Fluid-crystal transition

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

A second case to be considered is that of mixtures witli a small size ratio, <0.2. For a long time it was believed tliat such mixtures would not show any instability in tire fluid phase, but such an instability was predicted by Biben and Flansen [109]. This can be understood to be as a result of depletion interactions, exerted on the large spheres by tire small spheres (see section C2.6.4.3). Experimentally, such mixtures were indeed found to display an instability [110]. The gas-liquid transition does, however, seem to be metastable witli respect to tire fluid-crystal transition [111, 112]. This was confinned by computer simulations [113]. 

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 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]. 

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. 

Before moving on to our attempt to measure the complete double layer in a swollen propylammonium vermiculite with d = 43.6 A [18], we pause to note that (a) at ionic strengths relevant to cell fluids, namely c 0.12 M [19], the phase-transition temperature in the propylammonium vermiculite system is not so far away from our body temperature and (b) similar temperature-induced gel-crystal transitions are observed in many biochemical systems. An example is the deoxyhemoglobin molecule that causes sickle cell anemia [33], We also note that with both counterions, Tc decreases linearly with the logarithm of the salt concentration. 

The temperature of the hydration step should be above the gel-liquid crystal transition temperature of the lipid to allow the lipid to hydrate in its fluid phase with adequate agitation. Hydration time may differ slightly among lipid species and structures. We believe that good hydration prior to sonication makes the sizing process easier and improves the homogeneity of the preparation. 

In fluid bed granulation the feed melt concentration is 97%, but the product leaving the granulator has only 0.25% moisture. The water content of the solidifying melt must be significantly bwer than the possible water of hydration of the added magnesium nitrate (Mg (N03)2 6-7 H2O). The additive thus serves as an internal desiccant agent and shifts the 32 C crystal transition point to about 45 C. 

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... 

Only a narrow range of concentrations is found where fluid-crystal phase transition occurs. This can be explained by the fact that the depth of the potential is quite significant for small q it is about 4-5 kT just across the phase boundary. In retrospect it is even surprising that fluid-crystal phase transitions are observed at low volume fractions of colloids at small q for that reason. 

In 2000 Katayama et al. [35] conducted an in situ X-ray dillraction experiment on liquid phosphorus which strongly suggested a first order liquid-liquid phase transition between a molecular liquid and a polymeric liquid. This was particularly notable because not only did it occur in the stable liquid, as opposed to the more conunon metastable supercooled liquid, it also demonstrated coexistence of the two phases. This was clearly visible from a weighted sum of the diffraction patterns from either side of the transition compared to the diffraction pattern at the transition (Fig. 2.9). The coexistence was also apparent from in situ X-ray radiography [36] and had been suggested by ab initio MD simulations [54], However a further, more extensive, in situ X-ray diffraction investigation by Monaco et al. [53] established that the transition was in fact between a polymeric liquid and a molecular fluid the latter being the fluid form associated with the metastable white phosphorus crystal, which takes a molecular tetrahedral structure. Therefore while undoubtedly a first order liquid-fluid phase transition, phosphorus did not provide the first evidence for a first order liquid-liquid phase transition. 

The inclusion of cholesterol disturbs the crystalline structure of the gel phase, and the phospholipid chains are more mobile than in its absence. This prevents the crystallization of the hydrocarbon chains into the rigid crystalline gel phase. In the more fluid liquid crystalline phase, the rigid cholesterol molecules restrict the movement of the hydrocarbon chains. In consequence, the addition of cholesterol to lipid bilayers or lamellar mesophases gradually diminishes the gel-liquid crystal transition temperature and the enthalpy and broadens the DSC transition peak [72,73]. No transition can be detected by DSC at 50% cholesterol [73,74] (curve/of Fig. 7), which is the maximum concentration of cholesterol that can be incorporated before phase separation. However, laser Raman spectroscopic studies show that a noncooperative transition occurs over a very wide temperature range [75]. 

Like the fluid to crystal transition in the case of spherical colloidal particles, the isotropic-to-nematic transition in the case of rod-like particles is strongly influenced by the electrostatic repulsion between the particles. Onsager indicated that the effect of the electrostatic repulsion will be equivalent to an increase of the effective diameter of the particles. However, the electrostatic repulsion also depends on orientation and thus the... 

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