The Hard-Sphere Fluid-Crystal Transition

The hard sphere fluid-crystal transition plays an important role as a reference point in the development of theories for the liquid and solid states and their phase behaviour [10]. We consider it in some detail in the next section here the phase behaviour is relatively simple as there is no gas-liquid (GL) coexistence. After that we discuss the phase behaviour under the influence of the attraction caused by the depletion interaction now there is such GL transition. We illustrate the enrichment of the phase behaviour in the somewhat hypothetical system consisting of an assembly of hard spheres and (non-adsorbing) penetrable hard spheres. 

Following the work of Wood and Jacobson [7] and Alder and Wainwright [8], the location of the hard sphere fluid-crystal transition was determined from computer simulations by Hoover and Ree [11]. They found that the volume fractions of the coexisting fluid (f) and face centered cubic crystal ( ) are given by (f)f = von = 0.494 and = vqm = 0.545 at a coexistence pressure Pvo/kT = 6.l2. Here Vo = 4n/3)R, with R the radius of the hard sphere, is the hard sphere volume. As in Chap. 2, n = N/V refers to the number density of N particles in a volume V. 

We present a simple theoretical treatment of the hard sphere fluid-crystal transition that will also serve as a reference framework for our treatment of phase transitions in a system of colloids with depletion attraction. 

We start with the equation of state for the fluid phase of hard spheres interacting through (1.19). An accurate expression for hard spheres is the so called Camahan-Starling equation of state [12] which can be written in terms of the dimensionless pressure Pf as 

In Fig. 3.1 (left part) we compare the pressure given by the Carnahan-Starling equation of state (3.1) with computer simulations. We see that (3.1) is indeed very accurate. 

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In Fig. 4.3 state diagrams are plotted that were measured by Poon et al. [16, 17] for three size ratios q = Rg/R = 0.08,0.57 and 1. Here p is the polymer concentration relative to overlap, see (1.24). At 0polymer concentrations the mixtures appear as single-state fluid phases. At zero polymer content the hard-sphere fluid-crystal phase transition is found when the colloids occupy about half of the volume. Upon addition of polymer the fluid-crystal coexistence region expands for = 0.08 then a colloidal fluid at smaller volume fraction... 

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). 

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

Fig. 21. Entropy versus log-temperature diagram for the hard-sphere model. The solid curves give the computer simulation values for the supercooled fluid, glass, and crystal. The dashed curves have the following bases (a) a calculation from the virial equation using the known first seven coefficients and higher coefficients obtained from the conjectured closure (the plot corresponds quite closely with that calculated from the Camahan-Starling equation ) and (i>) an extrapolation of higher temperature behavior such as that used by Gordon et al., which implies a maximum in the series of virial coefficients. The entropy is defined in excess of that for the ideal gas at the same temperature and pressure. Some characteristic temperatures are identified 7, fusion point 7 , upper glass transition temperature T/, Kauzmann isoentropic point according to closure virial equation.

Experimentally, the hard-sphere phase transition was observed using non-aqueous polymer lattices [79, 80]. Samples are prepared, brought into the fluid state by tumbling and then left to stand. Depending on particle size and concentration, colloidal crystals then form on a time scale from minutes to days. Experimentally, there is always some uncertainty in the actual volume fraction. Often the concentrations are therefore rescaled so freezing occurs at ( )p = 0.49. The width of the coexistence region agrees well with simulations [11, 80]. 

The phase behaviour of such colloidal suspensions should be nearly the same as those of the hypothetical hard-sphere atomic system. Kirkwood [6] stated that when a hard sphere system is gradually compressed, the system will show a transition towards a state of long-range order long before close-packing is reached. In 1957, Wood and Jacobson [7] and Alder and Wainwright [8] showed by computer simulations that systems of purely repulsive hard spheres indeed exhibit a well-defined fluid-crystal transition. It has taken some time before the fluid-crystal transition of hard spheres became widely accepted. There is no exact proof that the transition occurs. Its existence has been inferred from numerical simulations or from approximate theories as treated in this chapter. However, this transition has been observed in hard-sphere-Uke colloidal suspensions [9]. 

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. 

Crystallization occurs for many common fluids, such as carbon tetrachloride, benzene, and cyclohexane, at pressures less than 200 MPa, thus their entire fluid range is rather limited. The supercritical noble gases significantly extend this range, achieving a maximum crystallization pressure for helium of P = 11.6 GPa. The viscosity increase prior to all such transitions is modest. The viscosity for a typical dense fluid is 1-100 mPa and this will increase by, at most, about three orders of magnitude. Experimentally, this viscosity and pressure regime is covered by many of the viscometers discussed below, and hard-sphere theory can explain most of the viscosity increase. 

Figure 2 Phase diagram of a fluid of parallel hard spherocylinders. Note that L is the length of the cylinders i.e., L = 0 corresponds to spheres. Here p is the number density divided by the number density at close packing. Top Monte Carlo simulation. (From Ref. 21.) Shaded areas indicate phase coexistence. The dashed line indicates a continuous transition. Bottom theoretical phase diagram of Ref. 19 resulting from the comparison of the free energies for = 3 (nematic), d = 2 (smectic), d = 1 (columnar), and d = 0 (crystal). Shaded areas again indicate phase coexistence. A subsequent revision of the Monte Carlo phase diagram was published in Ref. 22 replacing the columnar phase with a phase of a different symmetry.

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