Two- and three-phase equilibria in polydisperse Yukawa hard-sphere mixture. High temperature and mean spherical approximations

Phase behavior of the Yukawa hard-sphere polydisperse mixture with high degree of polydispersity is studied using high temperature approximation (HTA) and mean spherical approximation (MSA). We have extended and applied the scheme developed to calculate the phase diagrams of polydisperse mixtures described by the truncatable free energy models, i.e., the models with Helmholtz free energy defined by the finite number of the moments of the species distribution function. At high degree of polydispersity, several new features in the topology of the two-phase diagram have been observed: the cloud and shadow curves intersect twice and each of them forms a closed loop of the ellipsoidal-like shape with the liquid and gas branches of the cloud curve almost coinciding. Approaching a certain limiting value of the polydispersity index, the cloud and shadow curves shrink and disappear. Beyond this limiting value, polydispersity induces the appearance of the three-phase equilibrium at lower temperatures. We present and analyze corresponding phase diagrams together with distribution functions of three coexisting phases. In general, good agreement was observed between predictions of the two different theoretical methods, i.e., HTA and MSA. Our results confirm qualitative predictions for the three-phase coexistence obtained earlier within the framework of the van der Waals approach.


Introduction
Much of the efforts have been recently focused on the investigation of the effects of polydispersity on the phase behavior of the colloidal and polymeric fluids. Polydispersity is an intrinsic property of the colloidal and polymeric systems, since each of the particle is unique in its shape, size, charge, chain length, etc. In theoretical description, polydisperse fluids are usually treated as a system consisting of an infinite number of species with a species index ξ (polydispersity attribute) changing continuously [1]. Such a system is characterized by the distribution function of species f (ξ), with f (ξ)dξ being the fraction of the particles with the value of polydispersity attribute ξ in the range [ξ, ξ + dξ]. Several different theoretical methods for the phase behavior description of polydisperse fluids have been developed and used recently . Majority of the earlier studies are based on the van der Waals (vdW) [2, 4-6, 10, 17, 18] or Onsager [6,8,12,14] levels of description. Although such a description appears to be relatively successful in giving qualitative predictions for the phase behavior of polydisperse mixtures, it is not capable of providing quantitatively accurate results. The major challenge for a theoretical description of the phase behavior of polydisperse mixtures is due to the functional dependence of Helmholtz free energy of the system A on the distribution function f (ξ). As a result, phase equilibrium conditions are formulated in terms of the set of an infinite number of equations, e.g. chemical potential µ(ξ) for each value of the continuous species index ξ should be the same in each of the coexisting phases. Solution of such a set of equations for Helmholtz free energy of arbitrary form is next to impossible. However, there is a large number of systems that can be successfully described using the so-called truncatable free energy (TFE)

The model
We consider the fluid with interparticle interaction represented by the following multi-Yukawa hardsphere potential

Theory
Thermodynamic properties of the model at hand will be described here using HTA and MSA. Both theories have been used earlier [9,21] to study the phase behavior of polydisperse colloidal mixtures with the pair potential of the type specified above (2.1). Therefore, we will present here only the final expressions for the quantities needed in our phase equilibrium calculations. For more details of the HTA and MSA approaches we refer the reader to the original publications.

High temperature approximation
According to the HTA, Helmholtz free energy of the system A can be written as a sum of two terms: free energy of the reference system (A ref ) and the perturbation term describing the contribution to the free energy due to Yukawa potential (A 1 ): Here, A ref = A HS , where A HS is the free energy of the hard-sphere fluid, and for A 1 we have:
Here, we use Percus-Yevick approximation for the hard-sphere radial distribution function, since analytical expressions for its Laplace transform is known. All the rest of thermodynamical quantities can be obtained using the expression for Helmholtz free energy (3.1) and standard thermodynamic relations, e.g., differentiating A with respect to the density we get the expression for the chemical potential: , (3.4) and the expression for the pressure P of the system can be calculated invoking the following general relation: In the above expressions, A HS and µ(ξ) (HS) are calculated using the corresponding Mansoori et al. expressions [27].
Within the framework of the HTA approach, the model in question belongs to the class of 'truncatable free energy (TFE) models', i.e., the models with thermodynamic properties (Helmholtz free energy, chemical potential, pressure) defined by a finite number of generalized moments. In this study, we have the following moments:

Phase equilibrium conditions
We assume that at a certain temperature T , the system, which is characterized by the parent density ρ (0) and parent-phase distribution function f (0) (ξ) separates into q daughter phases with the densities ρ (1) , ρ (2) , . . . , ρ (q) , and q daughter distributions f (1) (ξ), f (2) (ξ), . . . , f (q) (ξ). Phase equilibrium conditions imply the equality of the pressure P and chemical potential µ(ξ) in each of q phases, i.e., Here, [ f (ξ)] denote the functional dependence of the corresponding quantity on the distribution function f (ξ). The phase separation is constrained by the conservation of the total number of the particles of each species ξ and by the conservation of the total volume occupied by the parent phase, i.e., is the ratio of the total number of the particles in the phase α to the total number of particles in the parent phase, υ (α) = 1/ρ (α) , (α = 1, 2, 3, . . . , q). Finally, the normalization of the distribution function f (α) (ξ) in equation (3.22) implies the conservation of the total number of the particles:  [2,[4][5][6]. In this study, we will follow the scheme developed by Xu and Baus [2,4,5] and present a set of equations (3.20)-(3.25) for two-and three-phase equilibria in terms of generalized moments (3.6)-(3.8). For polydisperse Yukawa hard-sphere mixture with size and interaction energy polydispersity, within the framework of the HTA thermodynamics of the system depends on the set of 9 generalized moments (3.6)-(3.8) and within the framework of MSA we have 10 generalized moments (B.2)-(B.5) and one MSA scaling parameter Γ (see Appendices I and II, respectively). In what follows we will conventionally denote these moments as m k and use the following definition: Note, that this set includes also the total number density of the system, i.e., m 0 (ξ) = 1 and m 0 = ρ.

Two-phase coexistence
In the case of the two-phase equilibrium (α = 1, 2), conditions (3.20)-(3.25) lead to the following set of equations in terms of the generalized moments m k : with k = 1, . . . , 8 for HTA and k = 1, . . . , 9 for MSA. In the latter case, this set of equations is supplemented by the following two equations for the scaling parameters Γ (1) and Γ (2) (3.31)

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Here, m (q) denote the set of all moments of the phase q, ex is the value of the excess chemical potential (to its ideal gas value) in the phase q. Solution of this two-phase problem [equations (3.27), (3.30) and (3.28), (3.29)], for a given temperature T , the density of the parent phase ρ (0) , and the parent species distribution function f (0) (ξ) gives the coexisting densities ρ (α) of the two daughter phases and corresponding species distribution functions f (α) (ξ), α = 1, 2.
The coexisting densities for different temperatures give binodals, which are terminated at a temperature for which the density of one of the phases is equal to the density ρ (0) of the parent phase. These termination points form the cloud and shadow coexisting curves, which intersect at the critical point, with the critical temperature T cr and the critical density ρ cr = ρ (1) = ρ (2) = ρ (0) . The cloud and shadow curves can be obtained as a special solution of the coexisting problem, when the properties of one phase are equal to the properties of the parent phase: assuming that the phase 2 is the cloud phase, i.e., ρ (2) = ρ (0) , and following the above scheme we will end up with the same set of equations, but with ρ (2) and f (2) (ξ) substituted by ρ (0) and f (0) (ξ), respectively.

Three-phase coexistence
In the case of three-phase equilibrium, the phase coexistence conditions (3.20)-(3.25) are reduced to the following set of equations: Here, Solution of this set of equations for a given temperature T , density of the parent phase ρ (0) , and parent species distribution function f (0) (ξ) gives the coexisting densities ρ (α) of the three daughter phases and the corresponding species distribution functions f (α) (ξ), α = 1, 2, 3.

Results and discussion
In this section we present numerical results for the phase behavior of polydisperse Yukawa hardsphere mixture. To complete description of the system, we have chosen log-normal distribution for the species distribution function f (ξ) of the parent phase, i.e., where I is the polydispersity index. Log-normal distribution is often used to describe polydisperse fluids [29,30]. In the monodisperse limit (I = 1), this distribution is represented by the Dirac delta-function, δ(ξ − 1). On the contrary, when I becomes very large (I ≫ 1), the above distribution becomes very wide, increasing the contributiion to the phase behavior of the particles with a large value of ξ. All calculations were carried out for the one-Yukawa version of the pair potential (2.1) with z = 1.8σ 0 . We consider polydispersity only in the strength (amplitude) of the pair potential A(ξ). In this case, we have chosen A(ξ) = A 0 ξ and σ(ξ) = σ 0 . Here, A 0 = 1 and σ 0 is the hard-sphere size for a monodisperse version of the model at I = 1, which is used as a distance unit. In what follows, the density ρ and temperature T are presented in reduced units, i.e., ρ * = ρσ 3 0 and T * = k B T /ǫ 0 .

Two-phase coexistence
We calculate the phase diagrams of the polydisperse Yukawa hard-sphere mixture at different values of polydispersity index I using HTA and MSA. In figure 1 (the upper panel), we show the liquid-gas phase diagram of the system in the limiting case of I = 1. In this case, the system is monodisperse, and the corresponding cloud and shadow curves coincide with binodals, and the critical point is located at their maximum. In addition, in the same figure we present the phase diagram obtained using the computer simulation method [31,32]. As one would expect, the agreement between computer simulations and theoretical results is reasonably good, with MSA predictions being a bit more accurate.
In further increase of polydispersity, several novel features appear. For small polydispersity, the system has only one critical point [9,11,15,16,21], which originates from the regular liquid-gas (LG) critical point of the corresponding monodisperse version of the system. With the polydispersity increase, there appears an additional critical point, which is induced by the polydispersity ( figure 2). This effect on the qualitative van der Waals level of description has been observed by Bellier-Castella et al. [2,4]. The second critical point, which we denote as polydisperse (P) critical point, is located at larger values of the density and at lower values of the temperature. With the appearance of the second critical point, the cloud and shadow curves intersect twice and each of them forms a closed loop of the ellipsoidal-like shape. With an increase of polydispersity, liquid and gas branches of the cloud curve approach each other. Here, the gas branch of the cloud curve has a negative slope, which indicates that under these conditions the system may show a reentrant phase behaviour.
In figure 2 we present the phase diagrams obtained by HTA and MSA theories, which include two critical points, cloud and shadow curves and critical binodals.

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Two-and three-phase equilibria Both approximations provide quantitatively close predictions for the phase behavior of the model. Polydispersity does not substantially change the relation between MSA and HTA results, i.e., HTA predicts slightly higher critical temperature and slightly lower critical density. We assume that similarly as in monodisperse case (figure 1) HTA is less accurate than MSA. Note that HTA describes correlations between the particles on the level of the reference system (in this study hard-sphere system) neglecting contribution from the attractive part of the potential to the structure.

Three-phase fractionation
As can be seen in subsection 4.1, at relatively large values of polydispersity index I in the phase behavior of our system, there appear several novel features, which are absent at lower values of I . In particular, at a limiting value of I (I = 1.032), the critical points disappear. This appears to be a sign for the possibility of the three-phase equilibria. For the system with the limiting value of polydispersity I = 1.032 with the temperature decrease at a certain temperature, the system is fractionated into three coexisting phases. We have studied this transition at three different values of polydispersity index I , i.e., I = 1.032, 1.036, 1.039. In figure 3 we show the phase diagrams of the system at the set values of I obtained using HTA (full line) and MSA (dotted line). T * (1) (I ) denotes the borderline between the one-phase and two-phase regions at polydispersity index I , T * (2) (I ) is the borderline between the two-phase and three-phase regions.
As can be seen in figure 3  phase.

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As noted above, the polydisperse system can be viewed as a mixture of an infinite number of components, each characterized by a continuous variable ξ. Thus, the Gibbs phase rule allows for a phase fractionation into, say q phases. Let the polydisperse system be fractionated into the (q − 1)-phase region with defined borderlines T * (1) . With both, the polydispersity index I and q increase with decreasing T * (q−1) (I ) , the (q)-phase region is fractionated. Note that in the monodisperse limit (I = 1), our system has only a one-phase (for T * > T * cr ) region and a two-phase (for T * < T * cr ) region.
We show how an initial log-normal parent-phase distribution, f (0) (ξ), fractionates into three daughter phases. The degree of fractionation depends on the temperature, and these three daughter distributions,  figure 4, the daughter distributions are reasonably well approximated by log-normal distributions. For low temperatures, larger differences are observed.
In general, both HTA and MSA give very close predictions for the three-phase behavior of the model at hand. The only noticeable difference is due to the upper value of the temperature range of the two-phase equilibrium stability.

Conclusions
In this paper we have studied the phase behavior of the Yukawa hard-sphere polydisperse mixture at a high degree of polydispersity using two theoretical methods: HTA and MSA. We confirm the appearance of the second critical point induced by polydispersity, which was observed earlier using qualitative van der Waals level of description [2,4]. With a further increase of polydispersity, several new features in the topology of the two-phase diagram emerge: the cloud and shadow curves intersect twice and each of them forms a closed loop of the ellipsoidal-like shape with liquid and gas branches of the cloud curve almost coinciding. Upon approaching a certain limiting value of the polydispersity index I (I = 1.032), the 23603-11 cloud and shadow curves shrink and disappear. We have studied the phase behavior of the model beyond this value of polydispersity. At lower values of the temperature and I 1.032, polydispersity induces the system to fractionate into three coexisting phases with different densities: low, intermediate and high. We present and analyze the corresponding phase diagrams together with distribution functions of three coexisting phases. In general, a good agreement was observed between predictions of the two different theoretical methods, i.e., HTA and MSA. Our results confirm the earlier qualitative predictions for the three-phase coexistence obtained in the framework of the van der Waals approach [4].