Size-dependent crystal growth

Other secondary effects, which are not exclusive to reactive crystallization, are size-dependent growth and growth rate dispersion. These effects may not be separable, but both can change the final mean particle size and PSD. Both are discussed in Sohnel and Garside (1992, pp. 103-105) and in Chapter 4 of this book. 

The simplifying assumption of size-independent crystal growth rate, utilized in the crystal population analyses so far, cannot always be justified. For one reason or another, many crystals exhibit size-dependent growth (section 6.2.7) and this leads to the generation of a different product CSD from that described previously under MSMPR conditions. 

Consider first the case where crystal growth rate depends on the crystal size (relaxing assumption (1)). Instead of the simplified equation (6.4.3b), we will have to work with equation (6.4.3a) ... 

Unlike melting and the solid-solid phase transitions discussed in the next section, these phase changes are not reversible processes they occur because the crystal stmcture of the nanocrystal is metastable. For example, titania made in the nanophase always adopts the anatase stmcture. At higher temperatures the material spontaneously transfonns to the mtile bulk stable phase [211, 212 and 213]. The role of grain size in these metastable-stable transitions is not well established the issue is complicated by the fact that the transition is accompanied by grain growth which clouds the inteiyDretation of size-dependent data [214, 215 and 216]. In situ TEM studies, however, indicate that the surface chemistry of the nanocrystals play a cmcial role in the transition temperatures [217, 218]. 

Crystal Morphology. Size, shape, color, and impurities are dependent on the conditions of synthesis (14—17). Lower temperatures favor dark colored, less pure crystals higher temperatures promote paler, purer crystals. Low pressures (5 GPa) and temperatures favor the development of cube faces, whereas higher pressures and temperatures produce octahedral faces. Nucleation and growth rates increase rapidly as the process pressure is raised above the diamond—graphite equiUbrium pressure. 

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 ... 

The dominant crystal size, is most often used as a representation of the product size, because it represents the size about which most of the mass in the distribution is clustered. If the mass density function defined in equation 33 is plotted for a set of hypothetical data as shown in Figure 10, it would typically be observed to have a maximum at the dominant crystal size. In other words, the dominant crystal size is that characteristic crystal dimension at which drajdL = 0. Also shown in Figure 10 is the theoretical result obtained when the mass density is determined for a perfectiy mixed, continuous crystallizer within which invariant crystal growth occurs. That is, mass density is found for such systems to foUow a relationship of the form m = aL exp —bL where a and b are system-dependent parameters. 

Such cases of curvature can arise due to so-called anomalous growth. A variety of mechanistic causes for this behaviour have been proposed which fall into two broad classes viz. growth rate dispersion and size-dependent crystal growth. Both classes... 

Growth rate fluetuations appear to inerease with an inerease in temperature and supersaturation leading to erystals of the same substanee, in the same solution at identieal supersaturation, exhibiting different growth rates this is thought to be a manifestation of the phenomenon of either size-dependent crystal growth or alternatively, growth rate dispersion. 

The data plotted in the figure clearly support the predicted positive dependence of crystal size on agitation rate. Precipitation in the crystal film both enhances mass transfer and depletes bulk solute concentration. Thus, in the clear film model plotted by broken lines, bulk crystal sizes are initially slightly smaller than those predicted by the crystal film model but quickly become much larger due to increased yield. Taken together, these data imply that while the initial mean crystal growth rate and mixing rate dependence of size are... 

Abegg, C.F., Stevens, J.D. and Larson, M.A., 1968. Crystal size distribution in continuous crystallizer when growth rate is size-dependent. American Institmte oj Chemical Engineers Journal, 41, 188. 

Garside, J. and Jancic, S.J., 1978. Prediction and measurement of crystal size distribution for size-dependent growth. Chemical Engineering Science, 4331. 

Mydlarz, I. and Jones, A.G., 1989. On modelling the size-dependent growth rate of potassium sulphate in an MSMPR crystallizer. Chemical Engineering Communications, 90, 47-56. 

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. 

These two lengths have been discussed in detail by Point et al. [93] and Dosiere et al. [94]. They study the size dependence of the growth rate of polyethylene for very small crystals using a decoration technique. The accuracy of their measurements is carefully considered, and they conclude that there is no size dependence of the growth rates for all length scales measured (>200 nm). Several other claims that there is no size dependence do not seem justified by the number and accuracy of the measurements involved. As shown below, a detailed investigation of these facts would be extremely useful and would enable limits to be placed on the magnitudes of i and g. 

There are obviously two steps involved in the preparation of crystal matter from a solution, the crystals must first form and then grow. The formation of a new solid phase either on an inert particle in the solution or in the solution itself is called nucleation. The increase in size of this nucleus with a layer-by-layer addition of solute is called crystal growth. Both nucleation and crystal growth have supersaturation as a common driving force. Unless a solution is supersaturated, crystals can neither form nor grow. The particle-size distribution of this weight, however, will depend on the relationship between the two processes of nucleation and growth. 

Some of the reports are as follows. Mizukoshi et al. [31] reported ultrasound assisted reduction processes of Pt(IV) ions in the presence of anionic, cationic and non-ionic surfactant. They found that radicals formed from the reaction of the surfactants with primary radicals sonolysis of water and direct thermal decomposition of surfactants during collapsing of cavities contribute to reduction of metal ions. Fujimoto et al. [32] reported metal and alloy nanoparticles of Au, Pd and ft, and Mn02 prepared by reduction method in presence of surfactant and sonication environment. They found that surfactant shows stabilization of metal particles and has impact on narrow particle size distribution during sonication process. Abbas et al. [33] carried out the effects of different operational parameters in sodium chloride sonocrystallisation, namely temperature, ultrasonic power and concentration sodium. They found that the sonocrystallization is effective method for preparation of small NaCl crystals for pharmaceutical aerosol preparation. The crystal growth then occurs in supersaturated solution. Mersmann et al. (2001) [21] and Guo et al. [34] reported that the relative supersaturation in reactive crystallization is decisive for the crystal size and depends on the following factors. 

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... 

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