Growth size dependency

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]. 

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 ... 

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. 

For eontinuous erystallizers the effeet of size-dependent growth and growth rate dispersion are diffieult to distinguish (Janse and de Jong, 1978 Randolph and White, 1977). 

Use of growth rate diffusivity (Randolph and White, 1977 Tavare and Gar-side, 1982) or size dependent growth (Abegg etal., 1968 Mydlarz and Jones, 1993) have both been proposed as alternative phenomenologieal means to deseribe the effeet of growth dispersion on erystal size distributions the latter being simpler mathematieally than the former, but in all probability both meehanisms ean oeeur. 

The size-dependent agglomeration kernels suggested by both Smoluchowski and Thompson fit the experimental data very well. For the case of a size-independent agglomeration kernel and the estimation without disruption (only nucleation, growth and agglomeration), the least square fits substantially deviate from the experimental data (not shown). For this reason, further investigations are carried out with the theoretically based size-dependent kernel suggested by Smoluchowski, which fitted the data best ... 

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. 

Sikdar, S.K., 1977. Size-dependent growth rate from curved log n(L) vs. L steady-state data. Industrial and Engineering Chemistry Fundamentals, 16, 390. 

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. 

Hence, the extent of non-inertial growth depends logarithmically on binder viscosity and the inverse of agitation velocity (Adetayo et al., 1995). Maximum granule size depends linearly on these variables. Also, the extent of growth has been observed to depend linearly on liquid loading y and,... 

Note that in the special case of size-independent growth, this term can be expressed as a closed function of the moments, i.e., G,t(c) = G(c)mk. Note also that when deriving Eq. (102) we have neglected the size-dependence of This is justified in turbulent flows and, in any case, to do otherwise would require a micromixing model that accounts for differential diffusion (Fox, 2003). 

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