- Email: [email protected]
Structural relaxation of As2S3 glass by length dilatometry
Journal of Non-Crystalline Solids 235±237 (1998) 527±533
Structural relaxation of As2S3 glass by length dilatometry Jirõ M alek
1
Joint Laboratory of Solid State Chemistry, Academy of Sciences of the Czech Republic & University of Pardubice, Studentsk a 84, Pardubice 530 09, Czech Republic
Abstract Structural relaxation of As2 S3 glass was studied by dilatometry. The Tool±Narayanaswamy±Moynihan (TNM) model was successfully applied to the quantitative description of isothermal experiments performed within 40°C below Tg . The activation energy of the relaxation process is identical within the experimental error with that of viscous ¯ow, which was found to be 267 kJ/mol. The non-linearity and non-exponentiality parameters were evaluated as x 0.31 and b 0.82, respectively. The temperature dependence of the normalized volume relaxation rate agrees well with the prediction based on the TNM model. This approach can be used for comparison of structural relaxation kinetics in various non-crystalline materials. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction When a glass-forming liquid is cooled into the glass-transition region, its macroscopic properties (volume, enthalpy, etc.) become time dependent because the time required for molecular structure rearrangement is close to the experimental time scale. The gradual approach of these properties to their equilibrium values is often referred to as the structural relaxation process, which re¯ects the time required for the ``glassy'' structure to rearrange into its new equilibrium con®guration. There are several theoretical concepts which have been the basis for a number of phenomenological models of structural relaxation. Probably the most frequently used approach is the Tool± Narayanaswamy±Moynihan (TNM) model [1±3], which has been successfully used for the description of annealing eects and dierent thermal 1 Tel.: +420-40-603-6145; fax: +420-40-603-6011; e-mail: [email protected].
0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 6 0 9 - 7
histories on relaxation behavior in many noncrystalline materials [4]. Most tests of the TNM model have been done not too far from equilibrium. However, Scherer in his remarkable paper [5] has shown that the TNM phenomenology is quite good for the description of volume relaxation data of oxide glasses down to about 100°C below Tg . The aim of this paper is to study the structural relaxation of As2 S3 glass far from equilibrium using high precision length dilatometry. Dilatometric relaxation data are then quantitatively described by the TNM model. If a glass is equilibrated at temperature T0 (usually near Tg ) and then suddenly cooled to temperature T, the length will change as shown in Fig. 1. Isothermal relaxation response can be expressed as the relative departure of actual specimen length l from the equilibrium length l1 , i.e. d (l ) l1 )/ l1 . The time dependence of d can be expressed by means of the stretched exponential relaxation function [3,4] ÿ 1 d t d0 exp ÿ nb ;
528
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
Fig. 1. (a) Schematic illustration of the length changes of a stabilized glass subjected to a temperature jump from temperature T0 to T. The dotted lines show length and ®ctive temperature changes of a glass during isothermal annealing. (b) The relaxation curve corresponding to isothermal annealing at a temperature T. The broken line shows the in¯ectional tangent.
where d0 is the initial departure from equilibrium. The parameter b (0 < b 6 1) decreases with an increase in the width of the distribution of relaxation times. The reduced time n is de®ned as [2]
Zt n 0
dt0 : s T ; t 0
2
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
The relaxation time s is de®ned as a function of temperature T and structure, represented by the ®ctive temperature Tf (see Fig. 1) Dh Dh 1 ÿ x : 3 s A exp x RT RTf Eq. (3) introduces both a non-linearity parameter x and the eective activation energy Dh . The ®ctive temperature changes with time according to the following equation: d n DT ; 4 Tf t T0 ÿ 1 ÿ d0 where DT T0 ) T. Eqs. (1)±(4) are sucient to describe the relaxational response of a glass subjected to a temperature jump to below Tg or to more complex thermal histories. In practice, however, they must be solved numerically [6,7].
2. Experimental The As2 S3 glass was prepared by synthesis from pure elements (5 N purity) in an evacuated silica ampoule by melting and homogenization at 950°C for a period of 12 h. The amorphous nature and the composition of the prepared ingot were checked by X-ray diraction and energy dispersive microanalysis. Dilatometric experiments were performed using a TMA CX02 instrument (R.M.I., Czech Republic) equipped with a capacitance displacement sensor. The sensor is controlled through a unique electronic system which ensures linearity to better than 0.1% (full scale), high sensitivity (0.01 lm) and very good baseline ¯atness over broad temperature and time scales. A rectangular specimen of about 4 ´ 4 ´ 10 mm3 polished to optical quality was used for the dilatometric measurements. The specimen was equilibrated in the dilatometer at a temperature of 200°C (well above Tg ) and then rapidly cooled to a temperature T, where an isothermal relaxation took place. The length changes during this process were recorded as a function of time. The ®rst reliable data were obtained after a time tini 5 min. The thermal history of the sample after each exper-
529
iment was then erased by heating to 230°C, and the experiment was repeated at dierent temperatures. The viscosity of the undercooled liquid in the glass transition range (108 ±1012 Pa s) was measured by penetration viscometry using the same equipment as described above. The method is based on a penetration of a quartz hemisphere into a ¯at specimen of a glass [8]. Viscosities from 105 to 108 Pa s were successfully measured by the parallel plate technique [9]. The accuracy of these viscosity measurement techniques has been veri®ed using NBS 711 standard glass.
3. Results Typical experimental results of isothermal structural relaxation measurements are shown in Fig. 2 as points. Experimental errors are comparable with the size of data points. Full lines in Fig. 2 were calculated by a ``curve ®tting'' technique using Eqs. (1)±(4) for the following set of TNM parameters: Dh 269 4 kJ molÿ1 ; x 0:31 0:02; b 0:82 0:02; ln A min ÿ66:2 0:3: The activation energy in Eq. (3) was in fact not taken as an adjustable parameter and was obtained independently. It is assumed that near the equilibrium (i.e., d @ 0) Eq. (3) is practically Arrhenian, and therefore the activation energy can be determined from the slope of log(tm ) vs. 1/T plot, as shown in Fig. 3. The tm is time corresponding to extrapolation of an in¯ectional slope of the isothermal relaxation curve to d 0 (see Fig. 1). The value of activation energy determined in this way is practically identical (within the limits of experimental errors) to the activation energy for the temperature dependence of the equilibrium viscosity of the undercooled liquid near Tg , as shown in Fig. 4.
530
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
Fig. 2. Isothermal relaxation data of As2 S3 glass at the temperatures indicated following a temperature jump from equilibrium at 200°C (points). The solid lines were calculated using Eqs. (1)±(4). The initial time (tini 5 min) is the time estimated for thermal equilibration of the dilatometer following the temperature jump.
Fig. 3. Logarithm of extrapolated equilibration time tm as a function of reciprocal temperature for structural relaxation data of As2 S3 glass. Points correspond to experimental data, and the solid line is a linear regression ®t to experimental data.
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
531
Fig. 4. Temperature dependence of the viscosity data of As2 S3 undercooled liquid. The straight line represents the best linear ®t to experimental data.
4. Discussion The agreement between isothermal experimental data and TNM prediction is reasonable. Data from Fig. 2 are shown in Fig. 5 on a reduced time scale. The solid line was obtained using Eq. (1) along with the calculated TNM parameters. The agreement is quite good for nearly all experimental data up to DT 40°C. However, careful examination reveals that there is small but noticeable difference at longer reduced time, particularly for lowest temperatures. This discrepancy can be eliminated assuming that the parameter b is a weak function of temperature which would lead to the conclusion that the hypothesis of thermorheological simplicity is not valid for large temperature departures from Tg [5,10]. The in¯ectional slope of the d(log t) plots can be de®ned as dd 5 bl ÿ d log t i and for an isotropic material is equal to one-third of the volume relaxation rate de®ned by Kovacs
[11] (i.e., bl @ bV /3). It is convenient [12] to express a ``normalized'' volume relaxation rate de®ned as follows: RF bl =Da;
6
where Da is the dierence between the linear thermal expansion coecient of the equilibrium undercooled liquid and the glass. Recently it was found [12] that RF can be expressed as a function of the TNM parameters in the following form: ÿ1 1:18 1 ÿ xh ; 7 RF b DT 2:303 where h Dh =RTg2 . Eq. (7) predicts increasing normalized volume relaxation rate with the magnitude of the temperature jump DT. For dilatometric experiments DT corresponds to T0 ) T under the assumption that the temperature jump is instantaneous. In fact, it is rather dicult to change the temperature so quickly and there is always a ®nite initial time tini needed to reach thermal equilibrium of a real sample. If T0 is too high (T0 Tg ) then immediately after the temperature jump the
532
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
Fig. 5. Normalized relaxation function for As2 S3 glass. Solid line was calculated using Eq. (1) for b 0.82. The symbols represent data points from Fig. 2 (for the reasons of clarity only every third point is plotted).
dilatometric relaxation response will be very fast. Consequently the ®ctive temperature of the sample Tf may change considerably. For these reasons it
seems to be more correct to de®ne the magnitude of the temperature jump in the following way DT [Tf (tini ) ) T] @ Tg ) T.
Fig. 6. The normalized relaxation rate as a function of DT for As2 S3 glass. Points correspond to length dilatometric data and the solid line was calculated using Eq. (7).
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
Fig. 6 shows the RF (DT) dependence (full line) calculated using Eq. (7) for the TNM parameters obtained by the curve ®tting method. Experimental RF (DT) data obtained from our dilatometric isothermal experiments using Eqs. (5) and (6) for Tg 188°C, Da 0.74 ´ 10ÿ4 Kÿ1 are plotted as points. It is seen that all these experimental data agree within the limits of experimental errors with the theoretical prediction for the TNM model as expressed by Eq. (7). The concept of normalized volume relaxation rate can be used for comparison of structural relaxation kinetics in various noncrystalline materials [12].
533
[4]. The experimentally determined temperature dependence of the normalized volume relaxation rate RF agrees well (within the limit of experimental error) with the prediction based on the TNM model (Eq. (7)).
Acknowledgements This work was supported by the Grant Agency of the Czech Republic under grant no. 203/96/ 0184.
5. Conclusions
References
It was found that the TNM model gives a good description of isothermal dilatometric relaxation data of As2 S3 glass in the temperature range up to 40°C below Tg . This value of the activation energy from analysis of the relaxational data is virtually identical to the activation energy for temperature dependence of the equilibrium viscosity of the undercooled liquid near Tg . The non-linearity parameter x 0.31 is considerably lower than the value reported for enthalpy relaxation of As2 Se3 glass (0.49) [13]. On the other hand, the non-exponentiality parameter b 0.82 is relatively high, which corresponds to narrow distribution of relaxation times in comparison with that for most polymers and some inorganic glasses
[1] A.Q. Tool, J. Am. Ceram. Soc. 29 (1946) 240. [2] O.S. Narayanaswamy, J. Am. Ceram. Soc. 54 (1971) 491. [3] C.T. Moynihan, A.J. Easteal, M.A. DeBolt, J. Tucker, J. Am. Ceram. Soc. 59 (1976) 12. [4] I.M. Hodge, J. Non-Cryst. Solids 169 (1994) 211. [5] G.W. Scherer, J. Am. Ceram. Soc. 69 (1986) 374. [6] I.M. Hodge, A.R. Berens, Macromolecules 15 (1982) 762. [7] G.W. Scherer, Relaxation in Glass and Composites, Wiley/ Interscience, New York, 1986. [8] R. Brueckner, D. Demharter, Glastech. Ber. 48 (1975) 12. [9] J. Wang, Mater. Lett. 31 (1997) 99. [10] O.V. Mazurin, Y.K. Startsev, Sov. J. Glass. Phys. Chem. 7 (1981) 274. [11] A.J. Kovacs, J. Polym. Sci. 30 (1958) 131. [12] J. M alek, S. Montserrat, Thermochim. Acta 313 (1998) 191. [13] A.J. Easteal, J.A. Wilder, R.K. Mohr, C.T. Moynihan, J. Am. Ceram. Soc. 60 (1977) 134.
Structural relaxation of As2S3 glass by length dilatometry Jirõ M alek
1
Joint Laboratory of Solid State Chemistry, Academy of Sciences of the Czech Republic & University of Pardubice, Studentsk a 84, Pardubice 530 09, Czech Republic
Abstract Structural relaxation of As2 S3 glass was studied by dilatometry. The Tool±Narayanaswamy±Moynihan (TNM) model was successfully applied to the quantitative description of isothermal experiments performed within 40°C below Tg . The activation energy of the relaxation process is identical within the experimental error with that of viscous ¯ow, which was found to be 267 kJ/mol. The non-linearity and non-exponentiality parameters were evaluated as x 0.31 and b 0.82, respectively. The temperature dependence of the normalized volume relaxation rate agrees well with the prediction based on the TNM model. This approach can be used for comparison of structural relaxation kinetics in various non-crystalline materials. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction When a glass-forming liquid is cooled into the glass-transition region, its macroscopic properties (volume, enthalpy, etc.) become time dependent because the time required for molecular structure rearrangement is close to the experimental time scale. The gradual approach of these properties to their equilibrium values is often referred to as the structural relaxation process, which re¯ects the time required for the ``glassy'' structure to rearrange into its new equilibrium con®guration. There are several theoretical concepts which have been the basis for a number of phenomenological models of structural relaxation. Probably the most frequently used approach is the Tool± Narayanaswamy±Moynihan (TNM) model [1±3], which has been successfully used for the description of annealing eects and dierent thermal 1 Tel.: +420-40-603-6145; fax: +420-40-603-6011; e-mail: [email protected].
0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 6 0 9 - 7
histories on relaxation behavior in many noncrystalline materials [4]. Most tests of the TNM model have been done not too far from equilibrium. However, Scherer in his remarkable paper [5] has shown that the TNM phenomenology is quite good for the description of volume relaxation data of oxide glasses down to about 100°C below Tg . The aim of this paper is to study the structural relaxation of As2 S3 glass far from equilibrium using high precision length dilatometry. Dilatometric relaxation data are then quantitatively described by the TNM model. If a glass is equilibrated at temperature T0 (usually near Tg ) and then suddenly cooled to temperature T, the length will change as shown in Fig. 1. Isothermal relaxation response can be expressed as the relative departure of actual specimen length l from the equilibrium length l1 , i.e. d (l ) l1 )/ l1 . The time dependence of d can be expressed by means of the stretched exponential relaxation function [3,4] ÿ 1 d t d0 exp ÿ nb ;
528
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
Fig. 1. (a) Schematic illustration of the length changes of a stabilized glass subjected to a temperature jump from temperature T0 to T. The dotted lines show length and ®ctive temperature changes of a glass during isothermal annealing. (b) The relaxation curve corresponding to isothermal annealing at a temperature T. The broken line shows the in¯ectional tangent.
where d0 is the initial departure from equilibrium. The parameter b (0 < b 6 1) decreases with an increase in the width of the distribution of relaxation times. The reduced time n is de®ned as [2]
Zt n 0
dt0 : s T ; t 0
2
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
The relaxation time s is de®ned as a function of temperature T and structure, represented by the ®ctive temperature Tf (see Fig. 1) Dh Dh 1 ÿ x : 3 s A exp x RT RTf Eq. (3) introduces both a non-linearity parameter x and the eective activation energy Dh . The ®ctive temperature changes with time according to the following equation: d n DT ; 4 Tf t T0 ÿ 1 ÿ d0 where DT T0 ) T. Eqs. (1)±(4) are sucient to describe the relaxational response of a glass subjected to a temperature jump to below Tg or to more complex thermal histories. In practice, however, they must be solved numerically [6,7].
2. Experimental The As2 S3 glass was prepared by synthesis from pure elements (5 N purity) in an evacuated silica ampoule by melting and homogenization at 950°C for a period of 12 h. The amorphous nature and the composition of the prepared ingot were checked by X-ray diraction and energy dispersive microanalysis. Dilatometric experiments were performed using a TMA CX02 instrument (R.M.I., Czech Republic) equipped with a capacitance displacement sensor. The sensor is controlled through a unique electronic system which ensures linearity to better than 0.1% (full scale), high sensitivity (0.01 lm) and very good baseline ¯atness over broad temperature and time scales. A rectangular specimen of about 4 ´ 4 ´ 10 mm3 polished to optical quality was used for the dilatometric measurements. The specimen was equilibrated in the dilatometer at a temperature of 200°C (well above Tg ) and then rapidly cooled to a temperature T, where an isothermal relaxation took place. The length changes during this process were recorded as a function of time. The ®rst reliable data were obtained after a time tini 5 min. The thermal history of the sample after each exper-
529
iment was then erased by heating to 230°C, and the experiment was repeated at dierent temperatures. The viscosity of the undercooled liquid in the glass transition range (108 ±1012 Pa s) was measured by penetration viscometry using the same equipment as described above. The method is based on a penetration of a quartz hemisphere into a ¯at specimen of a glass [8]. Viscosities from 105 to 108 Pa s were successfully measured by the parallel plate technique [9]. The accuracy of these viscosity measurement techniques has been veri®ed using NBS 711 standard glass.
3. Results Typical experimental results of isothermal structural relaxation measurements are shown in Fig. 2 as points. Experimental errors are comparable with the size of data points. Full lines in Fig. 2 were calculated by a ``curve ®tting'' technique using Eqs. (1)±(4) for the following set of TNM parameters: Dh 269 4 kJ molÿ1 ; x 0:31 0:02; b 0:82 0:02; ln A min ÿ66:2 0:3: The activation energy in Eq. (3) was in fact not taken as an adjustable parameter and was obtained independently. It is assumed that near the equilibrium (i.e., d @ 0) Eq. (3) is practically Arrhenian, and therefore the activation energy can be determined from the slope of log(tm ) vs. 1/T plot, as shown in Fig. 3. The tm is time corresponding to extrapolation of an in¯ectional slope of the isothermal relaxation curve to d 0 (see Fig. 1). The value of activation energy determined in this way is practically identical (within the limits of experimental errors) to the activation energy for the temperature dependence of the equilibrium viscosity of the undercooled liquid near Tg , as shown in Fig. 4.
530
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
Fig. 2. Isothermal relaxation data of As2 S3 glass at the temperatures indicated following a temperature jump from equilibrium at 200°C (points). The solid lines were calculated using Eqs. (1)±(4). The initial time (tini 5 min) is the time estimated for thermal equilibration of the dilatometer following the temperature jump.
Fig. 3. Logarithm of extrapolated equilibration time tm as a function of reciprocal temperature for structural relaxation data of As2 S3 glass. Points correspond to experimental data, and the solid line is a linear regression ®t to experimental data.
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
531
Fig. 4. Temperature dependence of the viscosity data of As2 S3 undercooled liquid. The straight line represents the best linear ®t to experimental data.
4. Discussion The agreement between isothermal experimental data and TNM prediction is reasonable. Data from Fig. 2 are shown in Fig. 5 on a reduced time scale. The solid line was obtained using Eq. (1) along with the calculated TNM parameters. The agreement is quite good for nearly all experimental data up to DT 40°C. However, careful examination reveals that there is small but noticeable difference at longer reduced time, particularly for lowest temperatures. This discrepancy can be eliminated assuming that the parameter b is a weak function of temperature which would lead to the conclusion that the hypothesis of thermorheological simplicity is not valid for large temperature departures from Tg [5,10]. The in¯ectional slope of the d(log t) plots can be de®ned as dd 5 bl ÿ d log t i and for an isotropic material is equal to one-third of the volume relaxation rate de®ned by Kovacs
[11] (i.e., bl @ bV /3). It is convenient [12] to express a ``normalized'' volume relaxation rate de®ned as follows: RF bl =Da;
6
where Da is the dierence between the linear thermal expansion coecient of the equilibrium undercooled liquid and the glass. Recently it was found [12] that RF can be expressed as a function of the TNM parameters in the following form: ÿ1 1:18 1 ÿ xh ; 7 RF b DT 2:303 where h Dh =RTg2 . Eq. (7) predicts increasing normalized volume relaxation rate with the magnitude of the temperature jump DT. For dilatometric experiments DT corresponds to T0 ) T under the assumption that the temperature jump is instantaneous. In fact, it is rather dicult to change the temperature so quickly and there is always a ®nite initial time tini needed to reach thermal equilibrium of a real sample. If T0 is too high (T0 Tg ) then immediately after the temperature jump the
532
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
Fig. 5. Normalized relaxation function for As2 S3 glass. Solid line was calculated using Eq. (1) for b 0.82. The symbols represent data points from Fig. 2 (for the reasons of clarity only every third point is plotted).
dilatometric relaxation response will be very fast. Consequently the ®ctive temperature of the sample Tf may change considerably. For these reasons it
seems to be more correct to de®ne the magnitude of the temperature jump in the following way DT [Tf (tini ) ) T] @ Tg ) T.
Fig. 6. The normalized relaxation rate as a function of DT for As2 S3 glass. Points correspond to length dilatometric data and the solid line was calculated using Eq. (7).
J. M alek / Journal of Non-Crystalline Solids 235±237 (1998) 527±533
Fig. 6 shows the RF (DT) dependence (full line) calculated using Eq. (7) for the TNM parameters obtained by the curve ®tting method. Experimental RF (DT) data obtained from our dilatometric isothermal experiments using Eqs. (5) and (6) for Tg 188°C, Da 0.74 ´ 10ÿ4 Kÿ1 are plotted as points. It is seen that all these experimental data agree within the limits of experimental errors with the theoretical prediction for the TNM model as expressed by Eq. (7). The concept of normalized volume relaxation rate can be used for comparison of structural relaxation kinetics in various noncrystalline materials [12].
533
[4]. The experimentally determined temperature dependence of the normalized volume relaxation rate RF agrees well (within the limit of experimental error) with the prediction based on the TNM model (Eq. (7)).
Acknowledgements This work was supported by the Grant Agency of the Czech Republic under grant no. 203/96/ 0184.
5. Conclusions
References
It was found that the TNM model gives a good description of isothermal dilatometric relaxation data of As2 S3 glass in the temperature range up to 40°C below Tg . This value of the activation energy from analysis of the relaxational data is virtually identical to the activation energy for temperature dependence of the equilibrium viscosity of the undercooled liquid near Tg . The non-linearity parameter x 0.31 is considerably lower than the value reported for enthalpy relaxation of As2 Se3 glass (0.49) [13]. On the other hand, the non-exponentiality parameter b 0.82 is relatively high, which corresponds to narrow distribution of relaxation times in comparison with that for most polymers and some inorganic glasses
[1] A.Q. Tool, J. Am. Ceram. Soc. 29 (1946) 240. [2] O.S. Narayanaswamy, J. Am. Ceram. Soc. 54 (1971) 491. [3] C.T. Moynihan, A.J. Easteal, M.A. DeBolt, J. Tucker, J. Am. Ceram. Soc. 59 (1976) 12. [4] I.M. Hodge, J. Non-Cryst. Solids 169 (1994) 211. [5] G.W. Scherer, J. Am. Ceram. Soc. 69 (1986) 374. [6] I.M. Hodge, A.R. Berens, Macromolecules 15 (1982) 762. [7] G.W. Scherer, Relaxation in Glass and Composites, Wiley/ Interscience, New York, 1986. [8] R. Brueckner, D. Demharter, Glastech. Ber. 48 (1975) 12. [9] J. Wang, Mater. Lett. 31 (1997) 99. [10] O.V. Mazurin, Y.K. Startsev, Sov. J. Glass. Phys. Chem. 7 (1981) 274. [11] A.J. Kovacs, J. Polym. Sci. 30 (1958) 131. [12] J. M alek, S. Montserrat, Thermochim. Acta 313 (1998) 191. [13] A.J. Easteal, J.A. Wilder, R.K. Mohr, C.T. Moynihan, J. Am. Ceram. Soc. 60 (1977) 134.