# Glass Transition, Crystallization of Glass-Forming Melts, and Entropy

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## Abstract

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## 1. Introduction

## 2. Glass and the Glass Transition

#### 2.1. Basic Definitions and Some Comments

#### 2.2. What Is the Right Deborah Number?

## 3. Residual Entropy of Glasses

#### 3.1. A Brief Overview of Some Recent Discussions

#### 3.2. Residual Entropy: A Simple Model

#### 3.3. On the Behavior of the Thermodynamic Coefficients in the Glass-Transition Range

## 4. Is the Kauzmann Paradox Really in Conflict with Basic Laws of Nature?

## 5. Summary of Results and Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**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 [64]). The reduced thermodynamic driving force, $\Delta g\left(T\right)/\Delta g\left({T}_{K}\right)$, is also shown; it has a maximum at the Kauzmann temperature, ${T}_{K}$ [65]. 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 [60], as shown here in the figure.

**Figure 2.**(

**a**) Dependence of the configurational entropy, ${S}_{conf}$, on the structural-order parameter, $\xi $, according to Equation (5). (

**b**) Equilibrium value, ${\xi}_{e}$, of the structural-order parameter as a function of reduced temperature, $\theta =(T/{T}_{m})$, according to Equation (6). (

**c**) Equilibrium value of the configurational entropy, ${S}_{conf}^{\left(e\right)}$, as a function of the reduced temperature, $T/{T}_{m}$, according to Equations (5) and (6).

**Figure 3.**Dependence of the structural-order parameter, $\xi $, on the reduced temperature, $\theta =(T/{T}_{m})$, for (

**a**) one particular and (

**b**) some set of heating and cooling rates. In cooling, the $\xi =\xi \left(\theta \right)$ curve decreases monotonically with decreasing temperature and approaches a constant value corresponding to the frozen-in state of a glass. The respective values of the residual or frozen-in configurational entropy are shown in dependence on cooling rate in (

**c**). The results are in full agreement with the ideas concerning non-zero values of the residual entropy in vitrification as developed in the last century and reconfirmed by the discussion reviewed in the preceding section.

**Figure 4.**Entropy production, $(1/R)({d}_{i}S/d\theta )$, in dependence on reduced temperature, $\theta =(T/{T}_{m})$ for (

**a**) one particular and (

**b**) some set of cooling and heating rates, ${q}_{\theta}$. With a decrease in the cooling rate, the glass transition temperature is shifted to lower temperatures [83,84], and the width of the glass transition region decreases [86]. In addition, in (

**c**), the total entropy produced in the respective cooling and heating runs is shown in dependence on the rate of change of temperature.

**Figure 5.**Calculated reduced heat capacity curves for various scanning rates (cooling rate is the same as the heating rate) to clarify the positions of differently defined glass transition temperatures. (

**a**) ${C}_{p}$ curves for three different cooling and heating rates (${q}_{\theta}={10}^{-4}$, ${10}^{-2}$, and 1·s${}^{-1}$). (

**b**) Visual definition of respective glass transition temperatures obtained: ${T}_{g,mpc}$, ${T}_{g,mph}$, ${T}_{g,overshoot}$, ${T}_{g,onset\phantom{\rule{0.166667em}{0ex}}cooling}$ and ${T}_{g,onset\phantom{\rule{0.166667em}{0ex}}heating}$. (

**c**) Calculated difference $\Delta T$ between ${T}_{g,onset\phantom{\rule{0.166667em}{0ex}}cooling}$ and ${T}_{g,overshoot}$ for a wider range of rates of change of temperature. A nearly linear dependence is observed.

**Figure 7.**Structural relaxation time, ${\tau}_{R}$; the induction time, ${\tau}_{ind}$, required to establish steady-state nucleation; and the average time of formation of a supercritical cluster at steady-state nucleation conditions, ${\langle \tau \rangle}_{ss}\cong 1/\left({J}_{ss}V\right)$, are shown in dependence on temperature (

**a**,

**c**). The average time of formation of the first supercritical nucleus, $\langle \tau \rangle $, can be expressed generally in a good approximation as the sum $\langle \tau \rangle \cong {\langle \tau \rangle}_{ss}+{\tau}_{ind}$ with ${\tau}_{ind}\cong {\tau}_{n}$ [129]. The dependence of $\langle \tau \rangle $ on temperature is illustrated in the lower part of (

**b**,

**d**). For temperatures near to the melting temperature, $\langle \tau \rangle $ is always determined by ${\langle \tau \rangle}_{ss}$. At the intersection of ${\langle \tau \left(T\right)\rangle}_{ss}$ with ${\tau}_{ind}\left(T\right)$, $\langle \tau \left(T\right)\rangle $ becomes dominated by the values of ${\tau}_{ind}$. Values of the parameters employed in the computation of the nucleation rates and related quantities are taken for $2{\mathrm{Na}}_{2}\mathrm{O}\xb71\mathrm{CaO}\xb73{\mathrm{SiO}}_{2}$ [131,132]. In (

**b**,

**d**), the Vogel temperature is replaced by the Kauzmann temperature (see also text).

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**MDPI and ACS Style**

Schmelzer, J.W.P.; Tropin, T.V.
Glass Transition, Crystallization of Glass-Forming Melts, and Entropy. *Entropy* **2018**, *20*, 103.
https://doi.org/10.3390/e20020103

**AMA Style**

Schmelzer JWP, Tropin TV.
Glass Transition, Crystallization of Glass-Forming Melts, and Entropy. *Entropy*. 2018; 20(2):103.
https://doi.org/10.3390/e20020103

**Chicago/Turabian Style**

Schmelzer, Jürn W. P., and Timur V. Tropin.
2018. "Glass Transition, Crystallization of Glass-Forming Melts, and Entropy" *Entropy* 20, no. 2: 103.
https://doi.org/10.3390/e20020103