# Effects of Structural Relaxation of Glass-Forming Melts on the Overall Crystallization Kinetics in Cooling and Heating

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## Abstract

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## 1. Introduction

## 2. Basic Equations

#### 2.1. Johnson–Mehl–Avrami–Kolmogorov (JMAK) Equation

#### 2.2. Description of the Rates of Nucleation and Growth and the Kinetics of Relaxation

#### 2.3. Possible Extensions

## 3. Results of Numerical Computations

## 4. Theoretical Analysis

#### 4.1. Some General Considerations

#### 4.2. Cold Crystallization Peak Temperature in Heating as a Function of the Heating Rate: Homogeneous Nucleation

#### 4.3. Crystallization Peak Temperature in Cooling as a Function of the Cooling Rate: Heterogeneous Nucleation

## 5. Summary of Results and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Characteristic Times of Relaxation and Crystallization: Some General Considerations

“Let us denote by ${T}_{k}$ the temperature at which the two kinds of barriers become equal. ${T}_{k}$ may be above or below the glass–transformation point, ${T}_{g}$, as defined in terms of the ’conventional’ duration of an experiment. If ${T}_{k}$ is above ${T}_{g}$, it will be impossible for the liquid to be studied as a liquid at temperatures between ${T}_{k}$ and ${T}_{g}$, other than by experiments of much shorter duration than the ’conventional’ one, since it will crystallize spontaneously during any experiments requiring more time than the average time between simple molecular jumps. According to the concepts presented in the previous part of this paper, adequate measurements under these circumstances would result in glass-like properties for the liquid. On the other hand, if ${T}_{k}$ is below ${T}_{g}$, it is still possible to distinguish between a glassy state and a true metastable liquid between ${T}_{g}$ and ${T}_{k}$. But below ${T}_{k}$ no such distinction is possible; the glass is then the only experimentally attainable form of the liquid …Accordingly, provided the free energy barriers vary with the temperature in the way that we have postulated, it is not permissible to extrapolate the curves in figures 3 to 6 indefinitely below ${T}_{g}$ and to infer thereby the existence of a ’thermodynamic’ glass–like transition”.

“Suppose that when the temperature is lowered a point is eventually reached at which the free energy barrier to crystal nucleation becomes reduced to the same height as the barriers to the simpler motions…At such temperatures the liquid would be expected to crystallize just as rapidly as it changed its typically liquid structure to conform to a temperature or pressure change in its surroundings…There are good theoretical reasons for believing in the existence of such a ‘pseudo-critical temperature’” ([54], pp. 220 and 247).

“In the past there has been a considerable amount of speculation concerning the existence of a critical point between crystalline and liquid states analogous to the critical point between liquids and gases. No experimental evidence for or against such a critical point has ever been found [75], though there is reason to believe that none is possible (Bernal [76]; but see Frenkel … [77]). It is apparent, however, that the behavior with which we are here concerned has a certain similarity to the behavior at a critical point in that here, as at a true critical point, the free energy barrier between the crystal and the liquid disappears. On the other hand, there is a fundamental difference in that the two states do not really merge and their free energies are decidedly different …, so that one cannot go reversibly from the one state to the other without a normal phase change” ([54], p. 248).

“The decrease of ${W}_{c}$ to zero is a failing of this particular model, since there is no point at which the liquid becomes unstable relative to the solid.”

“Nonetheless, it does indicate that a properly constructed density-functional model could describe the transition from a nucleation and growth mechanism to a spinodal transformation, which the CNT cannot do”.

## Appendix B. Application of the Lattice-Hole Model in the Computations and Some Possible Generalization of this Model

**Figure A1.**Determination of the degree, ${\alpha}_{3}$, and the rate, $d{\alpha}_{3}/dt$, of overall crystallization in dependence on time, t, in cooling (

**left side**) and heating (

**right side**) for sets of cooling (${q}_{c}$) and heating (${q}_{h}$) rates as shown in the figures. In contrast to Figure 3, here the effect of deviations in the state of the liquid from metastable equilibrium is accounted for. The computations are performed utilizing Equations (32) and (33).

**Figure A2.**Determination of the degree, ${\alpha}_{3}$, and the rate, $d{\alpha}_{3}/dt$, of overall crystallization in dependence on temperature, $\theta =T/{T}_{m}$, in cooling (left side) and heating (right side) for sets of cooling (${q}_{c}$) and heating (${q}_{h}$) rates as shown in the figures. In contrast to Figure 4, here the effect of deviations in the state of the liquid from metastable equilibrium is accounted for. The computations are performed utilizing Equations (32) and (33).

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**Figure 1.**Structural order parameter, $\xi $, and its equilibrium value, ${\xi}_{e}$, in terms of dependence on reduced temperature, $\theta =T/{T}_{m}$. (

**a**) Dependence of the equilibrium value of the structural order parameter for the whole range of temperatures between melting or liquidus temperature, ${T}_{m}$, and absolute zero as obtained in the framework of the lattice model employed here. (

**b**) Typical behavior of the structural order parameter, $\xi $, in dependence on temperature in the vicinity of the glass transition range if the liquid is cooled down and heated with the same constant rate of change in temperature. The dependencies $\xi \left(T\right)$ are shown by full curves if the system is cooled down (blue curve) and heated (red curve) with a constant rate (here taken as equal to $(dT/dt)=1.3$ K/s or $(d\theta /dt)={10}^{-3}{\mathrm{s}}^{-1}$); the dashed curve shows the equilibrium value of this parameter in the given range of temperature. The figure is taken from [21] (Creative Commons Attribution License).

**Figure 2.**Dependence of the steady-state nucleation rate, J, and the growth rate, u, on temperature in reduced coordinates, $\theta =T/{T}_{m}$, for the model employed here in the computations in the limiting case that the liquid is always in a metastable state.

**Figure 3.**Determination of the degree, ${\alpha}_{3}$, and the rate, $d{\alpha}_{3}/dt$, of overall crystallization in dependence on time, t, in cooling (

**left side**) and heating (

**right side**) for sets of cooling (${q}_{c}$) and heating (${q}_{h}$) rates as shown in the figures. In the presentation of the results of the numerical computations, we always express the heating rate in reduced variables as $q=d(T/{T}_{m})/dt$. Cooling is started at a temperature equal to $T={T}_{m}$, while heating is supposed to start at a temperature $T=({T}_{m}/2)$.

**Figure 4.**Determination of the degree, ${\alpha}_{3}$, and the rate, $d{\alpha}_{3}/dt$, of overall crystallization in dependence on temperature, $\theta =T/{T}_{m}$, in cooling (

**left side**) and heating (

**right side**) for sets of cooling (${q}_{c}$) and heating (${q}_{h}$) rates as shown in the figures.

**Figure 5.**Determination of the degree, ${\alpha}_{3}$, and the rate, $d{\alpha}_{3}/dt$, of overall crystallization in dependence on time, t, in cooling (

**left side**) and heating (

**right side**) for sets of cooling (${q}_{c}$) and heating (${q}_{h}$) rates as shown in the figures. In contrast to Figure 3, here the effect of deviations in the state of the liquid from metastable equilibrium is accounted for. The computations are performed utilizing Equations (58) and (59) with ${\mathsf{\Omega}}_{\mathsf{\Delta}g}=1$ and ${\mathsf{\Omega}}_{\sigma}=1$. The results obtained in such a way are shown by dashed curves; the curves shown in Figure 3 and Figure 4 are given for comparison as full curves again.

**Figure 6.**Determination of the degree, ${\alpha}_{3}$, and the rate, $d{\alpha}_{3}/dt$, of overall crystallization in dependence on temperature, $\theta =T/{T}_{m}$, in cooling (

**left side**) and heating (

**right side**) for sets of cooling (${q}_{c}$) and heating (${q}_{h}$) rates as shown in the figures. In contrast to Figure 4, here the effect of deviations in the state of the liquid from metastable equilibrium is accounted for. The computations are performed utilizing Equations (58) and (59) with ${\mathsf{\Omega}}_{\mathsf{\Delta}g}=1$ and ${\mathsf{\Omega}}_{\sigma}=10$. The results obtained in such a way are shown by dashed curves; the curves shown in Figure 3 and Figure 4 are given for comparison as full curves again.

**Figure 7.**Determination of the degree, ${\alpha}_{3}$, and the rate, $d{\alpha}_{3}/dt$, of overall crystallization in dependence on temperature, $\theta =T/{T}_{m}$, in cooling (

**left side**) and heating (

**right side**) for sets of cooling (${q}_{c}$) and heating (${q}_{h}$) rates as shown in the figures. In contrast to Figure 4, here the effect of deviations in the state of the liquid from metastable equilibrium is accounted for. The computations are performed utilizing Equations (58) and (59) with ${\mathsf{\Omega}}_{\mathsf{\Delta}g}=1$ and ${\mathsf{\Omega}}_{\sigma}=1$. The results obtained in such a way are shown by dashed curves; the curves shown in Figure 3 and Figure 4 are given for comparison as full curves again.

**Figure 8.**Determination of the degree, ${\alpha}_{3}$, and the rate, $d{\alpha}_{3}/dt$, of overall crystallization in dependence on time, t, in cooling (

**left side**) and heating (

**right side**) for sets of cooling (${q}_{c}$) and heating (${q}_{h}$) rates as shown in the figures. In contrast to Figure 3, here the effect of deviations in the state of the liquid from metastable equilibrium is accounted for. The computations are performed utilizing Equations (58) and (59) with ${\mathsf{\Omega}}_{\mathsf{\Delta}g}=1$ and ${\mathsf{\Omega}}_{\sigma}=10$. The results obtained in such a way are shown by dashed curves; the curves shown in Figure 3 and Figure 4 are given for comparison as full curves again.

**Figure 9.**Crystallization peak temperatures, ${\theta}_{p}={T}_{p}/{T}_{m}$, in dependence on cooling (blue) and heating (red) rates as obtained from the numerical computations. The results are given for nucleation and growth in metastable liquids (presented in Figure 3 and Figure 4) and accounting for deviations from metastability in the form as shown in Figure 5 and Figure 7 (${\mathsf{\Omega}}_{\sigma}=1$), respectively, and Figure 6 and Figure 8 (${\mathsf{\Omega}}_{\sigma}=10$).

**Figure 10.**Normalized steady-state nucleation rate, ${J}_{ss}/{J}_{ss}^{\left(max\right)}$, and normalized crystal growth rate, $u/{u}_{max}$, in dependence on reduced temperature, $T/{T}_{m}$. Here ${J}_{ss}^{\left(max\right)}$ is the maximum nucleation rate and ${u}_{max}$ is the maximum growth rate obtained via experiment; ${T}_{m}$ is the melting or liquidus temperature. The blue curve (1) shows the theoretical result when the kinetic term in the expression for the nucleation rate is determined via appropriate diffusion coefficients; the green curve (2) is drawn under the assumption of validity of the Stokes–Einstein–Eyring equation, allowing one to replace the diffusion coefficient with viscosity. Its wide coincidence with experimental data is reached by employing appropriate expressions for the curvature dependence of the surface tension (for details, see [43]). The reduced thermodynamic driving force, $\mathsf{\Delta}g\left(T\right)/\mathsf{\Delta}g\left({T}_{K}\right)$, is also shown; it has a maximum at the Kauzmann temperature, ${T}_{K}$ [68]. It is evident that crystallization occurs only in a relatively small temperature range. Typically, the maximum of the growth rate, ${T}_{max}^{\left(growth\right)}$, is located at temperatures much higher than the maximum of the steady-state nucleation nucleation rate [65,66,67], as shown here in the figure. The figure is taken from [56] (Creative Commons Attribution License).

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**MDPI and ACS Style**

Schmelzer, J.W.P.; Tropin, T.V.; Schick, C.
Effects of Structural Relaxation of Glass-Forming Melts on the Overall Crystallization Kinetics in Cooling and Heating. *Entropy* **2023**, *25*, 1485.
https://doi.org/10.3390/e25111485

**AMA Style**

Schmelzer JWP, Tropin TV, Schick C.
Effects of Structural Relaxation of Glass-Forming Melts on the Overall Crystallization Kinetics in Cooling and Heating. *Entropy*. 2023; 25(11):1485.
https://doi.org/10.3390/e25111485

**Chicago/Turabian Style**

Schmelzer, Jürn W. P., Timur V. Tropin, and Christoph Schick.
2023. "Effects of Structural Relaxation of Glass-Forming Melts on the Overall Crystallization Kinetics in Cooling and Heating" *Entropy* 25, no. 11: 1485.
https://doi.org/10.3390/e25111485