Crystal growth expressions

Size-dependent Crystal Growth. A number of empirical expressions correlate the apparent effect of crystal size on growth rate (30). The most commonly used correlation uses three empirical parameters to correlate growth rate with crystal size ... 

It has often been observed that the plot of ln(L) versus L results in curvature rendering the method of determining the growth rate from the slope strictly inappropriate, but ways to accommodate such deviations have also been proposed. Thus, if G = G(L) integration of equation 3.15 leads to the following expression for determining crystal growth rates (Sikdar, 1977)... 

Crystal growth rate may be expressed either as a rate of linear inerease of eharaeteristie dimension (i.e. veloeity) or as a mass deposition rate (i.e. mass flux). Expressed as a veloeity, the overall linear erystal growth rate, G (=dL/dt where L is the eharaeteristie dimension that is inereasing). The rate of ehange of... 

It was shown in Chapter 7 that the performance of continuous crystallizers is determined by the characteristics of a feedback loop relating the output performance expressed as crystal size distribution and to the feed concentration and residence time. Thus, an increase in crystallizer residence time, or decrease in feed concentration, reduces the working level of supersaturation. This decrease in supersaturation results in a decrease in both nucleation and crystal growth. This in turn leads to a decrease in crystal surface area. By mass balance, this then causes an increase in the working solute concentration and hence an increase in the working level of supersaturation and so on. There is thus a complex feedback loop within a continuous crystallizer, as considered in Chapter 7 and illustrated in Figure 8.11. 

Secondary nucleation is essentially a crystal growth process. Secondary nucleation occurs by the deposition of a stem of the polymer molecule on a preexisting crystal-face as shown in Figure 15. The overall rate of this process is given by the following expression [58],... 

As with nucleation, classical theories of crystal growth 3 20 2135 40-421 have not led to working relationships, and rates of crystallisation are usually expressed in terms of the supersaturation by empirical relationships. In essence, overall mass deposition rates, which can be measured in laboratory fluidised beds or agitated vessels, are needed for crystalliser design, and growth rates of individual crystal faces under different conditions are required for the specification of operating conditions. 

Because the rate of growth depends, in a complex way, on temperature, supersaturation, size, habit, system turbulence and so on, there is no simple was of expressing the rate of crystal growth, although, under carefully defined conditions, growth may be expressed as an overall mass deposition rate, RG (kg/m2 s), an overall linear growth rate, Gd(= Ad./At) (m/s) or as a mean linear velocity, // (= Ar/At) (m/s). Here d is some characteristic size of the crystal such as the equivalent aperture size, and r is the radius corresponding to the... 

Table 15.4. Mean over-all crystal growth rates expressed as a linear velocity ...

The specific surface area of a solid is the surface area of a unit mass of material, usually expressed as m g . There is an inverse relationship between surface area and particle size. Massive crystals of hematite from an ore deposit (e. g. specularite) may have a surface area 1 m g". As particle size/crystallinity is governed largely by the chemical environment experienced during crystal growth, the surface area of a synthetic iron oxide depends upon the method of synthesis and that of a natural one, upon the environment in which the oxide formed. 

The crystal growth rate has been found in many eases to be extremely rapid, more rapid than can be accounted for on the diffusion hypothesis thus Tammann (foe. cit.) found for benzophe-none a maximum crystallisation velocity of 2 4 mm. per minute (Walton and Judd, J. Phys. Ohem. xvni, 722,1914). Much higher values, e.g. 6840 mm. per minute for water and 60,000 mm. per minute for phosphorus (Gernez, O.R, xcv. 1278, 1882) have been recorded. In some cases the rate was found independent of the speed of rotation of the stirrer and occasionally the reaction velocity followed a bimolecular law instead of the simple unimolecular expression which holds true for solution. 

As can be seen from the expression for the driving force in terms of the chemical potential differences, which are related to the differences in temperature and concentration, the two transporting processes, heat transfer and mass transfer, are coupled in crystal growth. The degree of contribution from the respective transport process is determined by the degree of condensation of the environmental (ambient) phase. To grow crystals in a diluted ambient phase, a condensation process is required, and so mass transfer plays an essential role. The contribution of heat generated by crystallization in this case is small compared with that of the mass transfer. However, for crystallization in a condensed phase, such as a melt phase, heat transfer plays the essential role, and the contribution from the mass transfer will be very small, because the difference in concentration (density) between the solid and liquid phases is very small, smaller, say, than 1 or 2%. It is therefore necessary to classify the types of ambient phases and to be familiar with their respective characteristics from this standpoint. 

In Fig. 12.4 the velocity of crystal growth from the vapor is plotted as a function of the excess vapor pressure (P — Peq). When the surface acts as an ideal sink for incoming vapor atoms, the plot indicates that the velocity of growth should vary linearly with (P - Peq). When the sink efficiency is lower, the curve of v vs. (P - Peq) falls below the ideal curve. Use the results of Exercise 12.1 to demonstrate that the velocity of growth for the model employed there can be expressed in the form... 

Crystal growth from the vapor phase has been treated in Chapter 12. An expression for the net atom flux, Jv, gained at a macroscopically flat crystal surface during growth from the vapor has been obtained in Exercise 12.2 in the form of Eq. 12.27. To treat surfaces possessing nonuniform curvature, this relationship can be generalized in the form... 

Prove that all of the results obtained in Exercise 14.5 for crystal growth (including the basic differential equation, its solution, and the expression for the sink efficiency) also hold for crystal evaporation. 

The kinetics of transition from the liquid crystal to the fully ordered crystal of flexible, linear macromolecules was studied by Warner and Jaffe 38) on copolyesters of hydroxybenzoic acid, naphthalene dicarboxylic acid, isophthalic acid, and hydro-quinone. The analytical techniques were optical microscopy, calorimetry and wide angle X-ray diffraction. Despite the fact that massive structural rearrangements did not occur on crystallization, nucleation and growth followed the Avrami expression with an exponent of 2. The authors suggested a rod-like crystal growth. 

Further, substituting Eq. (12-8) into Eq. (12-3), integrating the resulting equation between t - 0 and t = tf and rearranging yields the expression for the overall crystal-growth rate coefficient as... 

Examination of Eqs. (12) to (16) reveals five distinct rate steps dissolution of B the reaction between A and B the generation of P nuclei in the liquid phase the mass transfer of dissolved P to the growing P crystals and the surface integration of the solute P into the crystal lattice (i.e., the crystal growth step). The relative importance of each of these steps can be characterized by a dimensionless number. For a reaction which is second order overall, and for nucleation and growth kinetics which can be represented by conventional power law expressions, we have... 

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