Asymmetric mixtures

As discussed in Sections 7.2.2 and 7.2.3, the studies of asymmetric mixtures are still quite scarce and so far restricted to few types of asymmetry and 

---

The MS closure results from s = 2. The HNC closure results from s = 1. In the latter two expressions, additional adjustable parameters occur, namely ( for the RY closure and for the BPGG version of the MS approximation. However, even when adjustable, these parameters cannot be chosen at will, as they should be chosen such that they eliminate the so-called thermodynamic inconsistency that plagues many approximate integral equations. We recall that a manifestation of this inconsistency is that there is a difference between the pressure as computed from the virial equation (10) and as computed from the compressibility equation (20). Note that these equations have been applied to a very asymmetric mixture of hard spheres [53,54]. Some results of the MS closure are plotted in Fig. 4. The MS result for y d) = g d) is about the same as the MV result. However, the MS result for y(0) is rather poor. Using a value between 1 and 2 improves y(0) but makes y d) worse. Overall, we believe the MS/BPGG is less satisfactory than the MV closure. 

Antiphase boundary (APB) conservative vacancy segregation at Arrhenius plot Asymmetrical mixtures Atomic-sphere approximation (ASA) ASA-LSDA... 

Figure 18. Different stages of the spinodal decomposition in an asymmetric mixture (0 = 0.5) t is the dimensionless time. The Euler characteristic is initially negative, which indicates that morphology is bicontinuous. After a certain time the Euler characteristic becomes positive, which indicates that the transition to dispersed morphology occurred. For a dispersed morphology the Euler characteristic equals twice the droplet number.

The above model assumes that both components are dynamically symmetric, that they have same viscosities and densities, and that the deformations of the phase matrix is much slower than the internal rheological time [164], However, for a large class of systems, such as polymer solutions, colloidal suspension, and so on, these assumptions are not valid. To describe the phase separation in dynamically asymmetric mixtures, the model should treat the motion of each component separately ( two-fluid models [98]). Let Vi (r, t) and v2(r, t) be the velocities of components 1 and 2, respectively. Then, the basic equations for a viscoelastic model are [164—166]... 

A shortcoming of the van der Waals classical mixing rules is that they are not applicable to the so-called asymmetric mixtures and to mixtures containing polar compounds. For that reason, different mixing rules have been proposed for the a-parameter, involving essentially a concentration dependence of the ky [36-39]. 

CO -benzene, and CO -n-decane. The critical densities and the corresponding compositions are plotted in Figure 1. The three hydrocarbons in order of higher to lower solubility in C0 were heptane, benzene, and decane. The measured binary diffusion coefficients or the decay rates of the order-parameter fluctuations at various temperatures and pressures are listed in Tables I, II, and III for CO -heptane, CO -benzene, and CO -decane systems respectively. In Figure 2, the critical lines of the three binary systems in the dilute hydrocarbon range are shown in the pressure-temperature space. dP/dT along the critical lines of CO.-heptane and CO -benzene systems are similar and lower than dP/dT along the critical line of CO -decane system, which indicates that C02 and decane form more asymmetric mixtures relative to CO with heptane or benzene. 

High-Pressure Vapor-Liquid Equilibria from a Corresponding States Correlation with Emphasis on Asymmetric Mixtures," I EC Process Design and Development, 1978, 17, 324. 

The view of this chapter is that a colloidal suspension is an extremely asymmetric mixture of large and small particles and that this mixture can be treated by the standard and well-developed theory of fluids. 

Plocker, U., Knapp, H. and Prausnitz, J. (1978) Ind. Eng. Chem. Proc. Des. and Dev. 17, 243. Calculation of high-pressure vapour-liquid equilibria from a corresponding-states correlation with emphasis on asymmetric mixtures. 

A two-parameter mixing rule is used with several cubic equations of state and is shown to be relatively successful in correlating the phase equilibrium behavior of biomolecules that cannot be correctly represented by conventional one-parameter mixing rules. The modification is related to the idea of local composition, which has been shown to improve the representation of the phase equilibrium in asymmetric mixtures. However, further improvement is still needed. 

This approach is very general. For example, it is not restricted to monodis-perse systems, and Krauth and co-workers have applied it successfully to binary [17] and polydisperse [18] mixtures. Indeed, conventional simulations of size-asymmetric mixtures typically suffer from jamming problems, in which a very large fraction of all trial moves is rejected because of particle overlaps. In the geometric cluster algorithm particles are moved in a nonlocal fashion, yet overlaps are avoided. 

J. G. Malherbe and S. Amokrane (1999) Asymmetric mixture of hard particles with Yukawa attraction between unhke ones a cluster algorithm simulation... 

For moderately asymmetric mixtures of nonpolar components, such as the mixture of methane with n-pentane, the conventional van der Waals mixing rules with no interaction parameter provide a sufficient description, and the EOS-G models... 

Panagiotopoulos, A. Z., and Reid, R. C., 1986. New mixing rules forcubic equations of state for highly polar asymmetric mixtures. ACS Symposium Series 300 American Chemical Society, Washington, D.C., pp. 571-582. 

Parameter will henceforth be referred to as the asymmetry of the model mixture, where xb > 1 characterizes a binary mixture in which the formation of B-B pairs is energetically favored, whereas for XB < L this is the case for A-A pairs. For the special case xb 1 the asymmetric mixture degenerates to the symmetric case previously studied in Refs. [84] and [85]. In addition, we define the selectivity of the solid surfaces by specifying Xs in Eq. (4.125d) in a fashion similar to xb in Eq. (4.125c). Hence, the parameter space of our model is spanned by the set , aBi s, Xb> X s -... 

Voutsas, E.C., Kalospiros, N.S., and Tassios, D.P., A combinatorial activity coefficient model for symmetric and asymmetric mixtures, Eluid Phase Equilibria, 109, 1, 1995. 

For strongly asymmetric mixtures (e.g., mixtures where the A-chains are stiff while the B-chains are flexible) the semi-grandcanonical approach is clearly not feasible, and one must work in a canonical ensemble where both the number of A-chains nA and the number of B-chains nB are fixed. However, the finite size scaling ideas for PL(M) as exposed above still can be exploited if one considers the order parameter M in L x L subsystems of a much larger system [267]. The usefulness of this concept was demonstrated earlier for Ising models and Len-nard-Jones fluids [268-271]. Gauger and Pakula [267] find an entropy-driven phase separation without any intermolecular interactions. 

Another important characteristic of the late stages of phase separation kinetics, for asymmetric mixtures, is the cluster size distribution function of the minority phase clusters n(R,T)dR is the number of clusters of minority phase per unit volume with radii between R and R + dR. Its zeroth moment gives the mean number of clusters at time x and the first moment is proportional to the mean cluster size. 

Separation of L-Mandelic Acid from Asymmetric Mixtures by Means of High-Pressure Crystallization... 

H. L. Friedman and C. V. Krishnan, Charge-asymmetric mixtures of electrolytes at low ionic strength, J. Phys. Chem. 78,1927 (1974). 

The SPH method provides an efficient way for the numerical simulations of the phase-separation phenomena in polymer blends. For instance, Okuzono used this approach to simulate a specific type of phase separation - the so-called viscoelastic phase separation - experimentally found in polymer solutions and dynamically asymmetric mixtures. Examining the effect of stress relaxation time on morphology of domains, it was shown that the more viscous phase forms network-like domains when the stress relaxation time is large. 

---