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On the heat dissipation function for dielectric relaxation phenomena in anisotropic media
$5.00 + 0.00 0020-7225/E 1992 Pergamon Press plc
Int. J. Engng Sci. Vol. 30, No. 3, pp. 305-315, 1992 Printed in Great Britain.
ON THE HEAT DISSIPATION FUNCTION FOR DIELECTRIC RELAXATION PHENOMENA IN ANISOTROPIC MEDIA-f LILIANA Department
GERRIT Department
RESTUCCIA
of Mathematics, University of Messina, Messina, Italy
ALFRED
KLUITENBERG
of Mathematics and Computing Science, Eindhoven University of Technology, Eindhoven, The Netherlands (Communicated
by G. A. MAUGIN)
Abstract-The heat dissipation function for polarizable anisotropic media in which phenomena of dielectric relaxation occcur is derived as a generalization of the heat dissipation function studied in the case that the media are isotropic. The methods of non-equilibrium thermodynamics are used. It is seen that the linearization of the theory leads to a dielectric relaxation equation for anisotropic polarizable media which has the form of a linear relation among the temperature, the components of the electric field vector and of the total polarization vector, the first derivatives with respect to time of the components of these vectors and of the temperature and the components of the second derivative with respect to time of the total polarization vector. It is shown that the heat dissipation function is due to irreversible phenomena of viscous flow, electric conduction and dielectric relaxation. Finally, the obtained results are applied to the particular cases of Debye media and De Groot-Mazur media.
1.
INTRODUCTION
In some previous papers [l-9] a theory for dielectric relaxation phenomena in polarizable continuous media was developed, which is based on the thermodynamics of irreversible processes [ 10-161. In the Refs [4-81 it was shown that, if several microscopic phenomena give rise to dielectric relaxation, it is possible to describe these microscopic phenomena with the aid of n macroscopic vectorial internal variables which are introduced as internal degrees of freedom in the entropy. It was demonstrated that these n vector fields can be considered as n contributions to the total specific polarization vector p so that this vector can be split in n + 1 parts, i.e. p = p(O)+
2
p(k),
(1.1)
k=l
where p(O) and pck) (k = 1, 2, . . . , n) are called partial specific polarization vectors. In the present paper we consider anisotropic polarizable media in which the polarization vector p is additively composed of two parts p(O) and p(l) p = p(O)+ p(l),
(l-2)
where we suppose that changes in p(O) and in p(l) are both irreversible processes [3,17]. We derive the explicit expression for the heat dissipation function due to the irreversible processes of viscous flow, electric conduction and dielectric relaxation. This expression is a generalization of the heat dissipation function derived in Ref. [9] for isotropic polarizable media. The methods of non-equilibrium thermodynamics are used. In Section 2 the entropy balance equation and the phenomenological equations are discussed. In Section 3 for the case that the equations of state may be linearized the dielectric relaxation equation for anisotropic polarizable media is derived. In Section 4 the heat dissipation function is obtained and in Sections 5 and 6 the particular cases of Debye media and De Groot-Mazur media are discussed. tThis work is supported by the Gruppo Nazionale per la Fisica Matematica of the Consiglio Nazionale delle Ricerche (C.N.R.), by the Nederlandse Organisatie Voor Wetenschappelijk Onderzoek (N.W.O.) and by the Department of Mathematics and Computing Science of the Eindhoven University of Technology. 305
L. RESTUCCIA
306
2. THE
and G. A. KLUITENBERG
GENERAL
FORMALISM
We consider anisotropic polarizable media in which the contributions of microscopic phenomena to the macroscopic polarization vector can be described by introducing one vectorial internal degree of freedom in the entropy. Thus, if u is the specific internal energy, E,~ the strain tensor and s the specific entropy we assume that s = s(u, Q3, p, I+“). (2.1) By virtue of (2.1) we obtain for the differential do of s the following expression Tds=du-v
i
r$‘de,B
_ E@q). dp + EC’). dp”‘,
Ly,fi=l
where Y = l/p is the specific volume. The temperature Eceq)and EC’)are given by
T-’ = $4~ E,~, @‘=-pT
and
- a k/3
(2.2)
T, the tensor r$J’ and the vector fields
p, p(l)),
(2.3)
(2.4)
s(u, QJ, p, p’%
Eceq)= - Td s(u, E,~, p, p’1’) dP
(2.5)
EC’)= T
(2.6)
Q3, p, p(I)).
In (2.3)-(2.6) t$’ is the equilibrium stress tensor, Eceq)is the equilibrium electric field and E(l) is the vectorial thermodynamic affinity conjugate to the internal variable p(l). We assume that r$J)=O
if
E,~=O,
p = p(l) = 0
and
T = To,
(2.7)
Eceq)= 0
if
.saS= 0,
p=p”‘=O
and
T= I&,,
(2.8)
EC’)= 0
if
saS = 0 ?
p=p”‘=O
and
T=T,,
(2.9)
where T,, is the temperature in the reference state. The strain tensor is measured with respect to this reference state (see Refs [4, 5, 71). The first law of thermodynamics may be written in the form [lo, 181 p$=
-div J(q)
E+pE$,
(2.10)
where Jcq) is the vector of the density of the heat flow, j@')is the vector of the density of the electric current and de,/dt is given by d$;($+z),
(a,p=1,2,3)
(2.11)
where we have supposed that the deformations and rotations of the medium are small from a kinematical point of view. Moreover, in (2.10) dldt is the material derivative with respect to time and u, is the cu-component of the velocity v of the medium. Rectangular axis-frame is used which is fixed in the space. It should be noted that (2.10) does not satisfy the requirements of the theory of relativity. However, we shall assume that the velocity of the medium with respect to the observer is small compared with the velocity of the light and in this case (2.10) is a good approximation for the first law. For the exact relativistic formulation of the first law for simple fluids and viscous polarizable and magnetizable fluid mixtures influenced by an electromagnetic field one may consult Kluitenberg [19] and Kluitenberg and De Groot [20,21].
Dielectric
relaxation
phenomena
307
Thermodynamic considerations of dielectric and magnetic phenomena are given in numerous papers and books (see for instance [22-291). From the first law (2.10) and the Gibbs relation (2.2) we obtain for the entropy balance p $ = -div( TIJ’*)) where a@) is the entropy production -I’-‘J(*) In this equation @
+ &I,
(2.12)
per unit volume and per unit time, given by
(Vi)dEap + pE(“) dp . grad T + 5 zap dt . dt + ,oE”’ 0,B=l
is the viscous stress tensor defined by (Vi)rap - tap - t$j’
(2.14)
and E(“) is the irreversible electric field defined by EC”)= E _ E(W)
(2.15)
In thermodynamic equilibrium r$’ equals the mechanical stress tensor tap (which occurs in the equation of motion and in the first law of thermodynamics) and the vector E@*) equals the electric field E of Maxwell’s equations. By virtue of the expression (2.13) for the entropy production and according to the usual procedure of non-equilibrium thermodynamics (see Ref. [3]) we have for polarizable anisotropic media the following phenomenological equations (2.16)
(2.17)
(2.18)
(2.19)
In (2.16)-(2.20) we have neglected for simplicity all cross effects except for possible cross effects among the two types of dielectric relaxation phenomena described by (2.16) and (2.17). Equations (2.18)-(2.20) are the well known forms for Ohm’s law for electric conduction, Fourier’s law for heat conduction and Newton’s law for viscous flow, respectively. The quantities L$f&, L[${$, . . . which occur in (2.16)-(2.20) are phenomenological tensors. Because of the symmetry of E,~ and t$$’ one can choose the tensor L$“$) so that it satisfies the symmetry relations (2.21)
(2.22) (2.23) (2.24)
308
L. RESTUCCIA and G. A. KLUITENBERG 3. LINEAR
EQUATIONS
OF STATE
EQUATION
FOR
AND
DIELECTRIC
ANISOTROPIC
RELAXATION
MEDIA
The free energy f is defined by
f =u-Ts.
(3.1)
Using expression (2.2) for the differential of the entropy we have 3
df = --s dT + Y c
r:J’ de,, + E@q). dp - EC’). dp”‘.
cr,fl=l
(3.2)
It follows that
f P-7Q,
s = - ET r$’ = p $-
Irs
(3.3)
p, p"'),
fG”>Q3, p, p”‘),
(3.4)
f(T, Q3, P,p"')
E(=9) =$
(35)
and E(l) = _ _
d f
dp(‘)
We postulate medium
(T
QJ, p, p”‘).
the following form for the specific free energy
(3.6)
f of an anisotropic dielectric
f =f(l)+fW 1
(3.7)
where
and
fC2) =; P.( i: ~gg&&J - 34lY + I5 ~gj2~P’,“pe”) 0,p=1 CT,fl=l + (T - T,)(kl @$A - gl +jPpr,).
(3.9)
Equation (3.8) characterizes the anisotropic Kelvin-Voigt media (see Ref. [30]). In (3.8) the tensors u,Byt and a+ do not depend on the temperature and on the strain tensor (i.e. they are constant) and they may be chosen so that they satisfy the symmetry relations %Yf=
~~~y5=~,~5y=~~~fy=~yl_IYs=~yssn=~syaS=~5y~a
%Yfi= aa,*
(3.10) (3.11)
Moreover, (P(T) is some function of the temperature and v. = l/p, is the specific volume in the reference state. In the following we replace v. by Y and p. by p. These quantities are supposed to be constants. In (3.9) the vectors ulyjm a&)= and the tensors u[s&, u&$&i are also supposed to be constants. It follows from (3.9) that these tensors may be chosen so that they satisfy the following symmetry relations u$&
= &&z
Cl&$, = U&,‘B~. The
tensors (3.10)-(3.13) reference state.
are determined
by the physical properties
(3.12) (3.13) of the medium in the
Dielectric relaxation phenomena
From (3.3)-(3.6)
and (3.7)-(3.9)
309
we obtain
(3.16) and
In (3.16) and (3.17) we have used the definitions P= pp
and
P(l)= pp(‘)
(3.18)
for the total polarization vector P and for the partial polarization vector P(l). We note that the assumptions (3.7)-(3.9) for the free energy! have led to the linear relations (3.15)-(3.17) among the stress tensor t$$, the strain tensor and the temperature and among the electric fields Efe9) and E(l) and the polarization vectors P and P(l) and the temperature, respectively. Equation (3.15) represents the Duhamel-Neumann law. Relations (3.15)-(3.17) are called equations of state. Now, we discuss the dynamical constitutive equations for dielectric relaxation in anisotropic media which follow from the theory if one assumes linear equations of state and constant phenomenological coefficients. If the mass density p is constant the phenomenological equations (2.16) and (2.17) for the irreversible dielectric relaxation phenomena may be written in the form
(3.20) where we have used the definitions (3.18) and (2.15). With the aid of the equations of state (3.16) and (3.17) equation (3.19) becomes (3.21) where (3.22) and
If the coefficients in (3.22) and (3.23) are constant, it follows from (3.21) that &@$=$Q$‘), provided all derivatives in (3.24) exist.
(3.24)
L. RESTUCCIA
310
and G. A. KLUITENBERG
On the other hand, using the equation of state (3.17), from (3.20) one gets
dP:” c3 hpyPt”
dt+
),=I
= Q’,“,
(3.25)
where
and
QS” =
tl
$,L$fjy~ +;, $$p$b\,(T
L{&&@,O,‘,Py +
- T,).
(3.27)
Multiplying both sides of equation (3.25) by c$ and summing over p we obtain with the aid of (3.24) (3.28) We assume that it is possible to define the inverse matrix (cg,j-’
such that (3.29)
Using (3.29) from (3.21) it follows that (3.30) Substituting (3.30) in (3.28) and using (3.23) and (3.27) it is possible to eliminate the internal vector field P(l) and to obtain the following explicit form for the dielectric relaxation equation
(3.31)
(3.37) Hence, it is seen that the linearization of the theory leads to a relaxation equation for anisotropic polarizable media which has the form of a linear relation among the temperature, the components of the electric field, the components of the total polarization vector, the time derivative of the temperature, the components of the first derivatives with respect to time of
Dielectric relaxation phenomena
311
the electric field and of the total polarization vector and the components of the second derivative with respect to time of the latter vector. An equation of the type (3.31) has been obtained in Ref. [31] using a microscopical model. Using (3.22), (3.23), (3.29) and (3.30) the equation of state (3.17) can be presented in the following form
Then with the help of (3.22), (3.35) and (3.37) from (3.38) one gets (3.39) where
Dzj=
From (3.38) components components The same
-
(3.41)
and (3.39) E(l) may be expressed as a linear function of the temperature, of the of the electric field, of the components of the total polarization and of the of the time derivative of the latter vector. remark holds for P(l) (see (3.22), (3.23) and (3.30)).
4. THE
HEAT
DISSIPATION
FUNCTION
Using (2.16)-(2.20) we obtain from (2.12), (2.13) and (3.18) the following balance equation for the specific entropy s
(4.1) We define the specific heat at constant deformation c(,) = ;Tu(T,
c(,) by
E,,s, p, I+~‘).
(4.2)
L. RESTUCCIA
312
By virtue of (3.1), (3.7)-(3.9)
and G. A. KLUITENBERG
and (3.14) the specific internal energy u is given by
3 +
2 n,p=1
(4.3)
a{i&d%$)
From this equation and (4.2) we obtain
From (3.14) and (4.4) it follows that
By virtue of (4.1) and (4.5) we obtain the following equation for the heat conduction
3
I
+ a~clc ,
d CY
T-lL$$jq)dT %J
+ cf’.
(4.6)
In (4.6) a@) is given by o(h) = o(P) + o(E) + o(R)7
(4.7)
where (4.8) @I
=
5
L$j”‘E,EB
(4.9)
a,@=1
and (4.10) The quantity a@) is called the heat dissipation function. CT(‘), aCE) and aor) represent the amount of heat dissipated per unit of volume and per unit of time by the irreversible phenomena of dielectric relaxation, of electric conduction and viscous flow, respectively. If aCP)vanishes, no dielectric relaxation phenomena occur and u(*) reduces to uCR)+ CT(~). Substituting (3.39) in (4.8) it is possible to eliminate the vector field EC’) conjugate to the internal variable P(l). We have
+ 2D$+‘(T - T,)] + i
Dc$E,,[ $
DB3,)P,,+2Df’(T
- T,)] + D’,4’Dg’(T - Tg)‘}.
(4.11)
y=l
From (4.7) and (4.9)-(4.11) it is seen that the heat dissipation function a@) has the form of a quadratic function of the components of the strain tensor, of the components of the electric field, of the components of the total polarization vector, of the components of the time derivatives of the latter vector and of the temperature.
Dielectric relaxation phenomena
313
From (4.8)-(4.10) and (3.38) it is possible to obtain the particular case of isotropic polarizable media discussed in Ref. [9] using the following properties of perfect isotropy for polar tensors of order four, two and one (vectors) (see Refs [lo, 15, 321) (4.12) One gets
ad = ad,,
and
a, =O.
and otR) = Q *$i
(d$J)2
(4.16)
+ 3n”(+ss2
(exactly obtained in Ref. [9]). In (4.16) the scalars r), and nv are the shear and the volume viscosity, respectively. Moreover, the definitions of the deviator and of the scalar part of the tensor sorS have been used. Equation (4.16) is the Rayleigh dissipation function well known from the theory of viscous flow. 5. DEBYE
MEDIA
If E = ECcq),EC”) vanishes (see (2.15)) and from (2.16) and (2.24) one gets L{$FLp= 0
and
L@& = -L&f& = 0.
(5.1)
Hence, the dielectric relaxation equation (3.31) becomes
where
(5.6)
x& = a$&. Equations (5.3)-(5.7) have been obtained (3.38) and (5.1) one gets
from (3.32)-(3.37)
using (5.1). Moreover,
(5.7) from
(5.8) where
(5.9) (5.10) (5.11)
314
L. RESTUCCIA and G. A. KLUITENBERG
Substituting (5.8) in (4.8) one obtains the heat dissipation function crCp)in this particular case.
+ y$, ~~~~~[ V$r L$$Eq + 2D$‘)(T - Z)] + ~~)~~)(~
- I;d’).
(5.12)
If the coefficients L{>& vanish the relaxation equation (5.2) becomes E, = u@&
+ a,$?&
(5.13)
Hence, the change in the electric field E is associated with changes in the ~larization vector P and in the temperature T, which is a well-known result for media without dielectric relaxation [331.
6. DE GROOT-MAZUR
MEDIA
In the case where
(see (2.24)) equations
(2.16) and (2.17) become
(6.2) and dp’,” _ o dt ’
(6.3)
From (6.3) it is seen that pf’) is constant and it can be supposed that p(l) = 0 (i.e. there is no internal vectorial degree of freedom). The dielectric relaxation equation (3.31) reduces to (6.4) where
(see (3.22), (3.26), (3.32)-(3.37)
x~Y~!E,~P = a@&3~
(6.5)
X$&V/3= L@$3l
(6.6)
xf& = a$$,,.
(6.7)
and (6.1)). The heat dissipation function a@‘) becomes (see
(4.8))
(6.8) Hence, this is the heat conduction.
dissipation
function
o
media
REFERENCES [If G. A. KLUITENBERG, Physica 68,75-92 (1973). [Z] G. A. KLUITENBERG, Physica 87A, 302-330 (1977). [3] G. A. KLUITENBERG, Physica 109A,91-122 (1981). [4] L. RESTUCCIA and G. A. KLUI’TENBERG, Physica AW, 157-182 (1988). [S] V. CIANCIO, J. Non-Equilibrium Thermodyn. 14,239~250 (1989). [6] L. RESTUCCIA and G. A. KLUMENBERG, J. Non-Equilibrium T~rmodyn.
15,335346
(1990).
without
Dielectric relaxation phenomena
315
[7] V. CIANCIO, L. RESTUCCIA and G. A. KLUITENBERG, J. Non-Equilibrium Thermodyn. 15, 157-171 (1990). [8] V. CIANCIO and L. RESTUCCIA, Physica A162,489-498 (1990). [9] L. RESTUCCIA and G. A. KLUITENBERG, to appear. [lo] S. R. DE GROOT and P. MAZUR, Non-Equilibrium Thermodynamics. North-Holland, Amsterdam (1962). [ll] J. MEIXNER and H. G. REIK, Thermodynamik der Irreversiblen Prozesse. Jfundbuch der Physik, Band III/2. Springer, Berlin (1959). [12] S. R. DE GROOT, Thermodynamics of Irreversible Processes. North-Holland, Amsterdam and Interscience, New York (1951). [13] I. PRIGOGINE, Etude Thermodynumique des Pht%omenes irreversibles. Dunod, Paris et Editions Desoer, Liege (1947). [14] I. PRIGOGINE, Introduction co Thermodynamics of Irreversible Processes. Interscience-Wiley, New York (1961). [15] G. A. KLUITENBERG, Plasticity and Non-Equilibrium Thermodynamics. CISM Lecture Notes. Springer, Berlin (1984). [16] I. GYARMATI, Non-Equilibrium Termodynamics, Springer, Berlin (1970). [17] C. J. F. BOTICHER and P. BORDEWIJK, Theory of Electric Polarization, Vol. II: Dielectrics in Time-Dependent Fields. Elsevier, Amsterdam (1978). [18] G. A. MAUGIN, Continuum Mechanics of Electromugnetic Solids. North-Holland, Amsterdam (1988). [19] G. A. KLUITENBERG, Relativistic thermodynamics of irreversible processes. Thesis, Leiden (1954). (201 G. A. KLUITENBERG and S. R. DE GROOT, Physicu 21, 148-168 (1955). 1211 G. A. KLUITENBERG and S. R. DE GROOT, Physicu 21, 169-192 (1955). [22j G. A. MAUGIN, ACCUMech. 35, l-70 (1980). 1231 G. A. MAUGIN. Phvsicu 81A. 454-468 (1975) c-[24] G. A. MAUGIN'and A. C. ERINGEN, j. Met. 16, 101-147 (1977). [25] W. J. CASPERS, Theory of Spin Relaxation. Interscience, New York (1964). [26] R. LENK, Browniun Motion and Spin Relaxation. Elsevier, Amsterdam (1977). [27] G. A. MAUGIN, Nonlinear Electromechanical Effects and Applications. Series of Lectures. World Scientific, Singapore (1985). [28] G. A. MAUGIN (Ed.), The Mechanical Behavior of Electromagnetic Solid Continua. Proc. of IUTAM-IUPAP Symposium, Paris, 1983. North-Holland, Amsterdam (1984). [29] G. A. MAUGIN and A. C. ERINGEN, Electrodynamics of Continua, Vol. I: Foundations and Solid Media; Vol. II: Fluids and Complex Media. Springer, New York, (1990). [30] G. A. KLUITENBERG and V. CIANCIO, Atti Act. Sci. Lett. Arti Pulermo Ser. IV XL (Parte I), 107 (1980-81). I311 G. A. MAUGIN. Arch. Mech. Scosow (Polund) 29. 143-159 (1977). [32] H. JEFFREYS, kurcesiun Tensors. Cambridge University Press (1957). [33] P. DEBYE, Polar Molecules. Dover, New York (1945). 1
(Received and accepted 9 July 1991)
Int. J. Engng Sci. Vol. 30, No. 3, pp. 305-315, 1992 Printed in Great Britain.
ON THE HEAT DISSIPATION FUNCTION FOR DIELECTRIC RELAXATION PHENOMENA IN ANISOTROPIC MEDIA-f LILIANA Department
GERRIT Department
RESTUCCIA
of Mathematics, University of Messina, Messina, Italy
ALFRED
KLUITENBERG
of Mathematics and Computing Science, Eindhoven University of Technology, Eindhoven, The Netherlands (Communicated
by G. A. MAUGIN)
Abstract-The heat dissipation function for polarizable anisotropic media in which phenomena of dielectric relaxation occcur is derived as a generalization of the heat dissipation function studied in the case that the media are isotropic. The methods of non-equilibrium thermodynamics are used. It is seen that the linearization of the theory leads to a dielectric relaxation equation for anisotropic polarizable media which has the form of a linear relation among the temperature, the components of the electric field vector and of the total polarization vector, the first derivatives with respect to time of the components of these vectors and of the temperature and the components of the second derivative with respect to time of the total polarization vector. It is shown that the heat dissipation function is due to irreversible phenomena of viscous flow, electric conduction and dielectric relaxation. Finally, the obtained results are applied to the particular cases of Debye media and De Groot-Mazur media.
1.
INTRODUCTION
In some previous papers [l-9] a theory for dielectric relaxation phenomena in polarizable continuous media was developed, which is based on the thermodynamics of irreversible processes [ 10-161. In the Refs [4-81 it was shown that, if several microscopic phenomena give rise to dielectric relaxation, it is possible to describe these microscopic phenomena with the aid of n macroscopic vectorial internal variables which are introduced as internal degrees of freedom in the entropy. It was demonstrated that these n vector fields can be considered as n contributions to the total specific polarization vector p so that this vector can be split in n + 1 parts, i.e. p = p(O)+
2
p(k),
(1.1)
k=l
where p(O) and pck) (k = 1, 2, . . . , n) are called partial specific polarization vectors. In the present paper we consider anisotropic polarizable media in which the polarization vector p is additively composed of two parts p(O) and p(l) p = p(O)+ p(l),
(l-2)
where we suppose that changes in p(O) and in p(l) are both irreversible processes [3,17]. We derive the explicit expression for the heat dissipation function due to the irreversible processes of viscous flow, electric conduction and dielectric relaxation. This expression is a generalization of the heat dissipation function derived in Ref. [9] for isotropic polarizable media. The methods of non-equilibrium thermodynamics are used. In Section 2 the entropy balance equation and the phenomenological equations are discussed. In Section 3 for the case that the equations of state may be linearized the dielectric relaxation equation for anisotropic polarizable media is derived. In Section 4 the heat dissipation function is obtained and in Sections 5 and 6 the particular cases of Debye media and De Groot-Mazur media are discussed. tThis work is supported by the Gruppo Nazionale per la Fisica Matematica of the Consiglio Nazionale delle Ricerche (C.N.R.), by the Nederlandse Organisatie Voor Wetenschappelijk Onderzoek (N.W.O.) and by the Department of Mathematics and Computing Science of the Eindhoven University of Technology. 305
L. RESTUCCIA
306
2. THE
and G. A. KLUITENBERG
GENERAL
FORMALISM
We consider anisotropic polarizable media in which the contributions of microscopic phenomena to the macroscopic polarization vector can be described by introducing one vectorial internal degree of freedom in the entropy. Thus, if u is the specific internal energy, E,~ the strain tensor and s the specific entropy we assume that s = s(u, Q3, p, I+“). (2.1) By virtue of (2.1) we obtain for the differential do of s the following expression Tds=du-v
i
r$‘de,B
_ E@q). dp + EC’). dp”‘,
Ly,fi=l
where Y = l/p is the specific volume. The temperature Eceq)and EC’)are given by
T-’ = $4~ E,~, @‘=-pT
and
- a k/3
(2.2)
T, the tensor r$J’ and the vector fields
p, p(l)),
(2.3)
(2.4)
s(u, QJ, p, p’%
Eceq)= - Td s(u, E,~, p, p’1’) dP
(2.5)
EC’)= T
(2.6)
Q3, p, p(I)).
In (2.3)-(2.6) t$’ is the equilibrium stress tensor, Eceq)is the equilibrium electric field and E(l) is the vectorial thermodynamic affinity conjugate to the internal variable p(l). We assume that r$J)=O
if
E,~=O,
p = p(l) = 0
and
T = To,
(2.7)
Eceq)= 0
if
.saS= 0,
p=p”‘=O
and
T= I&,,
(2.8)
EC’)= 0
if
saS = 0 ?
p=p”‘=O
and
T=T,,
(2.9)
where T,, is the temperature in the reference state. The strain tensor is measured with respect to this reference state (see Refs [4, 5, 71). The first law of thermodynamics may be written in the form [lo, 181 p$=
-div J(q)
E+pE$,
(2.10)
where Jcq) is the vector of the density of the heat flow, j@')is the vector of the density of the electric current and de,/dt is given by d$;($+z),
(a,p=1,2,3)
(2.11)
where we have supposed that the deformations and rotations of the medium are small from a kinematical point of view. Moreover, in (2.10) dldt is the material derivative with respect to time and u, is the cu-component of the velocity v of the medium. Rectangular axis-frame is used which is fixed in the space. It should be noted that (2.10) does not satisfy the requirements of the theory of relativity. However, we shall assume that the velocity of the medium with respect to the observer is small compared with the velocity of the light and in this case (2.10) is a good approximation for the first law. For the exact relativistic formulation of the first law for simple fluids and viscous polarizable and magnetizable fluid mixtures influenced by an electromagnetic field one may consult Kluitenberg [19] and Kluitenberg and De Groot [20,21].
Dielectric
relaxation
phenomena
307
Thermodynamic considerations of dielectric and magnetic phenomena are given in numerous papers and books (see for instance [22-291). From the first law (2.10) and the Gibbs relation (2.2) we obtain for the entropy balance p $ = -div( TIJ’*)) where a@) is the entropy production -I’-‘J(*) In this equation @
+ &I,
(2.12)
per unit volume and per unit time, given by
(Vi)dEap + pE(“) dp . grad T + 5 zap dt . dt + ,oE”’ 0,B=l
is the viscous stress tensor defined by (Vi)rap - tap - t$j’
(2.14)
and E(“) is the irreversible electric field defined by EC”)= E _ E(W)
(2.15)
In thermodynamic equilibrium r$’ equals the mechanical stress tensor tap (which occurs in the equation of motion and in the first law of thermodynamics) and the vector E@*) equals the electric field E of Maxwell’s equations. By virtue of the expression (2.13) for the entropy production and according to the usual procedure of non-equilibrium thermodynamics (see Ref. [3]) we have for polarizable anisotropic media the following phenomenological equations (2.16)
(2.17)
(2.18)
(2.19)
In (2.16)-(2.20) we have neglected for simplicity all cross effects except for possible cross effects among the two types of dielectric relaxation phenomena described by (2.16) and (2.17). Equations (2.18)-(2.20) are the well known forms for Ohm’s law for electric conduction, Fourier’s law for heat conduction and Newton’s law for viscous flow, respectively. The quantities L$f&, L[${$, . . . which occur in (2.16)-(2.20) are phenomenological tensors. Because of the symmetry of E,~ and t$$’ one can choose the tensor L$“$) so that it satisfies the symmetry relations (2.21)
(2.22) (2.23) (2.24)
308
L. RESTUCCIA and G. A. KLUITENBERG 3. LINEAR
EQUATIONS
OF STATE
EQUATION
FOR
AND
DIELECTRIC
ANISOTROPIC
RELAXATION
MEDIA
The free energy f is defined by
f =u-Ts.
(3.1)
Using expression (2.2) for the differential of the entropy we have 3
df = --s dT + Y c
r:J’ de,, + E@q). dp - EC’). dp”‘.
cr,fl=l
(3.2)
It follows that
f P-7Q,
s = - ET r$’ = p $-
Irs
(3.3)
p, p"'),
fG”>Q3, p, p”‘),
(3.4)
f(T, Q3, P,p"')
E(=9) =$
(35)
and E(l) = _ _
d f
dp(‘)
We postulate medium
(T
QJ, p, p”‘).
the following form for the specific free energy
(3.6)
f of an anisotropic dielectric
f =f(l)+fW 1
(3.7)
where
and
fC2) =; P.( i: ~gg&&J - 34lY + I5 ~gj2~P’,“pe”) 0,p=1 CT,fl=l + (T - T,)(kl @$A - gl +jPpr,).
(3.9)
Equation (3.8) characterizes the anisotropic Kelvin-Voigt media (see Ref. [30]). In (3.8) the tensors u,Byt and a+ do not depend on the temperature and on the strain tensor (i.e. they are constant) and they may be chosen so that they satisfy the symmetry relations %Yf=
~~~y5=~,~5y=~~~fy=~yl_IYs=~yssn=~syaS=~5y~a
%Yfi= aa,*
(3.10) (3.11)
Moreover, (P(T) is some function of the temperature and v. = l/p, is the specific volume in the reference state. In the following we replace v. by Y and p. by p. These quantities are supposed to be constants. In (3.9) the vectors ulyjm a&)= and the tensors u[s&, u&$&i are also supposed to be constants. It follows from (3.9) that these tensors may be chosen so that they satisfy the following symmetry relations u$&
= &&z
Cl&$, = U&,‘B~. The
tensors (3.10)-(3.13) reference state.
are determined
by the physical properties
(3.12) (3.13) of the medium in the
Dielectric relaxation phenomena
From (3.3)-(3.6)
and (3.7)-(3.9)
309
we obtain
(3.16) and
In (3.16) and (3.17) we have used the definitions P= pp
and
P(l)= pp(‘)
(3.18)
for the total polarization vector P and for the partial polarization vector P(l). We note that the assumptions (3.7)-(3.9) for the free energy! have led to the linear relations (3.15)-(3.17) among the stress tensor t$$, the strain tensor and the temperature and among the electric fields Efe9) and E(l) and the polarization vectors P and P(l) and the temperature, respectively. Equation (3.15) represents the Duhamel-Neumann law. Relations (3.15)-(3.17) are called equations of state. Now, we discuss the dynamical constitutive equations for dielectric relaxation in anisotropic media which follow from the theory if one assumes linear equations of state and constant phenomenological coefficients. If the mass density p is constant the phenomenological equations (2.16) and (2.17) for the irreversible dielectric relaxation phenomena may be written in the form
(3.20) where we have used the definitions (3.18) and (2.15). With the aid of the equations of state (3.16) and (3.17) equation (3.19) becomes (3.21) where (3.22) and
If the coefficients in (3.22) and (3.23) are constant, it follows from (3.21) that &@$=$Q$‘), provided all derivatives in (3.24) exist.
(3.24)
L. RESTUCCIA
310
and G. A. KLUITENBERG
On the other hand, using the equation of state (3.17), from (3.20) one gets
dP:” c3 hpyPt”
dt+
),=I
= Q’,“,
(3.25)
where
and
QS” =
tl
$,L$fjy~ +;, $$p$b\,(T
L{&&@,O,‘,Py +
- T,).
(3.27)
Multiplying both sides of equation (3.25) by c$ and summing over p we obtain with the aid of (3.24) (3.28) We assume that it is possible to define the inverse matrix (cg,j-’
such that (3.29)
Using (3.29) from (3.21) it follows that (3.30) Substituting (3.30) in (3.28) and using (3.23) and (3.27) it is possible to eliminate the internal vector field P(l) and to obtain the following explicit form for the dielectric relaxation equation
(3.31)
(3.37) Hence, it is seen that the linearization of the theory leads to a relaxation equation for anisotropic polarizable media which has the form of a linear relation among the temperature, the components of the electric field, the components of the total polarization vector, the time derivative of the temperature, the components of the first derivatives with respect to time of
Dielectric relaxation phenomena
311
the electric field and of the total polarization vector and the components of the second derivative with respect to time of the latter vector. An equation of the type (3.31) has been obtained in Ref. [31] using a microscopical model. Using (3.22), (3.23), (3.29) and (3.30) the equation of state (3.17) can be presented in the following form
Then with the help of (3.22), (3.35) and (3.37) from (3.38) one gets (3.39) where
Dzj=
From (3.38) components components The same
-
(3.41)
and (3.39) E(l) may be expressed as a linear function of the temperature, of the of the electric field, of the components of the total polarization and of the of the time derivative of the latter vector. remark holds for P(l) (see (3.22), (3.23) and (3.30)).
4. THE
HEAT
DISSIPATION
FUNCTION
Using (2.16)-(2.20) we obtain from (2.12), (2.13) and (3.18) the following balance equation for the specific entropy s
(4.1) We define the specific heat at constant deformation c(,) = ;Tu(T,
c(,) by
E,,s, p, I+~‘).
(4.2)
L. RESTUCCIA
312
By virtue of (3.1), (3.7)-(3.9)
and G. A. KLUITENBERG
and (3.14) the specific internal energy u is given by
3 +
2 n,p=1
(4.3)
a{i&d%$)
From this equation and (4.2) we obtain
From (3.14) and (4.4) it follows that
By virtue of (4.1) and (4.5) we obtain the following equation for the heat conduction
3
I
+ a~clc ,
d CY
T-lL$$jq)dT %J
+ cf’.
(4.6)
In (4.6) a@) is given by o(h) = o(P) + o(E) + o(R)7
(4.7)
where (4.8) @I
=
5
L$j”‘E,EB
(4.9)
a,@=1
and (4.10) The quantity a@) is called the heat dissipation function. CT(‘), aCE) and aor) represent the amount of heat dissipated per unit of volume and per unit of time by the irreversible phenomena of dielectric relaxation, of electric conduction and viscous flow, respectively. If aCP)vanishes, no dielectric relaxation phenomena occur and u(*) reduces to uCR)+ CT(~). Substituting (3.39) in (4.8) it is possible to eliminate the vector field EC’) conjugate to the internal variable P(l). We have
+ 2D$+‘(T - T,)] + i
Dc$E,,[ $
DB3,)P,,+2Df’(T
- T,)] + D’,4’Dg’(T - Tg)‘}.
(4.11)
y=l
From (4.7) and (4.9)-(4.11) it is seen that the heat dissipation function a@) has the form of a quadratic function of the components of the strain tensor, of the components of the electric field, of the components of the total polarization vector, of the components of the time derivatives of the latter vector and of the temperature.
Dielectric relaxation phenomena
313
From (4.8)-(4.10) and (3.38) it is possible to obtain the particular case of isotropic polarizable media discussed in Ref. [9] using the following properties of perfect isotropy for polar tensors of order four, two and one (vectors) (see Refs [lo, 15, 321) (4.12) One gets
ad = ad,,
and
a, =O.
and otR) = Q *$i
(d$J)2
(4.16)
+ 3n”(+ss2
(exactly obtained in Ref. [9]). In (4.16) the scalars r), and nv are the shear and the volume viscosity, respectively. Moreover, the definitions of the deviator and of the scalar part of the tensor sorS have been used. Equation (4.16) is the Rayleigh dissipation function well known from the theory of viscous flow. 5. DEBYE
MEDIA
If E = ECcq),EC”) vanishes (see (2.15)) and from (2.16) and (2.24) one gets L{$FLp= 0
and
L@& = -L&f& = 0.
(5.1)
Hence, the dielectric relaxation equation (3.31) becomes
where
(5.6)
x& = a$&. Equations (5.3)-(5.7) have been obtained (3.38) and (5.1) one gets
from (3.32)-(3.37)
using (5.1). Moreover,
(5.7) from
(5.8) where
(5.9) (5.10) (5.11)
314
L. RESTUCCIA and G. A. KLUITENBERG
Substituting (5.8) in (4.8) one obtains the heat dissipation function crCp)in this particular case.
+ y$, ~~~~~[ V$r L$$Eq + 2D$‘)(T - Z)] + ~~)~~)(~
- I;d’).
(5.12)
If the coefficients L{>& vanish the relaxation equation (5.2) becomes E, = u@&
+ a,$?&
(5.13)
Hence, the change in the electric field E is associated with changes in the ~larization vector P and in the temperature T, which is a well-known result for media without dielectric relaxation [331.
6. DE GROOT-MAZUR
MEDIA
In the case where
(see (2.24)) equations
(2.16) and (2.17) become
(6.2) and dp’,” _ o dt ’
(6.3)
From (6.3) it is seen that pf’) is constant and it can be supposed that p(l) = 0 (i.e. there is no internal vectorial degree of freedom). The dielectric relaxation equation (3.31) reduces to (6.4) where
(see (3.22), (3.26), (3.32)-(3.37)
x~Y~!E,~P = a@&3~
(6.5)
X$&V/3= L@$3l
(6.6)
xf& = a$$,,.
(6.7)
and (6.1)). The heat dissipation function a@‘) becomes (see
(4.8))
(6.8) Hence, this is the heat conduction.
dissipation
function
o
media
REFERENCES [If G. A. KLUITENBERG, Physica 68,75-92 (1973). [Z] G. A. KLUITENBERG, Physica 87A, 302-330 (1977). [3] G. A. KLUITENBERG, Physica 109A,91-122 (1981). [4] L. RESTUCCIA and G. A. KLUI’TENBERG, Physica AW, 157-182 (1988). [S] V. CIANCIO, J. Non-Equilibrium Thermodyn. 14,239~250 (1989). [6] L. RESTUCCIA and G. A. KLUMENBERG, J. Non-Equilibrium T~rmodyn.
15,335346
(1990).
without
Dielectric relaxation phenomena
315
[7] V. CIANCIO, L. RESTUCCIA and G. A. KLUITENBERG, J. Non-Equilibrium Thermodyn. 15, 157-171 (1990). [8] V. CIANCIO and L. RESTUCCIA, Physica A162,489-498 (1990). [9] L. RESTUCCIA and G. A. KLUITENBERG, to appear. [lo] S. R. DE GROOT and P. MAZUR, Non-Equilibrium Thermodynamics. North-Holland, Amsterdam (1962). [ll] J. MEIXNER and H. G. REIK, Thermodynamik der Irreversiblen Prozesse. Jfundbuch der Physik, Band III/2. Springer, Berlin (1959). [12] S. R. DE GROOT, Thermodynamics of Irreversible Processes. North-Holland, Amsterdam and Interscience, New York (1951). [13] I. PRIGOGINE, Etude Thermodynumique des Pht%omenes irreversibles. Dunod, Paris et Editions Desoer, Liege (1947). [14] I. PRIGOGINE, Introduction co Thermodynamics of Irreversible Processes. Interscience-Wiley, New York (1961). [15] G. A. KLUITENBERG, Plasticity and Non-Equilibrium Thermodynamics. CISM Lecture Notes. Springer, Berlin (1984). [16] I. GYARMATI, Non-Equilibrium Termodynamics, Springer, Berlin (1970). [17] C. J. F. BOTICHER and P. BORDEWIJK, Theory of Electric Polarization, Vol. II: Dielectrics in Time-Dependent Fields. Elsevier, Amsterdam (1978). [18] G. A. MAUGIN, Continuum Mechanics of Electromugnetic Solids. North-Holland, Amsterdam (1988). [19] G. A. KLUITENBERG, Relativistic thermodynamics of irreversible processes. Thesis, Leiden (1954). (201 G. A. KLUITENBERG and S. R. DE GROOT, Physicu 21, 148-168 (1955). 1211 G. A. KLUITENBERG and S. R. DE GROOT, Physicu 21, 169-192 (1955). [22j G. A. MAUGIN, ACCUMech. 35, l-70 (1980). 1231 G. A. MAUGIN. Phvsicu 81A. 454-468 (1975) c-[24] G. A. MAUGIN'and A. C. ERINGEN, j. Met. 16, 101-147 (1977). [25] W. J. CASPERS, Theory of Spin Relaxation. Interscience, New York (1964). [26] R. LENK, Browniun Motion and Spin Relaxation. Elsevier, Amsterdam (1977). [27] G. A. MAUGIN, Nonlinear Electromechanical Effects and Applications. Series of Lectures. World Scientific, Singapore (1985). [28] G. A. MAUGIN (Ed.), The Mechanical Behavior of Electromagnetic Solid Continua. Proc. of IUTAM-IUPAP Symposium, Paris, 1983. North-Holland, Amsterdam (1984). [29] G. A. MAUGIN and A. C. ERINGEN, Electrodynamics of Continua, Vol. I: Foundations and Solid Media; Vol. II: Fluids and Complex Media. Springer, New York, (1990). [30] G. A. KLUITENBERG and V. CIANCIO, Atti Act. Sci. Lett. Arti Pulermo Ser. IV XL (Parte I), 107 (1980-81). I311 G. A. MAUGIN. Arch. Mech. Scosow (Polund) 29. 143-159 (1977). [32] H. JEFFREYS, kurcesiun Tensors. Cambridge University Press (1957). [33] P. DEBYE, Polar Molecules. Dover, New York (1945). 1
(Received and accepted 9 July 1991)