Crystal growth rates

According to the kinetic theory, the nucleation controlled growth rate (G) of polymer crystals at the temperature Tc is described by the equation [30,31]  

However, when the contribution of the secondary nucleation term A is dominant (at low undercoolings), the diluent effect of the amorphous component must be taken into account, that is, an additional entropic effect to the nucleation barrier due to the reduced probability of extracting crystalline polymer sequences from the entangled melt at the growth front. Following the treatment of Boon and Azcue for a polymer-diluent mixture [35], the activation energy associated with the secondary nucleation in a miscible blend can be expressed as  

Since the last term on the right is negative, the value of A dffl is larger than the corresponding value of the plain crystalline polymer, which implies a decrease of growth rate in the blend, at the same AT (Fig. 10.4). Therefore, for a miscible blend. Equation (10.9) can be written in the form  

Plots of the term /(G) versus l/JTcAT are linear with a slope 0.1vfi 7e/5/jtB thus,by finding the values of b and 

The isothermal overall crystallization kinetics of polymer blends from the melt can be analyzed on the basis of the Avrami model [38, 39]  

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Over 50 acidic, basic, and neutral aluminum sulfate hydrates have been reported. Only a few of these are well characterized because the exact compositions depend on conditions of precipitation from solution. Variables such as supersaturation, nucleation and crystal growth rates, occlusion, nonequilihrium conditions, and hydrolysis can each play a role ia the final composition. Commercial dry alum is likely not a single crystalline hydrate, but rather it contains significant amounts of amorphous material. 

Increased pressures can lower the temperature at which crystallisation occurs. Experiments performed using Spectrosil (Thermal Syndicate Ltd.) and G.E. Type 204 (General Electric Company) fused siUcas (see Eig. 2) show that at pressures above 2.5 GPa (<25, 000 atm), devitrification occurs at temperatures as low as 500°C and that at 4 GPa (<40, 000 atm), it occurs at as low as 450°C (107). Although the temperatures and pressures were in the coesite-phase field, both coesite and quarts were observed. Both the devitrification rate and the formation of the stable phase (coesite) were enhanced by the presence of water. In the 1000—1700°C region at 500—4000 MPa (<5, 000-40,000 atm), a- and p-quarts were the primary phases. Crystal growth rates... 

The effects of a solvent on growth rates have been attributed to two sets of factors (28) one has to do with the effects of solvent on mass transfer of the solute through adjustments in viscosity, density, and diffusivity the second is concerned with the stmcture of the interface between crystal and solvent. The analysis (28) concludes that a solute-solvent system that has a high solubiUty is likely to produce a rough interface and, concomitandy, large crystal growth rates. 

If the crystallizer is now assumed to operate with a cleat feed (n = 0), at steady state (dn jdt = 0), and if the crystal growth rate G is invariant and a mean residence time T is defined as then the population balance can be written as... 

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... 

Figure 5.7 Effect of solution velocity on crystal growth rates after Miillin and Garside, 1967) number Sh = IcLjD, the partiele Reynolds number...

In addition to induction time measurements, several other methods have been proposed for determination of bulk crystallization kinetics since they are often considered appropriate for design purposes, either growth and nucleation separately or simultaneously, from both batch and continuous crystallization. Additionally, Mullin (2001) also describes methods for single crystal growth rate determination. 

Several authors have presented methods for the simultaneous estimation of crystal growth and nucleation kinetics from batch crystallizations. In an early study, Bransom and Dunning (1949) derived a crystal population balance to analyse batch CSD for growth and nucleation kinetics. Misra and White (1971), Ness and White (1976) and McNeil etal. (1978) applied the population balance to obtain both nucleation and crystal growth rates from the measurement of crystal size distributions during a batch experiment. In a refinement, Tavare and... 

The crystal growth rates can be directly determined from the second and third moment as described above. The calculated rates for calcium oxalate here are in the range 0.75 x 10 to 4.7 x 10 m/s. Literature values for the growth rate of calcium oxalate monohydrate vary considerably 1.08 x 10 m/s (Kavanagh, 1992), 3.4 X 10 to 5.0 x 10 m/s (Garside etal., 1982) and 2.8 x lO to 1.11 X 10 m/s (Nielsen and Toft, 1984). The values obtained from the experiments are therefore within the range of the literature data. It should be borne in... 

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... 

Bujac, P.B. and Mullin, J.W., 1969. A rapid method for the measurement of crystal growth rates in a fluidised bed crystallizer. Symposium on Industrial Crystallization. London, 1969. Rugby Institution of Chemical Engineers, pp. 121-129. 

Mulliii, J.W. and Garside, J., 1967. Crystallization of aluminium potassium sulphate a study in the assessment of crystallizer design data I Single crystal growth rates, II Growth in a fluidised bed. Transactions of the Institution of Chemical Engineers, 45, 285-295. 

Mydlarz, I. and Jones, A.G., 1990. On the estimation of size-dependent crystal growth rate functions in MSMPR crystallizers. Chemical Engineering Journal, 53, 125-135. 

O Hara, M. and Reid, R.C., 1973. Modelling Crystal Growth Rates from Solution. Englewood Cliffs Prentice-Hall. 

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

One of the more important uses of OM is the study of crystallization growth rates. K. Cermak constructed an interference microscope with which measurements can be taken to 50° (Ref 31). This app allows for study of the decompn of the solution concentrated in close proximity to the growing crystal of material such as Amm nitrate or K chlorate. In connection with this technique, Stein and Powers (Ref 30) derived equations for growth rate data which allow for correct prediction of the effects of surface nucleation, surface truncation in thin films, and truncation by neighboring spherulites... 

Comparison of crystal growth rate coefficients measured in ISC and FBC... 

Y. Wu, Y. Zhou, X. He and C. Bao, Crystal-growth rate of di-sodium phosphate in impinging stream crystallizer, a paper to be published. 

According to Hoffman s crystallization theory, a drop in the heat of fusion corresponds to an exponential decrease in nucleation and crystal growth rates [63]. Implicitly, the rate of crystallization is severely retarded by the presence of 3HV comonomer [64, 69, 72]. These low crystallization rates can hamper the melt processing of these copolymers since they necessitate longer processing cycle times. 

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