Unified approach for determining the enthalpic fictive temperature of glasses with arbitrary thermal history

Journal of Non-Crystalline Solids 357 (2011) 3230–3236

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Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l

Unified approach for determining the enthalpic fictive temperature of glasses with arbitrary thermal history Xiaoju Guo a, b, Marcel Potuzak a, John C. Mauro a,⁎, Douglas C. Allan a, T.J. Kiczenski a, Yuanzheng Yue b, c,⁎⁎ a b c

Science and Technology Division, Corning Incorporated, Corning, NY 14831, USA Section of Chemistry, Aalborg University, DK-9000 Aalborg, Denmark Key Laboratory for Glass & Ceramics, Shandong Polytechnic University, Jinan 250353, China

a r t i c l e

i n f o

Article history: Received 7 April 2011 Received in revised form 13 May 2011 Available online 14 June 2011 Keywords: Fictive temperature; Glass relaxation; Numerical simulation

a b s t r a c t We propose a unified routine to determine the enthalpic fictive temperature of a glass with arbitrary thermal history under isobaric conditions. The technique is validated both experimentally and numerically using a novel approach for modeling of glass relaxation behavior. The technique is applicable to glasses of any thermal history, as proved through a series of numerical simulations where the enthalpic fictive temperature is precisely known within the model. Also, we demonstrate that the enthalpic fictive temperature of a glass can be determined at any calorimetric scan rate in excellent agreement with modeled values. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The glass transition is a kinetic phenomenon with profound thermodynamic consequences [1]. As a liquid is supercooled through the glass transition region, it experiences a successive freezing of its configurational degrees of freedom [2,3]. This structural arrest leads to a change in second-order thermodynamic properties such as heat capacity and thermal expansion coefficient [4]. While most research has focused on dynamical aspects of supercooled liquids [5–7] and glass [7–10], the thermodynamic signatures of the glass transition have received somewhat less attention. In this paper we propose a technique for isolating the thermodynamic aspects of the glass transition based on differential scanning calorimetry (DSC). This technique enables a decoupling of the intrinsic thermodynamic state of a glass from the kinetic effects that inevitably occur during measurement [11,12]. We show that the approach is universally applicable to glasses independent of their thermal history and the measurement scan rate. The nonequilibrium state of a glass is typically described in terms of an additional order parameter known as the fictive temperature, Tf. However, multiple incompatible definitions of fictive temperature have been proposed in literature. As originally conceived by Tool [13], fictive temperature is the temperature from which a liquid must be rapidly quenched to produce a particular structural state of the glass

⁎ Corresponding author. ⁎⁎ Correspondence to: Y. Yue, Key Laboratory for Glass & Ceramics, Shandong Polytechnic University, Jinan 250353, China. E-mail addresses: [email protected] (J.C. Mauro), [email protected] (Y.Z. Yue). 0022-3093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2011.05.014

at low temperature. In other words, if a glass is brought to a temperature T = Tf, it will be in equilibrium. However, the pioneering crossover experiments of Ritland [14] showed that relaxation still occurs under these circumstances, demonstrating that T = Tf does not ensure equilibration of the glass. Narayanaswamy [15] subsequently overcame this problem by introducing additional order parameters in the form of multiple fictive temperatures, allowing for representation of the Ritland crossover effect. In the Narayanaswamy view, the use of multiple fictive temperatures “indicates that any nonequilibrium state is actually a mixture of several equilibrium states” [15]. This theory was recently disproven by Mauro et al., [16,17] who considered a statistical mechanical definition of fictive temperature distributions within the energy landscape description of glass-forming systems. An alternative definition to fictive temperature was proposed by Moynihan et al., [18,19] who adopted a macroscopic thermodynamic approach to the problem of glass relaxation. Rather than interpreting fictive temperature in a microscopic structural sense, Moynihan defined Tf as “simply the structural contribution to the value of the macroscopic property of interest expressed in temperature units” [19]. This interpretation bypasses the problems with a microscopic definition by instead focusing on the macroscopic thermodynamics of the glass. This is the definition of fictive temperature considered in the current paper. Although the fictive temperature calculation was performed only for inorganic oxides in Moynihan's paper, the definition and determination of fictive temperature has also been widely discussed in the polymers literature [20–22]. In addition to the above structural and thermodynamic definitions of fictive temperature, the difference between fictive and equilibrium temperatures has been used to characterize the irreversible relaxation process by Nieuwenhuizen [23] and Garden et al. [24]. However, in this paper we

X. Guo et al. / Journal of Non-Crystalline Solids 357 (2011) 3230–3236

treat the fictive temperature as determined by a single thermodynamic property, viz., the enthalpy. While Moynihan offers a comparatively simple definition of Tf, the measurement of Tf for glasses having different thermal histories is complicated by structural relaxation during measurement. A calorimetric area-matching (i.e., enthalpy-matching) approach for determining the thermodynamic fictive temperature of a glass was proposed by Moynihan et al. [18] based on differential scanning calorimetry (DSC), in which the isobaric heat capacity (Cp) of a sample is measured at 1 atm along a linear temperature path. One single Cp upscan curve is utilized in this approach. The fictive temperature derived using this method reflects the structural contribution to the enthalpy of a glass, i.e., the temperature at which the equilibrium liquid has the same structural enthalpy as the glass sample. The Moynihan routine requires that the upscan rate of the DSC be of similar magnitude as the cooling rate of the initially formed glass, usually 10 or 20 K/min. Moynihan et al. [18] also determined the Tf of the B2O3 glass cooled at 80 K/min in DSC based on the area matching approach with an upscan rate of 10 K/min. Due to this mismatch of cooling and upscan rates, they observed a small exothermic peak on the Cp curve, the area of which was considered during the Tf determination procedure. However, the Moynihan routine cannot be used for rapidly cooled (hyperquenched) glasses where the cooling rate (q) is much faster than any possible scan rate in a laboratory DSC. For instance, Fig. 1 shows the heat capacity measured using DSC of basaltic glass fiber hyperquenched at 10 6 K/s. As shown in this figure, the measured heat capacity from the DSC corresponds to the glassy heat capacity only at temperatures lower than 550 K. The measured heat capacity at temperatures higher than 550 K is a mixture of the glassy heat capacity and an exothermal peak due to relaxation. The exothermal peak is so broad and intense that the extrapolation of the glassy heat capacity to high temperature cannot be determined from this single upscan. For example, both the red and blue curves shown in Fig. 1 may be considered as possible extrapolations of glassy heat capacity. However, the fictive temperature determined based on these two extrapolated curves would differ greatly. A method for determining Tf of hyperquenched (q ≈ 10 6 K/s) glass fibers was proposed by Yue et al. [25,26] and independently by Velikov et al. [10] based on two consecutive Cp curves at the same upscan rate. The first upscan yields the Cp curve of the initially formed hyperquenched (HQ) glass, Cp1(T), where the precise quench rate may not be known. This first upscan is followed by downscan of the sample at a cooling rate equal to the heating rate of the upscan. A second upscan of the DSC is then performed on this “rejuvenated” glass using the same scan rate as previously. The integrated difference between

1.6

Cp (J/g-K)

1.4 1.2

?

1.0

3231

the heat capacity of the rejuvenated glass, Cp2(T), and the HQ glass, Cp1(T), gives the enthalpy difference due to the thermal history difference of the two scans, which can then be used to determine the fictive temperature of the initially formed glass. The Yue method is manifested by Eq. (4) in Ref. [25] and theoretically can be applied for determining Tf of a glass with any thermal history. However, it should be stressed that the graphical method shown in Fig. 2 of Ref. [25] is only applicable for the glass with a fictive temperature Tf higher than Teq (the onset temperature of the liquid equilibrium), since only in this case the integration term Cp liquid–Cp glass in Eq. (4) of Ref. [25] is equal to the term (Cp2–Cp1). In the case of Tf b Teq the abovementioned two terms are not interchangeable. Ref. [25] did not graphically show how to apply its Eq. 4 in determining the Tf of the glass in the case of Tf b Teq. It should be mentioned that the Yue method was incorrectly applied by several other scientists for the case of Tf b Teq [27]. In addition, the Yue et al. method requires that the second upscan should be made at the standard rate of 10 K/min in order to determine the standard glass transition temperature Tg that is needed to find the Tf of a glass. The goal of this paper is to present a general, unified approach for the calorimetric determination of fictive temperature both graphically and numerically. 2. Generalized method for determination of fictive temperature In Fig. 2 we present a general routine for rigorous determination of fictive temperature for glasses of any thermal history (e.g., annealed or hyperquenched) under isobaric conditions. Our method reveals a thermodynamic signature of glass that is independent of kinetic effects due to changes in DSC upscan rate [11,12]. As with the Yue et al. approach, [25,26] the technique involves comparing Cp1(T) and Cp2(T) from two consecutive upscans of the DSC at the same heating rate. The thermal history of the glass prior to measurement need not be known. The two upscans are interposed by a downscan. Unlike previous methods, there is no restriction on the particular value of the DSC upscan rate. The new technique involves three area-matching steps, illustrated in Fig. 2(a)–(c). The first step involves calculation of Tf2, the fictive temperature of the rejuvenated glass. Theoretically, the fictive temperature of any glass is independent of the heating rate during upscan, which is also proved by our experiment and the new technique as shown below. Thus, the heating rate for the second upscan does not need to be the same as the previous cooling rate. However, in order to make the problem simple, the upscan rate is set to be equal to the previous downscan rate in the development of the routine. The value of Tf2 must satisfy the area-matching condition of Fig. 2(a) to ensure that the structural enthalpy of the glass is equal to that of the equilibrium liquid at the fictive temperature. This step is similar to that proposed previously by Moynihan et al. [18]. The second step is to calculate the enthalpy difference between the initially formed and rejuvenated glass, as depicted in Fig. 2(b). This step is identical to that proposed previously by Yue et al. [25,26] The final step of the calculation, shown in Fig. 2(c), involves calculating the fictive temperature of the initially formed glass, Tf1, by:    ∞  ∫ Cp liquid −Cp glass dT = ∫ Cp2 −Cp1 dT:

Tf 1 Tf 2

ð1Þ

0

0.8 0.6

400

600

800

1000

1200

T (K) Fig. 1. Representative measured heat capacity curve (solid line) of hyperquenched glasses where the heating rate for the DSC upscan is 20 K/min. The dashed lines and the question mark show the uncertainty of extrapolating the glass heat capacity encountered when the second standard upscan is missing.

The graphic in Fig. 2 is drawn for the case of an initially quenched glass, but the routine is equally applicable for annealed glasses, where the initial cooling rate is slower than the upscan rate of the DSC. This situation is depicted in Fig. 3 for an experimentally measured sample of NIST SRM 710A, a standard soda lime silicate glass obtained from Schott AG. The glass was annealed for several hours at its annealing point (818 K, equal to the glass transition temperature), followed by a very slow rate cooling to room temperature over the course of a few

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Fig. 3. Measured heat capacity curves of annealed NIST SRM 710A soda lime silicate glass using a DSC upscan rate of 10 K/min. (a) Decomposition of the heat capacity signal into glassy and excess contributions. (b) Application of the area-matching procedure to determine Tf1 and Tf2.

Fig. 2. Graphical representation of the three-step procedure for determining the calorimetric fictive temperatures Tf1 and Tf2. (a) First, calculate the fictive temperature of the rejuvenated glass, Tf2, on the second upscan using the illustrated area-matching. (b) Second, calculate the area between the two DSC upscan curves, which gives the difference between the enthalpies of as-formed (H1) and rejuvenated glasses (H2). (c) Finally, determine the fictive temperature of the as-formed glass, Tf1, by area-matching using Eq. (1).

days. DSC measurements were conducted on optically polished 24-mg disc-shaped samples in a platinum pan under an argon atmosphere (40 ml/min flow rate). Fig. 3(a) shows the experimental heat capacity curves on the first and second upscans. This figure shows that the overshoot of the Cp curve between Tg and Teq during the first upscan is greatly enhanced compared to that of the second upscan. The origin of this difference is that on the first upscan the glass starts in a much deeper minimum of the potential enthalpy surface [4] due to its higher degree of annealing compared to the glass that is subjected to the second upscan. Theoretically, the enhancement of this overshoot is the direct consequence of the non-exponentiality and nonlinearity of the glass relaxation process [28]. The excess heat capacity is obtained by subtracting the heat capacity of the glass from the measured heat capacity curve. The heat capacity of the glass is extrapolated by using a three-parameter Maier–Kelly empirical expression [29]. The heat capacity of the glass is predominantly determined by the atomic vibrations, whereas the excess heat capacity is largely determined by the number of accessible configurations within the time scale of the

measurement. A detailed discussion about the vibrational and configurational contributions to the excess heat capacity is given by several other authors [30,31,11]. Fig. 3(b) shows the area-matching technique of Fig. 2 applied to the annealed glass. Please note that in the annealed case both of the integrals in Fig. 3 are negative, since Cp2(T) ≤ Cp1(T) and Tf1 b Tf2. To further demonstrate the ability of the three-step area-matching routine, we repeat the experiment using four different DSC scan rates for an identical glass. The experimental heat capacity curves are shown in Fig. 4, it is clear shown four different upscan rates give the same fictive temperature, which shows that the area-matching routine successfully isolates the thermodynamic value of Tf1 independent of the kinetic factors during measurement. The fictive temperature of the annealed sample is found to be 766 ± 2 K, or about 11 K below the strain point. (Strain point is defined as the 10 13.5 Pa-s isokom temperature, i.e., the temperature at which the equilibrium shear viscosity is 10 13.5 Pa s.) Fig. 5 shows the application of the generalized method to annealed hyperquenched basaltic glass fibers. The Cp curves of both the partially annealed HQ basaltic glass fibers and the rejuvenated fibers (i.e., standard glass) are shown in Fig. 5. The fibers were obtained using the centrifugal process described elsewhere [9]. The partially annealed HQ basaltic fibers were obtained by heat treating the as-made fibers at 773 K for 96 h. Both Cp curves were measured using 20 K/min. The rejuvenated fibers refer to those subjected to the first upscan and downscan at 20 K/min. First, we need to stress the features of the heat capacity curves shown in Fig. 5. A striking pre-endothermic peak appears before the exothermic peak appears, which is followed by the primary glass transition peak. The origin of this peak has been discussed in detail elsewhere [9,32,33]. This pre-peak must be taken into account when determining the average fictive temperature of the annealed HQ

X. Guo et al. / Journal of Non-Crystalline Solids 357 (2011) 3230–3236

a

glasses. Otherwise, an erroneous Tf value of such glasses would be obtained. In order to determine the Tf values of the partially annealed HQ glass, the area (enthalpy) of the pre-endotherm should be subtracted from that of the exotherm. As shown in Fig. 5, the fictive temperature of the partially annealed HQ basaltic glass fiber is 958 K which is about 6 K higher than the fictive temperature of a 20 K/min-cooled basaltic glass fiber.

Heat Capacity (J/g-K)

2.6 Scan Rate (K/min):

2.4

10

2.2

15

20

25

2.0 1.8 1.6

3. Numerical validation

1.4 1.2 1.0 780

First Upscan 800

820

840

860

880

900

920

940

Temperature (K)

Heat Capacity (J/g-K)

b Scan Rate (K/min):

1.6

10

15 20

25

1.4

1.2 1.0 780

Second Upscan 800

820

840

860

880

900

920

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Temperature (K)

c Fictive Temperature (K)

3233

800 780 760

(Mean = 766.0 K, St. Dev. = 2.0 K) 740 720 5

10

15

20

25

30

DSC Scan Rate (K/min) Fig. 4. Scan rate dependence of the annealed SRM 710A glass on the (a) first and (b) second upscans. (c) The calculated values of Tf1 using the area-matching method are independent of scan rate.

1.6

-

=

Cp (J/g-K)

Tf2 Tf1 1.4

Next we present a thorough validation of the area-matching technique using a series of simulated DSC “virtual experiments” for a model glass. The purpose of these simulations is to validate the area-matching technique for determination of fictive temperature against the precisely known values of fictive temperature within the model. The simulations are performed using the Mazurin–Startsev algorithm [34] for evolution of fictive temperature under isobaric conditions. The model incorporates stretched exponential relaxation expressed as a Prony series with twelve terms. The number of terms is chosen to ensure accurate representation of the stretched exponential, while the Prony series is adopted to allow for convenient numerical solution with simple exponentials. Calculations are performed under ambient pressure with stretching exponents ranging from β= 3/7 to 1, the former value being characteristic of long-range forces [35] and the latter being the upper limit for simple exponential relaxation. The model uses an overall rescaling of relaxation time proportional to a nonequilibrium viscosity given by the MAP model [36] where the equilibrium viscosity is given by the MYEGA expression [37]. The parameters of the model are published in row 7 of Table 1 in Mauro et al. [36], wherein this same model is used for evolution of Tf with thermal cycling. The simulated glasses are formed by linearly cooling an initially equilibrated liquid from 1498 to 298 K. We consider initial cooling rates (q) between 10−3 and 106 K/s to cover the full range of realistic thermal histories from very slowly annealed to hyperquenched. The initially cooled glasses are then subjected to two DSC upscans between 298 and 1498 K, with an intervening downscan at the same rate. The DSC scan rates are varied from 10−3 to 103 K/s, well beyond the realistic limits of a laboratory DSC. The high final temperature is chosen to ensure equilibration of the liquid for even the fastest simulated DSC scans considered. Fig. 6(a) compares the derived values of Tf1 using the area-matching method versus the known as-formed values of Tf1 given by the model. Here we show results using a stretching exponent of β = 3/5, the characteristic value for short-range forces [35]. The standard deviation of error between the derived and actual fictive temperature is 0.046 K. We found similarly excellent agreement for all other values of β. The results in Fig. 6(a) confirm that the area-matching method of Fig. 2 is applicable for all thermal histories and yields Tf1 values that are independent of DSC scan rate, effectively separating the thermodynamic signature of the glass from the kinetic aspects of DSC measurement. The impact of cooling rate on fictive temperature, as elucidated by Moynihan [18,38], can be seen in the values of Tf2 for the second upscan, plotted in Fig. 6(b).

1.2

4. Discussion 1.0

0.8

700

800

900

1000

T (K) Fig. 5. Fictive temperature determination of annealed HQ basaltic glass fiber by the generalized area-matching method. The blue line represents the measured isobaric heat capacity (Cp) curves of the HQ basaltic glass fibers subjected to an annealing process at 773 K for 96 hours, which was obtained using a DSC upscan rate of 20 K/min. The red line represents the standard Cp curves of the same sample subjected to a prior downscan at 20 K/min, which was obtained using a DSC upscan rate of 20 K/min.

The Moynihan routine, Yue routine, and the generalized routine proposed in this paper are all based on Moynihan's macroscopic definition of fictive temperature. Thus, for one heat capacity curve, if the same extrapolated glassy heat capacity and the liquid heat capacity are used, it could be expected that the fictive temperature from these three techniques should be the same. Furthermore, Eq. (4) in Ref. [25] for the Yue routine is identical to Eq. (1) in this paper if Tf2 = Tg. Thus, we can say that the Yue routine is a special case of the approach proposed in this paper. However, the graphical technique shown in Fig. 2 of Ref. [25] is only valid when Tf N Teq. For the glasses whose fictive temperature is lower than Teq, according to Eq. (4) in

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log[q (K/s)]

a

1250 Symbols: Calculated Values from Area-Matching Lines: Actual Fictive Temperatures

Fictive Temperature of Initially Formed Glasses (K)

1200

6 5

1150

4 1100

3 2

1050

1 1000

0 -1

950

2

St. Dev. = 0.046 K

3 900 -3

-2

-1

0

1

2

3

Log[DSC Scan Rate (K/s)]

Fictive Temperature of Rejuvenated Glass (K)

b 1100 1050 1000 950 900 -3

-2

-1

0

1

2

3

Log[DSC Scan Rate (K/s)] Fig. 6. (a) Results of Tf1 calculations using the simulated DSC with initial cooling rates between 10−3 and 106 K/s and scan rates between 10−3 and 103 K/s. The derived values of Tf1 using area-matching (symbols) are in excellent agreement with the actual fictive temperatures from the relaxation model (lines). The missing points in the lower-right corner of the figure are due to computational limitations using the Mazurin–Startsev algorithm, which produced near-vanishing time steps in these particular cases. (b) The kinetic effect of cooling rate on the glass transition is evident in the value of Tf2, which varies linearly with the logarithm of DSC scan rate, as expected from the work of Moynihan et al. [18].

Ref. [25], the graphical method should be different from the one used in Fig. 2 which will be specified in the Appendix of this paper. In comparison with the previous approaches, the present generalized one addresses the following crucial aspects: First, the present approach incorporates both the Moynihan and the Yue routines. In addition, it uses any reference temperature Tf2 for determining the Tf1 of glasses with any thermal histories. Although in principle the Yue approach (i.e., Eq. (4) in Ref. [25]) can be used to determine the fictive temperature of the glasses with Tg b Tf b Teq, e.g., the graphical method has not been specified in Ref. [25]. The fictive temperature determination of partially aged HQ glass is shown in Fig. 5. Second, it is experimentally proved that the fictive temperature of a glass does not depend on the upscan rate in DSC. In addition, neither the first downscan rate nor the second upscan rate will influence the derived fictive temperature of the interested glass. Furthermore, the second upscan rate does not need to be the same as the first downscan rate. Third, using numerical simulations where the exact fictive temperature of the glass is known, we have shown that the calorimetric areamatching technique reproduces the known fictive temperature with

excellent accuracy, independent of the upscan rate used during measurement. Finally, we emphasize a fundamental difference between the often confused quantities of fictive temperature (Tf) and glass transition temperature (Tg). The latter is an inherently kinetic property, defined as the temperature at which the glass-forming system has a structural relaxation time equal to an external observation time scale [2,18] (typically taken as 100 s for molecular glasses, and 30–40 s for oxide glasses, or at a fixed shear viscosity of 10 12 Pa s). The fictive temperature, in contrast, is an inherently thermodynamic property. Our technique illustrated in Fig. 2 enables this thermodynamic order parameter to be determined independently of the kinetic effects during measurements. Such decoupling of thermodynamics and kinetics is necessary for building a complete and coherent understanding of the glassy state. Further discussion of calorimetric techniques for determination of Tg is given by Yue et al. [25]. However, the derived fictive temperatures by this approach are sensitive to the details of the experimentally measured heat capacity curves. Any property which may influence the enthalpy of the glassy state could also influence the derived fictive temperature. In the case of glass fibers, the following aspects should be considered: First, the cooling path is never an instantaneous quench from a fully equilibrated liquid. Thus, there is not just a single fictive temperature but rather a distribution of fictive temperatures. The final fictive temperature evaluated using our technique is an average of these different fictive temperatures. Second, there could be a small fictive temperature gradient in the cross-section of the glass. However, for fibers this gradient is so small that it can be neglected according to our previous finite element simulations [39] since the fibers studied here are rather thin (i.e., diameters range from 3 to 15 μm). Third, the fibers undergo tension in their axial direction during the fiber forming. This kind of tension can be frozen into the glass structure due to fast quenching rate. Consequently, additional excess energy besides the hyperquenching induced excess energy is trapped in the fibers. When the fibers are heated in DSC, the additional energy is released and contributes to the exotherm peak area [40]. However, the tension excited energy is significantly smaller than the hyperquenching excited energy. Thus, the area-matching results are influenced by tension only to a limited extent. Furthermore, in this work we studied discontinuous basaltic glass fibers, which are subject to significantly smaller tension (~1 MPa) compared to continuous fibers (up to 70 MPa depending on the fiber diameter) [41]. Therefore the influence of the tension on the Tf results is negligible. As discussed in this paper, the excess Cp can be regarded as a constant for most oxide glasses since the liquid Cp is nearly constant and the extrapolated glass Cp is close to a constant above the glass transition. However, organic glass formers like polymer systems could show an increase or a decrease of the liquid heat capacity with temperature, depending on the nature of the systems. Therefore, more attention should be paid to the calculation of excess heat capacity when considering different glass systems. Finally, it should be mentioned that here we have tested the general routine only under ambient pressure. It is expected that the method could be applicable to the glasses under other pressures. However, the technique is not applicable under varying pressures (i.e., non-isobaric conditions), where a fictive pressure may also be considered. The determination of Tf values of the glass as a function of the pressure has been discussed elsewhere [42–44]. 5. Conclusions We have proposed a generalized routine for determining the fictive temperature of a glass both graphically and numerically. The routine unifies graphical aspects of previous work by Moynihan et al. [18,19] and Yue et al. [25,26] and is applicable to glasses with any

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thermal history. Using numerical simulations where the exact fictive temperature of the glass in known, we have shown that the calorimetric area-matching technique reproduces the known fictive temperature with excellent accuracy, independent of the upscan rate used during measurement. Using our technique we accurately determined the fictive temperatures of glasses where the complicated endotherm and exotherm peaks shown in the Cp curves, for instance, (1) highly annealed glasses; (2) drawn glasses [45] and (3) annealed hyperquenched glasses. Acknowledgment We thank Adam J. Ellison of Corning Incorporated for his everlimitless supply of enlightening discussions. Appendix A As described in the Introduction and Discussion parts, the graphical technique presented for the Yue et al. approach [25] is only applicable for glasses where the fictive temperature is higher than Teq. In order to avoid incorrect application of Yue's approach, a universal graphical method which agrees strictly with Eq. (4) is preferred. First, we will prove that the graphical approach in Ref. [25] is not applicable for the partially annealed HQ basaltic glass fibers. The fictive temperature calculation of annealed basaltic glass fibers is given in Fig. A1(a). The heat capacity curves shown here are the same as the ones in Fig. 5. The value of Tf1 is found to be 969 K based on the graphical method in Ref. [25], which is about 11 K higher than the value from our generalized method. The discrepancy of the fictive temperature determined by these two graphical approaches is too large to be attributed to numerical error. In order to clarify this difference, we start from the foundation of Yue et al. approach, Eq. (4) in Ref. [25]. As stated above, Eq. (4) in Ref. [25] is a special case where Tf2 = Tg, thus, the deduction will be performed using Eq. (1) of this paper. The enthalpy trapped in the partially annealed fiber is about 2.2 J/g K more than that in the standard glass. Thus, the right side of Eq. (1) is

The glassy heat capacity, Cp glass(T = 952 K)= 1.036 J/g K and Cp glass (T = 969 K)= 1.038 J/g K. In order to simplify the problem, Cp glass is approximately taken as 1.037 J/g K. Thus,

    ∫ Cp liquid −Cp glass dT≈ Tf 1 −Tf 2 ð1:4−1:037Þ = 2:2

Tf 1 Tf 2

ðA2Þ

Tf 1 = Tf 2 + 6 = 958K; which gives the same value as the generalized approach as shown in Fig. 5. Besides the typical example shown in Fig. 5, we will further specify the application regime of the graphical approach in Ref. [25]. Taking the curves shown in Fig. A1 as an example, the equation describing the curves shown in Fig. A1 should be ∞     Cp2 −Cp glass dT = ∫ Cp2 −Cp1 dT:

Tf 1

∫ 0

ðA3Þ

0

If the graphical approach is applicable for glass having a certain fictive temperature, the left side of Eq. (A3) should be equal to the left side of Eq. (1),

∫

Tf 1     Cp2 −Cp glass dT = ∫ Cp liquid −Cp glass dT:

0

Tf 2

Tf 1

ðA4Þ

When the fictive temperature of the interested glass, Tf1, is lower than Tcross as shown in Fig. A2, Eq. (A4) can be rewritten as

Tf 2

∫



Tf 1  Tf 1     Cp2 −Cp glass dT + ∫ Cp2 −Cp glass dT = ∫ Cp liquid −Cp2 dT

0

  ∫ Cp2 −Cp1 dT = 2:2ðJ=g KÞ: ∞

3235

Tf 2

  + ∫ Cp2 −Cp glass dT

Tf 2

Tf 1

ðA1Þ

0

ðA5Þ

Tf 2

∫

Tf 2 

Tf 1    Cp2 −Cp glass dT = ∫ Cp liquid −Cp2 dT:

0

Tf 2

Since Tf2 is a constant, Eq. (A5) is correct only for one certain Tf1 value in the regime where the initial fictive temperature of the glass Tf1 is lower than Tcross.

Figure A1. Fictive temperature determination of annealed hyperquenched (HQ) basaltic glass fiber by area-matching method proposed by Yue et al. [25,26] The blue line represents the measured isobaric heat capacity (Cp) curves of the HQ basaltic glass fibers subjected to an annealing process at 773 K for 96 h, which was obtained using a DSC upscan rate of 20 K/min. The red line represents the standard Cp curves of the same sample subjected to a prior downscan at 20 K/min, which was obtained using a DSC upscan rate of 20 K/min.

Figure A2. Schematic representation of the characteristic temperatures shown in the DSC curve. The heat capacity shown here is the excess heat capacity, as discussed in the text.

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When the fictive temperature is located in Tcross ~ Teq, Eq. (A4) should be Tf 2

∫ 0



Tf 1  Tcross     Cp2 −Cp glass dT + ∫ Cp2 −Cp glass dT + ∫ Cp2 −Cp liquid dT Tf 1

+ ∫



Tf 2

Tcross Tf 1

+ ∫

Tcross

Tcross    Cp liquid −Cp glass dT = ∫ Cp liquid −Cp2 dT Tf 2



ðA6Þ

Tf 1    Cp2 −Cp glass dT + ∫ Cp liquid −Cp glass dT:

Tcross

Tcross

Cancelling the same terms showing in both sides of the equation, we obtain Tf 2

∫

Tf 1  Tcross      Cp2 −Cp glass dT + ∫ Cp2 −Cp liquid dT = ∫ Cp liquid −Cp2 dT:

0

Tcross

Tf 2

ðA7Þ The right side of Eq. (A7) is a constant. The area calculated in Eq. (A7) is used when we determine the fictive temperature of the rejuvenated glass as shown in Fig. 2(a), which can be described by Tf 2

∫

Teq  Tcross      Cp2 −Cp glass dT + ∫ Cp2 −Cp liquid dT = ∫ Cp liquid −Cp2 dT:

0

Tcross

Tf 2

ðA8Þ This means Eq. (A7) is only correct when Tf1 = Teq. When the fictive temperature of the initial glass is higher than Teq, Cp2 = Cp liquid, Eq. (A6) is always correct and rigorous. Therefore, the graphical method in Ref. [25] is valid only when Tf1 N Teq. For the glasses whose fictive temperature is lower than Teq, the generalized method is applicable and it is the exact graphical way of Eq. (1). References [1] C.A. Angell, J.M. Sare, E.J. Sare, J. Phys. Chem. 82 (1978) 2622. [2] J.C. Mauro, P.K. Gupta, R.J. Loucks, J. Chem. Phys. 126 (2007) 184511. [3] J.C. Mauro, R.J. Loucks, P.K. Gupta, J. Phys. Chem. A 111 (2007) 7957.

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