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Thermal transport near glass transition in AgI-AgPO3 glasses
SOLID STATE ELSEYIER
Solid State Ionics 86-88
(1996)
463-466
Thermal transport near glass transition in AgI-AgPO,
IONICS
glasses
Y. Hiki”‘“, H. Takahashib, Y. KogureC “Tokyo Institute of Technology, Emeritus, 39-3-303 Motoyoyogi, Shihuya-ku, Tokyo 151, Japan hApplied Physics Group, Faculty of Engineering, Ibaraki University, Nakanarusawa, Hitachi 316, Japan ‘Teikyo University of Science and Technology, Yamanashi 409-01, Japan
Abstract The thermal conductivity, the thermal diffusivity and the heat capacity of AgI-AgPO, glasses were simultaneously determined during the annealing process of the materials below the glass transition temperature Tg by using an improved hot-wire method developed by the authors. The three thermal quantities showed relaxational behaviors during the annealing, and the relaxation times were determined from the annealing measurement. For example, in (AgI),,(AgPO l),,.,; TB = IOK’C, the relaxation times were around 1000-30 min at annealing temperatures of 50-98°C. The observed behavior was considered to be due to a structural relaxation in the glass. Keywords: Thermal transport; Glass transition temperature; Glasses
1. Introduction The present authors have developed an improved hot-wire method, called the multicurrent method, to measure the thermal conductivity K, the thermal diffusivity D, and the heat capacity C, of materials simultaneously and accurately [l]. The method was applied for studying the temperature dependences of these quantities below and above the glass transition temperature T, for AgI-AgPO, glasses [2]. The research is now further extended to study the time variation of the thermal properties below T, when the temperature is changed and kept constant. Through such an annealing experiment we intend to investigate relaxation phenomena in glasses. The relaxation of heat capacity C is directly related to the structural relaxation of the material. Now consider the thermal transport by phonons, and *Corresponding
0167-2738/96/$15.00
P/I
author.
Copyright 01996
SOl67-2738(96)00176-2
remember the gas kinetics relation for phonon gas, K = (l/3) Cvl, where C is the heat capacity per unit volume, v and 1 are the velocity and mean free path of phonons. Using this formula and the definition D = KIC, the relation Dd is obtained. We expect to obtain more information from the observed relaxations of C and D which are independent static and dynamic thermal properties.
2. Experimental The details of principle and practice of the hotwire method for measuring the thermal conductivity and the thermal diffusivity of glasses have been described elsewhere [l]. An outline of the method is as follows. In a fused-quartz cylindrical specimen cell, a thin Pt heating wire with two attached Pt potential leads is stretched along its axis. There is an open window on the upper wall of the cell, and a
Elsevier Science B.V. All rights reserved
Y. Hiki et al. I Solid State lonics 86-88(1996)463-466
464
melted specimen is poured into the cell through the window. A constant electric current is supplied to the heating wire, and the voltage between the potential leads is measured. The heating wire enclosed inside the specimen plays two roles: as an electric heating element and a resistance thermometer. When electric current I is suddenly applied to the wire, the temperature of the wire logarithmically increases with time I [3]. The change in the resistance R between the potential leads is R -R,,
= (aR,/47rL~)I’
ln[(ilDla’c)t]
= Al’ In t + BI’; A =
(ffR,,/‘hrL~),
B=
(cuR,/hL~)
ln(4Dla’c),
(1)
where R,, is the initial value of R, (Y is the temperature coefficient of resistivity of the wire, L is the distance between the potential leads, a is the radius of the wire, c =exp y= 1.781 where y is Euler’s constant, and K and D are the thermal conductivity and diffusivity of the specimen. In our (multicurrent I) method [l], multivalues of constant current I = 20, 40, 60, 80, 100 and 110 mA are used and the resistance R for each current value I is measured. The current is applied for a very short time, 30 ms, and the time interval between successive measurements with different current values is 10 s. As shown by Eq. (I ), (R-R,,) is proportional to I’. We obtained a linear R vs. I2 relation, and from the extrapolated value of R to zero current, the parameter R, is determined. Then the large current, 110 mA, is applied for a longer time, 2.5 s, and the time dependence of the resistance is observed. The R vs. t data are taken at an interval of 25 ms, and 101 data points are obtained for the run. From the observed linear (R-R,)) vs. In t relation, two constants A and B in Eq. (1) are determined. By using the values of R,, A and B, the thermal conductivity K and the thermal diffusivity D are obtained. The heat capacity per unit volume is also determined by the relation C = KID.
3. Results First, examples of raw data of the multicurrent hot-wire method to determine the thermal conduc-
tivity K and the thermal diffusivity D will be shown. In Fig. la, the R vs. I* data are plotted, to determine the R, values. A reasonable linear relation can be seen between the two quantities, and the extrapolation to obtain R, can accurately be made. The results of the experiment on the time changes of resistance are shown in Fig. 1b. An acceptable linear relationship holds between the two quantities, and we can use the data to determine reliable values of the parameters A and B. After a prolonged annealing at room temperature, the temperature of the specimen is increased to Ts and is kept constant. Then the hot-wire experiment starts to determine the three parameters R,,, A and B as functions of the annealing time t,. The raw data are somewhat scattered, and smoothing procedures are necessary. The smoothed parameter values are used to determine the experimental K, D, C vs. ta relations. Examples of the results of the annealing
1.2870
1.2865
o ta= 60 min
1.2860
1.2855
1.2650
0
5
10 1’ (W3 A’)
5OOr
,
(b) 400 -
T,
r
I
= 74.6 “c
I = 110 mA
Fig. I. (a) R vs. I’ data to determine R,, in Eq. (1) for (AgI),, ~(AgPO,),, i at various annealing times t, at an annealing temperature T,. (b) dV [x(R-R,,)] vs. In t data to determine A and B in Eq. (I ).
Y. Hiki et al. I Solid State lonics 86-88 3.32
I
,
3.26 - (‘)
,
,
T,=74.6
I
I
I
,
,
“C
2.0 1.9
0
120
240 t,
360
480
600
bin)
Fig. 2. Changes of the thermal conductivity K, thermal diffusivity D. and heat capacity C with annealing time temperature
T,, for
(AgI),, ,(AgPO,),,,.
The
td at
an annealing
solid curves
are
results of the parameter fits using Eq. (2).
exy-r:ment are shown in Fig. 2, where the measured thermal conductivity K, thermal diffusivity D, and derived heat capacity C are plotted against the annealing time t,. The annealing behaviors of the three quantities seem to obey relaxational processes with an initial rapid change followed by a gradual approach to an equilibrium. We here simply assume a single relaxation, and data fittings are carried out by using the formula
At:,) = a + b exp[-(t,
+ t,,)/r],
(2)
where f = K, D or C, the annealing times is t,, and a, t,, and r are fitting parameters. Here T is the relaxation time and t,) means the initiation time, representing the time when the relaxation process has been initiated before the hot-wire experiment was started. We do not start the experiment immediately after the specimen temperature is changed to T,. The fitted value of the relaxation time, which is the most important quantity, is shown to be insensitive to the t,, value. Thus we fix the value as t, = 40 min, which is typical in the present experiments, and determine h,
(1996) 463-466
465
other parameters. The solid curves in Fig. 2 represent the results of the parameter fits, and the fittings seem to be agreeable. In analyses of relaxation phenomena, the Kohlrausch-Williams-Watts (KWW) formulation is sometimes used to take into account the distribution of relaxation, where the time dependence of apparent relaxation adopted is of the form of exp[-(t/r)‘]; O< /?z 1. We also tried this formulation. In Fig. 2c, as examples, two fittings for /3 = 1.0 and 0.5 are shown, and we cannot differentiate which is the better fitting. Therefore, the single relaxation was presently used. Here we note the relaxation times r(C) and 7(D) which respectively represent independent static and dynamic thermal properties. It was found that the two values were always nearly equal. For example, for (AgI),,, (AgPO,),,, at T, = 74.6”C; the values are 7(C)= 320 min and 7(D)=367 min. Furthermore, the amounts of relaxation of the above two quantities were shown to be close to each other. For example, from Fig. 2b and c the following values are reduced as representative amounts: [{C(480 min) - C(0 min)}/ C(0 min>]= -0.09 and [{D(480 min)-D(0 min)}/ D(0 min)]=O.l 1.
4. Discussion The relaxation time r(C) is determined, and in Fig. 3 the logarithmic values are plotted against the inverse of the annealing temperature T, (in Kelvin) normalized by the glass transition temperature T,. The results are for (AgI),(AgPO,), ox glasses with x=0.3 (T,= 108°C) and x=0.5 (T,=83”C). The T, values are those determined by the usual DTA method for the same glass specimens. The data for the two kinds of glasses shown in Fig. 3 are on a common straight line, when the data at the lowest temperature is ignored. (The experiment cannot accurately be performed at this temperature, and the data might be reserved.) The above result suggests that the observed relaxation of thermal activation type is due to a structural change of the glass forming network AgPO, and is not sensitive to the AgI content. The activation energy calculated from the slope of the line is E= 145 kJ/mol. The relaxation time at T,= T, indicated by the arrow in the
Y. Hiki et al. I Solid State tonics 86-88
466
6.0 0
x=0.3
l
x=0.5 /
5.0
2 L
1
4.0
b a
0
/P
1
/
w 3.0
/+
!1
2.0
’
I
0.95 1.00
I
/
1
1.05
1.10
1.15
1.20
1 j
1.25
Tg/T, Fig. 3. Relaxation time 7 in logarithmic scale plotted against inverse of annealing temperature T, in Kelvin normalized by glass transition temperature Te for (AgI),(AgPO,), I glasses, x=0.3
and 0.5. figure is 380 s.Interpretation for these quantitative results remains a future problem to be considered. Some of the studies by other authors concerning the relaxation phenomena in glasses below T, will be cited here. Hanaya et al. [4] observed the enthalpy relaxation in AgI-AgPO, by an adiabatic calorimeter, and the values of relaxation time were 7= 60110 ks. Pascheto et al. [5] used their differential scanning calorimetry data, and obtained relaxation times in AgI-AgPO, after carrying an elaborated data analysis. The observed activation energy for the relaxation was around E =450 kJ/mol. These studies seem to be related to the structural relaxation char-
(1996) 463-466
acterized by the long relaxation time and the large activation energy. There were some mechanical relaxation (internal friction and modulus change) studies at low temperatures for glasses. Liu and Angel1 [6J observed a relaxation peak in AgI-AgPO, with relaxation time values 7 = l-lo-” s and an activation energy E = 15 kJ/mol. The E value was in agreement with the activation energy for Ag ion motion determined electrically. Carini et al. [7] observed an activation energy E = 30 kJ/mol in AgIAg,O-B,O, glasses. These studies are concerned with the relaxation due to ion migration, not to the structural relaxation. Finally, it is noted that, in the present experiment, the heat capacity C decreases and the thermal diffusivity D (ccphonon mean free path) increases with annealing time. The relaxation times and amounts of relaxation are nearly the same for both of the quantities. This means that the elements contributing to the extra heat capacity and the elements originating the phonon scattering could be the same.
References [II H. Takahashi, Y. Hiki and Y. Kogure, Rev. Sci. Instrum. 65
(1994) 2901. PI Y. Hiki, H. Takahashi and Y. Kogure,
Solid State Ionics 70/71 (1994) 362. 131H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Univ. Press, Oxford, 1959). [41 M. Hanaya, M. Nakayama and M. Oguni, J. Non-Cryst. Solids 172-174 (1994) 608. [51 W. Pascheto, M.G. Parthun, A. Hallbrucker and G.P. Johari, J. Non-Cryst. Solids 171 (1994) 182. 161C. Liu and C.A. Angell, J. Non-Cryst. Solids 83 (1986) 162. [71 G. Carini, M. Cutroni, M. Federico, G. Galli and G. Tripodo, Phys. Rev. B 30 (1984) 7219.
Solid State Ionics 86-88
(1996)
463-466
Thermal transport near glass transition in AgI-AgPO,
IONICS
glasses
Y. Hiki”‘“, H. Takahashib, Y. KogureC “Tokyo Institute of Technology, Emeritus, 39-3-303 Motoyoyogi, Shihuya-ku, Tokyo 151, Japan hApplied Physics Group, Faculty of Engineering, Ibaraki University, Nakanarusawa, Hitachi 316, Japan ‘Teikyo University of Science and Technology, Yamanashi 409-01, Japan
Abstract The thermal conductivity, the thermal diffusivity and the heat capacity of AgI-AgPO, glasses were simultaneously determined during the annealing process of the materials below the glass transition temperature Tg by using an improved hot-wire method developed by the authors. The three thermal quantities showed relaxational behaviors during the annealing, and the relaxation times were determined from the annealing measurement. For example, in (AgI),,(AgPO l),,.,; TB = IOK’C, the relaxation times were around 1000-30 min at annealing temperatures of 50-98°C. The observed behavior was considered to be due to a structural relaxation in the glass. Keywords: Thermal transport; Glass transition temperature; Glasses
1. Introduction The present authors have developed an improved hot-wire method, called the multicurrent method, to measure the thermal conductivity K, the thermal diffusivity D, and the heat capacity C, of materials simultaneously and accurately [l]. The method was applied for studying the temperature dependences of these quantities below and above the glass transition temperature T, for AgI-AgPO, glasses [2]. The research is now further extended to study the time variation of the thermal properties below T, when the temperature is changed and kept constant. Through such an annealing experiment we intend to investigate relaxation phenomena in glasses. The relaxation of heat capacity C is directly related to the structural relaxation of the material. Now consider the thermal transport by phonons, and *Corresponding
0167-2738/96/$15.00
P/I
author.
Copyright 01996
SOl67-2738(96)00176-2
remember the gas kinetics relation for phonon gas, K = (l/3) Cvl, where C is the heat capacity per unit volume, v and 1 are the velocity and mean free path of phonons. Using this formula and the definition D = KIC, the relation Dd is obtained. We expect to obtain more information from the observed relaxations of C and D which are independent static and dynamic thermal properties.
2. Experimental The details of principle and practice of the hotwire method for measuring the thermal conductivity and the thermal diffusivity of glasses have been described elsewhere [l]. An outline of the method is as follows. In a fused-quartz cylindrical specimen cell, a thin Pt heating wire with two attached Pt potential leads is stretched along its axis. There is an open window on the upper wall of the cell, and a
Elsevier Science B.V. All rights reserved
Y. Hiki et al. I Solid State lonics 86-88(1996)463-466
464
melted specimen is poured into the cell through the window. A constant electric current is supplied to the heating wire, and the voltage between the potential leads is measured. The heating wire enclosed inside the specimen plays two roles: as an electric heating element and a resistance thermometer. When electric current I is suddenly applied to the wire, the temperature of the wire logarithmically increases with time I [3]. The change in the resistance R between the potential leads is R -R,,
= (aR,/47rL~)I’
ln[(ilDla’c)t]
= Al’ In t + BI’; A =
(ffR,,/‘hrL~),
B=
(cuR,/hL~)
ln(4Dla’c),
(1)
where R,, is the initial value of R, (Y is the temperature coefficient of resistivity of the wire, L is the distance between the potential leads, a is the radius of the wire, c =exp y= 1.781 where y is Euler’s constant, and K and D are the thermal conductivity and diffusivity of the specimen. In our (multicurrent I) method [l], multivalues of constant current I = 20, 40, 60, 80, 100 and 110 mA are used and the resistance R for each current value I is measured. The current is applied for a very short time, 30 ms, and the time interval between successive measurements with different current values is 10 s. As shown by Eq. (I ), (R-R,,) is proportional to I’. We obtained a linear R vs. I2 relation, and from the extrapolated value of R to zero current, the parameter R, is determined. Then the large current, 110 mA, is applied for a longer time, 2.5 s, and the time dependence of the resistance is observed. The R vs. t data are taken at an interval of 25 ms, and 101 data points are obtained for the run. From the observed linear (R-R,)) vs. In t relation, two constants A and B in Eq. (1) are determined. By using the values of R,, A and B, the thermal conductivity K and the thermal diffusivity D are obtained. The heat capacity per unit volume is also determined by the relation C = KID.
3. Results First, examples of raw data of the multicurrent hot-wire method to determine the thermal conduc-
tivity K and the thermal diffusivity D will be shown. In Fig. la, the R vs. I* data are plotted, to determine the R, values. A reasonable linear relation can be seen between the two quantities, and the extrapolation to obtain R, can accurately be made. The results of the experiment on the time changes of resistance are shown in Fig. 1b. An acceptable linear relationship holds between the two quantities, and we can use the data to determine reliable values of the parameters A and B. After a prolonged annealing at room temperature, the temperature of the specimen is increased to Ts and is kept constant. Then the hot-wire experiment starts to determine the three parameters R,,, A and B as functions of the annealing time t,. The raw data are somewhat scattered, and smoothing procedures are necessary. The smoothed parameter values are used to determine the experimental K, D, C vs. ta relations. Examples of the results of the annealing
1.2870
1.2865
o ta= 60 min
1.2860
1.2855
1.2650
0
5
10 1’ (W3 A’)
5OOr
,
(b) 400 -
T,
r
I
= 74.6 “c
I = 110 mA
Fig. I. (a) R vs. I’ data to determine R,, in Eq. (1) for (AgI),, ~(AgPO,),, i at various annealing times t, at an annealing temperature T,. (b) dV [x(R-R,,)] vs. In t data to determine A and B in Eq. (I ).
Y. Hiki et al. I Solid State lonics 86-88 3.32
I
,
3.26 - (‘)
,
,
T,=74.6
I
I
I
,
,
“C
2.0 1.9
0
120
240 t,
360
480
600
bin)
Fig. 2. Changes of the thermal conductivity K, thermal diffusivity D. and heat capacity C with annealing time temperature
T,, for
(AgI),, ,(AgPO,),,,.
The
td at
an annealing
solid curves
are
results of the parameter fits using Eq. (2).
exy-r:ment are shown in Fig. 2, where the measured thermal conductivity K, thermal diffusivity D, and derived heat capacity C are plotted against the annealing time t,. The annealing behaviors of the three quantities seem to obey relaxational processes with an initial rapid change followed by a gradual approach to an equilibrium. We here simply assume a single relaxation, and data fittings are carried out by using the formula
At:,) = a + b exp[-(t,
+ t,,)/r],
(2)
where f = K, D or C, the annealing times is t,, and a, t,, and r are fitting parameters. Here T is the relaxation time and t,) means the initiation time, representing the time when the relaxation process has been initiated before the hot-wire experiment was started. We do not start the experiment immediately after the specimen temperature is changed to T,. The fitted value of the relaxation time, which is the most important quantity, is shown to be insensitive to the t,, value. Thus we fix the value as t, = 40 min, which is typical in the present experiments, and determine h,
(1996) 463-466
465
other parameters. The solid curves in Fig. 2 represent the results of the parameter fits, and the fittings seem to be agreeable. In analyses of relaxation phenomena, the Kohlrausch-Williams-Watts (KWW) formulation is sometimes used to take into account the distribution of relaxation, where the time dependence of apparent relaxation adopted is of the form of exp[-(t/r)‘]; O< /?z 1. We also tried this formulation. In Fig. 2c, as examples, two fittings for /3 = 1.0 and 0.5 are shown, and we cannot differentiate which is the better fitting. Therefore, the single relaxation was presently used. Here we note the relaxation times r(C) and 7(D) which respectively represent independent static and dynamic thermal properties. It was found that the two values were always nearly equal. For example, for (AgI),,, (AgPO,),,, at T, = 74.6”C; the values are 7(C)= 320 min and 7(D)=367 min. Furthermore, the amounts of relaxation of the above two quantities were shown to be close to each other. For example, from Fig. 2b and c the following values are reduced as representative amounts: [{C(480 min) - C(0 min)}/ C(0 min>]= -0.09 and [{D(480 min)-D(0 min)}/ D(0 min)]=O.l 1.
4. Discussion The relaxation time r(C) is determined, and in Fig. 3 the logarithmic values are plotted against the inverse of the annealing temperature T, (in Kelvin) normalized by the glass transition temperature T,. The results are for (AgI),(AgPO,), ox glasses with x=0.3 (T,= 108°C) and x=0.5 (T,=83”C). The T, values are those determined by the usual DTA method for the same glass specimens. The data for the two kinds of glasses shown in Fig. 3 are on a common straight line, when the data at the lowest temperature is ignored. (The experiment cannot accurately be performed at this temperature, and the data might be reserved.) The above result suggests that the observed relaxation of thermal activation type is due to a structural change of the glass forming network AgPO, and is not sensitive to the AgI content. The activation energy calculated from the slope of the line is E= 145 kJ/mol. The relaxation time at T,= T, indicated by the arrow in the
Y. Hiki et al. I Solid State tonics 86-88
466
6.0 0
x=0.3
l
x=0.5 /
5.0
2 L
1
4.0
b a
0
/P
1
/
w 3.0
/+
!1
2.0
’
I
0.95 1.00
I
/
1
1.05
1.10
1.15
1.20
1 j
1.25
Tg/T, Fig. 3. Relaxation time 7 in logarithmic scale plotted against inverse of annealing temperature T, in Kelvin normalized by glass transition temperature Te for (AgI),(AgPO,), I glasses, x=0.3
and 0.5. figure is 380 s.Interpretation for these quantitative results remains a future problem to be considered. Some of the studies by other authors concerning the relaxation phenomena in glasses below T, will be cited here. Hanaya et al. [4] observed the enthalpy relaxation in AgI-AgPO, by an adiabatic calorimeter, and the values of relaxation time were 7= 60110 ks. Pascheto et al. [5] used their differential scanning calorimetry data, and obtained relaxation times in AgI-AgPO, after carrying an elaborated data analysis. The observed activation energy for the relaxation was around E =450 kJ/mol. These studies seem to be related to the structural relaxation char-
(1996) 463-466
acterized by the long relaxation time and the large activation energy. There were some mechanical relaxation (internal friction and modulus change) studies at low temperatures for glasses. Liu and Angel1 [6J observed a relaxation peak in AgI-AgPO, with relaxation time values 7 = l-lo-” s and an activation energy E = 15 kJ/mol. The E value was in agreement with the activation energy for Ag ion motion determined electrically. Carini et al. [7] observed an activation energy E = 30 kJ/mol in AgIAg,O-B,O, glasses. These studies are concerned with the relaxation due to ion migration, not to the structural relaxation. Finally, it is noted that, in the present experiment, the heat capacity C decreases and the thermal diffusivity D (ccphonon mean free path) increases with annealing time. The relaxation times and amounts of relaxation are nearly the same for both of the quantities. This means that the elements contributing to the extra heat capacity and the elements originating the phonon scattering could be the same.
References [II H. Takahashi, Y. Hiki and Y. Kogure, Rev. Sci. Instrum. 65
(1994) 2901. PI Y. Hiki, H. Takahashi and Y. Kogure,
Solid State Ionics 70/71 (1994) 362. 131H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Univ. Press, Oxford, 1959). [41 M. Hanaya, M. Nakayama and M. Oguni, J. Non-Cryst. Solids 172-174 (1994) 608. [51 W. Pascheto, M.G. Parthun, A. Hallbrucker and G.P. Johari, J. Non-Cryst. Solids 171 (1994) 182. 161C. Liu and C.A. Angell, J. Non-Cryst. Solids 83 (1986) 162. [71 G. Carini, M. Cutroni, M. Federico, G. Galli and G. Tripodo, Phys. Rev. B 30 (1984) 7219.