Phase separation in binary hard-core mixtures: An exact result.

We show that certain lattice models for a binary mixture of hard particle mixtures can be mapped onto a one-component lattice gas or, equivalently, onto an Ising model. The repulsive interaction between unlike particles in the mixture leads to an attractive nearest-neighbour interaction in the one-component lattice gas. In particular, we have found a lattice model of a binary hard-core mixture that maps onto the Ising model with nearest neighbor interactions. The existence of a phase transition in the Ising model provides a direct proof of the occurence of a rst-order, entropy driven demixing transition in the hard-core mixture on a lattice. The same mapping can be extended to a lattice polymer in solution. The athermal polymer-solvent mixture now maps onto the Flory-Huggins lattice model. This result leads to a very simple interpretation of the entropic contribution to the interaction parameter in the Flory-Huggins theory. 2 In a recent Letter, Biben and Hansen [1] have used an approximate analytical theory for dense uid mixtures to show that asymmetric binary hard-sphere uids should phase separate if the sizes of the two spheres are su ciently dissimilar (typically, if the size ratio is less than 0.2). In this Letter we show how to construct simple lattice models of binary hard-core mixtures. By transforming to a grand-canonical ensemble, we can map these binary hard-core mixtures onto one-component lattice gases with attractive nearest neighbor interactions. This mapping is of interest because one component lattice gases can, in turn, be mapped on Ising-like models for which the phase behavior is known exactly. To illustrate our approach, rst consider a trivial model-system, namely a square lattice with at most one particle allowed per square. Apart from the fact that no two particles can occupy the same square, there is no interaction between the particles. For N sites, the grand-canonical partition function is: =X fnig exp[ Xi ni] (1) The sum is over all allowed sets of occupation numbers fnig and is the chemical potential. Next, we include \small" hard particles that are allowed to sit on the links between the large particles (see g 1). Small particles are excluded from the edges of a square that is occupied by a large particle. For a given con guration fnig of the large particles, one can then exactly calculate the grand canonical partition function of the small particles. Let M = M (fnig) be the number of free spaces accessible to the small particles. Then clearly: small(fnig) = M Xl=0 M !zl s l!(M l)! = (1 + zs)M(fnig); (2) where zs is the fugacity of the small particles. M can be written as: M (fnig) = 2N 4Xi ni + X ninj: (3) The second sum is over nearest neighbor pairs and comes from the fact that when two large particles touch, one site is doubly excluded. The total partition function for the \mixture" is: mixture =X fnig exp[( 4 log(1 + zs))Xi ni + [log(1 + zs)]X ninj] (4) This is simply a one-component lattice gas-Ising model with a shifted chemical potential and an e ective nearest neighbor attraction with a well strength log(1+ zs)= . As is well known, this lattice model can again be transformed to 3 a 2-D Ising spin model that can be solved in the zero eld case [2, 3]. In the language of our mixture model, no external magnetic eld means: (1 + zs)2 = zl; (5) where zl = exp , the large particle fugacity. The order-disorder transition in the Ising model then corresponds to phase separation in the language of our model. This demixing is purely entropic, just like the transition predicted by [1] for the hard sphere mixture. In fact, the mapping described above can also be carried through when energetic interactions between the large particles are included. However, for the sake of clarity, we will restrict ourselves to athermal hard-core mixtures. Of course, there is a wide variety of lattice models for hard-core mixtures that can be mapped onto one-component systems with e ective attraction. one can de ne all sorts of other models on which the two types of particles reside. The model discussed above is only special in the sense that it can be mapped onto a model that is exactly solvable. One important question that is raised by the work of Biben and Hansen [1] is whether the demixing transition is of the uiduid or the uid-solid type. The phase transition in the square-lattice model that we discussed above provides no answer to this question, as there is no distinction between \liquid" and \solid" in a lattice-gas on a square, or simple-cubic lattice. There are, however, slightly more complex lattice models that do have a distinct solid and uid phase. An example is a mixture of large and small hard-hexagons on a triangular lattice. This model can be solved exactly in the limit that only small or large hexagons are present. In the latter case, Baxter [4] has shown that the system undergoes a uid-solid transition. To our knowledge, the phase behavior of the mixture cannot be computed analytically. However, we have performed preliminary computer simulations on this model that show a clear demixing transition. Thus far, however, we have not found evidence of a uiduid transition in this system. Yet, from the above mapping, it is immediately obvious that a uiduid transition does, in fact, occur in another hard-core mixture, namely that of a mixture of hard-core monomers and polymers on a lattice. To this end, we consider, once again our model of a mixtures of large and small hard squares (cubes, in 3D) on a square (cubic) lattice, i.e. the model that could be mapped onto the 1-component lattice gas with nearest neighbor interactions. We now construct \polymers" by connecting N large squares (cubes), while the solvent is represented by the small particles. A grand-canonical summation over all con gurations of the small particles, yields a very simple expression for the partition function of the polymers, namely Zpolymers =X fnig exp[J=kBT X ninj] (6) with J kBT log(1 + zs) and where the sum is over all acceptable (i.e. nonoverlapping) con gurations of the hard-core polymers. However, eqn. 6 is pre4 cisely the expression for the partition function of the Flory-Huggins latticemodel [5]. This model has been studied extensively both using approximateanalytical theories, in particular the famous Flory-Huggins theory and mod-i cations thereof (for a critical review see [6]) and, more recently, by directnumerical simulations (see e.g. [7] ). However, in those cases, the coupling con-stant J was interpreted as a purely energetic interaction, whereas the model thatwe consider is completely athermal. We can now translate the results that havebeen obtained for the Flory-Huggins lattice-gas model directly to our hard-coremixture. In particular, the existence of a rst-order uiduid phase separationin this model, provides direct proof that a purely entropic demixing transi-tion exists in an hard-core polymer-solvent mixture. Finally, we note that theFlory-Huggins theory for polymer solutions yields the following (approximate)expression for the free-energy a polymer solution:Fmix=kBT = =N log + (1 ) log(1 ) + (1 )(7)where is the fraction of the volume occupied by polymer, while the parameteris related to the coupling constant J of the original lattice model, by=If, as was assumed in the original Flory-Huggins theory, J is due to energeticinteractions, then should vary as 1=T . However, in the present (extreme) in-terpretation of the same lattice-gas model, the parameter would be completelyindependent of temperature. In fact, there is a large body of experimental datathat shows that, for many polymer solutions, has a large, if not dominant,temperature-independent part. This would suggest that excluded volume ef-fects do indeed contribute considerably to . It should be noted that, althoughour mixture model maps onto the Flory-Huggins model, it is not identical at amicroscopic level. In particular, we keep the chemical potential rather than thedensity of the solvent constant. It is, in fact, well known (see e.g. [6]) that formixtures of a polymer with a compressible solvent, the parameter is no longerpurely energetic. Equation 6 provides an intuitively clear explanation for thise ect.5 References[1] T. Biben and J.P. Hansen, Phys.Rev.Lett.66, 2215 (1991)[2] T.D. Lee and C.N. Yang, Phys. Rev.87, 410 (1952)[3] L. Onsager, Phys. Rev.65, 117 (1944)[4] R.J. Baxter \Exactly Solved Models in Statistical Mechanics", AcademicPress, London, (1982)[5] P.J.Flory, Principles of Polymer Chemistry, Cornell University Press,Ithaca (1953).[6] K.F. Freed and M.G. Bawendi J.Phys.Chem, 93, 2194 (1989)[7] K. Kremer and K. Binder, Computer Physics Reports, 7, 259 (1988)6 Figure 1: Lattice gas with two sizes of hard particles. The overlappingcon gurations are not allowed.