Evolution of an ensemble of spherical particles in metastable media with allowance for their unsteady-state growth rates

Abstract

The process of particle nucleation and growth at the initial and intermediate stages of bulk crystallization in metastable liquids is studied. An integrodifferential model of balance and kinetic equations with corresponding boundary and initial conditions is formulated with allowance for non-stationary temperature/concentration field around each evolving particle. The model is solved using the saddle-point technique in a parametric form. The particle-radius distribution function, supercooling/supersaturation of liquid, total number of particles in liquid and their average size are found analytically. The melt supercolling (solution supersaturation) decreases with time due to the latent heat of phase transformation released by evolving crystals. As this takes place, the particle-radius distribution function is bounded by the maximal size of crystals and shifts to larger crystal radii with time as a result of particle nucleation and growth.

Appendices

Apendix A

Coefficients and functions entering the analytical solution are given by

$$\begin{aligned} a_0(z)= & {} -\Gamma \left( \frac{1}{4}\right) h(0, t) H(0) , \\ a_1(z)= & {} 4 \Gamma \left( \frac{1}{2}\right) H(0)\left[ H(0)\left( \frac{\partial h}{\partial \nu }\right) _{\nu =0}+h(0, t) H^{\prime }(0)\right] , \\ a_2(z)= & {} -8 \Gamma \left( \frac{3}{4}\right) H^2(0)\left[ 2 H^{\prime }(0)\left( \frac{\partial h}{\partial \nu}\right) _{\nu =0}\right. \\{} & {} \left. +H(0)\left( \frac{\partial ^2 h}{\partial \nu^2}\right) _{\nu =0}+h(0, t) H^{\prime \prime }(0)\right] , \\ a_3(z)= & {} \frac{32}{3} H^3(0)\left[ 3 H^{\prime \prime }(0)\left( \frac{\partial h}{\partial \nu}\right) _{\nu =0}+h(0, t) H^{\prime \prime \prime }(0)\right. \\{} & {} \left. +3H^{\prime }(0)\left( \frac{\partial ^2 h}{\partial \nu^2}\right) _{\nu =0}+H(0)\left( \frac{\partial ^3 h}{\partial \nu^3}\right) _{\nu =0}\right] , \\ h(0, t)= & {} \frac{z}{\gamma _*^2}(1 - z^2)^2 ,\ \ \left( \frac{\partial h}{\partial \nu}\right) _{\nu =0} = -\frac{2z^2}{\gamma _*}(1-z^2)\\ & \quad +\frac{(1-z^2)^2}{2\gamma _*} , \\ \left( \frac{\partial ^2 h}{\partial \nu^2}\right) _{\nu =0}= & {} 5z^3 - 3z ,\ \ \left( \frac{\partial ^3 h}{\partial \nu^3}\right) _{\nu =0} = \frac{15}{2}\gamma _*z^2 - \frac{3}{2}\gamma _* , \\ \psi _0= & {} \Gamma \left( \frac{1}{4}\right) \frac{\varkappa b_1}{4\gamma _*^3 p^{-1/4}} \left( z^2 - z^4 + \frac{z^6-1}{3} \right) , \\ \psi _1= & {} 2\Gamma \left( \frac{1}{2} \right) \frac{b_1 \varkappa ^2}{\gamma _*^2 p^{1/2}} \\{} & {} \times \left( \frac{1}{6} + \frac{2z}{19} - \frac{z^2}{2} -\frac{4z^3}{19} + \frac{2z^5}{19} + \frac{z^4}{2} - \frac{z^6}{6} \right) , \\ \psi _2= & {} \Gamma \left( \frac{3}{4}\right) \frac{b_1 \varkappa ^3}{\gamma _* p^{3/4}} \\{} & {} \times \left( \frac{1}{25} + \frac{z}{15} - \frac{1}{4}z^2 - \frac{2}{15}z^3\right.\\ & \left. + \frac{14}{25}z^4 - \frac{1}{15}z^5 - \frac{17}{100}z^6 \right) , \\ \psi _3= & {} \frac{4}{3p} \left( -\frac{3}{20} - \frac{39}{100}z - \frac{3}{5}z^2 - \frac{13}{20}z^3\right.\\ & \left. + \frac{13}{10}z^4 + \frac{1}{100}z^5 - \frac{2}{5}z^6 \right) , \\ z= & {} z(x_1(t)) = 1 - \frac{\gamma _*}{2}x_1(t), \quad x_1(t) = \int \limits _0^t \varpi (t_1) \textrm{d} t_1 . \end{aligned}$$

Apendix B

The i-th approximations for the particle-radius distribution function, the total number of particles, and their average size are defined by the following expressions

$$\begin{aligned} \Psi _i(x_1(t),s)= & {} \frac{1}{\sqrt{1-\gamma _*s}} E_{1i}^{-1}(t,s) \exp \left[ -\frac{p(1-E_{1i}^2)}{E_{1i}^2}\right] , \\ E_{1i}(x_1(t), s)= & {} 1 - b_1\int \limits _0^{x_1(t)+\frac{2\sqrt{1-\gamma _*s}}{\gamma _*} - \frac{2}{\gamma _*}} \sum \limits _{k=0}^{i} p^{-(k+1)/4}a_k(x_2) \textrm{d}x_2 , \\ N_i(x_1(t))= & {} \frac{1}{l_0^3} \int \limits _0^{s_m(x_1(t))} \Psi _i(x_1(t), s) \textrm{d}s, \\ \bar{L}_i(x_1(t))= & {} l_0 \int \limits _0^{s_m(x_1(t))}s\Psi _i\textrm{d}s \left( \int \limits _0^{s_m(x_1(t))}\Psi _i\textrm{d}s \right) ^{-1} , \end{aligned}$$

where $i = 0,1,2,....$ and

$$\begin{aligned} n_i(x_1(t)) = \frac{N_i(x_1(t))}{N_3(x_*)}, \quad u_i(x_1(t)) = \frac{\bar{L}_i(x_1(t))}{\bar{L}_3(x_*)} . \end{aligned}$$