Nucleation and evolution of spherical crystals with allowance for their unsteady-state growth rates

Abstract

The growth dynamics of a spherical crystal in a metastable liquid is analyzed theoretically. The unsteady-state contributions to the crystal radius and its growth rate are found as explicit functions of metastability level Δ and time t. It is shown that the fundamental contribution to the growth rate represents the time independent solution of a similar temperature conductivity problem (Alexandrov and Malygin 2013 J. Phys. A: Math. Theor. 46 455101) whereas the next unsteady-state contribution is proportional to ∆ 2 t. On the basis of these explicit unsteady-state solutions, the process of transient nucleation and growth of spherical crystals in a metastable system is theoretically studied at the intermediate stage of phase transformation. A complete analytical solution for the particle-radius distribution function and metastability level is constructed with allowance for the Weber-Volmer-Frenkel-Zel'dovich and Meirs kinetic mechanisms. It is shown that the obtained unsteady-state contribution to the crystal growth rate plays an important role in the nucleation process and drastically changes the particle-radius distribution function.

On the theory of the unsteady-state growth of spherical crystals in metastable liquids

“…where √ s k = iμ k , μ k satisfies the equation μ k cos μ k = sin μ k , and 'i' is the imaginary unit. Note that ψ 3 (τ ) coincides with the solution of thermally controlled crystal growth found in [12] in the case of single-component systems (C 0 = 0). Now the first equation (2.10) can be easily integrated so that…”

“…If we restrict ourselves to the main contribution to the crystal radius and its first correction, one can obtain the following dimensional expressions for R(t) and V(t) from (2.18): mC(1, 0) represents the system supercooling. If we formally put C 0 = 0, expressions (2.19) transform to the corresponding solutions for one-component systems [12]. Note that the growth rate V(t) of crystals in binary melts is less than the rate of their growth in singlecomponent systems (m > 0, k < 1).…”

On the theory of crystal growth in metastable systems with biomedical applications: protein and insulin crystallization

“…The growth rate of crystals V is a function of time and different heat and mass transfer parameters if crystals grow in supercooled and supersaturated liquids. The analytical theories developed for the thermally controlled growth [46], diffusionally controlled growth [47] and crystal growth in binary melts [48] give the following expressions for dimensional (V) and dimensionless (U) velocities…”

“…An important point is that expressions (3.16) derived for crystal growth in metastable one-component liquids take into account the non-stationary fluctuations of the temperature (concentration) field caused by the release of latent heat of crystallization (absorption and displacement of the dissolved impurity) on the surfaces of growing spherical particles. Specifically, we note that, if it is necessary to describe the growth rates of crystals more accurately, corrections of higher orders of smallness can be taken into account (for details, see [46][47][48]). The first term U = w on the right-hand side of expression (3.16) represents the growth processes in steady-state temperature (concentration) fields and does not describe some fluctuations in the particle growth rates.…”

Phase transformations in metastable liquids combined with polymerization

“…Note that the rate of crystal growth is often independent of their radius r (see, among others, [23][24][25]).…”