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Some aspects of the theory of thermally activated processes in dielectric and viscoelastic materials
Journal of Electrostatics, 3 (1977) 43--51
43
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
SOME ASPECTS OF THE T H E O R Y OF T H E R M A L L Y ACTIVATED PROCESSES IN DIELECTRIC AND VISCOELASTIC MATERIALS
BERNHARD
GROSS
Instituteof Physics and Chemistry of S~o Carlos, Universityof S~o Paulo, S~o Carlos, S.P. (Brazil)
Summary With reference to some recent criticism, it is shown in section 1 that the conventional expression for the thermally stimulated dielectric depolarization current is correct. The problem of a distribution of relaxation times versus a distribution of activation energies is discussed in section 2. It is shown that multiple TSC peaks are compatible with a single activation energy if the system is given by a ladder structure. In viscoelasticity, as in dielectric behavior, the energy loss per cycle is frequently found to be constant over a wide interval of frequencies. This behavior can be interpreted in terms of a complex structure o f loss free and lossy elements containing a single temperature-activated parameter as discussed in section 3.
1. Analysis of thermally stimulated depolarization currents Recently, the validity of the conventional expression for the thermally activated depolarization current in dielectrics [1] has been questioned [2,3]. In view of the wide use of this expression, a short discussion of the question seems to be justified (c.f. [4,5] ). The isothermal decay o f the time-dependent c o m p o n e n t of a polarization P in a shorted sample is described by
dP(t)/dt + P(t)/r = O,
(1)
where r is a temperature-dependent parameter, i.e. r = r(T). Equation (1) is a differential equation with a constant coefficient, the (isothermal) solution of which is
P( t ) = P ( O}exp (--t /r )
(2)
This result applies at any constant temperature. When the temperature during the measurement changes, however, r becomes a function which depends implicitly on time, i.e. r = r IT(t)] = r*(t). Then eqn. (1) is a differential equation with a time-dependent coefficient, whose solution is t
P(t) = P(0)exp [-- f ds/T*(s)]. 0
(3)
44 According to Scaife [2], the fundamental differential equation for isothermal conditions should be written ~P(t,T)/~ t + P(t,T)/~ = O,
(4)
which indicates that this equation applies to different, but constant, temperatures. While the introduction of a constant parameter, i.e. T, in the expression f o r P is not routinely used, eqn. (4) is formally correct. When, however, T becomes a function of time, Scaife does not generalize eqn. (4) by replacing aP/a t by d P / d t , but introduces the identity: ~P/~ t = d P / d t -- (~P/~ T ) d T / d t ,
(5)
and takes from the isothermal solution, eqn. (2): aP/~ T = --P(O )exp(--t/T )d(1/~)/dT.
(6)
Since eqn. (2) is only valid under isothermal conditions, this expression must, in the end, again led to an identity. Substitution of eqns. (5) and (6) into eqn. (4) gives d P / d t + P/~ + P t d ( 1 / T ) / d t = 0,
(7)
where the explicit dependence on d / d T has disappeared. Equation (7) can be written d P / d t + P d ( t / 7 ) / d t = O,
(S)
giving again the solution P = Poexp(--t/T),
(9)
where, as according to Scaife, one would have r = T [T(t)] = r*(t). That is, the non-isothermal solution is obtained from the isothermal solution by replacing the constant parameter r by the time-dependent parameter r *(t). If this procedure was correct, it would provide a new m e t h o d for solving differential equations with time-dependent coefficients: Take the solution of the equation with constant coefficients, replace the constant coefficients by the corresponding time-dependent coefficients, and y o u would have the solution of the general time-dependent equation. The theory of Bessel functions and the like would become unnecessary. In reality, eqn. (9) applies only under isothermal conditions: eqn. (5) is not a generalization, but an identity, and the procedure leading to expression (6), i.e. deriving a partial differential from the isothermal solution is not in line with established mathematical practice. We conclude that eqn. (3) remains valid. The isothermal discharge current is given by J = J(O)exp(--t/T),
(10)
45
which is the solution of the differential equation dJ/dt + J/r = 0.
(11)
One might ask w h y this expression is n o t taken as the starting point, instead of eqn. (1). Integration for the non-isothermal case gives t
J = J0 exp [-- f ds/r(s)].
(12)
0
But this cannot be the correct solution, because in the absence of any conduction current, the total current must be a total differential, i.e. J = A d D / d t , and obey a conservation law, A being the electrode area and D the dielectric displacement. The correct differential equation for the current follows from eqn. (1), with J = AdP/dt, giving dJ/dt + J/r + (J/r)dr/dt = 0,
(13)
or, after integration, t
J = (Qo/r)exp [-- f ds/r(s)].
(14)
0
This example illustrates the fact that the non-isothermal differential equation might contain terms which depend on time derivatives and thus do n o t show up in the isothermal equation. On these grounds, eqn. (11) is ruled out, b u t eqn. (1) is n o t necessarily validated. 2. Multiple activation energies and TSC spectra The investigation o f thermally activated currents has been preceded by the study of isothermal effects over a wide range of temperatures. Sixty years ago, it was already found that currents measured at different, b u t constant temperatures could be transformed into each other by an affine transformation of coordinates. This is the "law of corresponding states" established by Wagner [6]. It applies also to curves of loss factor and absorption capacitance. It can be shown that it is equivalent to the so-called time-temperature superposition principle [7]. An analogous situation exists in viscoelasticity. Viscoelasticity response functions (creep function, relaxation function, compliance, elastic modulus) can frequently be superposed by an affine transformation of coordinates and this has allowed the construction of "master" curves extending over an extremely wide range of temperatures [8]. Materials which obey such a law have been called "theologically simple" [9]. The dielectric and the viscoelastic relaxation effects have been interpreted for a long time in terms of a distribution of relaxation times. The validity of a "law of corresponding states" or time--temperature superposition principle (TTSP) imposes the condition that all relaxation times must depend on temperature in the same way. This is incompatible with a distribution of activation energies [10,11]. One would perhaps conclude, therefore, that dielectrics which exhibit multiple TSC peaks cannot obey the TTSP. Recurring
46
to viscoelastic theory, we shall try to show that this is n o t necessarily so and that multiple TSC peaks might be compatible with a single activation energy. A successful theory of viscoelastic behavior of cross-linked polymers based on Brownian motion of isolated flexible chains has been developed by Rouse [12]. We shall n o t enter into the details of the theory since it has been shown to be formally equivalent to a model in which p o l y m e r molecules are represented by springs moving in a viscous medium [13,14]. This equivalent model is shown in Fig. 1; it is a uniform ladder network. The fundamental importance of ladder networks for the representation and analysis of linear systems has been shown by Cauer [15], 70 years after Maxwell b u t 50 years ago from the present. They complete the representation by the Maxwell and Wagner--Voigt models in the formal treatment of electrical networks, dielectrics, and viscoelastic media. When the structure of the ladder network is given, known procedures allow the determination of the response functions [16].
E/dn
Fig. 1. Ladder structure representation of Rouse's theory.
An instantaneous deformation (elongation or shear) s0 of a viscoelastic material generates an instantaneous stress o which subsequently relaxes. If is the relaxation function, and permanent flow is excluded, the relation between stress and deformation can be written o = s0 [2E + ~ ( t ) ] .
(15)
For the uniform ladder network equivalent to Rouse's theory, one obtains [17] e = So
2 E + I~
1
(2~,/N)exp(--tSn)
+ (E/N)exp(--ts~
, n = 1,2,3..,
(16)
where N is the molecular weight and sn =
(1/r)4 sin 2 ( n T r / 2 ~ ) ,
(17)
which for n < < N gives s n = (1/r)n 2 rr2/N ~ ,
(18)
is the damping constant, E the elastic modulus, a n d f = l I E t h e spring constant. Within the present context, it is necessary to express the same situation in electrical terms using one of the electro-mechanical analogies [18]. We select the analogy in Table 1 which is structure-conservative (the electrical system a n d r = 77/E; ~
47
TABLE 1 Electro-mechanical analogy Electrical
Mechanical
current J voltage V capacitor C resistor R
stress a deformation dashpot 1/~ spring 1/E
has the same structure as the mechanical one and is n o t its dual form), b u t which is n o t energy-conservative (a lossless electrical parameter corresponds to a lossy mechanical one and a lossy electrical parameter corresponds to a lossless mechanical one. Then the short-circuit discharge current corresponding to eqn. (16) is J-- V ( Q N / r ) e x p ( - 4 U r ) + Y~ ( Q n / r ) e x p ( - ( t / r ) 4 sin2nlr/2_.&r) .
(19)
1
The relaxation parameter r is thermally activated and given by r = roexp(u/ kT) where u is an activation energy and k is Boltzmann's constant. The current is a total differential, as required from general principles. It is given in terms of a discrete spectrum of relaxation times. The temperature dependence of all relaxation times is the same; therefore the TTSP and the "law of corresponding states" apply, since only a single activation energy is involved. It thus follows that a single activation energy can give a spectrum of relaxation times. For small values of n, i.e. for the first terms of the spectrum, the preexponential factor ro is increased to N:r0. Since N might reach values of 10 4 while T0 is of the order of 1013 s, the apparent escape frequency is v = 1/N:r0 < 10 -21 H z This might explain the extremely small values sometimes found for the apparent pre-exponential factor. The thermally activated current is given by a series of expressions of the form of eqn. (14). Since successive relaxation times are rather widely spaced, each term can give a separate TSC peak when the heating rate is low. Thus, a single activation energy can give multiple peaks in the TSC spectrum. In conclusion, the example shows that it is possible to construct systems which have a single activation energy and simultaneously a complex TSC spectrum. In viscoelasticity, such systems do exist. Whether their electrical equivalent is also found in nature might still be open to conjecture. Nevertheless, there is sufficient evidence to indicate that care should be taken in the interpretation of TSC curves; a peak in the TSC spectrum might n o t necessarily correspond to a separate activation energy and the pre-exponential factor, as determined from the shape of the peak, is n o t necessarily identical with the true pre-exponential factor r0. Results of non-isothermal experiments should
48 be compared with those of isothermal measurements and their compatibility be checked. 3. The constant loss mechanism
It is observed that many high polymers have a dielectric loss angle that is constant over a considerable range of frequencies, or a mechanical loss angle that is frequency-independent down to extremely low frequencies [19]. This behavior is in agreement with the Curie--Von Schweidler relaxation function ( t ) = k t - " which has also been found v~id over several orders of magnitude. However, this relaxation function cannot be generally valid because its integral, i.e. the stored charge, diverges. Neither can the constancy of loss per cycle down to the lowest frequencies be explained in simple terms, since from purely mathematical principles of network theory it follows that the loss per cycle must be an odd function of frequency. Thus, if one requires the function to be a finite constant down to the lowest frequencies, it must exhibit a jump at zero frequency and cannot be very simple. In view of this situation, early investigators interpreted the Curie--Von Schweidler law in terms of a distribution of relaxation times that leads to a convergent expression for the charge. However, the association of such a distribution with a distribution of independent D e b y e mechanism with time constants extending over many orders of magnitude might also appear unsatisfactory. Similarly, replacement of the distribution of time constants b y a distribution of activation energies has lead to an uncomfortably broad range of activation energies. For this reason, Jonscher has recently developed a very interesting dielectric theory that accepts the constancy of the loss factor and considers it as the manifestation of a universal mechanism [20]. In viscoelasticity, Pelzer [21] has expressed a somewhat similar view, replacing the distribution of independent relaxation times and its equivalent system of independent Maxwell-Debye models by a ladder structure of the Cauer type. Such a procedure has been suggested by the success of the Rouse theory. However, to account for an approximately constant loss, the uniform ladder structure of Rouse's theory has to be generalized. Pelzer started from general network theory in order to determine what kinds of behavior approximating frequency-independent loss are compatible with fundamental principles and can be realized as physical structures. While Pelzer discusses in detail also the dielectric loss, we shall limit ourselves here to the problem as it appears in viscoelasticity. For a sinusoidally-varying stress, the deformation is given by = (G' -- iG")o,
(20)
where G is the complex elastic compliance. For the loss factor to be constant, G" must be constant over a sufficiently wide interval. A function which immediately suggests itself for this purpose is the complex logarithm, G = k In (1 + 1/zr),
(21)
49
where z is a complex frequency and 1/T a sufficiently large real constant. Development for z = io~ = o~ exp(i~/2) gives G = (k/2)ln(1 + 1/co:T 2) -- i k arccot ¢o~.
(22)
In view of the properties of the function arccot, this gives G" ~_ const, for cot < < 1. It is n o t e w o r t h y that this expression for G" is the same as in Gevers' theory [22]. Conventional methods [23] enable the determination of the distribution function of time constants belonging to the logarithmic function; as can be expected, it is essentially a box-shaped function. Here, however, we are interested in realizing the given function in the form of a ladder structure. This is possible by means of the well-known continued fraction expansion of the logarithm,
(1/4)In(l+4z) =z_~-+z_~(1/2) +z_~3-+ z[ (1/4) +z~5-+ z~(1/6)
(23)
where we have used the conventional notation z/(l+z) = ~ + z. Putting T = T i E a n d k = l I E one obtains
(24)
G = (l/4E)ln(1 + 4.EI,Tz) or
V = ( l / 4 E ) [4E ~[~+ 4E - -
+ 4El 3"z + 4E ](1/4) + 4E [ 5 . z + 4E ](1/6) + . . . ]
(25)
The terms of this expansion can be identified with the parameters of a ladder structure of springs and dashpots with progressively stronger spring and damping constants. This structure of unequal elements is shown in Fig. 2. A model of this kind would result if we imagine that the cross-linking of the polymer chains leads to connected groups that taper from a bulky central region and thin o u t at the outer region of each group where fewer chains are cross-connected parallel to one another. Each spindle-like group is immersed in the matrix o f molecules which are not cross-linked with it and act merely as a viscous fluid. The molecular chains possess entropy elasticity, and thus many chains connected in parallel (central region of the group) are equivalent to a stronger spring and their greater number, as compared with the outer region, causes a higher friction (larger damping disc). Thus, qualitatively, the behavior shown in Fig. 2 does n o t appear unlikely for high polymers. The expression (21) for the elastic compliance, as well as the equivalent ladder n e t w o r k representation, contain only a single temperature activated parameter, v/z. T and ,7. Therefore, the system can be expected to obey the time--temperature superposition principle. The steady-state response functions of electrical and mechanical systems containing a finite n u m b e r of parameters are given by rational functions. Since all rational functions can be expanded into finite continuous fractions, all such response functions can be represented b y finite ladder structures of the Cauer type. The infinite continuous fraction expansion o f the logarithm
50
2f
4f
6f
/ 1 1 /
-
OR, IN SIMPLE FORM
C
11
2
[3
Fig. 2. Ladder structure representation of Pelzer's constant loss theory (f = 1/E)o
provides an example for the representation of response functions of systems with continuously distributed parameters by means of infinite ladder structures of the Cauer type. Continuous fraction expansions exist for many transcendental functions [24]. Insofar as they contain only positive elements, they can all be "realized" by physical systems. Viscoelastic theory contains many other examples of this type [25]. Therefore, while all response functions can be interpreted in terms of distributions of time constants, in many cases the alternative representation by means of ladder structures appears attractive, bringing out features (like the validity of the TTSP) which might not be immediately apparent otherwise.
Acknowledgement The author gratefully acknowledges the concession of a research grant by F u n d a ~ o pelo Amparo da Ci~ncia do Estado de S~o Paulo.
References 1 C. Bucci and R. Fieschi, Phys. Rev. Lett., 12 (1964) 16. 2 B.K.P. Scaife, J. Phys. D: Appl. Phys., 7 (1974) L 171. 3 B.K.P. Scaife, J. Phys. D: Appl. Phys., 8 (1975) L 72. 4 J. van Turnhout, J. Phys. D: Appl. Phys., 8 (1975) L 69. 5 B. Gross, J. Phys. D: Appl. Phys., 8 (1975) L 127. 6 K.W. Wagner, Elektrotech. Z., 36 (1915) 135, 163. 7 B. Gross, J. Appl. Phys., 40 (1969) 3397. 8 J.D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 1961, p. 203. 9 F.R. Schwarzl and A.J. Staverman, J. Appl. Phys., 23 (1952) 838. 10 B. Gross, J. Electrochem. Soc., 115 (1968) 376. 11 J. van Turnhout, Thermally Stimulated Discharge of Polymer Eleetrets, Elsevier, Amsterdam, 1975, p. 41. 12 P.E. Rouse, J. Chem. Phys., 21 (1953) 1272. 13 B. Gross and R.M. Fuoss, J. Polym. Sci., 19 (1956) 39. 14 J.D. Ferry, ref. 8, p. 166. 15 W. Cauer, Arch. Elektrotech., 17 (1926) 355.
51
16 B. van der Pol and H. Bremmer, Operational Calculus, Cambridge Univ. Press, Cambridge, 1955. 17 B. Gross, J. Polym. Sci., 20 (1956) 123. 18 G. Kegel, Kolloid Z., 135 (1954) 125. 19 C.G. Garton, Trans. Faraday Soc., A 42 (1946) 56. 20 A.K. Jonscher, Nature, 253 (1975) 717. 21 H. Pelzer, J. Polym. Sci., 25 (1957) 51. 22 J. van Turnhout, ref. 11, p. 72. 23 B. Gross, Mathematical Structure of Theories of Viscoelasticity, Herrnann, Paris, 1953. J. Appl. Phys., 18 (1947) 212; ibid., 19 (1948) 257. 24 B. Gross and E.P. Braga, Singularities of Linear System Functions, Elsevier, Amsterdam, 1961. 25 R.S. Marvin and H. Oser, J. Res. Natl. Bur. Stand., 66B (4) (1962) 171.
43
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
SOME ASPECTS OF THE T H E O R Y OF T H E R M A L L Y ACTIVATED PROCESSES IN DIELECTRIC AND VISCOELASTIC MATERIALS
BERNHARD
GROSS
Instituteof Physics and Chemistry of S~o Carlos, Universityof S~o Paulo, S~o Carlos, S.P. (Brazil)
Summary With reference to some recent criticism, it is shown in section 1 that the conventional expression for the thermally stimulated dielectric depolarization current is correct. The problem of a distribution of relaxation times versus a distribution of activation energies is discussed in section 2. It is shown that multiple TSC peaks are compatible with a single activation energy if the system is given by a ladder structure. In viscoelasticity, as in dielectric behavior, the energy loss per cycle is frequently found to be constant over a wide interval of frequencies. This behavior can be interpreted in terms of a complex structure o f loss free and lossy elements containing a single temperature-activated parameter as discussed in section 3.
1. Analysis of thermally stimulated depolarization currents Recently, the validity of the conventional expression for the thermally activated depolarization current in dielectrics [1] has been questioned [2,3]. In view of the wide use of this expression, a short discussion of the question seems to be justified (c.f. [4,5] ). The isothermal decay o f the time-dependent c o m p o n e n t of a polarization P in a shorted sample is described by
dP(t)/dt + P(t)/r = O,
(1)
where r is a temperature-dependent parameter, i.e. r = r(T). Equation (1) is a differential equation with a constant coefficient, the (isothermal) solution of which is
P( t ) = P ( O}exp (--t /r )
(2)
This result applies at any constant temperature. When the temperature during the measurement changes, however, r becomes a function which depends implicitly on time, i.e. r = r IT(t)] = r*(t). Then eqn. (1) is a differential equation with a time-dependent coefficient, whose solution is t
P(t) = P(0)exp [-- f ds/T*(s)]. 0
(3)
44 According to Scaife [2], the fundamental differential equation for isothermal conditions should be written ~P(t,T)/~ t + P(t,T)/~ = O,
(4)
which indicates that this equation applies to different, but constant, temperatures. While the introduction of a constant parameter, i.e. T, in the expression f o r P is not routinely used, eqn. (4) is formally correct. When, however, T becomes a function of time, Scaife does not generalize eqn. (4) by replacing aP/a t by d P / d t , but introduces the identity: ~P/~ t = d P / d t -- (~P/~ T ) d T / d t ,
(5)
and takes from the isothermal solution, eqn. (2): aP/~ T = --P(O )exp(--t/T )d(1/~)/dT.
(6)
Since eqn. (2) is only valid under isothermal conditions, this expression must, in the end, again led to an identity. Substitution of eqns. (5) and (6) into eqn. (4) gives d P / d t + P/~ + P t d ( 1 / T ) / d t = 0,
(7)
where the explicit dependence on d / d T has disappeared. Equation (7) can be written d P / d t + P d ( t / 7 ) / d t = O,
(S)
giving again the solution P = Poexp(--t/T),
(9)
where, as according to Scaife, one would have r = T [T(t)] = r*(t). That is, the non-isothermal solution is obtained from the isothermal solution by replacing the constant parameter r by the time-dependent parameter r *(t). If this procedure was correct, it would provide a new m e t h o d for solving differential equations with time-dependent coefficients: Take the solution of the equation with constant coefficients, replace the constant coefficients by the corresponding time-dependent coefficients, and y o u would have the solution of the general time-dependent equation. The theory of Bessel functions and the like would become unnecessary. In reality, eqn. (9) applies only under isothermal conditions: eqn. (5) is not a generalization, but an identity, and the procedure leading to expression (6), i.e. deriving a partial differential from the isothermal solution is not in line with established mathematical practice. We conclude that eqn. (3) remains valid. The isothermal discharge current is given by J = J(O)exp(--t/T),
(10)
45
which is the solution of the differential equation dJ/dt + J/r = 0.
(11)
One might ask w h y this expression is n o t taken as the starting point, instead of eqn. (1). Integration for the non-isothermal case gives t
J = J0 exp [-- f ds/r(s)].
(12)
0
But this cannot be the correct solution, because in the absence of any conduction current, the total current must be a total differential, i.e. J = A d D / d t , and obey a conservation law, A being the electrode area and D the dielectric displacement. The correct differential equation for the current follows from eqn. (1), with J = AdP/dt, giving dJ/dt + J/r + (J/r)dr/dt = 0,
(13)
or, after integration, t
J = (Qo/r)exp [-- f ds/r(s)].
(14)
0
This example illustrates the fact that the non-isothermal differential equation might contain terms which depend on time derivatives and thus do n o t show up in the isothermal equation. On these grounds, eqn. (11) is ruled out, b u t eqn. (1) is n o t necessarily validated. 2. Multiple activation energies and TSC spectra The investigation o f thermally activated currents has been preceded by the study of isothermal effects over a wide range of temperatures. Sixty years ago, it was already found that currents measured at different, b u t constant temperatures could be transformed into each other by an affine transformation of coordinates. This is the "law of corresponding states" established by Wagner [6]. It applies also to curves of loss factor and absorption capacitance. It can be shown that it is equivalent to the so-called time-temperature superposition principle [7]. An analogous situation exists in viscoelasticity. Viscoelasticity response functions (creep function, relaxation function, compliance, elastic modulus) can frequently be superposed by an affine transformation of coordinates and this has allowed the construction of "master" curves extending over an extremely wide range of temperatures [8]. Materials which obey such a law have been called "theologically simple" [9]. The dielectric and the viscoelastic relaxation effects have been interpreted for a long time in terms of a distribution of relaxation times. The validity of a "law of corresponding states" or time--temperature superposition principle (TTSP) imposes the condition that all relaxation times must depend on temperature in the same way. This is incompatible with a distribution of activation energies [10,11]. One would perhaps conclude, therefore, that dielectrics which exhibit multiple TSC peaks cannot obey the TTSP. Recurring
46
to viscoelastic theory, we shall try to show that this is n o t necessarily so and that multiple TSC peaks might be compatible with a single activation energy. A successful theory of viscoelastic behavior of cross-linked polymers based on Brownian motion of isolated flexible chains has been developed by Rouse [12]. We shall n o t enter into the details of the theory since it has been shown to be formally equivalent to a model in which p o l y m e r molecules are represented by springs moving in a viscous medium [13,14]. This equivalent model is shown in Fig. 1; it is a uniform ladder network. The fundamental importance of ladder networks for the representation and analysis of linear systems has been shown by Cauer [15], 70 years after Maxwell b u t 50 years ago from the present. They complete the representation by the Maxwell and Wagner--Voigt models in the formal treatment of electrical networks, dielectrics, and viscoelastic media. When the structure of the ladder network is given, known procedures allow the determination of the response functions [16].
E/dn
Fig. 1. Ladder structure representation of Rouse's theory.
An instantaneous deformation (elongation or shear) s0 of a viscoelastic material generates an instantaneous stress o which subsequently relaxes. If is the relaxation function, and permanent flow is excluded, the relation between stress and deformation can be written o = s0 [2E + ~ ( t ) ] .
(15)
For the uniform ladder network equivalent to Rouse's theory, one obtains [17] e = So
2 E + I~
1
(2~,/N)exp(--tSn)
+ (E/N)exp(--ts~
, n = 1,2,3..,
(16)
where N is the molecular weight and sn =
(1/r)4 sin 2 ( n T r / 2 ~ ) ,
(17)
which for n < < N gives s n = (1/r)n 2 rr2/N ~ ,
(18)
is the damping constant, E the elastic modulus, a n d f = l I E t h e spring constant. Within the present context, it is necessary to express the same situation in electrical terms using one of the electro-mechanical analogies [18]. We select the analogy in Table 1 which is structure-conservative (the electrical system a n d r = 77/E; ~
47
TABLE 1 Electro-mechanical analogy Electrical
Mechanical
current J voltage V capacitor C resistor R
stress a deformation dashpot 1/~ spring 1/E
has the same structure as the mechanical one and is n o t its dual form), b u t which is n o t energy-conservative (a lossless electrical parameter corresponds to a lossy mechanical one and a lossy electrical parameter corresponds to a lossless mechanical one. Then the short-circuit discharge current corresponding to eqn. (16) is J-- V ( Q N / r ) e x p ( - 4 U r ) + Y~ ( Q n / r ) e x p ( - ( t / r ) 4 sin2nlr/2_.&r) .
(19)
1
The relaxation parameter r is thermally activated and given by r = roexp(u/ kT) where u is an activation energy and k is Boltzmann's constant. The current is a total differential, as required from general principles. It is given in terms of a discrete spectrum of relaxation times. The temperature dependence of all relaxation times is the same; therefore the TTSP and the "law of corresponding states" apply, since only a single activation energy is involved. It thus follows that a single activation energy can give a spectrum of relaxation times. For small values of n, i.e. for the first terms of the spectrum, the preexponential factor ro is increased to N:r0. Since N might reach values of 10 4 while T0 is of the order of 1013 s, the apparent escape frequency is v = 1/N:r0 < 10 -21 H z This might explain the extremely small values sometimes found for the apparent pre-exponential factor. The thermally activated current is given by a series of expressions of the form of eqn. (14). Since successive relaxation times are rather widely spaced, each term can give a separate TSC peak when the heating rate is low. Thus, a single activation energy can give multiple peaks in the TSC spectrum. In conclusion, the example shows that it is possible to construct systems which have a single activation energy and simultaneously a complex TSC spectrum. In viscoelasticity, such systems do exist. Whether their electrical equivalent is also found in nature might still be open to conjecture. Nevertheless, there is sufficient evidence to indicate that care should be taken in the interpretation of TSC curves; a peak in the TSC spectrum might n o t necessarily correspond to a separate activation energy and the pre-exponential factor, as determined from the shape of the peak, is n o t necessarily identical with the true pre-exponential factor r0. Results of non-isothermal experiments should
48 be compared with those of isothermal measurements and their compatibility be checked. 3. The constant loss mechanism
It is observed that many high polymers have a dielectric loss angle that is constant over a considerable range of frequencies, or a mechanical loss angle that is frequency-independent down to extremely low frequencies [19]. This behavior is in agreement with the Curie--Von Schweidler relaxation function ( t ) = k t - " which has also been found v~id over several orders of magnitude. However, this relaxation function cannot be generally valid because its integral, i.e. the stored charge, diverges. Neither can the constancy of loss per cycle down to the lowest frequencies be explained in simple terms, since from purely mathematical principles of network theory it follows that the loss per cycle must be an odd function of frequency. Thus, if one requires the function to be a finite constant down to the lowest frequencies, it must exhibit a jump at zero frequency and cannot be very simple. In view of this situation, early investigators interpreted the Curie--Von Schweidler law in terms of a distribution of relaxation times that leads to a convergent expression for the charge. However, the association of such a distribution with a distribution of independent D e b y e mechanism with time constants extending over many orders of magnitude might also appear unsatisfactory. Similarly, replacement of the distribution of time constants b y a distribution of activation energies has lead to an uncomfortably broad range of activation energies. For this reason, Jonscher has recently developed a very interesting dielectric theory that accepts the constancy of the loss factor and considers it as the manifestation of a universal mechanism [20]. In viscoelasticity, Pelzer [21] has expressed a somewhat similar view, replacing the distribution of independent relaxation times and its equivalent system of independent Maxwell-Debye models by a ladder structure of the Cauer type. Such a procedure has been suggested by the success of the Rouse theory. However, to account for an approximately constant loss, the uniform ladder structure of Rouse's theory has to be generalized. Pelzer started from general network theory in order to determine what kinds of behavior approximating frequency-independent loss are compatible with fundamental principles and can be realized as physical structures. While Pelzer discusses in detail also the dielectric loss, we shall limit ourselves here to the problem as it appears in viscoelasticity. For a sinusoidally-varying stress, the deformation is given by = (G' -- iG")o,
(20)
where G is the complex elastic compliance. For the loss factor to be constant, G" must be constant over a sufficiently wide interval. A function which immediately suggests itself for this purpose is the complex logarithm, G = k In (1 + 1/zr),
(21)
49
where z is a complex frequency and 1/T a sufficiently large real constant. Development for z = io~ = o~ exp(i~/2) gives G = (k/2)ln(1 + 1/co:T 2) -- i k arccot ¢o~.
(22)
In view of the properties of the function arccot, this gives G" ~_ const, for cot < < 1. It is n o t e w o r t h y that this expression for G" is the same as in Gevers' theory [22]. Conventional methods [23] enable the determination of the distribution function of time constants belonging to the logarithmic function; as can be expected, it is essentially a box-shaped function. Here, however, we are interested in realizing the given function in the form of a ladder structure. This is possible by means of the well-known continued fraction expansion of the logarithm,
(1/4)In(l+4z) =z_~-+z_~(1/2) +z_~3-+ z[ (1/4) +z~5-+ z~(1/6)
(23)
where we have used the conventional notation z/(l+z) = ~ + z. Putting T = T i E a n d k = l I E one obtains
(24)
G = (l/4E)ln(1 + 4.EI,Tz) or
V = ( l / 4 E ) [4E ~[~+ 4E - -
+ 4El 3"z + 4E ](1/4) + 4E [ 5 . z + 4E ](1/6) + . . . ]
(25)
The terms of this expansion can be identified with the parameters of a ladder structure of springs and dashpots with progressively stronger spring and damping constants. This structure of unequal elements is shown in Fig. 2. A model of this kind would result if we imagine that the cross-linking of the polymer chains leads to connected groups that taper from a bulky central region and thin o u t at the outer region of each group where fewer chains are cross-connected parallel to one another. Each spindle-like group is immersed in the matrix o f molecules which are not cross-linked with it and act merely as a viscous fluid. The molecular chains possess entropy elasticity, and thus many chains connected in parallel (central region of the group) are equivalent to a stronger spring and their greater number, as compared with the outer region, causes a higher friction (larger damping disc). Thus, qualitatively, the behavior shown in Fig. 2 does n o t appear unlikely for high polymers. The expression (21) for the elastic compliance, as well as the equivalent ladder n e t w o r k representation, contain only a single temperature activated parameter, v/z. T and ,7. Therefore, the system can be expected to obey the time--temperature superposition principle. The steady-state response functions of electrical and mechanical systems containing a finite n u m b e r of parameters are given by rational functions. Since all rational functions can be expanded into finite continuous fractions, all such response functions can be represented b y finite ladder structures of the Cauer type. The infinite continuous fraction expansion o f the logarithm
50
2f
4f
6f
/ 1 1 /
-
OR, IN SIMPLE FORM
C
11
2
[3
Fig. 2. Ladder structure representation of Pelzer's constant loss theory (f = 1/E)o
provides an example for the representation of response functions of systems with continuously distributed parameters by means of infinite ladder structures of the Cauer type. Continuous fraction expansions exist for many transcendental functions [24]. Insofar as they contain only positive elements, they can all be "realized" by physical systems. Viscoelastic theory contains many other examples of this type [25]. Therefore, while all response functions can be interpreted in terms of distributions of time constants, in many cases the alternative representation by means of ladder structures appears attractive, bringing out features (like the validity of the TTSP) which might not be immediately apparent otherwise.
Acknowledgement The author gratefully acknowledges the concession of a research grant by F u n d a ~ o pelo Amparo da Ci~ncia do Estado de S~o Paulo.
References 1 C. Bucci and R. Fieschi, Phys. Rev. Lett., 12 (1964) 16. 2 B.K.P. Scaife, J. Phys. D: Appl. Phys., 7 (1974) L 171. 3 B.K.P. Scaife, J. Phys. D: Appl. Phys., 8 (1975) L 72. 4 J. van Turnhout, J. Phys. D: Appl. Phys., 8 (1975) L 69. 5 B. Gross, J. Phys. D: Appl. Phys., 8 (1975) L 127. 6 K.W. Wagner, Elektrotech. Z., 36 (1915) 135, 163. 7 B. Gross, J. Appl. Phys., 40 (1969) 3397. 8 J.D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 1961, p. 203. 9 F.R. Schwarzl and A.J. Staverman, J. Appl. Phys., 23 (1952) 838. 10 B. Gross, J. Electrochem. Soc., 115 (1968) 376. 11 J. van Turnhout, Thermally Stimulated Discharge of Polymer Eleetrets, Elsevier, Amsterdam, 1975, p. 41. 12 P.E. Rouse, J. Chem. Phys., 21 (1953) 1272. 13 B. Gross and R.M. Fuoss, J. Polym. Sci., 19 (1956) 39. 14 J.D. Ferry, ref. 8, p. 166. 15 W. Cauer, Arch. Elektrotech., 17 (1926) 355.
51
16 B. van der Pol and H. Bremmer, Operational Calculus, Cambridge Univ. Press, Cambridge, 1955. 17 B. Gross, J. Polym. Sci., 20 (1956) 123. 18 G. Kegel, Kolloid Z., 135 (1954) 125. 19 C.G. Garton, Trans. Faraday Soc., A 42 (1946) 56. 20 A.K. Jonscher, Nature, 253 (1975) 717. 21 H. Pelzer, J. Polym. Sci., 25 (1957) 51. 22 J. van Turnhout, ref. 11, p. 72. 23 B. Gross, Mathematical Structure of Theories of Viscoelasticity, Herrnann, Paris, 1953. J. Appl. Phys., 18 (1947) 212; ibid., 19 (1948) 257. 24 B. Gross and E.P. Braga, Singularities of Linear System Functions, Elsevier, Amsterdam, 1961. 25 R.S. Marvin and H. Oser, J. Res. Natl. Bur. Stand., 66B (4) (1962) 171.