statistical weights before. Free chains: Since free chains cannot have a segment in 1_. ri 1 or M, we have, Gf(z, 1, k) SG(z, k), 0, S< z < M z 1, M Gf(z, s, k) subject to the conditions (425) (426) (427) Gf(z, s, k)= 0, for all s, z 1, M. Adsorbed chains: We define similar expressions for adsorbed chains as for free chains. For instance, the statistical weight of a chain adsorbed to surface '1' is given by GI(z, 1, k) { G(z, k), 0, z 1, M otherwise (428) G(z, s, k) GI(1, s,2)  SGm(z) g(z,k) (G(z',s G.()5l4, 2) 1,1)), if z / 1, k (/ 2 , 1) + G(2, sl 1)} (429) (430) subject to the conditions G (M, s, k) 0; G (1,s, 4) 0, for all s, k. (431) In equation (431), the term Gm(1) g(1, 2) Y14 Gf(2, s 1,1) would occur in the recursive expression for Gf(1, s, 2) according to the notation. However, it actually corresponds to conformations of smers with only the end segment adsorbed. Hence it is added to Ga(1, s, 2). Gm(z) g(z, k) (G'(z', s 1, 1)) Bridged chains: The recursive relations for bridged chains are very similar to those for the adsorbed chains. For instance, the statistical weight of a chain forming a bridge with the last chain end leaving surface '1' is given by G1(z,, k) = 0, z> 1 (432) zs, k) G(z) gz, k) Gz', 1,1), if z k / 2 (433) G(1,s,2) Gm(1)5g(1, 2) { (2, ,1) + G(2, 1,1) + G(2, 1,}34) 1,4 subject to the conditions G (M, s, k) 0; G (1, s, 4) 0, for all s, k. (435) Again, the terms Gm(1) g(1,2) Y14 G (2, s1,1) and G,(1) g(1,2) :14 G(2, s1,1) would occur in the recursive expressions for G (1, s, 2) and Gj(1, s, 2) respectively. However they actually correspond to bridged chains with the last chain end at surface '1'. Hence they are added to Gb(1, s, 2). Now the segment densities of adsorbed chains and bridged chains are found by writing similar composition rules. As an example, the volume fraction of the segments which form tails from surface '1' is given by zr 62b Ga(z,s,k)Zl Y kG(z,r S+1,) (z) 2 C m (436) s=1 k=l G ) Since a chain has two ends, the prefactor 2 is added to the equation. Once we have the statistical weights of free, adsorbed and bridged chains, we can also estimate the average number and sizes of loops, tails, trains and bridges. Details of these are provided in Schel iP' 'u and Fleer's paper [34]. To give an illustration, we consider the average number and size of loops. In a loop, both the ends are adsorbed on the same surface. Therefore, we consider the statistical weight of an adsorbed r mer having the .g,,, ,.I s in 7.';., '2' and the gn, ,.. s + 1 in 7.';., '1' adsorbedd), with the rmer adsorbed at least once before yi,, ,., s. This is equal to the product Gal (2, r s + 1, 4) >kt4 Gal (2, s, k). We normalize the statistical weight with the statistical weight of an adsorbed rmer to obtain the probability of finding an adsorbed rmer having the segment s in lir '2' and the segment s + 1 in 1, r '1' adsorbedd), with the rmer adsorbed at least once before segment s. This will be equal to the fraction of adsorbed chains with loops ending in 'i,,n ,,i s. Summing over all the possible values of s gives the average number of loops per adsorbed chain, ni. The average number of loops per adsorbed chain adsorbedd on surface '1') n,l is then given by G Z3 a{(2,r s + 1,4) EZkcG(2, s, k)} Gn' a(r) G.(2) (437) M 6 G ) = G z, r, k). (438) z=1 k 1 The average size of the loops in the chains adsorbed on surface '1' la, is then given by (a ,1 (439) '1,l where (,1 is the fraction of loops, O0 is the amount of polymer adsorbed on the surface '1', and are given by ', = a (440) 0" ,tr ,(z) + ,ta(z)}. (441) Loops are formed on surface '1' by chains in groups '2' and '4', i.e., by chains adsorbed on surface '1' with or without forming bridges. Therefore, an expression can be written for the average number of loops on surface '1' per bridging chain, n, as explained in Schel i' .1 and Fleer's paper [34]. The average size of such loops can be calculated using an expression similar to Equation (439). The average number of loops on surface '1' per unit area, ni, is then given by n= Lnil + nb,l f} (442) where f is the fraction of chains adsorbed on surface '1', fb is the fraction of bridging chains with the last chain end leaving from surface '1', and are given by 0 (443) OCb fb (444) fl >7 z The average size of loops adsorbed on surface '1' 11,1 is then obtained as a weighted sum of the two average sizes l", and l/,, and is given by a a a+ lb b ,b S+ nl fb (445) nif + bI 11 4.2.5 Estimation of Interaction Forces A detailed derivation of the thermodynamics OF interacting polymer li. rs and the method of calculation of polymerinduced forces was presented in C'! lpter 3. Also, a brief summary of the method of estimating polymerinduced forces using Monte Carlo simulations was presented in C'! lpter 2. 4.3 Results and discussion We shall now examine what changes or improvements one observes concerning the structure of the adsorbed l. ri and the forces resulting from the interaction between two adsorbing (polymerbearing) surfaces when anisotropy is introduced in the meanfield formalism. Our primary focus here will be on the results of SCAFT (with nonreversal chain statistics analogous to SF2) relative to its isotropic version, SF2. The structure and isotropic stresses in the system can also be obtained exactly using simulations, as we have discussed elsewhere [57]. We shall compare the predictions of the structure of adsorbed 1I, r and the interaction forces by SCAFT and SF2 with some simulation results. The simulation results used in the following discussion are based on a lattice Monte Carlo technique in which we model the polymer chains as selfavoiding walks (SAW's) and sample the chain statistics using a modification of the configurational bias algorithm of Siepmann and Frenkel [64] due to de Joannis [43]. We use periodic boundary conditions in the x and y directions and consider two impenetrable, adsorbing surfaces confining the Il i~ in the zdirection. The surface segment interaction is considered only in the .,.i i:'ent Ii Vr, as shown in Equation (45). All results have been generated for chains of 200 segments, in a good solvent with a simplecubic lattice of size L/a = 40 in the x and y directions 4 where a is the lattice spacing. In this work, we focus on the introduction of anisotropy in the meanfield theory and attempt to understand what improvement it offers. An attempt is made to relate the force between the two adsorbing surfaces to the structure of the adsorbed 1lv r. 0.5 C 1 *SCAFT S* SF S10F 0.4 SCAFT SF V Simulations 0.2 0 SCAFT  *(z) 0.3 4(z) 10o2 S= 0.75 .. 10' =I.0 v 0.1 r = 200 r=200 v v s = 1.0 s =1.0 0.0 I I I 10 T"" " 0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 z/a z/a Figure 42: Overall segment density distribution: Comparison of lattice meanfield results (SF1, SF2, SCAFT) with lattice Monte Carlo simulations. (a) H/a = 5.0. (b) H/a = 40.0. 4.3.1 Structure of the Adsorbed Layers We first focus on the overall segment density distribution. Segment densities are reported, following the normal convention, as the fraction of a lattice li r occupied by segments (i.e., area fraction). As an example, Figure 2 presents the overall segment density for the case of surface coverage F = 0.75 and H/a = 5.0 (Figure 42a) and for F = 1.0 and H/a = 40 (Figure 2b). The predictions of SF1 (i.e., based on the firstorder Markov statistics for the chains) are also shown in Figures 42a and 42b. The corresponding segment densities of loops, tails, trains and bridges for F = 0.75 and H/a = 5.0 are given in the appendix (Table Bl). Some more results on the overall segment densities and segment densities of loops, tails, trains and bridges are available 4 The numerical meanfield theories considered here are onedimensional, i.e., the x and y directions extend to infinity, and variations are considered only in the zdirection. Hence in the meanfield context, the lattice size of (L/a)= 40 specified in the x and y directions simply implies that 1600 lattice sites are considered per lIr (normal to the x y plane. in the appendix (see Tables B2 and B3) for different surface separations and surface coverages. Based on the above results, first one observes that, for H/a = 5.0, while the predictions of the overall segment density of SF1 agree well with the simulations, those of SF2 differ consistently. Since SF2 prevents segment backfolding, it permits a lower number of allowable conformations near the surface than SF1, which results in reduced segment densities near the surface (where the concentration is high) and higher value of densities away from the surface. This results in SF2 predicting higher segment densities for the bridges, loops and tails and lower densities for trains as compared to SF1 and simulations. Introduction of anisotropy significantly improves the predictions. SCAFT partially corrects SF2 in the appropriate manner. The overall segment density profiles show a surprising agreement with the simulations for H/a = 5.0. However, for H/a = 40.0, it is clear that the corrections due to anisotropy are insufficient to capture the interactions between the segments accurately. Even SCAFT predicts an order of magnitude higher segment densities near the center. In general, the correction due to anisotropy in the mean field is twofold: Near the surface: In typical situations near an adsorbing surface one would expect a significant density of trains under strongly adsorbing conditions (i.e., (z = 1)) in which all segments are oriented in the same direction (i.e., parallel to the surface). Introduction of anisotropy will cause more bonds to orient along the surface, thus giving rise to an increase in the number of segments forming trains. This is confirmed by the increase of segment densities of trains (relative to SF2) observed in Figures 42a and 42b. This improvement also gives rise to an excellent agreement between SCAFT and simulations in the overall segment density near the surface. Far from the surface: Anisotropy also leads to an underprediction of the segment density of bridges compared to SF2 and simulations. Far from the adsorbing surfaces, the bond orientations are almost random and hence the introduction of anisotropy (ordering) only serves to increase the 'rigidity' of the chain, i.e., decrease the number of allowable conformations for that part of the chain, resulting in a lower density compared to isotropic meanfield predictions. Further, restricted equilibrium requires that the total amount of polymer chains between the surfaces remain constant. Therefore, an increase in the segment density near the surface will have to be accompanied by a corresponding decrease away from the surface. 030 8 V Simulations = 0.75 Simulations s SCAFT r=200 *SCAFT, 1.' SFF X=I. 2.i _. 2 1 SF A O 20 r 0.75 , e r=200 ." C 0 o ; ' 0) 10 2 4 6 8 10 12 02 4 6 8 10 12 H/a H/a (a) (b) Figure 43: Number and size distribution of bridges, as a function of H/a, for a con stant adsorption energy, X = 1.0 kT and F = 0.75 (a) Average number of bridges, nbr, (b) Average size of bridges, lbr. SCAFT also predicts lower segment densities of tails and loops 5 relative to SF2 but higher values compared to the simulations. Again, one observes that the correction due to anisotropy leads to an improvement but the results still differ from those of the simulations possibly because of the meanfield approximation in the introduction of anisotropy. The results obtained for the number and size of bridges, loops and tails confirm the above observations. Sample results are shown in Figures 43a and 43b for the number and size of bridges (for F = 0.75). Further results for the number and size of bridges, loops and tails as functions of F and H/a are available in Appendix A (figures A 1 through A 12). SCAFT predicts lower number of bridges than SF2 and simulations, but bridges of larger average size than the simulations. This difference is pronounced at larger separation distances. The net result is an underprediction of the segment densities contributed by bridges. 4.3.2 Interaction Forces One of the central issues that is not clearly understood is the relationship between the structure of the adsorbed li. r and the resulting forces of interaction between two adsorbed 1i rv. Evidently, the work done in compressing two lV. is (and hence the interaction force) depends on the rearrangement of the bridges, loops, tails and trains 5 This could explain why SCAFT predicts lower repulsion than SF2 in Figures 45a. upon compression. While bridges generally contribute to an attractive interaction be tween the surfaces, interactions among tails, bridges and loops result in steric repulsion. It is of interest to examine the effects of bond correlations on the balance between the two competing factors in determining the interaction between two adsorbed l1, rs. Further, as noted earlier, an evaluation of the performance of meanfield approximations would be incomplete without an examination of the forces of compression due to the interaction between two 1i, r. Here, we shall consider the effect of the introduction of anisotropy in the mean field on polymerinduced forces. At low surface coverages, one essentially encounters a singlechain regime. There fore the system can be analyzed as a collection of single chains. When the surfaces are close to each other, at strong adsorption conditions, the chains form bridges between the surfaces. These bridges are loosely stretched, and as a first approximation, can be considered to behave as Hookean springs L;]'' As the surface separation increases, the bridges become strongly stretched, and then one can consider the system to be a collection of nonlinear elastic tethers [62], with each tether governed by the Pincus law of elasticity [65]. In this singlechain regime, the interaction between the two surfaces is predominantly attractive, with the exception of very low separations, where steric interactions between the segments of the chain lower the attraction. As the surface coverage increases, the chains are packed closer and closer that beyond the proximal regions of the l.. ri, the system is in a semidilute regime, and steric interactions between different sections of a chain and between chains become important. Under such conditions, the force of interaction between the surfaces crosses over from attraction to repulsion. Such behavior has also been qualitatively observed in experiments (see for instance, Fleer et al. [12]). 6 In this work, the distribution of bridges and snapshots of simulations are shown for a single chain of 1000 segments, to illustrate the importance of bridgebridge steric interactions, especially at weak adsorption energies, and how such interactions p1 l a vital role in systems with multiple chains. Schenll i' i1 and Fleer [34] have examined this problem in the lattice meanfield approximation, and have been able to reproduce the bridgingattractionandsteric repulsion force profile qualitatively. They used the firstorder Markov approximation to model the chains in the lattice (SF1). What is surprising about their results is that the meanfield theory appears to provide reasonable predictions in the singlechain regime, which is counterintuitive. In C'!i pter 3 we have discussed the correct calculation of forces of compression using lattice meanfield theory, and have reinterpreted the Scheul, i. iFleer results. Here, we show that even with the introduction of anisotropy in the mean field, the lattice meanfield theory fails to capture any significant bridging attraction. We shall restrict our discussions to the effects of polymer 1?v r anisotropy on the predictions of force of compression of the polymer 1.r . We first examine the force per unit area (the excess normal stress in the system) as a function of surface separation H/a, for surface coverages F = 0.25 through 1.25, and for an adsorption energy X = 1.0 kT, in Figures 44a through 44f. At low coverages, for the extent of bridging observed (see for instance Figures 43a and 43b and Figure A 1 in Appendix A), meanfield theories predict nearly negligible attractive forces. Further, the meanfield results, even considering bondorientation effects, are significantly more repulsive than predicted by the simulations. We shall revisit these observations towards the end of this section and draw further conclusions. An interesting observation to be made from these figures is that at high coverages, SCAFT and SF2 predict a repulsive force with a power law behavior (shown only for F = 1.25), the exponent being independent of the coverage. The exponents are 3.67 and 3 for SCAFT and SF2, respectively. To further illustrate and understand the limitations of meanfield theories, we examine the force as a function of F for a fixed surface separation H/a = 4.5, shown in Figure 45a. The results of SF1 (with the correct calculation of forces) are also included for comparison. It is clear that at low coverages, there is seldom any attraction predicted by any of the meanfield formulations. On the other hand, simulations show that at low coverages, where the surface consists of isolated polymer chains, there is a linear increase in the attractive force with a corresponding increase in the coverage. S2/ *SCAFT, r=0.25  SF2, r = 0.25 CSimulations, = 0.25 r =200 ; X, = 1.0 2 3 4 5 6 7 8 9 1C 4 6 10 12 0 S SCAFT, F = 0.5 4 SF2, F = 0.5 USimulations, F = 0.5 Sr = 200 Xs =1.0 2 3 4 5 6 7 8 9 10 H/a  SCAFT, F = 1.0  SF r = 1.0 *Simulations, r =1.0 S r = 200 1 Xs = 1.0 e#....m  2 4 6 8 10 12 14 16 H/a SCAFT, F =1.25 0 0 SF2, =1.25 0* Simulations, F = 1.25 S0o r=200 0 = 1.0 o 0 0 " '\o . i . 2 4 6 8 10 12 14 Figure 44: F= 1.25. (a) F 0.25. (b) r 0.5. (c) F 0.75. (d)r 1.0. (e) F 0.25 0.05 0.35 1.125. (f) 0.25 0.6 . 0.2 SCAFT 0.5 ,SCAFT 0.2 SF SF I * *SF S Simulations 0.4 /a= 0. 15 04~H/a =5.0 H/a=4.5 r =200 r= 200 0.3 =0.74 0.1 X. I : / 02 5 / 0.2 0.05 a 0.05 0 .1 .. .. . 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 r ric F F/F (a) (b) Figure 45: Force per unit area f at a fixed separation of H/a = 4.5, (a) as a function of surface coverage F, for adsorption energy X, = 1.0 kT, (b) as a function of rescaled surface coverage F/Fo, for rescaled adsorption energy X, Xc = 0.74. At high coverages, we observe that SF2 predicts higher repulsion than SCAFT. This is counterintuitive, since the introduction of anisotropy serves to increase the 'rigidity' of the chains. For this reason, one would expect a polymer 1v.r in an anisotropic meanfield to be less compressible than one in isotropic meanfield. However, mean field theories underestimate the saturation coverage because the excludedvolume interactions are implemented in a meanfield sense and that the chain statistics permit overlap and crossover. It is found that the saturation coverages for simulations [66], SF1, SF2 and SCAFT are, respectively, 1.2, 0.7, 0.72 and 0.92. This explains why SF2 predicts less compressible l, rs than SCAFT, contrary to intuition. Moreover, due to the differences in the chain statistics and the meanfield assump tions, one can expect that the critical adsorption energy 7 Xc will be different for the different approximations, namely, SF1, SF2 and SCAFT, and SAWsimulations [66]. 7 The critical adsorption energy Xc is the effective adsorption energy for which the energetic attraction of the segments to the surface is compensated exactly by the en tropic repulsion arising from the conformational restrictions imposed by the presence of surface. Therefore, at Xs = Xc, the overall segment density profile is flat. The critical adsorption energy is estimated as the adsorption energy at which the surface excess 0" (= (z( ) wb), where Qb is the bulk concentration) is zero. 81 0.05 9.5 SSCAFT o 9 0 SCAFT a S 0.04 SF SF SF S8.5 * H/ a=5.0 H/a = 5.0 [ 0 r=200 r =200 0.03 X.= 0.74 m = 0.74 E 2 .N E El 7.5 Z 0.02  0 75 0) 6.5 Q 0.01 0 5.5 . 0 0.5 1 1.5 2 0 0.5 1 1.5 2 F/F FlF 0 0 (a) (b) Figure 46: Number and size distribution of bridges, as a function of rescaled surface coverage P/F0, for a constant rescaled adsorption energy, X, Xc= 0.74, and H/a = 5.0. (a) Average number of bridges, nbr, (b) Average size of bridges, lbr. Our calculations show that the critical adsorption energies are, respectively, 0.18, 0.22, 0.21 and 0.26. Due to the differences in oF and Xc, it is instructive to examine the results for the same relative surface saturation and relative adsorption energies. There fore, we plot the force per unit area as a function of rescaled coverage F/Fo for a fixed H/a and a rescaled adsorption energy (Xs Xc) in Figure 45b. The results are similar to those shown in Figure 45a, except that at high coverages, SCAFT is now seen to predict higher repulsive forces, as would be expected. The average number and size of bridges, loops and tails are plotted as a function of F/Fo for a given (X, Xc). Figures 46a and 46b show the number and size of bridges. The number and size of loops and tails are available in the supplemental information (see Figures A4 11 through A4 14). An enlarged view of the results at low coverages, illustrated in Figures 44a, 4 4b, 45a, 45b, shows that SCAFT and SF2 do predict some attraction, even though quantitatively negligible compared to the corresponding simulation results. As stated earlier, in a highly undersaturated regime, the chain conformations are dictated essentially by singlechain statistics. The assumption of meanfield is expected to be highly inaccurate here. For chains in a lattice, constructed with a firstorder Markov statistics, with random mixing approximation in the lateral dimensions, the following expression has been used [67] to estimate the radius of gyration R,, expressed as a 82 10   SCAFT 8  SF I  Pincus Law SSimulations / 6 P H/a = 4.5 r=200 4 0s = 1.0  4 2 o. '0  ..0 2 . , 2  I  I  I  I  I  I  0 0.2 0.4 0.6 0.8 1 1.2 1.4 F Figure 47: Force per bridge fb, as a function of F for a fixed wall separation H/a = 4.5. multiple of the length of a step in the lattice, R (r/6) ( + D) (1 D), (446) where r is the chain length, and D is the lattice coordination number. For a chain length of 200 segments in a cubic lattice, the radius of gyration is about 5.7 lattice units. Based on the estimates of rms thickness of the adsorbed l. r, one can estimate that singlechain statistics prevails up to a surface coverage of F r 0.5. Therefore, the observed attraction then corresponds to that of an ideal nonreversal chain (SF2), and a weighted ideal nonreversal random walk (SCAFT). We noted earlier that at low coverages the system is expected to behave as a collection of single bridges with negligible interactions among bridges. Therefore, the magnitude of the attractive force is expected to increase linearly with surface coverage, as seen in the simulation results. The number of bridges in this regime decreases exponentially and the length of the bridges increases linearly 8 with H/a. In order to see if the system behaves as a collection of elastic tethers (governed by Pincus law of elasticity [65]) in the strongly stretched region, we plot the force per bridge as a function of F in Figure 47. At very low coverages, one does observe a qualitatively similar behavior, though the magnitude of the force is much less than the Pincus law estimate and the predictions from the simulations. The preceding observations indicate that the meanfield theories are severely limited in their ability to provide a reasonable quantitative estimate of the attraction between interacting polymer l1. r at low coverages, under restricted equilibrium. They predict qualitatively expected behavior at higher coverages, where steric interactions are significant. The attempt to improve meanfield predictions by the introduction of anisotropy seems to work in the prediction of structural details of the adsorbed l.,r, but is clearly wanting in the prediction of forces. To obtain any significant improvement in the predictions of the theories of polymer adsorption, one may have to go beyond the constraints of the mean field. Further, inadequate chain statistics (which one often resorts to in order to retain manageable propagation relations) introduce further limitations in the theoretical predictions. Even though the elimination of backfolding leads to a significant improvement, as shown in this work and by Jimenez et al. [57] and Simon and Ploehn [68], this is merely the simplest of the refinements that may be necessary. A careful examination is needed to study the limitations due to chain statistics. 4.4 Concluding Remarks In summary, we have presented an examination of the effects of the orientational anisotropy of polymer chains in a physisorbed 1 v.r on meanfield predictions of struc ture of the lirv and the resulting forces of interactions between two lIi r. In addition to chain statistics, anisotropic effects are expected to be among the most important elements of meanfield theories having an impact on the predictions. Our objective here s This type of behavior has been ii. 1. .1 by Ji et al. [62], who show that the elas ticity of such tethers is nonlinear and is governed by the Pincus law of elasticity [65]. has been to present at least a preliminary examination of what improvements can be expected through the introduction of anisotropy. The results show that introduction of anisotropy does improve the segment density distribution close to the adsorbing surface. However, far away from the surface, meanfield theories consistently predict higher densities than simulations. It is also found that meanfield theories fail to provide useful results at lower coverages on the forces of interaction. A linear force profile, qualitatively consistent with the scaling predictions of Ji et al. [62], is observed at low surface coverages, even though there are quantitative discrepancies. Both the meanfield formulations predict the crossover from net attraction to repulsion to occur at similar coverages (F/Fo ~ 0.5). In the above comparisons, we have used the structural properties and interaction forces computed using Monte Carlo simulations as a reference. However, a caveat is in order in this respect, namely that there is no clear way to compare rigorously the meanfield theories with exact calculations (e.g., simulations). In order to be able to make any conclusive comments on the reliability of meanfield theories even for the simple problem of homopolymer adsorption, one has to consider many issues. First, an agreement between a meanfield formulation ( i, SFi) and simulations for only certain properties (F~, segment densities) does not necessarily imply that the formulation will be equally good for all properties. Secondly, the fact that improving the chain statistics leads to improved segment densities in certain regions (as we noted in the previous section) does not guarantee that similar improvements can be expected in the overall performance. That is, agreement in certain respects could occur because of fortuitous cancellation of errors, while the errors could compound each other in other respects. Thirdly, it is unclear currently as to what is the right way (or, the best way) to compare meanfield calculations with simulations. Intuitively, it appears reasonable to compare the results for the same rescaled coverage, F/F0, and rescaled adsorption energy, (X, Xc). However, whether such a rescaling is sufficient (or, even valid) remains unclear. This issue deserves further attention, since the interpretations could be misleading if the right comparisons are not made. On the other hand, despite such uncertainties, the use of simulations as benchmarks remains the only option to examine 85 the reliability of the meanfield theories, since a direct comparison with experiments only introduces additional uncertainties. CHAPTER 5 BENDING OF SEMIFLEXIBLE POLYMERS 5.1 Introduction This chapter concerns the problem of bending of a single semiflexible polymer modeled as an incompressible elastic rod, under two different conditions: one end of the rod is clamped with respect to both position and orientation while the other end is clamped with respect to position but has a flexible orientation (the I ,riI,, i,:,,,: problem), and both ends of the rod are clamped with respect to position but have flexible orientations (the doublehinge problem). 5.2 Wormlike Chains as Slender Elastic Rods In C'!i pter 1 (see Section 1.1.3), we discussed the wormlike chain model for semiflexible polymers. Here we state the connection between the wormlikechain model, a microscopic description of semiflexible polymers, and a macroscopic continuum description based on the theory of elasticity, as elastic rods. For detailed derivations, the interested reader is referred to C'!i pter 3 of Yamakawa [2]. The probability distribution of finding the end segment (corresponding to s L) of a wormlike chain at position R given that the first segment is at origin and has a specified orientation uo for L > 0 is the Green's function. Once again, r(s) and u(s) may be regarded as Markov random processes on the proper time scale of s (or L). This Green's function satisfies a Schrodingerlike equation analogous to the behavior of a rigid electric dipole in an electric field. The solutions for such equations are known. The 'Lagrangian' for this problem is given by L c +k u, (51) kT where U [u(s)]2, (52) 2 