Fluid-crystal coexistence

3 Phase Transitions of Hard Spheres Plus Depletants Basics 

Hard spheres freezing melting Bernal glass 

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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... 

Figure C2.6.10. Phase diagram of colloid-polymer mixtures polymer coil volume fraction jiri n vs particle volume fraction ( ). (a) Narrow attractions, 5/a = 0.1. Only a fluid-crystal transition is present. Tie lines indicate coexisting phases, (b) Longer range attractions, 5/a = 0.4. Gas, liquid and crystal phases (G, L and C) are present, as well as a critical point (CP). The three-phase triangle is shaded (reproduced with permission from [99]. Copyright 1992 EDP Sciences).

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. 

Fig. 3.1 The pressure of hard spheres. The curves are the Carnahan—Starling exjaession (3.1) for a fluid < 0.494) and the cell model result (3.12) for an fee crystal (solid curves, (j) > 0.545). The closed symbols are Monte Carlo computer simulation results [13]. The two open symbols cmrespond to the fluid-solid coexistence from simulation [11], the dotted line is the themetical result (see Sect. 3.2.3)...

In Fig. 4.3 we also plot the (equilibrium) binodals using FVT outlined in Chap. 3 for hard spheres plus penetrable hard spheres with diameters of 2Rg. Qualitatively, the phase diagram topology is quite well predicted. For q = 0.08, only equilibrium fluid, crystal and fluid + crystal regions are found and predicted. Both for q = 0.57 and 1 the phase diagram contains fluid, gas, liquid and crystalline (equilibrium) phases. In the different unmixing regions one now finds gas-liquid coexistence with a critical point, three-phase gas-liquid-crystal and... 

So far, reference has only been made to equilibrium systems, but, as discussed earlier, at very high particle volume fractions, non-equilibrium states may appear, in this case glass states. Similarly, when systems are quenched rapidly into a two-phase, fluid-solid coexistence region, the equilibrium state for the solid phase ought to be a soMd crystal, but very frequently colloidal gels are formed instead. Understanding such non-equilibrium behaviour is one of the challenges in modern colloidal science. 

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]. 

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

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]. 

Zone two can be defined by the absence of montmorillonite and by the tie-line mica-opal (Figure 37). Zone one, which contains montmorillonite shows the coexistence of feldspar and montmorillonite (Figure 37a). Trona and halite found in the sediments are considered to indicate higher alkalinity and alkali content of the pore fluids that effected the crystallization of the feldspar "facies" in zone two at the lake center. Here the evaporated fluids became more concentrated. 

The type of crystalline structure that is formed depends on the concentration of the particles as well as the magnitude of the Debye-Hiickel thickness. For large Debye-Hiickel thicknesses a body-centered cubic crystal is formed, whereas for smaller values a face-centered cubic crystal is preferred. An example of the latter observed experimentally in a dispersion of latex spheres is shown in Figure 13.3. Note that this crystallization phenomenon is analogous to crystallization of simple atomic fluids, as is evident from Figure 13.3a, which shows the coexistence of a crystal with a liquidlike structure. 

Thermodynamic considerations. A rigorous thermodynamic analysis shows that empirical rules which consider bonding forces of ions in crystalline phases alone are invalid. It is necessary to compare binding forces of ions in a mineral and the medium from which that mineral crystallized. For transition elements, this requires information about relative CFSE s of the cations in coexisting minerals, silicate melts, aqueous solutions and hydrothermal fluids. 

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