Statistical Approach to Crystal Nucleation in Glass-Forming Liquids

Abstract

In this work, methods of description of crystal nucleation by using the statistical approach are analyzed. Findings from classical nucleation theory (CNT) for the average time of formation of the first supercritical nucleus are linked with experimental data on nucleation in glass-forming liquids stemming from repetitive cooling protocols both under isothermal and isochronal conditions. It is shown that statistical methods of lifetime analysis, frequently used in medicine, public health, and social and behavioral sciences, are applicable to crystal nucleation problems in glass-forming liquids and are very useful tools for their exploration. Identifying lifetime with the time to nucleate as a random variable in homogeneous and non-homogeneous Poisson processes, solutions for the nucleation rate under steady-state conditions are presented using the hazard rate and related parameters. This approach supplies us with a more detailed description of nucleation going beyond CNT. In particular, we show that cumulative hazard estimation enables one to derive the plotting positions for visually examining distributional model assumptions. As the crystallization of glass-forming melts can involve more than one type of nucleation processes, linear dependencies of the cumulative hazard function are used to facilitate assignment of lifetimes to each nucleation mechanism.

Crystal Nucleation and Growth in Cross-Linked Poly(ε-caprolactone) (PCL)

“…Extending previously obtained results, in [28] the average time of formation of the first supercritical nucleus in cooling and heating was specified. These results allow one to determine the time and temperature when the nucleation-growth processes become of importance.…”

Crystal Nucleation and Growth in Cross-Linked Poly(ε-Caprolactone) (PCL)

“…Extending previously obtained results, in [29] the average time of formation of the first supercritical nucleus in cooling and heating was specified. These results allow one to determine the time and/or temperature when the nucleation-growth processes become of importance.…”

Theory of crystal nucleation of glass‐forming liquids: Some new developments

“…As such time scale, we will employ (as it is conventionally done, see Refs. 28, 31) the average time, $$\langle \tau \rangle$$, of formation of the first supercritical crystal nucleus. As shown recently in Ref.…”

“…In recent years, we addressed several new developments in the description of the phase transformation kinetics partly specific for crystallization of glass-forming liquids. In particular, we analyzed the relation between the time of formation of the first supercritical nucleus, time lag in nucleation, and steady-state nucleation rate; 14 formulated simple relations for the dependence of the surface tension on temperature and pressure generalizing the Stefan-Skapski-Turnbull rule; 15,16 performed an analysis of the conditions for applicability of the Tolman equation to the description of the curvature dependence of the surface tension in crystal nucleation; formulated expressions for the Tolman parameter for crystallization caused by variations of temperature; derived simple relations for the specification of the Tolman parameter for crystallization caused both by variation of temperature and pressure; [17][18][19] performed an analysis of the Kauzmann paradox and Kauzmann's suggestion of the existence of a kinetic spinodal to prevent it 20 and its correlation with crystallization processes; [21][22][23] advanced estimates of the maxima of nucleation, growth, and overall crystallization rates in dependence on pressure and their correlation with different measures of fragility and the glass transition pressure, 24 formulated the concept of a Kauzmann pressure and performed an analysis of its importance with respect to pressure-induced crystallization; 16 derived some general results concerning possible applications of the nucleation theorem to crystallization; 25 demonstrated the necessity of incorporation of self-consistency corrections in the specification of the work of critical cluster formation in nucleation; 26 advanced a description of anisotropic nucleation, growth, and ripening under stirring; 27 analyzed statistical methods in the description of crystal nucleation; 28 and performed an analysis of possible effects of the glass transition and the accompanying its structural relaxation of the supercooled liquid on the crystallization kinetics in the vicinity and below the glass transition temperature. [29][30][31][32][33][34] In the present paper, we would like to advance two of these developments giving, in particular, also an answer to one of the open problems as described in Ref.…”