Predicting the Thermodynamic Ideal Glass Transition Temperature in Glass-Forming Liquids

The Kauzmann temperature TK is a lower limit of glass transition temperature, and is known as the ideal thermodynamic glass transition temperature. A supercooled liquid will condense into glass before TK. Studying the ideal glass transition temperature is beneficial to understanding the essence of glass transition in glass-forming liquids. The Kauzmann temperature TK values are predicted in 38 kinds of glass-forming liquids. In order to acquire the accurate predicted TK by using a new deduced equation, we obtained the best fitting parameters of the deduced equation with the high coefficient of determination (R2 = 0.966). In addition, the coefficients of two reported relations are replaced by the best fitting parameters to obtain the accurate predicted TK, which makes the R2 values increase from 0.685 and 0.861 to 0.970 and 0.969, respectively. Three relations with the best fitting parameters are applied to obtain the accurate predicted TK values.


Introduction
If crystallization can be avoided by sufficiently rapid cooling, a supercooled liquid will become a glassy state at glass transition temperature T g , at which the viscosity of the supercooled liquid is typically 10 12 Pa s (10 poise = 1 Pa s) [1][2][3][4][5]. Liquid-glass transitions are generally observed in various supercooled liquids, including molecular liquids, ionic liquids, metallic liquids, oxides, and chalcogenides [5,6]. Figure 1 shows the temperature dependence of the entropy difference between various supercooled liquids and their crystalline phases [5,7]. With temperature decreases, their entropic surplus is consumed, and the glass transition sets in when the slope of the curve changes. For lactic acid, its glass transition temperature T g has been marked in Figure 1. Its curve can then be extrapolated to the Kauzmann temperature T K , at which point ∆S will vanish. In other words, the entropy of the supercooled liquid equals the entropy of its crystalline counterpart at T K . Below T K , the entropy of the supercooled liquid will become less than that of its crystalline phase. However, it is difficult to see how a disordered and nonperiodic liquid has a lower entropy than a periodic crystal of the same density [8]. As a consequence, Kauzmann temperature T K is a lower limit of glass transition temperature (i.e., the thermodynamic ideal glass transition temperature) and the supercooled liquid will condense into glass, having the same entropy as the perfect crystal at T K [8].
The glass transition temperature T g plays an important role in liquid-glass transition. T g has significant thermophysical properties for predicting glass-forming ability (GFA) and the stability of glass formers. Thermodynamically, the lowest value of T g is the Kauzmann temperature T K for a certain glass-forming liquid. In other words, the Kauzmann temperature T K is the lowest temperature at which a supercooled liquid can exist. The Kauzmann temperature T K is studied, which is beneficial to understand the nature of glass transition, and to find a correlation between the measured glass respectively. (Adapted from Ref. [5,7]).
The glass transition temperature Tg plays an important role in liquid-glass transition. Tg has significant thermophysical properties for predicting glass-forming ability (GFA) and the stability of glass formers. Thermodynamically, the lowest value of Tg is the Kauzmann temperature TK for a certain glass-forming liquid. In other words, the Kauzmann temperature TK is the lowest temperature at which a supercooled liquid can exist. The Kauzmann temperature TK is studied, which is beneficial to understand the nature of glass transition, and to find a correlation between the measured glass transition temperature Tg and the thermodynamically ideal glass transition temperature TK. In addition, it can be seen that fragility parameter m is related to Tg and TK (see below). The fragility parameter m is applied to describe the degree of departure from an Arrhenius relation of the temperature dependence of viscosity. That is, Tg and TK can also be applied to describe the temperature dependence of viscosity in glass-forming liquids. Therefore, studying the temperature TK is a classic problem in amorphous materials. TK temperatures in various glass-forming liquids have been calculated, such as in metallic liquids [4,[9][10][11][12][13][14][15][16], molecular liquids [6,17], ionic liquids [6,17], and oxides [6,12]. The Kauzmann temperature TK will be predicted by Tg and m in glass-forming liquids.

Expressions of Predicting TK
Universally, TK can be acquired by [5,9,11,14,18]: where ΔSm is the entropy of fusion at the melting point Tm, and Δcp l-c (T) is the specific heat capacity (Adapted from Ref. [5,7]).

Expressions of Predicting T K
Universally, T K can be acquired by [5,9,11,14,18]: where ∆S m is the entropy of fusion at the melting point T m , and ∆c p l-c (T) is the specific heat capacity difference between the supercooled liquid and its crystalline counterpart. If ∆S m and ∆c p l-c (T) are acquired, T K will be calculated. The entropy of fusion ∆S m , can be obtained by ∆S m =∆H m /T m , where ∆H m is the heat of fusion, which can be obtained by the integration of the melting peak [19]. The so-called "step method", which consists of heating the sample to a certain temperature with a constant rate, and then annealing isothermally during each step, can be applied to determine the specific heat capacity of the sample on heating, in reference to the specific heat capacity of a standard sapphire [4,15]. The data of c p (T) sample can be calculated by the following equations [4,15]: where c p (T) sample and c p (T) sapphire are the specific heat capacity of sample and sapphire, respectively, m i the mass, µ i the mole mass, and Q * i the heat flux. Meanwhile, the temperature dependence of the specific heat capacity c p liquid (T) of the supercooled liquid can be expressed as [4,11,15]: where R is gas constant. The specific heat capacity c p crystal (T) of the crystal can be expressed as [4,11,15]: The parameters of expressions for c p liquid (T) and c p crystal (T) can be determined by fitting the data measured in steps in reference to sapphire. Therefore, the specific heat capacity difference between the supercooled liquid and its crystalline counterpart can be calculated by Equations (3) and (4), with the known parameters. T K can be calculated by the above formulas so far. From the above analysis, acquiring the Kauzmann temperature T K is cumbersome and time-consuming. Therefore, the easily obtained parameters are applied to predict T K . The Angell's fragility parameter m, based on viscosity or relaxation time, is defined as [1, [20][21][22][23]: A similar fragility has been defined as [17]: where m min = log 10 (η g /η 0 ). η g denotes viscosity (typically 10 12 Pa s) at glass transition temperature T g . η 0 is the high temperature limit of viscosity, which can be determined by the following equation [2,24]: where h is Planck's constant, N A is Avogadro's number, ρ is the density of the liquid and M is the molar mass. The η 0 value is about set as 10 −5 Pa s [2,17,[24][25][26]. Thus, generally, the log 10 (η g /η 0 ) value is equal to 17. The expressions related to T K have been studied, and they can be utilized to calculate T K , which make calculation simpler. T K as a function of T g and Angell's fragility parameter m has been reported, and the expression can be described by [1, 16,17,25]: From Equation (8), T K can be expressed as: The other expression of T K as a function of T g and m can be expressed as [1, 16,17,25]: Equation (10) is transformed into: Materials 2020, 13, 2151 Furthermore, a new expression of predicting T K as a function of T g and m is also deduced by us, and this expression is derived as follows. Another expression of the Kauzmann temperature is presented by [27]: Additionally, m can be calculated by the following Equation [27]: where Λ a is the constant and equals 40. ∆c p l-g (T g )=c p liquid (T g )-c p glass (T g ) is the specific heat capacity difference between the supercooled liquid and its glass state at T g . When ∆c p l-g (T g ) is replaced by ∆c p l-c (T g ), the numerical factor would increase from 40 to 43, but the quality of the correlation remains unchanged, where ∆c p l-c (T g )=c p liquid (T g )-c p crystal (T g ) is the specific heat capacity difference between the supercooled liquid and its crystalline counterpart at T g [27]. Hence, ∆c p l-g (T g ) is replaced by ∆c p l-c (T g ), and m can be expressed as: where Λ b is the constant and equals 43. The ratio T m /T g is about constant Λ c , which equals 3/2 [27][28][29][30].
Plugging this T m /T g relation into Equation (14): From Equation (12) and Equation (15), we obtain: The expanded Equation (16) can be expressed by: It can be seen that these expressions of predicting T K are expressed as the function of T g and m from Equations (9), (11), and (17). Because T g and m have been reported for a lot of glass-forming liquids, predicting T K will be made simpler and more convenient by the above formulae.

Methods
As can be seen from the above, Equations (9), (11), and (17) can be applied to predict T K . In order to obtain accurate T K values, the coefficient of determination, R 2 is applied to evaluate the accuracy of the predicted T K . In statistics, generally, R 2 is defined as: R 2 = 1−SS res /SS tot , where SS res is the sum of squares of residuals and SS tot is the total sum of squares. R 2 is a statistical measure of how well the predicted T K values approximate the reported T K values. The higher is the R 2 value (0 ≤ R 2 ≤ 1), the more accurate is the predicted T K . The predicted T K values perfectly fit the reported T K when R 2 equals 1.

Results and Discussion
The values of the glass transition temperature T g , Angell's fragility parameter m, and the Kauzmann temperature T K for various glass formers are listed in Table 1. Figure 2a shows the predicted Kauzmann temperature T K cal1 , according to Equation (9) at m min = 17. Many reported T K values for various glass formers do not fall on the curve of the predicted T K cal1 . Meanwhile, the R 2 value of this correlation equals 0.685, which is relatively low. It indicates that the predicted T K cal1 values by using Equation (9) at m min = 17 are inaccurate. Although the log 10 (η g /η 0 ) (i.e., m min ) value is generally equal to 17, the viscosity change in the glass transition is approximately two orders of magnitude [16,18]. Therefore, the log 10 (η g /η 0 ) value is considered to have a range from 15 to 17 [16]. In fact, generally, the η 0 value is set as about 10 −5 Pa s, but η 0 values have differences in some amorphous materials [31]. This will cause a change of the log 10 (η g /η 0 ) value as well. In our previous study, the log 10 (η g /η 0 ) value is considered to have a range from 14 to 18 [32]. As a result, the m min value slightly fluctuates. In order to obtain the most accurate predicted Kauzmann temperature by using Equation (9), we regard the m min value as a fitting parameter, which has no restrictions, and can be an arbitrary value. Therefore, we obtain the best fit and the most accurate predicted Kauzmann temperature T K cal1* by using Equation (9), when m min equals 9.96. Although there is a difference between this value (m min = 9.96) and the m min value obtained by η g and η 0 of the amorphous materials, and this value may not have a physical meaning, the most accurate T K cal1* by using Equation (9) at m min = 9.96 can be obtained. Our purpose is to make the Kauzmann temperature accurately predictable, so it is feasible that the most accurate predicted Kauzmann temperature T K cal1* values are obtained by using Equation (9) at m min = 9.96. Figure 2b shows the predicted Kauzmann temperature T K cal1* , according to Equation (9) at m min = 9.96.
From Figure 2, it can be seen that the R 2 value greatly increases from 0.685 to 0.970 when the predicted values obtained by using Equation (9) at m min = 17 are replaced by those obtained by using Equation (9) at m min = 9.96. It indicates that the accuracy of the predicted values obtained by using Equation (9) at m min = 9.96 are greatly improved. The predicted values obtained by using Equation (9) at m min = 17 and 9.96 have also been listed in Table 1 for convenience in comparing the predicted (T K cal1 and T K cal1* ) values with the reported T K values. Figure 3a illustrates the predicted Kauzmann temperature T K cal2 by Equation (11) at m min = 17.
Compared to Figure 2a, the predicted Kauzmann temperature T K cal2 values (R 2 = 0.861) obtained by Equation (11) at m min = 17 are more accurate than those obtained by Equation (9) at m min = 17. In order to obtain the most accurate predicted Kauzmann temperature by using Equation (11), we also regard the m min value as the fitting parameter. Therefore, we obtain the best fit and the most accurate predicted Kauzmann temperature T K cal2* by using Equation (11), when m min equals 11.50. Figure 3b shows the predicted Kauzmann temperature T K cal2* , according to Equation (11) at m min = 11.50. From Figure 3, it can be seen that the R 2 value increases from 0.861 to 0.969 when the predicted values obtained by using Equation (11) at m min = 17 are replaced by those obtained by using Equation (11) at m min = 11.50. The predicted values obtained by using Equation (11) at m min = 17, and 11.50 are also listed in Table 1.
Materials 2020, 13, x FOR PEER REVIEW 5 of 10 predicted TK values approximate the reported TK values. The higher is the R 2 value (0 ≤ R 2 ≤ 1), the more accurate is the predicted TK. The predicted TK values perfectly fit the reported TK when R 2 equals 1.

Results and Discussion
The values of the glass transition temperature Tg, Angell's fragility parameter m, and the Kauzmann temperature TK for various glass formers are listed in Table 1. Figure 2a shows the predicted Kauzmann temperature TK cal1 , according to Equation (9) at mmin = 17. Many reported TK values for various glass formers do not fall on the curve of the predicted TK cal1 . Meanwhile, the R 2 value of this correlation equals 0.685, which is relatively low. It indicates that the predicted TK cal1 values by using Equation (9) at mmin = 17 are inaccurate. Although the log10(ηg/η0) (i.e., mmin) value is generally equal to 17, the viscosity change in the glass transition is approximately two orders of magnitude [16,18]. Therefore, the log10(ηg/η0) value is considered to have a range from 15 to 17 [16]. In fact, generally, the η0 value is set as about 10 −5 Pa s, but η0 values have differences in some amorphous materials [31]. This will cause a change of the log10(ηg/η0) value as well. In our previous study, the log10(ηg/η0) value is considered to have a range from 14 to 18 [32]. As a result, the mmin value slightly fluctuates. In order to obtain the most accurate predicted Kauzmann temperature by using Equation (9), we regard the mmin value as a fitting parameter, which has no restrictions, and can be an arbitrary value. Therefore, we obtain the best fit and the most accurate predicted Kauzmann temperature TK cal1* by using Equation (9), when mmin equals 9.96. Although there is a difference between this value (mmin = 9.96) and the mmin value obtained by ηg and η0 of the amorphous materials, and this value may not have a physical meaning, the most accurate TK cal1* by using Equation (9) at mmin = 9.96 can be obtained. Our purpose is to make the Kauzmann temperature accurately predictable, so it is feasible that the most accurate predicted Kauzmann temperature TK cal1* values are obtained by using Equation (9) at mmin = 9.96. Figure 2b shows the predicted Kauzmann temperature TK cal1* , according to Equation (9) at mmin = 9.96. From Figure 2, it can be seen that the R 2 value greatly increases from 0.685 to 0.970 when the predicted values obtained by using Equation (9) at mmin = 17 are replaced by those obtained by using Equation (9) at mmin = 9.96. It indicates that the accuracy of the predicted values obtained by using Equation (9) at mmin = 9.96 are greatly improved. The predicted values obtained by using Equation (9) at mmin = 17 and 9.96 have also been listed in Table 1 for convenience in comparing the predicted (TK cal1 and TK cal1* ) values with the reported TK values.   shows the predicted Kauzmann temperature TK cal2* , according to Equation (11) at mmin = 11.50. From Figure 3, it can be seen that the R 2 value increases from 0.861 to 0.969 when the predicted values obtained by using Equation (11) at mmin = 17 are replaced by those obtained by using Equation (11) at mmin = 11.50. The predicted values obtained by using Equation (11) at mmin = 17, and 11.50 are also listed in Table 1. A new formula (Equation (17)) has been deduced in the above introduction, whose expression is also a function of Tg and m. In the literature, Λb equals 43 [27] and Λc (the ratio Tm/Tg) is equal to about 3/2 [27][28][29][30]. We plug these values into Equation (17), and the curve of the predicted TK new is shown in Figure 4a. The R 2 value of this correlation equals 0.801. In order to obtain the most accurate predicted Kauzmann temperature by using Equation (17), we also regard Λb and Λc values as the fitting parameters. The best fit of the experimental data yields Λb = 18.47 and Λc = 1.12. Plugging the fitted values into Equation (17): new* K 1. 12 23.17 20.69 Figure 4b shows the predicted Kauzmann temperature TK new* , according to Equation (18). From Figure 4, it can be seen that the R 2 value increases from 0.801 to 0.966 when the predicted values obtained by using Equation (17) at Λb = 43 and Λc = 3/2 are replaced by those obtained by using Equation (17) with the best fitting parameters (i.e., Equation (18)). The predicted values obtained by using Equation (17) at Λb = 43 and Λc = 3/2 and using Equation (18) are also listed in Table 1.  Table 1. The values of T g , T K , and m for various glass-forming liquids. The data (numbers 19-38) were taken from Ref. [17,33].

Glass Formers
Tg A new formula (Equation (17)) has been deduced in the above introduction, whose expression is also a function of T g and m. In the literature, Λ b equals 43 [27] and Λ c (the ratio T m /T g ) is equal to about 3/2 [27][28][29][30]. We plug these values into Equation (17), and the curve of the predicted T K new is shown in Figure 4a. The R 2 value of this correlation equals 0.801. In order to obtain the most accurate predicted Kauzmann temperature by using Equation (17), we also regard Λ b and Λ c values as the fitting parameters. The best fit of the experimental data yields Λ b = 18.47 and Λ c = 1.12. Plugging the fitted values into Equation (17): Figure 4b shows the predicted Kauzmann temperature T K new* , according to Equation (18).
From Figure 4, it can be seen that the R 2 value increases from 0.801 to 0.966 when the predicted values obtained by using Equation (17) at Λ b = 43 and Λ c = 3/2 are replaced by those obtained by using Equation (17) with the best fitting parameters (i.e., Equation (18)). The predicted values obtained by using Equation (17) at Λ b = 43 and Λ c = 3/2 and using Equation (18) are also listed in Table 1.

Conclusions
The Kauzmann temperature TK in 38 kinds of amorphous materials have been predicted. Meanwhile, we regard the mmin value as the fitting parameter to improve the accuracy of predicting TK values. The coefficient of determination R 2 values increase from 0.685 and 0.861 to 0.970 and 0.969, respectively, when the coefficients of two reported relations are replaced by the best fitting parameters. In addition, a new formula of predicting TK values with R 2 = 0.966 is deduced. Therefore, three equations with the best fitting parameters have relatively high R 2 values, which indicates that they can be applied to obtain the accurate predicted TK values.

Conclusions
The Kauzmann temperature T K in 38 kinds of amorphous materials have been predicted. Meanwhile, we regard the m min value as the fitting parameter to improve the accuracy of predicting T K values. The coefficient of determination R 2 values increase from 0.685 and 0.861 to 0.970 and 0.969, respectively, when the coefficients of two reported relations are replaced by the best fitting parameters. In addition, a new formula of predicting T K values with R 2 = 0.966 is deduced. Therefore, three equations with the best fitting parameters have relatively high R 2 values, which indicates that they can be applied to obtain the accurate predicted T K values.