Igniting homogeneous nucleation

Transient homogeneous nucleation is studied in the limit of large critical sizes. Starting from pure monomers, three eras of transient nucleation are characterized in the classic Becker-Döring kinetic equations with two different models of discrete diffusivity: the classic Turnbull-Fisher formula and an expression describing thermally driven growth of the nucleus. The latter diffusivity yields time lags for nucleation which are much closer to values measured in experiments with disilicate glasses. After an initial stage in which the number of monomers decreases, many clusters of small size are produced and a continuous size distribution is created. During the second era, nucleii are increasing steadily in size in such a way that their distribution appears as a wave front advancing towards the critical size for steady nucleation. The nucleation rate at critical size is negligible during this era. After the wave front reaches critical size, it ignites the creation of supercritical clusters at a rate that increases monotonically until its steady value is reached. Analytical formulas for the transient nucleation rate and the time lag are obtained that improve classical ones and compare very well with direct numerical solutions.

Figure 1

(a) Scaled activation energy $g_k∕g_m$ as a function of the scaled size $k∕k_c$. (b) Scaled dimensionless density $r=\rho e^{\alpha }$ as a function of the scaled dimensionless monomer density $r_1={\rho }_1{e}^{\alpha }$ for the equilibrium distribution (solid line). Data correspond to liquid iron at maximum undercooling (dot-dashed line), whereas for disilicate glass, $\rho \approx {\rho }_1$ (solid line).

Figure 2

(a) Comparison of $s_n\left(t\right)$ evaluated (at different times) from the numerical solution of the discrete equations (26) to the asymptotic result (54) (solid line). (b) $K\left(T\right)$ calculated from Eq. (41) with $K\left(0\right)={ϵ}^3$ (solid line) is compared to the numerically obtained position of the wave front. Data correspond to disilicate glass at $820\mathrm{K}$. All variables are written in dimensionless units.

Figure 3

(a) Evolution of the dimensionless flux at critical size $j\left(t\right)$, and (b) number of clusters surpassing critical size $N_c\left(t\right)$ as a function of dimensionless time for disilicate glass at $820\mathrm{K}$, $k_c=34$. Solid lines correspond to numerical results, dashed lines to the approximation given by Eq. (58), dot-dashed lines to the linearization approximation (67), and dotted lines to the approximation (C8) corresponding to linearizing the equations for $K\left(T\right)$ and $A\left(T\right)$ as in Appendix .

Figure 4

Same as in Fig. 3 for disilicate glass at $703\mathrm{K}$, $k_c=18$.

Figure 5

(a) Evolution of the dimensionless flux at critical size $j\left(t\right)$, and (b) number of clusters surpassing critical size $N_c\left(t\right)$ as a function of time (in dimensionless units) for disilicate glass at $820\mathrm{K}$, $k_c=34$. Solid lines correspond to numerical results, dashed lines to the approximation given by Eqs. (76, 77, 78, 79), and dot-dashed lines to Eqs. (81, 82, 83).

Figure 6

Same as Fig. 5 for disilicate glass at $703\mathrm{K}$, which has a critical size $k_c=18$.