Hard spheres phase separation

Thus g is negative at low densities as is the molecular excess volume = d jdp. A high density estimate of g is obtained by (1) assuming that the triplet correlation function of any particles 1, 2,3 is such that particles 2 and 3 are uncorrelated except for the fact that they may not overlap, and (2) the use of Eqs. (68) and (45) for the pair correlation function on contact of the hard spheres of diameter 1. This approximate computation indicates that the sign of g (and v ) undergoes an inversion as the density increases. This implies that a mixture of hard spheres will separate into two phases at sufficiently high p and low T, e.g., the predicted miscibility gap for p = 0.1 occurs for n 1.1. ... 

Biben T and Flansen J P 1991 Phase separation of asymmetrio binary hard-sphere fluids Phys. Rev. Lett. 66 2215-18... 

The first molecular dynamics simulation of a condensed phase system was performed by Alder and Wainwright in 1957 using a hard-sphere model [Alder and Wainwright 1957]. In this model, the spheres move at constant velocity in straight lines between collisions. All collisions are perfectly elastic and occur when the separation between the centres of... 

T. Biben, J.-P. Hansen. Phase separation of asymmetric binary hard-sphere fluids. Phys Rev Lett (5(5 2215-2218, 1991. 

As in previous theoretical studies of the bulk dispersions of hard spheres we observe in Fig. 1(a) that the PMF exhibits oscillations that develop with increasing solvent density. The phase of the oscillations shifts to smaller intercolloidal separations with augmenting solvent density. Depletion-type attraction is observed close to the contact of two colloids. The structural barrier in the PMF for solvent-separated colloids, at the solvent densities in question, is not at cr /2 but at a larger distance between colloids. These general trends are well known in the theory of colloidal systems and do not require additional comments. 

Figure 4. SEC calibration curve for silica sol separation (hard sphere particles, single pore size column) (1), Column PSM-1500 (8.9 fim), 30 X 0.78 cm mobile phase O.IU Na.HPO -NaH.PO, pH 8.0...

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

S. Sanyal, N. Easwear, S. Ramaswamy, and A.K. Sood Phase Separation in Binary Nearly-Hard-Sphere Colloids Evidence for the Depletion Force. Europhys. Lett. 18, 107 (1993). 

Second, since the entire enterprise is constmcted so as to locate points (of phase equilibrium) at which the free-energy difference vanishes, in NIRM one is inevitably faced with the task of determining some very small number by taking the difference between two relatively large numbers. This point is made more explicitly by the hard-sphere data in Table I. One sees that the difference between the values of the free energy [37] of the two crystalline phases is some four orders of magnitude smaller than the separate results for the two phases, determined by ESM. Of course one can see this as a testimony to the remarkable care with which the most recent recent ESM studies have been carried out [34]. Alternatively, one may see it as a strong indicator that another approach is called for. 

Recent experimental studies (1-3), on systems of sterically stabilized colloidal particles that are dispersed in polymer solutions, have highlighted the role played by the free polymer molecules. These experiments are particularly relevant because the systems chosen are model dispersions in which the particles can be well approximated as monodisperse hard spheres. This simplifies the interpretation of the data and leads to a better understanding of the intcrparticle forces. DeHek and Vrij (1, 2) have added polystyrene molecules to sterically stabilized silica particles dispersed in cyclohexane and observed the separation of the mixtures into two phases—a silica-rich phase and a polystyrene-rich phase—when the concentration of the free polymer exceeds a certain limiting value. These experimental results indicate that the limiting polymer concentration decreases with increasing molecular weight of... 

FIG. 14 Phase diagram of a system of hard-spheres between two parallel walls. In the three-dimensional limit (/c -> oo) the system is fluid-like for 3D < 0.5. When the walls separation is comparable to the particles size (/c 1) the system can undergo disorder-order phase transition. Adapted from Chavez-Paez et al. [39]. 

One interesting consequence of these equations, which we will call the Henderson-Chan (HC) formulae, is that they predict phase separation when X2 is small [21]. Lebowitz and Rowlinson [22] showed that Eqs. (37) and (41) predicted that fluid hard sphere mixtures were miscible under all circumstances. This prediction has been unchallenged until recently. 

One interesting feature of the HC formulae is that phase separation is predicted when there is a small concentration of large spheres [21]. There is some earlier numerical data from integral equations [52-54] indicating that phase separation occurs in asymmetric hard sphere mixtures. This may shed light on recent experimental results [55-65] who have observed evidence of phase instability in colloidal suspensions. 

We fitted two parameters in this equation, namely dus and the product eflOir to achieve the best agreement with the Bender EOS in the pressure range up to 300 MPa, which approximately corresponds to the maximal smoothed density in the pore of 1 nm width at SO MPa in the bulk phase. In the case of supercritical fluids it is not possible to evaluate Efl- and Off separately because we cannot invoke additional information such as the surface tension. By this reason the hard sphere diameter was taken equal to the collision diameter. The parameters fitted for 298.IS K are listed in Table 1. 

For the adhesive hard-sphere model, the theoretical phase diagram in the Tg-0 plane has been partially calculated (Watts et al. 1971 Barboy 1974 Grant and Russel 1993). According to this model, there is a critical point r. c = 0.0976 below which the suspension is predicted to phase separate into a phase dilute in particles and one concentrated in them (see Fig. 7-4). The particle concentration at the critical point of this phase transition is 0c = 0.1213. This phase transition is analogous to the gas-liquid transition of ordinary... 

Figure 7.4 Phase diagram for adhesive hard spheres as a function of Baxter temperature rg. The solid line is the spinodal line for liquid-liquid phase separation (the dense liquid phase is probably metastable), the dot-dashed line is the freezing line for appearance of an ordered packing of spheres, and the dashed line is the percolation transition. (Adapted from Grant and Russel 1993, reprinted with permission from the American Physical Society.)...

---