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Forces and fluid structure between patterned solutes : the influence of solvent phase behaviour Overduin, Sarah Danielle 2005

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Forces and Fluid Structure Between Patterned Solutes: The Influence of Solvent Phase Behaviour by Sarah Danielle Overduin B . S c , Brock University, 2000 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Chemistry)  T H E UNIVERSITY OF BRITISH C O L U M B I A October 4, 2005 © Sarah Danielle Overduin, 2005  11  Abstract The behaviour of planar surfaces and spherical solutes immersed in a liquid near liquidvapour or liquid-liquid coexistence is examined using computer simulation and integral equation theories. Of interest is the structure of the solvent, and the resulting forces between the solutes. B o t h uniform and chemically patterned solutes are examined. Grand canonical Monte Carlo calculations are used to investigate the phase behaviour of a binary mixture of Lennard-Jones particles (species A and B) confined between planar, parallel, chemically patterned plates. Attention is focussed on the influence of surfaceinduced transitions on the net force acting between the plates. In addition to the stable and metastable bulk states that play a crucial role for homogeneous surfaces, for certain patterns and surface separations a bridge phase analoguous to that recently reported for one-component systems is observed. We find that the separation at which bridge formation occurs is limited by the unfavourable interfacial tension between the A-rich and 5-rich regions of the fluid. It is found that bridge phase formation leads to strongly attractive plate-plate forces that are equal in magnitude to those observed for homogeneous surfaces. The effect of surfactant particles on the confined mixture is also examined. We show that these liquid bridges can be extended by reducing the interfacial liquid-liquid tension when surfactant particles are added to the system. In addition, other fluid structures that are not observed in the binary fluid can be stabilized. We give a qualitative discussion of the surface-surfactant induced liquid structures and examine in detail the associated forces acting between the plates. Isotropic and anisotropic hypernetted-chain ( H N C ) integral equation theories are used to obtain the interaction of solutes both near and far from the solvent liquid-vapour or liquid-liquid coexistence. Uniform and chemically patterned (patched) solutes are considered, and the influences of particle and patch sizes are investigated. Solvophilic (or A-philic, in the case of mixtures) and solvophobic (A-phobic) solutes (or patches) are examined. Near liquid-vapour coexistence in the one-component fluid, drying-like behaviour occurs between solvophobic solutes (patches) of sufficient size. This gives rise to relatively long-ranged attractive forces that are strongly orientation dependent for the patched solute particles. Similar results are obtained in the yl-rich mixture; "drying" of species A, and "wetting" by species B occurs near ^4-phobic solutes (patches) in an A-rich fluid. In a 73-rich fluid, "wetting" by species A and drying of species B occurs between A-philic solutes (patches). We also report grand canonical Monte Carlo results for a pair of uniform solutes and demonstrate that the anisotropic H N C theory gives qualitatively correct solvent structure in the vicinity of the solutes. Comparison with previous simulations also shows that the solute-solute potentials of mean force given by the anisotropic theory are more accurate (particularly at small separations) than those obtained using the isotropic method.  iii  Contents Abstract  ii  Contents  iii  List of Tables  v  List of Figures  vi  List of symbols  ix  Acknowledgements  x  1  Introduction  1  2  Theoretical background 2.1 Statistical mechanics of 2.2 F l u i d phase stability  6 6 11  fluids  3  F l u i d s c o n f i n e d b e t w e e n c h e m i c a l l y p a t t e r n e d surfaces 3.1 Introduction 3.1.1 Summary of related work 3.2 Models and Simulation Methods 3.3 Binary mixtures 3.3.1 Chemically Homogeneous Walls 3.3.2 Chemically patterned surfaces 3.4 Surfactant-stabilized liquid structures 3.4.1 B u l k Systems 3.4.2 Confined Systems: Fluid Structure 3.4.3 Confined Systems: Forces between Surfaces 3.5 Summary  4  I n t e g r a l e q u a t i o n c a l c u l a t i o n s for u n i f o r m a n d p a t t e r n e d p a r t i c l e s 4.1 Survey of previous work 4.2 Model Systems 4.3 Theoretical Approach 4.4 Results for solutes in a one-component fluid 4.4.1 Stability of the solvent 4.4.2 Uniform Solutes 4.4.3 Patched Solutes  14 14 15 15 19 19 21 29 30 34 41 46 . .  49 50 51 53 59 59 60 69  Contents  5  4.5  Results for solutes in a two-component fluid 4.5.1 A-rich mixtures 4.5.1.1 Stability of the solvent mixture 4.5.1.2 Uniform and patched solutes 4.5.2 5-rich mixtures 4.5.2.1 Stability of the solvent mixture 4.5.2.2 Uniform and patched solutes  4.6  Summary  Summary and Conclusions  iv 76 76 76 78 95 95 96 106 107  Bibliography  110  A Legendre transform used for solution of the anisotropic H N C theory .  115  B Force between particles  117  V  List of Tables 3.1 3.2  Values of the interaction parameters X 3 for the different surfactant models. Properties of the stable and metastable state using a (lOcrx 10crx20cr) system with homogeneous walls. The densities and mole fractions are obtained at the midplane. The numbers in this table can be regarded as the bulk values. a/  16  21  vi  List of Figures 3.1 3.2 3.3 3.4 3.5 3.6  Example of patterning on the surface of the plates Density profiles of the stable and metastable states of a fluid between homogeneous surfaces Density and mole fraction profiles at the midplane for various values of £ i . The three-dimensional density profile between chemically patterned surfaces A comparison of the net pressure obtained via the mechanical and virial routes The mole fraction, XA, in the midplane region, the reduced surface tension, /?7<7 , and the reduced net pressure, fiP cr , as functions of the surface separation Mole fraction profiles at the midplane for different separations Density and mole fraction profiles at the midplane between multistriped surfaces The three-dimensional density profile between surfaces with surfaces having free moving patterns Reduced densities of species A, B and C midway between homogeneous surfaces The reduced densities and mole fractions in "bulk" systems containing surfactant 1 The reduced densities and mole fractions in "bulk" systems containing surfactant 3 Density profiles of species A and B in systems containing surfactant 1 with pn'c = - 0 . 2 and Ry!A = -2.61 Density profiles of species A and B in systems containing surfactant 1 with Pfi'c = 0.0, Pn'A = -2.61 and /VB = -0.174 Density profiles of species A and B in systems containing surfactant 3 . . . Cross-sections of the reduced density and mole fraction near one surface and midway between the two surface Density profiles of species A (dark) and B (light) in systems containing surfactant 3 Cross-sectional density profiles of all three species averaged over a slab of width lcr located midway between the surfaces Reduced densities of species A, B and C in systems containing surfactant 3 The net pressure as a function of the surface separation h for systems containing surfactants Density profiles of CA CR and of the surfactant center The surface tension as a function of the surface separation, h, for systems containing surfactant 3 2  3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22  net  17 20 22 23 24  3  25 26 . 28 29 31 32 33 35 36 37 38 40 41 42 43 45 46  List of Figures  4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31  vii  Schematic showing the arrangement of the particles embedded in the surface of a hard sphere 51 Schematic of the coordinate system used in the solution of the anisotropic HNC equations 58 The mechanical stability indicators for a pure A fluid at different temperatures 60 The force between uniform solutes in a one-component liquid 61 The force between uniform solutes in a one-component liquid 62 The force between solvophobic solutes 64 Contour plots of the distribution function g,4-cc(?~> COS 9) 65 The distribution function gA-cc^, cosf? = 1) for different densities 66 The distribution function gA-cc{f, cos 8 = 1) for different temperatures . . 67 The distribution function gA-cc(r, cos 8 = 1) for different separations . . . 68 The distribution function gA-cc(r, cos 8 = 1) for different solute sizes . . . 69 The reduced force between hard solutes of differing size 70 Comparison distribution function gA-cc(.r, cosd = 1) obtained from different theories 71 Contour plots of the distribution function gA-cc{ , cos 8) f ° solutes with a solvophobic patch 72 Contour plots of the distribution function, gA-ccix, cos 8), for solutes having a solvophobic patch 73 The distribution function, gA-ccir,cosd = 1) between patched solutes . . . 74 The force between patched solutes 75 Material and mechanical stability indicators for the A-rich mixture . . . . 77 The solvent-solute distribution functions, gBc(r,cos8) (top) and gAc{cos8) between uniform solutes 78 The reduced force between uniform solutes 79 Contour plot of the distribution functions gA-ccir, cos 0) 81 Contour plot of the distribution functions gB-cc( , cosd) 82 Distribution functions, gA-cc{ , cos8 = 1), between the solutes for different densities 83 Distribution functions, gB-cc( , cos8 = 1), between the solutes for different densities 84 Comparison of distribution functions, gA-cc{ , cos 8 = 1), obtained from the two theories 85 Distribution functions, gA-cc( ,cos8 = 1), between the solutes in a system with p\ = 0.6 86 Distribution functions, gB-cc{' ,cos8 = 1), between the solutes in a system with p\ - 0.6 87 Comparison of the distribution functions, gA-cc( , cos 8 = 1), obtained using the two theories, between patched solutes 88 Comparison of the distribution functions, gB-cc(f,cosd = 1), obtained using the two theories, between patched solutes 89 Convergence of the distribution function, gB-cc( ,cosd = 1), using the isotropic H N C theory 90 The reduced force between solutes calculated using the anisotropic HNC theory 91 r  r  r  r  r  r  r  r  r  r  viii  List of Figures  4.32 The reduced force between solutes calculated using the anisotropic HNC theory 92 4.33 Comparison of the convergence of the reduced force between solutes calculated using the isotropic HNC theory 93 4.34 Material and mechanical stability indicators in the 5-rich mixture 96 4.35 Distribution functions gAc(f,cos8) and gBc( ,cosd) around a uniform Aphilic solute in the B-rich mixture 97 4.36 Contour plot of the distribution function gA-cc(r,cosd) for patched solute in the 73-rich mixture 98 4.37 Comparison of the distribution functions gA-cc(f, cos 9) obtained using the anisotropic and isotropic HNC theories 99 4.38 The reduced force between solutes calculated using the anisotropic HNC theory 101 4.39 Distribution function, gA-cc{r,cos9 = 1), between the solutes for different patch sizes 102 4.40 Distribution function, g -cc{r,cos9 = 1), between the solutes for different patch sizes 103 4.41 Distribution functions, gA-ccir, cos9 = 1), for different values of R . . . . 104 4.42 Distribution functions, gR-cc(r, cos9 = 1), for different values of R . . . . 105 r  B  A. 1 Coordinate system showing the positions of two solvent particles with respect to the origin  115  B. l The coordinate system used in the derivation of the force  117  ix  List of symbols Chemical potential  a  7  Surface tension  V  Volume  P  Pressure  T H  Temperature The Hamiltonian  j^trans  Translational kinetic energy-  fC  Rotational kinetic energy  u  Total potential energy  u(12)  Pair potential  k  Boltzmann's constant  rot  N  B  h  Planck's constant (Ch.  z  separation Activity (Ch. 2); otherwise, the z-coordinate  2); otherwise, surface  Grand partition function Phase space probability density  f  N  Potential due to the external field c(12)  Direct correlation function  /i(12)  Total correlation function  3(12) 6(12)  Pair distribution function  p (X")  n-particle density  n  Bridge functions Density of species u Kirkwood G-factors  XT  Isothermal compressibility  S(k) S (k)  Structure factor  T Q R D  Helmholtz free energy  cc  Concentration-concentration structure factor Gibbs free energy Distance between solute centres Diameter of the solute Mole fraction of species v Length of period of the surface pattern  tl  a F(R)  Twice the width of the surface stripe attracting species A Characteristic diameter of a solvent particle Force between solutes  X  Acknowledgements I would like to thank my supervisor, Gren (a.k.a. Prof. Patey), for all his help and encouragement throughout these last five years, and for many interesting discussions about running, work, politics, running, health, the weather, Newfoundland, running, birthdays, hiking, running, theology, death, jobs and running, among other things. I would also like to thank Liam, who has been tremendously helpful, as well as extremely distracting. Finally, I would like to thank the many members, including honorary (i.e., Roman), of the lab, the coffee people from the I A M , westgrid, and chemistry, and all the friends I have met here, who have made life, and work, fun. Oh, and also to my family - Thanks for your support!  i  Chapter 1 Introduction The behaviour of solutes immersed in a liquid depends on the direct interaction between the solute particles, and on indirect effects due to their interactions with all other solution components. In the absence of electrostatic interactions, the correlations in a liquid are usually short-ranged. However, near a phase transition (e.g., liquid-liquid or liquid-vapour) long-ranged correlations can occur. For example, the presence of a surface can induce a wetting or drying transition. In this thesis we examine the effect of adding large, uniform and chemically patterned solute particles to a fluid near coexistence. Furthermore, the effect of confinement, by chemically patterned surfaces, on the fluid phase behaviour is investigated. The development of the surface forces apparatus (SFA) in the 1970's [1] allowed researchers to directly measure the forces between surfaces in liquids over length scales ranging from 0.1 nm to 1/j.m. The unexpected discovery of extremely long-ranged, strongly attractive forces between hydrophobic surfaces in water was made shortly thereafter [2-5]. The origins of these forces have since been debated. In the experiments, long-ranged attraction was observed as the surfaces were brought together. At some small separation, the liquid between the surfaces evaporated and the surfaces jumped into contact. However, when the plates were subsequently pulled apart, vapour was observed between the plates well beyond the separation at which evaporation occurred while they were being brought together. These results became the focus of much investigation, both theoretical and experimental, in part because of the implications to biological processes. What was puzzling about the results was the presence of attractive forces before evaporation occurred. A n early theory suggested the forces were due to changes in the solvent structure at the surfaces [2]. According to this theory, the orientational order of water at the surface resulted in long-ranged correlations in the ordering of water molecules. However, simulations, and other experimental results, showed that water ordering near a surface extends at most over several molecular diameters, not hundreds of molecular diameters. Long-ranged forces have also been measured in solvents other than water [6, 7], suggesting the phenomena is not directly related to the molecular structure of the solvent. Another theory suggested electrostatic correlations [8] were responsible for the long-ranged forces. Although electrostatic effects can give rise to long-ranged effects in a fluid, further careful experimental analysis showed that these were not the origins of the forces measured using the SFA [9-15]. In 1993, Berard et al. [16] published grand canonical Monte Carlo (GCMC) simulation results that demonstrated that a liquid confined between infinite, planar plates becomes metastable with respect to the vapour phase at a certain separation. The forces between the plates were found to be attractive when the intervening liquid became metastable. At a smaller, critical separation, evaporation between the plates occurred, coinciding with a jump in the strength of the attraction. The separation at which the liquid became unstable increased as the liquid neared coexistence. A simple mean field analysis [16]  Chapter 1. Introduction  2  showed that the surfaces effectively "shift" the chemical potential of the confined liquid. A t the critical separation the effective chemical potential of the confined fluid is equal to the bulk coexistence chemical potential, and at still smaller separations it equals that of the spinodal. Beyond this separation, spontaneous cavitation occurs, as the liquid is no longer metastable between the surfaces. The presence of a metastable state can also explain the hysteresis observed i n the S F A experiments. A fluid "trapped" i n a metastable state must overcome a free energy barrier i n order to reach the stable state; hence the presence of the liquid state up to the spinodal separation. In contrast, as the surfaces are pulled apart from contact, the vapour can persist up to a separation where it ceases to be a stable or metastable state. Other simulations were later performed and confirmed these results [17]. The critical separation, D , below which only the vapour phase is stable, can be estimated for macroscopic surfaces by [11], c  2j  lv  L) ~ —: c  (1.1)  P | P-l -  Here p is the density of the bulk liquid | /j/ — u | is the difference i n the chemical potentials of the liquid I and vapour v phases, and 7/„ is the liquid-vapour surface tension. Near coexistence the difference between the chemical potentials of the liquid and vapour phases approaches zero, and D becomes large. Liquid water, at ambient conditions, is relatively near liquid-vapour coexistence and the critical separation is predicted to be ~ lOOnm [18]. This theory does not give any information about the forces between surfaces confining a metastable liquid. There is a simple thermodynamic explanation for the occurrence of attractive forces between surfaces that induce a phase transition. The grand potential of a liquid between two surfaces can be written n = -PV + jA, (1.2) v  c  where P is the pressure, V is the volume, 7 is the solid-liquid surface tension, and A is the area of the surfaces i n contact with the fluid. A stable state occurs when Q, is at the global minimum. W h e n the surfaces are far enough apart that the area is much smaller than the volume, the —PV term in the above expression is dominant. The pressure of a metastable bulk state must be smaller than that of the stable bulk state. In fact the pressure of any liquid between the surfaces that differs from the bulk must be less than the bulk pressure, provided the surfaces are far enough apart that surface-induced structure effects are not important. Thus, changes in the liquid density that occur when either the liquid between the surfaces becomes metastable, or a transition to the vapour occurs, give rise to attractive forces. The thermodynamic arguments used above indicate that attractive forces due to solvent phase behaviour are not limited to hydrophobic solutes i n water. In general, they can occur whenever confining surfaces induce a phase transition. Indeed, attractive forces between surfaces in nonaqueous media have also been observed [6, 7]. A s an example, measurements using the atomic force microscope have shown that long-ranged attractive forces occur between solvophobic surfaces in many organic solvents [6]. In addition, aggregation in colloidal systems has been linked both experimentally [19] and theoretically [20] to liquid-liquid demixing behaviour. Experiments using the S F A have shown that the  Chapter 1. Introduction  3  forces between hydrophilic surfaces immersed in pure octamethyltetrasiloxane are oscillatory [7]. However, when trace amounts of water were added, the forces became attractive at separations up to 140nm. A t a smaller critical separation, there was a jump in the size of the attraction, coincident with the formation of a water-rich liquid between the plates. A t still smaller separations, the force became oscillatory due to the packing of water layers. Simulations done on confined mixtures have demonstrated that, surfaces that favour the minority species can cause a phase transition between the surfaces at some separation [21]. This is particularly true when the bulk is near liquid-liquid coexistence, and a fluid rich in the minority species corresponds to the metastable state. The jump in the force occurring at the critical separation is comparable to the difference between the pressures of the stable and metastable bulk phases. The separation at which the transition occurs depends on how close the bulk fluid is to coexistence. Although the phase behaviour of the fluid does account for the simulation results, there is still debate [15] as to whether or not the long-ranged forces in the experiments are due to a metastable confined liquid. Nevertheless, it is clear that the phase behaviour of the fluid can give rise to long-ranged attraction between confining surfaces. The possibility that solvent phase behaviour could lead to such long-range forces in liquids received widespread attention [16-18, 21-33] In particular, possible implications to biological systems were immediately recognized [18]. Hydrophobic interactions had long been believed to play an important role in biological processes such as protein folding [34]. Clearly if the phase behaviour of water plays a part in such systems, the size of proteins, or hydrophobic parts of a protein, must be large enough to generate such a transition. E q (1.1) applies only to macroscopic objects, although related expressions can be used to estimate the force between smaller objects [11, 35]. Strong attractive forces do not occur between all sizes of hydrophobic molecules in water. A t low mole fraction, small hydrophobic solutes are more likely surrounded by water molecules than by other hydrophobic molecules [18, 36]. The question then is, how large does a hydrophobic particle have to be in order to generate attractive forces due to "drying" behaviour. Weeks et al. [18] developed a meanfield theory that predicts that surface drying occurs between solutes with length scales on the order of one nanometer. This prediction compares well with simulation results of hydrophobic solutes in water [17, 25, 37]. The theory applied to a Lennard-Jones (LJ) fluid, also agrees with the simulations of a confined L J fluid [22]. Integral equation studies were also done with hydrophobic spherical solutes immersed i n water [27, 29, 38]; it was observed that forces between solutes twice the size of the water molecules are attractive. A l l of these results suggest that the length scales present i n proteins are large enough to generate surface "drying" and consequently give rise to attractive forces. These results certainly do not prove that forces resulting from surface "drying" effects are important in protein folding, nor do they provide any detail as to the role they may play; however, they demonstrate that the length scale present i n such systems is consistent with such an effect. The range of the forces between nanometer-sized solutes is much smaller than those occurring between macroscopic surfaces. In addition, the applicability of thermodynamic arguments for "drying" near small surfaces is still unclear. The theories and simulations show only that the force between small surfaces is attractive, and that this attraction occurs due a decrease in the density of the intervening water. Strictly speaking, discussion of metastable states and thermodynamic stability is not appropriate for finite-sized systems.  Chapter 1. Introduction  4  From the simulation results it was observed that attractive forces occur before the liquid-vapour transition due to the presence of the confined metastable liquid. However, the strength of the attraction increases suddenly when the vapour transition occurs. Some researchers have investigated the kinetics of this transition i n order to determine whether or not it would be kinetically feasible in biological systems. M u c h work is still being done on this problem; however, many of the predictions thus far suggest evaporation between the plates is kinetically viable on the timescales required for protein folding [26, 31]. Simulations were done on hydrophobic polymer collapse in water [23]. The results suggest the formation of a vapour bubble nucleates the collapse, and that this step is the rate determining step. These results lend further support to the idea that surface drying plays an important role in protein folding. Surface-induced demixing transitions have also received significant attention [19, 20, 28, 39-42]. This work has relevance to a variety of physical systems including aggregation in colloidal suspensions, as well as i n biological systems. The formation of liquid bridges between a protein and lipid particles in a two-dimensional model has been observed in computer simulations [39, 40, 43]. The bridges were accompanied by attractive forces. Integral equation theories [28, 44] and density functional theories [20] have also shown that demixing-like behaviour can occur between two finite sized solutes. In particular Kinoshita demonstrated, using approximate theories, that the addition of trace amounts of a "hydrophilic" component to a system containing large "hydrophilic" solutes immersed in a "hydrophobic" solvent resulted in strong attractive forces [27, 33, 38, 45]. Similarly, the addition of a "hydrophobic" component to a system containing "hydrophobic" surfaces immersed in a "hydrophilic" solvent, also resulted in an increased attraction between the surfaces [28]. From the above discussion it is clear that, whether or not surface induced phase behaviour is important to biological processes, it can lead to the association of solute particles immersed in a liquid. In this work, we examine the effect of solvent phase behaviour on interactions between uniform and patterned solutes or surfaces, and establish relationships between the relative length scales and the strength and range of the resulting forces. In addition, the structure of the fluid due to confining surfaces is investigated. As described above, confining surfaces that favour the minority species in a mixture can cause demixing to occur. It is interesting to consider what happens when the surfaces are patterned such that only some of their parts favour the minority species. For example, the surfaces can be patterned with alternating stripes that favour the major and minor solution components. Provided the area of a stripe preferring the minority species is large enough, a liquid bridge joining the stripes on the two surfaces will form. B y varying the width of the stripes, information relating the relevant length scales can be obtained. The formation of a liquid bridge is accompanied by the formation of an energetically unfavourable liquid-liquid interface separating the alternating phases. Therefore, one would expect the length of the bridge to be limited by this interface. In an attempt to increase the separation at which bridge formation occurs, a small amount of surfactant was added to the system. Surfactant particles consist of two parts that favour each of the alternating phases and, consequently, preferentially occupy the interface, reducing the interfacial tension. The work presented here is the first to demonstrate that surfactants can be used to stabilize liquid structures formed due the solvent phase behaviour between chemically patterned surfaces. To study  Chapter 1. Introduction  5  these confined systems, both with and without surfactant, G C M C simulation techniques are used. G C M C simulations are well suited to the study of confined liquids near demixing coexistence because they allow for large fluctuations i n the density and composition of the system. The structure of the solvent fluid and the forces between the surfaces are considered in detail. Apart from implications for large particles in solution, fluid phase behaviour between chemically patterned surfaces is also of interest as a means of producing ordered liquid structures. The behaviour of spherical solutes immersed in a liquid near either a liquid-liquid or a liquid-vapour transition is also examined in this thesis. Chemically patterned (patched) particles are of particular importance because completely uniform particles rarely occur i n nature. The behaviour of patterned particles in a liquid can lead to self-assembly as has been shown experimentally [46] and using simulations [47]. Here the influence of solvent phase behaviour on the interactions between patterned solutes is investigated for the first time. Although simulations can be used to study large solutes immersed in a liquid, extensive computational effort is required. Instead, we use integral equation theories, which, though approximate, have been successfully used to study systems away from coexistence with large length scale discrepancies [48-50]. We compare results from two integral equation methods: the anisotropic hypernetted chain ( H N C ) approximation, and the isotropic H N C approximation. In the former, the solvent structure is directly obtained around a pair of solutes, in contrast with the latter, where the solvent structure is only obtained around a single solute. Therefore, we expect the anisotropic H N C approximation to provide more accurate results. This is the first application of the anisotropic H N C theory to a system near coexistence. One of the goals here is to see how well the theories compare and determine whether either method can be used to successfully study the behaviour of large solutes in fluids near coexistence. ^ The remainder of this thesis is organized as follows. A n overview of the theoretical background is presented in Chapter 2. The results are presented i n Chapters 3 and 4. In Chapter 3, results for fluid mixtures confined between chemically patterned surfaces obtained using G C M C simulations are examined, and in Chapter 4 integral equation results for uniform and patterned solutes in one and two-component liquids are discussed. Finally, the results and conclusions are summarized in Chapter 5.  6  Chapter 2 Theoretical background 2.1  Statistical mechanics of fluids  In this section, we show how the simulation and theoretical methods used in this work are derived from statistical mechanics. The Hamiltonian of a system consisting of N identical particles can be written as H = K,%  ans  + U (X ),  +  (2.1)  N  N  where K% and KT^ are the translational and rotational kinetic energies, and UN{X ) is the potential energy. The positions and orientations of the N particles are represented by X = {Xi,X , ...,XN}, where Xi represents the position r = (r ,r ,r i) and orientation u>i = {4>i,0i,Xi)- The angles (fa, 6i,Xi) are the Euler angles described elsewhere [51]. In the following discussion, we assume that the fluid behaves classically, and that the particles are rigid, such that the potential energy of the fluid depends only on the centres of mass of the particles, and on their orientations. In the work presented in the following chapters, the fluid species interact through either spherically or cylindrically symmetric potentials. For notational simplicity, here we consider a one-component system of identical particles, noting that the generalization to the mixture case is straightforward [52, 53]. If pairwise additivity is assumed, the potential energy can be written, ANS  N  N  xi  2  U (X ) N  N  = Y u(u ,u ,T - Tj) f  Ji  j  i  i<j  yi  Z  + J>TO,  (2.2)  i  where ^(Xj) is the potential acting on particle i due to the presence of an external field, and u(u>i,Uj,Ti — Tj) is the intermolecular potential between particles i and j. The aim of equilibrium statistical mechanics is to relate the macroscopic observables of a system, such as the pressure, density and internal energy to the interactions among particles. The thermodynamic properties of a system can be completely described by some minimum set of variables; for example fixing the chemical potential of each species, \x , the volume, V, and the temperature, T determines the state of the system. The collection of all possible configurations having the same JJL , V and T is known as the grand canonical ensemble. In this ensemble, the number of particles is not fixed, making it particularly useful for the problems of interest i n this thesis; near a liquid-vapour or a liquid-liquid phase transition, large spatial fluctuations in the density (or composition) can occur due to the presence of a solute particle. The grand partition function of the one-component system described above can be written, v  u  E  =E  (J^JN)  JP ex  [-PU (X )} N  N  dX , N  (2.3)  7  Chapter 2. Theoretical background  where f3 = (ksT)^ , z = A exp[0u] is the activity, and A = A A results from integration over the translational and rotational degrees of freedom. The constants A and A are given by [53], - 3  1  3  3  r o t  r a n s  rot  trans  - -  A  =  U ^ J  (2  -  4)  and  v  -  (wi) •  '  (2 5)  Here h is Planck's constant, k is Boltzmann's constant, T is the temperature, m is the mass of the particle and I is its moment of inertia. Each configuration represents a point in phase space. The probability that the system is in a particular configuration is given by the phase space probability density, B  /  W  , , ^ - t « £ ! a .  (  ,  6 )  The idea of GCMC simulations is to generate a series of random configurations having the same values for /x, V and T, with a probability consistent with the phase space probability density. Thermodynamic properties can then be obtained as averages over the configurations, oo  (A) = J2  „  A{X )f{X -N)dX . N  N  N  (2.7)  The grand potential is an important thermodynamic property and can be written in terms of the grand partition function, n = -k TlnZ, B  (2.8)  In the absence of an external potential (due, for example, to the presence of confining surfaces), fi = — PV. Provided the simulation system is large enough, exact results can be obtained for a given model using the GCMC method. When dealing with systems having multiple length scales (e.g., large solute particles, small solvent particles), the simulations become computationally difficult. However, in the limit that the solute particles are infinitely large, they can be approximated by infinite planar surfaces. For this case, only the fluid between the surfaces need be simulated, significantly reducing the required computational effort. In this thesis, simulations are primarily used to investigate the infinite solute limit. Systems with large, finite-sized solutes remain a challenge to simulate. Further details about the GCMC simulation method are given in chapter 3. Integral equation theories have been developed to obtain the particle-particle (or pair) distribution functions. In general, solution of these equations requires much less computational effort than is required for simulations; thus the theories are ideal for studying systems that have multiple length scales. A disadvantage of integral equation methods is that, in general, approximations must be introduced in order to solve the equations. Nevertheless, these theories have been successfully applied to a number of systems. There are several types of integral equation theories. Some examples include those based on the Born-Green-Yvon hierarchy, and those based on the Ornstein-Zernike (OZ) equation. In  Chapter 2.  Theoretical background  8  this thesis, theories based on the O Z equation are used. The O Z equation can be related to the grand partition function as presented below; more detailed discussions on the origins of integral equation theories can be found elsewhere [52, 54-56]. The n-particle density function is proportional to the probability of finding any n particles in the spatial and orientational phase element S(Xi — X[)5(X — X' ) • • • 5(X — X' ) regardless of the positions of the remaining N — n particles. Here 5(Xi — X'A is the Dirac delta function. The n-particle density function can be expressed as an average quantity, 2  '  pin){x n)  =  (w^ny  6{Xl  -  x  xS(X, -X[)---  '  l  )  6  {  i{X  X  -  2  x  '  2  )  '  • •  6  {  X  n  2  -  K  - X' ).  n  n  n  )  (2.9)  n  In the presence of an external field ^(X), where z(X) = zexp[—ftty^X)], the n-particle density can be written as a functional derivative of the grand partition function with respect to the function z(X), Jn) n\  _  (Y  9  {  X  Z  ( l)  ^ nE  • - • ( n)  X  Z  -  )  X  S  SziX,) • • • 5z(x y  ( 2  n  -  1 0 )  The two-particle (or pair) density is particularly useful, in part because of its relation to the structure factor, a quantity measurable by, for example, neutron scattering experiments [54]. Adopting the notation Xi = i the singlet and pair densities can be written explicitly, 9  E  K ]  6z(l)  5lnz(l)'  [ Z A L )  and 9  [ l Z )  -  E  6z(l)6z{2Y  { 2  -  U )  For a homogeneous system, the singlet density is a constant, p ^ ( l ) = p/u) = (A'') /(uV). Defining the pair distribution function g^>(\2) = u> p^ \l2)/p [53], the structure factor can be written, 2  S ( k ) = l + p Jduij  2  2  du J dr [g \\2] - l ] e x p ( i k - r ) . {2  l2  2  (2.13)  The structure factor can be related to thermodynamic properties as is discussed in the next section. Note that the pair distribution function is simply the probability of finding a particle with position and orientation X , given there is a particle with position and orientation X\. We note in E q . (2.11) that the singlet density can be written as the first logarithmic derivative. The second logarithmic derivative gives the pair density plus some extra terms 2  5pM(l)  <5 ln£ 2  0(12)  <yinz(l)<51nz(2)  S]RZ(2) =  />)(!, 2 ) - p « ( l ) p « ( 2 ) +p  ( 1 )  ( l ) 5 ( l - 2).  (2.14)  9  Chapter 2. Theoretical background  The direct pair correlation function can be defined [56] as c<>(12) = 2  5pW(2)  1  8p^(l)  p«(l) V  (2)  1 }  <51nz(l)  (2.15)'  ^ ( 2 ) '  This can be rearranged to give an expression for C7 (12), _1  ! ^  Substituting Q and  (  5lnz(l) 1  2  p 2 )  ) =  =  1  ( 2 )  ^(1)^(2)  ~  C  (  1  2  )  '  ( 2  '  1 6 )  into the definition of a functional inverse [56], J d3Cy- (13)c7(32) = 6(1 - 2),  (2.17)  1  and rearranging gives p  (2)  ( 1 2  = -p^(l)p^(2)c^(12) + p (l) Jd3c^(13) W  )_ (i)(i)p(i)(2) p  x [p^(3)p (2) - p (1)  ( 2 )  (32)] .  (2.18)  Using the definition of the total pair correlation function h^(12) = g^(12) — 1 and the fact that for a homogeneous, isotropic fluid, p ^ ( l ) = p/ui and p( )(12) = p g^ (12) / LU [53] we obtain, 2  hS (12) = c 2)  (2)  (12) + ^  J  d3c (13)/i (32), (2)  (2)  2  2  (2.19)  which is the OZ equation for a one-component system. This expression can readily be generalized for a multicomponent system,  C( ) = S( ) + E ~, I ^ ( 1 3 ) ^ ( 3 2 ) , 12  C  12  (2.20)  where the subscripts a, (3 and u represent the species of particles 1, 2 and 3, respectively. In the above relations, LU = J did and will depend on the symmetry of the problem: for spherical particles LU = 1, for linear particles to = 4.7T and for general symmetry, LU = 8TT . The OZ equation can be viewed as the definition of the direct correlation function, and is an exact expression. Using Eq. (2.15) as the definition, as was done here, allows us to see how the direct correlation function relates to the grand partition function. To solve the OZ equation a closure relation is needed. Several closure relations have been derived from functional expansions of c^ ^(12). The hypernetted-chain (HNC) closure is one of the most popular of these types of closures, and has been used for a variety of systems. The HNC can be derived from a functional Taylor expansion of ln[p^ (2)(1) / z(2)} about p^(3) = P/LU. To do this, we consider the external field ^(X) to be due to a particle at position X L . The potential acting on a particle at X due to the external field is then ^(X ) = u(12). The presence of the particle at position X± perturbs the system, and the 2  2  2  2  Chapter 2. Theoretical background  10  singlet density becomes, p + Ap - \2). This quantity is proportional to the probability of finding a particle, at X , given that there is a particle at position X\. In other words, p + A p ^ ' ( 2 ) = pgr( )(12). Likewise, the change in the logarithm of the density divided by z{2) is, ln[p/z(2)] + Aln[pW(2)/z(2)] - ln[p^( )(12)/z(2)]. Expanding ln[p( )(2)/z(2)] gives (  1  2  2  2  Aln[pW(2)/z(2)]  =  J d3  1  51n[pW(2)/z(2)] 6pW(3)  A p W ( 3 ) + ... (2.21)  d3c (23)Ap (3). (2)  =  pJ  (1)  (2.22)  d?>c \2?>)hP\n). {2  The left side of the equation is A l n [p«(2)/z(2)]  pg^ (12) z(2)  = ln  -ln  .Z-  <7 (12) _exp(-/3u(12))_ (2)  ln =  ln[ ( >(12)] + /fo(12).  (2.23)  2  5  Combining Eqs. (2.22) and (2.23) with the O Z equation (Eq. (2.19)) and rearranging we obtain the H N C closure, c (12) = h {\2) - l n ^ > ( l 2 ) - (3u{l2). (2)  {2)  2  (2.24)  Again, this expression can readily be generalized to multicomponent systems, „(2)/  J ( 1 2 ) = /$(12) - ln($(12) - ^ ( 1 2 ) .  (2.25)  Other closures can also be derived from functional Taylor expansions of, for example, p >(l)/z(2) about p (2) = p (Percus-Yevick closure) and of p ( l ) about V(X) = 0 (mean spherical approximation) [55]. The only approximation made i n the derivation of these closure relations is in the truncation of the density functional expansion. The closures can also be obtained from diagramatic expansions [54, 55]. We do not provide the details of this derivation here, but it is useful to point out that using this method, an exact closure can be obtained, (1  (1)  ( 1 )  c (12) = hS \l2) - ln </ )(12) - pu(12) + 6(12), (2)  2  2  (2.26)  where 6(12) is the bridge function. In general, the bridge function cannot be calculated and some approximation is required. It is clear from E q . (2.26) that the approximation made to obtain the H N C closure is that 6(12) = 0. The H N C has been used in conjunction with the O Z equation and results for single and multicomponent liquids using this theory generally compare reasonably well with simulation results. In this work, we are interested in determining how the the H N C approximation performs for solutions containing a large solute at infinite dilution and one or two smaller species. We also use the related anisotropic H N C closure (to be described in more detail in Chapter 4) and compare the results with those obtained from the isotropic H N C equations. Note that i n the following sections, we refer always to the pair correlation functions, and so we omit the superscript (2).  Chapter 2. Theoretical background  2.2  11  Fluid phase stability  A major focus of this thesis is the interactions occurring between large solute particles due to solvent phase behaviour. It is useful to examine how the stability of a fluid phase is related to both thermodynamic quantities and the pair correlation functions. More detailed discussions on fluid stability can be found elsewhere [57, 58]. When two phases coexist the pressure, temperature and chemical potentials of each species i n both phases are equal. That is, for phases 1 and 2, 7\ = T , Pi = P and Pu\ = P~v2 for all species v. The coexistence, or binodal, line defines the boundary where the fluid ceases to be stable as a single phase. Beyond coexistence, the fluid can persist as a single phase as long as the single phase corresponds to a metastable state. Eventually, the fluid can no longer exist as a single phase; the line separating the metastable fluid from the unstable fluid is called the spinodal. A t the spinodal, spontaneous phase separation occurs and density (or composition) fluctuations i n the fluid become infinite, resulting i n long-range correlations. The integral of the pair correlation function h p(\2) for a liquid mixture is, 2  2  a  G  a  0  uj j  =  j ° J  duJl  dt  2  (2.27)  0( ' ' ' ' ^  dTl2ha  Vl2  u  1  u  2  The Kirkwood G-factors, G p [59], diverge when the correlations become long ranged, and these divergences can be related to divergences in various material and mechanical response functions [57]. In a pure liquid the isothermal compressibility, XT, which is a mechanical response function, - 1 dp_ (2.28) X T = v r dV T a  diverges at the liquid-vapour spinodal. The compressibility can be written i n terms of a Kirkwood G-factor, or the structure factor (see E q . (2.13)),  I±£_£Si=°), P  (,  2 9 )  P  where the subscripts of G have been omitted since we are dealing with a pure system. The stability of a fluid phase is determined by the free energy of the system. Before continuing, we introduce the differentials of the Helmholtz, J , and the Gibbs, Q, free energies for a binary mixture, 7  dF/N-df  = -sdT - Pdv + =  ^p dx , v  u  —sdT — Pdv + (pp — p )dxp a  (2.30)  and dG/N = dg = vdP - sdT + (p,p - p )dxp. a  (2.31)  Here v = V/N is the molar volume, s = S/N is the molar entropy and xp is the mole fraction of species j3. Note that for this discussion, g denotes the molar Gibbs free energy,  Chapter 2. Theoretical background  12  not the pair distribution function. In a pure system, the isothermal compressibility can be related to the free energy by [57], d f\ 7dv T 7JJ  _ P  2  = XT — •  2  T  ( 2  3 2  )  A mechanical instability in the pure fluid is signaled by a change in the concavity of / , and the mechanical stability indicator can be written,  A similar quantity can be used to signal a mechanical instability in a binary mixture [57], »  J  d  y  < 0,  /  2  (2.34)  T,np-fi  a  where / ' = / — xpipp — p ), and df = —sdT — pdv — xpd{pp — p ). The left-hand side of Eq. (2.34) is inversely proportional to the isothermal compressibility at constant T and - P-0, a  a  m  -  (2.35,  Other isothermal compressibilities can also be obtained (for example, at constant xp); however, divergences in these are only sufficient conditions for fluid instability, and not necessary conditions. X T , ^ - ^ can also be written in terms of G /s, a  a  k-BTxT^-w  =  + XpGp/3 + 2px x,3G f3] .  + P \ \ <XOL  1  X G  a  (2.36)  a  Concentration fluctuations in a mixture can be described by the concentration-concentration structure factor. For a binary mixture, this is written as [60] S (k = 0) = x x cc  a  [1 + px xp(G  p  a  + Gpp + Gap)],  aa  (2.37)  where x is the mole fraction of species a. Note the similarity to the expression for the structure factor given in Eq. (2.13). The concentration-concentration structure factor is related to the Gibbs free energy by, a  / &g\  _  Nk T B  {dx ) - (k  = oy  2  Tp  Scc  ( 2  -  3 8 }  A change in the concavity of g signals the instability of the fluid with respect to a material change. Thus, the second derivative of g with respect to mole fraction can be used as a material stability indicator. When d 2 g  < 0,  1  dX^y  rp  p  (2.39)  Chapter 2. Theoretical background  13  the fluid is stable with respect to material fluctuations. It is convenient to calculate G p from the integral equation methods, and from these, a  the stability criteria can then be obtained to determine whether or not the fluid is near a phase transition. However, we note that the integral equations become unstable while the stability indicators are still finite, and therefore cannot exactly predict the spinodal [58]. Nevertheless, the instability in the equations does occur near to the spinodal in most cases.  14  Chapter 3 Fluids confined between chemically patterned surfaces 3.1  Introduction  In this chapter we present G C M C simulation results for a binary liquid mixture confined between chemically patterned surfaces. A s discussed in Chapter 1, the presence of confining surfaces can induce a phase transition at some critical separation. This suggests that chemically patterned surfaces, where alternating areas of the surfaces favour the two different bulk phases, will induce a transition to a bridge phase, provided the area of the surface favouring the metastable phase is large enough. Suppose, for example that a liquid containing a hydrophobic component with trace amounts of water is confined between two surfaces patterned with alternating hydrophilic and hydrophobic stripes. A t some surface separation, one might expect the formation of a water-rich bridge between the hydrophilic surfaces. Moving the mixture closer to a demixing transition should facilitate bridge formation. The question then arises as to how large the hydrophilic surface area must be for bridge formation to occur at a given surface separation. In this chapter, this question is addressed for a simple liquid mixture. It was previously noted that a phase transition in a confined fluid is accompanied by strong attractive forces between the surfaces (see Chapter 1). One might expect then, that the forces occurring at the onset of bridge formation will also be attractive. However, since a phase similar to the bulk metastable phase can only occur between parts of the surfaces, the behaviour of the force is not immediately clear. Here the perpendicular force between the surfaces when bridge formation occurs is calculated for the first time, and analyzed in the context of the pressures of the stable bulk phases. A factor limiting the formation of liquid bridges between the surfaces is the surface tension due to the formation of a liquid-liquid interface. Due to their amphiphilic nature, surfactant particles can minimize this surface tension by preferentially occupying the interface. In this chapter we consider the effect of the addition of a small number of surfactant particles to liquid mixtures confined between chemically patterned surfaces. We are interested in whether the surfactant can increase the separation at which bridge formation occurs. In addition, we investigate other structures that can be stabilized at the surface by the surfactant. We are interested only in systems containing very small amounts of surfactant, such that the phase behaviour of the bulk binary mixture is only minimally altered.  Chapter 3. Fluids confined between chemically patterned surfaces  3.1.1  15  S u m m a r y of related work  The formation of a liquid bridge phase, composed of alternating higher and lower density fluid regions, was first established in one component systems [61-67] using lattice-gas models, mean-field theories and G C M C simulations. In these studies, it was observed that, depending on the surface separation and the chemical nature of the surfaces, three different phases could occur between chemically patterned surfaces: the liquid, the vapour and the bridge phase. In addition, the formation of other surface structures was noted, including 'droplets', consisting of liquid condensed near strongly attractive areas of the surfaces, and 'vesicles", consisting of vapour pockets near weakly attractive areas of the surfaces [64]. In this work, we also discuss the presence of similar types of fluid structures when surfactant particles are present. The surface separation at which bridge formation occurs between striped surfaces was found to be similar to the width of the stripe [61, 65]. Later analysis of a simple continuum model of a fluid confined between two walls, showed that this relationship between surface separation and stripe width arises because the favourable wall-liquid surface tension must compensate for the unfavourable surface tension that results from forming two liquid-liquid interfaces [68]. Although the analysis was done for a continuum system, the relationship is expected to hold for microscopic systems. Bridge formation has also been noted between geometrically corrugated surfaces. Notably, Curry et al. [69] demonstrated the formation liquid-solid bridge phases between such surfaces. Thus, the surface patterning need not be chemical in nature for bridge formation to occur. In addition, the formation of bridges is not restricted to fluid phases. In fact, it is expected that any confinement induced transition will give rise to the formation of bridge phase, if the surface patterning is appropriate.  3.2  Models and Simulation Methods  G C M C simulations were performed in cells of dimension L X Ly X (L = h). Periodic boundary conditions were used in the x and y directions and two planar surfaces of infinite thickness and area were placed at z = 0 and z = h. The fluids studied contain either species A and B, or species A, B and C, where C is a surfactant particle consisting of two spherically symmetric particles, CA and CR, whose centers are separated by one a. The interactions between particles of type a and j3 (A, B, CA and CB) have the generalized Lennard-Jones form x  z  Ua,{r)=AcJ-f{^f-\ y  (3.1)  ap  with "length" and energy parameters o~ $ and e g, respectively. The strength of the attractive interaction is determined by the coefficients e p and \ p. Following previous work [21] we select t A = £AB = ^BB = e, O A = O B = O B = \ A = 1 and X B = A = 0.5. These parameters were chosen to roughly mimic a water/hydrocarbon mixture. If the interactions are written in the standard L J form, then the parameters obey the relationships, BB = AB = ° AA °BB = °~AB = 2 °~AA- Thus, the A particles are slightly smaller and interact with each other through a more strongly attractive potential. The values of A Q / 3 for the three different surfactant models discussed are summarized in Table 3.1: a  a(  a  A  E  E  A  2  5  e  a  n  d  A  1/6  B  a  A  A  B  B  Chapter 3. Fluids confined between chemically patterned surfaces  16  Table 3.1: Values of the interaction parameters X p for the different surfactant models. a  surfactant  ACUA  1 2 3  1.3 1.3 1.3  ^c c  ^C B  A  B  0.6 0.6 0.5  ^C C  A  B  0 0 0.75  0 0 0  B  0 0 0.5  surfactants 1 and 2 have identical interparticle interactions and represent the "simplest" surfactant with only two attractive interactions XC A > 0, and XC B > 0, while surfactant 3 includes attractive interactions between surfactant particles. For all three models = e and XQ B = ^C A = ^c c = 0. The values of A for surfactant 3 were chosen to qualitatively mimic the interactions of a particle having a hydrophobic tail and a polar head group in a water/hydrocarbon mixture. Particles of species v, with centers a distance Z from the wall (W), interact with each surface v i a A  A  A  B  B  A  f  oo  ,Z<0,  where ay/v and ew are length and energy parameters for the surface-particle interactions, C (x; £) is the patterning function, and £ is the pattern length. We note here, that i n a strict sense, the inverse fourth power law used in our model is valid for an infinite, planar, homogeneous surface. However, we would expect it to remain a reasonable form at least for larger surface areas. Furthermore, employing the same wall-particle interaction throughout allows us to isolate size and pattern effects without the added complication of a varying potential function. In all cases owv — 1) £\VB = c\vc 0, and for surfactants 1 and 3, wc — 0. A l l other values of ew» depend on the surface patterning. Note that surfactants 1 and 2 differ only in their interaction with the surfaces. Following Henderson [70], we use a striped pattern shown in F i g . 3.1 with £ (x;£) defined by v  =  B  e  A  C(z;0=  E n=—oo  #(*-^ + f)ff(^ + f-*), ^  '  ^  '  where n is an integer that accounts for the periodicity, £i determines the amount of coverage at fixed £, and H is the Heaviside step function. Unless otherwise specified f = 10cr for all systems studied in this work. For pair potential calculations a cylindrical cutoff, s = 5cr was used; the potential was explicitly calculated between particle i and all other particles within a cylinder of radius s , centered on i. Unless otherwise stated, s was set to 5a. Long-range corrections to the energy and pressure were included in some simulations done with homogeneous walls (i.e., no chemical patterning). The expression for the energy correction, U , was obtained as described previously [16]. Briefly, the pair distribution function is assumed to approach 1 beyond the cutoff distance, and the singlet distribution function, p^(z ;h), is approximated c  c  c  c  2  (3-  Chapter 3. Fluids confined between chemically patterned surfaces  17  Figure 3.1: Example of patterning on the surface of the plates. The length of one period in the pattern is £, and the ratio of £ / 2 is the width of the stripe which attracts species A. Dotted lines represent periodic boundary conditions and solid lines represent a boundary between values of ty/Ax  by the G C M C density profiles when U = 0. The density profiles were obtained from at least 10 configurations sampled on a grid of Az = O.Olcr. Taking the asymptotic limit of the pair potential we have c  7  u (r) = c  4«7 7r J2 2  ( K(i)P  {0)  (z \h) 2  (  f  - j ) dz , 2  (3.4)  where v{i) is the species of particle i. A n analogous expression is obtained for the pressure correction. Long-range corrections were not calculated for systems with chemically patterned walls because these effects were found to be small in the homogeneous wall case, and should not influence qualitative results for such systems. Since the model solutions considered are obviously simplified, our objective is to elucidate general qualitative features, and not an accurate determination of quantitative details. Configurational chemical potentials, //„, as described by Adams [71, 72], were used for the G C M C calculations, p! = /i„ - pl + k T ln = k TB -k TlnV/a ,  a  deal  v  B  3  Pu  3  B  u  {  j  B  where p is the chemical potential of species u, p]f is the ideal contribution to the chemical potential, p is the average number density of species v and B is the original B eal  v  u  v  Chapter 3. Fluids confined between chemically patterned surfaces  18  parameter of Adams. Simulations were performed with p! , p' , V, T and h fixed. Insertions, translational and rotational displacements (for surfactant particles) were performed in the usual way [72]. In addition, identity exchanges whereby a particle of type A is converted to one of type B or vice versa were used to improve sampling. Typically, at the state parameters considered here, insertion and deletion moves of particles of type A and B have an acceptance rate of ~ 10%, which is comparable to the identity exchanges. The rate of acceptance and deletions for surfactant particles is ~ 2%, which is adequate but means rather long G C M C runs are necessary to obtain well converged density profiles for simulations containing surfactant particles. Typically, for a given system, an equilibration run of ~ 6 x 10 particle moves (a move can be any of those described above) was required; this was then followed by a further run of ~ 10 — 10 moves over which averages were taken. A  B  6  6  9  The properties of interest are the density profiles of the fluid and the forces acting between the surfaces. A grid with Ax = Az — O.lcr in the x and z directions, was used to obtain the density profiles. Some simulations were performed with a finer grid ( A z = O.Olcr) within O.lcr of the surfaces of the plates. A t equilibrium, the pressure of a liquid is constant throughout; therefore we can calculate the pressure at any layer in the slit. The force, Fk, normal to the plates, acting on an infinitesimal area, Ak, of the surface at location k, can be divided into two terms: the force due to the momentum transfer to that area of the plate, and the force due to the attractive interaction between the surface and the particles perpendicular to k. We have then, h  (3.6) where p„(0) is the density at the surface (assuming identical walls, pv(0) is equal to p (h), and is evaluated by extrapolating the final average density profiles at either surface.) The pressure between the plates is the total force acting on the plate divided by the area of the plate v  (3.7) where the sum is over the area elements, k, k^ is the number of elements within n£ and A ^ is the area L x n£. This expression is identical to the contact theorem [73-75], with the exception that, the nonuniformity of the plates necessitates summing the forces over n£, rather than an arbitrary area. For both uniform and patterned plates, the force must be calculated over a sufficiently large area of the plate to obtain statistically smooth results. Typically, the area of the plate in a simulation cell (L x L ) is used. For some cases it was useful to perform the density extrapolations for the attractive and inert areas separately, due to the variation in steepness of the density profiles at the wall for these two areas. A t the attractive regions of the wall, the density profiles become quite steep, and the function p (z)exp(/3^v(z)) was sometimes used in the extrapolation, while for the hard wall regions, a linear extrapolation was sufficient. n  y  x  y  L/  The pressure, P, and surface tension, 7, can also be calculated from the grand potential, fi, which is given generally by fi =  -py  A+  + 7  7  i  L ,  (3.8)  19  Chapter 3. Fluids confined between chemically patterned surfaces  where 7 = h(P — P ), JL = L h(P — P ), A = L L is the area of the solid surfaces and L = L . Here P , P , and P = P are the elements of the pressure tensor. We use the virial route to calculate these quantities as discussed by Henderson [70]. Briefly, the leading-order change in the grand potential for the displacement field, e, is given by zz  x  yy  xx  y  yy  yy  xx  x  y  zz  (5Q)  T41  = ( > ^ {-k TV  • et + e • V U ) ) ,  B  (3.9)  t  where the sum on i is over all particles and U is the total potential energy including both particle-particle and wall-particle interactions. We calculate the pressure and surface tension as follows t  P=  ^  W  .  ^  ( _ 3  5A  v  1 0  )  '  For chemically patterned walls, to calculate the pressure, one must make a deformation to the volume such that in addition to the surface area and L, the patterning, £, remains fixed, as discussed previously. Similarly, the deformation made when calculating the surface tension, must be chosen such that, the volume, L, and the patterning of the plates remain fixed. Choosing displacement fields of e = e(0,0, z) for the pressure, and e = e(0, y, 0) for the surface tension, in the limit that e approaches 0, we obtain, 1 /.  ^^,  \  i<j  Y}zi%ii)  \ i<j  - Y^ZiKv  2  13  + (h~ Zi)  (h - )]^,  (3.12)  Zi  v  (Zi) + (h- zt)  (h - zi)]^.  (3.13)  We note here that the pressure calculation requires a sum over all particle interactions. This allows us to determine which interactions give rise to specific features of the pressure.  3.3 3.3.1  Binary mixtures C h e m i c a l l y Homogeneous Walls  It is useful to briefly consider the chemically homogeneous case. Following earlier work, we use the reduced temperature, T* = k T/e = 1.15 together with j3p' = —2.61 and B  A  A version of this section has been published, S.D. Overduin and G . N . Patey J . Chem. Phys. 117, 3391 (2002). l  Chapter 3. Fluids confined between chemically patterned surfaces  20  f3p! = —0.174. The grand potential of the bulk fluid with these parameters has two minima; the global minimum indicates that the i?-rich phase is the stable state, whereas the A-rich phase is metastable. For this and similar systems, it has been shown that, at separations smaller than some critical value, surfaces that favour the metastable state can induce a phase transition essentially from the stable to the metastable bulk state. This is analogous to surface wetting and drying observed in one-component systems [16], and B  Figure 3.2: Density profiles of the stable and metastable states for systems (lOcr x 10<r x h) with homogeneous walls. Results are shown for h = 10CT an 6a. can be thought of as the wetting of a surface by one species accompanied by the drying of the other. For this to occur for the present model and state parameters, A must be attracted to the surfaces more strongly than B. This is achieved by setting twA = — 3e and CWB = 0. The total reduced density, pa = ( A + P s ) c , profiles perpendicular to the plates for h = 20a and 6a are shown in Fig. 3.2 Both stable and metastable profiles are included. For this system the phase transition occurs at h ~ 13cr. For h = 20a, the fluid in 3  3  P  Chapter 3. Fluids confined between chemically patterned surfaces  21  the midplane region has the properties of the relevant stable or metastable bulk state, and the pressure is essentially the bulk pressure. The densities, mole fractions and pressures obtained for h = 20a are given in Table 3.2. We note that at the smaller separation, 6a, the midplane region is structured and, consequently, the fluid properties differ somewhat from the corresponding bulk states. Table 3.2: Properties of the stable and metastable state using a (10cr x 10a x 20a) system with homogeneous walls. The densities and mole fractions are obtained at the midplane. The numbers in this table can be regarded as the bulk values.  BPa pa  3  3  x  B  Stable  Metastable  0.539 0.327 0.881  0.468 0.685 0.981  Clearly for simulations with h < 12>a, the stable state between the walls depends on the strength and nature of the wall-particle interactions. We investigate the effects of chemical patterning at separations where attractive homogeneous surfaces induce a phase transition. Several questions are of interest. These include, the influence of patterning on the critical separation below which transitions occur, the nature of the surface-stabilized phases, and the strength and range of the resulting force acting between the plates.  3.3.2  C h e m i c a l l y p a t t e r n e d surfaces  In all calculations, the state parameters T*, Bp and j3p' were as given above. Also, the values e A = —3e and ewB = 0 were used throughout. We begin by considering a system that is 10a x 10cr x 6a (these are the x y z dimensions) with £ = 10a and varying values of £i. Total density and mole fraction, X , profiles at the midplane are shown in Fig. 3.3, and a three-dimensional plot of a particular system is given in Fig. 3.4. £i > 1.2£. From Fig. 3.3, we see that as the width of the attractive stripe is increased from zero, A-rich "bridges" joining both surfaces are formed. We note that this behaviour is analogous to the "bridge phase" that occurs in one-component systems, first described by Rocken and Tarazona [61], and more recently studied in detail by Schoen and coworkers [62-64]. For the smaller stripe widths, the 5-rich regions (adjacent to the inert surfaces areas) have a composition close to that of the stable bulk phase, whereas the composition of the yl-rich regions (adjacent to the attractive stripes) differs from that of both the stable and metastable bulk states. For the larger stripe widths, the reverse situation occurs with the composition of the ^4-rich regions becoming similar to that of the metastable bulk state. We are interested in the net pressure A  B  W  A  P  n e t  = P - P  b u l k  ,  (3.14)  where P is the pressure of the bulk solution given in Table 3.1. P determines the force acting between the walls, and for the homogeneous case the surface-induced transition b u l k  n e t  Chapter 3. Fluids confined between chemically patterned surfaces  22  Figure 3.3: Density and mole fraction profiles at the midplane for (10a x 10cr x 6cr) systems with £ = 10a and various values of £i. Results are shown for ^ = 0 (open squares), 3o (filled squares), 4o (open triangles), 5cr (filled triangles), 6a (open diamonds), 7a (filled diamonds), 9a (open circles) and 10a (filled circles). leads to a strong attractive interaction [21]. We wish to determine the forces associated with the bridge phase described above. In Section 3.2, two possible routes to the pressure are discussed; a mechanical route analogous to the contact theorem often applied in the homogeneous case, and a virial route based on the grand potential. Apart from discrepancies of numerical origin, both routes should give identical results. The net pressures obtained using the different routes are compared in F i g . 3.5. We see that both methods give qualitatively similar results, and we emphasize that the total pressures actually calculated differ by not more than ~ 5%. The discrepancies that do exist could come from several sources. For example, the extrapolations required in the mechanical route are a potential source of small but  Chapter 3. Fluids confined between chemically patterned surfaces  Figure 3.4: The three-dimensional density profile for a (10a x 10<r x 6a) system with £ = 10a and £i = 5a. systematic errors. Also, it is possible that the periodic boundary conditions and potential truncation scheme employed in the simulations have slightly different influences on the virial and mechanical pressure estimates. Finally, it is worth mentioning that as a further check on the mechanical route we simulated a system with one homogeneous inert surface and one striped surface, and calculated the average force at each wall. Consistent estimates of the pressure were obtained. From F i g . 3.5, it is apparent that the formation of a bridge phase is accompanied by a net attractive force acting between the walls. Initially, the net pressure decreases with increasing a minimum is reached at « 0.6, the net pressure then increases until = 1 (i.e., full coverage). The existence of a minimum is somewhat surprising at first sight because one might have expected that the strongest attraction would occur at full coverage. However, for h = 6a the attractive forces lead to an increase in total density for > 0.6, and this in turn increases the pressure between the walls. Results for the larger separation, h = 10a (£ = 20a), are also plotted in F i g . 3.5, and here we see that a minimum does not occur; the strongest attraction occurs near full coverage. However, we do note that the strongest attractive forces achieved are of similar magnitude for both separations. It is of interest to examine the phase behaviour and pressure as a function of the surface-  23  Chapter 3. Fluids confined between chemically patterned surfaces  24  Figure 3.5: A comparison of the net pressure obtained via the mechanical (open triangles) and virial (filled triangles) routes as a function of (i.e., the surface coverage) for (10a x 10(7 x 6cr) systems with £ = 10u. Results obtained by the virial route (filled circles) for a (20cr x 10cr x 10cr) system with £ = 20<r are also included. surface separation, h. To that end, calculations were carried out for £ = 10a with fixed £i and varying h. Results obtained for £i = 5<7, 6a and 7 a are shown in F i g . 3.6. The mole fractions plotted are the average values obtained over a slat of width la centered at the midplane. The onset of bridge formation is signaled by relatively rapid changes in concentration in the midplane region, together with a marked drop in the net pressure to more strongly attractive values. The reduced surface tension, ftya , calculated by means of Eq. (3.13) is also included in Fig. 3.6. We note that as the plates are brought together the surface tension rapidly becomes more negative as the bridge phase is formed. Of course, this is as we would expect since it is precisely this favourable surface tension contribution to fl that stabilizes the bridge phase at smaller separations. Bridge formation as a function 2  Chapter 3. Fluids confined between chemically patterned surfaces  25  Figure 3.6: The mole fraction, XA, in the midplane region, the reduced surface tension, /?7<r , and the reduced net pressure, (3P o , as functions of the surface separation, h/o, for (10a x 10a x h) systems with £ = 10a. Results are included for £i = 7a (open triangles), ^ = 6a (filled circles) and £i = 5a (open squares). The lines are drawn to guide the eye. 2  net  3  of separation is further illustrated in F i g . 3.7, where midplane XA profiles are plotted for £i = 6a. The formation of A-rich bridges as h is decreased is evident in the plot. Returning to F i g . 3.6, we see that, as one might expect, the range of the attractive interaction decreases with decreasing stripe width, £i. However, the magnitude of the attractive interaction remains approximately constant. We note that for the fully covered case (£i = £) the transition essentially from the stable to the metastable bulk state occurs at h « 13a, and that (3P o « —0.1 following the transition [21]. We also note (Fig. 3.6) that for the patterned case the pressure does not drop as sharply as a function of h as it does for homogeneous surfaces. This can be understood by considering that for the homogeneous case essentially only two phases (the bulk stable and metastable states) are net  3  Chapter 3. Fluids confined between chemically patterned surfaces  26  Figure 3.7: Mole fraction profiles at the midplane for different separations for a (10a x 10a x h) system with £ = 10a and £i = 6a. Results are included for h = 2.5a (open squares), h = 3a (filled squares), h — 4a (open triangles), h = 5a (filled triangles), h = 8a (open circles) and h = 10a (filled circles). involved, and even with finite size effects one observes a relatively sharp transition from one to the other at a particular value of h. The situation for patterned surfaces is different in that the bridge phase exists only between the walls, and the bridges differ in density and composition (see F i g . 3.7) for different separations. Thus one does not observe a single sharp transition between two phases, but rather a series of different bridges and the corresponding pressures as h is varied. We do note that the maximum attraction appears to be achieved when the composition of the A-rich bridges approaches that of the metastable bulk state. Also, the pressure drops become sharper for wider stripes that stabilize bridges closer in composition to the metastable bulk. The systems discussed above are characterized by patterns with "wide" stripes in the  Chapter 3. Fluids confined between chemically patterned surfaces  27  sense that £1 and £ are roughly of the order of h. It is of interest to consider the other limit where the stripes are "narrow" with respect to the surface separation. Therefore, calculations were carried out for systems with h = 6a, £ = 2a and a range of £ i . For £i < la, no phase change occurs and the densities and concentrations at the midplane remain close to those of the stable bulk state. A relatively sharp transition occurs at £i ~ 1.2cr accompanied by a drop in pressure comparable with that found for homogeneous surfaces. However, in this case a bridge phase does not form, nor does one obtain the metastable bulk state. Rather, one finds something in between with structural features reflecting the pattern on the surfaces extending over the entire distance between the walls. Midplane profiles for £i = 1.2<7 are shown in Fig. 3.8, and the surface-induced structure is evident in the density. We note that for £i = 1.2cr it is possible to "fit" just two particles into a single stripe and this leads to the sharply peaked pattern in the density profile. However, it must be remarked that the "sharp" stripes employed in our model are physically less realistic when £ and £i are of the order of a particle diameter. O n these length scales any physical pattern would not be as sharp as we assume, and this would tend to discourage any long-range, surface-induced structural features. A l l results discussed above involved patterns consisting of rectangular stripes. It is of interest to ask to what extent our observations depend on the shape of the attractive surface areas. Therefore, we have also investigated patterns consisting of attractive circular "spots" applied on a square lattice with the centers separated by 10a. Note that for this pattern the attractive regions of the surfaces are finite in all directions. A s for the striped case, bridges and accompanying strong attractive forces are observed for this system. Furthermore, for both spots and stripes the transitions occur at approximately the same fractional attractive surface area. Thus, not surprisingly, the size rather than the shape of the attractive parts of the surface appears to be the important factor in bridge formation. As noted in our introduction, we are not only interested in the forces between patterned surfaces, but also in the possible relevance of our results for the interaction of mesoscopic "particles" (e.g. colloids, polymer segments etc.) in binary fluids near coexistence. W i t h this in mind, we have carried out some further analysis. A n obvious question to consider is to what extent, if any, our observations are influenced by the fact that multiple bridges are formed in our systems. One can perhaps guess the answer by noting from Fig. 3.3 that there appears to be little "interference" between adjacent bridges. Nevertheless, we have investigated this further for the h = 6a case by increasing £ while holding £ fixed until the A-vich. bridges are separated essentially by stable bulk fluid. This demonstrates that the formation of bridges depends strongly on £i and h, but not on £. Also, if the bridges are truly independent of each other, then the net pressure should be inversely proportional to £ (i.e., P = C / £ ) . We have verified that this is in fact the case as £ becomes large; for example, for h = 6a and £i = 6a, P is linear in £ for £ > 15a. These observations strongly imply that bridges and the corresponding forces could occur between immersed macroparticles or parts thereof. In the following chapter, we look explicitly at systems containing macroparticles. x  net  net  - 1  A l l of the results discussed above were for surfaces with commensurate patterning. Of course, in solution macroparticles are free to move and can rotate, align etc. as they see fit. Therefore, it is of interest to enquire if bridges are the stable thermodynamic state if the stripes are free to move. That is, if the stripes are free to take any position on the surface,  Chapter 3. Fluids confined between chemically patterned surfaces  28  0.8 tf*  »A  A  I 0. 0.4  0.2 1 0.8  i< -  0 6  X  0.4  0.2  0  x/a Figure 3.8: Density and mole fraction profiles at the midplane for multistriped, (10a x 10a x 6a) systems with £ = 2a. Results are shown for £ = l a (open triangles) and £i = 1.2a (filled triangles). The structure observed in the profiles is due to particle packing (see text). x  will the possibility of forming a bridge drive them to align. Calculations were carried out for systems with h = 6o~, £ = 20a and £j = 2a and 6a. Although £ and £i are fixed, the patterns are not required to be commensurate, and random stripe displacements along the surface were included as one of the G C M C moves. For £i = 2a, the stripes do not align and are completely indifferent to their relative positions on the wall. This is consistent with the fact that for h = 6a, stripes of this width do not form significant bridges even in the commensurate case. In contrast, for £ = 6a the stripes do align, driven by the formation of A-rich bridges as illustrated in Fig. 3.9. It can be seen from Fig. 3.9 that the bridges are similar to those in the static commensurate system, except that the positional fluctuations tend to reduce the structure in the x direction giving smoother profiles. This suggests that x  Chapter 3. Fluids confined between chemically patterned surfaces  29  Figure 3.9: The three-dimensional density profile for a (20a x lOo x 6a) system with £ = 20a and £ i = 6a. In this calculation the stripes on both walls were free to move independently, but are aligned by bridge formation. The profiles are plotted using a coordinate system that places the origin at (x\ — X2)/2, where X\ and x are the x coordinates of stripes 1 and 2. 2  macroparticles might well rotate or align such that bridges are formed; the accompanying attractive forces could then lead to coagulation in colloidal systems, or folding of some sort in the case of polymers.  3.4  Surfactant-stabilized liquid structures  2  A version of this section has been published, S.D. Overduin and G.N. Patey J. Chem Phys 119 8676 (2003). 2  Chapter 3. Fluids confined between chemically patterned surfaces  30  We briefly overview the results for binary mixtures discussed in Section 3.3 before examining surfactant-containing systems. Following previous work, we use the reduced temperature T* = k T/e = 1.15 and chemical potentials for species A and B,/3p' = —2.61 and (3pJ = —0.174, unless otherwise stated. There are two minima in the grand potential for bulk systems with these parameters; the global minimum corresponds to the stable B-rich state and the second minimum corresponds to the metastable A-rich state. Decreasing the chemical potential of species B moves the system towards coexistence which occurs at (3u.' ~ —0.522 [21]. A t coexistence, the mole fractions of species A in the two phases are 0.08 and 0.816, demonstrating that the mixture is well below the critical temperature. Where two bulk phases can exist, a small perturbation to the system can induce a transition from the stable bulk phase to the metastable bulk phase. For example, at (3p' = —0.174, the A-rich phase becomes stable in the presence of confining surfaces that interact favourably with species A. Similarly, chemically patterned surfaces (described in Section 3.2) can induce a transition to a bridge phase when ^ ~ h (see Section 3.3). The relationship between £i and h arises due to the formation of liquid-liquid interfaces; the favourable solid-liquid interface compensates for the unfavourable liquid-liquid interface when the areas of the two are of comparable size. One would expect, therefore, that reducing the liquid-liquid interfacial tension by the addition of surfactant would result in the formation of liquid bridges at larger surface separations. This is indeed observed in the present work, as described below. B  A  B  B  B  3.4.1  Bulk  Systems  In our model, the confined fluid is in thermodynamic equilibrium with the corresponding bulk system. Therefore, we must know the bulk properties in order to understand the effect of surfactant in confined systems. Although a variety of complex phases are possible in surfactant mixtures, it is useful to think of the addition of a small amount of surfactant as a "perturbation" to the binary bulk system. A binary mixture must be near demixing coexistence for the formation of a bridge phase to occur. We are, therefore, primarily interested in any perturbation of the binary demixing transition due to the presence of surfactant. In order to maintain numerical consistency, "bulk" calculations are performed in systems with inert surfaces at sufficiently large separations (h = 20c) that the fluid in the middle region has no surface-induced structure. Typical density profiles for all three components are shown in F i g . 3.10; we see that surface induced structural effects extend no further than ~ 5cr from the surfaces. We consider systems containing surfactant 1 and surfactant 3, noting that surfactants 1 and 2 are identical in the bulk case. Figs. 3.11 and 3.12 show how the densities and mole fractions of species A and B change with varying surfactant chemical potential. Our surfactant models interact more strongly with species A than with species B. Consequently, as (3p' increases, the density of species A increases, whereas that of species B simultaneously decreases (Figs. 3.11 and 3.12). The surfactant increases the solubility of species B in species A and vice versa. A t some values of (3p' , mixtures of nearly equivalent mole fractions of species A and B are possible, due to the surfactant's ability to increase the solubility of each species in the other. This is somewhat akin to increasing the temperature of the binary mixture, such as to move the system closer to the demixing critical point. Consequently, we might expect c  c  Chapter 3. Fluids confined between chemically patterned surfaces  31  Figure 3.10: Reduced densities of species A, B and C midway between homogeneous surfaces separated by 20a. One system (top) contains surfactant 1 with (3pJ = —0.2, and the other (bottom) contains surfactant 3 with j3p' — —1.35. In both systems f3p' — —2.61 and (3p' = -0.174. c  c  A  B  that, for a fixed temperature, a phase change in a system with surfactant will have a smaller associated change in composition. The transition from a 5-rich to an A-rich fluid occurs over a smaller range of (3p' for surfactant 3 than for surfactant 1, due to the attractive CA — CB and CB — CB interactions. In the binary mixture {(3p' = —2.61 and (3p' = —0.174), decreasing the chemical potential of species B results in a phase change [21]. From Figs. 3.11 and 3.12, we see that a phase change also occurs in surfactant systems when (3p' is varied at fixed (3u-cc  A  B  B  Chapter 3. Fluids confined between chemically patterned surfaces  32  Figure 3.11: The reduced densities (top) and mole fractions (bottom) of species A (squares), B (circles), and C (triangles) in "bulk" systems containing surfactant 1, as functions of (3p! (right) and Bp! (left). In the plots on the left, (3p! is held fixed at -0.2, and in the plots on the right, (3p' is fixed -0.174. "Bulk" properties are calculated from systems confined between inert surfaces at a separation of 20a, in order to maintain numerical consistency.  B  c  c  B  The chemical potentials chosen for surfactants 1 and 2 are (3p' = —0.2 and —1.35, respectively. Note that due to finite-size effects, the transitions are not sharp but are signaled by relatively fast crossovers from 5-rich to A-rich states. The values (3p! = —0.2 and —1.35 correspond to systems where bridge formation occurs at extended separations in confined geometry when f3p! = —0.174) (see below). Surfactant 3 has a smaller effect on the binary demixing behaviour than surfactant 1; with surfactant 3 the transition from a fi-rich to an c  c  B  Chapter 3. Fluids confined between chemically patterned surfaces  33  Figure 3.12: The reduced densities (top) and mole fractions (bottom) of species A (squares), B (circles), and C (triangles) in "bulk" systems containing surfactant 3, as functions of f3p' (right) and (3p' (left). In the plots on the left, (3p' is held fixed at -1.35, and in the plots on the right, (3p' is fixed -0.174. "Bulk" properties are calculated from systems confined between inert surfaces at a separation of 20a, in order to maintain numerical consistency.  B  c  c  B  A-rich fluid occurs over a much smaller range of 0p' , than with surfactant 1. Although the pressure of surfactant-containing systems is greater than that of binary systems, in both cases it decreases with an increasing concentration of species B. B  Chapter 3. Fluids confined between chemically patterned surfaces  3.4.2  34  Confined Systems: F l u i d Structure  Liquid bridges form in mixtures near two-phase coexistence, when confined between chemically patterned surfaces with £ ~ h, as discussed above. W h e n surfactant is added to the system this "constraint" is removed, and bridges form at separations up to five times larger than £ i . For example, in confined systems with pattern parameters £ i = 2a and £ = 10a, bridge formation in the absence of surfactant occurs at h = 3cr, while in the presence of surfactant, bridge formation occurs at h = 15a. The formation of these surfactantstabilized bridges does still require that the chemical potentials of species A and B be near their binary coexistence values. To show this we vary Bp! in a confined system containing surfactant 1, keeping Bp! and Bp! fixed; as (3p! decreases, the bridge becomes weaker and eventually breaks. A n example of this is shown in F i g . 3.13. This demonstrates that the presence of surfactant in a mixture confined between chemically patterned surfaces can only enhance bridge formation when the system is near demixing coexistence. x  B  A  c  B  As discussed above, adding surfactant to the mixture changes the fluid composition. Therefore, we expect the fluid structure of bridge systems containing surfactant to differ from the fluid structure of binary bridge systems. We examine in detail three mixtures containing surfactant 1 and two mixtures containing surfactant 3, noting that surfactants 1 and 2 behave nearly identically despite the CA - surface attraction of surfactant 2; for surfactant 1, Bp! = 0, —0.2, —0.5, and for surfactant 3, Bp! = —1.35 and —1.32. In all three systems containing surfactant 1, the bulk densities of species A and B are of similar magnitude; when Bp! = 0, the fluid is richer in species A, when Bp! = —0.2 and —0.5, it is richer in species B. Species B is dominate in both surfactant 3 systems considered, and the difference in the densities of species A and B is larger than in the surfactant 1 case. Although all three surfactants do extend the length of the liquid bridges formed in the presence of chemically-patterned surfaces, they also change the structure of the confined fluid and the dependence of bridge formation on the patterning parameters £ and £ i . Some results illustrating this for surfactant 1 are shown in F i g . 3.14. The bulk fluid containing surfactant 1 with Bp! = 0 is richer in species A. A t a separation of 10a, the composition of the fluid in the middle region is similar to the bulk composition when the surfaces are unpatterned and inert, as shown in Fig. 3.14 (a). In binary A — B mixtures where species A is dominate when confined by inert surfaces, surface-patterning does not induce bridge formation. We also note that in such a binary system with inert surfaces, the 5-rich fluid regions would not extend more than ~ l a from the surfaces. W h e n the surface is patterned with £x = l c , the fluid composition in the middle region is nearly unaffected, while regions of the ^4-rich fluid protrude from the surface [Fig. 3.14 (b)], similar to what occurs in the binary mixture. However, increasing £ 1 , we see the formation of A-rich liquid bridges perpendicular to the attractive surface areas [Fig. 3.14 (c) and (d)]. In systems with this surface separation (10a) and £ = 2a, the reduced density of species A in the bridges (at the midplane) is ~ 0.25, which is more than double its bulk value of ~ 0.11, and that of species B is ~ 0.10, much lower than its bulk value of ~ 0.16. The surfactant density is greatest in the interface, and is oriented with the CA end directed into the A-rich bridge (not shown). The CA end of surfactant 1 interacts attractively with species A, and the CB end interacts attractively with species B. A l l other surfactant 1 interactions are repulsive (Table 3.1). Consequently, the C ends pointing away from the bridge, induce an increase in the density c  c  c  c  c  x  B  Chapter 3. Fluids confined between chemically patterned surfaces  35  Figure 3.13: Density profiles of species A (dark) and B (light) in systems containing surfactant 1 with (3p' = —0.2 and f3p' = —2.61. The chemical potential of species B is varied such that PUB = - 0 . 1 7 4 in (a), J3p = - 0 . 1 0 0 in (b) and (3p = - 0 . 0 0 in (c). c  A  B  B  of species B between the A-rich bridges giving rise to the alternating dark(A-rich)-light(.Brich) bridges illustrated in Fig. 3.14 (c) and (d). This can only occur when the separation between adjacent bridges is sufficiently small. Therefore, for this particular system, if the attractive stripes are separated by large distances (large £), the distinct dark-light bridges will disappear in favour of structure similar to that shown in Fig. 3.14 (b) for & = \o. We note that this behaviour only occurs for some surfactants and state parameters; for example, in the surfactant 3 system, increasing the attractive regions of the surfaces does not result in the formation of bridges when the fluid in the middle region is already rich  Chapter 3. Fluids confined between chemically patterned surfaces  36  Figure 3.14: Density profiles of species A (dark) and B (light) in systems containing surfactant 1 with /3p' = 0.0, /3p' = —2.61 and flp' = —0.174. The patterning parameters £ are (a) 0, (b) lo, (c) 2a and (d) 6cr. c  A  B  t  in species A. Bridges also form in surfactant 1 cases at chemical potentials where the bulk fluid is richer in species B {/3p' = —0.2 and —0.5). In these systems, structures such as those observed in Fig. 3.14 (b) do not occur, however, the bridges do resemble those shown in Fig. 3.14 (c) and 3.14 (d) as discussed below. Of course, in all cases the bridges will eventually disappear as the surface separation is increased weakening the influence of the attractive stripes. Some results for systems containing surfactant 3 with h = lOo are shown Fig. 3.15. For 3p' = —1.35, we do not observe bridge formation for ^ = la at this separation, rather the c  c  Chapter 3. Fluids confined between chemically patterned surfaces  37  Figure 3.15: Density profiles of species A (dark) and B (light) in systems containing surfactant 3 with /3p' = -2.61, (3p' = -0.174 pp' = -1.35 in (a) and (b) and f3p' = -1.32 in (c) and (d). The patterning parameter £i is \o in (a) and (c) and 2o in (b) and (d). A  B  c  c  fluid in the middle region remains rich in species B and similar in composition to the bulk [Fig. (3.15a)]. However, for £i = 2cr, distinct A-rich bridges are formed [Fig. 3.15 (b)] and persist in the £ —» oo limit. However, the situation is somewhat different for (3p' — —1.32. Here the situation for £i = ler is the same as that discussed above [Fig. (3.15c)], but for £i = 2<7 the structure obtained depends on £. For example, for the situation shown in Fig. 3.15 (d) we see that we do not obtain distinct dark-light bridges, but instead the Arich regions expand and join in the middle forming structures that resemble those discussed above for surfactant 1 [see Fig. 3.14 (b)]. This is a consequence of the fact that the CA~CB c  Chapter 3. Fluids confined between chemically patterned surfaces  38  Figure 3.16: Cross-sections of the reduced density (top) and mole fraction (bottom) near one surface and midway between the two surfaces of surfactant 1 systems with Bp! = -0.5 and h = 6a. The values of £j are 6a (triangles), 2a (squares), and l a (circles). c  and CB — CB interactions of surfactant 3 are attractive. As a result, when surfactant 3 aligns with the CA end directed into A-rich fluid regions, the CB end of the surfactant attracts both surfactant and species B, leading to an excess of A throughout the middle region. Of course, as £ —> oo, the fluid perpendicular to the inert areas of the surface must approach the 5-rich bulk composition, and the distinct dark-light bridge pattern must be recovered in this limit. The density and mole fraction of species A in the bridges depends on this is illustrated in Fig. 3.16 for a particular surfactant 1 system. As we would expect, wider stripes give  Chapter 3. Fluids confined between chemically patterned surfaces  39  rise to higher densities and mole fractions of species A in the bridge, both near the surfaces and at the midplane. The density of species A near the surface is always greater than in the middle due to the direct attractive interaction of this species with the surfaces. We now examine in more detail the formation and structure of liquid bridges as a function of surface separation, focusing on systems with £1 — 2o. We discuss differences and similarities among the model surfactants for the chemical potentials where bridge formation is observed. Some typical results for a system containing surfactant 3 are illustrated in Fig. 3.17. W h e n the fluid mixture is confined between surfaces separated by a distance larger than required for bridge formation, elongated A-rich "bubbles" extending from the attractive regions of the surfaces [Fig. 3.17 (a)] are observed for all three surfactants. As the surfaces move towards each other the bubbles overlap forming a bridge [Fig. 3.17 (b)]. We note that unlike the corresponding zero surfactant cases, the A-rich bridges now tend to "bulge" beyond the region defined by the surface stripes. However, one must be careful because the profiles plotted in Fig. 3.17 are not very sensitive to details of the concentration and can be a little misleading; from the more precise plots given in Fig. 3.18 we see that the density of A is actually much greater perpendicular to the stripe than it is in the bulging region. Also, as the separation is decreased the surfactant particles become more rigidly aligned perpendicular to the interface (Fig. 3.18) and the liquid-liquid interfaces become sharper. There is an increase of species A in the bridge, both near the surfaces and at the midplane. The density of species A near the surface is always greater than in the middle due to the attractive interaction between this species and the surfaces. Surfactants 1 and 2 behave nearly identically despite the fact that the CA end of surfactant 2 is attracted to the surface stripes, whereas the surfaces are completely inert towards surfactant 1. Surfactant 2 does have a higher density than surfactant 1 near the surfaces, however, this does not appear to significantly influence other structural properties. As pointed out above, surfactant 3 does behave somewhat differently from the other two surfactants due to the attractive CA — CB and CB — C interactions. Midplane densities of the three fluid components are shown in Fig. 3.18 for systems containing surfactant 1 and surfactant 3. We note, that for surfactant 3, the midplane densities are not consistent with the "bulk" values even at a separation of 20cr. Given that there is little structure evident in the midplane cross sectional densities at 20cr, this is surprising at first sight. However, if we examine the density profiles as a function of z [see F i g . 3.19 (a)] we observe that at h — 20(7, the densities are still slowly varying in the midplane region. This has a significant effect on the net pressure obtained for patterned walls with surfactant 3, as discussed below. It is also apparent from F i g . 3.18 that the liquid bridges formed with surfactant 3 have a higher density of species A and a lower density of species B than those formed with surfactant 1. The density of surfactant 3 in the interface is also greater than that of surfactant 1 in comparable systems. A t a separation of 6o the results obtained with surfactants 1 and 3 are very similar; species A is well localized in the bridge, the surfactant C in the interface, and species B in the regions between bridges. In F i g . 3.19 (b) and (c) we see how the density of species A increases gradually in the middle region, while that of species B decreases. This demonstrates that although bridges form at h ~ 15a, the bridge becomes richer in species A as the surface separation decreases. B  To conclude this discussion, we point out that other types of patterning also give rise to liquid bridge formation. For example, in binary and one-component systems, surfaces  Chapter 3. Fluids confined between chemically patterned surfaces  40  Figure 3.17: Density profiles of species A (dark) and B (light) in systems containing surfactant 3 with Pp' = - 2 . 6 1 , /3(i' = -0.174, (3p' = - 1 . 3 5 in (a) and (b) and fip' = -1.32 in (c) and (d). In all systems £i is 2cr and the surface separations are (a) 20cr, (b) 16a, (c) 15a and (d) 8a. A  B  c  c  patterned with circular attractive areas induce bridge formation. W h e n surfactant is added to such systems the length of the liquid bridges can be extended, analogous to the extension of bridges in systems with stripes. This demonstrates that liquid bridges can also be extended when the patterning is finite in all directions. The properties of these cylindrical bridges are similar to those induced by stripe patterns, and their composition varies with circle size and surface separation.  Chapter 3. Fluids confined between chemically patterned surfaces  t  •  •  •  •  <  0  -5  .  x/a  .  .  .  .  t .  .  .  .  5-5  .  .  .  .  0  41  .  5  X / G  Figure 3.18: Cross-sectional density profiles of all three species averaged over a slab of width l a located midway between the surfaces. In all systems shown 6p! = —2.61 and Bp! = —0.174. Results for surfactant 1 with Bp! = —0.2 are plotted on the right and those for surfactant 3 with Bp! = —1.35 on the left. The curves marked with circles are for systems with inert walls separated by 20a. In all other systems £i = 2a and the symbols indicate different surface separations as follows: squares (6cr), triangles (10a), diamonds (15a), inverted triangles (17a) and right-facing triangles (20a) A  B  c  c  3.4.3  C o n f i n e d Systems: Forces between Surfaces  It is interesting to examine the forces acting between the surfaces as the fluid structure changes with changing surface separation. In confined binary mixtures near demixing  Chapter 3. Fluids confined between chemically patterned surfaces  42  Figure 3.19: Reduced densities of species A, B and C in systems containing surfactant 3 with £ i = 2cr and the chemical potentials are as in F i g . 3.18. The densities are averaged over the width of the attractive stripe. The plots are for the surface separations (a) 20cr, (b) 15a and (c) 8cr. coexistence, the transitions induced by the presence of confining surfaces are accompanied by strong attractive forces [21]. The attractive forces arising due to bridge formation when the surfaces are chemically patterned were, to a good approximation, found to scale linearly with the area of the attractive regions. This indicates that for the binary systems the changes i n force with surface separation are simply related to changes i n the structure and composition of the liquid bridges. This in turn suggests that forces of similar origin may influence the interaction of macroparticles immersed in such mixtures.  Chapter 3. Fluids confined between chemically patterned surfaces  43  The addition of surfactant increases the complexity of the system, and in general the simple scaling relationship between the area of the surface pattern and the force does not hold as well. Nevertheless, for £ > 2a scaling does approximately apply over certain separation ranges; to make this apparent the net pressure results for £i = 2a shown below (Fig. 3.20) are scaled by the area ratio, F = . The net pressure, P = P - P , x  n e t  b u l k  Figure 3.20: The net pressure as a function of the surface separation h for systems containing surfactants 1 (diamonds), 2 (triangles), with £i = la (a) and £i = 2a [(b)F = In all systems Bp! = —2.61, Bp! = -0.174; for surfactants 1 and 2, Bp! = —0.2 and for surfactant 3 Bp! = —1.35. Results are also included for homogeneous surfaces that are inert for £x = la (a) and attractive to species A for £i = 2a (b). These results are denoted by circles for surfactant 1 and by right-facing triangles for surfactant 3. A  c  B  c  Chapter 3. Fluids confined between chemically patterned surfaces  44  where P is taken to be the pressure obtained with inert walls 20a apart, is plotted in F i g . 3.20. Results for several systems containing different surfactants are included. Unlike the zero surfactant case, we see that the initial formation of liquid bridges is not accompanied by strong attractive forces when surfactant is present. Furthermore, the behaviour of P as a function of surface separation when £ = l a differs significantly from that when ^ > a, despite the fact that liquid bridges are observed in both cases. The net pressure with £i = l a is nearly identical to that between homogeneous inert surfaces both with [Fig. 3.20 (a)] and without (not shown) surfactant, where no liquid bridges are formed. This demonstrates that for ^ = l a changes in pressure at small surface separations are due to changes in the density of species B, and not to bridge formation. The net pressure for surfaces with £ i = 2a [Fig. 3.20 (b)], and £ i = 6a (not shown) in systems containing surfactant 1 and surfactant 2, has the same qualitative features as those confined by homogeneous attractive surfaces. In these systems the forces do become attractive, but the surface separation at which this occurs (~ 10a) does coincide with the distance at which ^4-rich bridges are first formed (~ 14a). Rather, the attractive forces appears to be associated with the overlap of the surfactant "layers" as the surfaces are brought together. The density profiles of the ends, CA, CB, and center, C , of surfactant 1 are plotted in F i g . 3.21. We note that the surfactant structures extending from the surfaces overlap at h 10a, which corresponds to the separation at which attractive forces are seen in the surfactant 1 results plotted in F i g . 3.20 (b). Homogeneously attractive surfaces show a similar attractive force at slightly smaller separations [Fig. 3.20 (b)]. This too appears to be associated with the overlap of surface-induced surfactant layers. b u l k  n e t  x  Attractive forces between patterned surfaces are also observed in surfactant 3 systems, however, for £i = 2a, they are observed well before bridge formation occurs [Fig. 3.20 (b)]. The attractive forces observed at large separation (e.g. 20a) are due to the fact that, as discussed above, the densities in the middle region are still slowly varying and have not reached their bulk values (see Figs. 3.18 and 3.19). Simple scaling clearly does not work at these separations. However, at smaller surface separations (h < 11a) the most significant changes with separation occur in the bridge composition and structure. Thus at the smaller separations the net pressure approximately scales with the attractive surface area. For all three surfactants, with £ > 2a the attractive interaction reaches a minimum at / i ~ 6a [Fig. 3.20 (b)]. A t smaller separations the net pressure decreases and eventually becomes positive. This does not resemble the behaviour in surfactant free systems and is likely due to the additional length scale introduced by the surfactant. Noting that species A will almost always "coat" the attractive surface areas, surfactant particles cannot be oriented perpendicular to each surface without directly interacting with each other when h < 6a. This results in a less favourable arrangement of particles, giving rise to a greater pressure. The peaks in the net pressure observed at smaller separations are due to particle packing, and are also observed in the absence of surfactant. x  The surface tension 7 associated with the liquid-solid interfaces is plotted as a function of h in F i g . 3.22. Results for surfactant 3 (those for surfactants 1 and 2 are similar) and several values of £1 are shown. A s expected, 7 is negative except at very small separations. We note that, at least for narrow stripes, the surface tension does not scale with stripe area; 7 ( ^ i / 0 exhibits a significant dependence on ^ and, at larger h, becomes more negative as £1 becomes larger. This behaviour is not surprising: the surfactant will displace some  Chapter 3. Fluids confined between chemically patterned surfaces  45  Figure 3.21: Density profiles of CA (light gray), CB (white) and of the surfactant center (dark gray) in systems containing surfactant 1 with (3p' = —2.61, (3p' = —0.174 and (3p' = —0.2. In all systems £ i = 2a and the surface separations are: (a) 6cr, (b) 10a and (c) llcr. One can see that the surfactant "bubbles" begin to overlap at a separation of 10a. A  B  c  of species A from the stripe area, leading to an increase in 7, and this effect will be more important for smaller £1. Finally, we see from Fig. 3.22 that for all systems 7 becomes essentially constant for h > 8a, indicating that the surface tension is determined by the fluid surface "layers" that are approximately 4a thick.  Chapter 3. Fluids confined between chemically patterned surfaces  46  Figure 3.22: The surface tension as a function of the surface separation, h, for systems containing surfactant 3 (b). The symbols represent £i = la (diamonds), £i = 2a (squares), and £i = £ = 10a (circles). The chemical potentials and the scaling factor F are as in F i g . 3.20.  3.5  Summary  In this chapter, we have used G C M C calculations to investigate the phase behaviour of binary mixtures or ternary mixtures, in which one of the components behaves as surfactant, confined between chemically patterned plates, but in equilibrium with a stable bulk fluid. Particular attention is focused on the plate-plate forces associated with surface-induced phase transitions, and on the structure of the fluid in the bridges. For practical reasons,  Chapter 3. Fluids confined between chemically patterned surfaces  47  a simple L J model was employed in our calculations, but since we are dealing with very general phase-change-related phenomena, we would fully expect our results to be relevant for real physical systems. A l l that is needed is a mixture relatively near liquid-liquid coexistence, and surfaces or macroparticles parts of which attract one species more than the other. Most of our calculations involved patterns consisting of parallel stripes (attractive to one species) of width, £x, and repeated along the surface with a period, £. Not surprisingly, the structure of the system and the net plate-plate pressure was found to depend on the relative sizes of £1, £ and the surface-surface separation, h. In the binary case for h < 3o, one finds strong structural features in the fluid that reflect particular patterns as well as particle packing constraints. Hence, thinking simply in terms of phase behaviour is clearly not sufficient in this regime. Therefore, the following generalizations are valid at larger separations where a description in terms of phase behaviour is more appropriate. Given this proviso, in the absence of surfactant particles, as the striped surfaces are brought together three types of behaviour are observed. If the surface coverage is high (£1 ~ £), then the system behaves much as in the homogeneous case; at some separation a transition basically from the stable to the metastable bulk state occurs with an accompanying attractive force between the walls. If the surface coverage is too low to induce a transition between the bulk phases, then the relationship between £1 and h becomes crucially important. If £1 < < h, then no transition occurs and the stable bulk state remains stable between the plates. The most interesting behaviour occurs when £1 ~ h. In this case a bridge phase forms below a particular separation. We show that, while the separation necessary for bridge formation is generally smaller than that needed to induce a transition with homogeneous surfaces, the accompanying net plate-plate force is attractive and equally strong. Furthermore, bridges form even as approaches zero, and their contribution to the net force is linear in the number of bridges in this limit. This implies that similar "bridges" and associated forces can occur for large, but finite-sized particles in solution. Calculations for patterns consisting of circles arranged on a square lattice revealed similar bridge structures and analogous changes in the net pressure. Thus, our observations do not appear to have a strong dependence on the shape of the attractive surface patches. This also has obvious significance for the relevance of our results to arbitrary macroparticles. Further in the macroparticle context, it was demonstrated that even if the stripes on each surface are free to move independently, if £1 ~ h they align to form bridges. Of course, macroparticles in solution are free to move, rotate, and in the polymer case, reconfigure. Therefore, it is clearly important that attractive patches do not have to be brought together in a commensurate fashion to form bridges. Rather, our results suggest that the possibility of forming bridges could direct the ordering of nonuniform macroparticles in solution, and that the resulting forces could contribute to aggregation or configurational changes. This is explicitly investigated in the following chapter. We show that the dependence of bridge formation on the patterning parameters can be rather dramatically altered by the addition of relatively small amounts of surfactant. The surfactant acts to reduce the unfavourable liquid-liquid interfacial tension, effectively "decoupling" h and £1 and allowing bridges to form at much larger separations. Indeed, with surfactant bridges are observed at separations that are up to ten times larger than the stripe width. This is particularly striking for very narrow stripes (£1 « o) where in  Chapter 3. Fluids confined between chemically patterned surfaces  48  the absence of surfactant bridges are only possible at very small separations where the structure and forces acting between the plates are dominated by particle packing effects. We note that surfactant-stabilized bridge extension is not limited to the striped patterns explored in detail in this section; analogous effects are observed for the cylindrical bridges resulting from circular surface patterns. In addition to extending the range of liquid bridges, the combination of chemically patterned plates, a liquid mixture near demixing coexistence, and a surfactant can lead to other interesting structures. For instance, when the surfaces are too far apart for bridges to exist, extended bubbles of a particular "phase" form adjacent to the attractive surface areas. These structures can be viewed essentially as surface-induced "micelles" and can be of different shape depending on the geometry of the attractive surface patches. Also, unlike the zero surfactant case, liquid structures that do not consist simply of repeated independent bridges can now be obtained. For example, for certain surfactants and state parameters, the surface-to-surface bridges are themselves joined by bridges parallel to the plates giving pattern that could be described as repeated crosses. Clearly, the addition of surfactant significantly extends the types of liquid structures possible in confined systems. We have also investigated the net pressure acting between the plates as a function of surface separation when surfactant particles are present. In contrast to the binary systems, where strong attraction occurs on bridge formation, for the surfactant-containing systems, the behaviour of the net pressure is less predictable. For narrow stripes there is a strong dependence on £1; bridges associated with £1 ~ lcr have little effect on the pressure. For £1 > 2cr long-range attractive forces can occur, but are in general considerably weaker than the zero-surfactant case. Moreover, at large separations, the force does scale with stripe width, but depends on other factors such as fluid ordering perpendicular to the plates. Given these observations, while one would expect the addition of surfactant to influence the forces between immersed macroparticles, exactly how this would influence association or aggregation is not as predictable as in surfactant-free mixtures.  49  Chapter 4 Integral equation calculations for uniform and patterned particles The formation of a bridge phase between infinite, chemically-patterned surfaces, described in the previous chapter, suggests that association between large solutes occurs on length scales similar to the solute size. In this chapter, we use integral equation theories to explicitly examine the behaviour of uniform and patterned spherical solutes immersed i n a liquid near either liquid-vapour or liquid-liquid coexistence. We are interested in the solvent structure, and corresponding forces that occur between the solutes. Clearly, macroscopic solutes, will behave much like infinite planes, and the results obtained in the previous section are immediately applicable, for similarly patterned spheres. However, when the solutes are small, but still large compared to the solvent particles, (for example with diameters of five to ten times the diameter of the solvent particles), the behaviour may be different. Although the formation of a bridge phase between patterned planes does suggest that smaller particles will associate, the length scales on which this will occur are less obvious. For example, the liquid-liquid interface which limited the length of bridges between chemically patterned surfaces, is much less well defined when dealing with finite-sized spheres. In addition, we expect the curvature of the surfaces to influence the results. Indeed, even in macroscopic systems curvature can be important; for example, true wetting or drying transitions do not occur around the surface of sphere, although such transitions can occur in the presence of a planar surface [76]. Although simulations can be used to study systems with large size asymmetries (i.e., large solute particles, small solvent particles), and we compare with some such results in this section, it is computationally demanding. In contrast, integral equation theories can readily be used to solve such problems; the caveat is that integral equation theories require approximations. Improving the approximations often increases the computational requirements, mitigating the usefulness of the theory as an efficient method. Nevertheless, these theories have successfully been used to study systems containing large solutes. In this chapter we examine the behaviour of large spherical solutes in a liquid near either liquid-vapour or liquid-liquid coexistence. We are especially interested in chemically patterned (patched) spheres, which have not been considered previously and where both the intersolute distance and the orientations of the solutes influence the structure. The results obtained using two different integral equation methods are discussed and where possible, compared with simulation results. The effects of solute size, patch size, and solvent coexistence on the interactions between the solutes are examined.  Chapter 4. Integral equation calculations for uniform and patterned particles  4.1  Survey of previous work  W i t h i n the last ten years (particularly within the last five), much work has been done using integral equation theories to study the behaviour of large solutes in liquids [2729, 33, 38, 42, 45, 77-81]. Notably, Kinoshita and coworkers have investigated such systems using a variety of approximations [27-29, 33, 38, 42, 45, 77, 78, 81]. The H N C closure relation is one of the most used approximations in integral equation theory. It has proved to be accurate for many simple systems, including some with large size asymmetries. However, failures have been noted, especially when large fluctuations in density can occur, such as is the case near phase transitions and in supercritical fluids [79]. Various researchers have found the results could be improved in these regions by including some bridge diagrams [28, 82, 83]. Using the so-called H N C - P a d e theory, in which certain types of bridge diagrams are exactly calculated using Monte Carlo integration techniques, Kinoshita [28] investigated the behaviour of both "hydrophobic" and "hydrophilic" uniform particles in water containing trace amounts of another smaller solute. Simple model potentials that roughly mimic the interaction between water and the hydrophobic or hydrophilic surface were chosen. The results show that the addition of either a hydrophobic, or hydrophilic smaller solute can result in either increased attraction when the surfaces were hydrophobic, or decreased repulsion, when the surfaces were hydrophilic. This occurred when the solvent-small solute mixture was both near and away from liquid-liquid coexistence, although the change in force was much more dramatic near coexistence. The author also used the reference interaction site model ( R I S M ) , to examine the structure of the fluid between two spherical surfaces. The results show that the addition of the small solute either enhances the solvent depletion between hydrophobic surfaces, by preferentially occupying the region between the solutes, or reduces the enrichment of the solvent between two hydrophilic solutes, by pulling solvent particles away from the region between the surfaces, thus giving rise to the increased attraction (decreased repulsion) in the force. We are interested in similar systems; namely, both uniform and patched solutes immersed in pure and mixed solvents. Although the H N C - P a d e theory does appear to give reasonably accurate results, it is a more computationally expensive method. This is especially true for a system containing patched solutes. The inclusion of other bridge diagrams in the closure, as in the nonlocal integral equation theories, has proved useful for studying surface-induced phase transitions [82, 83]. As an example, the hydrostatic H N C closure ( H H N C ) , obtained from density functional expansions of the bulk correlation functions, was successfully used to study condensation between two planar surfaces [82, 83]. A n anisotropic variation of this method was also used to study nanoparticles in a supercritical fluid. It was found that the results obtained using this theory were in near quantitative agreement with computer simulation results, in contrast to the isotropic H N C theory, which did not even agree qualitatively with the simulations [84]. The anisotropic H N C method, a special case of the anisotropic nonlocal theory, was first used to study ionic systems [85], and has since been used to study nanoparticles in supercritical one and two-component liquids [48, 79]. A n advantage of the method is that the distribution functions are directly obtained around a pair of solutes rather than only  50  Chapter  4. Integral equation calculations for uniform and patterned particles  51  around a single solute. Thus the structure between the solutes can be obtained. This can also be done using RISM, however for patched solutes with cylindrical symmetry, the anisotropic equations are more convenient. Furthermore, since the anisotropic HNC equations are identical to the isotropic HNC except that two solutes rather than one are treated, the physical significance of the differences between the results of the two methods can easily be deciphered. We expect the inclusion of two solutes to be important near a solvent phase transition, given the evaporation that was observed to occur between plates (see Chapter 3). The effect of confinement can be significant, even at large intersolute separations when the solvent is near liquid-liquid or liquid-vapour coexistence. In this chapter we investigate the behaviour of solutes due to solvent phase behaviour and the effect of confining solutes on the solvent structure. The goal of this work is twofold: to investigate the relevant length scales (solute size and patch size) at which phase-like behaviour is important, and to compare and contrast the isotropic and anisotropic H N C theories applied to these problems.  4.2  Model Systems  Three different systems were studied. The majority of the results reported in this section were obtained for System I described below. The other two systems were used to compare our theoretical results with previously published molecular dynamics (MD) simulation results (System II), and to investigate solute size effects (System III). In System I, the solvent consists of either one or two components that interact through the generalized L J potential described in Section 3.1. The solute model in this system, consists of a hard core of diameter D, with n smaller " L J " particles embedded in its surface as depicted in Fig. 4.1. In our formulation, the n surface particles are evenly spaced on s  s  Figure 4.1: Schematic showing the arrangement of the particles embedded in the surface of a hard sphere. The solvent-solute potential is obtained for this model in System I. rings that are centered about the z-axis and lie on a sphere of diameter D' = D — 0.2a. For this model, D = 10a in all cases considered in this work. The number of rings is determined  Chapter 4. Integral equation calculations for uniform and patterned particles  52  to be the maximum integer number that can fit on the sphere, with a space of no less than one cr between adjacent rings. Similarly, the number of particles on a given ring is the maximum number that can fit with a space of no less than one o between their centers. The surface particles are either all identical ("spherically symmetric"), or are of two types such that the solute has a patch. For numerical convenience, we only consider cylindrically symmetric solutes. The full potential between the solute centered at the origin, with its symmetry axis along the z-axis, and a given solvent particle of species a is oo, < c ( r  ' ° C  r<(D + a)/2  fe) -M$) 12  £?AJT^e  S S ) =  r > (D + o)/2 '  ^  where = rj(r, D j ) is the distance between a solvent particle and the j t h surface particle, r = (r,cosd,d>) and D j = (Dj,cos6j,<fij) are the vectors describing the positions of the solvent and surface solute particles respectively, and A.,- is a parameter that controls the interaction between the solvent and surface particles. We have used a simplified notation, r = r i , since the position and orientation of the solute, particle 2, are fixed at r = 0 and 0,2 = #2 = 0, and the solvent, particle 1, has spherical symmetry and does not depend on orientation f^. The integral over 0 is performed to force cylindrical symmetry. The potential u (r, cos 9) can be represented by an expansion in Legendre polynomials, P/(cos#), such that 2  s  aC  M  u  s aC  (r, cos 6) ~ u (r, cos 9) = ^  P (cos 9)u (r) l  aC  ;  aC  (4.2)  where the expansion coefficients are given by 21 4-  1  «LcW = — y  r  1  ^cos^(cosc?K (r,cosc?). c  , (4.3)  It is useful to describe the potential with a Legendre polynomial expansion for two reasons. First, expanding the O Z equation and H N C in terms of expansion coefficients of the correlation functions allows the use of a numerically efficient algorithm to solve the integral equations, as discussed in the following section. The second reason is the smoothing that occurs in the 9 direction when higher order terms i n the expansion are omitted. This is advantageous because it maintains consistency with the averaging in the cf> direction, discussed above. This model was chosen because it accurately describes the long-range potential decay, and because it naturally describes patched particles, without imposing an artificial cutoff in cos 9. The long-range decay of the solvent-solute potential is proportional to r if the solute is a single, L J particle, and to r ~ , if the solute is an infinite, planar surface composed of L J particles. If the solute size is between these two limits, the potential decay is proportional to r~ where 4 < r < 6. Explicitly including surface particles in the solute model, as described above, ensures correct long-range behaviour of the potential. -  6  4  T  System II is considered to allow comparison with previous M D simulation results by Shinto et al. [86]. In this system, the solvent-solvent interaction is given by the shifted Lennard-Jones potential,  u{r) = u {r) - u {rT\ LJ  LJ  (4-4)  Chapter 4. Integral equation calculations for uniform and patterned particles 53 where uu(r) is as i n Chapter 3 and r The solute-solvent potential is,  cut  oo, u (r) = { $ ( r ) - $ ( r 0, AC  D  is the cutoff distance. In our work r  cut  = 2.5a.  r <ro r < r < r + rg** r > r + rg"  + rg"),  D  (4.5)  D  D  where <p(r) = 27re  0.4  10  a  a" 3 A r ( r - rjg + 0.61Ar)  r- r  D  (4.6)  3  r-£, = D / 2 , A r = a / V 2 and rg * is the cutoff distance. In this work r g = 0.987a, which corresponds to a purely repulsive potential. This interaction is derived for a particle interacting with the (100) face of a face centered cubic planar solid through the L J potential [87]. For System III, the solvent model is as i n System I. The solute-solvent interaction is described by the hard sphere potential, 1  ut  oo, 0,  u ci.r) A  r<(D + o-)/2 r>(D + o)/2  (4.7)  where D is the diameter of the solute. This is the simplest model of a solvophobic particle, and was chosen to isolate solute size effects.  4.3  Theoretical Approach  The isotropic and anisotropic H N C theories are described in this section. We first describe the isotropic theory for a fluid containing only spherically symmetric particles, and then for a fluid containing patched solutes. The O Z equation is presented for the three cases of interest: i) both particles 1 and 2 have spherical symmetry, ii) particle 1 has spherical symmetry and particle 2 has cylindrical symmetry, and iii) both particles 1 and 2 have cylindrical symmetry. In all systems investigated, the solvent particles have spherical symmetry, and the infinitely dilute solute has either spherical or cylindrical symmetry. Finally, we discuss the anisotropic theory and show how it is related to the isotropic theory. We recall from Chapter 2 that the O Z equation is 7 ^ ( 1 2 ) = Y,^  J  b-( ) + -( )l 13  c  13  C  ^(32)A  ( -8) 4  where we have introduced the continuous function n p(12) = h p(l2) — c @(12), for numerical convenience. Fourier transforming the convolution integral i n E q . (4.8) [53], removes the integral over positions dr , to give a  a  a  3  ^ ( 1 2 ) = J2 ^  J [M  1 3  ) + 5^(13)] c ^ ( 3 2 ) d f i  3  (4.9)  Chapter 4. Integral equation calculations for uniform and patterned particles  54  where c (12) = J c (12) exp [zk • r ] dr , Q/3  a0  12  (4.10)  12  and r i 2 = ri — r . W h e n all the interactions between fluid components can be described by spherically symmetric potentials (i.e., the interaction potentials are independent of fi), the O Z equation simplifies to an algebraic equation i n Fourier space, 2  ^ ( 1 2 ) = ^Pu [fjaA ) + Cau(k)] c~, (k).  (4.11)  k  p  In the above equation, the scalar k is used since all directions are equivalent in the spherically symmetric case. The Fourier-space coefficients can be conveniently calculated using the fast Fourier transform method [88]. We only consider infinitely dilute solute systems (i.e., pc —»• 0). In this limit, the O Z equation decouples, so that the solvent-solvent pair distribution functions are independent of the solvent-solute and solute-solute pair functions, and the solvent-solute pair functions are independent of the solute-solute pair functions. A n iterative procedure can be used to solve the O Z equation in conjunction with the HNC closure, c p(r) = h (r) - lng (r) - Bu p(r), a  a/3  a0  (4.12)  a  noting that, for spherically symmetric particles, the correlation functions depend only on the distance between the centers of the particles, r =| r | = | ri — r |. In all systems that we investigate, the solvent particles interact with each other through spherically symmetric potentials. Since the O Z equation decouples, we use the method just described to solve for the solvent-solvent correlation functions i n all cases, and use these functions as input to solve for the solvent-solute functions. We also consider fluids containing patched solutes, that do not interact with the solvent particles through a spherically symmetric potential. In this case, the integral over orientation (dQ ) i n E q . (4.9) remains. We follow the expansion method first derived by B l u m and Toruella [89] to obtain an expression for the O Z equation i n terms of expansion coefficients of the correlation functions. We begin by writing the solvent-solute correlation functions, c c(12) and n c(12) as expansions i n Legendre polynomials, 12  2  3  a  a  c (12) =  J2c c(r)Pi(cosd), l  aC  a  i  (12) = ^vLci^PiicosB).  VaC  (4.13)  i  In order to represent the functions exactly, an infinite number of terms i n the expansions are required. However, one is often able to obtain converged results for a finite number of terms. We note that the Legendre polynomials expansion can be used because the solute particles have cylindrical symmetry, and the solvent particles have spherical symmetry. For general symmetry, an expansion i n rotational invariants is required [89]. The coefficients in the Legendre expansion can be obtained through an expression analogous to E q . (4.3). The Fourier transform of the expansion in r-space, also gives an expansion in Fourier space, c (12) =  Y,~ °c(k)Pi(cose), c  aC  i  (4.14)  Chapter 4. Integral equation calculations for uniform and patterned particles  55  where the coefficients are given by the Hankel transform, /•oo  c {k) = Am / Jo l  r ji{kr)c {r)dr.  1  2  a0  (4.15)  l  aC  The function ji(kr) is the spherical Bessel function of order I [90]. It is numerically efficient to use the fast Fourier transform method. However, the fast Fourier transform method can only be used directly if I = 0 or 1. In order to write the Hankel transform i n terms of these two Bessel functions, we use the hat transformation [89, 91],  -s: -aC  (s)  •  for I  ds,  0  c (s)  even,  (4.16)  for / odd,  (4.17)  t > 0,  (4.18)  t > 0,  (4.19)  l  ds,  aC  where Ff(x) and Pf(x) are the polynomials t t  r _y-i 2i, f  x  +  t  i +  i\(t-i)\(i  i=0  ^24+3( ) — 2  ly  + ±)\  X  i ! ( * - i ) ! ( i + §)!  i=0  p  e 0  = p° = o.  (4.20)  In the previous expressions the non-integer factorial g! = T(q + 1) is used. The complete Hankel transforms can then be calculated from the hat transforms using POO  -aC  (k) = 4?r / r j (kr)c (r)dr, Jo 2  l  Q  aC  f o r  I even,  (4.21)  for I odd.  (4.22)  POO  ac( ) = 4m  c  r ji(kr)cf (r)dr,  k  2  aC  Jo  The O Z equation for the solvent-solute correlation functions can be written as an algebraic equation in terms of the expansion coefficients, n , l  a/3  (4.23) These coefficients in fc-space can be back transformed using, 1 V () l  s  aC  2^J  ==  f°°  io(ks)v ( ) >  k2 o  l  k  aC  dk  f o r  1  >  even  (4.24)  Chapter 4. Integral equation calculations for uniform and patterned particles 56  €c(s) = 7T2 / 2 7 r  fc J'i(**)J&c(fc)dfc, 2  Jo  f  °  r  I  odd,  (4.25)  and  vLc(r) = v *c(r) l  - i f s P, (-) 7/i (s)ds, 2  e  for /  c  Jo  r  v  s  {^j fj^ds,  r&c(r) = ff (r)-±£s Ff 3  aC  even,  (4.26)  /  for Z odd,  (4.27)  In order to solve for the expansion coefficients of the correlation functions, the isotropic H N C closure must also be written i n terms of expansion coefficients. This was first done by Fries and Patey [91], and the expression is, c (r) = J>< l  aC  r  + c%{r)] (^f^)  dr - (3u (r),  (4.28)  l  aC  where  » i  C  W = - « W +  fccW  (4-29)  and  We note that the expansion coefficients are only coupled through the H N C closure. In summary, to solve the solvent-solute correlation functions, the following procedure is followed: i)  A n initial guess is made for rj {r) and c (r).  ii)  The H N C is used to obtain the new coefficients,  iii)  The two-step forward Hankel transform is done to obtain  iv)  The O Z equation is used to obtain  v)  The two-step back Hankel transform is done to obtain rf^ {r).  vi)  In order to obtain numerical convergence, the old and new coefficients are mixed with a mixing parameter, K, such that rf {r) — Kn ^ {r) + (1 — K)n (r) and c (r) = ^ ( r ) + (1 - K)c {r).  l  aC  aC  c ^ (r). l  w  c ^ (k). l  w  if^ {k). w  w  l  aC  l  w  l  aC  l  aC  aC  Since the solute particles have cylindrical symmetry, the solute-solute correlation function can be written as a spherical harmonic series expansion, Vcc(12) = Y Vcc{r)$ {Sl Sl ) mnl  J  mnl  u  2  1  (4.31)  Chapter 4. Integral equation calculations for uniform and patterned particles  57  where  (4.32)  Yt{^i)Y-^{0 ). 2  In the above equations, fij = (9^) represents the orientation of particle i, and Yj£* is the spherical harmonic of order m, p!. Again, for particles of arbitrary symmetry, an expansion in rotational invariants is required; the spherical harmonic expansion can be used in this case because the solutes have cylindrical symmetry. Following the argument used above for the solvent-solute correlation functions, the Fourier transform fjcc( 12) can also be written as a spherical harmonic expansion, the coefficients of which can be obtained from the solvent-solute correlation functions through the OZ equation,  vg&{k) = {2l + l)(™  n  )^2pAVcM  l Q  + ^M}^c(k).  Q  (4.33)  The coefficients in r-space can be obtained using the same two-step back transform procedure described above. The solvent-mediated force between two solutes can be calculated from the solute-solute correlation function,  F  {  R  )  _ _ Stnl^ca(r)]  _ -  kBT  d k e T  J!^l,  (4.34)  where the HNC closure [Eq. (4.12)] was used to obtain the second equality and R is the distance between the two solute particles. The isotropic HNC provides an approximate method to directly determine the structure of the solvent density around a single solute particle. However, as discussed in Chapter 3, the solvent structure between two large solutes can be dramatically different than the solvent structure around a single solute. This is especially true when the solvent is near a fluid-fluid (liquid-liquid or liquid-vapour) phase transition. To obtain the density profiles around two solutes separated by a distance R along the z-axis (see Fig. 4.2), the superposition approximation can be used, 9a-cc{r, cos(9)  = g (r)g c(r aC  a  - R),  (4.35)  where g c is obtained from the isotropic HNC method. This method approximates the solvent structure between two solutes, and we expect that, in the absence of long-range correlations, the method will provide reasonably accurate results. Near solvent phase coexistence, long-range correlations are present. Therefore, it would be better to use a method that directly accounts for the presence of two solutes when obtaining the solvent-solute correlation functions. The anisotropic HNC theory provides such a method. In the anisotropic theory, the total potential for a solvent particle of species a = A or B, interacting with both solutes is u _ (v) = u {r) + u (r - R). (4.36) a  a  ClC2  aCl  aC2  Treating the pair C\C as a single solute (from now on we use the subscript CC to refer to the pair of solutes) at infinite solute dilution (pec 0 ) the OZ equation again decouples, and the solvent-solvent correlation functions are identical to those obtained from the 2  Chapter 4. Integral equation calculations for uniform and patterned particles 58  X  A  z  Figure 4.2: Schematic of the coordinate system used in the solution of the anisotropic HNC equations. When the solutes have patches, the patches are centered about R; only the orientations of the two solutes that preserve the cylindrical symmetry are considered in this work. isotropic method. To solve for the a — CC correlation functions, we combine Eqs. (4.8) and (4.12),  (4.37) Although the convolution integral in the above equation could be solved using the expansion method of Blum and Torruella [89], the large number of expansion coefficients required to describe the solvent-solute potential, when the solute consists of two C particles, makes the problem numerically difficult. In addition, the accuracy of such an expansion would depend on the separation between C i and C , complicating the comparison of results obtained for different values of R. Instead, the convolution integral can be simplified by taking the Legendre transform of the functions in the integrand, to obtain [92] 2  9a-cc{r, cos 9; R)  exp  - Bu -.cc (r, cos 9, R) a  (4.38) For a pair of patched solutes, we consider only those orientations that preserve cylindrical symmetry. We expect the three orientations for which this restriction applies to be representative of the most and least preferred arrangements. The force between the two solute particles can be calculated from the solvent-solute density profile (see Appendix A or Attard for derivation [8]), ()  F  r  = J2-P»  d  r  V  K-cc{r; R)] o,_cc(r; R),  (4.39)  Chapter 4. Integral equation calculations for uniform and patterned particles 59  where V represents the gradient with respect to r. The isotropic equations were solved using 8192 points in r with A r = 0.01a. The Legendre expansion coefficients of the potential were obtained using Gaussian quadrature. The back hat transforms can be problematic at small r because of the r~ and r terms in Eqs. (4.16) and (4.17). To deal with this it was sometimes necessary to fit the integrand in Eq. (4.16) or (4.17) to a polynomial and then perform the integration analytically. Considerable computational effort is needed to solve the anisotropic equations, since the fast Fourier transform method cannot be used to deal with the convolution integral in the OZ equation. Eq. (4.38) was solved iteratively using 200 angular grid points and 800 spatial grid points with A r = 0.05a. A n accurate representation of the bulk functions, Ci/j/(0, typically requires a finer grid than that used to calculate h _cc{ ), particularly when r and r' are large. Following Attard [93], the discrete "quick" Legendre transform is employed to calculate h -cc{r). Gaussian quadrature is used for the transformation of both c (r) and h _cc( )- To accelerate convergence of the iterative sequences, we followed the procedure formulated by Pulay ([94]). In this chapter we also discuss some simulation results for spherical solutes of the type described for System III. The simulations were performed using G C M C techniques, described in Chapter 3 and detailed elsewhere [72]. Simulations of the pure L J solvent were performed in a cubic cell of volume V = L , where L = 10a. Simulations of systems containing 2 solutes, each having a diameter of D = 3a, fixed in space such that their centers are separated by a distance R = 6a, were performed in a cubic cell with L = 17a. Types of moves employed included translations, creations and destructions of the solvent particles. To avoid sampling a metastable state, it was sometimes necessary to "trap" the system by preventing the density from falling below some minimum value. This was especially important for simulations done near liquid-vapour coexistence. 3  - 4  r  u  v  r  vv  u  3  4.4 4.4.1  Results for solutes in a one-component fluid  1  S t a b i l i t y of t h e solvent  We are interested in the solvent response to the addition of solute as the liquid-vapour transition is approached. Therefore, we wish to carry out calculations at state parameters that are close to liquid-vapour coexistence of the bulk solvent. For the L J fluid estimates of the critical point and coexistence curve have been obtained using both computer simulations [58, 95, 96] and various integral equation approximations [58]. Smit [95] reports the critical temperature and density to be T* = k T /e = 1.316 ±0.006 and p* = p a = 0.304 ±0.006. Our calculations are carried out along the isotherm T* = 1.2, for which the coexisting liquid and vapour densities have been accurately determined by Valleau [96] to be 0.5705(20) and 0.0970(20), respectively. Estimates of the critical point and coexistence curves have been reported [58] for the HNC and related integral equation theories. However, it is now known [16, 58] that the H N C theory does not show true critical or spinodal behaviour. As the liquid-vapour spinodal is approached, the isothermal compressibility becomes larger and 3  B  c  c  c  A version of this section has been submitted for publication, S.D. Overduin and G.N. Patey J. Chem. Phys. submitted July, 2005. J  Chapter 4. Integral equation calculations for uniform and patterned particles 60  Figure 4.3: The stability indicator, XT (see Eq. (2.33)) plotted for different temperatures, T* = 1.2 (plus signs), T* = 1.25 (circles) and T* = 1.3 (squares). larger as expected, but does not diverge before the HNC equations become numerically unstable. In the present calculations at T* = 1.2 we were able to obtain numerically stable HNC solutions for p* > 0.488. The isothermal compressibility (see Eq. (2.33) is plotted in Fig. 4.3. The lowest density for which we report results is p* = 0.49. At this point the isothermal compressibility given by the HNC theory is large (see Fig. 4.3), and it is reasonable to view this value as close to, but not at, the spinodal. For the truncated and LJ fluid (System II) we encountered no instability in the integral equations at the temperature of interest T* — 1.2, indicating that the system is supercritical at this temperature. Note that this is consistent with simulations which give [95] T* = 1.085 ± 0.005 for this model. 4.4.2  U n i f o r m Solutes  In order to compare the isotropic and anisotropic HNC methods we calculate the force between solute pairs using Eqs. (4.34) and (4.39), respectively. For System I, the force  Chapter  4. Integral equation calculations for uniform and patterned particles  between two solvophobic solutes [i.e., X — 0.1 in E q .  61  (4.1)] is shown for both methods  in Fig. 4.4. A t T* = 1.2 and p* = 0.6, both methods agree well for surface separations  o F  "  •10 F > V *  0  ;  I"  -20 -25 §  2  4  6  8  10  12  14  (R~D)/G Figure 4.4: The force between uniform solutes with D = 10a, that are either solvophobic (top), or solvophilic (bottom) in System I . The solid and thick dotted curves were obtained using the isotropic equations and are for the solvent densities p* = 0.49 and p* = 0.60. respectively. The plus signs and crosses were obtained using the anisotropic equations and represent results for p* = 0.49 and p* = 0.60, respectively. The lines through the anisotropic results are to guide the eye. (R—D)/o > 2a. At smaller separations, the isotropic theory predicts a much more strongly attractive force than the anisotropic method. At a density near the spinodal (p* = 0.49), the different methods agree only when the surface separation is large ( > 10a). At smaller separations, the anisotropic theory now predicts a stronger attraction than the isotropic approach. We note that the force at contact is smaller for the lower density. This is not unexpected; the higher density gives rise to a larger "pressure" difference between the  Chapter 4. Integral equation calculations for uniform and patterned particles  (R-D)/c Figure 4.5: The force between uniform solutes with D = 10a, that are solvophobic in a liquid of density p* = 0.49 (top), or p* = 0.60 (bottom) in System I. The solid curves were obtained using the isotropic equations, the crosses were obtained using the isotropic theory in conjunction with the super position approximation, and the plus signs were obtained using the anisotropic theory. The lines through the superposition and anisotropic results are to guide the eye. depleted density region confined by the solutes and the higher density near the outside solute surfaces. We would also expect the range of the force between two solvophobic solutes to increase as the liquid becomes unstable. Both HNC approaches do predict this behaviour, as shown in the inset of Fig. 4.4, however, a much more dramatic effect is obtained from the anisotropic equations. The forces between two solvophilic solutes is oscillatory at short ranges (see Fig. 4.4). The theories agree well with each other except at very short separations (R < la). At these separations the isotropic theory predicts more attractive forces than the anisotropic theory for both p* = 0.49 and p* = 0.6. In fact when  62  Chapter 4. Integral equation calculations for uniform and patterned particles 63 p* = 0.49 the anisotropic equations predict a repulsive force at contact. The force can also be calculated from E q . (4.39) by using the isotropic equations together with the superposition approximation to obtain the density around two solute particles. If the density profiles obtained with superposition are exactly the same as those obtained through the anisotropic theory, the forces calculated in this way would be identical. The forces calculated the three possible ways are shown in F i g . 4.5. We see that, in both high and low density cases, there is better agreement between the forces calculated from the anisotropic theory and the superposition route at small separations than with the force calculated using the isotropic theory and E q . (4.34). However, at larger separations, the force calculated from the superposition route differs from the results of the other two methods, even when the other two agree with each other. We are mostly interested in the behaviour of the solutes at larger separations, and since the superposition method does not appear to give accurate results in this region, for the remainder of this thesis we will limit our discussion of forces to the those obtained using the other two routes. In view of the differences in the results from the two methods, the question arises as to which is more accurate. We would expect the anisotropic method to be superior because it accounts directly for the presence of two solutes. Simulations provide a good test of the integral equation method, and simulation results have been reported [86] for System II at T* = 1.2, p* = 0.5925 , and with rg * = 0.987a. A s noted above, System II is supercritical at this temperature and the integral equations were readily solved for the simulation parameters. The forces given by both H N C methods are compared with the simulation results in F i g . 4.6. Both methods agree well with the simulations when (R — D)/a > 2a. A t smaller separations, the isotropic equations predict a more strongly attractive force as was observed in System I. The anisotropic theory also overestimates the magnitude of the forces near contact, however, it performs much better than the isotropic method. Slightly better agreement between simulation and theory is also achieved for the forces between solvophilic particles using the anisotropic method, as demonstrated by Egorov and Rabani [48]. 4  A major advantage of the anisotropic H N C method is that it directly calculates the density profiles around a pair of solutes, providing insight into the structural changes in the solvent as the solutes approach each other. A n overview of the solute-induced solvent structure is given by the contour plots of gA-cc(r, cos 6) shown in Fig. 4.7. These plots are for hard sphere solutes (System III), but very similar "pictures" are obtained for System I. We shall first describe results for the large solutes (D = 10a), and consider the solute size dependence below. Far from coexistence, the structure around the solutes is simply due to the packing of solvent particles. A s the system approaches coexistence, large densitydepleted layers form around and between the solutes. Density profiles between System I solutes are plotted in F i g . 4.8. A t p* = 0.49, the density between the solutes drops to less than 80% of the bulk value when R = 21a. In contrast, the density profiles around solvophilic particles exhibit only a weak dependence on the bulk density. Similar effects are observed with variations in temperature; for a given density, as the temperature moves away from coexistence, the density depletion around the solute decreases as shown in F i g . 4.9. The density around the solute is not significantly affected by changes in temperature at the higher density (p* = 0.6). This is because, over the range of temperatures investigated, the high density system remains far from the  Chapter 4. Integral equation calculations for uniform and patterned particles 64  20 10 h  -60 ' 0  ' 1  ' 2  1  3  (R-D)/o  1  1  4  5  Figure 4.6: The force between solvophobic solutes with diameter D = lOo, in System II with p* = 0.5925 and T* = 1.2 obtained using simulations (plus signs), the anisotropic H N C (crosses) and the isotropic H N C (solid curve). The simulation data was obtained from the literature [86]. The line through the anisotropic results is to guide the eye. coexistence temperature. The decrease in density between the solutes is greater at smaller solute separations, as shown in F i g . 4.10. When the surfaces are separated by ~ 2a, the density drops to only 30% of its bulk value at p* = 0.49. This corresponds to a density of p* m 0.15, which is close to the density of the coexisting vapour phase (p* ~ 0.1). We also see density depletion between the solutes and dewetting-like behaviour at p* = 0.6. Farthest from coexistence (p* = 0.7), density depletion does occur between the solutes at small surface separations, however, the depletion is much less pronounced than at the lower densities. As noted above, hard-sphere solutes were use to investigate solute size effects in an unambiguous way. Contour plots for solutes of diameter l a are shown in F i g . 4.7. We see  Chapter 4. Integral equation calculations for uniform and patterned particles  65  Figure 4.7: Contour plots of the distribution function gA-cc{i\ cos 6) for a system containing hard sphere solutes (System III) at p* = 0.49 (top), p* = 0.6 (middle) and p* = 0.7 (bottom). In the panels on the left the solute diameter is D = la and on the right D = 10<7. that near the spinodal, even these small particles induces a slight decrease in the solvent density in their vicinity. However, for p* > 0.6, the surface dewetting disappears for these solutes. For larger solutes, layers of reduced density are observed when p* > 0.6, as shown in Fig. 4.7 for the D = 10a case. At p* = 0.6, a solute with D ~ 3cr is required to induce depletion of the solvent. Figure 4.11 shows g -cc(r, cos 9) between two solutes with surfaces separated by 8a. Near the spinodal, the density between the solutes decreases as the solute diameter increases to 4a. For larger solutes, the density at the surface continues to decrease with increasing solute size, but the density in the middle region remains nearly constant. At higher densities, the solvent depletion between the solutes is smaller, and for the surface separation 8a, less dependent on solute size. At shorter surface separations the changes in density with solute size are larger, as one would expect. A  Chapter 4. Integral equation calculations for uniform and patterned particles  Figure 4.8: The distribution function gA-cc{r, cos 6 = 1) taken along z for: p* = 0.49, dotted curve, p* = 0.55, dark dashed curve, p* = 0.6, light dashed curve, and p* = 0.7, solid curve. The top and bottom panels show results for solvophilic and solvophobic solutes, respectively. In all cases R/cr = 21. We have also calculated the intersolute forces for a range of hard sphere solutes. The results are shown in Fig. 4.12. It is worth noting that (consistent with the density contour plots discussed above) for p* = 0.49 and 0.60 the forces between even rather small solutes exhibit sensitivity to the solvent phase behaviour. We see also, that for the higher density systems p* = 0.6 and 0.7, the force between the smallest solute D = lo, is slightly repulsive at short separations, whereas, for all larger solutes, including D — 2o, the force is attractive at all separations. These results are consistent with those obtained by Kinoshita [29] for solutes in water. In order to further test the anisotropic theory we carried out a limited number of GCMC simulations [72] for System III. Solutes of diameter 3cr were considered because solutes of  66  Chapter 4. Integral equation calculations for uniform and patterned particles 67  3  2.5 2 CD  (fi O  o  CD  1.5 1  r  jf  i|  0.5 0 1  T  CD GO O O  CD  1.2 1 0.8 0.6 0.4 0.2 0 •0.2  -j  0  r  i  r  T  i  '  i  5 10 15 20 25 30 35 40  r/a Figure 4.9: The distribution function gA-cc{f, cos 9 = 1) taken along z for: T* = 1.2, solid curve, T* = 1.25, light dashed curve, T* = 1.3, dark dashed curve, T* — 1.4, dotted curve, and T* = 1.6, dash-dotted curve. In the top panel, p* = 0.6, and in the bottom panel, p* = 0.49. In all cases R/o = 17. this size are large enough to induce significant density depletion around their surfaces, but small enough that simulations can be performed in a reasonable time. Two state points, P^conf _ _ 29 i —2.8 corresponding to bulk densities p* = 0.609 and 0.631, respectively, were considered. The configurational chemical potential, p f, is convenient to use and is related to the full chemical potential by, p = p — u. -f fc^Tlnp*, where p is the ideal contribution. Qualitatively, the solvent correlation functions obtained (Fig. 4.13) agree reasonable well with the anisotropic integral equation estimates. This is significant because it verifies that liquid-like, rather than vapour-like densities, are stable between the solutes, at least when D = 3cr. For p* = 0.6, the theory does appear to overestimate the contact value, and two factors likely contribute to this discrepancy. First, we note that the a n c  con  conS  ldeal  ldeal  Chapter 4. Integral equation calculations for uniform and patterned particles  II  CD  (/) O  CJ  1.2 1 0.8 0.6 0.4 0.2 0 1.2  ll  © w  o  CJ  II  CD  in O  o  1 0.8 0.6 0.4 0.2 0  M L  J T  1.2 1 0.8 0.6 0.4 0.2 0 -8 -6 -4 -2  0  I  2  1  4  l  6  8  (r-R/2)/o  Figure 4.10: The distribution function gA-cc(y, cos 9 = 1) taken between the two solutes along the z direction for the solute separations R/o = 13,17,21 and 25. The densities are p* = 0.49 (top), p* = 0.6 (middle) and p* = 0.7 (bottom). simulation values were obtained by averaging over a cylinder of diameter 0.02a and height 0.05a; therefore, we might expect some underestimation of the contact value. Secondly, since the theory is approximate, theoretical and GCMC calculations carried out at the same bulk density and temperature are likely not the same "distance" from the corresponding coexistence curves, and this could have a large effect at contact. For comparison we plot simulation results for a system slightly farther from coexistence, p* — 0.631. Note that the contact value then increases, and the results more closely resembles the theoretical plot. hulk  68  Chapter 4. Integral equation calculations for uniform and patterned particles  2 1.8 -  0.4 fc i i i i i - 4 - 3 - 2 - 1 0 1 2 3 4  -I  (r-R/2)/o Figure 4.11: The distribution function gu-cc( ,cos6 — 1) taken along the z direction for hard-sphere solutes of different size, and with surface separation (R — D)/a = 8. The solvent densities are: p* = 0.49 (top), p* = 0.7 (middle) and p* = 7 (bottom). The curves represent the solutes sizes D/a = 1,2,3,4,6 and 10 in descending order. r  4.4.3  Patched  Solutes  The anisotropic H N C method was applied to patched solutes using System I. In F i g . 4.14, contour plots of gA-cc are shown for solutes that have a solvophobic patch covering 50% of their surface. Near the spinodal, density-depleted solvent layers form around the solvophobic patch and extend up to ~ lOcr into the solution. If the solvophobic patches face each other (top panel), the density between the solutes is depleted, similar to the uniform solute case. W h e n the solvophilic parts of the solutes face each other (middle panel), there is a slight depletion around the pair of solutes, except directly between them. The density depletion due to the solvophobic patches extends around the pair of solutes. If the  69  Chapter 4. Integral equation calculations for uniform and patterned particles 70  Figure 4.12: The reduced force between hard sphere solutes (System II) for p* = 0.49 (top), p* = 0.6 (middle) and p* = 0.7 (bottom). The curves represent different solute sizes, D = la (plus signs), D = 2a (crosses), D = 4a (stars), D = 6a (open squares) and D = 10a (filled squares). solvophobic patch of one solute faces the solvophilic side of the other (bottom panel), the density is again depleted around both solutes. At p* = 0.6, the situation is similar but the density depletion is significantly more localized. Patches of different size are of obvious interest, and contour plots of gA-cc{r, cos 8) for solutes having smaller solvophobic patches covering 10% and 25% of their surfaces are shown in Fig. 4.15. The particles are oriented such that the solvophobic patches are facing each other. As one might expect, the density depletion extends less far into the liquid as the patch size is reduced. Nevertheless, the 25% patches do have a large effect on the density between the solutes. As the solvent moves away from coexistence the effect of the solvophobic patch decreases significantly.  Chapter 4. Integral equation calculations for uniform and patterned particles  1.6 1.4 1.2 p  1  II  x>  S  0.8  co  0.6  o  0.4 0.2 0 0  2  4  6  r/o  8  10  Figure 4.13: The distribution function gA-cc( , # = 1) between two hard-sphere solutes of diameter D = 3o, obtained using the anisotropic HNC (solid curve) for a density, p* — 0.6 and G C M C simulations for densities p* = 0.609 (dotted curve), and p* = 0.631 (dashed curve). r cos  It is instructive to examine the solvent distribution function between solutes together with the resulting intersolute forces. From Fig. 4.16, it can be seen that for 50% patches facing each other, the density profile between the solutes is nearly identical to the density between two fully solvophobic solutes (see Fig. 4.8). For solutes with smaller patches (10% or 25% of the surface is solvophobic), the density depletion is not as large, but is still significant for small separations. Similarly, the forces between the solutes are identical to those occurring between fully solvophobic solutes when the solvophobic patch is large (> 50%) (see Fig. 4.17). For smaller patches, the forces are weaker, but remain qualitatively similar to those observed in the fully solvophobic case. If the solvophilic sides of the solutes are facing, the density between them is similar to  71  Chapter  4. Integral equation calculations for uniform and patterned particles  72  Figure 4.14: Contour plots of the distribution function gA-ccifi cos 6) for solutes with a solvophobic patch covering 50% of the surface. In the top two panels the solvophobic patches are facing each other, in the middle two the solvophilic sides are facing each other, and in the bottom two panels the solvophobic patch of one solute faces the solvophilic side of the other. The density in the panels on the left is p* = 0.6 and on the right p* = 0.49. In all cases R — 21a. the density between two fully solvophilic solutes, except when the solvophobic patches are large ( > 75%) (Fig. 4.16). For this orientation, the forces resemble the fully solvophilic case if the solvophobic patches are small. For larger solvophobic patches, the forces tend towards the fully solvophobic result, especially when the intersolute separation is large (Fig. 4.17). However, at smaller separations the forces are much weaker than the fully solvophobic case even for 75% patches. If the solutes are oriented such that the solvophobic and solvophilic parts face each other, the densities near the surfaces tend to resemble those for the corresponding fully covered  Chapter  4. Integral equation calculations for uniform and patterned particles  73  Figure 4.15: Contour plots of the distribution function, gA-cc(f, cos#), for solutes having a solvophobic patch covering 10% (left) or 25% (right) of the surface. The densities of the solvent are p* = 0.49 (top), p* — 0.6 (middle) and p* = 0.7 (bottom). In all cases the solvophobic patches are facing each other and R = 17'o. particle. Between the particles the density decreases as the solvophobic patch increases in size (Fig. 4.16). The forces between the solutes in this orientation are much like those that occur when the solvophilic sides of the solutes face each other; the solutes with small patches behave like the fully solvophilic solutes, and the solutes with large patches behave more like the fully solvophobic case. Finally, we note that the variation of the distribution functions and forces with orientation and patch size show qualitatively similar trends both near and away from coexistence. However, away from coexistence the magnitude of the variations is much smaller. The stronger forces observed when the solvophobic patches face each other indicate a tendency for orientational alignment. Near coexistence, alignment would occur even when the sur-  Chapter 4. Integral equation calculations for uniform and patterned particles  Figure 4.16: The distribution function, gA-ccir, cos 9 = 1) taken along the z direction, for solutes having patches covering 0% (solid curve), 25% (light dashed curve), 50% (thick dashed curve), 75% (dotted curve) and 100% (dash-dotted curve) of the surfaces. The density in the panels on the right is p* = 0.6, and on the left p* = 0.49. In all cases R = 17 o. The solutes are oriented with the solvophobic patches facing each other (top), facing away from each other (middle), and with the solvophobic patch of one solute facing the solvophilic side of the other (bottom). faces of the solutes are separated by up to ~ 10cr, depending on the size of the patch. To summarize, in this section we have investigated the solvent density profiles about uniformly solvophilic, uniformly solvophobic, and patched solutes. The effective forces acting between the solutes were also calculated. Results obtained using the anisotropic and isotropic HNC methods are compared with each other, and (in limited cases) with computer simulations. Away from coexistence, both methods agree well except at small separations, where the anisotropic results were found to be in better accord with simulations. Near  74  Chapter 4. Integral equation calculations for uniform and patterned particles  Figure 4.17: The force between solutes having solvophobic patches covering 0% (plus signs), 25% (empty squares), 50% (stars), 75% (crosses) and 100% (filled squares) of the surfaces. The density in the panels on the right is p* = 0.6 and on the left p* = 0.49. The solutes are oriented with the solvophobic patches facing each other (top), facing away from each other (middle), and with the solvophobic patch of one solute facing the solvophilic side of the other (bottom). coexistence, and particularly near the spinodal, the methods differ with the anisotropic approach predicting much longer ranged forces. Density profiles given by the anisotropic theory were in qualitative agreement with G C M C results obtained for one particular system near coexistence. We find that surface "drying" of a solvophobic solute immersed in liquid depends on the state parameters of the solvent and on the size of the solutes. Near the spinodal even small solutes (D = la) can induce drying-like behaviour. The depleted density layers of the fluid extend out several solvent particle diameters before merging into the bulk. Away  75  Chapter 4. Integral equation calculations for uniform and patterned particles from the spinodal, but still in the vicinity of coexistence, drying is only observed for larger solutes (D > 3<T). Far from coexistence, no drying is observed. The density-reduced layers overlap as two large solvophobic solutes approach each other in a solvent near coexistence, creating a depleted region between the two. The solute-solute interaction is strongly influenced by this effect, in the case of both uniform and patched solutes. A s the solvent approaches liquid-vapour coexistence, the force acting between the solutes becomes much longer ranged and continues to increase in range up to the spinodal. For the patched case, the force depends on the patch size and on the orientation of the solutes. The strongest forces occur when the solvophobic patches face each other and the solutes will tend to prefer this orientation. In general, the force decreases with decreasing patch size, but for solutes of diameter 10cr they are important for patches that cover 25% of the solute surface, and remain significant down to 10% coverage.  4.5  Results for solutes in a two-component fluid  In this section we consider System I solutes, with D = 10a, immersed in a solvent mixture of species A and B. We discuss first systems rich in species A, containing only a small amount of species B, and compare results with those obtained for the one-component systems described in the previous section. We then discuss mixtures rich in species B, making reference to the simulation results described in Chapter 3. B o t h the isotropic and anisotropic theories are used to investigate patched and spherically symmetric solutes.  4.5.1  A-rich mixtures  4.5.1.1  Stability of the solvent mixture  In the previous section it was shown that long-range attractive forces occur between solutes immersed in a liquid near liquid-vapour coexistence. It is interesting to consider what happens when a small amount of co-solvent, species B, is also present. Before discussing results for systems containing solute particles, it is useful to briefly examine the stability of the A - B mixtures. In a two-component liquid, both liquid-liquid and liquid-vapour transitions can occur. W h e n either transition occurs, there is a change in both the density, and the composition of the fluid. A s mentioned in Section 4.2, the spinodal and binodal curves cannot be accurately determined from the H N C theory, since the theory becomes numerically unstable near the spinodal. However, by monitoring the material and mechanical stability indicators, we can determine an approximate spinodal curve. The mechanical and material stability indicators can be written as second derivatives of the free energy. The system must obey the criteria given in Eqs. (2.39) and (2.34) for the solution to be stable (see Chapter 2 for a discussion about the stability indicators). In this section we consider two cases: in case I, the total reduced density of the solvent is p\ = (PA + PB)O = 0.49, and in case II the total solvent density is p* = 0.6. These densities were chosen to compare with the one-component systems discussed above. Since we consider only systems containing small amounts of species B, the total density is similar to the density of species A in each case. The mechanical and material stability indicators for the two cases are plotted in F i g . 4.18. In the one-component system, p\ = 0.49 3  76  Chapter 4. Integral equation calculations for uniform and patterned particles 7 7  Figure 4.18: Material (5^(0), represented by triangles) and mechanical {xr, -^ , represented by squares) stability indicators for a system with p* = 0.49 (top) and p* = 0.6 (bottom). hA  B  corresponds to a system near the liquid-vapour spinodal. We see from Fig. 4.18, that both S (0) and X r ^ ^ - w increasing as x increases, but that the mechanical stability indicator is diverging faster, indicating the transition is primarily a liquid-vapour transition. The total density of the system is fixed; as the density of species B increases, the density of species A decreases. We recall from our previous discussion on one-component fluids that the HNC theory becomes unstable beyond p* = p* = 0.488. In case I, solutions can be obtained from the HNC theory up until XB = 0.003 or, equivalently, p\ ~ 0.488. Thus at this density, the presence of species B appears to have very little effect on the phase behaviour of species A. In contrast to case I, the liquid phase in case II is stable over a much larger range of XB- The material stability indicator becomes large faster than the mechanical stability indicator, indicating that the transition is a demixing transition. We point out that the demixing transition is also accompanied by a change in the total density. Solutions can be obtained using the HNC theory for XB < 0.024 or p* « 0.586, which is well above a r e  cc  B  A  A  Chapter  4. Integral equation calculations for uniform and patterned particles  78  the density at the liquid-vapour spinodal in a pure A system. We cannot obtain a stable solution to the integral equations for larger values of x , with either density From the stability indicators, we can see that the mixtures are approaching either the liquid-vapour (p* = 0.49) or the liquid-liquid (p* = 0.6) spinodal. B  t  4.5.1.2  Uniform and patched solutes  The addition of a single A-phobic solute (i.e., a solute that has only a weak attraction to species A) to the solvent mixture, results in depletion of species A from the fluid surrounding the solute, as occurred in the one-component solvent (see Fig. 4.19). At the same time  Figure 4.19: The solvent-solute distribution functions, g c(r,cosO) (top) and (^(cosfl) (bottom) around a single, spherically symmetric A-phobic (Aj = 0.1 for all particles embedded in the solute surface) solute in an A-rich mixture. In the panels on the left, p* = 0.6 (case II) and on the right p\ = 0.49 (case I). The curves represent different mole fractions: in case II, XB = 0 (red), XB = 0.001 (green) and XB = 0.01 (blue); in case I x = 0 (red), x = 0.001 (green) and x = 0.003 (blue). B  t  B  B  B  there is an increase in the density of species B in this region of the fluid. In both high  Chapter  4. Integral equation calculations for uniform and patterned particles  79  Figure 4.20: The reduced force between uniform A-phobic solutes for case I (top) and case II (bottom). Each curve without symbols represents a force calculated using the isotropic HNC method for different mole fractions: in case I, XB = 0.0 (red), XB = 0.001 (pink); in case II, x = 0.0 (red), x = 0.001 (light blue), XB = 0.01 (yellow). The curves with symbols represents the force calculated using the anisotropic equations: in case I, XB = 0.0 (green), XB = 0.001 (dark blue); in case II, XB — 0.0 (green), x = 0.001 (dark blue), x = 0.01 (pink). B  B  B  B  and low density cases species B occupies the fluid regions depleted in species A . In case I, these regions extend much farther from the surface. Consequently, there is an enrichment of species B over a larger volume in case I. However, it is interesting to note that near the surface the enrichment in species B is much greater in case II. This again demonstrates that the mixture in case II is near a demixing transition. The presence of species B disrupts the attractive interactions between particles of species A more in case II than in case I, owing to the higher density of species A in the former.  Chapter 4. Integral equation calculations for uniform and patterned particles 80 As the mole fraction of species B increases in case II, there is a greater reduction in the density of species A and enhancement of species B near the solute surface. This is especially evident when x = 0.01 [see F i g . 4.19 (left panels)]. The integral equations become unstable on further addition of species B (i.e., when x > 0.01 in case II), when the vl-phobic solute is present. Since the solute density is infinitely small, the instability cannot be due to true divergences in either the structure factor or the isothermal compressibility. Instead, the equations appear to become unstable due to numerical instabilities arising from a local surface "wetting" transition. We note that, in contrast to the planar case, a first-order wetting transition cannot occur around a sphere, no matter the size, due to the increase in the surface area of the liquid-liquid (or liquid-vapour) interface [76]. B  B  The situation for case I is somewhat different. Solutions to the isotropic integral equations in the presence of an A-phobic solute could be obtained for all stable bulk mixtures (i.e., x < 0.003) in this case. A s noted, when x = 0.003, the density of species A nears that of the liquid-vapour spinodal (p ~ 0.488). It appears then that "dewetting" at the surface of a solute is numerically less challenging for the integral equations. The changes in the solvent-solute distribution functions in case I with increasing mole fraction are small. However, the changes do extend over a long range ( > 8a from the solute surface), as shown in F i g . 4.19 (right panels). Of interest are the changes in force with the addition of species B to the A-nch fluid. In Fig. 4.20, the forces between uniform solutes are calculated for different mole fractions. In case I, there is little change in the forces between ^4-phobic solutes with increasing x , although as was noted in the discussion on one-component systems above, there are differences between the force calculations of the two theories [see F i g . 4.20 (top)]; the anisotropic theory predicts much longer ranged forces. In case II, the forces between Aphobic solutes for x = 0 and 0.001 are similar, and in these cases the isotropic and anisotropic results compare well, except when R — D < 2a [see F i g . 4.20 (bottom)]. In contrast, when x = 0.01, the force becomes strongly attractive at a larger separation, R — D ~ la, and oscillatory at small separations. The oscillations in the force arise due to the packing of B particles between the solutes. We note that, for case II with x = 0.01 we were not able to obtain stable solutions to the anisotropic equations for R — D < 9a. The instability in the theory suggests that the system is near a phase transition. B  B  A  B  B  B  B  The force between j4-philic solutes are oscillatory at shorter separations for both case I and case II, and are similar to the forces obtained for the one-component systems. The two theories compare well in all cases except when R — D < 2a. There are some small changes as the mole fraction of species B increases at small separations in case I. For example, the magnitude of the repulsion in the oscillating region decreases slightly. A reduction in the force between solvophilic solutes with the addition of a small amount of a solvophobic component was also observed using the HNC-Pade integral equation theory. We now turn our discussion to the solvent structure between patterned (or patched) solutes. Here results obtained for the solvent structure between solute particles using the anisotropic theory are presented first, followed by comparison with results obtained using the isotropic theory. Finally, the forces that arise between the solute due to the solvent structure are discussed. T h e solutes are identical to those described in the one-component case, having yl-phobic patches. Note that species B "sees" the solute as uniform (i.e., Xj = 0.1 for all n embedded surface particles in u [see E q . (4.1)]. Depletion of species A s  BC  Chapter  4. Integral equation calculations for uniform and patterned particles  81  and enrichment of species B occurs next to the A-phobic patches of the solutes (see Figs.  30 f-  Figure 4.21: Contour plot of the distribution functions gA-cc(r, cos 6) in systems with, R = 17a, p* = 0.49 (left) and p* = 0.6 (right). Three different orientations are plotted: in the top, the yl-phobic patches face each other; in the middle, the A-philic sides of the solutes face each other; in the bottom, the ^4-philic side of one solute faces the ^-phobic patch of another. In all cases x = 0.001 and the A-phobic patch covers 25% of the surface. The distribution functions were obtained using the anisotropic HNC method. t  B  4.21 and 4.22). When the fluid is near liquid-vapour coexistence (case I), the fluid depleted in species A extends out to > 10a from the surface of the solute. In both case I and case II, the depletion of species A around the A-phobic patches of the solute is similar to that occurring around the solvophobic patches of a solute immersed in the one-component fluid (compare Figs. 4.14 and 4.21, noting the plots are for different separations). When the A-philic solute sides face each other, there is a slight increase in the density of species A between the particles. In case I, the depletion due to the A-phobic patches extends around  Chapter  4. Integral equation calculations for uniform and patterned particles  82  Figure 4.22: Contour plot of the distribution functions gB-cc(f, cos#), in systems with R = 17er, pi = 0.49 (left) and p\ = 0.6 (right). The three different solute orientations plotted are as in Fig. 4.21. In all cases x = 0.001, and the A-phobic patch covers 25% of the surface. The distribution functions were obtained using the anisotropic HNC method. B  the A-enriched fluid between the solutes (see Fig. 4.21). When the A-phobic patches are facing, the density of species A between the solutes decreases; in case I, the fluid depleted in species A even extends around the A-rich fluid near the A-philic sides of the surfaces [see Fig. 4.21 (middle right panel)]. Similarly, when the A-phobic patch of one solute faces the A-philic side of the other, depletion of A occurs near the A-phobic patches and partly extends around the A-philic sides of the surfaces [see Fig. 4.21 (bottom panels)] The depletion of species A near the A-phobic patches is accompanied by the enrichment of the fluid in species B (see Fig. 4.22), as was observed around a single solute. The fluid rich in species B also extends as far from the surface as does the depletion of species A. It is interesting to note that, in case II, the enrichment of species B actually occurs  Chapter  4. Integral equation calculations for uniform and patterned particles  83  around both A-phobic and A-philic parts of the solutes, except immediately adjacent to the A-philic sides, as shown in the panels on the left side of Fig. 4.22.  Figure 4.23: Distribution functions (gA-cc(r,cos6 = 1)) between the solutes for different densities p* = 0.6 (left) and p* = 0.49 (right) obtained using the anisotropic HNC method. The A-phobic patches cover 100% (red), 90% (green), 75% (dark blue), 50% (pink), 25% (light blue) 10% (yellow) and 0% (black) of the solute surfaces. Three different orientations are plotted: in the top, the A-phobic patches face each other; in the middle, the A-philic sides face each other; in the bottom, the A-philic side of one solute faces the A-phobic patches of another. In all cases x = 0.001 and R = 17c. t  B  Looking more closely at the density profiles between the solutes for the two cases with the same mole fraction of species B, x = 0.001, we see that, in case I, there is a greater increase in the depletion of species A between solutes with the A-phobic patches facing, than in case II [see Fig. 4.23 (top panels)]. In both cases, the depletion between patched solutes with the A-phobic patches facing, is similar to that between fully A-phobic solutes. B  Chapter  4. Integral equation calculations for uniform and patterned particles  -2  0  2  (r-R/2)/rj  -2  0  84  2  (r-R/2)/o  Figure 4.24: Distribution functions (g -cc{f-. cos 6 = 1)) between the solutes for different densities p* = 0.6 (left) and p* = 0.49 (right) obtained using the anisotropic HNC method. The curves and solute orientations are as in Fig. 4.23. In all cases %B = 0.001 and R. = YJo. B  except when the patches are small (i.e., covering 10% or less of the solutes). Even for this small patch size, some depletion of species A does occur. When the ^4-philic sides of the solutes are facing, the density between them is much as it is when the solutes are fully A-philic, except when the A-phobic patches are large (i.e., covering 90% or more of the solutes). When the A-phobic patch of one solute faces the A-philic side of the other, the density of species A is only depleted near the A-phobic surfaces, thus resulting in an interface between the yl-enriched and ^-depleted fluids. For all orientations, changes in the density of species A with patch size are greater in case I than in case II (see Fig. 4.23). The density of species B between the solutes also strongly depends on solute orientation (see Fig. 4.24). In both cases I and II, there is a dramatic increase in the density of species B between two solutes with the ^4-phobic patches facing. For the separation, R = YJo  Chapter  4. Integral equation calculations for uniform and patterned particles  - 2 - 1 0  1  85  2  (r-R/2)/o  Figure 4.25: The distribution functions, g -cc(r, cos 9 = 1) (dashed curve) and gs-cccos9 = 1) (solid curve), between the solutes calculated using the anisotropic HNC theory. The A-phobic patch covers 75% of the surface and the solutes are oriented with the A-philic sides facing. In this system, p* = 0.6, x = 0.01 and R = 21a. A  t  B  (shown in Fig. 4.24), the density of species B midway between the solutes is approximately four times greater than the bulk in case I, and two to three times greater than the bulk in case II. In case II, the density of species B near the solute surfaces increases to more than twenty times its bulk value. Even when the A-phobic patches cover only 10% of the solute surfaces, there is a large increase in the density of species B between the solutes, although the increase is less significant than that occurring between solutes with larger patches. When the patches cover 25% or more of the solute surfaces, the increase in the density of species B is nearly identical to that occurring between uniform A-phobic solutes. When the solutes have the opposite orientation, with facing A-philic sides, a dramatic increase in species B only occurs for patches covering 90% or more of the solutes. It is interesting to note that, even when the A-philic sides are facing, the fluid between the solutes is enriched in species £?, provided the A-phobic patches cover 50% or more of the surfaces. For example, shown in Fig. 4.25, are the solvent-solute distribution functions for this orientation when the patches cover 75% of the surfaces and R = 21a. The fluid is enriched in species B and slightly depleted in species A, over the entire distance except near the surfaces. A similar, though smaller, effect occurs when the patches cover 50% of the surfaces. Thus, the presence of Aphobic patches affects the density near the A-philic sides of the solutes.  Chapter  4. Integral equation calculations for uniform and patterned particles  (r-R/2)/o  86  (r-R/2)/a  Figure 4.26: Distribution functions (gA-cc (r,cosO = 1)) between the solutes in a system with Pj = 0.6 with mole fractions XB = 0.01 (left) and XB = 0.001 (right). In all cases the A-phobic patch covers 25% of the solute surface. Each curve represents a different separation: R = 2lo (red), R = 19cr (green), R = 17a (dark blue). Three different solute orientations are plotted: in the top, the A-phobic parts of the solutes face each other; in the middle, the A-philic patches face each other; in the bottom, the A-philic: patch of one solute faces the A-phobic part of another. All curves were obtained using the anisotropic HNC theory. Using the anisotropic HNC theory, we compare changes to the fluid structure with changes in the composition of the solvent, and with changes in the separation between solutes in case II. In Figs. 4.26 and 4.27, the solvent-solute distribution functions are plotted for two different mole fractions and three different solute separations. There is an increased depletion of species A between two solutes with the A-phobic patches facing, as the mole fraction of species B increases [see Fig. 4.26 (top panels)]. At the same time, the  Chapter  4. Integral equation calculations for uniform and patterned particles  (r-R/2)/o  87  (r-R/2)/o  Figure 4.27: Distribution functions, gB-ccir,cos0 = 1), between the solutes in a system with p\ = 0.6 with mole fractions x = 0.01 (left) and x = 0.001 (right). In all cases the A-phobic patch covers 25% of the solute surface. The curves and solute orientations are as in Fig. 4.26. A l l curves were obtained using the anisotropic HNC theory. B  B  density of species B between the solutes increases with respect to the bulk [see Fig. 4.27 (top panels)]. In fact, the density midway between the solutes reaches nearly four times the bulk density when x = 0.01 and R = \7o. We note also that at higher mole fractions, the densities of species A and B is more dependent on patch size. For example, when x = 0.01 the increase in the densities between the solutes when the A-phobic patches cover 10%> or 25% of the solute surfaces is much less than for patches covering 50% or greater of the surfaces (see Fig. 4.26 and 4.27). The enrichment of species B between the A-phobic patches of the solutes occurs because the addition of B particles moves the system nearer demixing coexistence. The A-phobic patches favour the B particles, driving a demixinglike transition between the solutes similar to what occurs between planar surfaces [21]. B  B  Chapter  4. Integral equation calculations for uniform and patterned  particles  88  However, we note that for the separation of R = 17a (shown in Fig. 4.27), although the fluid between the solutes is richer in species B than is the bulk fluid, the density of species A is still greater than that of species B in this region.  1 I  _-|  1  I  I  -  1  3  1  I  -  2  1  I  -  1  r  I  0  I  1  2  I  3  (r-R/2)/a Figure 4.28: Distribution functions, gA-cc(r, cos# = 1), between the solutes calculated using the anisotropic HNC theory (solid curve) and the isotropic HNC theory together with superposition (dashed curve) when the A-phobic patch covers 25% of the solute surfaces. Three different orientations are plotted: in the top, the A-phobic parts of the solutes face each other; in the middle, the A-philic patches face each other; in the bottom, the A-philic patch of one solute faces the A-phobic part of another. In all cases pl = 0.6, XB = 0.01 and R = 17a. The largest changes in the fluid with changes in separation occur when the A-phobic patches face each other. For this orientation, the depletion of A between the surfaces increases with decreasing separation (Fig. 4.26); at the same time, there is an increase in the density of species B. At a separation of 15cr, there is a very large increase in the  Chapter  4. Integral equation calculations for uniform and patterned particles  89  Figure 4.29: Distribution functions, gB-Cc{r, cos# = 1), between the solutes calculated using the anisotropic HNC theory (solid curve) and the isotropic HNC theory together with superposition (dashed curve) when the A-phobic patch covers 25% of the solute surfaces. Three different orientations are plotted: in the top, the A-phobic parts of the solutes face each other; in the middle, the A-philic patches face each other; in the bottom, the A-philic patch of one solute faces the A-phobic part of another. In all cases p* = 0.6, x = 0.01 and R = 17a. t  B  density of species B, especially when XB = 0.01. As mentioned, we were unable to obtain stable solutions to the anisotropic equations for separations, R < l7o when x = 0.01 and the A-phobic patch covers 25% of the solutes. This suggests that at this separation, the fluid confined between the solutes undergoes a "demixing transition". This cannot be a true demixing transition since the solutes are finite-sized; nevertheless, the sudden change in density at some critical separation is akin to a phase change between infinite, planar surfaces. When XB = 0.001, the solutions to the anisotropic equations can be obtained B  Chapter  4. Integral equation calculations for uniform and patterned particles  90  Figure 4.30: Distribution functions, gB-cc(r, cos6 = 1), between the solutes calculated using the isotropic H N C theory together with superposition with different numbers of expansion coefficients: M = 14 (red), M , = 13 (green), Mi = 12 (dark blue), Mi = 11 (pink) and Mi = 10 (light blue). Results are shown for A-phobic patches covering 25% (top left), 10% (top right), 50% (bottom left), and 75% (bottom right). In all cases, the A-philic parts of the solutes are facing each other, p* = 0.6, x = 0.001 and R = 17a. t  B  for R > 14a, suggesting that, for this mole fraction, a "demixing transition" occurs at a smaller separation. Although the most dramatic changes in fluid structure occur when the A-phobic patches are facing, changes also occur for the other orientations as the separation decreases. This is especially true when XB = 0.01. In this case we notice that there is an increase in the density of species B (and to a lesser extent, a decrease in the density of species A) between the solutes for both other orientations [see Fig. 4.27 (middle and bottom panels)]. It appears that the increased density of species B near the A-phobic patches draws B particles in all around the solutes. The results discussed so far were obtained using the anisotropic H N C theory. The solvent-solute distribution functions, g -cc(r, cos 9), can also be obtained from the isotropic a  Chapter  4. Integral equation calculations for uniform and patterned  particles  91  Figure 4.31: The reduced force between solutes calculated using the anisotropic H N C theory. The A-phobic patches cover 100% (red), 90% (green), 75% (dark blue), 50% (pink), 25% (light blue) 10% (yellow) and 0% (black) of the solutes surfaces. Three different orientations are plotted: in the top, the A-phobic parts of the solutes face each other; in the middle, the A-philic patches face each other; in the bottom, the A-philic: patch of one solute faces the A-phobic part of another. In all cases pi = 0.6 and XB = 0.001. HNC method, through the superposition approximation [Eq. (4.35)]. In Figs. 4.28 and 4.29 the solvent-solute distribution functions are plotted using the two methods, for case II with XB = 0.001 and solutes having A-phobic patches covering 25% of the surfaces. From these figures, the most noticeable differences occur in the density of species B for all orientations (Fig. 4.29) and in the density of species A when the A-phobic patches are facing (Fig. 4.28). The superposition approximation predicts less depletion of species A between the solutes when the A-phobic patches are facing, and greater enrichment of species B for all orientations. The isotropic equations were solved by doing a Legendre polynomial  Chapter 4. Integral equation calculations for uniform and patterned particles 92  Figure 4.32: The reduced force between solutes calculated using the anisotropic HNC theory. The curves are as in Fig. 4.31. In all cases p\ = 0.6 and x = 0.01. B  expansion of the distribution functions [see Section 4.3, Eq. (4.13)]. The number of terms kept in the expansion must, be finite; thus, we must impose a truncation. This introduces some inaccuracy. In principle, one could continue to increase the number of expansion terms, until converged results are obtained. However, numerical difficulties arise with the hat transform, and in particular the back hat transform, for high order terms. In this work Mi terms are kept in the expansions of the distribution functions [Eq.(4.13)], and we are unable to obtain results for Mi > 14. In Fig. 4.30, results obtained for different values of Mi are plotted for the density of species B between the solutes with the A-philic sides facing. The density midway between the solutes is converged for all patch sizes when Mi > 11. However, near the surfaces, there are still significant differences as M / increases from 13 to 14. Furthermore, the convergence for some patch sizes appears to be oscillatory. Of the patch sizes shown, convergence appears to be the worst when 50% of  Chapter  4. Integral equation calculations for uniform and patterned particles  93  Figure 4.33: The reduced force between solutes calculated using the isotropic HNC theory [Eq. (4.34)] for a system having density p\ = 0.6 and mole fraction, x = 0.001 (left) or XB = 0.01 (right). The red and green curves are for A-phobic and A-philic solutes. The remaining curves are for solutes with an A-phobic patch covering 25% of the surfaces. The light blue, pink and dark blue curves were obtained using M/ = 10, Mi = 12 and Mi = 14. B  the solutes are A-phobic. We point oul thai the number of terms kepi in the expansion of the distribution functions need not be the same as in the solvent-solute potential. In order to maintain consistency with the anisotropic: method, the number of terms in the solventsolute potential is always M = 10 [see Eq. (4.2)]. This orientation (A-philic sides facing) was found to be, by far, the most sensitive to the number of expansion terms. In addition, the density of species A appears to be converged when Mi = 11, for all orientations. From these results, we can be confident that the differences between the two theories midway between the solutes for both species at all orientations, are real (see Figs. 4.28 and 4.29), and not due to problems with convergence.  Chapter 4. Integral equation calculations for uniform and patterned particles 94 The depletion of species A and enrichment of species B between the A-phobic parts of the solute is accompanied by attractive, long-ranged forces. The size, range and sign of the force depend on the orientation of the solute particles with respect to each other, as well as on the size of the patches. The forces, calculated using the anisotropic H N C , between the solutes are plotted for case II with XB — 0.001 in Fig. 4.31. We note again that for this system, solution of the anisotropic equation could not be obtained for small separations when the A-phobic patches of the two solutes face each other. Thus, in Fig. 4.31, the force is only shown when R — D > 3o. When the A-phobic parts face each other, the force is long-ranged and attractive [see Fig. 4.31 (top)]. Attractive forces are observed even when the size of the A-phobic patches covering the solutes is only ~ 10%. When the A-phobic parts face away from each other, the force is not attractive, and becomes oscillatory at small separations [see Fig. 4.31 (middle)]. The force between the patched particles in this orientation is similar to the force between uniform, A-philic solutes, except when the A phobic patches are large, covering 75 to 90% of the solute surfaces. For these large patches, the forces are attractive. This is due to the dewetting of species A , and simultaneous wetting of species B, that occurs all around the solutes when the patches are large (see Fig. 4.25) . W h e n the A-phobic patch of one solute faces the A-philic side of the other, the forces are similar to those between the solutes when the A-philic sides are facing. These results are consistent with those observed in the one-component systems discussed above. The repulsive force for this orientation is due to the unfavourable "interface" occurring between the A-enriched (£?-depleted) and A-depleted (5-enriched) fluid layers near the two different solutes. Similar results are obtained when XB — 0.01. A s pointed out, we were not able to obtain stable solutions to the anisotropic integral equations for all separations for large A-phobic patches. The forces for those separations where solutions were obtained, are shown in F i g . 4.32. It is interesting to note that stable solutions can be obtained when the A-philic sides face each other, for much smaller separations than when the A-phobic patches are facing. This is further evidence that the instability in the equations is related to phase behaviour between the solutes. The results are qualitatively similar to those obtained for lower mole fraction (x = 0.001); however, there is an obvious increase in the range of the force when XB = 0.01, as is expected given the system is nearer liquid-liquid coexistence. B  The force between the solutes can also be calculated from the isotropic theory using E q . (4.34). The results are shown for case II with x = 0.001 and x = 0.01 in Fig. 4.33 when 25% of the solute surfaces are A-phobic. The number of expansion coefficients appears to be sufficient, at least to obtain the qualitative behaviour (the oscillations in the force when A-phobic patches are facing appear to be decreasing as the M increases). When the A-phobic patches are facing, the force is attractive, and qualitatively similar to the results obtained using the anisotropic theory, as is the case for the force between uniform solutes in this system (see Fig. 4.20). However, differences between the two theories are apparent for the other orientations. When the A-philic sides are facing, the isotropic H N C theory predicts that the forces between patched solutes are much stronger than those between uniform A-philic solutes, particularly when x = 0.01. In this case, the forces become strongly repulsive, even at relatively large separations (i.e., when R ~ 15cr). When the A-philic side of one solute faces the A-phobic patch of the other, the forces are predicted by the isotropic theory to be nearly as attractive as when the A-phobic patches are facing [see B  B  t  B  Chapter 4. Integral equation calculations for uniform and patterned particles 95 Fig. 4.33 (bottom)]. Attraction between the solutes is also predicted for this orientation when the patch size is larger. In contrast, the anisotropic theory predicts a repulsive force for this orientation except when the A-phobic patch covers 25% or less of the solute surfaces. To summarize, the addition of a small amount of solute B to a system containing a A-phobic solute, results i n the formation of B enriched fluid regions between the A phobic area of the solutes. When the A-phobic patches face each other, strong, long-range attractive forces accompany the changes i n density. There is an increase i n the range of the force with increasing x . This is because the system moves nearer, either a liquidvapour transition (case I), or a liquid-liquid transition (case II). While the isotropic H N C approximation in conjunction with the superposition approximation does give qualitatively similar results to those obtained using the anisotropic theory for most solute orientations, there are differences i n the B — CC distribution functions when the A-phobic patches are facing. Solution to the anisotropic equations could not be obtained for all orientations and separations when species B was present. This is especially true when the A-phobic surfaces face each other, and the intersolute separation is relatively small (depending on XB and the size of the patches). These results suggest that, at small separations, a demixing transition between the solutes occurs. B  4.5.2  jB-rich m i x t u r e s  4.5.2.1  Stability of the solvent mixture  We now consider systems rich i n species B containing a small amount of species A . The solvent mixture i n these systems is similar to that employed i n the simulations discussed in Chapter 3. For all results i n this section, the temperature is T* = 1.15, and the total reduced density is p* = p cr + p o — 0.325, as was used for the simulations. The density can also be written i n terms of the L J parameters, and as such, is a measure of the "packing fraction", g; in this case is g = 2~2„pvO~„ = 0.443 (see Chapter 3 for details about the mixture parameters). In terms of pack fraction, the density is somewhat higher, since the B-type particles are slightly larger than the A-type particles. From our simulation results, and from simulation results obtained previously [21], we know that the mixture is near demixing coexistence when a small amount of species A is present. For example, i n an open system when XB = 0.881, a metastable bulk state exists, i n which the density is approximately 0.685, and the mole fraction of species B is very low (x = 0.013) [21]. The presence of the metastable state was exploited i n the simulations described i n Chapter 3. 3  t  A  3  B  J  B  We can also examine the behaviour of the material and mechanical stability indicators to determine how near the system is to coexistence. In F i g . 4.34, the stability indicators, Xr,iJ, -fi and S (k = 0) are plotted. The material stability indicator (S (k = 0)) is rapidly increasing as the mole fraction of species B decreases. In contrast, the mechanical stability indicator (XT,H -H )I while increasing, is much smaller. These results, along with the simulation results reported i n Chapter 3, indicate that the mixture is near a demixing transition, and not near a liquid-vapour transition. A  B  cc  cc  A  b  Chapter 4. Integral equation calculations for uniform and patterned particles  Figure 4.34: Stability indicators for the J5-rich mixture with p\ = (p + p )o = 0.325. The triangles represent the material stability indicator, S (0) and the squares represent the mechanical stability indicator, XT,H -U 3  A  B  cc  A  4.5.2.2  B  Uniform and patched solutes  As in the previous section the solutes are patterned with A-philic and A-phobic sides; species B interacts with the solutes through a uniform potential. However, the strength of the A — CC attraction is reduced (e c = 0.5). This is to reduce the strong surface wetting by species A that occurs when the solutes are more strongly A-philic. With the reduced attraction, we are able to obtain solutions from the anisotropic equations over a greater range of XB and separation. Since species A is now the minority species, we refer to the A-philic part of a solute as the patch. We briefly discuss the solvent density around a single, uniform A-philic solute immersed in the mixture. The density of species A near the solute increases as the mole fraction of species B decreases, while the density of species B near the surface decreases (see Fig. 4.35). Unlike the A-rich fluids where the minority component, species B, is excluded from the bulk and is thus forced near the solute surfaces because of the strong interactions between A-type particles, in the B-rich fluid, species A (now the minority component), only prefers A  96  Chapter 4. Integral equation calculations for uniform and patterned particles  Figure 4.35: Distribution functions g c{r, cos9) (top) and gscir, cos 9) (bottom) around a uniform. A-philic solute. The different curves represent different mole fractions: .//< - 1.0 (red), x = 0.97 (green), x = 0.881 (dark blue), x = 0.8 (pink). A  B  B  B  to be near the solute surface because of its A-philic nature. Similarly, in the absence of species A , or when only a small amount of v4-type particles is present, the A-philic solute is solvated by the B particles [Fig. 4.35 (top)]. This is due to the weak interactions between B particles; in the simulation results presented in Chapter 3, we see that even a hard wall is slightly solvated by the 5-rieh fluid. The changes in density near the solute do not extend as far from the surface as was observed near the A-phobic solute in an A-rich fluid. Thus, one might expect the range of the forces between two solutes to be shorter in this case. In the remainder of this chapter we will discuss results pertaining to the behaviour of two solutes immersed in the mixture. The contour plots in Fig. 4.36 provide an overview of the changes in the density of species A that occur around the solutes for different orientations. Nearer coexistence, the changes to the density around the solutes are greater, as was the  97  Chapter 4. Integral equation calculations for uniform and patterned particles  Figure 4.36: Contour plot of the distribution function gA-cc(r, cos9) for mole fractions XB = 0.8 (left), and XB = 0.97 (right). In the top panels, the A-phobic parts of the solutes face each other; in the middle panels the A-philic patch of the solutes face each other; in the bottom panels, the A-philic patch of one solute faces the A-phobic patch of the other. In all cases the A-phobic patches cover 50% of the solutes. case in the A-rich fluids discussed above. For both mole fractions shown in Fig. 4.36, the density of species A increases with respect to the bulk density near the A-philic parts of the solutes, and decreases near the A-phobic: patches. When the A-phobic patches are facing, there is a depletion of species A from the fluid between the solutes when XB = 0.8 [see Fig. 4.36 (top left side)]. Likewise, when the A-philic sides of the solutes are facing, there is an increase in the density of species A between the solutes [see Fig. 4.36 (middle)]. Finally, when the A-philic side of one solute faces the A-phobic patch of the other, there is an interface separating the A-enriched and A-depleted regions of the fluid between the solutes. We note that changes in density near the solutes for these systems are much weaker  98  Chapter 4. Integral equation calculations for uniform and patterned particles  than those occurring in the A-rich fluid. It is useful to compare the densities obtained from the isotropic HNC method in conjunction with the superposition approximation with those obtained from the anisotropic  -  2  -  1  0  1  2  (r-R/2)/o Figure 4.37: Comparison of the distribution functions gA-cc(?, cos 6) obtained using the anisotropic HNC theory (red curve) and the isotropic HNC theory in conjunction with the superposition approximation (green curve). Three different patch orientations are shown: in the top panel, the A-phobic parts of the solutes face each other; in the middle panel the A-philic patch of the solutes face each other; in the bottom panel, the A-philic: patch of one solute faces the A-phobic patch of the other. In all cases the A-phobic patches cover 50% of the solutes, x = 0.001. and R = 15a. B  HNC method. The distribution functions from the two methods are compared in Fig. 4.37 for three different patch sizes. Qualitatively, the results from the two methods compare well. However, there are quantitative differences. For all three patch sizes shown, the anisotropic theory predicts a higher density of species A between the solutes. The differ-  99  Chapter 4. Integral equation calculations for uniform and patterned particles ences midway between the solutes are largest when the A-phobic patches cover 50% or less of the surface. However, a greater density of species A in contact with the solutes is predicted by the superposition approximation when the A-phobic patch covers 90% of the solutes as shown in the bottom panel of Fig. 4.37. Both theories show clearly that the density between the A-philic sides of the solutes is much greater than in the bulk. Although we do not have simulation results to compare with directly, we expect that the anisotropic results are more accurate, given the improvement observed for the one-component systems, and the fact that the anisotropic theory directly accounts for the presence of two solute particles. In F i g . 4.38, the force between the solutes calculated using the anisotropic theory is shown for different patch sizes. When the A-phobic patches face each other, the forces between the solutes are oscillatory, and only become strongly attractive when the separation between the solutes is very small R — D < 2a. There are almost no differences in the forces as the size of the A-philic part of the solute increases, except of course when they are uniformly A-philic [see Fig. 4.38 (top)]. In addition, the forces are similar for the two mole fractions shown, x = 0.881 and XB = 0.97. When the A-philic sides of the solutes face each other, the forces become attractive and longer ranged, especially when the composition nears that at the liquid-liquid coexistence. The changes in the force with changes in patch size are also large near liquid-liquid coexistence for this orientation [see F i g . 4.38 (middle)]. W h e n 50% or more of the solutes are A-philic, the forces between the solutes are much as they are when the solute is uniformly A-philic. For smaller A-philic parts of the solute, the forces are still attractive and long-ranged, but the strength of the attraction is somewhat decreased. There are also variations in the force with patch size when the A-philic side of one solute faces the A-phobic patch of the other and the solvent is near demixing coexistence (XB = 0.881). A t this mole fraction, the forces between the patched solutes become more repulsive (or less attractive at very short separations), than they are between uniformly A-phobic solutes. This is likely due to the unfavourable "surface tension" arising between the A-enriched and A-depleted fluid. The solvent structure between the solutes giving rise to the forces is now examined in more detail. The results discussed in the remainder of this chapter were all obtained using the anisotropic theory. The distribution functions between the solutes, obtained at different mole fractions are compared first. The enrichment of species A between the solutes with the A-philic sides facing each other is much greater for a lower mole fraction of species B (XB = 0.8), than for a higher mole fraction (x = 0.97), as shown in Fig. 4.39 (middle). For both mole fractions, we notice that for solutes with only a small A-philic part (the A-phobic patch covers 90% of the surface), the density increases near the surface of the solute, but does not extend throughout the distance between the solutes. When the A-philic parts are larger, there is an increase in the density of species A midway between the solutes (see F i g . 4.39). For patches where at least half of the solutes are A-philic, the increase in density of species A is as great as when the solutes are fully A-philic, and nearly as great when 25% of the solute surfaces are A-philic. The increased attraction between the solutes in this orientation occurs due to the increase of species A between the solutes, similar to what was observed between chemically patterned surfaces. For the separation shown in F i g . 4.39 (R = 15cr), the change in density is not as great as when bridge formation occurs between B  f  B  100  Chapter  4. Integral equation calculations for uniform and patterned particles  101  Figure 4.38: The reduced force between solutes calculated using the anisotropic HNC theory, for a system with pi = 0.325 and x — 0.881 (left) and %B = 0.97 (right). The A-phobic patches cover 100% (red), 75% (green), 50% (dark blue), 25% (pink) and 0% (light blue) of the solute surfaces. Three different orientations are plotted: in the top, the A-phobic parts of the solutes face each other; in the middle, the A-philic patches face each other; in the bottom, the A-philic patch of one solute faces the A-phobic part of another. B  the A-philic stripes on the surfaces. We will examine changes with separation below. When the A-philic sides face away from each other, the density of species A between the solutes is much as it is when the solutes are fully A-phobic, even when only 10% of the solute is A-phobic. This is true for both mole fractions shown in the top panels of Fig. 4.39. When the A-phobic patch of one solute faces the A-philic side of the other, the density near the A-philic side is similar to that between fully A-philic solutes, and near the A-phobic side is similar to that between fully A-phobic solutes. Thus, in this region, there is a steep change in the density between the two solutes. This is reflected in the increased  Chapter 4. Integral equation calculations for uniform and patterned particles  -2  0  2  -2  0  (r-R/2)/a  2  (r-R/2)/a  Figure 4.39: Distribution function, gA-cc(r, cos 9 = 1), between the solutes for different A-phobic patches covering 100% (red), 90% (green), 75% (dark blue), 50%. (pink), 25% (light blue), 10% (yellow) and 0% (black) of the solute surfaces. In the top panels, the A-phobic parts of the solutes face each other; in the middle panels the A-philic patch of the solutes face each other; in the bottom panels, the A-philic patch of one solute faces the A-phobic patch of the other In the panels on the left x = 0.8, and on the right x = 0.97. B  B  repulsion between the solutes observed in Fig. 4.38. We see from the plots of g -cc(' ,cos9 = 1) (Fig. 4.40) that the changes to the density of species B are much less than those to the density of species A . In fact, when x = 0.97, there is only a small change in the density very near the solute surface. Nearer coexistence (x = 0.8), the changes are larger, especially near the surfaces of the solutes. When the A-philic sides of the solutes face each other, the density of B is similar to the density between two fully A-philic solutes except when the A-phobic patches cover 90% of the solute surfaces. W h e n the A-phobic patches of the solutes face each other, the density of r  B  B  B  102  Chapter 4. Integral equation calculations for uniform and patterned particles  CD o o o o CD  CD o o o o CD  II CD CT) O CJ  o o OQ  -2  0  (r-R/2)/a  2  -2  0  (r-R/2)/a  Figure 4.40: Distribution function, (jB-cc(r,cos$ = 1), between the solutes for different patch sizes. The curves and solute orientations are as in Fig. 4.39. In the panels on the left XB = 0.8, and on the right x = 0.97. B  species B is similar to the density between two fully A-phobic solutes for all patch sizes investigated. Finally, when the A-philic side of one solute faces the A-phobic patch of the other, the density of species B near the A-philic parts is similar to that near a fully A-philic solute, and near the A-phobic part, is similar to that near a fully A-phobic solute. We notice that the changes to the density of species A and species B are much smaller than those occurring in the A-rich fluids, even though, from our simulation results, we might expect the density of species A to reach around ten times its bulk value. The reason for the discrepancy is due perhaps to the curvature of the solute. In the case of chemicallypatterned surfaces, the width of a stripe required for an A-rich bridge to form is similar to the separation between the surfaces. In the case of spherical solutes, we would expect the curvature of the surface to reduce the distance between the solutes required for bridges to  103  Chapter 4. Integral equation calculations for uniform and patterned particles  - 3 - 2 - 1 0 1 2 3 - 3 - 2 - 1 0 1 2 3  (r-R/2)/o  (r-R/2)/a  Figure 4.41: Plots of the distribution functions, QA-CC^, cos(9 = 1) for different values of R: R = 14a (solid curve), R = 15a (long dashed curve), R = 17a (short-dashed curve), R = 19a (dotted curve). In the plots on the left XB = 0.8, and on the right x = 0.97. In the top panels, the A-phobic patches of the solutes face each other; in the middle panels the A-philic side of the solutes face each other; in the bottom panels, the A-philic side of one solute faces the A-phobic patch of the other. In all cases the A-phobic patches cover 50% of the solute surfaces. B  form. What we see is that the density of the A-rich fluid is lower between spherical solutes than between A-philic: areas on a planar solute, suggesting that the curvature weakens the effect of the A-philic: surface. This is reasonable since the liquid-liquid interface is much broader with spherical particles than with planar surfaces. The separation between the solutes determines the density of the two species between the A-philic sides of the solute. When the A-philic patches of the solutes face each other, the density of species A becomes greater as the separation becomes smaller (see Fig. 4.41).  104  Chapter 4. Integral equation calculations for uniform and patterned particles  CD  o  CJ  O o  CO  CD  cn O o  o o  m CD  CD  CO o  CJ o o  I  GG CD  - 3 - 2 - 1 0 1 2 3 - 3 - 2 - 1 0 1 2 3  (r-R/2)/a  (r-R/2)/a  Figure 4.42: Plots of the distribution functions, g -cc(r, e,os6 = 1), for different values of R. The curves and solute orientations are as in Fig. 4.41. In all cases the A-phobic patches cover 50% of the solute surfaces. B  At the same time the density of species B decreases with decreasing separation, although the amount of the decrease is relatively small (see Fig. 4.42). The density of species A decreases with decreasing separation when the A-phobic patches of the solutes face each other, while the density of species B remains nearly constant, with differences arising only from the packing of particles between the surfaces. When the A-philic patch of one solute faces the A-phobic patch of another, the changes in the density of species A and species B reflect the two solute sides. The trends in the density changes occur for x = 0.97 and x = 0.8, but are more pronounced when x = 0.8. In particular, when the A-philic sides face each other, the density of species A is more that twice its bulk value. We note that for x = 0.8, we were not able to obtain stable solutions to the anisotropic H N C equations for separations less than 14cr, when A-philic sides of the solutes face each other. B  B  B  B  105  Chapter 4. Integral equation calculations for uniform and patterned particles or the A-philic side of one solute faces the A-phobic side of the other. This occurred when the A-philic sides of the solutes covered 50% or more of the surfaces. The instability in the anisotropic method may be an indication that the increase in the density of species A between the solutes at this separation, becomes very great, and essentially a "wetting transition" occurs. From our simulation results, we would expect there to be a large jump in density between the solutes. We note that the instability in the equations is likely numerical in nature and not related to divergences in the global concentration-concentration structure factors, since the transition is not a true transition in the thermodynamic sense, as is it confined to the regions between the solutes. Nevertheless, the instability may point to a "demixing transition".  4.6  Summary  In this section, the behaviour of large spheres immersed in liquids near coexistence was examined using the anisotropic and isotropic integral equation theories. In a one-component fluid, better agreement with simulation for calculations of the force was obtained using the anisotropic theory when the intersolute separation is small. A t larger separations, the force between uniform solutes calculated from the two theories compare well with each other when the system is far from a liquid-vapour phase transition; however, near liquidvapour coexistence, the anisotropic theory predicts longer ranged attraction. We expect the anisotropic theory to provide more accurate results, since the contribution from both solutes is accounted for directly. For patched solutes, the two theories predict qualitatively similar solvent density profiles between the solutes. The isotropic theory combined with superposition does appear to overestimate the wetting of the surfaces by species B, in the A-rich mixtures, and underestimate the dewetting of species A when the A-phobic solutes are facing. Similarly, differences are also observed in B-rich systems, where the anisotropic theory predicts a higher density of species A between solutes with the A-philic sides facing. In both A-rich and 5-rich systems, attraction between the solutes occurs when the sides of the solutes favouring the minority species (or the vapour) are facing, and the solutes will tend to favour this orientation. The closer the system is to either a liquid-liquid or liquid-vapour transition, the longer the range of the forces between the solutes. Stable solutions to the anisotropic equations cannot be obtained for small separations when the A-phobic patches are facing, in A-rich systems, or the A-philic patches are facing in 5-rich systems. The range of the force and the strength of the attraction between solutes depends on the solute size, and on the patch size. In A-rich systems, the attraction between solutes with the A-phobic patches covering 25% or more of the surfaces was comparable to that between uniform A-phobic solutes. In contrast, the attraction was weaker, though still significant, when the patches covered only 10% of the solutes. Differences became more pronounced as the system neared either liquid-vapour or liquid-liquid coexistence. Attraction was observed between all hard-sphere solutes with diameters of 2o or more. The range of the force increased as the size of the particle increased.  106  107  Chapter 5 Summary and Conclusions In this thesis, both the interactions between solutes due to solvent phase behaviour, and the effect of confining solutes (surfaces) on solvent phase behaviour, were examined. The latter was explicitly studied in Chapter 3 using G C M C simulations, and the former, in Chapter 4, using integral equation theories. One purpose of this work was to determine the pertinent length scales at which phase behaviour effects become important. This is relevant both to the types of phases that can exist between chemically patterned confining surfaces, and to the behaviour of uniform and patterned solutes in a liquid. A second purpose was to explore the effect of solvent phase behaviour on the spatial and orientational alignment of solute particles. B o t h the alignment of infinite, patterned surfaces, and of finite-sized patterned particles were considered. Results obtained using G C M C simulations showed that the phase behaviour of a confined liquid can be controlled by chemical patterning on the surface. In fact, patterned surfaces can give rise to the formation of a bridge phase, a phase that cannot exist in the bulk. The bridge phase consists of alternating A-rich and 73-rich fluid regions corresponding to the metastable and stable bulk phases. Both "striped" and "spotted" surfaces can induce the formation of a bridge phase. Thus, altering the geometry of the surface pattern allows one to control the structure of the confined liquid. The phase behaviour of the confined fluid can also be used to organize the patterning on the surfaces. For example, alignment of stripes on opposite surfaces occurs due to the formation of a bridge phase. The separation at which bridge formation occurs is limited by the liquid-liquid interface. In order for a bridge to form, this unfavourable surface tension must be compensated for by a favourable solid-liquid surface tension. Hence, there is a relationship between the surface separation and the length scale of the surface patterning (e.g., width of a stripe) that must be obeyed in order for bridge formation to occur; the surface area of the favourable solidliquid interface must be at least as large as the area of the liquid-liquid interface between the alternating A-rich and B-rich fluid regions. This connection between length scales is consistent with that found for bridge formation in a one-component system [61, 63]. The formation of a bridge phase is accompanied by strong attractive forces between the surfaces. It was found that the strength of the force is linearly proportional to the number of bridges, provided they are adequately separated, such that the fluid between two adjacent bridges has the composition of the bulk fluid. This demonstrates that the force between the surfaces is mostly due to the force between the surface stripes (or spots) and suggests that similar forces would occur between particles due to phase-like behaviour of the solvent. The strength of the force is related to the difference in pressure between the stable and metastable bulk phases. When the stripe width (spot diameter) is relatively large compared to the separation between stripes (spots), the size of the attractive force is nearly equal to this difference, as is also the case when homogeneous surfaces induce a phase transition [21].  Chapter 5. Summary and Conclusions  108  The relationship between surface separation and patterning length can be altered by the addition of surfactant particles which preferentially occupy the liquid-liquid interface, thereby reducing the surface tension, and allowing the formation of longer bridges. Indeed, in the presence of surfactant, bridge formation can occur at separations up to ten times the stripe width. The lengthening of the bridges is most dramatic when the stripes are narrow (~ 1 — 2cr). The forces due to bridge formation were found to be much less predictable, and highly model dependent i n surfactant-containing systems. B o t h attractive and repulsive forces were observed for the models investigated, and the forces appear to be due partly to bridge formation, and partly to the structure of the surfactant perpendicular to the surfaces. In addition to liquid bridges, the presence of surfactants also stabilizes other surface-induced structures, including extended bubbles. Thus, the combination of a liquid near demixing coexistence, chemically patterned surfaces and surfactants can give rise to complex, ordered liquid structures. The behaviour of large particles immersed in a liquid near liquid-vapour or liquid-liquid coexistence was studied directly using integral equation theories. Solvent profiles around uniformly A-philic, uniformly A-phobic, and patched solutes were investigated. The effective forces acting between the solutes were also calculated. Results obtained using the anisotropic and isotropic H N C methods were compared with each other, and (in limited cases) with computer simulations. Away from coexistence, both methods agree well, for uniform solutes, except at small separations, where better agreement with simulation was found using the anisotropic method. Near liquid-vapour coexistence, and particularly near the spinodal, the methods differ with the anisotropic approach predicting much longer ranged forces. Density profiles obtained from the anisotropic theory were in qualitative agreement with G C M C results obtained for one particular system near liquid-vapour coexistence. As expected, we find that surface "drying" of an A-phobic solute immersed in a onecomponent liquid depends on the size of the solute and the state parameters of the solvent. Near the spinodal even rather small solutes (D = l u ) can induce drying-like behaviour. The layers of lower density fluid extend out several solvent particle diameters before merging into the bulk. Away from the liquid-vapour spinodal, but still i n the vicinity of coexistence, drying was only observed for larger solutes (D > 3a). Far from coexistence, no drying was observed. Similar results were observed in liquid mixtures. Surface "drying" of species A occurred around a uniformly A-phobic solute in an A-rich fluid. The drying was accompanied by the "wetting" of the surface by species B. Species B is excluded from the A-rich fluid and therefore, preferentially occupies the A-depleted region next to the solute. Likewise, "drying" of species B around a uniformly A-philic solute immersed in a 5-rich fluid also occurs. In this case, the "drying" occurs due to the enrichment of species A near the surface. Both "drying" and "wetting" effects are most significant in fluids near either liquid-vapour, or liquid-liquid coexistence. As two large A-phobic solutes in an A-rich fluid, approach each other i n a solvent near coexistence, the A-reduced layers, and in the mixture, B-enhanced layers overlap and create an A-depleted (and 5-enriched) region between the pair. For both uniform and patched solutes this strongly influences the solute-solute interaction. T h e force acting between the solutes becomes much longer ranged as the solvent approaches liquid-vapour or liquid-liquid  Chapter 5. Summary and Conclusions  109  coexistence, and continues to increase in range up to the spinodal. For the patched case, the force depends on the patch size and on the orientation of the solutes. The strongest attractive forces occur when the A-phobic patches face each other and the solutes will tend to prefer this orientation. In general, the force decreases with decreasing patch size, but for solutes of diameter 10cr they are important for patches that cover 25% of the solute surface, and remain significant down to 10% coverage. The interactions between A-philic solutes in a B-rich mixture are qualitatively similar. In this case the forces are most attractive when the A-philic patches are facing. While the results from the two different methods are qualitatively similar in most cases, quantitatively, they differ dramatically for some systems. This is especially true for the forces calculated via the two methods between patched solutes. B o t h methods do predict attractive forces between the solutes in an A-rich fluid when the A-phobic patches are facing. However, when the A-philic sides are facing, the forces between patched solutes are predicted to be much more strongly repulsive by the isotropic theory than by the anisotropic theory. When the A-philic side of one solute faces the A-phobic side of the other, the forces calculated using the two methods differ even qualitatively for some patch sizes. The isotropic equations predict attractive forces for all patch sizes with this orientation, while the anisotropic equations predict the forces to be repulsive, except when the A-phobic patches are very large. We note that, where comparison with simulation was possible, the anisotropic equations were found to be more accurate, both in this thesis, and in other previous work [79]. In addition, we expect the anisotropic equations to be more accurate, given the effect of two solutes is directly accounted for. Thus, given the large quantitative differences (and in some cases qualitative difference) found here, one must be careful in interpreting results obtained from the isotropic theory, especially in the case of patched solutes. In this work we have shown that solvent phase behaviour can be used to control the interactions of larger solutes. It is interesting to consider how this could be explored experimentally. The surface force balance [97] and the lateral force microscope [98] have been used to measure shear forces between two uniform, polymer-coated surfaces. One could use these instruments to look at the shear force between non-uniform surfaces. Based on the simulation results here, and on simulations of the shear force between patterned surfaces [68], one might expect a long-ranged shearing force in systems where bridge formation could occur. It has been observed that the shear force in propanol-water mixtures between uniform surfaces can be dramatically altered when partial demixing occurs [98]. A s discussed in Chapter 3, attractive forces also occur between hydrophilic surfaces immersed in an organic liquid (octamethyltetrasiloxane) containing trace amounts of water due to a confinement-induced demixing transition [7]. If bridge formation were to occur, one would expect to observe changes in the lateral or shear force with changes in the relative positions of the surfaces. We note also that it has been observed experimentally that alignment of large (micro to millimeter scale) particles in water can occur due to "evaporation" near the hydrophobic areas of the solutes [99]. 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(4.37), 5a-cc(ri;R)  =  exp - /3u _ c(n, R ) + J~] P a  C  / dr c (|r! -  x /  2  a7  7  r |)(c7 -cc(r2; 2  7  (A.l)  R)-l),  we can rewrite the integral (which we call y) in spherical coordinates, and using h = g — 1 we obtain, y  =  dr c (\r -T \)h _cc{r2\R)  J  2  /•27T  =  J  oll  l  2  1  P+ \  d(p J 2  1>  dcos6 J rldr c (ri,r , 2  2  a7  cos8 )h^cc(r ,  2  12  2  cos6 ;R). 2  (A.2)  Keeping i n mind that the particles at position 1 and 2 are both solvent particles (i.e., neither is at the origin) the angle between them, 8\ , can be written in terms of 8i, 8 and (f> , where we choose 4>i = 0 (see F i g . A . l ) , using the geometric relation, 2  2  2  cos #12 = cos 8\ cos 8 + sin #i sin 8 cos § . 2  2  2  (A.S)  Figure A . l : Coordinate system showing the positions of two solvent particles with respect to the origin.  Appendix A. Legendre transform used for solution of the anisotropic HNC theory 116 Writing h and c as Legendre expansions, we obtain f27T  ?  =  / -  /• + !  # 2 / rfcosc? / r  7 0  ~  2  / J  2  f  ir J] ^(r ,r )P((cos^ )^/ !;_ (r ;P)P (cos^)  OO  dxh j r\dr  x J  2  1 2  l  c c  2  7 l  n=0  ^ h % _ ( r ; R)  l  ai  ri  2  cc  2  n=0  1=0  J  1  OO  c ( ,r )  2  °  C  2  1=0  J  /. 7T  =  2  2  dcos9 P (cos9 )P (cos9 ). 2  l  12  n  (A.4)  2  The spherical harmonic addition theorem [51],  + 2J2  P(cos0 ) = Pi(cos^)P (cos^ ) 12  z  2  m=l  f ~ ^'Pr(cose^Pr(cos9 ) m  c o s ( m 0 ) (A.5)  2  \  l  +  2  ) -  m  can then be substituting into E q . (A.4). Noting that the sum of associated Legendre polynomials disappears due to integration over the interval [0,27r] of cos(mc6 ), we obtain 2  OO  OO  r\dr J2 2  /  c (n,r ) l  ai  (=0  x y  £  2  ^_  c c  ( r i ; P)P/(cos6 ) X  (A.6)  71=0  dcos9 P (cos9 )P (cos9 ). 2  l  2  n  (A.7)  2  Using the orthogonal property of Legendre polynomials,  r  2  +1  J  dcos9 P (cos9 )P (cos9 ) 2  l  2  n  = ^-^5 ,  2  (A.8)  l<n  we obtain, oo  /  ^  2  £  c ^ n , r )h\_ {r -R)P (cos 2  cc  2  9,).  t  (A.9)  Substituting y into E q . ( A . l ) we get g -cc (n, cos 91; P ) = a  exp [ - 3u - c (r , cos 0 , P ) a  +  C  x  X  E^E TTT ' p  ( c o s  2  ^  x Jr dr c (r r )h ^ (r R)], 2  l  2  2  l  ay  u  2  which is the expression presented in C h . 4 (Eq.(4.38)).  cc  2]  l )  (A.10)  117  Appendix B Force between particles The discussion here follows closely that presented by A t t a r d [8]. The force between two solute particles can be calculated as the negative derivative Of the free energy with respect to the separation R, F  {  R  )  --8R-  -Z^-dR'  (  R  1  )  where F is the Helmholtz free energy. The configurational integral is given by Z  N  = J  e- «dY 0n  • • • dv .  l  N  Here the Hamiltonian is defined as 7i = UN+W, solvent particles is given by,  (B.2)  where potential due to interactions between  N  u  N  = ^2 u(r ). l  (B.3)  ii  The external potential due to the two solutes separated by R (see F i g . B . l ) is given by,  *iv = 2 i ( » ) ^ ( r - R ) . i n  r  +  i  The derivative of the configurational integral is the following  Figure B . l : The coordinate system used in the derivation of the force.  (B.4)  Appendix B. Force between particles  dZ -dR  d  N  =  Je~ dx 16  •••dr  mN  dR  118  x  N  =  / E ^T^"'' '' ••'*»/  - r,).  1  (B.5)  In the last equality, we multiplied the right-hand side of the equation by 1 = J dr*S(r* — r,). Multiplying the numerator and the denominator by and rearranging, we obtain,  The above expression can be written in terms of the solvent density p(r*), since  = P(r").  (B.7)  A n expression for the force can be obtained by substituting for dZ^/dR in E q . ( B . l ) , F(R) = - J dvp{v)^-  (B.8)  In our case, the potential \I/(r) = u i ( r ) + u (r — R). Using r = r — R, we can write 2  d\T/(r)  _  2  du (r ) _ du (r ) dr 2  dR  2  2  =  2  2  2  dR  dr  dR  -V[iz (r )]-R, 2  (B.9)  2  where R is a unit vector in the direction of R. Substituting the last equality into E q . (B.8), gives an expression for the force F(R) = - J dvp(v)V [u (r )} • R, 2  2  (B.10)  which, can be rewritten i n terms of an integral over r , 2  F(R) = - J dr p(r )V [u (r )] • R. 2  2  2  2  (B.ll)  Here V is the gradient with respect to r . This expression can readily be generalized for mixtures, 2  F(R)  = -E/  D R  W 2)V r  [u (r)] • R, 27 l  2  (B.12)  Appendix B. Force between particles  119  where 7 represents the solvent species. This equation is equivalent to that presented in Ch. 4 (Eq. 4.39). Since R defines the z-axis, the dot product in Eq. ( B . l l ) yields, V  W  r )].R 2  =  *|M  _  du (r 2  (B.13)  cos9 ) dr dr dz  2)  2  2  2  du (r , cos9 ) dcos9 d cos 9 dz  |  2  2  2  2  (B 14)  2  2  2  Using the relations r\ = x\ + y + z\ and z — r cos 9 , we obtain an expression for the force, 2  2  F(R)  =  2  2  2TT J r\dr J dcos 9 p(r , cos 9 ) 2  2  du {r ,cos9 )  2  2  du (r ,cos9 ) f sin 9 ^ . 9< a - r' — , . (B.15) cos or 0 cos 9 \ r ' Here we have written the force between two solutes in a one-component solvent; the extension to multicomponent solvents is straightforward. Since the solvent-solute potential (u(r , cos 9 )) is discontinuous at ra, in principle we need to include the contribution at contact. However, in Systems I and III, the potential is repulsive at very small separations ensuring that the density, p(r , cos9 ) is negligible at the contact separation r^ = (D + o)/2. In System II, the solutes are hard spheres, and therefore the contribution from contact is the only term in the force. We can rewrite Eq. B.5 in the following way, 2  2  2  2  n  2  2  2  2  1  n  1  2  2  2  2  2  2  H  = -Z J N  P(r)c^« A  (l _ ^ ( 0 )  d r  2  .  (B.16)  Since the solute particles interact with the solvent particles through the hard-sphere potential,  *M -{Zm.  (B-17)  the quantity 1 — e *( ) can be written as sum of two step functions, _/3  r  1-  _ / 3 e  *  ( r )  = H(r) + H(r - R).  (B.18)  Here, the step function is defined by the following,  ={j;  ;t : r  t  (B.i9)  Using the above, we can take the derivative, jL  (  l  _ -Wr)) e  =  A(tf (r) + rd  fUr-R))  =  V*ff (r-R)-5j  =  - | - ^ 2 - r  rd  ri  r  2  d  ) ,  (B.20)  120  Appendix B. Force between particles  where we have used r • R = rRcos 8 , and the fact that the gradient of the step function is 2  2  X7 H (r) R  rd  = -8(r-r ). r  (B.21)  d  Using Eqs. (B.16),(B.20) and (B.l), we obtain F(R)  =  2iTk T  J \cose (r )e-^^dr (^^5(r -r ) j  =  2irk Tr J d cos 9 p(r , cosf? )  =  2-KksTr , J d cos 6 p{r , cos 6 ) cos 8 .  +  B  S  2P  2  B  d  2  d  2  2  2  R r  2  d  2  Rr  2  (B.22)  2  2  d  2  2  From this expression, we see that the force between the two hard sphere solutes is related to the density of the solvent at contact. In the limit that the solutes are infinitely far apart, the solvent density around each hard sphere solute is independent of cos# , and the force between the solutes approaches zero. When the solutes are near each other, the presence of the second solute alters the density around the first (i.e. the density around each solute is no longer uniform), and there is a force between the solutes. 2  

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