The Liquid-Solid Transition

The simplest model for an atomic assembly is to consider the atoms as hard spheres with a radius a. Computer simulations have been used to describe the physical behaviour of such assemblies as the number density is changed. At low number densities the assembly is a fluid and the hard spheres diffuse in a gaseous fashion. There are three degrees of freedom corresponding to kBT(2 for each orthogonal translational direction. At intermediate densities the motion of an individual sphere becomes more complex. Some of the time it will move inside a transient cage of 

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Patterns of Relaxation Near the Liquid-Solid Transition. 170... 

Viscoelastic Properties at and Around the Liquid-Solid Transition. . 189... 

Fig. 4. Schematic of the divergence of the longest relaxation time as the liquid-solid transition is approached from either side...

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

This most simple model for the relaxation time spectrum of materials near the liquid-solid transition is good for relating critical exponents (see Eq. 1-9), but it cannot be considered quantitatively correct. A detailed study of the evolution of the relaxation time spectrum from liquid to solid state is in progress [70], Preliminary results on vulcanizing polybutadienes indicate that the relaxation spectrum near the gel point is more complex than the simple spectrum presented in Eq. 3-6. In particular, the relation exponent n is not independent of the extent of reaction but decreases with increasing p. 

The liquid-solid transition for these systems seems to have the same features as for chemical gelation, namely divergence of the longest relaxation time and power law spectrum with negative exponent. 

The effective hard sphere diameter may be used to estimate the excluded volume of the particles, and hence the low shear limiting viscosity by modifying Equation (3.56). The liquid/solid transition of these charged particles will occur at... 

Let us first consider the liquid-solid phase transformation. At the melting point (or more appropriately, fusion point for a solidification process), liquid and solid are in equilibrium with each other. At equilibrium, we know that the free energy change for the liquid-solid transition must be zero. We can modify Eq. (2.11) for this situation... 

Although this article is mainly concerned with the liquid/solid transition temperature and the effect of increasing temperature on the tensile properties of the solid state, we recognize the importance of studies on the effect of increasing temperature on the viscosity of the liquid plastisol. The latter studies, which indicate the well known initial lowering of viscosity followed by a rapid rise in viscosity, have an important bearing on the use of PVC pastes for such applications as rota-... 

Examination of Cast Film. The film can now be examined both before and after stripping from foil. A simple test was devised to determined the transition point from liquid to nonliquid state. A palette knife was drawn along the foil from the end where some of the plastisol was still liquid. Plastisol flowed easily in front of the knife until it came to a point where the plastisol was no longer spreadable. At this point, which was sharply defined, it was impossible to push the blade further. The temperature at which this occurs has been called the liquid/solid transition point (Figure 5). 

The experimental data do not indicate the existence of a critical point for the liquid-solid transition. 

In 1949, Turnbull and Fisher extended to the liquid-solid transition the earlier work of Becker and Doring on homogeneous nucleation of the gas-liquid transition. Their work presumed the existence in the liquid of a steady-state distribution of small crystallites which were taken to be approximately spherical in shape. The free energy of a crystallite containing i atoms was assumed to have the form... 

The natural first question to ask is whether the crystal-liquid surface free energy can be measured experimentally by some method that is independent of nucleation kinetics. In gas-liquid nucleation studies, for example, it is routine to measure the surface tension of the liquid and to use its equality with the gas-liquid surface free energy to make predictions of nucleation rates and compare them with experiment. For the liquid-solid transition, the situation is quite different, however. This is true first because the surface tension and the surface free energy are no longer strictly equal due to the possible existence of strains in the crystal. The second reason is that measurements of liquid-solid free energies or interfacial tensions are by no means simple to devise or carry out, and so are available only in certain special cases. These limited experimental data are summarized in this section. 

Gunton subsequently extended Lunger sfield theory of nucleation to the liquid-solid transition, within the context of the density functional approach of Harrowell and Oxtoby. In this way they were able to calculate the preexponential factor in the nucleation and obtained results in agreement with qualitative expectations from the classical theory. 

The paucity of direct exjjerimental information on energy transfer for the liquid-solid transition precludes one from drawing any general conclusions, but the unity of the observations (Table IV) for systems as different as V - V, T transfer in H2/D2 mixtures and HCl in solution in xenon, which show essentially the same relaxation time in liquid or high-temp>erature solid, tempt sjyeculation, all the more as comparable results are obtained in glasses or adsorbates. ... 

One of the problems with VMC is that it favors simple states over more complicated states. As an example, consider the liquid-solid transition in helium at zero temperature. The solid wave function is simpler than the liquid wave function because in the solid the particles are localized so that the phase space that the atoms explore is much reduced. This biases the difference between the liquid and solid variational energies for the same type of trial function, (e.g. a pair product form, see below) since the solid energy will be closer to the exact result than the liquid. Hence, the transition density will be systematically lower than the experimental value. Another illustration is the calculation of the polarization energy of liquid He. The wave function for fully polarized helium is simpler than for unpolarized helium because antisymmetry requirements are higher in the polarized phase so that the spin susceptibility computed at the pair product level has the wrong sign ... 

In membrane filtration, water-filled pores are frequently encountered and consequently the liquid-solid transition of water is often used for membrane pore size analysis. Other condensates can however also be used such as benzene, hexane, decane or potassium nitrate [68]. Due to the marked curvature of the solid-liquid interface within pores, a freezing (or melting) point depression of the water (or ice) occurs. Figure 4.9a illustrates schematically the freezing of a liquid (water) in a porous medium as a fimction of the pore size. Solidification within a capillary pore can occur either by a mechanism of nucleation or by a progressive penetration of the liquid-solid meniscus formed at the entrance of the pore (Figure 4.9b). 

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