Nucleation rate temperature dependence

It should be noted here that Equation (7.19a) and Equation (7.19b) are readily obtained from the Avrami equation (Eq. 7.6b) by simple integration by parts. Denoting the growth rate and nucleation rate temperature dependencies as g(T) and /(T), respectively, and assuming that the temperature T is a linear function of time t, one obtains the following expressions for expectancy E T), for the temperature-dependent nucleation rate and for the instantaneous nucleation, respectively ... 

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

The nucleation rate is, in fact, critically dependent on temperature, as Fig. 8.3 shows. To see why, let us look at the heterogeneous nucleation of b.c.c. crystals at grain boundaries. We have already looked at grain boundary nucleation in Problems 7.2 and 7.3. Problem 7.2 showed that the critical radius for grain boundary nucleation is given by... 

Application of Eqs. (21)-(27) to the calculations of the nucleation rates J for various alloy models revealed a number of interesting results, in particular, sharp dependence of J and embryo characteristics on the supersaturation, temperature, interaction radius, etc. These results will be described elsewhere. 

The uniformity of film thickness is dependent upon temperature and pressure. The nucleation rate rises with pressure, such that at pressures above atmospheric the high rate of nucleation can lead to comparatively uniform oxide films, while increase in temperature reduces the density of oxide nuclei, and results in non-uniformity. Subsequently, lateral growth of nuclei over the surface is faster than the rate of thickening until uniform coverage is attained, when the consolidated film grows as a continuous layer ... 

The temperature dependence of the spreading rate is generally small and can often be neglected against that of the nucleation rate (but see Sect. 3.6.3). 

Reid et al. [ 1.12] described the effect of 1 % addition certain polymers on the heterogeneous nucleation rate at-18 °C the rate was 30 times greater than in distilled, microfiltered water and at -15 °C, the factor was still 10 fold hogher. All added polymers (1 %) influenced the nucleation rate in a more or less temperature-dependent manner. However, the authors could not identify a connection between the polymer structure and nucleation rate. None the less it became clear that the growth of dendritic ice crystals depended on to factors (i) the concentration of the solution (5 % to 30 % sucrose) and (ii) the rate at which the phase boundary water - ice crystals moved. However, the growth was found to be independent of the freezing rate. (Note of the author the freezing rate influences the boundary rate). 

The temperature dependence of the reaction was studied, and the activation energy of the reaction was calculated to be approximately 100 kj mol The exponent n was found to lie in the range 1-2, which is consistent with a 2D diffusion controlled reaction mechanism with deceleratory nucleation. The rate of reaction increases markedly with the amount of water added to the LDH with very small amounts of water added, the deintercalation process does not go to completion. This effect is a result of the LiCl being leached into solution. An equilibrium exists between the LDH and gibbsite/LiCl in solution. The greater [LiCl], the further to the LDH side this lies. 

Measurements were undertaken of the solubility of each phase in acid solutions, of the growth rate of gypsum crystals and the dissolution rate of hemihydrate. The growth rate depends on the square of the supersaturation and on temperature with an activation energy of 64 kJ/mol. The nucleation rate appears to vary linearly with supersaturation. 

The nucleation rate increased from 65°C to 70°C and dropped from 70°C to 80°C. Thus 70°C seems to be the optimum temperature for maximum nucleation. Published work on alumina trihydrate by Misra and White (5) and Brown (9 10) revealed that the nucleation rate decreases with increasing temperature, at greater than 70 C by the former but from 50 to 75°C by the latter. This nucleation rate dependence on temperature differs with normal chemical reaction where the reaction rate increases with increase in temperature. It is not clear whether then-studies at different temperatures in the published work were conducted at constant initial absolute supersaturation (AC7C ) for all the temperatures studied or at constant initial concentration. The latter would account for the higher nucleation rates obtained at lower temperatures as the AC/C is higher at lower temperatures since C decreases with temperature. 

At a given temperature, the crystal surface area has a bigger contribution to the nucleation rate than AC. The low dependence of the nucleation rate on the concentration driving force implies that the secondary nucleation of alumina trihydrate is removal-limited. 

It is remarkable that the predictions of classical nucleation theory without any consideration of polymer connectivity are borne out in experiments. At higher supercooling, deviations are expected because of temperature dependence of the nucleation rate prefactor. 

Figure 4-2 Calculated nucleation rate for Vc = 46 x 10 m /mol, E = 250 kj/mol, A5m-c = 50 J-K -moP, Al = 5 x 10 m, the equilibrium temperature of 1500K for (a) and (b), and the equilibrium pressure of 3 GPa for (c). (a) The dependence of crystal nucleation rate on the interface energy. Note that for a small change in interface energy from 0.300 to 0.295 J/m, the peak nucleation rate increases by more than one order of magnitude. If the interface energy changes from 0.3 to 0.2 J/m, the peak nucleation rate would increase by 17 orders of magnitude, (b) The nucleation rate of crystal and melt as a function of temperature, (c) The nucleation rate of crystal and melt as a function of pressure.

Bertram, A. K., and J. J. Sloan, Temperature-Dependent Nucleation Rate Constants and Freezing Behavior of Submicron Nitric Acid Dihydrate Aerosol Particles under Stratospheric Conditions, J. Geophys. Res., 103, 3553-356f (f998a). 

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