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

Abstract

The Kauzmann temperature T_K 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 T_K. Studying the ideal glass transition temperature is beneficial to understanding the essence of glass transition in glass-forming liquids. The Kauzmann temperature T_K values are predicted in 38 kinds of glass-forming liquids. In order to acquire the accurate predicted T_K 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 T_K, 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 T_K values.

1. 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 10¹² 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 ΔS 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 transition temperature T_g and the thermodynamically ideal glass transition temperature T_K. In addition, it can be seen that fragility parameter m is related to T_g and T_K (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, T_g and T_K can also be applied to describe the temperature dependence of viscosity in glass-forming liquids. Therefore, studying the temperature T_K is a classic problem in amorphous materials. T_K 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 T_K will be predicted by T_g and m in glass-forming liquids.

2. Expressions of Predicting T_K

Universally, T_K can be acquired by [5,9,11,14,18]:

$$\Delta S_m={\displaystyle {\int }_{T_K}^{T_m}\frac{\Delta c_p^{l-c}(T)}{T}}dT$$

(1)

where ΔS_m is the entropy of fusion at the melting point T_m, and Δc_p^l-c(T) is the specific heat capacity difference between the supercooled liquid and its crystalline counterpart. If ΔS_m and Δc_p^l-c(T) are acquired, T_K will be calculated. The entropy of fusion ΔS_m, can be obtained by ΔS_m=ΔH_m/T_m, where ΔH_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]:

$$c_p{(T)}_{\mathrm{sample}}=\frac{Q_{sample}^{\ast }-Q_{pan}^{\ast }}{Q_{sapphire}^{\ast }-Q_{pan}^{\ast }}\times \frac{m_{\mathrm{sapphire}}\times {\mu }_{\mathrm{sample}}}{m_{\mathrm{sample}}\times {\mu }_{\mathrm{sapphire}}}\times c_p{(T)}_{\mathrm{sapphire}}$$

(2)

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 $Q_i^{\ast }$ 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]:

$$c_p(T)=3R+a\cdot T+b\cdot T^{-2}$$

(3)

where R is gas constant. The specific heat capacity c_p^crystal(T) of the crystal can be expressed as [4,11,15]:

$$c_p(T)=3R+c\cdot T+d\cdot T^2$$

(4)

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]:

$$m=\frac{d\lg (\eta )}{d(T_g/T)}|{}_{T=T_g}=\frac{d\lg (\tau )}{d(T_g/T)}|{}_{T=T_g}=\frac{D{T}_0{T}_g}{{(T_g-T_0)}^2\ln (10)}$$

(5)

A similar fragility has been defined as [17]:

$$m_S=\frac{d\left[\frac{\lg (\eta (T))/{\eta }_0^{}}{\lg (\eta (T_g)/{\eta }_0)}\right]}{d(T_g/T)}|{}_{T=T_g}=\frac{m}{m_{\min }}$$

(6)

where m_min = log₁₀(η_g/η₀). η_g denotes viscosity (typically 10¹² Pa s) at glass transition temperature T_g. η₀ is the high temperature limit of viscosity, which can be determined by the following equation [2,24]:

$${\eta }_0=h{N}_{\mathrm{A}}\rho /M$$

(7)

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 η₀ value is about set as 10⁻⁵ Pa s [2,17,24,25,26]. Thus, generally, the log₁₀(η_g/η₀) 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]:

$$m_s=\frac{m}{m_{\min }}=\frac{T_g^{}}{T_g^{}-T_K^{}}$$

(8)

From Equation (8), T_K can be expressed as:

$$T_K=T_g-m_{\min }{T}_g/m$$

(9)

The other expression of T_K as a function of T_g and m can be expressed as [1,16,17,25]:

$$m_s=\frac{m}{m_{\min }}=\frac{T_g^2+T_K^2}{T_g^2-T_K^2}$$

(10)

Equation (10) is transformed into:

$$T_K=T_g\times {[(m-m_{\min })/(m+m_{\min })]}^{1/2}$$

(11)

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]:

$$T_K=T_m{(1+\frac{\Delta H_m}{T_g\Delta c_p^{l-c}(T_g)})}^{-1}$$

(12)

Additionally, m can be calculated by the following Equation [27]:

$$m_{}={\Lambda }_a\frac{\Delta c_p^{l-g}(T_g)}{\Delta S_m}$$

(13)

where Λ_a is the constant and equals 40. Δc_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 Δc_p^l-g(T_g) is replaced by Δc_p^l-c(T_g), the numerical factor would increase from 40 to 43, but the quality of the correlation remains unchanged, where Δc_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, Δc_p^l-g(T_g) is replaced by Δc_p^l-c(T_g), and m can be expressed as:

$$m={\Lambda }_b\frac{\Delta c_p^{l-c}(T_g)}{\Delta S_m}$$

(14)

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):

$$m={\Lambda }_b{\Lambda }_c{T}_g\frac{\Delta c_p^{l-c}(T_g)}{\Delta H_m}$$

(15)

From Equation (12) and Equation (15), we obtain:

$$T_K={\Lambda }_c{T}_g{(1+\frac{{\Lambda }_b{\Lambda }_c}{m_{}})}^{-1}={\Lambda }_c{T}_g(\frac{m}{m_{}+{\Lambda }_b{\Lambda }_c})$$

(16)

The expanded Equation (16) can be expressed by:

$$T_K={\Lambda }_c{T}_g-{\Lambda }_b{\Lambda }_c^2\frac{T_g}{m_{}+{\Lambda }_b{\Lambda }_c}$$

(17)

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.

3. 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, R² is applied to evaluate the accuracy of the predicted T_K. In statistics, generally, R² is defined as: R² = 1−SS_res/SS_tot, where SS_res is the sum of squares of residuals and SS_tot is the total sum of squares. R² is a statistical measure of how well the predicted T_K values approximate the reported T_K values. The higher is the R² value (0 ≤ R² ≤ 1), the more accurate is the predicted T_K. The predicted T_K values perfectly fit the reported T_K when R² equals 1.

4. 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. 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 (K)	m	TK (K)	TKcal1 (K)	TKcal1* (K)	TKcal2 (K)	TKcal2* (K)	TKnew (K)	TKnew* (K)
1	Mg65Cu25Y10	404 [4]	50 [34]	320 [4]	266.64	323.52	283.53	319.65	264.63	320.06
2	Pd77.5Cu6Si16.5	637 [34]	73 [34]	560 [16,35]	488.66	550.09	502.47	543.44	507.28	555.91
3	Cu47Ti34Zr11Ni8	673 [34,36]	59 [34]	537 [36]	479.08	559.39	500.30	552.42	482.27	558.08
4	Zr41.2Ti13.8Cu12.5Ni10Be22.5	625 [34,37]	46 [34]	558 [6,13]	394.02	489.67	424.04	484.12	390.27	482.86
5	Zr46.75Ti8.25Cu7.5Ni10Be27.5	590 [13,34]	46 [34]	560 [13]	371.96	462.25	400.30	457.01	368.42	455.82
6	SiO2	1480 [38]	25 [34]	876 [38]	473.60	890.37	645.92	900.08	620.11	907.07
6	SiO2	1452 [34,38]	25 [34]	876 [38]	464.64	873.52	633.70	883.05	608.38	889.91
7	GeO2	816 [34,38]	21 [34]	418 [38]	155.43	428.98	264.75	441.17	300.63	460.41
8	Pd40Ni40P20	578 [3,13]	46 [3,13]	500 [9,13]	364.39	452.85	392.15	447.72	360.92	446.55
9	La55Al25Ni20	491 [13,34,39]	42 [34]	337 [10,13]	292.26	374.56	319.61	370.73	290.45	368.45
9	La55Al25Ni20	470.3 [10]	42 [34]	337 [10,13]	279.94	358.77	306.14	355.10	278.21	352.91
10	La55Al25Ni15Cu5	472 [13,34,39]	37 [34]	318 [10,13]	255.14	344.94	287.25	342.25	258.09	339.07
10	La55Al25Ni15Cu5	449.3 [10]	37 [34]	318 [10,13]	242.86	328.35	273.44	325.79	245.68	322.76
11	La55Al25Ni10Cu10	467 [13,34,39]	35 [34]	332 [10,13]	240.17	334.11	274.76	331.99	246.41	328.74
11	La55Al25Ni10Cu10	440.6 [10]	35 [34]	332 [10,13]	226.59	315.22	259.23	313.22	232.48	310.16
12	La55Al25Ni5Cu15	459 [13,34,39]	42 [34]	304 [10,13]	273.21	350.15	298.78	346.57	271.52	344.44
12	La55Al25Ni5Cu15	435 [10]	42 [34]	304 [10,13]	258.93	331.84	283.16	328.44	257.32	326.43
13	La55Al25Ni5Cu10Co5	466 [13,16,34,39]	37 [16,34]	363 [13,16]	251.89	340.56	283.60	337.90	254.81	334.76
13	La55Al25Ni5Cu10Co5	439.1 [10]	37 [16,34]	363 [13,16]	237.35	320.90	267.23	318.39	240.10	315.44
14	Zr46(Cu4.5/5.5Ag1/5.5)46Al8	703 [14]	49 [14,16]	671 [14,16]	459.10	560.10	489.51	553.47	455.25	553.63
15	Zr46Cu46Al8	715 [14,16]	43 [14,16]	596 [14,16]	432.33	549.39	470.67	543.58	429.00	540.69
16	Zr44Ti11Ni10Cu10Be25	620 [16,40]	39 [16]	504.5 [16]	349.74	461.66	388.61	457.52	350.43	453.73
17	Pd43Ni10Cu27P20	582 [15,16]	65 [12,16]	532 [15,16,41]	429.78	492.82	445.28	486.71	438.19	494.47
17	Pd43Ni10Cu27P20	576 [11]	65 [12,16]	447 [11,12]	425.35	487.74	440.69	481.69	433.67	489.37
18	Au77Ge13.6Si9.4	294 [16]	85 [12,16]	199 [16]	235.20	259.55	240.05	256.58	250.74	264.83
19	2-metylpentane	80.5	58	58	56.91	66.68	59.52	65.85	57.17	66.46
20	Butyronitrile	100	47	81.2	63.83	78.81	68.47	77.90	63.23	77.77
21	Ethanol	92.5	55	71	63.91	75.75	67.20	74.81	63.86	75.28
22	n-propanol	102.5	36.5	73	54.76	74.53	61.88	73.97	55.56	73.27
23	Toluene	126	59	96	89.69	104.73	93.67	103.42	90.29	104.49
24	1-2 propan diol	172	52	127	115.77	139.06	122.50	137.36	115.16	137.81
25	Glycerol	190	53	135	129.06	154.29	136.26	152.40	128.55	153.06
26	Triphenil phospate	205	160	166	183.22	192.24	184.26	190.76	219.15	203.31
27	Orthoterphenyl	244	81	200	192.79	214.00	197.18	211.50	203.75	217.68
28	m-toluidine	187	79	154	146.76	163.42	150.28	161.50	154.42	165.98
29	Propylene carbonate	156	104	127	130.50	141.06	132.28	139.61	144.43	145.73
30	Sorbitol	266	93	226	217.38	237.51	221.10	234.91	235.60	243.71
31	Selenium	307	87	240	247.01	271.85	251.87	268.78	264.45	277.79
32	ZnCl2	380	30	250	164.67	253.84	199.85	253.71	180.95	251.90
33	As2S3	455	36	265	240.14	329.12	272.43	326.77	244.48	323.64
34	CaAl2Si2O8	1118	53	815	759.40	907.90	801.76	896.78	756.43	900.63
35	Propilen glycol	167	52	127	112.40	135.01	118.94	133.37	111.81	133.81
36	3-Methyl pentane	77	36	58.4	40.64	55.70	46.10	55.30	41.37	54.77
37	3-Bromopentane	108	53	82.5	73.36	87.70	77.45	86.63	73.07	87.00
38	2-methyltetrahydrofuran	91	65	69.3	67.20	77.06	69.62	76.10	68.51	77.31

. 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 R² 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 log₁₀(η_g/η₀) (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 log₁₀(η_g/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(η_g/η₀) value as well. In our previous study, the log₁₀(η_g/η₀) 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 η₀ 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 R² 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 (R² = 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 R² 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.

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 R² 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):

$$T_{\mathrm{K}}^{\mathrm{new}*}=1.12T_g-23.17\frac{T_g}{m+20.69}$$

(18)

Figure 4b shows the predicted Kauzmann temperature T_K^new*, according to Equation (18). From Figure 4, it can be seen that the R² 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.

5. 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 R² 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 R² = 0.966 is deduced. Therefore, three equations with the best fitting parameters have relatively high R² values, which indicates that they can be applied to obtain the accurate predicted T_K values.