Crystal nucleation theory

However, application of nucleation theory to extended and chain-folded crystals did not reveal consistent results in that the ere values as determined by equation (3a) are very different and the ratio of surface free energy of folded to extended chain crystals was about 12. This ratio size was inconsistent with other values determined for the melting points of the various crystals. Nucleation theory when applied to the growth rates of the extended chain crystals implied that multiple surface nucleation, i.e., Kyi of equation (3a) was involved, but when applied to the growth rates of folded chain crystals at high supercooling, uni-molecular nucleation, i.e., Ky was involved,... 

Following crystal nucleation theory, consider the nucleation of a small crystal from the melt [34,36]. The random coils in the melt gather to form a small nucleus of a size a b l, as shown in Figure 5.9. We define the surface energy of the ab plane as and that of the al plane as <7s.The change in free energy A4> for crystallization from the melt is ... 

Qualitative examples abound. Perfect crystals of sodium carbonate, sulfate, or phosphate may be kept for years without efflorescing, although if scratched, they begin to do so immediately. Too strongly heated or burned lime or plaster of Paris takes up the first traces of water only with difficulty. Reactions of this type tend to be autocat-alytic. The initial rate is slow, due to the absence of the necessary linear interface, but the rate accelerates as more and more product is formed. See Refs. 147-153 for other examples. Ruckenstein [154] has discussed a kinetic model based on nucleation theory. There is certainly evidence that patches of product may be present, as in the oxidation of Mo(lOO) surfaces [155], and that surface defects are important [156]. There may be catalysis thus reaction VII-27 is catalyzed by water vapor [157]. A topotactic reaction is one where the product or products retain the external crystalline shape of the reactant crystal [158]. More often, however, there is a complicated morphology with pitting, cracking, and pore formation, as with calcium carbonate [159]. 

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

The visible crystals that develop during a crystallization procedure are built up as a result of growth either on nuclei of the material itself or surfaces of foreign material serving the same purpose. Neglecting for the moment the matter of impurities, nucleation theory provides an explanation for certain qualitative observations in the case of solutions. 

In spite of these obstacles, crystallization does occur and the rate at which it develops can be measured. The following derivation will illustrate how the rates of nucleation and growth combine to give the net rate of crystallization. The theory we shall develop assumes a specific picture of the crystallization process. The assumptions of the model and some comments on their applicability follow ... 

Models used to describe the growth of crystals by layers call for a two-step process (/) formation of a two-dimensional nucleus on the surface and (2) spreading of the solute from the two-dimensional nucleus across the surface. The relative rates at which these two steps occur give rise to the mononuclear two-dimensional nucleation theory and the polynuclear two-dimensional nucleation theory. In the mononuclear two-dimensional nucleation theory, the surface nucleation step occurs at a finite rate, whereas the spreading across the surface is assumed to occur at an infinite rate. The reverse is tme for the polynuclear two-dimensional nucleation theory. Erom the mononuclear two-dimensional nucleation theory, growth is related to supersaturation by the equation. 

Sawada et al. [110] and the authors of this Chapter [104,111] have proposed another theory, the bundle-like nucleation theory, for the mechanism of ECC formation. Both groups of workers suggested that crystallization under high pressure starts from partially extended-chain nucleation rather than from the folded-chain nucleation as proposed by Hikosaka [103,104]. This theory was established on the basis of the following facts ... 

The layout of this article is as follows Section 2 considers the equilibrium aspects of the crystals whilst Sects. 3 and 4 explain the growth theories, divided into nucleation and non-nucleation theories, respectively. Finally, Sect. 5 provides an overview and suggests future lines of investigation. 

Although specific calculations for i and g are not made until Sect. 3.5 onwards, the mere postulate of nucleation controlled growth predicts certain qualitative features of behaviour, which we now investigate further. First the effect of the concentration of the polymer in solution is addressed - apparently the theory above fails to predict the observed concentration dependence. Several modifications of the model allow agreement to be reached. There should also be some effect of the crystal size on the observed growth rates because of the factor L in Eq. (3.17). This size dependence is not seen and we discuss the validity of the explanations to account for this defect. Next we look at twin crystals and any implications that their behaviour contain for the applicability of nucleation theories. Finally we briefly discuss the role of fluctuations in the spreading process which, as mentioned above, are neglected by the present treatment. 

The growth rate of many crystals is often observed to depend upon temperature in a manner consistent with nucleation theories. If measurements are made on growth from solutions of different concentrations then, at equivalent thicknesses, the dependence of growth rate upon concentration may be determined. Equations (3.16) and (3.17) can be used to predict the concentration dependence of this nucleation approach. 

Finally we would like to draw attention to low molecular weight results and their analysis on the basis of surface nucleation theory. The theory was originally developed for infinitely long chains and cannot easily be applied to extended or once-folded chain crystallization. Therefore any discrepancies in this area would not be surprising and would not discredit the theory at higher molecular weights. 

A drastic departure from nucleation theory was made by Sadler [44] who proposed that the crystal surface was thermodynamically rough and a barrier term arises from the possible paths a polymer may take before crystallizing in a favourable configuration. His simulation and models have shown that this would give results consistent with experiments. The two-dimensional row model is not far removed from Point s initial nucleation barrier, and is practically identical to a model investigated by Dupire [35]. Further comparison between the two theories would be beneficial. 

Transfomation from a meta-stable phase, such as supersaturated solution, to a thermodynamically more favorable phase requires first the crystal nucleation of a germ of the new phase. According to the classical nucleation theory, the volume nucleation rate J (cm" sec ), describing the number of nuclei(i.e., a critical germ) formed per volume per time, is given by ... 

In this work, we developed the safeguard active-set method by modifying the active-set method for thermodynamic equilibrium in order to include the classical nucleation theory. At tn, assume that the partition ( (r ), M(t ), N(t ) and the crystallization time tciyst(t ) forM(t ) are known. For a new feed vector and RH at Vu compute W(tn+i), M(t i), N(t + )) and tciyst(t +i) as follows ... 

Here our interest is in the application of homogeneous nucleation theory to produce the comprehensive plots of meta-stable crystallization. Fig. 1 illustrates the meta-stable efflorescence paths(solid lines) of (NH4)2S04 and (NH4)3H(S04)2 particles as a function of RH with the decreasing rate of ARH = 0.005 min with the deliquescence paths(O). Fig. 2 shows the expectation time of the aqueous particle composed of (NH4)2S04 and H2SO4... 

Crystallization can be divided into three processes the primary nucleation process, the growth process, and the overgrowth process. The growth process is mainly controlled by the secondary nucleation mechanism. The steady (stationary) primary and secondary nucleation mechanisms of atomic or low molecular weight systems have been well studied since the 1930s by applying the classical nucleation theory (CNT) presented by Becker and Doring, Zeldovich, Frenkel and Turnbull and Fisher and so on [1-4]. 

Hikosaka presented a chain sliding diffusion theory and formulated the topological nature in nucleation theory [14,15]. We will define chain sliding diffusion as self-diffusion of a polymer chain molecule along its chain axis in some anisotropic potential field as seen within a nucleus, a crystal or the interface between the crystalline and the isotropic phases . The terminology of diffusion derives from the effect of chain sliding diffusion, which could be successfully formulated as a diffusion coefficient in our kinetic theory. 

As described earlier, Doi s kinetic theory leads to a prediction that the SD is triggered by extension of unoriented crystalline sequences prior to crystal nucleation. In order to confirm this prediction the conformational change... 

Crystallization conditions can often be manipulated to favor the nucleation of alternate crystal forms. A metastable polymorph of metformin hydrochloride has been isolated using capillary crystallization techniques, and subsequently studied using thermal microscopy [24]. Calculations based on classical nucleation theory indicated that a metastable form could be obtained using high degrees of... 

In classical nucleation theory the Gibbs energy of a nucleus is considered as the sum of contributions from the bulk and the surface. Let us consider nucleation of a spherical crystal from its liquid below its melting temperature at 1 bar. The difference in Gibbs energy between a nucleus with radius r and its liquid is... 

Figure 3 shows a plot of the volume normalized nucleation time constant as a function of isothermal crystallization temperature for PEO droplets, taken from the work of Massa and Kalnoki-Veress [84]. As expected, droplets of different volumes have the same value of r V. The inset in Fig. 3 is a plot consistent with classical nucleation theory (see Eqs. 1, 4) only the last four data points correspond to the work of Massa and Kalnoki-Veress. The first... 

Several refinements of our experiments could test these theories further. By measuring etch pit densities as well as pit dimensions on sequentially-etched crystals, nucleation rate data and pit growth data could be collected, yielding information about the rate-limiting steps and mechanisms of dissolution. In addition, since the critical concentration is extremely dependent on surface energy of the crystal-water interface (Equation 4), careful measurement of Ccrit yields a precise measurement of Y. Our data indicates an interfacial energy of 280 + 90 mjm- for Arkansas quartz at 300°C, which compares well with Parks value of 360 mJm for 25°C (10). Similar experiments on other minerals could provide essential surface energy data. 

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