Binary hard spheres

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

Oykstra M, van Roi] R and Evans R 1999 Direot simulation of the phase behaviour of binary hard-sphere mixtures test of the depletion potential desoription Phys. Rev. Lett. 82 117-20... 

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

R. Dickman, P. Attard, V. Simonian. Entropic forces in binary hard sphere mixtures. J Chem Phys 707 205-213, 1997. 

T. Biben, P. Bladon, D. Frenkel. Depletion effects in binary hard-sphere fluids. J Phys Condens Matter 2.T0799-10821, 1996. 

The direct correlation function is k dependent and is calculated explicitly through the binary hard sphere expression (86)... 

The theory described above is applied to the binary hard-sphere system with the size ratio (Jijci = 0.2 and the concentration of small particles ci = 0.5. Figure 1 shows the diffusion constants D, of small (s = 1) and big (s = 2) particles versus the total packing fraction 9. The dotted lines show the power-law fit D = f o 9 — fj with 7=1.31 and 2.36 for small and big particles, respectively. The diffusion constant of the big particles is found to vanish at 95 =0.52, which is close to the liquid-glass transition point (9=0.516) of a one-component hard-sphere system, while Di becomes zero at 9 =0.53 (> 9b). This means that for 9b < 9 < 9x there exists a new phase (delocalized phase) with mobile small particles diffusing through the voids of a glassy structure formed by the immobile big particles. 

In this section we will briefly review the collision model for binary hard-sphere collisions using the notation in Fox Vedula (2010). The change in the number-density function due to elastic hard-sphere collisions (Boltzmann, 1872 Cercignani, 1988 Chapman Cowling, 1961 Enksog, 1921) obeys an (unclosed) integral expression of the form ... 

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

DFT studies of binary hard-sphere mixtures predate the simulation studies by several years. The earliest work was that of Haymet and his coworkers [221,222] using the DFT based on the second-order functional Taylor expansion of the Agx[p]- Although this work has to some extent been superceded, it was a significant stimulus to much of the work that followed both with theory and computer simulations. For example, it was Smithline and Haymet [221] who first analyzed the Hume-Rothery rule in the context of hard sphere mixture behavior and who first investigated the stability of substitutionally ordered solid solutions. The most accurate DFT results for hard-sphere mixtures have come from the WDA-based theories. In particular the results of Denton and Ashcroft [223] and those of Zeng and Oxtoby [224] give qualitatively correct behavior for hard spheres forming substitutionally disordered solid solutions. 

A critical review of these results as well as several other novel approaches to theories of mixtures evaluated for mixtures of hard spheres has been presented. by Salsbuig and Fickett. These authors also compare the various theories numerically particularly for the diameter ratio 5/3 used by Smith for his preliminary (low density) Monte Carlo calculations of binary hard sphere mixtures. Salsburg and Fickett conclude for these systems that... 

The hard sphere interaction causes a hquid to sohd transition at a solution volume fraction of 0.545, as described above with leferenee to Figure 3.4. The resulting sohd has an fee lattice. This is shghtly, about 0.001 kT, more stable than its elose packed counterpart at 0.74, the hep lattice. Binary hard sphere systems can form five different lattices of different stoichiometry depending on the size ratio. 

The PY equation for a binary hard sphere liquid has been considered by Lebowitz (1964). Lebowitz obtains the Laplace transforms of rg (r) exactly, and these have been inverted and evaluated numerically by Throop and Bearman (1965). However, contact between theory and experiment is most readily established in q space and so, following Enderby and North (1968), we return to an intermediate step in Lebowitz s argument in which the direct correlation functions are rigorously derived. These are as follows ... 

P p2) using a binary hard-sphere model and obtained P p2)l uW which agrees well with the measurement. The data in Fig. 4 thus demonstrate that the binary hard-sphere model" can indeed describe the depletion effect in our colloid/PEP mixture. 

In order to close these expressions for particulate pressures, we also need equations for the variance of total particle volume concentration in an assemblage of particles belonging to the two different types. For an arbitrary polydisperse particulate pseudo-gas, variances of partial volume concentrations for different particles can be evaluated on the basis of the thermodynamical theory of fluctuations. According to this theory, these variances are expressible in terms of the minors of a matrix that consists of the cross derivatives of the chemical potentials for particles of different species over the partial number concentrations of such particles [39]. For a binary pseudo-gas, these chemical potentials can be expressed as functions of number concentrations using the statistical theory of binary hard sphere mixtures developed in reference [77]. However, such a procedure leads to a very cumbersome and inconvenient final equation for the desired variance. To simplify the matter, it has been suggested in reference [76] to ignore a slight difference between this variance and the similar quantity for a monodisperse system of spherical particles of the same volume concentration. This means that the variance under question may be approximately described by Equation 7.4 even in the case of binary mixtures. 

In 1964 Lebowitz and Rowlinson [6] showed that, within the Percus-Yevick treatment of hard sphere fluids [7], binary hard sphere mixtures are completely miscible for aU concentrations and size ratios. This proof was later extended by Vrij [8] to hard sphere mixtures with an arbitrary number of components. Up till 1990, it was indeed generally beUeved that hard sphere mixtures do not phase... 

B. B. Laird and A. D. J. Haymet, Calculation of the entropy of binary hard sphere mixtures from pair correlation functions. J. Chem. Phys., 97, 2153,1992. 

Example We consider the case of e classical ideal gas which consists of particles of mass m, with the only interaction between the particles being binary hard-sphere collisions. The particles are contained in a volume 2, and the gas has density n = N/Q. In this case, the energy e, of a particle, with momentum p and position r, and its multiplicity gi are given by... 

Kozina A, Sagawe D, Diaz-Leyva P, Bartsch E, Palberg T (2012) Polymer—enforced crystal-hzation of a eutectic binary hard sphere mixture. Soft Matter 8 627-630... 

RESULTS FOR COMPLEX SYSTEMS 5.1 Binary Hard Spheres... 

Figure 5 Density profiles for the binary hard sphere fee [100] crystal-melt interface plotted on a fine scale. The dashed, vertical line is the Gibbs dividing surface defined in Section 2.2. The dotted grid is commensurate with the lattice planes in the bulk crystal and is included to better visualize the expansion of the lattice constant in the interfacial region. (Reprinted by permission of the American Institute of Physics from Davidchack and Laird )...

The effect of a structured surface on the crystallization of hard-sphere colloids has been extensively studied in experiments [87, 88, 89, 90], These experiments indicate that crystallization on a template is induced at densities below freezing. This finding is supported by computer simulations of hard spheres in contact with a patterned substrate, by Heni and Lowen [91], These simulations indicate that surface freezing already sets in 29% below the coexistence pressure. Furthermore the effect of a surface on crystallization has also been studied in mixtures of binary hard-spheres [92, 93] and colloid-polymer mixtures [94, 95, 96], In both systems surface crystallization was found to take place before bulk fluid-solid coexistence. In the systems studied in Refs. [92, 93, 94, 95, 96], depletion forces favor the accumulation of the larger component on the wall, and this should facilitate surface crystallization [97]. 

Foffi, G., Gotze, W., Sciortino, R, Tartaglia, R, and Voigtmann, Th. 2004. a-relaxation processes in binary hard-sphere mixtures. Phys. Rev. E 69 011505. 

FIGURE 8.1 Phase diagram of binary hard spheres with a small-to-large size ratio of 0.82. The phase diagram is shown in the composition x, reduced pressure p, representation, where x = N /iN + Nj) is the number fraction of small spheres. The labels Tccl and fccs denote the fee crystals of large and small particles, respectively. (From Hynninen AP et al. 2007. Nature Materials 6 202-205. With permission.)... 

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