Perfect fractional crystallization

There are two ways of imagining a process of perfect fractional crystallization. 

Figure 4.8 Fraction of amorphous polyethylene as a function of time for crystallizations conducted at indicated temperatures (a) linear time scale and (b) logarithmic scale. Arrows in (b) indicate shifting curves measured at 126 and 130 to 128°C as described in Example 4.4. [Reprinted with permission from R. H. Doremus, B. W. Roberts, and D. Turnbull (Eds.) Growth and Perfection of Crystals, Wiley, New York, 1958.]...

The first theoretical considerations concerning n (p) and G (p) of concentrated 3-D emulsions and foams were based on perfectly ordered crystals of droplets [4,5,15-18]. In such models, at a given volume fraction and applied shear strain, all droplets are assumed to be equally compressed, that is, to deform affinely under the applied shear thus all of them should have the same shape. Princen [15,16] initially analyzed an ordered monodisperse 2-D array of deformable cylinders and concluded that G = Qiox(p < (/), and that G jumps to nearly the 2-D Laplace pressure of the cylinders at the approach of ( > = 100%, following a ( — dependence. 

Semiconductors like silicon or germanium are an intermediate case. Their electrons are not as tightly bound as in insulators so that at any given time a small fraction of them will be mobile. In a perfect germanium crystal, for instance at 25°C, about 3 x 1019 electrons per m3 are free. This corresponds to a concentration of 5 x 10-8 M or 50 nM. It is much lower than the concentration of charge carriers (cat- and anions) in an aqueous electrolyte solution. Despite this small concentration, the conductivities are of the same order of magnitude, because the electrons in a semiconductor are typically 108 times more mobile than ions in solution. 

These two equations are plotted in Fig. 25, from which it is seen that in a perfectly mixed crystallizer the dominant-size fraction (mode) appears at l/L = 3. Saeman illustrated his analysis with data from large-scale vacuum crystallization of ammonium nitrate (M9, SI). 

So-called superlattices of 5mn alkylthiolate protected silver particles having truncated octahedral shapes as well as thiol-stabilized 5 6 mn gold particles can be obtained from solution. Fractional crystallization is a very usual method to separate chemical compounds from other compounds and impurities. Mixtures of thiolate stabilized gold nanoparticles between 1.5 and 3.5 nm could successfully be fractioned into real monodisperse species containing 140, 225, 314, and 459 atoms. 2D assemblies have also become available of these fcc-structured nanoparticles. The decisive criterion to successhilly fraction and crystallize metal nanoparticles is to protect them perfectly by strongly bound ligand molecules in order to avoid coalescence. 

Figure 5 Calculated fractional crystallization trends for group IIIAB iron meteorites using several different types of models. The simple fractional crystallization models assume a perfectly mixed liquid whereas the other four models assume different types of imperfect mixing (reproduced by permission of the Meteoritical Society from Meteorit.

The Avrami model (19,20) states that in a given system under isothermal conditions at a temperature lower than V. the degree of crystallinity or fractional crystallization (70 as a liinction of time (t) (Fig. 11) is described by Equation 5. Although the theory behind this model was developed for perfect crystalline bodies like most polymers, the Avrami model has been used to describe TAG crystallization in simple and complex models (5,9,13,21,22). Thus, the classical Avrami sigmoidal behavior from an F and crystallization time plot is also observed in TAG crystallization in vegetable oils. This crystallization behavior consists of an induction period for crystallization, followed by an increase of the F value associated with the acceleration in the rate of volume or mass production of crystals, and finally a metastable crystallization plateau is reached (Fig. 11). 

In a somewhat similar manner i-xylene can be separated from a mixture of m- and /i-xylene this binary system forms a eutectic. Carbon tetrachloride produces an equimolecular solid compound with /i-xylene, but not with 0- or m-xylene. Egan and Luthy (1955) reported on a plant for the production of pure -xylene by crystallization meta-para- xylene mixtures in the presence of carbon tetra-chloride. Up to 90 per cent of the para- isomer was recovered by distillation after splitting the separated solid complex. The meta- isomer was recovered by fractionally crystallizing the CCU-free mother liquor. Perfect separation of /i-xylene is not possible, because the ternary system CCU/m-xylene/CCU -xylene forms a eutectic, but fortunately the concentration of the complex CCI4 /i-xylene in this eutectic is very low. Several commercial clathration processes for the separation of m-xylene from Cg petroleum reformate fractions using a variety of complexing agents have been operated (Sherwood, 1965). 

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. 

Due to excluded volume and optimal space-filling constraints, these numbers need to be scaled by about q/N jox, where q is the effective coordination number (which is, e.g., q 10.0 for a perfect fee crystal). For large M, the relative fraction rjnter of the inter-chain... 

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