Crystal growth rate determination

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. 

Ang, H-M. and Mullin, J.W. (1979) Crystal growth rate determinations from desupersaturation measurements nickel ammonium sulphate hexahydrate. Transactions of the Institution of Chemical Engineers, 57, 237-242. 

Garside, J., Mullin, J.W. and Das, S.N. (1973) Importance of crystal shape in crystal growth rate determinations. Industrial and Engineering Chemistry Process Design and Development, 12, 369-371. 

In order to evaluate the application of modulated-temperature differential scanning calorimetry (M-TDSC) to the study of the crystallisation kinetics of semicrystalline polymers, isothermal crystallisation kinetics in poly(e-caprolactone)-SAN blends are investigated. The temperature dependence of d In G/dT (G =crystal growth rate), determined by M-TDSC agrees approximately with previous experimental data and theoretical values. These were obtained from direct measurements of spherulite growth rate by optical microscopy. Here, theoretical and M-TDSC experimental results show that the d In G/dT versus temperature plots are not sensitive to the noncrystalline component in the poly(e-caprolactone)-SAN blends. 15 refs. 

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)... 

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 overall rate of crystallization is determined by both the rate of nuclei formation and by the crystal growth rate. The maximum crystal growth rate lies at temperatures of between 170 and 190 °C [71, 72], as does the overall crystallization rate [51, 61, 75], The former is measured using hot stage optical microscopy while the latter is quantified by the half-time of crystallization. Both are influenced by the rate of nucleation on the crystal surface and the rate of diffusion of polymer chains to this surface. It has been shown that the spherulite growth rate decreases with increasing molecular weight due to the decrease in the rate of diffusion of molecules to this surface [46, 50, 55, 71, 74],... 

Monodisperse oxide particles were only formed at the narrow concentration range in alcohol solution. Monodisperse particles, which have submi-crometer order with narrow size distribution, were amorphous and hydrated. The spherical morphology was retained after the crystallization. The rate-determining step of particle growth was polynuclear layer growth of first order from Nielsen s chronomal analysis. 

It is this large and continuous variability in bulk composition coupled with the fact that crystal structures may be different for the same anhydrous or hydrous bulk composition which makes zeolite identification so difficult (see Breck, 1970, for example and Deer, et al., Vol. 4, 1962). The factors determining which species of zeolite will crystallize are undoubtedly complex, involving such variables as the chemical activity of dissolved ionic species, crystal growth rate and ease of nucleation however, certain patterns of mineral paragensis can be discerned through a survey of the literature. 

To ensure that all the overall crystal-growth rate coefficients are measured under the conditions without nucleation, the metastable region of the solution has to be determined first and therefore the solubility and super solubility need to be measured. 

Fig. 14 Binary phase diagram for C246H494 in octacosane. The top curve shows the equilibrium liquidus for extended-chain crystals, and the bottom line the metastable liquidus for once-folded crystals. Experimental dissolution temperatures are fitted to the Flory-Huggins equation with / = 0.15 (solid lines). Vertical dotted lines (a) and (b) indicate the concentrations at which the growth rates were determined as a function of Tc in [29]. Horizontal dotted lines indicate the temperatures at which the rates were determined in [45] as a function of concentration. G(c) at Tc = 106.3 °C, measured along line (c), is shown in Fig. 12. The shading indicates schematically the crystal growth rate (black = fast), and the dashed line the position of the growth rate minimum...

Fig. 33 Determination of side-surface free energyfrom chainlength dependence of extended-chain crystal growth rates G, (left) and Ga (right) from the melt using Eq. 5 [30]...

The rate of crystal growth is determined by the degree of supersaturation, the rate of molecular diffusion to the crystal surface, and the time required for TAG molecules to fit into the growing crystal lattice (Mulder and Walstra, 1974 Walstra, 1987). Compared to nucleation, the driving force required for crystal growth is relatively low (Sato et al, 1989). However, in a multicomponent fat, the supersaturation for each TAG is small (Walstra, 1998). This fact, combined with competition between similar molecules for the same sites in a crystal lattice, means that milk fat crystallization is especially slow (Skoda and van den Tempel, 1967 Knoester et al, 1968 Grail and Hartel, 1992). 

Optical microscopy (inverted microscope) Study crystallization processes in situ Monitor transformations in suspensions Determine transformation times Screen and characterize additive/solvent interactions with specific crystal faces Identify nucleation mechanisms Measure crystal growth rates... 

As we have shown, crystal growth from a liquid melt is accomplished by establishing an equilibrium between the solid crystal and the liquid melt. By displacing the equilibrium in slight favor of the solid, the crystal will grow. The crystal growth rate is determined by two factors ... 

Where do the steps come from and what is the rate controlling factor in determining the crystal growth rate The goal of crystal growth theories is to try to answer these two questions. 

Burton-Cabrera-Frank (BCF) Model. The models discussed in the previous section all require two-dimensional nucleation events for a new layer to start. These models fail to account for observed crystal growth rates at low supersaturations and are unsatisfying in the sense that they make crystal growth a noncontinuous process with the formation of a critical size two-dimensional nucleus the rate-determining step. A basis for a model in which the steps are self-perpetrating was put forward by Frank (1949). Frank s idea was that dislocations in the crystal are the source of new steps and that a type of dislocation known as a screw dislocation could... 

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