Entropic forces in dilute colloidal systems

Transcription

1 Entroic forces in dilute colloidal systems R. Castañeda-Priego Instituto de Física, Universidad de Guanajuato, Lomas del Bosque 103, Col. Lomas del Camestre, León, Guanajuato, Mexico A. Rodríguez-Lóez and J. M. Méndez-Alcaraz Deartamento de Física, Cinvestav, Av. IPN 2508, Col. San Pedro Zacatenco, México, Distrito Federal, Mexico Received 10 March 2006; ublished 25 May 2006 Deletion forces can be accounted for by a contraction of the descrition in the framework of the integral equations theory of simle liquids. This aroach includes, in a natural way, the ects of the concentration on the deletion forces, as well as energetic contributions. In this aer we systematically study this aroach in a large variety of dilute colloidal systems comosed of sherical and nonsherical hard articles, in two and in three dimensions, in the bulk and in front of a hard wall with a relief attern. We show by this way the form in which concentration and geometry determine the entroic interaction between colloidal articles. The accuracy of our results is corroborated by comarison with comuter simulations. DOI: /PhysRevE I. INTRODUCTION Comlex fluids are comosed of a large amount of articles of many different secies. For examle, a clean aqueous susension of olystyrene sheres contains, obviously, the olystyrene sheres and the water molecules, but also ions dissociated from the surface of the sheres, called counterions, and at least two different secies of salt ions. The size of the first ones ranges from about some tens of nanometers, u to several microns. The other articles are of subnanometer dimensions. When looking at this susension by means of light scattering, for examle, only the big sheres seem to be resent in the system, since they are the only ones able to affect the hotons of wavelengths of the order of micrometers. Therefore, in this examle the exerimental access to the system is limited, and we can only observe a art of it. The hysics of this art, however, is determined by all the existing articles, observed or not. Regardless of the kind of condensed matter system that we might be able to imagine, and of the exerimental method that we might roose in order to look inside it, the conclusion is always the same: We can observe only a art of the system, but the hysics of this art is also determined by the rest. Physicists deal with the restricted nature of the exerimental observations by develoing contracted ictures, or ective theories, containing arameters related to the unobserved variables. The basic concet behind this kind of aroach is the ective air interaction otential between the observed articles. Deletion forces are a articular case of this kind of interaction, which describe the hase of a great variety of colloidal and olymeric systems with an imortant excluded volume contribution to the free energy. The term deletion forces originally refers to the attraction between two colloidal articles arising when macromolecules are ut into the susension 1,2. This attraction results from the exulsion of added macromolecules from the ga between two aroaching articles, giving rise to an imbalance between the osmotic ressures inside the ga and outside it and, therefore, to a decrease of the free energy of the system when the articles come together. This henomenon has been successfully studied in the usual case that the system can be modeled as a binary mixture of hard sheres, PACS numbers: y, Gy as in the case of uncharged globular macromolecules and sherical colloidal articles, by means of erturbations theory 3,4, density functional theory 5 7, and integral equations theory 8. Moreover, some attemts to account for olymer nonideality beyond the hard sheres model have also been reorted, using extensions of the theoretical frameworks mentioned above 8 11, as well as self-consistent mean field calculations 12. Indeed, deletion forces can be strongly affected if van der Waals or coulombic interactions are resent in the system. With regard to the relative merits of each aroach, it can be said that erturbations theory straightforwardly allows for the incororation of some geometrical ects on the deletions forces, as in the case of a mixture of sherical and nonsherical colloidal articles 13,14, but makes it very difficult to include concentration ects and energetic contributions in the theoretical scheme 4. The aroaches based on the density functional theory DFT incororate otherwise, additionally to some geometrical ects, as in the case of a colloidal mixture in front of a curved hard wall 15,16, concentration ects in a natural way. However, the energetic contributions to the deletion forces are only very hard to cature. Finally, the aroach based on the integral equations theory IET catures both concentration ects and energetic contributions in a natural way, and, as we show in this work and in a revious aer, also the ects of the geometry of the comonents of the colloidal system 17. In addition, IET seems to work better than DFT for systems in even dimensions 18,19. The method based on the DFT works better in extreme situations, as in the case of very large differences in the size of the colloidal articles, but the aroach based on the IET is far easier to imlement, both schemes roviding an excellent quantitative descrition of the deletion forces, when they are comared with comuter simulations. Actually, the distinction we make between density functional theory and integral equations theory is quite artificial. In fact, IET can be obtained from DFT 20. Nevertheless, once the equations of IET are obtained from DFT, the methods to work with them differ from the methods of DFT to such an extent that both are very often identified in the literature as different routes /2006/735/ The American Physical Society

2 CASTAÑEDA-PRIEGO, RODRÍGUEZ-LÓPEZ, AND MÉNDEZ-ALCARAZ The integral equations theory reresents a natural starting oint for the imlementation of the contraction formalism from which the deletion forces are determined. The basic idea is that deletion forces are a secial case of the more general ective interactions resulting from a contraction of the descrition of liquid mixtures. Therefore, if certain comonents of a mixture are not exlicitly considered, their influence on the structure of the remaining articles has to be included in the ective interaction otential between the latter ones. This is obtained by demanding the satial distribution of the observed articles to be the same as in the original mixture. Technically, this is done by rewriting the Ornstein-Zernike equation for the original mixture as an ective Ornstein-Zernike equation for the remaining articles, and connecting it with the ective interaction otential by means of an aroriate closure relation. This idea was first imlemented in order to calculate the interaction between two charged macroions immersed in a bath of small counterions and salt ions 21, as well as in further aroaches to the same roblem 22,23. We consider here the case of large hard sheres immersed in a bath of smaller sherical and nonsherical hard articles in order to describe deletion forces of only entroic nature. In this aer we systematically study a large variety of dilute colloidal systems comosed of sherical and nonsherical articles, in two and in three dimensions, in the bulk and in front of a hard wall with a relief attern. We show by this way the form in which concentration and geometry determine the entroic interaction between colloidal articles, leading to surrising results. For instance, the design of surfaces of entroic otential. After a brief introduction, we summarize in Sec. II the integral equations theory of liquid mixtures. Then, the contraction of the descrition rocedure and its meaning are exlained in Sec. III. Here, some comuter simulation results for two-dimensional binary mixtures of hard disks are shown in order to clarify some of the involved ideas. In Sec. IV the general formulation of deletion forces in multicomonent systems is resented, and the resulting equations are exlicitly written for dilute systems. In Sec. V the articular cases of two- and three-dimensional binary and ternary mixtures are discussed. Section VI deals with colloidal mixtures in front of a flat, or concave, or convex hard wall. The results obtained in Sec. VI are carefully worked u in Sec. VII to a method of entroic engineering, which allows the design of surfaces of entroic otential. The ects of the geometry of the comonents of the system are addressed in Sec. VIII, where three-dimensional binary mixtures of hard sheres and hard sherocylinders are studied. Section IX deals with concentration ects and energetic contributions, as far as it is ossible to obtain information about from the dilute limit in our equations. Finally, the aer ends with a section of conclusions. II. INTEGRAL EQUATIONS THEORY In an homogeneous colloidal mixture of sherical secies, the total correlation h ij r between a article of secies i and a article of secies j, searated by the distance r, can be exressed in a logical, almost grahical form by the exansion h ij r = c ij r + n c ik rc kj r rdr kv k=1 + n k n c ik rc kl rc lj r r rdrdr lvv k,l=1 +, 1 where the function c ij r reresents the direct correlation between articles i and j, defined in such a way that h ij r includes all ossible correlations involving all the N = i=1 N i articles in the volume V; N i is the number of articles of secies i and n i =N i /V its bulk number density. The second term of the right hand side of Eq. 1 reresents all links between two articles mediated by a third one. The third term of the right hand side reresents all links between two articles mediated by two other articles, and so forth. In highly dilute systems, the direct correlation function reduces to the air interaction otential u ij r between the articles, so far they do not overla, scaled with the thermal energy; c ij r= u ij r/k B T 24. The minus sign guarantees that the total correlation function at short distances becomes negative ositive for reulsive attractive interactions. Equation 1 is a geometrical series and can therefore be rewritten as the well known Ornstein-Zernike equation OZ 24, h ij r = c ij r + n c ik rh kj r rdr, 2 kv k=1 which needs a closure relation linking the structure functions h ij r and c ij r with the air interaction otential u ij r for its solution. This closure relation has the following general form 24 c ij r = u ij r + h ij r ln1+h ij r + b ij r, 3 where =1/k B T and b ij r is a structure function known as bridge function. Further aroximations for b ij r are necessary in order to close the scheme. One of the most successful closure relations is an emirical one roosed by Rogers and Young RY 25, which has the form c ij r = ex u ij r1+ ex ijrf ij r 1 f ij ij r 1. r 4 Here, ij r=h ij r c ij r is the indirect correlation function and f ij r=1 ex ij r is the mixing function. The mixing arameter ij is obtained by demanding thermodynamic consistency of the solution of OZ. This can be illustrated by remembering the contribution of Biben and Hansen 26,27, where they studied asymmetric binary mixtures =2 of hard sheres by means of OZ+RY. The mixing arameter was obtained in their work from the assumtion that it scales with the contact distance between articles, ij =/ ij, and by fixing by demanding the same value of the isothermal

3 ENTROPIC FORCES IN DILUTE COLLOIDAL SYSTEMS comressibility as calculated from the comressibility equation of state 24, P 2 =1 n x i x j c ij 0, 5 n T i,j=1 and from the differentiation of the virial equation of state 24, P n =1+2n 3 2 i,j=1 x i x j 3 ij g ij + ij. 6 Here, ij = i + j /2 is the contact distance between articles, i is the diameter of the articles of secies i, n = i=1 n i is the total bulk density, x i =n i /n=n i /N is the molar fraction of secies i, g ij r=h ij r+1 is the artial radial distribution function, P is the osmotic ressure in the system, and c ij 0 denotes the Fourier transform c ij q of c ij r evaluated at q = 0. Since the closure relations are aroximations, their use in calculating thermodynamic roerties will lead, in general, to different results, when the roerties are determined by using different routes. Instead, by imlementing RY the value of can be fixed by demanding the restoration of thermodynamic consistency, at least artially, as seen above. Biben and Hansen were mainly interested in the behavior of the long wavelength limit q 0 of the concentrationconcentration structure factor S cc q, given by 28 S cc q = x 2 2 S 11 2x 1 x 2 S 12 q + x 2 1 S 22 q, 7 since its divergence signals the end of thermodynamic stability with resect to hase searation, as can be seen from the equation of state 28 S cc 0 = Nk B T 2 G/x 1 2 N,P,T. Here, S ij q=x i ij +nx i x j h ijq is the artial structure factor and G the Gibbs free energy of the system. By using OZ in order to write S ij q in terms of c ij q, Eq. 7, together with the condition S cc 0, leads to the exression n 1 c n 1n 2 c n 2 c 22 0 =1 9 for the sinodal curve characterizing the hase searation. It means that the articles in the system demix when Eq. 9 is fulfilled. Biben and Hansen found that it may occur for size ratios 1 / 2 5. Indeed, this rediction has been corroborated by several authors, by means of exeriments and of comuter simulations They observed that the segregated big articles condense in a hase whose density is larger than the density of the same secies in the homogeneous system. This result will hel us to understand the meaning of the contraction of the descrition exlained in the next section. It is widely acceted that binary hard sheres in three dimensions exhibit a gas-liquid searation for a certain range of size rations, but all of which seems to be reemted by the freezing transition, i.e., the gas-liquid transition is metastable with resect to freezing III. CONTRACTION OF THE DESCRIPTION As seen in the revious section, the structure of an homogeneous colloidal mixture of sherical secies is given by the OZ equation 24 h ij q = c ij q + n k c ik qh kjq, k=1 10 written here in the Fourier sace. This equation owns a very imortant, but not much exloited roerty: It does not change its form when only a art of the system can be observed, i.e., its form is invariant under contractions of the descrition. Let us suose, without lost of generality, that we can only observe the articles of secies 1 in the mixture of comonents. In a light scattering exeriment, for examle, this condition can be imosed by the wavelength of the light, if it is comarable to 1 but very much larger than the diameters of the other comonents. The other secies are there, but we suose to ignore that fact. Therefore, we describe the structure of the observed articles by means of the monodiserse OZ equation h 11 q = c 11 q + n 1 c 11 qh 11q. 11 Since h 11q is a quantity which can be measured, it has to be the same in both Eqs. 10 and 11, then the articles have the same structure regardless of our ability to distinguish the different kinds of comonents. The other secies not longer aear as searated secies in Eq. 11. Instead, their ects on the structure of comonent 1 are included in c 11 q. This fact is denoted by the suerindex. The new ective direct correlation function can be obtained in terms of the direct correlation functions of the original mixture by rewriting Eq. 10 in the monodiserse form 11, i.e., exloiting the invariance of the form of the OZ equation under contractions of the descrition. This leads to 8 c 1i qn i c i1 q q = c 11 q + 1 n i c ii q c 11 + i1 i1 j1,i c 1i qn i c ij qn j c j1 q 1 n i c ii q1 n j c jj q + 12 In addition, the ective interaction otential u 11 r can be obtained from Eqs. 11 and 12 by using a closure relation of the general form 24 c 11 r = u 11 r + h 11 r ln1+h 11 r + b 11 r, 13 as we will see in the next section. Equation 12 can be written in closed form by using matrix notation 11,22. Its resent form, however, makes its alication easier in the following sections, since it straightforwardly takes a very simle closed form in the case of binary and ternary mixtures. Once the descrition is reduced, we can also calculate the thermodynamic roerties of the system comosed of only the observed secies. The isothermal comressibility T, for examle, is given by

4 CASTAÑEDA-PRIEGO, RODRÍGUEZ-LÓPEZ, AND MÉNDEZ-ALCARAZ 1 T =1 n 1 c The exression for the sinodal curve characterizing a hase transition between two hases of different densities can be obtained from Eqs. 12 and 14, together with the condition T. For binary mixtures =2 it leads to n 1 c n 1n 2 c n 2 c 22 0 =1, 15 which exactly agrees with Eq. 9 for the sinodal instability with resect to hase searation in the original binary mixture. When the condition 9 is fulfilled, the articles in the mixture segregate. Equation 15 tells us that this henomenon looks like a gas-liquid hase transition when only the articles of secies 1 are observed, which agrees with the exerimental observations in asymmetric binary mixtures of hard sheres, as well as with comuter simulation results for the same systems this gas-liquid hase transition is, however, metastable with resect to freezing 38. In general, the thermodynamic roerties of the original mixture are not all catured by the reduced descrition. However, the roerties deending only on the structure of the surviving secies, which is invariant under contractions of the descrition, can still be obtained from the reduced system, as in the case of Eqs. 9 and 15. In the original mixture the articles get the structure described by h ij r due to their thermal agitation and to the interactions given by u ij r, i, j=1,...,. In the case where we can only observe the articles of secies 1, they get the same structure h 11 r as in the original mixture, but due to the thermal agitation and to the ective interaction otential u 11 r. The latter can be obtained, at a molecular level, from the integration of the average forces calculated by fixing two articles of secies 1, searated by the distance r, and by integrating the linear moment exchange between all the articles of the other secies and the two fixed articles of secies 1. We carried out this calculation in a two-dimensional binary mixture =2 of hard disks by means of molecular dynamics simulations MD, for the cases of infinite dilution of secies 1 n 1 0, size ratios 1 / 2 =5 and 10, and concentrations 2 =0.2 and 0.4. Here, i =n i 2 i /4 is the twodimensional filling fraction of secies i. The comuter simulations were erformed in a quadratic simulation box with standard eriodic boundary conditions, using the Verlet algorithm 39. The number of articles used in this work for 1 / 2 =5 was N 1 =2 and N 2 = for 2 = For 1 / 2 =10 we used N 1 =2 and N 2 = for 2 = The length L of the simulation box was adjusted to give the rescribed density of the system according to the relation L 2 =N 2 /n 2. At the beginning of the simulation, the two larger hard disks were laced at the center of the simulation box, searated by the distance r, remaining fixed in that osition. The smaller hard disks were then laced randomly in the simulation box in a nonoverlaing configuration and allowed to move according to the Verlet algorithm, until equilibrium was reached. Further configurations were generated to calculate the average linear moment exchange. The same rocedure was reeated for different values of r in FIG. 1. The figure shows the deletion otential u 11 r for two-dimensional binary mixtures of hard disks in the infinite dilute limit of articles of secies 1, 1 0, for the size ratio 1 / 2 =5, and filling fractions 2 =0.2 and 0.4 of the articles of secies 2. The oen symbols were obtained from molecular dynamics simulations. The full lines corresond to the results obtained from the MSA-PY aroximation, the dashed lines to the results obtained from the HNC-PY aroximation, and the dotted lines to the dilute limit in our equations. order to get the average force in the whole range of significant searations. The ective interaction otential results from the integration of the average force. The length L was always large enough for the smaller disks to get their bulk structure on the border of the simulation box. Therefore, the larger articles at the center of one cell do not feel the other articles of the same secies in the other cells. The ective interaction otentials calculated by MD are shown in Figs. 1 and 2 by the oen symbols. The original binary mixtures are comosed of hard disks of two different diameters. Their interactions are only of exclude volume tye and, therefore, the free energy of the mixtures is only of entroic nature. If we are not able to observe the articles of secies 2, although they are resent in the system, which we FIG. 2. The figure shows the deletion otential u 11 r for two-dimensional binary mixtures of hard disks in the infinite dilute limit of articles of secies 1, 1 0, for the size ratio 1 / 2 =10, and filling fractions 2 =0.2 and 0.4 of the articles of secies 2. The oen symbols were obtained from molecular dynamics simulations. The full lines corresond to the results obtained from the MSA-PY aroximation, the dashed lines to the results obtained from the HNC-PY aroximation, and the dotted lines to the results obtained from the dilute limit in our equations

5 ENTROPIC FORCES IN DILUTE COLLOIDAL SYSTEMS suose to ignore, we observe a monodiserse susension comosed of articles of secies 1 interacting with the otential u 11 r shown in Figs. 1 and 2, which is long-ranged. Thus, the free energy of the ective monodiserse system contains a finite contribution from the excess energy. This constitutes an interesting result: One can go from an entroic system to an energetic system by ignoring a art of it. Alternatively, we can also say that the original binary mixture is an entroic icture of the monodiserse system, the latter in the sense that the articles of secies 1 have the same structure in the entroic mixture as in the monodiserse susension with interaction u 11 r. This assumtion was corroborated by other authors 40, who erformed comuter simulations for three-dimensional binary mixtures of hard sheres, for either the entroic mixture and the contracted monodiserse susension. The radial distribution function g 11 r they found was the same in both cases. Although our simulations were done for the infinite dilute limit of the observed articles, n 1 0, this does not affect the generality of our results. Indeed, the contraction of the descrition is a very general concet which can be alied even in the case that we can only see two of the susended articles. The rest of them, of the same and of different secies, determines the ective interaction otential between the two observed articles, as addressed in Ref. 41. In that secial case, when working with monodiserse systems, u 11 r and the otential of mean force w 11 r= ln1+h 11 r are the same. The attractive well at contact shown in Figs. 1 and 2 is due to the deletion ect arising from the exulsion of small articles from the ga between the two large articles. As exected from the Asakura-Oosawa aroximation AO 1,2, its deth increases with 2 and with 1 / 2. Simultaneously, a otential barrier develos in front of the attractive well, and the interaction becomes more long-ranged, oscillating around zero at larger searations. This is due to the correlations between the contracted articles, which are not included in AO. We now develo a method able to cature such ects. IV. ENTROPIC POTENTIALS The ective interaction otential u 11 r is obtained from Eq. 13, where c 11 r is given by the inverse Fourier transform of Eq. 12, and b 11 r needs further aroximations. If we take, for examle, b 11 r=0,wegetu 11 r from the hyernetted chain aroximation HNC 24: u 11,HNC r =+for r 1 = c 11 r + h 11 r ln1+h 11 r for r In the mean sherical aroximation MSA 24 we get u 11,MSA r =+for r 1 = c 11 r for r One more aroximation must still be done in order to obtain the structure functions h ij r and c ij r of the original mixture, so that we can evaluate c 11 r. Since the hard-core interaction between overlaing articles never changes, next we only try with the nonoverlaing art of the interaction otentials, i.e., with r 1. In the case of binary mixtures c 11 r becomes 1 c 11 r = c 11 r + F n 2c 12 2 q 1 n 2 c 22 q, 18 where the symbol F 1 X denotes the inverse Fourier transform of X. If we take, for examle, the Percus-Yevick aroximation PY 24, c ij r = ex u ij r ij r +1 ij r 1, 19 for the binary mixtures of hard disks of the simulations shown in Figs. 1 and 2, we get the results shown in the same figures for u 11,HNC-PY r and u 11,MSA-PY r. In the notation u 11,X-Y r the symbol X stands for the aroximation used for b 11 r in Eq. 13, and the symbol Y for the aroximation used in the calculation of the structure functions of the original mixture. The theoretical results and the MD data qualitatively agree in the whole range of significant searations. There are, however, areciable quantitative differences in the deth of the attractive well in the contact region and in the height of the reulsive barrier in front of it. HNC-PY considerably underestimates the deth of the deletion well, and MSA-PY overestimates it a little. Both aroximations, HNC-PY and MSA-PY, underestimate the height of the reulsive barrier. These differences, however, are rather similar to those normally observed when studying the structure of liquids 42. Actually, the determination of the accuracy of the aroximations done in the integral equations theory of liquids is often an emirical task. Therefore, we can seak only a osteriori about the accuracy of our aroximations for u 11 r. This leads to some unexected results. For examle, if we take RY for h ij r and c ij r, and MSA for b 11 r, we get an ective interaction otential, u 11,MSA-RY r, which is much more inaccurate than u 11,MSA-PY r, although RY is a better aroximation than PY for the structure of the original mixture this result is not shown in the Figs. 1 and 2, but in Ref. 18. The structure functions h ij r and c ij r of the original mixture were obtained in this aer from the numerical solution of Eq. 2 by means of a five arameters version of the Ng algorithm 43. MSA-PY seems to be a very good aroximation. This can be understood from Eqs. 3 and 13, which together lead to u 11 r = u 11 r + c 11 r c 11 r + b 11 r b 11 r. 20 This result means that u 11 r is given by u 11 r lus the correlations terms c 11 r c 11 r+b 11 r b 11 r. Ifweneglect the difference between the bridge functions, not the bridge functions self, we get the exression

6 CASTAÑEDA-PRIEGO, RODRÍGUEZ-LÓPEZ, AND MÉNDEZ-ALCARAZ 1 u 11 r = u 11 r F n 2c 12 2 q 21 1 n 2 c 22 q, which exactly agrees with u 11,MSA-PY r for the case of an original mixture of hard articles, where MSA and PY are equivalent. Thus, MSA-PY in Figs. 1 and 2 results from neglecting only the difference b 11 r b 11 r. Moreover, if we take RY instead of PY for the original mixture in 21, we obtain results for u 11 r which are still more accurate than u 11,MSA-PY r this result is not shown in Figs. 1 and 2. However, such imrovement becomes relevant only close to the instability described by Eq. 15. The idea that u 11 r can be seen as a erturbation around u 11 r has been used by several authors in the ast and is well exlained in Ref. 44 by following a rather different method than the one used in this aer. HNC-PY, on the other hand, looks really bad in comarison with the simulation data. This can be understood by taking n 1 0 in Eq. 11, which together with Eq. 16 leads to u 11,HNC r= ln1+h 11 r. This agrees with the definition of the otential of mean force w 11 r between articles of secies 1 note that this is no longer true for the concentrated systems. Therefore, the results for u 11,HNC-PY r shown in Figs. 1 and 2 corresond to the otential of the mean force obtained from PY, what we have corroborated by comaring u 11,HNC-PY r with ln1+h 11 r. Moreover, from Eqs. 11 and 13 we get u 11 r = w 11 r + n c 11 rh 11 r rdr + b 11 r. 1V 22 This result means that u 11 r is given by w 11 r lus the correlation terms n 1 V c 11 rh 11 r rdr+b 11 r. If we neglect the bridge function we recover u 11,HNC r in the limit case n 1 0. Thus, HNC-PY in Figs. 1 and 2 results from neglecting b 11 r. Moreover, the difference between u 11,HNC-PY r and the simulation data in those figures allows for the evaluation of b 11 r. This could be an icient method to evaluate bridge functions through the ective interactions instead of the structure functions, which involve the inversion of the OZ equation and thereby introduce very large errors at small distances. We will, however, not reort further in this direction at the moment. In very diluted systems the virial-like exansion c ij r =,,...,=0 n 1 n 2 n c,,..., ij r of the direct correlation functions can be made in Eq. 21. In the case of binary mixtures of hard sheres it leads to u 11 r = n 2 F 1 c 12 0,0 qc 21 0,0 q = n c 0,0 12 rc 0,0 21 r rdr, 2V for n 1 0, u to linear terms in n 2. Here, we used 1 n 2 c 22 q 1 =1+n 2 c 22 q+n 2 2 c 22 2 q+ and the convolution theorem for Fourier transforms. In addition, it is known that 24 c 0,0 ij r = 1forr ij = 0 for r ij. 25 Equation 23, oreq.24, can also be obtained from Eq. 13, or from Eq. 16, by taking h 11 r1 remember that we are trying with r 1 and neglecting b 11 r, as well as from Eq. 17. Following those routes we do need to take the additional aroximation c 0,1 11 r=0. Moreover, the leading terms of the bridge functions are of quadratic order in the density 24 and, therefore, their exclusion of the dilute limit in our equations is exact. The integral in Eq. 24 accounts for the volume V d of the region in the ga between the articles of secies 1, searated by the distance r, from which the articles of secies 2 are excluded due to their simultaneous overla with both articles of secies 1. This result can, therefore, be written as u 11 r= k B TN 2 /VV d. Since the term in arenthesis corresonds to the ressure of an ideal gas comosed of articles of secies 2, the ective interaction otential is equivalent to the decrement of the free energy of the ideal gas when its volume is increased by V d. This corresonds to the Asakura-Oosawa AO aroximation for the deletion forces 1,2,8. This result means that the dilute limit of our aroach catures the exact dilute limit of the deletion interaction otential. In the following sections we generalize this result to mixtures of more than two comonents, as well as to nonsherical geometries, and use the symbol u 11,AO r when working in that limit case. As we can see from Eqs. 13 and 16 21, an evaluation of u 11 r requires a comlete knowledge of the structure functions c ij r and h ij r of the original mixture. To roceed along these lines makes sense when the contraction of the descrition is imosed, for examle, by exerimental techniques unable to detect all the comonents. Then, Eqs. 13 and allow for an interretation of the results in terms of models including the exerimentally invisible secies. On the other hand, our aroach aarently makes no sense when the goal of the contraction of the descrition is to simlify the mathematical roblem. Fortunately, this is not the case. Aart from the lot we can learn about deletion forces just evaluating u 11 r in the cases in which the comlete roblem can be solved, we also find that simle aroximations for c ij r and h ij r could lead to useful exressions for u 11 r. Equations 23 and 24, for examle, do not use a recise inut for c ij r and h ij r, and they still work fine for some cases in which the contracted articles are really not diluted, as we show in the next sections. Indeed, the generation of hierarchies, like the one defined by successive contractions of the descrition, is reresentative of the usual thinking of hysicists working on many-body roblems. This allows for simle aroximations at a higher level of descrition which conduce to accurate results in a more reduced level of the hierarchy. V. BINARY AND TERNARY MIXTURES Let us be very meticulous at the beginning of this section in order to exlain the method of evaluating entroic otentials in the dilute limit. In a three-dimensional binary mixture

7 ENTROPIC FORCES IN DILUTE COLLOIDAL SYSTEMS of hard sheres the Fourier transform of c 0,0 12 r is 45 c 12 0,0 q = 4 q 0 rc 0,0 12 rsinqrdr = 4 q 3 q 12 cosq 12 sinq 12. Therefore, Eq. 23 takes the form u 11,AO r = 8n 2 q 12 cosq 12 sinq 12 2 r 0 q 5 sinqrdq, which can be analytically integrated to give u 11,AO r = r r for 1 r 1 + 2, and 0 for larger distances. Here, 2 =n /6 is the three-dimensional filling fraction of secies 2, and = 1 / 2 the size ratio. The contact value is u 11,AO + 1 = 2 1+3/2; the deletion attraction increases either with 2 or. Equation 28 can also be obtained from Eq. 24, rewritten as u 11,AO r = 2n 2 r = 2n r r 12 r+r c 0,0 12 rc21 rdrr r 0,0 rdr drr r 12 rdr, 29 or from a geometrical sketch by evaluating the volume V d of the region in the ga between the articles of secies 1 from which a article of secies 2 is excluded due to its simultaneous overla with both articles of secies 1, as done by Asakura and Oosawa 1,2. The first exression in Eq. 29 is obtained by going to olar coordinates, after integrating over the angle 46. The limits of integration in the second exression of Eq. 29 corresond to the simultaneous overlaing condition of a article of secies 2 with both articles of secies 1, as required by Eq. 25. In the secial case seen in the revious aragrah the three routes, Eqs. 23 and 24, as well as the AO rocedure, lead, more or less straightforwardly, to the analytical exression given in Eq. 28. However, the first two routs quickly lead to extremely comlicated integrals, and the AO rocedure becomes intractable, when trying with more sohisticated geometries. Nevertheless, Eq. 24 rovides a numerical scheme which can be easily imlemented in every case. By constructing an satial grid around the overlaing region, we can check the simultaneous overlaing condition in every oint of the grid. If the condition is fulfilled we add a volume element to the corresonding discretization of the integral in Eq. 24 and ass to the next oint of the grid. In the oosite case, we only ass to the next oint of the grid, and so forth. Let us now try this idea in the case of twodimensional binary mixtures of hard disks, where it is already hard to get an analytical exression for the integral involved in Eq. 23. The Fourier transform of c 0,0 12 r is 45 c 12 0,0 q =20 rc 0,0 12 rj 0 qrdr J 1 q 12 =2 12, 30 q where J m qr is the Bessel function of order m. Equation 23 takes the form u 11,AO r = 2 2 J 0 qrj 2 1 q n dq. 20 q 31 We evaluate this integral numerically. The AO rocedure allows in this case for a straightforward evaluation of V d, leading to u 11,AO r = r 1+2cos r r for 1 r 1 + 2, and 0 for larger distances. Here, 2 =n /4 is the two-dimensional filling fraction of secies 2, and = 1 / 2 the size ratio. We can also take the grid route by constructing a square grid of ste = 2 /10 in order to evaluate numerically the integral in Eq. 24 in the form described in the revious aragrah. We also took = 2 /20, but the result does not areciably differ from the one obtained with the larger ste. All routes lead to the same result, which corresonds to the dotted lines in Figs. 1 and 2. In some ractical situations it should be convenient to have simle exressions for the otential at contact. In that case, Eq. 32 can be aroximately written as u 11,AO /1.89 1/ /2 +O 3/2. This is a very good aroximation for 2. From the Figs. 1 and 2 we can see that the dilute limit only catures the interaction otential in the range going from the contact, r= 1, to the case in which a article of secies 2 exactly fit in the ga between the two articles of secies 1, r= For larger distances the double overla is not longer ossible. That is also the reason why the AO limit does not cature the reulsive barrier growing in front of the attractive contact well. This barrier can only be exlained including the ects of a second article of secies 2, which makes it ossible for the formation of overlaing bridges between the articles of secies 1 around the searation r= We will come back to this oint in Sec. IX. From Figs. 1 and 2 we can also see that the contact values of u 11,HNC r and u 11,AO r are very similar; both values considerably underestimate the deth of the deletion well at contact. However, they become very accurate for filling fractions 2 of the order of 10%, or below it this result is not shown in Figs. 1 and 2, which are values often met in exerimental

8 CASTAÑEDA-PRIEGO, RODRÍGUEZ-LÓPEZ, AND MÉNDEZ-ALCARAZ situations. In the next sections we only use the grid route based on Eq. 24 in order to evaluate u 11,AO r, but let us first work a little with ternary mixtures. U to this oint we have assumed that we can only observe the articles of secies 1 in a mixture of sherical comonents. The other secies are there, but we choose to ignore that fact. Therefore, we can describe the structure of the observed articles by means of the ective monodiserse OZ Eq. 11. Let us now assume that we can observe the articles of secies 1 and 2 in the same mixture of sherical comonents. In this new situation the contraction of Eq. 2 leads to the ective OZ equation for a binary mixture, h q = c q + 2 =1 n c qh q, 33 written here in the Fourier sace. The ective direct correlation function c q is found to be given by q = c q + c + i, i, j,,i c i qn i c i q 1 n i c ii q c i qn i c ij qn j c j q 1 n i c ii q1 n j c jj q Here, the greek indexes run over the observed secies, i.e.,,=1,2. The c 11 r from Eq. 12 does not corresond to the c 11 r from Eq. 34, since the first one is a more contracted member of the hierarchy than the latter. The first one can be obtained after the contraction of secies 2 in Eq. 33, which leads to Eq. 18, but having the structure functions of Eq. 33 as inut. By using 34 in those exressions, Eq. 12 can be recovered. To roceed along these lines allows for the generation of hierarchies defined by successive contractions of the descrition. However, in this aer we no longer reort on that question. In the secial case of ternary mixtures we can choose from the contraction of two comonents, or of only one. If we contract two comonents we get Eqs. 11 and 12 again, but with =3, which add some terms to Eq. 18. The contraction of only one comonent corresonds to the case described by Eqs. 33 and 34, in which two secies can be observed. In that case, the ective crossed interaction otential is given by 1 u 12 r = u 12 r n 3 F c 13qc 32 q 1 n 3 c 33 q, 35 if we neglect the difference between the crossed bridge functions b 12 r b 12 r. In the case of systems comosed of hard articles, in the infinite dilute limit of observable secies, n 1 0 and n 2 0, u to linear terms in the density of the contracted secies, n 3,weget u 12,AO r = n 3 F 1 c 13 0,0 qc 32 0,0 q = n c 0,0 13 rc 0,0 32 r rdr. 3V The integral in Eq. 37 accounts for the volume V d of the region in the ga between two articles of secies 1 and 2, searated by the distance r, from which a article of secies 3 is excluded due to its simultaneous overla with both articles of secies 1 and 2. Thus, the dilute limit in our equations also catures the AO aroximation in the case of ternary mixtures. In three-dimensional systems Eqs. 36 and 37 can be analytically evaluated to give 8 u 12,AO r = r r r/ r for 12 r , and 0 for larger distances. Here, i = i / 3 and = /2. This result can also be straightforwardly obtained from the AO rocedure. The contact value of Eq. 38 is u 12,AO + 12 = / Therefore, at fixed 2, the deletion attraction increases with 1 u to the asymtotic value u 12,AO + 12 = when 1. As we will see in the next section, this case corresonds to a binary mixture in front of a flat hard wall. In two-dimensional systems Eq. 36 becomes J 0 qrj 1 q 13 J 1 q 32 u 12,AO r = n dq. 30 q 39 We evaluate this integral numerically. The AO rocedure allows also in this case for a straightforward evaluation of V d, leading to 1 u 12,AO r = cos r r r cos r r r , 40 for 12 r , and 0 for larger distances. Here, r 1 = 3 /4r4r/ and r 2 =2r/ 3 r 1. We also take the grid route based on Eq. 37 by constructing a square grid of ste = 3 /10. All routes lead to the same results, which corresond to the dotted lines in Fig. 3. The deletion attraction increases with 1, at fixed 2, until reaching the asymtotic value corresonding to a flat hard wall when 1. Indeed, u 12,AO r can be interreted as the deletion interaction otential between a convex hard wall of

9 ENTROPIC FORCES IN DILUTE COLLOIDAL SYSTEMS If we assume that we can only see the wall and the articles of secies 1, the concentration rofile of this secies in front of the wall is described by the ective monodiserse inhomogeneous OZ equation 11 h w1 q = c w1 q + n 1 h w1qc 11 q, 43 with FIG. 3. The figure shows the wall-article deletion otential u w1,ao x for a two-dimensional binary mixture of hard disks in the dilute limit in our equations. The dotted lines corresond to the case of convex hard walls with scaled radius of curvature R/ 2 =5 and 10. They can also be interreted as the dilute limit of the ective interaction otential u 12,AO r in an homogeneous ternary mixture of hard disks with 2 3, 1 2, 2R 1, and x + 1 /2 r see Sec. V. The full line corresonds to the case of a hard flat wall R, and the dashed lines to the case of concave hard walls with scaled radius of curvature R/ 2 =5 and 10. The arameters of the binary mixture in front of the walls are 1 0, 2 =0.3, and 1 / 2 =5. radius of curvature 1 /2 and articles of secies 2 immersed in a bath of smaller articles of secies 3. We try further with wall ects in the next section. c wi qn i c i1 q q = c w1 q + 1 n i c ii q c w1 + i1 j1,i i1 c wi qn i c ij qn j c j1 q 1 n i c ii q1 n j c jj q + 44 and c 11 q is given by Eq. 12. The other secies are there, but we choose to ignore that fact, which is exactly what forces us to develo an ective descrition of the system. In the case of binary mixtures Eq. 44 leads to neglecting the difference b w1 r b w1 r 1 u w1 r = u w1 r n 2 F c w2qc 21 q 1 n 2 c 22 q, and, therefore, to u w1,ao r = n 2 F 1 c w2 0,0 qc 21 0,0 q VI. IN FRONT OF A HARD WALL The concentration rofile of a colloidal mixture of sherical comonents in front of a wall is described by the inhomogeneous OZ equation 24, h wi q = c wi q + n j h wjqc ji q, j=1 41 which has to be comlemented with a closure relation of the general form 24 c wi r = u wi r + h wi r ln1+h wi r + b wi r. 42 Here, h wi r is the wall-article total correlation function, c wi r the wall-article direct correlation function, u wi r the wall-article interaction otential, b wi r the wall-article bridge function, and c ij r the same direct correlation function of the bulk susension aearing in Eqs. 2 and 3. The wall-article structure functions deend on the ositional vector r and, therefore, their Fourier transforms on the wave vector q, since the wall breaks the isotroy of the system. The roerties of Eqs. 41 and 42 have been carefully studied in a series of aers ublished in the last ten years, for systems of charged and uncharged comonents We will use here their invariance under contractions of the descrition in order to get the ective wall-article interaction otential in systems comosed of hard colloidal articles in front of a hard wall. = n c 0,0 w2 rc 0,0 21 r rdr 2V 47 in systems of hard articles, in the infinite dilute limit of secies 1, n 1 0, u to linear terms in n 2. As it can be seen from Eq. 47, the AO aroximation is again catured by the dilute limit of our equations. Equation 46 is still more comlicated to evaluate than Eq. 23, due to its deendence on the wave vector. In the case of flat hard walls, however, Eq. 47 and the AO rocedure allow for a straightforward evaluation of u w1,ao r, which leads to u w1,ao x = x 2 2 in three-dimensional systems, and to u w1,ao x = tan 1 1 2x/ x/ x/ x/ in two-dimensional systems, both for 1 /2x 1 /2+ 2 and 0 for larger distances. Here, = 1 / 2 and x is the distance between the surface of the wall and the center of articles of secies 1. The contact values are

10 CASTAÑEDA-PRIEGO, RODRÍGUEZ-LÓPEZ, AND MÉNDEZ-ALCARAZ u w1,ao + 1 /2= and u w1,ao + 1 / / /2 for 2, resectively. Equations 48 and 49 can also be obtained from the limit 1, at fixed 2, of Eqs. 38 and 40, resectively. The result for the two-dimensional case is shown in Fig. 3 by the full line for a binary mixture with 2 =0.3, and 1 / 2 =5. The deletion attraction increases with the radius of curvature R of the convex walls, reaching an extreme value for flat walls R. We now consider binary mixtures of hard sheres in front of concave hard walls of radius of curvature R in the dilute limit. Although it is ossible to get an analytical exression for u w1,ao x also in this case, we refer to work in the following with the numerical evaluation of Eq. 47 by constructing a square grid of ste = 2 /10 around the double overlaing region. In the case of two-dimensional systems the latter is defined by the conditions R x2 2 + y FIG. 4. The figure shows the contact wall-article deletion otential u w1,ao x,z for a binary mixture of hard sheres, in the dilute limit, in front of a hard wall with a linear ste edge of height h=10 2 along the y axis. The uer level of the ste is in the region with x0 and z=h, on a lane arallel to the xy lane, and the lower level in the region with x0 and z=0,onthexy lane. The arameters of the binary mixture in front of the wall are 1 0, 2 =0.3, and 1 / 2 =5. VII. SURFACES OF ENTROPIC POTENTIAL x2 + x R 2 + y , 50 which have to be fulfilled simultaneously. Here, x 2,y 2 and R x,0 are the coordinates of the articles of secies 2 and 1, resectively, referred to the center of curvature of the concave wall. We check the overlaing conditions 50 in every oint of the grid. If both conditions are fulfilled simultaneously we add a volume element to the integral in Eq. 47. In the oosite case, we ass to the next oint of the grid. The results obtained by this way are shown in Fig. 3 by the dashed lines. As we can see from that figure, the more concave the wall, the larger the deletion attraction. The revious results indicate that the susended articles in front of a wall are ushed to the wall due to the deletion interaction, and that the amlitude of such interaction can be maniulated by changing the arameters of the susension, such as article size and concentration, and of the wall, such as its curvature. This occurs although we are working with hard articles and hard walls, whose interaction is only reulsive and short-ranged. The origin of the long-ranged deletion forces is that the system increases its entroy when the larger articles are laced close to the wall, letting free the volume of the bulk for the smaller articles, which are very much more numerous. Moreover, the entroy of the smaller articles may become still larger when the larger articles are ut in an ordered array. Indeed, this henomenon has been already used in order to generate selfassembled structures, as in the crystallization of rotein susensions 52 or in the selective adsortion of colloidal articles on structured walls 53,54. Regarding the latter, there is strong evidence for nonadsorion of hard sheres on flat hard walls However, as we will see in the next section, the situation looks very different when the wall has a relief attern. When the walls have a relief attern combining concave and convex regions the deletion otential at contact defines a otential landscae that we will call here surface of entroic otential. One of the simlest cases, in which we can calculate this roerty, is when an otherwise flat wall has a linear ste edge of height h along the y-axis. Let us consider a hard wall with a ste edge of height h=10 2, and, in front of it, a binary mixture of hard sheres with arameters 1 0, 2 =0.3, and 1 / 2 =5. These values are the same as in Fig. 3, but a direct comarison is not ossible since we are trying with a three-dimensional system this time. The uer level of the ste is in the region with x0 and z=h, ona lane arallel to the xy-lane, and the lower level in the region with x0 and z=0,onthexy-lane. Note that the meaning of the variable x differs from the one reviously used in this aer. In the infinite dilute limit of articles of secies 1, u to linear terms in n 2, the wall-article deletion otential is given by Eq. 47, with the aroriate overlaing conditions for c 0,0 w2 r and c 0,0 12 r. In order to evaluate this equation at contact we follow the grid route by constructing a cubic grid of ste = 2 /10 over the wall. Then, we check in every oint of the grid the condition of simultaneous overlaing of a article of secies 2 with the wall and with a articles of secies 1 always in contact with the wall. The latter is then moved along the x-axis in order to scan the region around the ste edge. The results we found for u w1,ao x,z are shown in Fig. 4. The line reresents a contact scanning of the wall, beginning on the uer level and ending on the lower level of the ste. Far away from the edge the scanning article only see a flat wall and, therefore, the value u w1,ao x0,h=u w1,ao x0,0= 4.8 can be also obtained from the contact limit of Eq. 48. Closing the edge from the left, the scanning article first feels a force oosite to its motion, arallel to the wall, and a some weaker attrac