Perfect equilibrium crystallization

The rate (or kinetics) and form of a corrosion reaction will be affected by a variety of factors associated with the metal and the metal surface (which can range from a planar outer surface to the surface within pits or fine cracks), and the environment. Thus heterogeneities in a metal (see Section 1.3) may have a marked effect on the kinetics of a reaction without affecting the thermodynamics of the system there is no reason to believe that a perfect single crystal of pure zinc completely free from lattic defects (a hypothetical concept) would not corrode when immersed in hydrochloric acid, but it would probably corrode at a significantly slower rate than polycrystalline pure zinc, although there is no thermodynamic difference between these two forms of zinc. Furthermore, although heavy metal impurities in zinc will affect the rate of reaction they cannot alter the final position of equilibrium. 

The higher the pressure over equilibrium, the higher the diamond nucleation and growth rate and the smaller and less perfect the crystal. Lower synthesis temperatures favor cubes and higher ones, octahedra. Suitable control of these variables permits the growth of selected types of... 

Fig. 9. Two-parameter ordering phase diagram for a system of 500 identical hard spheres (Truskett et ai, 2000 Torquato et ai, 2000). Shown are the coordinates in structural order parameter space (r, ) for the equilibrium fluid (dot-dashed), the equilibrium FCC crystal (dashed), and a set of glasses (circles) produced with varying compression rates. Here, r is the translational order parameter from (26) and is the bond-orientational order parameter Q( from (25) normalized by its value in the perfect FCC crystal ( = Each circle...

Polymer crystallization has been described in the framework of a phase field free energy pertaining to a crystal order parameter in which = 0 defines the melt and assumes finite values close to unity in the metastable crystal phase, but = 1 at the equilibrium limit (23-25). The crystal phase order parameter (xj/) may be defined as the ratio of the lamellar thickness (f) to the lamellar thickness of a perfect polymer crystal (P), i.e., xlr = l/P, and thus it represents the linear crystallinity, that is, the crystallinity in one dimension. The free energy density of a polymer blend containing one crystalline component may be expressed as... 

In addition to the ambiguities inherent to the physical concept, the determination of thermodynamic quantities such as the latent heat and the volume change at the transition is often hampered by the fact that the crystalline state of chain molecules is quite complex. The polymer crystals are usually polycrystalline and coexist with the disordered amorphous domain. An accurate estimation of the equilibrium melting temperature defined for a perfectly aligned crystal requires great effort [5,18,19]. At the melting temperature, equilibrium usually exists between the liquid and somewhat imperfect crystalline phases. 

State of perfect (or nearly perfect) equilibrium. Moreover the separation or crystallisation gives rise to an establishment of mutual equilibrium between the two liquid layers, or between the crystals and saturated solution. 

The equilibrium crystals just discussed had to be grown in the hexagonal condis phase area of the phase diagram (Fig. 5.156) to achieve the chain extension by subsequent annealing of the initially grown chain-folded crystals. On crystallization from the melt in the orthorhombic phase area, the initially produced lamellar crystals also do not correspond to the crystals that one melts in a later experiment, although the crystal perfection is less than in the condis phase area. Figure 6.23 illustrates the... 

Maintaining perfect equilibrium while cooling is one end of a complete spectrum of possibilities. The other end of the spectrum is that crystals form, but always completely out of equilibrium. This end of the spectrum involves an infinite number of cases and so is rather difficult to discuss in a finite number of words. A subset of these possibilities is the case where crystallization produces crystals in equilibrium with the liquid, as required by the diagram, but after forming, they do not react with the liquid in any way. This is called surface equilibrium (because the liquid is at all times in equilibrium with the surface of the crystals) or fractional crystallization, and is a model process just as much as is equilibrium crystallization. It is also used in connection with liquid-vapor processes (fractional distillation fractional condensation), as well as isotope fractionation processes. 

The melting enthalpy of perfect polymer crystals, AH, can be obtained using DSC and a hot stage equipped with a polarizing microscope, when the equilibrium melting point of the sample is known. The procedure for the measurement of AHf is recommended as follows ... 

The melting enthalpy of perfect polymeric crystals can be measured by DSC. When crystalline polymers absorb a small amount of the solvent, the equilibrium melting temperature of the pure polymeric crystals is depressed from to... 

If the substance in the state of interest is a hquid or gas, or a crystal of a different form than the perfectly-ordered crystal present at zero kelvins, the heating process will include one or more equilibrium phase transitions under conditions where two phases are in equilibrium at the same temperature and pressure (Sec. 2.2.2). For example, a reversible heating process at a pressure above the triple point that transforms the crystal at 0 K to a gas may involve transitions from one crystal form to another, and also melting and vaporization transitions. 

We obtain the same result if a species present in one phase is totally excluded from another. For example, solvent molecules of a solution are not found in a pure perfectly-ordered crystal of the solute, undissociated molecules of a volatile strong acid such as HCl can exist in a gas phase but not in aqueous solution, and ions of an electrolyte solute are usually not found in a gas phase. For each such species absent from a phase, there is one fewer amount variable and also one fewer relation for transfer equilibrium on balance, the number of independent variables is still 2-1-5. 

The perfectly ordered crystal is one with the lowest free enCTgy. Since a certain amount of lattice disorder can be tolerated at equilibrium, it does not necessarily represent the crystal with pafect intmial orda-. 

Let us briefly address the different periods and begin with the nucleation step. In principle, crystallization can start at the equilibrium melting point i.e. the temperature where the chemical potentials of a monomer in the melt, and in a perfect infinite crystal composed of extended chains, are equal... 

When such tables became widely available they were used to compute equilibrium constants for reactions whose equilibrium constants were known experimentally. Most agreed very well, but there were some discrepancies, indicating that for some substances % r=0 k was not 0.00, but some small but significant value. Analysis of this problem led to the conclusion that for the substances that formed perfectly regular crystals at 0 K, % r=o k was indeed 0.00, but for substances that could have some randomness in the crystal it was not. The easiest example to visualize is CO. This is a linear molecule, with the atoms at either end more or less the same size. In a perfect crystal the molecules would be aU lined up head to tad, i.e., CO-CO-CO-. However, the difference between the two ends is small enough that when CO freezes there is some randomness in the orientation, for example, CO-CO-OC-CO-OC-CO-OC. This introduces randomness—imperfection—into the crystal, so that its entropy at 0K>0. 

An edge dislocation in such a. structure is shown schematically in Fig. 3.4, where the plane of molecules shown contains the slip direction and is normal to the slip plane, the trace of which is AB. Figure 3.4(a) shows the molecular arrangement in the perfect, undeformed crystal. Since the crystal structure provides a periodic field of force, molecules will slip from one equilibrium position to the next under a shear stress, so that the Burgers vector will be... 

Systems involving an interface are often metastable, that is, essentially in equilibrium in some aspects although in principle evolving slowly to a final state of global equilibrium. The solid-vapor interface is a good example of this. We can have adsorption equilibrium and calculate various thermodynamic quantities for the adsorption process yet the particles of a solid are unstable toward a drift to the final equilibrium condition of a single, perfect crystal. Much of Chapters IX and XVII are thus thermodynamic in content. 

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. 

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