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Characteristic temperatures of enthalpy relaxation in glass
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Journal of Non-Crystalline Solids 354 (2008) 1112–1118 www.elsevier.com/locate/jnoncrysol
Characteristic temperatures of enthalpy relaxation in glass Yuan-Zheng Yue
*
Section of Chemistry, Aalborg University, Aalborg, Denmark Available online 19 November 2007
Abstract The glass transition temperature, Tg, directly measured by differential scanning calorimetry at 10 K/min is compared with the Tg indirectly determined by fitting viscosity data to a viscosity model for oxide glasses. The results show good match between the two Tg values. A standard, unified approach for measuring Tg is proposed. Characteristic temperatures of enthalpy relaxation in glass are defined, and the relationships between these temperatures are illustrated by performing aging and calorimetric experiments on hyperquenched glasses. The features of the energy release peak, the endothermic pre-peak, and the real glass transition are discussed with respect to their physical origins. Ó 2007 Elsevier B.V. All rights reserved. PACS: 61.43.Fs; 61.20.Lc; 64.70.Pf; 66.20.+d Keywords: Calorimetry; Enthalpy relaxation; Glass transition; Fragility; Viscosity
1. Introduction The relaxation time of any liquid towards equilibrium has a unique dependence on temperature. In glass science, the dynamics and the physical properties are often compared between different glass systems. Yet, this is only meaningful if the comparison is made at a universal characteristic temperature (e.g. the glass transition temperature Tg) or at a temperature scaled by a characteristic temperature, usually by Tg (i.e. Tg/T). For instance, the dynamic features of various liquids are well distinguishable by the use of the fragility concept based on the Tg/T dependence of the relaxation time [1]. Tg is the most useful, characteristic, dynamic temperature of a liquid. However, the value of Tg depends both on its definition and on the method of determination, which have not yet been unified in glass literature. Therefore, different Tg values for same glass composition are often found in literature. To make the determination of Tg more accurate and reasonable, carefully performed measurements have been reported in
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0022-3093/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.11.027
[2,3]. The enthalpy-matching approach turned out to be a precise method for determining Tg, which is based on the heat capacity versus temperature curve obtained from calorimetric upscan [2]. Besides the characteristic temperature Tg, it is also important to define and determine other characteristic temperatures such as the fictive temperature (Tf), the onset temperature of energy release (Tz), and the onset temperature of the endothermic aging pre-peak (Tpp) [4–14]. Tz and Tpp are encountered when hyperquenched glass (HQG) or aged HQG is upscanned in a calorimeter [4–12]. The relationships between these characteristic temperatures have not yet been fully clarified. Although the fictive temperature itself describes neither the details of the structure nor the details of the thermal history of glasses, its relation to the cooling rate does provide information on the dynamic features of a liquid. This relation may be used for determining the activation energy of viscous flow and the fragility index of the liquid [12]. Tf is a parameter that reflects enthalpy and physical properties (e.g. density, refraction index, etc.) of a glass. Tf of a glass varies with both the cooling rate and the aging degree [3,5,12,13]. In particular, when a HQG glass is aged and then calorimetrically scanned, it exhibits changes in Tf. The structure of the glass
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corresponds to that of its liquid at T = Tf. A glass with high Tf, i.e. a HQG, is a good object for studying the vibrational dynamics and potential energy landscape [6]. The Tf of a HQG can be calculated from the heat capacity data using the method proposed in [5,11,12]. For glass fiber industry, Tf is a useful parameter to monitor and control the stability of the fiber spinning process, since Tf is sensitive to both drawing and cooling conditions. This paper focuses on two aspects. First, Tg values determined by a calorimetric method are compared with those determined using a viscometric method. Based on this, a standard method for determining Tg is proposed and recommended to glass community. Second, the relationships between the various characteristic temperatures of enthalpy relaxation are elucidated for two hyperquenched oxide glasses by performing aging and differential scanning calorimetric (DSC) measurements. The endothermic pre-peak of an aged HQG is compared with the glass transition peak concerning their physical origins. Distinguishing between these two peaks is important for clarification of glass transition of glasses [4–12]. Determination of the characteristic temperatures is an essential step for studying the dynamics, the potential energy, and the structural evolvement in both liquid and glass. In addition, unifying both the definition and the determination method for Tg and Tf is useful for establishing a unified technological standard for evaluating and controlling the density, the refractive index, the enthalpy level, and hence the stress level of glass products. A unified method for Tg determination makes a precise, reasonable comparison in fragility and other dynamic properties between different glass systems possible. 2. Experimental
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Table 1 Comparison between the Tg values estimated by fitting the viscosity data to Eq. (1), Tg,vis and Tg measured by DSC, Tg,DSC Glass
Tg,vis (K)
Tg,DSC (K)
DTga (K)
CAS1 CAS2 CAS3 CAS4 CAS5 CAS6 CAS7 CAS8 CAS9 CAS10 Anorthite Wollastonite Gehlenite NaPoLi CMP BAS1 BAS2 NIST710a Diopside
1044 1055 1067 1078 1087 1066 1078 1085 1094 1103 1128 1037 1112 516 798 899 946 819 985
1043 1057 1065 1075 1084 1066 1075 1082 1093 1104 1128 1040 1110 516 799 902 943 822 986
1 2 2 3 3 0 3 3 1 1 0 3 2 0 1 3 3 3 1
The uncertainty of Tg,vis and Tg,DSC are ±2 and ±1 K, respectively. a Deviation DTg = Tg,vis Tg,DSC.
capacity (Cp) curves were obtained using the evaluation method described in [11,12]. The Cp curve of the first upscan reflects the enthalpy level of a fresh sample with unknown thermal history (i.e. a sample cooled at an unknown rate prior to the first upscan), whereas the Cp curve of the second upscan reflects the enthalpy level of a sample with a defined thermal history (i.e. a sample cooled at the standard rate of 10 K/min prior to the second upscan). The standard Tg values of all glasses were obtained from the DSC second upscans.
2.1. Viscometric and calorimetric measurements
2.2. Hyperquenching-aging-scan experiments
Both viscometric and DSC experiments were performed on 19 inorganic glasses (17 silicate and 2 phosphate glasses), the chemical composition of which are given in [15–21]. Table 1 lists the Tg values of the 19 glasses. ‘CAS’ refers to the calcium aluminosilicate glasses with different CaO/SiO2 and CaO/Al2O3 ratios [15,16]. ‘NaPoLi’ refers to sodium lithium metaphosphate (Na2O Æ Li2O Æ 2P2O5) glasses. ‘BAS1’ refers to the glass made from a basaltic melt. ‘NIST710a’ refers to the standard glass made by National Institute of Science and Technology. ‘CMP’ refers to the calcium metaphosphate glass. The viscosity data for the low-temperature range were measured using the parallel-plate (servo-hydraulic test machine from MTS company) [22], beam-bending (Baehr VIS 401) [23], and penetration (Baehr DIL 802 V) viscometers [24], whereas viscosity data for the high-temperature range were obtained using concentric cylinder viscometers (Paar Physica and Brookfield model RTVD) [15–21]. All the samples underwent two runs of up- and downscans at 10 K/min in argon in a DSC (Netzsch STA 449C Jupiter). The heat
To determine the characteristic temperatures of glasses and liquids, the hyperquenching-aging-DSC scan experiments were conducted on two types of glass systems. The first one is a basalt-like glass system (in short, BAS2) used for making insulation materials – stone wool, the composition is given in [4]. BAS2 is obtained from a mixture of several types of rocks with basalt as a dominant ingredient. The hyperquenched BAS2 (in short, HQ-BAS2) is obtained from the BAS2 melts using the cascade fiber-spinning process. The second system is the calcium metaphosphate glass (CaO Æ P2O5) (CMP). Two types of hyperquenched samples (in short, HQ-CMP) were achieved by means of both continuous fiber drawing and cascade fiber-spinning, which yield hyperquenching rates of 104 and 108 K/min, respectively. The cooling rates were calculated using the method proposed in [12]. One HQ-BAS2 sample was aged at 723 K for 55 days, whereas another was aged at 773 K for 55 days. They were both scanned twice by a DSC at 10 K/min. The first scan exhibits the enthalpy relaxation of the aged HQ-BAS2,
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while the second scan shows the enthalpy relaxation of the ‘standard’ glass (i.e. the glass cooled at a standard rate of 10 K/min). The HQ-CMP samples were not subjected to any aging process. 3. Results Fig. 1(a) and (b) show both the logarithm of viscosity (log g) and the heat capacity (Cp), respectively, as a function of temperature (T) for glass BAS1. The Cp curve is measured at 10 K/min during the second upscan. The solid curve in Fig. 1(a) is obtained by fitting the viscosity data to the Avramov–Milchev (AM) model log g ¼ A þ BðT g =T ÞF ;
ð1Þ
where A and B are constants (A + B = 12) and F is the fragility index. The reasons why Eq. (1) is applied here are as follows. First, for oxide glasses, Eq. (1) fits viscosity data better than the Vogel–Fulcher–Tamman (VFT) and the Adam–Gibbs (AG) models [15]. Second, both the standard Tg and the fragility index F can be directly derived from Eq. (1). The detailed discussion about Eq. (1) is given in [12,15,25]. From Fig. 1(a), it is seen that the quality of 14 12
data fit by Eq. (1)
Tg,vis
8
a)
6 4 2 0 1.8
upscan downscan
Cp (Jg-1K-1)
1.6 1.4
1200
measured data linear fit
1.2 1.0 0.8 600
b)
Tg,DSC
800
1000
1200
1400
1000
Tg,DSC (K)
log (Pa s)
10
the fit is satisfying. Tg values can be derived by using Eq. (1). The obtained Tg is denoted Tg,vis (Fig. 1(a)). The standard Tg obtained by DSC is denoted Tg,DSC. It is defined as the onset temperature of the glass transition peak, i.e. the temperature of the intersection between the extrapolated line of the glass Cp and the extrapolated line of the rapidly rising Cp (Fig. 1(b)). The Cp curve is measured at the standard upscan rate of 10 K/min. Before the upscan, the sample must be cooled also at 10 K/min. The Tg,DSC corresponds to the inflection point of the downscan Cp curve (see the dotted lines in Fig. 1(b)). The uncertainty of both Tg,vis and Tg,DSC are ±2 and ±1 K, respectively. Fig. 1 shows a direct correspondence between the characteristic viscosity (1012 Pa s) and the calorimetric Tg. This link exists not only for basaltic glasses, but also for other oxide glasses (Table 1 and Fig. 2). As shown in Table 1 the maximum difference in Tg values between Tg,vis and Tg,DSC is rather small, i.e. equals ±3 K. Fig. 2 shows an excellent linear fitting of the Tg,vis versus Tg,DSC relationship with a correlation coefficient of 0.9996. The dashed curve is the fitting curve. In Fig. 3 two common methods for determining Tg are illustrated. Fig. 3(a) shows the typical enthalpy relaxation of HQ-BAS2, which is manifested by the deviation of the heat capacity (Cp1) of the hyperquenched glass (first upscan) from the heat capacity (Cp2) of the ‘standard’ state (second upscan). The second upscan is regarded as the standard upscan, since both the heating rate (qh) and the prior cooling rate (qc-pre) are 10 K/min. As shown in Fig. 3(a), the fictive temperature Tf is determined using the enthalpy-matching method, a detailed description of which is given in [5,11,12]. Fig. 3(b) demonstrates a well-known enthalpy-matching method used for determining Tg [2]. A comparison between Fig. 3(b) and (c) shows that the Tg determined by the enthalpy-matching method is in accordance with Tg,DSC. Such a correspondence is a general feature of the oxide
800
T (K) Fig. 1. Direct correlation between the characteristic viscosity (1012 Pa s) and the calorimetric glass transition temperature (Tg) for the BAS1 glass. (a) The logarithm of the viscosity (log g) as a function of temperature (T). The triangles represent the data measured using the parallel-plate compression viscometer (in the lower temperature range) and the concentric cylinder viscometer (in the higher temperature range). The solid curve is obtained by fitting the experimental data to Eq. (1). (b) The heat capacity (Cp) as a function of temperature. The solid curve is the Cp recorded during the second upscan, whereas the dashed curve represents the second downscan Cp.
600
400 400
600
800
1000
1200
Tg,vis (K) Fig. 2. Comparison between the glass transition temperatures measured by DSC at 10 K/min, Tg,DSC, and those obtained from the fit of viscosity data to Eq. (1), Tg,vis, for 19 inorganic glasses [5–11].
Y.-Z. Yue / Journal of Non-Crystalline Solids 354 (2008) 1112–1118
a)
Te
B
Cpliq
= DE
=
1.4
B 1.2
Tz
Cp2
1.0
Cpg
A
Cp1
0.8
Cp (Jg-1K-1)
1.6
Tg
b) =
1.4 1.2
Tz
standard
1.0 0.8 1.6
c)
upscan 1 upscan 2 (standard)
1.4
Tpp
Tcross
1.2 1.0 0.8 400
600
800
1000
1200
T (K) Fig. 3. Determination of various characteristic temperatures during enthalpy relaxation measured by DSC on both non-aged and aged HQBAS2 glasses. (a) Determination of fictive temperature (Tf) by using the enthalpy-matching method, and of the onset temperature (Tz) of energy release from the HQ-BAS2. Cp,liq is the heat capacity of the liquid, whereas Cp,g is the heat capacity of the glassy state. (b) High-temperature part (i.e. above Tg) of Fig. 3(a), which includes the left borderline (the line for determining Tg) and the right borderline (the line for determining Tf). The way of placing the left borderline is described in Ref. [1]. (c) The onset temperature of the aging-induced pre-peak, Tpp, [4] and the standard glass transition temperature (Tg) for the glass that is first hyperquenched and then annealed at 723 K for 55 days. Tcross is the distinction temperature between the pre-peak and the energy release peak. The heating and cooling rates for the DSC measurements are 10 K/min. The vertical dashed line illustrates the correspondence between the Tg value determined by the enthalpy-matching method and the onset Tg value. Fig. 3(a) and (b) are re-plotted from [11].
Tg
where Cp,liq and Cp,g are the heat capacities of the liquid (above Tg) and the extrapolated glassy state, respectively. Tf can be estimated using Eq. (2). As shown in Fig. 4, DE increases with the cooling rate of the glass. Fig. 5 shows a large enhancement of the pre-peak of the aged HQ-BAS2 by extending the aging time (ta) from 8 to 55 days. As a consequence, the onset temperature of the pre-peak, Tpp, shifts to a higher temperature (i.e. from 797 to 928 K). The pre-peak nearly overlaps with the real Tg peak. If the aging time was sufficient, the Tpp would approach the real Tg, and the pre-peak would be fully superimposed on the real glass transition peak. Fig. 6 shows that for the HQ-BAS2 the Tpp increases with increasing aging temperature, Ta. In addition, Fig. 6 implies the existence of the energetic and structural heterogeneities, since the lower energy regimes (represented by the pre-peak) and the higher energy regimes (repre-
1.6
1.4
0.15
Cp,exc (Jg-1K-1)
A
quently aged at temperatures below the real Tg. The prepeak shown in Fig. 3(c) is caused by the aging process. Fig. 4 demonstrates that the onset temperature (Tz) of energy release for the HQ-CMP depends on the prior cooling rate. Tz decreases with increasing cooling rate. The inset in Fig. 4 shows the excessive heat capacity (Cp,exc = Cp2 Cp1) as a function of temperature (see Fig. 3(a)). Te is the characteristic temperature, at which the quenchedin energy is completely released, i.e. Cp1 = Cp2. At Te the liquid returns to internal equilibrium. The integral of the Cp,exc over T represents the amount of energy stored in the glass during hyperquenching, DE (see hatched areas in Fig. 3(a)) Z Te Z Tf DE ¼ ðC p2 C p1 ÞdT ¼ ðC p;liq C p;g ÞdT ; ð2Þ
Tf
Cp (Jg-1K-1)
1.6
1115
1.2
Te
108 K/min 0.10 0.05
Tz1
Tz2
105 K/min
0.00 400
600
800
T (K)
Tz2 1.0
Tz1 Standard 108 K/min 105 K/min
0.8
glasses if both the prior downscan rate and the upscan rate are equal to 10 K/min. Fig. 3(c) shows another characteristic temperature, i.e. the so-called shadow glass temperature Tpp, which is the onset temperature of the endothermic pre-peak (in the literature there are different names for the pre-peak [26], e.g. excess endothermic peak [8–10], sub-Tg peak [27], shadow Tg peak [4,6] and pre-endotherm [7,28]). The pre-peak can only occur if a glass is first hyperquenched and subse-
300
400
500
600
700
800
900
T (K) Fig. 4. Heat capacity (Cp) as a function of temperature for the CMP glasses cooled at the rates: 10 K/min (solid curve), 105 K/min (dashed curve), and 108 K/min (dot-dashed curve). Inset: the excessive heat capacity as a function of temperature. Tz1 and Tz2 are the onset temperatures of energy release. All Cp curves were measured at a DSC upscan rate of 10 K/min. The glass cooled at 10 K/min is termed as standard glass.
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1.8
ta=8 days ta=55 days standard
Cp (Jg-1K-1)
1.6
1.4
Tpp2 Tpp1
1.2
Tg 1.0 600
700
800
900
1000
T (K) Fig. 5. Comparison between the glass transition peak of the ‘standard’ BAS2 and the pre-peak of the HQ-BAS2 aged at 773 K for periods of 8 and 55 days, respectively. The DSC up and downscans were performed at 10 K/min.
0.3
Cp,exc (Jg-1K-1)
0.2
Ta=723 K
exo 0.1
Tcross Tcross
0.0
-0.1
Ta=773 K
ta=55 days
600
700
800
900
1000
1100
T (K) Fig. 6. The excessive heat capacity (Cp,exc) of the HQ-BAS2 aged at different temperatures for 55 days. The DSC upscans were performed at a rate of 10 K/min.
sented by the energy release peak) co-exist in the same glass. 4. Discussion 4.1. Glass transition temperature (Tg) From Eq. (1), log g = A + B, when T = Tg. The constant A + B depends on the heating rate (qh) and the prior cooling rate (qc-pre) used to measure Tg by DSC. When Tg is measured at qh = qc-pre = 10 K/min, A + B equals 12, and this is confirmed by Figs. 1 and 2. The overall relation between cooling rate and viscosity is described in [12]. As shown in Fig. 2, there is an excellent agreement between Tg,vis at the viscosity of 1012 Pa s and Tg,DSC measured at qh = qc-pre = 10 K/min. This is reflected by the linear
relation Tg,vis = Tg,DSC with the correlation coefficient of 0.999. The agreement means that fitting viscosity data of oxide glasses to Eq. (1) can be precisely performed by fixing A + B at 12, i.e. by using the characteristic viscosity of g = 1012 Pa s at Tg. Since the 19 glass systems in Table 1 involve a wide range of fragilities (i.e. fragility index F of Eq. (1) ranges from 2.7 to 5.4, which corresponds to a range of Angell fragility index m [1] from 83 to 167), it is expected that the relation Tg,vis = Tg,DSC should apply to other inorganic glass systems. The further importance of the relation Tg,vis = Tg,DSC lies in the following aspects. First, Tg,vis can be obtained by measuring the viscosity and then by fitting the data with Eq. (1). Reversely, if Tg,DSC is measured, the viscosity at Tg,DSC is naturally known, i.e. equal to 1012 Pa s. Second, the entire log g T relationship for the glasses with high crystallization tendency may be obtained by fitting Eq. (1) to the high-temperature viscosity data (usually above the liquidus temperature) and to the 1012 Pa s – Tg,DSC point without measuring the low-temperature viscosities (above Tg). The measurements of the low T viscosities are impossible for glass systems that have a strong tendency to crystallize. Third, the viscosity model log g = A + (12 A)(Tg/T)F can be used directly to construct the fragility plot of glass systems. Fourth, the Tg measured at qh = qc-pre = 10 K/min is the most suitable characteristic temperature for scaling the temperatures of a fragility plot, since for the fragility plot, the Tg, in the literature, usually assumed to correspond to the viscosity of 1012 Pa s. The present work (see Figs. 1 and 2) shows that such an assumption is only reasonable, if Tg is assigned as the onset temperature of the glass transition during the DSC upscan determined at qh = qc-pre = 10 K/min. The viscosity at the onset temperature determined at qh = qc-pre 5 10 K/min would deviate from 1012 Pa s, since the onset temperature depends on both qh and qc-pre [12]. According to the above analysis, Tg,DSC (at 10 K/min) and Tg,vis (at 1012 Pa s) are equal and may both be used as the standard Tg. The experimental methods used to determine both the Tg,DSC and the Tg,vis can be applied as the standard method for determining Tg. This standardization of Tg makes a direct comparison between the glass transition temperatures of different glass systems possible. Here it should be noted that glass transition occurs over a range of temperatures, not at a single temperature. The Tg defined in this work is a well representative temperature for the glass transition, i.e. the onset temperature associated with the viscosity of 1012 Pa s. From glass literature it is known that even for the same glass, the measured Tg values are quite diversified, since various experimental methods are used and different heating and prior cooling rates are used in different measurements. For instance, most Tg measurements reported are performed at the heating rates 1–30 K/min that corresponds to viscosities of 1013–1011.5 Pa s. However, problems caused by the diversity in Tg could be avoided using the method presented in the present work.
Y.-Z. Yue / Journal of Non-Crystalline Solids 354 (2008) 1112–1118
4.2. Fictive temperature (Tf) and onset temperature for energy release (Tz) Figs. 3 and 4 illustrate that the enthalpy of the HQ-BAS2 begins to decrease when the temperature exceeds the onset temperature (Tz) of release of the excess enthalpy (i.e. the hatched area between the Cp2 and Cp1 curves in Fig. 3(a)). The excess energy (or excess enthalpy) is not completely released until the temperature reaches the equilibrium temperature Te (see Figs. 3 and 4). When a liquid at Te is cooled at 10 K/min to a temperature well below Tg, it becomes a glass with a specific Tf, i.e. the standard Tg defined above, which can be observed during DSC reupscan at 10 K/min. The excess energy is attributed to the freezing-in of the excited configurational and vibrational states at Tf. The most excited states are so unstable that a temperature equal to Tz (well below Tg), and hence the lowest kinetic energy 1/2kbTz (where kb is the Boltzmann constant), can bring the glass to an energy level corresponding to the real temperature. As shown in Fig. 4, the onset temperature (Tz) of the energy release of the HQ-CMP decreases with increasing cooling rate. This implies that Tz decreases with increasing the fictive temperature Tf, since Tf increases with increasing the cooling rate. The dependence of Tz on Tf involves three situations. First, when a liquid is cooled at a rate higher than the standard rate of 10 K/min, Tf is higher than Tz. Second, when a liquid is cooled at 10 K/min, Tf and Tz merges to the standard Tg. Finally, when a liquid is cooled at a rate below 10 K/min, Tz does not exist, since no energy release takes place. Instead, an enhancement of the Tg-peak would occur, i.e. a deficient heat capacity would exist in the glass transition region, and therefore, Tf should be lower than Tg. As mentioned previously, the equilibrium temperature Te is the temperature at which the enthalpy release is completed. Fig. 4 shows that the Tz (423 K) of the HQ-CMP (cooled at 108 K/min) is much lower than the ideal glass temperature T0 (630 K) that is obtained by fitting the VFT equation to the viscosity data of the CMP melt. According to [29,30], T0 is approximately equal to the Kauzmann temperature Tk (a thermodynamic, characteristic temperature) [31]. This means that the Tz can be far below Tk, if the quenching rate of a liquid is high enough. From the relation Tz < Tk, it can be inferred that the hyperquenching process of a liquid involves not only a configurational but also a vibrational freezing-in process, since Tk is the temperature at which the configurational entropy vanishes [31]. Finally, two important questions still remain open: (a) what is the quantitative relationship between Tz and Tf and (b) are there limiting values for both Tz and Tf, respectively. 4.3. The onset temperature of the endothermic pre-peak (Tpp) The Tpp fundamentally differs from the glass transition temperature (Tg) with respect to their physical sources.
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The pre-peak is a consequence of thermal aging and is attributed to the energetic and structural heterogeneities of glass. The pre-peak is a manifestation of the non-exponential nature of glass relaxation. In detail, pre-peaks can be regarded as the aging-induced ‘shadow’ glass transitions of the short relaxation time components (‘microglasses’) of the macroglass [4]. But, the real glass transition is a consequence of the slowing-down of the primary, cooperative relaxation process (the so-called a relaxation) upon cooling, or speeding-up of the a relaxation process upon heating. An abrupt change in the configurational entropy occurs during the glass transition upon heating or cooling. The size of the cooperative rearrangement regions rapidly increases when the temperature is lowered towards the glass transition [32]. Distinguishing the endothermic pre-peak transition from the glass transition is essential for understanding both the structural relaxation of hyperquenched glasses during aging process, and the structural heterogeneity in glass. The pre-peak shows a recovery feature under certain aging and DSC-scan conditions [28]. Fig. 3(b) shows a crossover temperature Tcross between the pre-peak and the energy release peak, which depends on aging time and temperature (see also Figs. 5 and 6). The pre-peak is particularly pronounced for a rapidly cooled glass compared with a slowly cooled glass, if they both are aged under same conditions (i.e. same aging temperature Ta and time ta). The Tpp and Tg peak would overlap, as Ta or ta increase sufficiently. If a slowly cooled glass is aged at a temperature below Tg for long time, an enhancement of the Tg peak can be observed on the DSC upscan curve. This indicates that the aging process leads to a decrease in Tf. The difference value between Tpp and Tg depends both on the remaining excess energy of the HQG, and on the non-exponentiality of relaxation. It is evident that Tpp increases with Ta and ta. According to [27], Tpp should equal Ta. However, as shown in Fig. 5, that equality does not always apply, and hence it is not a universal feature of the pre-peak as stated in [27]. Furthermore, the height of pre-peak is rather sensitive to degree of aging, e.g. it is even measurable after only 8 min aging on HQ-BAS2 at Ta/Tg = 0.87 [26]. 5. Conclusions For the oxide glasses, the temperature at a viscosity of 1012 Pa s (Tg,vis), which is obtained by fitting the viscosity–temperature data to the AM model, is in accordance with the onset temperature of the glass transition measured calorimetrically at the prior downscan and upscan rates of 10 K/min, Tg,DSC. Both Tg,vis and Tg,DSC should be adopted and used as the standard Tg and as the scaling temperature of the fragility plot. The onset temperature (Tz) of the energy release decreases with increasing the fictive temperature of hyperquenched glasses. However, it is unclear how Tz is quantitatively related to Tf. The origin of the pre-peak is fundamentally different from that of the real glass transition.
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Acknowledgments The author would like to thank M. Solvang and S.L. Jensen for useful discussions. Part of this work is financially supported by Rockwool International A/S. References [1] C.A. Angell, Science 267 (1995) 1924. [2] C.T. Moynihan, A.J. Easteal, M.A. Debolt, J. Tucker, J. Am. Ceram. Soc. 59 (1976) 12. [3] G.W. Scherer, J. Am. Ceram. Soc. 67 (1984) 504. [4] Y.Z. Yue, C.A. Angell, Nature 427 (2004) 717. [5] V. Velikov, S. Borick, C.A. Angell, Science 294 (2001) 2335. [6] C.A. Angell, Y.Z. Yue, L.M. Wang, J.R.D. Copley, S. Borick, S. Mossa, J. Phys.: Cond. Mat. 15 (2003) 1051. [7] Y.Z. Yue, S.L. Jensen, J. de C. Christiansen, Appl. Phys. Lett. 81 (2002) 2983. [8] H.S. Chen, C.R. Kurkjian, J. Am. Ceram. Soc. 66 (1983) 613. [9] H.S. Chen, A. Inoue, J. Non-Cryst. Solids 61&62 (1984) 805. [10] A. Inoue, T. Masumoto, H.S. Chen, J. Non-Cryst. Solids 83 (1986) 297. [11] Y.Z. Yue, J. de C. Christiansen, S.L. Jensen, Chem. Phys. Lett. 357 (2002) 20. [12] Y.Z. Yue, R. von der Ohe, S.L. Jensen, J. Chem. Phys. 120 (2004) 8053;
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[20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
Y.Z. Yue, R. von der Ohe, S.L. Jensen, J. Chem. Phys. 121 (2004) 11508. D.J. Plazek, K.L. Ngai, in: J.E. Mark (Ed.), Physical Properties of Polymers, Handbook, 1996, p. 139. O.S. Narayanaswamy, J. Am. Ceram. Soc. 54 (1971) 491. M. Solvang, Y.Z. Yue, S.L. Jensen, D. Dingwell, J. Non-Cryst. Solids 336 (2004) 179. M. Solvang, Y.Z. Yue, S.L. Jensen, D. Dingwell, J. Non-Cryst. Solids 351 (2005) 499. Y.Z. Yue, Phos. Res. Bull. 13 (2002) 39. Y.Z. Yue, J. Non-Cryst. Solids 345 (2004) 523. Y.Z. Yue, S.L. Jensen, J. de C. Christiansen, in: Proceedings of the XIX International Congress on Glass, Edinburgh, Scotland, 1–6 July 2001, p. 311. M. Solvang, Y.Z. Yue, J.deC. Christiansen, S.L. Jensen, Glass Technol. C 43 (2002) 340. Y.Z. Yue, Unpublished data. R. Bru¨ckner, Y.Z. Yue, A. Habeck, Glass Sci. Technol. 67 (1994) 114. Y.Z. Yue, G. Carl, C. Ru¨ssel, Glass Sci. Technol. 71 (1999) 67. K.-U. Hess, D.B. Dingwell, S.L. Webb, Am. Miner. 80 (1995) 297. A. Avramov, J. Non-Cryst. Solids 238 (1998) 6. Y.Z. Yue, C.A. Angell, Nature 435 (2005) E1. G.P. Johari, J. Chem. Phys. 119 (2003) 2935. Y.Z. Yue, Phys. Chem. Glasses 46 (2005) 354. C.A. Angell, J. Res. Nat. Inst. Stand. Technol. 102 (1997) 171. D. Kivelson, G. Tarjus, J. Phys. Chem. B 105 (2001) 6620. W. Kauzmann, Chem. Rev. 43 (1948) 219. G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139.
Journal of Non-Crystalline Solids 354 (2008) 1112–1118 www.elsevier.com/locate/jnoncrysol
Characteristic temperatures of enthalpy relaxation in glass Yuan-Zheng Yue
*
Section of Chemistry, Aalborg University, Aalborg, Denmark Available online 19 November 2007
Abstract The glass transition temperature, Tg, directly measured by differential scanning calorimetry at 10 K/min is compared with the Tg indirectly determined by fitting viscosity data to a viscosity model for oxide glasses. The results show good match between the two Tg values. A standard, unified approach for measuring Tg is proposed. Characteristic temperatures of enthalpy relaxation in glass are defined, and the relationships between these temperatures are illustrated by performing aging and calorimetric experiments on hyperquenched glasses. The features of the energy release peak, the endothermic pre-peak, and the real glass transition are discussed with respect to their physical origins. Ó 2007 Elsevier B.V. All rights reserved. PACS: 61.43.Fs; 61.20.Lc; 64.70.Pf; 66.20.+d Keywords: Calorimetry; Enthalpy relaxation; Glass transition; Fragility; Viscosity
1. Introduction The relaxation time of any liquid towards equilibrium has a unique dependence on temperature. In glass science, the dynamics and the physical properties are often compared between different glass systems. Yet, this is only meaningful if the comparison is made at a universal characteristic temperature (e.g. the glass transition temperature Tg) or at a temperature scaled by a characteristic temperature, usually by Tg (i.e. Tg/T). For instance, the dynamic features of various liquids are well distinguishable by the use of the fragility concept based on the Tg/T dependence of the relaxation time [1]. Tg is the most useful, characteristic, dynamic temperature of a liquid. However, the value of Tg depends both on its definition and on the method of determination, which have not yet been unified in glass literature. Therefore, different Tg values for same glass composition are often found in literature. To make the determination of Tg more accurate and reasonable, carefully performed measurements have been reported in
*
Tel.: +45 9635 8522; fax: +45 9635 0558. E-mail address: [email protected]
0022-3093/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.11.027
[2,3]. The enthalpy-matching approach turned out to be a precise method for determining Tg, which is based on the heat capacity versus temperature curve obtained from calorimetric upscan [2]. Besides the characteristic temperature Tg, it is also important to define and determine other characteristic temperatures such as the fictive temperature (Tf), the onset temperature of energy release (Tz), and the onset temperature of the endothermic aging pre-peak (Tpp) [4–14]. Tz and Tpp are encountered when hyperquenched glass (HQG) or aged HQG is upscanned in a calorimeter [4–12]. The relationships between these characteristic temperatures have not yet been fully clarified. Although the fictive temperature itself describes neither the details of the structure nor the details of the thermal history of glasses, its relation to the cooling rate does provide information on the dynamic features of a liquid. This relation may be used for determining the activation energy of viscous flow and the fragility index of the liquid [12]. Tf is a parameter that reflects enthalpy and physical properties (e.g. density, refraction index, etc.) of a glass. Tf of a glass varies with both the cooling rate and the aging degree [3,5,12,13]. In particular, when a HQG glass is aged and then calorimetrically scanned, it exhibits changes in Tf. The structure of the glass
Y.-Z. Yue / Journal of Non-Crystalline Solids 354 (2008) 1112–1118
corresponds to that of its liquid at T = Tf. A glass with high Tf, i.e. a HQG, is a good object for studying the vibrational dynamics and potential energy landscape [6]. The Tf of a HQG can be calculated from the heat capacity data using the method proposed in [5,11,12]. For glass fiber industry, Tf is a useful parameter to monitor and control the stability of the fiber spinning process, since Tf is sensitive to both drawing and cooling conditions. This paper focuses on two aspects. First, Tg values determined by a calorimetric method are compared with those determined using a viscometric method. Based on this, a standard method for determining Tg is proposed and recommended to glass community. Second, the relationships between the various characteristic temperatures of enthalpy relaxation are elucidated for two hyperquenched oxide glasses by performing aging and differential scanning calorimetric (DSC) measurements. The endothermic pre-peak of an aged HQG is compared with the glass transition peak concerning their physical origins. Distinguishing between these two peaks is important for clarification of glass transition of glasses [4–12]. Determination of the characteristic temperatures is an essential step for studying the dynamics, the potential energy, and the structural evolvement in both liquid and glass. In addition, unifying both the definition and the determination method for Tg and Tf is useful for establishing a unified technological standard for evaluating and controlling the density, the refractive index, the enthalpy level, and hence the stress level of glass products. A unified method for Tg determination makes a precise, reasonable comparison in fragility and other dynamic properties between different glass systems possible. 2. Experimental
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Table 1 Comparison between the Tg values estimated by fitting the viscosity data to Eq. (1), Tg,vis and Tg measured by DSC, Tg,DSC Glass
Tg,vis (K)
Tg,DSC (K)
DTga (K)
CAS1 CAS2 CAS3 CAS4 CAS5 CAS6 CAS7 CAS8 CAS9 CAS10 Anorthite Wollastonite Gehlenite NaPoLi CMP BAS1 BAS2 NIST710a Diopside
1044 1055 1067 1078 1087 1066 1078 1085 1094 1103 1128 1037 1112 516 798 899 946 819 985
1043 1057 1065 1075 1084 1066 1075 1082 1093 1104 1128 1040 1110 516 799 902 943 822 986
1 2 2 3 3 0 3 3 1 1 0 3 2 0 1 3 3 3 1
The uncertainty of Tg,vis and Tg,DSC are ±2 and ±1 K, respectively. a Deviation DTg = Tg,vis Tg,DSC.
capacity (Cp) curves were obtained using the evaluation method described in [11,12]. The Cp curve of the first upscan reflects the enthalpy level of a fresh sample with unknown thermal history (i.e. a sample cooled at an unknown rate prior to the first upscan), whereas the Cp curve of the second upscan reflects the enthalpy level of a sample with a defined thermal history (i.e. a sample cooled at the standard rate of 10 K/min prior to the second upscan). The standard Tg values of all glasses were obtained from the DSC second upscans.
2.1. Viscometric and calorimetric measurements
2.2. Hyperquenching-aging-scan experiments
Both viscometric and DSC experiments were performed on 19 inorganic glasses (17 silicate and 2 phosphate glasses), the chemical composition of which are given in [15–21]. Table 1 lists the Tg values of the 19 glasses. ‘CAS’ refers to the calcium aluminosilicate glasses with different CaO/SiO2 and CaO/Al2O3 ratios [15,16]. ‘NaPoLi’ refers to sodium lithium metaphosphate (Na2O Æ Li2O Æ 2P2O5) glasses. ‘BAS1’ refers to the glass made from a basaltic melt. ‘NIST710a’ refers to the standard glass made by National Institute of Science and Technology. ‘CMP’ refers to the calcium metaphosphate glass. The viscosity data for the low-temperature range were measured using the parallel-plate (servo-hydraulic test machine from MTS company) [22], beam-bending (Baehr VIS 401) [23], and penetration (Baehr DIL 802 V) viscometers [24], whereas viscosity data for the high-temperature range were obtained using concentric cylinder viscometers (Paar Physica and Brookfield model RTVD) [15–21]. All the samples underwent two runs of up- and downscans at 10 K/min in argon in a DSC (Netzsch STA 449C Jupiter). The heat
To determine the characteristic temperatures of glasses and liquids, the hyperquenching-aging-DSC scan experiments were conducted on two types of glass systems. The first one is a basalt-like glass system (in short, BAS2) used for making insulation materials – stone wool, the composition is given in [4]. BAS2 is obtained from a mixture of several types of rocks with basalt as a dominant ingredient. The hyperquenched BAS2 (in short, HQ-BAS2) is obtained from the BAS2 melts using the cascade fiber-spinning process. The second system is the calcium metaphosphate glass (CaO Æ P2O5) (CMP). Two types of hyperquenched samples (in short, HQ-CMP) were achieved by means of both continuous fiber drawing and cascade fiber-spinning, which yield hyperquenching rates of 104 and 108 K/min, respectively. The cooling rates were calculated using the method proposed in [12]. One HQ-BAS2 sample was aged at 723 K for 55 days, whereas another was aged at 773 K for 55 days. They were both scanned twice by a DSC at 10 K/min. The first scan exhibits the enthalpy relaxation of the aged HQ-BAS2,
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while the second scan shows the enthalpy relaxation of the ‘standard’ glass (i.e. the glass cooled at a standard rate of 10 K/min). The HQ-CMP samples were not subjected to any aging process. 3. Results Fig. 1(a) and (b) show both the logarithm of viscosity (log g) and the heat capacity (Cp), respectively, as a function of temperature (T) for glass BAS1. The Cp curve is measured at 10 K/min during the second upscan. The solid curve in Fig. 1(a) is obtained by fitting the viscosity data to the Avramov–Milchev (AM) model log g ¼ A þ BðT g =T ÞF ;
ð1Þ
where A and B are constants (A + B = 12) and F is the fragility index. The reasons why Eq. (1) is applied here are as follows. First, for oxide glasses, Eq. (1) fits viscosity data better than the Vogel–Fulcher–Tamman (VFT) and the Adam–Gibbs (AG) models [15]. Second, both the standard Tg and the fragility index F can be directly derived from Eq. (1). The detailed discussion about Eq. (1) is given in [12,15,25]. From Fig. 1(a), it is seen that the quality of 14 12
data fit by Eq. (1)
Tg,vis
8
a)
6 4 2 0 1.8
upscan downscan
Cp (Jg-1K-1)
1.6 1.4
1200
measured data linear fit
1.2 1.0 0.8 600
b)
Tg,DSC
800
1000
1200
1400
1000
Tg,DSC (K)
log (Pa s)
10
the fit is satisfying. Tg values can be derived by using Eq. (1). The obtained Tg is denoted Tg,vis (Fig. 1(a)). The standard Tg obtained by DSC is denoted Tg,DSC. It is defined as the onset temperature of the glass transition peak, i.e. the temperature of the intersection between the extrapolated line of the glass Cp and the extrapolated line of the rapidly rising Cp (Fig. 1(b)). The Cp curve is measured at the standard upscan rate of 10 K/min. Before the upscan, the sample must be cooled also at 10 K/min. The Tg,DSC corresponds to the inflection point of the downscan Cp curve (see the dotted lines in Fig. 1(b)). The uncertainty of both Tg,vis and Tg,DSC are ±2 and ±1 K, respectively. Fig. 1 shows a direct correspondence between the characteristic viscosity (1012 Pa s) and the calorimetric Tg. This link exists not only for basaltic glasses, but also for other oxide glasses (Table 1 and Fig. 2). As shown in Table 1 the maximum difference in Tg values between Tg,vis and Tg,DSC is rather small, i.e. equals ±3 K. Fig. 2 shows an excellent linear fitting of the Tg,vis versus Tg,DSC relationship with a correlation coefficient of 0.9996. The dashed curve is the fitting curve. In Fig. 3 two common methods for determining Tg are illustrated. Fig. 3(a) shows the typical enthalpy relaxation of HQ-BAS2, which is manifested by the deviation of the heat capacity (Cp1) of the hyperquenched glass (first upscan) from the heat capacity (Cp2) of the ‘standard’ state (second upscan). The second upscan is regarded as the standard upscan, since both the heating rate (qh) and the prior cooling rate (qc-pre) are 10 K/min. As shown in Fig. 3(a), the fictive temperature Tf is determined using the enthalpy-matching method, a detailed description of which is given in [5,11,12]. Fig. 3(b) demonstrates a well-known enthalpy-matching method used for determining Tg [2]. A comparison between Fig. 3(b) and (c) shows that the Tg determined by the enthalpy-matching method is in accordance with Tg,DSC. Such a correspondence is a general feature of the oxide
800
T (K) Fig. 1. Direct correlation between the characteristic viscosity (1012 Pa s) and the calorimetric glass transition temperature (Tg) for the BAS1 glass. (a) The logarithm of the viscosity (log g) as a function of temperature (T). The triangles represent the data measured using the parallel-plate compression viscometer (in the lower temperature range) and the concentric cylinder viscometer (in the higher temperature range). The solid curve is obtained by fitting the experimental data to Eq. (1). (b) The heat capacity (Cp) as a function of temperature. The solid curve is the Cp recorded during the second upscan, whereas the dashed curve represents the second downscan Cp.
600
400 400
600
800
1000
1200
Tg,vis (K) Fig. 2. Comparison between the glass transition temperatures measured by DSC at 10 K/min, Tg,DSC, and those obtained from the fit of viscosity data to Eq. (1), Tg,vis, for 19 inorganic glasses [5–11].
Y.-Z. Yue / Journal of Non-Crystalline Solids 354 (2008) 1112–1118
a)
Te
B
Cpliq
= DE
=
1.4
B 1.2
Tz
Cp2
1.0
Cpg
A
Cp1
0.8
Cp (Jg-1K-1)
1.6
Tg
b) =
1.4 1.2
Tz
standard
1.0 0.8 1.6
c)
upscan 1 upscan 2 (standard)
1.4
Tpp
Tcross
1.2 1.0 0.8 400
600
800
1000
1200
T (K) Fig. 3. Determination of various characteristic temperatures during enthalpy relaxation measured by DSC on both non-aged and aged HQBAS2 glasses. (a) Determination of fictive temperature (Tf) by using the enthalpy-matching method, and of the onset temperature (Tz) of energy release from the HQ-BAS2. Cp,liq is the heat capacity of the liquid, whereas Cp,g is the heat capacity of the glassy state. (b) High-temperature part (i.e. above Tg) of Fig. 3(a), which includes the left borderline (the line for determining Tg) and the right borderline (the line for determining Tf). The way of placing the left borderline is described in Ref. [1]. (c) The onset temperature of the aging-induced pre-peak, Tpp, [4] and the standard glass transition temperature (Tg) for the glass that is first hyperquenched and then annealed at 723 K for 55 days. Tcross is the distinction temperature between the pre-peak and the energy release peak. The heating and cooling rates for the DSC measurements are 10 K/min. The vertical dashed line illustrates the correspondence between the Tg value determined by the enthalpy-matching method and the onset Tg value. Fig. 3(a) and (b) are re-plotted from [11].
Tg
where Cp,liq and Cp,g are the heat capacities of the liquid (above Tg) and the extrapolated glassy state, respectively. Tf can be estimated using Eq. (2). As shown in Fig. 4, DE increases with the cooling rate of the glass. Fig. 5 shows a large enhancement of the pre-peak of the aged HQ-BAS2 by extending the aging time (ta) from 8 to 55 days. As a consequence, the onset temperature of the pre-peak, Tpp, shifts to a higher temperature (i.e. from 797 to 928 K). The pre-peak nearly overlaps with the real Tg peak. If the aging time was sufficient, the Tpp would approach the real Tg, and the pre-peak would be fully superimposed on the real glass transition peak. Fig. 6 shows that for the HQ-BAS2 the Tpp increases with increasing aging temperature, Ta. In addition, Fig. 6 implies the existence of the energetic and structural heterogeneities, since the lower energy regimes (represented by the pre-peak) and the higher energy regimes (repre-
1.6
1.4
0.15
Cp,exc (Jg-1K-1)
A
quently aged at temperatures below the real Tg. The prepeak shown in Fig. 3(c) is caused by the aging process. Fig. 4 demonstrates that the onset temperature (Tz) of energy release for the HQ-CMP depends on the prior cooling rate. Tz decreases with increasing cooling rate. The inset in Fig. 4 shows the excessive heat capacity (Cp,exc = Cp2 Cp1) as a function of temperature (see Fig. 3(a)). Te is the characteristic temperature, at which the quenchedin energy is completely released, i.e. Cp1 = Cp2. At Te the liquid returns to internal equilibrium. The integral of the Cp,exc over T represents the amount of energy stored in the glass during hyperquenching, DE (see hatched areas in Fig. 3(a)) Z Te Z Tf DE ¼ ðC p2 C p1 ÞdT ¼ ðC p;liq C p;g ÞdT ; ð2Þ
Tf
Cp (Jg-1K-1)
1.6
1115
1.2
Te
108 K/min 0.10 0.05
Tz1
Tz2
105 K/min
0.00 400
600
800
T (K)
Tz2 1.0
Tz1 Standard 108 K/min 105 K/min
0.8
glasses if both the prior downscan rate and the upscan rate are equal to 10 K/min. Fig. 3(c) shows another characteristic temperature, i.e. the so-called shadow glass temperature Tpp, which is the onset temperature of the endothermic pre-peak (in the literature there are different names for the pre-peak [26], e.g. excess endothermic peak [8–10], sub-Tg peak [27], shadow Tg peak [4,6] and pre-endotherm [7,28]). The pre-peak can only occur if a glass is first hyperquenched and subse-
300
400
500
600
700
800
900
T (K) Fig. 4. Heat capacity (Cp) as a function of temperature for the CMP glasses cooled at the rates: 10 K/min (solid curve), 105 K/min (dashed curve), and 108 K/min (dot-dashed curve). Inset: the excessive heat capacity as a function of temperature. Tz1 and Tz2 are the onset temperatures of energy release. All Cp curves were measured at a DSC upscan rate of 10 K/min. The glass cooled at 10 K/min is termed as standard glass.
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1.8
ta=8 days ta=55 days standard
Cp (Jg-1K-1)
1.6
1.4
Tpp2 Tpp1
1.2
Tg 1.0 600
700
800
900
1000
T (K) Fig. 5. Comparison between the glass transition peak of the ‘standard’ BAS2 and the pre-peak of the HQ-BAS2 aged at 773 K for periods of 8 and 55 days, respectively. The DSC up and downscans were performed at 10 K/min.
0.3
Cp,exc (Jg-1K-1)
0.2
Ta=723 K
exo 0.1
Tcross Tcross
0.0
-0.1
Ta=773 K
ta=55 days
600
700
800
900
1000
1100
T (K) Fig. 6. The excessive heat capacity (Cp,exc) of the HQ-BAS2 aged at different temperatures for 55 days. The DSC upscans were performed at a rate of 10 K/min.
sented by the energy release peak) co-exist in the same glass. 4. Discussion 4.1. Glass transition temperature (Tg) From Eq. (1), log g = A + B, when T = Tg. The constant A + B depends on the heating rate (qh) and the prior cooling rate (qc-pre) used to measure Tg by DSC. When Tg is measured at qh = qc-pre = 10 K/min, A + B equals 12, and this is confirmed by Figs. 1 and 2. The overall relation between cooling rate and viscosity is described in [12]. As shown in Fig. 2, there is an excellent agreement between Tg,vis at the viscosity of 1012 Pa s and Tg,DSC measured at qh = qc-pre = 10 K/min. This is reflected by the linear
relation Tg,vis = Tg,DSC with the correlation coefficient of 0.999. The agreement means that fitting viscosity data of oxide glasses to Eq. (1) can be precisely performed by fixing A + B at 12, i.e. by using the characteristic viscosity of g = 1012 Pa s at Tg. Since the 19 glass systems in Table 1 involve a wide range of fragilities (i.e. fragility index F of Eq. (1) ranges from 2.7 to 5.4, which corresponds to a range of Angell fragility index m [1] from 83 to 167), it is expected that the relation Tg,vis = Tg,DSC should apply to other inorganic glass systems. The further importance of the relation Tg,vis = Tg,DSC lies in the following aspects. First, Tg,vis can be obtained by measuring the viscosity and then by fitting the data with Eq. (1). Reversely, if Tg,DSC is measured, the viscosity at Tg,DSC is naturally known, i.e. equal to 1012 Pa s. Second, the entire log g T relationship for the glasses with high crystallization tendency may be obtained by fitting Eq. (1) to the high-temperature viscosity data (usually above the liquidus temperature) and to the 1012 Pa s – Tg,DSC point without measuring the low-temperature viscosities (above Tg). The measurements of the low T viscosities are impossible for glass systems that have a strong tendency to crystallize. Third, the viscosity model log g = A + (12 A)(Tg/T)F can be used directly to construct the fragility plot of glass systems. Fourth, the Tg measured at qh = qc-pre = 10 K/min is the most suitable characteristic temperature for scaling the temperatures of a fragility plot, since for the fragility plot, the Tg, in the literature, usually assumed to correspond to the viscosity of 1012 Pa s. The present work (see Figs. 1 and 2) shows that such an assumption is only reasonable, if Tg is assigned as the onset temperature of the glass transition during the DSC upscan determined at qh = qc-pre = 10 K/min. The viscosity at the onset temperature determined at qh = qc-pre 5 10 K/min would deviate from 1012 Pa s, since the onset temperature depends on both qh and qc-pre [12]. According to the above analysis, Tg,DSC (at 10 K/min) and Tg,vis (at 1012 Pa s) are equal and may both be used as the standard Tg. The experimental methods used to determine both the Tg,DSC and the Tg,vis can be applied as the standard method for determining Tg. This standardization of Tg makes a direct comparison between the glass transition temperatures of different glass systems possible. Here it should be noted that glass transition occurs over a range of temperatures, not at a single temperature. The Tg defined in this work is a well representative temperature for the glass transition, i.e. the onset temperature associated with the viscosity of 1012 Pa s. From glass literature it is known that even for the same glass, the measured Tg values are quite diversified, since various experimental methods are used and different heating and prior cooling rates are used in different measurements. For instance, most Tg measurements reported are performed at the heating rates 1–30 K/min that corresponds to viscosities of 1013–1011.5 Pa s. However, problems caused by the diversity in Tg could be avoided using the method presented in the present work.
Y.-Z. Yue / Journal of Non-Crystalline Solids 354 (2008) 1112–1118
4.2. Fictive temperature (Tf) and onset temperature for energy release (Tz) Figs. 3 and 4 illustrate that the enthalpy of the HQ-BAS2 begins to decrease when the temperature exceeds the onset temperature (Tz) of release of the excess enthalpy (i.e. the hatched area between the Cp2 and Cp1 curves in Fig. 3(a)). The excess energy (or excess enthalpy) is not completely released until the temperature reaches the equilibrium temperature Te (see Figs. 3 and 4). When a liquid at Te is cooled at 10 K/min to a temperature well below Tg, it becomes a glass with a specific Tf, i.e. the standard Tg defined above, which can be observed during DSC reupscan at 10 K/min. The excess energy is attributed to the freezing-in of the excited configurational and vibrational states at Tf. The most excited states are so unstable that a temperature equal to Tz (well below Tg), and hence the lowest kinetic energy 1/2kbTz (where kb is the Boltzmann constant), can bring the glass to an energy level corresponding to the real temperature. As shown in Fig. 4, the onset temperature (Tz) of the energy release of the HQ-CMP decreases with increasing cooling rate. This implies that Tz decreases with increasing the fictive temperature Tf, since Tf increases with increasing the cooling rate. The dependence of Tz on Tf involves three situations. First, when a liquid is cooled at a rate higher than the standard rate of 10 K/min, Tf is higher than Tz. Second, when a liquid is cooled at 10 K/min, Tf and Tz merges to the standard Tg. Finally, when a liquid is cooled at a rate below 10 K/min, Tz does not exist, since no energy release takes place. Instead, an enhancement of the Tg-peak would occur, i.e. a deficient heat capacity would exist in the glass transition region, and therefore, Tf should be lower than Tg. As mentioned previously, the equilibrium temperature Te is the temperature at which the enthalpy release is completed. Fig. 4 shows that the Tz (423 K) of the HQ-CMP (cooled at 108 K/min) is much lower than the ideal glass temperature T0 (630 K) that is obtained by fitting the VFT equation to the viscosity data of the CMP melt. According to [29,30], T0 is approximately equal to the Kauzmann temperature Tk (a thermodynamic, characteristic temperature) [31]. This means that the Tz can be far below Tk, if the quenching rate of a liquid is high enough. From the relation Tz < Tk, it can be inferred that the hyperquenching process of a liquid involves not only a configurational but also a vibrational freezing-in process, since Tk is the temperature at which the configurational entropy vanishes [31]. Finally, two important questions still remain open: (a) what is the quantitative relationship between Tz and Tf and (b) are there limiting values for both Tz and Tf, respectively. 4.3. The onset temperature of the endothermic pre-peak (Tpp) The Tpp fundamentally differs from the glass transition temperature (Tg) with respect to their physical sources.
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The pre-peak is a consequence of thermal aging and is attributed to the energetic and structural heterogeneities of glass. The pre-peak is a manifestation of the non-exponential nature of glass relaxation. In detail, pre-peaks can be regarded as the aging-induced ‘shadow’ glass transitions of the short relaxation time components (‘microglasses’) of the macroglass [4]. But, the real glass transition is a consequence of the slowing-down of the primary, cooperative relaxation process (the so-called a relaxation) upon cooling, or speeding-up of the a relaxation process upon heating. An abrupt change in the configurational entropy occurs during the glass transition upon heating or cooling. The size of the cooperative rearrangement regions rapidly increases when the temperature is lowered towards the glass transition [32]. Distinguishing the endothermic pre-peak transition from the glass transition is essential for understanding both the structural relaxation of hyperquenched glasses during aging process, and the structural heterogeneity in glass. The pre-peak shows a recovery feature under certain aging and DSC-scan conditions [28]. Fig. 3(b) shows a crossover temperature Tcross between the pre-peak and the energy release peak, which depends on aging time and temperature (see also Figs. 5 and 6). The pre-peak is particularly pronounced for a rapidly cooled glass compared with a slowly cooled glass, if they both are aged under same conditions (i.e. same aging temperature Ta and time ta). The Tpp and Tg peak would overlap, as Ta or ta increase sufficiently. If a slowly cooled glass is aged at a temperature below Tg for long time, an enhancement of the Tg peak can be observed on the DSC upscan curve. This indicates that the aging process leads to a decrease in Tf. The difference value between Tpp and Tg depends both on the remaining excess energy of the HQG, and on the non-exponentiality of relaxation. It is evident that Tpp increases with Ta and ta. According to [27], Tpp should equal Ta. However, as shown in Fig. 5, that equality does not always apply, and hence it is not a universal feature of the pre-peak as stated in [27]. Furthermore, the height of pre-peak is rather sensitive to degree of aging, e.g. it is even measurable after only 8 min aging on HQ-BAS2 at Ta/Tg = 0.87 [26]. 5. Conclusions For the oxide glasses, the temperature at a viscosity of 1012 Pa s (Tg,vis), which is obtained by fitting the viscosity–temperature data to the AM model, is in accordance with the onset temperature of the glass transition measured calorimetrically at the prior downscan and upscan rates of 10 K/min, Tg,DSC. Both Tg,vis and Tg,DSC should be adopted and used as the standard Tg and as the scaling temperature of the fragility plot. The onset temperature (Tz) of the energy release decreases with increasing the fictive temperature of hyperquenched glasses. However, it is unclear how Tz is quantitatively related to Tf. The origin of the pre-peak is fundamentally different from that of the real glass transition.
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