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Use of the temperature wave method for investigating thermal relaxational processes in polymers
Polymer Science U.S.S.R. Vol. 29, No. 2, pp. 476-.479, 1986 Printed in Poland
0032-3950/86 $10.00+ .00 • 1988 Pergamon Journals Ltd.
METHODS OF INVESTIGATION USE OF THE TEMPERATURE WAVE METHOD FOR INVESTIGATING THERMAL RELAXATIONAL PROCESSES IN POLYMERS* Y'u. 1. POLIKARPOV M. I. Kalinin Polytechnical Institute, Leningrad
(Received 11 April 1986) The method of temperature waves is used for the first time to determine the complex thermophysical characteristics of materials, and the advantages of the method are considered. In the case of flat temperature waves, the solution of the linear problems of heat conductivity, making due allowance for the finite character of the rate of heat dissipation, provides a means of deriving expressions for calculating the moduli and arguments of complex thermophysical characteristics. In the region of the glass transition temperature of polyvinyl acetate (PVA) its thermophysical characteristics are complex quantities depending on the frequency of the temperature vibrations, which reflects the relaxational character of the heat accumulation and transfer processes associated with segmental mobility.
IT IS known [1-5] that relaxational processes have a significant effect on the thermophysical properties of polymers, and it is most expedient to study the laws governing the distribution and absorption of heat associated with the dynamics of macromolecules, when the temperature changes are harmonic [5, 6]. Furthermore, the periodic heating calorimeters used for this purpose fail to provide for determining all the complex thermophysical charactoristics (CTPC), and have various disadvantages [5], which hinder measurements and decrease the accuracy in determining the complex heat t capacity. In this work the feasibility of applying the method of flat temperature waves to determining CTPC (complex temperature conductivity a* = ao exp jg,, thermal activity b* = be exp jg~, therma 1 conductivity A*=2o expjgx, and the thermal capacity of unit volume C* = Co expjgc), associated with segmental mobility in polymers is considered. Apart from the advantages ennumerated in previous publications [7-9], the given method provides a relatively simple means of obtaining CTPC relations as a function of the absolute temperature and of the frequency of the temperature vibrations of investigating the changes in these characteristics during annealing, vitrification (softening) or crystallization (melting), and of separating the effect of non-linear and inherent effects on CTPC, and of studying thermal relaxation processes close to the equilibrium state. An undoubted advantage of this method is the fact that it provides a means of using the linear theory of heat conductivity for determining CTPC in regions of phase and relaxational transitions, where these characteristics are markedly dependent on the temperature, since the amplitudes of the temperature vibrations can be assumed fairly small. Allowing for the fact that the rate of propagation of heat is linear [10, 11 ], and that the thermo-
* Vysokomol. soyed. A29: No. 2, 424-426, 1987. 476
Thermal relaxational processes in polymers
477
physical characteristics can be complex quantities, a theory of the method is developed for obtaining the theoretical relations for the three most preferred measuring cells represented in Fig. 1. From a solution of the unidimensional homogeneous linear problem of thermal conductivity for the propagation of flat temperature waves in an isotropic semi-infinite body (Fig. la) in a stable harmonic process, the following theoretical equations are obtained for the modulus ao and argument (phase) 6, of the complex temperature conductivity, i.e.
ao = coh 21 [ln2 ( gzl ~t ) + tp" ]
( 1)
6, = n/2 - 2 arctan [e/In (#z/81) ],
(2)
where 09 is the angular velocity of the temperature vibrations; 92 and al are the temperature vibration amplitudes at the input surface and at a depth h of the semi-infinite body; and ~ is the phase shift between those vibrations.
b
a
FIG. 1. Measuring cells:
c
/ - t e s t specimen, 2 - s t a n d a r d material.
A body can be considered practically semi-infinite in which the propagating flat wave is almost completely damped at a distance equal to its length. The distance Ls in which the amplitude of the temperature wave is decreased by a factor of N will be defined by the equation LN=
in N/[~/~o COS(zrl4
-
ODE)].
:5
1
,, zTo
s/o
s5o T~K
FIG. 2. Values of 1 -- ca, 2, 3 - Co, and 4, 5 - ao for PVA as a function of temperature. The frequencies of the temperature vibrations are 2, 5 - 0 . 0 2 and 3, 4 - 0 . 0 1 Hz; ,92 =0.3 K.
478
Y u . I. P O L I K A R P O V
In the case of complex temperature conductivity it is impossible to allow for the effect on it of heat exchange from the specimen side face, and the formulae given by Filippov [7] were used• The equation ao/cos 6, =coh2/[2~ ln(02/01)] was thus found, which is a criterion for the applicability of eqns. (I) and (2), from which the values of ao and 6a are calculated, and which shows that heat exchange can be ignored. The advantage of using the measuring cell under consideration is that it provides for simple analysis of the interconnection of the CTPC under consideration and the quantities undergoing measurement. To determine all the CTPC it is preferable to use the relative method of flat temperature waves, which has many advantages compared with the absolute method [8]. Figure lb shows a measuring cell consisting of the semi infinite body under investigation making thermal contact with a standard unrestricted plate, on the opposite side of which harmonic temperature vibrations of amplitude 0a are established. In this case the CTPC are calculated by means of eqn. (1) and (2), and also the equations b = b, [(Oa/Oz)expl~,- cosh (4)eola¢ d) ]/smh (x/Jco/a¢ d) 2" = b* ~/a-*
and
C* =
b*/4~,
(3) <4)
where j=,,/-----I, a~ and b, are the temperature conductivity and the thermal activity of the standard material; d is the thickness of the standard plate; ~u i~ the phase shift between the temperature vibrations of amplitude 01 and 02. For both cells under consideration a semi-infinite body is required (L ~ 30--50 ram), made from the test material. This disadvantage is removed on using the cell shown in Fig. Ic. In this cell the thin plate test specimen is placed between an endless semi-infinitebody and at the plate, on the opposite surface of which harmonic temperature vibrations of amplitude 0z are established, and makes thermal contact with them. In this case, the theoretical relations have the form (b*) 2= b~ {(~92figl)(exp/~) [(Oa/32)(expjc/) - cosh(~/jog/a, (0z/0t)2(expj2:p)- 1 exp(4~h) •
=
(02/0~)(expj~) {(b*/b,)
d)]/smh(4jog/ac d) }- 1
+ [(~3/02)(expj~u) - cosh(~/j~-/a~ d)]/sinh ( 4 J ~ ) } (b*/be) + 1
(5)
(6)
where 02 and 0a are the amplitudes of the temperature vibrations on the contact surface of the test plate of thickness h with the standard plate and the semi-infinite body respectively. Analysis of eqn. (3)--(6) shows that to attain the maximum accuracy in determining CTPC it is essential that b,~<[ b'l, and Itanh~/j~/aa*ht •] tanh~/f~-:acdlz 1. To measure the CTPC of the polymers an automated setup was constructed which provided a means of establishing harmonic temperature vibrations with non-linear distortion of less than 3 % and amptitude 0"05-5 K over the frequency range 10-3-1 Hz, and also of measuring, with errors less than 1 ~ , both the amptitudes of the temperature vibrations exceeding 0.02 K, and also the phase differences between these vibrations, on the surfaces of a flat source contacting the surface of the test specimen (Fig. la) or a standard plate (Fig. lb and lc). As a result, the accuracy in determining the CTPC moduli was 3-5 %, and in determining the arguments 0.02-radians. The material used for the control tests was PVA of MM 600,000, pzgs K= 1121 kg/m a and a glass transition temperature of 301 K. Ebonite was used as the standard, the temperature conductivity and heat capacity of which were determined by the temperature wave method, using the cell described in Fig. la, and differential scanning calorimetry. Figure 2 shows the temperature relations for the moduli of Co= Co/P29s K and ao of the complex heat capacity and temperature conductivity of PVA and also, for comparison, the dynamic specific heat capacity ca as a function of temperature, as determined at a heating rate of 16 dog/rain. It can be seen that in the glass transition region of PVA co and ao depend on the frequency of the temperature vibrations, which is determined by re-
Pavel Ignatevich Zubov
479
axational thermal processes. Accordingly, it is shown that the method under consideration provides a moans of determining the effect of segmental mobility on the CTPC of polymers. The authors wish to express their sincere thanks to Ye. V. Kuvshinskii for his constant attention and valuable advice during this work.
Translated by N. STANDEN REFERENCES 1. Yu. K. GODOVSKII, Teplofizicheskie metody issleodovaniya polimerov (Thermophysical Methods of Investigating Polymers), Moscow, 1976 2. L. N. NOVICHENOK and Z. P. SHULMAN, Teplofizicheskie svoistva polimerov (Thermophysical Properties of Polymers). Minsk, 1971 3. G. G. MAMEDALIEVA, D. Sh. ABDINOV and G. M. ALIEV, Dokl. Akad. Nauk SSSR 190: 1393, 1970 4. M. G. KULKARNI and R. A. MASHELKAR, Polymer 22: 867, 1981 5. I-L GOBRECHT, K. HAMANN, and G. WIIk,ERS, J. Phys. E. 4: 21, 1971 6. L HATTA and A. J. IKUSHIMA, Japan J. Appl. Phys. 20: 1995, 1981 7. L. P. FILIPPOV, Izmerenie teplovykh svoistv tverdykh i zhidkih metallov pri vysokikh temperaturakh (Measurement of the Thermal Properties of Solids and Molten Metals at High Temperatures). Moscow, 1967 8. L. P. FILIPPOV, Issledovanie teploprovodnosti zhidkostei (Investigation of the Thermal Conductivity of Liquids). Moscow, 1970 9. R. P. YURCHAK and A. A. MAGAKHED, Vestn. Moscow Oos. Univ. Fizika, astronomiya 20: 41, 1979 10. A. V. LYKOV, Teoriya teploprovodnosti (Theory of Thermal Conductivity). p. 599, Moscow, 1967 II. P. VERNOTTE, Compt. rend. Acad. sci. 246: 339, 1958
Polymer Science U.S.S.R. Vol. 29, No. 2, pp. 479-481, 1986
Printed in Poland
0032-3950/86 $10.00 +.00 0 1988 Pergamon Jouraala Ltd.
PAVEL IGNATEVICH ZUBOV* (1906-1980 On October 11 1986 at the age of 81, after a long serious illness, Professor Pavel Ignatevich Zubov, honoured scientist and technologist of the RSFSR, and Doctor of chemical science, terminated his considerable teaching endeavours in the field of the physical and colloid chemistry of polymers. P. 1. Zubov was born on January 29, 1906 in Kalache, Voronezh region into a peasant family. Professor Zubov received his general education in this town, and up to 1926 worked in various Soviet establishments, and held various elected posts in the Young Communist League: secretary of a rural cell, and then head of the propaganda of the district committee of the Al-Union Lenin Young Communists League.
0032-3950/86 $10.00+ .00 • 1988 Pergamon Journals Ltd.
METHODS OF INVESTIGATION USE OF THE TEMPERATURE WAVE METHOD FOR INVESTIGATING THERMAL RELAXATIONAL PROCESSES IN POLYMERS* Y'u. 1. POLIKARPOV M. I. Kalinin Polytechnical Institute, Leningrad
(Received 11 April 1986) The method of temperature waves is used for the first time to determine the complex thermophysical characteristics of materials, and the advantages of the method are considered. In the case of flat temperature waves, the solution of the linear problems of heat conductivity, making due allowance for the finite character of the rate of heat dissipation, provides a means of deriving expressions for calculating the moduli and arguments of complex thermophysical characteristics. In the region of the glass transition temperature of polyvinyl acetate (PVA) its thermophysical characteristics are complex quantities depending on the frequency of the temperature vibrations, which reflects the relaxational character of the heat accumulation and transfer processes associated with segmental mobility.
IT IS known [1-5] that relaxational processes have a significant effect on the thermophysical properties of polymers, and it is most expedient to study the laws governing the distribution and absorption of heat associated with the dynamics of macromolecules, when the temperature changes are harmonic [5, 6]. Furthermore, the periodic heating calorimeters used for this purpose fail to provide for determining all the complex thermophysical charactoristics (CTPC), and have various disadvantages [5], which hinder measurements and decrease the accuracy in determining the complex heat t capacity. In this work the feasibility of applying the method of flat temperature waves to determining CTPC (complex temperature conductivity a* = ao exp jg,, thermal activity b* = be exp jg~, therma 1 conductivity A*=2o expjgx, and the thermal capacity of unit volume C* = Co expjgc), associated with segmental mobility in polymers is considered. Apart from the advantages ennumerated in previous publications [7-9], the given method provides a relatively simple means of obtaining CTPC relations as a function of the absolute temperature and of the frequency of the temperature vibrations of investigating the changes in these characteristics during annealing, vitrification (softening) or crystallization (melting), and of separating the effect of non-linear and inherent effects on CTPC, and of studying thermal relaxation processes close to the equilibrium state. An undoubted advantage of this method is the fact that it provides a means of using the linear theory of heat conductivity for determining CTPC in regions of phase and relaxational transitions, where these characteristics are markedly dependent on the temperature, since the amplitudes of the temperature vibrations can be assumed fairly small. Allowing for the fact that the rate of propagation of heat is linear [10, 11 ], and that the thermo-
* Vysokomol. soyed. A29: No. 2, 424-426, 1987. 476
Thermal relaxational processes in polymers
477
physical characteristics can be complex quantities, a theory of the method is developed for obtaining the theoretical relations for the three most preferred measuring cells represented in Fig. 1. From a solution of the unidimensional homogeneous linear problem of thermal conductivity for the propagation of flat temperature waves in an isotropic semi-infinite body (Fig. la) in a stable harmonic process, the following theoretical equations are obtained for the modulus ao and argument (phase) 6, of the complex temperature conductivity, i.e.
ao = coh 21 [ln2 ( gzl ~t ) + tp" ]
( 1)
6, = n/2 - 2 arctan [e/In (#z/81) ],
(2)
where 09 is the angular velocity of the temperature vibrations; 92 and al are the temperature vibration amplitudes at the input surface and at a depth h of the semi-infinite body; and ~ is the phase shift between those vibrations.
b
a
FIG. 1. Measuring cells:
c
/ - t e s t specimen, 2 - s t a n d a r d material.
A body can be considered practically semi-infinite in which the propagating flat wave is almost completely damped at a distance equal to its length. The distance Ls in which the amplitude of the temperature wave is decreased by a factor of N will be defined by the equation LN=
in N/[~/~o COS(zrl4
-
ODE)].
:5
1
,, zTo
s/o
s5o T~K
FIG. 2. Values of 1 -- ca, 2, 3 - Co, and 4, 5 - ao for PVA as a function of temperature. The frequencies of the temperature vibrations are 2, 5 - 0 . 0 2 and 3, 4 - 0 . 0 1 Hz; ,92 =0.3 K.
478
Y u . I. P O L I K A R P O V
In the case of complex temperature conductivity it is impossible to allow for the effect on it of heat exchange from the specimen side face, and the formulae given by Filippov [7] were used• The equation ao/cos 6, =coh2/[2~ ln(02/01)] was thus found, which is a criterion for the applicability of eqns. (I) and (2), from which the values of ao and 6a are calculated, and which shows that heat exchange can be ignored. The advantage of using the measuring cell under consideration is that it provides for simple analysis of the interconnection of the CTPC under consideration and the quantities undergoing measurement. To determine all the CTPC it is preferable to use the relative method of flat temperature waves, which has many advantages compared with the absolute method [8]. Figure lb shows a measuring cell consisting of the semi infinite body under investigation making thermal contact with a standard unrestricted plate, on the opposite side of which harmonic temperature vibrations of amplitude 0a are established. In this case the CTPC are calculated by means of eqn. (1) and (2), and also the equations b = b, [(Oa/Oz)expl~,- cosh (4)eola¢ d) ]/smh (x/Jco/a¢ d) 2" = b* ~/a-*
and
C* =
b*/4~,
(3) <4)
where j=,,/-----I, a~ and b, are the temperature conductivity and the thermal activity of the standard material; d is the thickness of the standard plate; ~u i~ the phase shift between the temperature vibrations of amplitude 01 and 02. For both cells under consideration a semi-infinite body is required (L ~ 30--50 ram), made from the test material. This disadvantage is removed on using the cell shown in Fig. Ic. In this cell the thin plate test specimen is placed between an endless semi-infinitebody and at the plate, on the opposite surface of which harmonic temperature vibrations of amplitude 0z are established, and makes thermal contact with them. In this case, the theoretical relations have the form (b*) 2= b~ {(~92figl)(exp/~) [(Oa/32)(expjc/) - cosh(~/jog/a, (0z/0t)2(expj2:p)- 1 exp(4~h) •
=
(02/0~)(expj~) {(b*/b,)
d)]/smh(4jog/ac d) }- 1
+ [(~3/02)(expj~u) - cosh(~/j~-/a~ d)]/sinh ( 4 J ~ ) } (b*/be) + 1
(5)
(6)
where 02 and 0a are the amplitudes of the temperature vibrations on the contact surface of the test plate of thickness h with the standard plate and the semi-infinite body respectively. Analysis of eqn. (3)--(6) shows that to attain the maximum accuracy in determining CTPC it is essential that b,~<[ b'l, and Itanh~/j~/aa*ht •] tanh~/f~-:acdlz 1. To measure the CTPC of the polymers an automated setup was constructed which provided a means of establishing harmonic temperature vibrations with non-linear distortion of less than 3 % and amptitude 0"05-5 K over the frequency range 10-3-1 Hz, and also of measuring, with errors less than 1 ~ , both the amptitudes of the temperature vibrations exceeding 0.02 K, and also the phase differences between these vibrations, on the surfaces of a flat source contacting the surface of the test specimen (Fig. la) or a standard plate (Fig. lb and lc). As a result, the accuracy in determining the CTPC moduli was 3-5 %, and in determining the arguments 0.02-radians. The material used for the control tests was PVA of MM 600,000, pzgs K= 1121 kg/m a and a glass transition temperature of 301 K. Ebonite was used as the standard, the temperature conductivity and heat capacity of which were determined by the temperature wave method, using the cell described in Fig. la, and differential scanning calorimetry. Figure 2 shows the temperature relations for the moduli of Co= Co/P29s K and ao of the complex heat capacity and temperature conductivity of PVA and also, for comparison, the dynamic specific heat capacity ca as a function of temperature, as determined at a heating rate of 16 dog/rain. It can be seen that in the glass transition region of PVA co and ao depend on the frequency of the temperature vibrations, which is determined by re-
Pavel Ignatevich Zubov
479
axational thermal processes. Accordingly, it is shown that the method under consideration provides a moans of determining the effect of segmental mobility on the CTPC of polymers. The authors wish to express their sincere thanks to Ye. V. Kuvshinskii for his constant attention and valuable advice during this work.
Translated by N. STANDEN REFERENCES 1. Yu. K. GODOVSKII, Teplofizicheskie metody issleodovaniya polimerov (Thermophysical Methods of Investigating Polymers), Moscow, 1976 2. L. N. NOVICHENOK and Z. P. SHULMAN, Teplofizicheskie svoistva polimerov (Thermophysical Properties of Polymers). Minsk, 1971 3. G. G. MAMEDALIEVA, D. Sh. ABDINOV and G. M. ALIEV, Dokl. Akad. Nauk SSSR 190: 1393, 1970 4. M. G. KULKARNI and R. A. MASHELKAR, Polymer 22: 867, 1981 5. I-L GOBRECHT, K. HAMANN, and G. WIIk,ERS, J. Phys. E. 4: 21, 1971 6. L HATTA and A. J. IKUSHIMA, Japan J. Appl. Phys. 20: 1995, 1981 7. L. P. FILIPPOV, Izmerenie teplovykh svoistv tverdykh i zhidkih metallov pri vysokikh temperaturakh (Measurement of the Thermal Properties of Solids and Molten Metals at High Temperatures). Moscow, 1967 8. L. P. FILIPPOV, Issledovanie teploprovodnosti zhidkostei (Investigation of the Thermal Conductivity of Liquids). Moscow, 1970 9. R. P. YURCHAK and A. A. MAGAKHED, Vestn. Moscow Oos. Univ. Fizika, astronomiya 20: 41, 1979 10. A. V. LYKOV, Teoriya teploprovodnosti (Theory of Thermal Conductivity). p. 599, Moscow, 1967 II. P. VERNOTTE, Compt. rend. Acad. sci. 246: 339, 1958
Polymer Science U.S.S.R. Vol. 29, No. 2, pp. 479-481, 1986
Printed in Poland
0032-3950/86 $10.00 +.00 0 1988 Pergamon Jouraala Ltd.
PAVEL IGNATEVICH ZUBOV* (1906-1980 On October 11 1986 at the age of 81, after a long serious illness, Professor Pavel Ignatevich Zubov, honoured scientist and technologist of the RSFSR, and Doctor of chemical science, terminated his considerable teaching endeavours in the field of the physical and colloid chemistry of polymers. P. 1. Zubov was born on January 29, 1906 in Kalache, Voronezh region into a peasant family. Professor Zubov received his general education in this town, and up to 1926 worked in various Soviet establishments, and held various elected posts in the Young Communist League: secretary of a rural cell, and then head of the propaganda of the district committee of the Al-Union Lenin Young Communists League.