Disorder-order transition, hard-spheres

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

FIGURE 11.10 Gibbs free energy, G, versus osmotic pressure, II, for a suspension of hard spheres showing the intersection of the disordered and the ordered curves corresponding to the disorder-order transition with 4a-a%/3 = 0.74. Adapted from Cast et al. [63]. Reprinted with permission Academic Press. 

This is strong evidence for assuming that dispersions of ideal hard spheres would be expected to show a transition in the viscous behaviour between

short time selfdiffusion coefficient Z)s. This still shows a significant value after the order-disorder transition. The problem faced by the rheologist in interpreting hard sphere systems is that at high concentrations there is... 

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]. 

A liquid state theory has been developed on the basis of an ideal liquid, which is a hard-sphere liquid. Usually, thus, a random disordered structure of liquid has been assumed. This is the basis for the description of liquid by the two-body density correlator, or the radial distribution function g r). Recent studies indicate this picture is not sufficient even for a hard-sphere liquid [46,47], The assumption of a disorder structure of a liquid is always correct as the zeroth order approximation. However, we believe that a physical description beyond this is prerequisite for understanding unsolved fundamental problems in a liquid state, which include thermodynamic and kinetic anomalies of water type liquids, liquid-liquid transition, liquid glass transition, and crystal nucleation. 

Although the thermodynamic analysis of weak flocculation and colloidal phase separation, given above, illustrates the basic principles, some of the details are incorrect, in particular for more concentrated dispersions. One missing feature is the prediction of an order/disorder transition in hard sphere dispersions (for which Vmin is 0), where, at equilibrium, a colloidal crystal phase is predicted to coexist with a disordered phase over a narrow range of particle volume fractions (ip), that is, 0.50 < tp < 0.55 (Dickinson, 1983). In molecular hard-sphere fluids this is known as the Kirkwood-Alder transition , and is an entropy-driven effect. 

Hard-sphere colloids are defined as particles that repel only when in contact. When the volume fraction ( ) (a ratio between the volume of all colloidal particles and the total volume of the sample) of hard-sphere colloidal suspensions increases to 4> = 0.494, the system starts to form crystals and completely crystallizes at 4> = 0.545 (Figure 13.4). Between ( ) = 0.494 and ( ) = 0.545, the system is in the coexistence phase. Thus, at ( ) = 0.545, there is a transition from a disordered arrangement of particles, similar to a liquid, to a crystalline ordered packing. Colloidal crystals possess long-range order that results in beautiful opalescence. 

Bresme F, Vega C, and Abascal JLF. 2000. Order-disorder transition in the solid phase of a charged hard sphere model. Physical Review Letters 85 3217-3220. 

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