A thermodynamic approach to heat and electric conduction in solids

Plrysicu

l2lA

(1983) 552-562

North-Hollmd

A THERMODYNAMIC

Publishing

APPROACH

CONDUCTION

ReceiLcd

lIeat and electric conduction thermodynamics. of the reciprocal

relations.

with microscopic

dynamical

analogy

TO HEAT

14 March

equations

of thi\ equation

ELECTRIC

19X3

in a rlgid solid IS lnvestlgated

A non-local

AND

IN SOLIDS

This allows us to apply directly

compared

Co.

in the framework

the original

con\titutive

equation

ofcxtendcd

of Onsager

Irreversible

to the analysir

for the heat flux previousI)

used for the analysis

Icad\ to the posGbility

procedure

obtained

IS

of Fecond sound in solids. A hydro-

of a turbulent

phonon

110~ under wmc

spccitic phyGc;ll condition\.

I. Introduction

One of the main problems

related

to the description

of non-equilibrium

systems

is the choice of fundamental variables. While in equilibrium this problem is well known. and there are diverse equivalent formulations, this is not the case in non-equilibrium.

where

no general

irreversible thermodynamics, librium. i.e.. the conserved

criterion

exists

for this choice.

In classical

fundamental variables are the same as in equiones. Many possible different choices arise in non-

equilibrium. One may take as fundamental variables. for instance, the temperature and the temperature gradient, as is usually done in rational thermodynamics. or the temperature

and its time derivative.

This choice has a direct physical

it has a definite

experimental

In extended

the classical

variables

meaning.

are complemented

irreversible

by the dissipative

sense and

thermodynamics, fluxes obtaining

a

generalization of the entropy and the entropy flux’ ‘). Different works have been produced developing such a procedure and discussing the convenience or not of such an approach’ “). While this theory has been applied widely to hydrodynamic problems, few works have been published for solid conductors’ “). In this work, we first treat extended irreversible thermodynamics (EIT) in a rigid heat and electric conductor, and we explore some of the consequences which are derived from this formalism. In this way the theory gets a larger held of application and gains in completeness. Rather than stating the more general possible formulation of the theory, our aim is to show how one can apply in a direct way the original formulation of Onsager in order to obtain the reciprocal 037%4371~83~000~0000,!$03.00

(’ 1983 North-Holland

HEAT

AND

ELECTRIC

CONDUCTION

IN SOLIDS

553

relations. In this way, one may avoid some of the criticisms which have been pointed out on the usuaiderivation of these relationsrO). Afterwards, we look for physical relations that involve the parameters which appear in the theory. With this objective we compare our results, in the case of heat conduction only, with equations deduced from Boltzmann equation describing second-sound propagation. In this case a physical interpretation of the parameters is obtained and a hydrodynamical analogy is proposed. The plan of the paper is as follows. Heat and electric conduction in rigid solids is investigated in the framework of extended irreversible thermodynamics in section 2. In section 3 we pay attention to the Onsager reciprocal relations, whereas in section 4 we look for a physical interpretation of some of the generalized equations by studying some aspects of second-sound propagation. Finally in section 5 a hydrodynamical analogy is developed and the possibility of a turbulent flow of phonons is examined.

2. Extended irreversible thermodynamics The system studied is a solid conductor, rigid and isotropic. Whereas in classical thermodynamics the entropy per unit mass s is a function of U, the internal energy per unit mass, and of c,, the electron number density per unit mass, in EIT a non-equilibrium entropy is postulated which depends on U, c, and on the dissipative fluxes, q (heat flux) and J, (electric current). When one studies a system in an equilibrium state, this non-equilibrium entropy reduces to the classical one because both q and J, vanish. If s is differentiable enough, the following Gibbs equation is obtained ds = 0-I du - 0 -‘PI dc, + T-‘tl(a,q + a,J,)*dq where T is the equilibrium absolute temperature, be identified below, and

+ T-‘c(y,q

+ Q,)*dJ,,

(2.1)

cli and y, coefficients which will

0 - ‘(u, c,, 4, J,) = (8s /au )‘& I, 9

(2.2)

- Q-‘rlG4 ce, 434

(2.3)

= wwU.q.,~~

Note that in analogy with classical theory 8 is a generalized non-equilibrium temperature and r,t is a generalized non-equilibrium electrochemical potential which, when developed in series up to second order in the fluxes, have the following form &‘(u, c,, q, J,) = T--‘(u, c,) + ~(i%-‘/~q2)q2 + (8’0 -‘/aqaJ,)q

.J, ,

+ +(i?20-‘/M:)J;

(2.4)

J.E. LLEBOT

554

et al.

8-“l(U, c,, q, J,) = T-‘p,(u, c,) + ;(a’0 -‘tj/dq2)q2 + &m -‘v/aJ;)J; + (a2e-‘~/aqaJe)q where p? is the equilibrium appear

*J, )

electrochemical

(2.5)

potential.

in eqs. (2.4) and (2.5) have been evaluated

heat conductoP)

and in an electric conductor”).

Some of the terms which

in the case of a metallic In both studies,

rigid

these terms are

very small compared with the values of both T-’ and T-‘,u, and therefore we will omit them here and we will consider only the classical local equilibrium values for the temperature and the electrochemical potential. However, those terms may be important if one studies the fluctuations of the dissipative fluxes in a nonequilibrium state6). From eq. (2.1) we have two supplementary order write (WW,,,,~

= T-‘v(a,q

GWJcL,,,

=

T-

‘u (y’q

+ +

a241

YZJJ

equations

of state that up to second

(2.6)

9

(2.7)

3

where v is the specific volume and c(, and yi depend on u and c,. Because of the equality of crossed derivatives, from eqs. (2.6) and (2.7) it is obtained ~1~= yl. Considering the balance equations for u and c, pti = -v-q

+J;E,

(2.8)

-V-J,

pi’,= and taking

(2.9)

the time derivative

p.j + V.(T-‘q

of (2.1)

-,u,T-‘J,)

= q.(VT-’

one easily gets + T-b14

+ J; { T-‘E

+ T-‘y,&)

- V(p,T-‘)

+ Tm’cr24+ T-‘y24

,

(2.10) where E is the electric

field and p the mass density.

Eq. (2.10) can be viewed as an entropy balance if we identify with the classical expression for the entropy flux as J, = Tm’q - pLeT-‘J e and the entropy g = q *(VT-’

production

J, in accordance

(2.11) g

+ T-‘cr,tj + T-‘y,&

+ J; { T-‘E

- V&T-‘)

+ Tp’cr24 + T-‘y24} .

(2.12)

The entropy production (2.12) has the structure of a bilinear form. In order to obtain the simplest equations for 4 and J, compatible with the definite positive character of (2.12) we may develop the generalized thermodynamic forces in terms

HEAT

AND

of q and J,. Consequently,

ELECTRIC

CONDUCTION

for q and J, may be written

the evolution

555

IN SOLIDS

in compact

form as (2.13) with M given by M = T(cr,y, - a2y,))’ and N a column

Y2Pll

-

YIP21

WI2

-

YIP22

~lPu,l

-

a2Pll

ElP22

-

@,A2

(2.14)

vector

N = T(aly2 - a,y,)-’

T-‘y,E

- yzVT-’ - ylV(,u,T-‘)

r,VT-’

- a,T-‘E

(2.15)

+ alV(peT-‘) >

together with the restrictions upon the possible values of the phenomenological coefficients c(,, arising from the positive character of Q, which can be written PII

3

0;

P22

3

0;

PllP22

3

ah2

+

(2.16)

P2d2.

Equations as (2.13b(2.15) for the evolution of J, and even more general ones are usually found in plasma physics. Here, we are considering a solid at rest, we omit viscous effects and we restrict our attention to linear equations, so that the supplementary terms appearing in such generalized Ohm equations are not relevant in this case. In order for eqs. (2.13k(2.15) to be completely determined we should know the expressions of the coefficients pi,, a, and y,. We determine pli by comparing eqs. (2.13t(2.15) in the stationary case with the usual phenomenological equationsr2). (2.17)

VT = - (l/A)4 + (t/A)(rr + A.6,

where i is the heat conductivity power, rz the Peltier coefficient identifications are obtained PII

=

1112 =

(l/~T2); (-

ll~T2)(n

~22

= +

at zero electrical current, t the thermo-electric and R the isothermal resistivity. The following

(R/T) A);

+ ~21

WT’){(P, =

(l/~T’W

-

cT)(n -

On the other side, in view of eqs. (2.13E(2.15), for a, and y2 in previous papers7x8), we consider aI = - r,~c~,T;

(2.18)

+ A) - R)Je 3

E - VA = (c/A)q - {(tlG(n

a2 = -r,p,,T;

PL,)

+

A)}

;

and of the identification

yl = - r3,u12T;

(2.19)

.

y2 = -r,p,,T,

taken

(2.20)

where t, are relaxation times, which for i = 1,4 are ass.ociated to the heat and electric conduction, whereas for i = 2,3 they are linked to the cross effects between

J.E. LLEBOT

5%

et al.

both phenomena. The identification of ~1~and y, has been taken by similarity with the corresponding one for tl, and y2. On the other side, one can see that this identification is dimensionally correct, all the numerical terms being included in the relaxation times. The relaxation times in general are difficult to measure. However, when studying the propagation of thermal waves with high-frequency thermometry in dielectric solids’3), a relation is found between r, and quantities such as density, thermal conductivity and velocity of propagation of thermal waves, which can be measured experimentally. Other possible determinations of r, can be found in the literature on the basis of measurements of thermal conductivities and the second sound in NaF14), and from acoustic attenuation experimentsi5). Determinations of r4 can be attained from measurement of the electric conductivity’6). In section 4 more details about these topics are presented. Finally, from (2.1) and from the corresponding expressions for tl, and yI a generalized Gibbs equation is obtained in the explicit form ds = T-‘du

- peTm’ dc, - (T,o/I>T*)q *dq - (z,v/3,T*)(tT - p,)J;dq

+ (vPT*)(~

+ pe)q

x (71+ c(e))Jc . dJ, .

*dJc- hW){R

+ (~/~T)(PL, - CT) (2.21)

We want to point out the inconsistency that, whereas constitutive equations of hyperbolic character, i.e., with finite velocities for the propagation of thermal and electric signals, have been used before, no generalization of the Gibbs equation was used.

3. Onsager reciprocal relations In the classical derivation of the Onsager reciprocal relations, the fluxes are required to be time derivatives of state variables. This requirement is not always accomplished in systems where vectorial or tensorial fluxes are present. Such is the case we are dealing with: neither the heat flux nor the electric current are time derivatives of state variables. In the classical theory12), when studying heat conduction in anisotropic rigid solids this difficulty is circumvented in the following way: the entropy production reads dS/dt =

s k

q-VT_‘dV.

(3.1)

After integrating by parts, neglecting thermal dilatation and taking into account that the energy for an adiabatically isolated system of constant volume and shape

HEAT

remains

constant,

dS/dt

AND

ELECTRIC

up to the second

= -(p/T;)

CONDUCTION

order

55-l

IN SOLIDS

in T one obtains

AT(&/&)dI’,

where AT = T(r) - T,,, being

T(r) the temperature

in r,,, a reference point. Expression considered as the flux and (- pT;‘At)

?u/iit = - (p/T;)

(3.2)

K(r, r’)AT(r’)

in r and To the temperature

(3.2) is in a bilinear form, where au/at is as the thermodynamical force. If it is stated

dr’ ,

(3.3)

s ,

the Onsager relations have the form K(r, r’) = K(r’, r). This, in fact, implies the symmetry of the thermal conductivity tensor. The fundamental objections to this procedure come from the legitimacy of going from eq. (3.1) to eq. (3.2). We will not discuss this subject here. (For more details see ref. 10.) In the framework of EIT we do not find this difficulty in applying the classical Onager’s treatment. Indeed, in an equilibrium state, where there is not a temperature gradient, nor an electric field, nor a gradient in the electrochemical potential, the vector N in eq. (2.13) vanishes. Furthermore one can write eqs. (2.6) and (2.7) in a matrix form (3.4) with g given by (3.5) The classical theory one-time correlation (aa)

states that if a set of varables of these variables satisfies

a verifies eq. (3.1) and if the

= kg-‘,

(3.6)

where a is the vector (J?) and k the Boltzmann constant, then, the principle microscopic reversibility implies that the matrix defined as L=M.g-’ is symmetrical. we are led to

of

(3.7) In our case, with g and M from eqs. (3.5) and (2.14)

respectively,

558

J.E. LLEBOT

and therefore -

to P,~ = pZ, that is equivalent

(n + PC)= (CT- A)

to

(3.9)

1

so that the Thomson

relation

the equality

the cross derivatives

between

et al.

ET = - z is recovered

in this case. For another

of s implies

part,

that CI~= “J,, or in view of

(2.20) - z2(tT - pL,)= zj(n + A.)

(3.10)

In order for eqs. (3.9) and (3.10) to be compatible we get 7* = 73. Expressions similar to eq. (3.10) are found in the literature”) in the framework of the hidden variables theory. In their work the authors assume z2 = 73 in order to be able to recover the Thomson relation. In our case, we get the equality from the standard original development of the classical Onsager theory. Also, some authors have assumed that the equality of the crossed derivatives of non-equilibrium entropy leads directly to Onsager relations4). As we have shown, this is not exactly true. The equality of crossed derivatives leads to eq. (3.10), so that the microscopic argument of time reversal symmetry cannot be avoided at all even in a more general theory than the classical one. Let us finally note that the three new parameters 7,, z2 and 74 introduced in the theory, are related to the dissipative coefficients 2, ~7 and rr through the fluctuation-dissipation theorem which links equilibrium fluctuations of the thermodynamic fluxes with the dissipative coefficients. Therefore, when this theorem is considered6%‘), the number of independent parameters of the generalized theory is the same as in the classical theory.

4. A physical insight: second sound in solids In the following we shall restrict ourselves to a rigid solid where only heat conduction takes place. Starting from a linearized Boltzmann equation in the Callaway approximation in three dimensions, Guyer and Krumhansl18) heat flux at low temperatures of the form 4 + 7,‘q

+ fpcc;VT

have derived

- $c;znjV2q + 2V(Vsq)}

= 0,

an equation

for the

(4.1)

where 7, is the relaxation time of the resistive collisions (umklapp scattering, impurity scattering), z,, is the relaxation time of the normal (momentumconserving) collision, c is the specific heat per unit volume and c,, the first sound velocity. Equations such as (4.1) can be obtained in the framework of EIT in a direct way from a macroscopic analysis. According to this theory, an expression for c

HEAT AND ELECTRIC CONDUCTION

559

IN SOLIDS

is obtained”‘) (T = -V(O-‘q where

-J,)+q(W-‘+

CI is a coefficient

equilibrium

depending

temperature

temperature

p(u) = - ;d(w-'T-*)/du the procedure

VB-’ + T-la4

on the internal

energy,

and

the non-

+ fl(u)q2,

T the equilibrium

Following

only

(4.2)

given by*)

O-+4, q) = T-‘(u) being

T-‘atj),

(4.3) and fl a functiorP)

given by

. outlined

(4.4) in section

2, it can be stated

that

= poq + p’V2q + p2V(V*q),

(4.5)

where, as in section 2 the coefficients p1 are phenomenological coefficients. Now we have included some nonlocal terms, as V*q and V(V ‘4). Taking into account the identifications obtained in section 2, we can write CI =

-z(llT)-‘;

p. = (T*E,)-‘;

pl = - r&iT2)-1;

p2 = - r;(iT*)-’

,

(4.6)

where r, and r, are parameters assimilated to correlation lengths of transversal and longitudinal perturbations of the heat flux. Taking into consideration eqs. (4.5) and (4.6) one gets”) r# + q = - AVT + RT2V{ /3(u)q*} + r;V*q + r;V(V.q)

(4.7)

For the moment we drop the term corresponding to non-equilibrium corrections to the non-equilibrium temperature, whose influence will be studied in the following section. We obtain an equation which is very similar to the one obtained by Guyer and Krumhansl.

If we compare

both equations,

we obtain

the following

results

This identification is important because parameters such as r, A, r. and rl can be identified in function of magnitudes which have an experimental determination. Moreover, the identification of E, agrees with the expression for heat conductivity obtained by Wannier from a kinetic theory analysis’6). Note also that eq. (4.7) is a generalization of the Maxwell-Cattaneo equation and that both eqs. (4.7) and (4.1) have been obtained from very different bases and methods. In previous works”), an equation similar to (4.6) was obtained and it was compared with a model for high-frequency thermometry”). Here, the comparison is more direct than in ref. 19.

560

J.E. LLEBOT

5. A hydrodynamical In this section order

analogy

we deal with a possible

corrections

et al

experimental

to the non-equilibrium

section

were neglected

analogy

of the present

for simplicity. equations.

temperature,

consequence which

In this way we outline

Though

many

of second-

in the previous

a hydrodynamical

hydrodynamical

models

for the

flow of phonons have been proposed, the characteristic feature of this treatment is the presence of the nonlinear terms in the temperature, which may lead to a turbulent flow. If we consider

that S+CK while 3./r, r$s

4 = - (i/T)VT

+ (i/z)T2V{fi(u)q2}

and r:/t

remain

finite, eq. (4.7) reads

+ (r$z)V2q + (rf/T)V(V.q)

This assumption states in view of (4.8) that T,-+ rj, i.e., the resistive very rare. In the stationary case, the latter equation becomes (r$5)V2q - (ti/T)VT

=

(n/r)T2V{ /cqu)q2) ,

(5.1) processes

are

(5.2)

where the last term in eq. (5.1) vanishes from the balance law for the internal energy. Eq. (5.2) is formally very similar to the dynamic equation describing the flux of an incompressible fluid in the stationary state

pvv -vp =pv*vv, where v is the velocity,

(5.3)

p the density, p the pressure

and p the viscosity

of the fluid.

From eq. (5.3) the Reynolds number is defined, which shows a relation between the viscous forces (/AV’V) and the inertia1 ones (pv -Vu). Low Reynolds numbers indicate that viscous forces predominate upon inertia1 ones and the Poiseuille flow is obtained. When the flow reaches Reynolds numbers larger than a certain critical value, the flow becomes turbulent. For these values, the inertia1 nonlinear term is representative

and plays an important

role.

Similarly, if j(u) is constant, we observe a complete analogy between eqs. (5.2) and (5.3). In this case, we could define an adimensional number analogous to the Reynolds one, as a quotient between the convective term p(u)(A/r)T2Vq2 and the diffusive one (r$T)V2q. This can be written as

being I a characteristic transversal length of the system (in analogy with the diameter of the tube in hydrodynamics). For low values of A, i.e., when the

HEAT AND ELECTRIC CONDUCTION

IN SOLIDS

561

non-linear term has not importance compared with (r$r)V*q, the phonon flow shows the characteristics of the Poiseuille flow, V2q = $pcz,‘VT.

(5.5)

This equation can be obtained from (4.7) when r,-+co, in the stationary case when the term V{/?(u)q2) has been neglected. When the heat flux increases and A takes higher values the phonon flow will probably show a turbulent pattern. In this way A would be an index for the phonon flow. The critical value for the heat flux may be evaluated by using the same order of magnitude (103) for A than for the hydrodynamical phenomena, since it depends only on the structure of the equation. It is given by (5.6) The observation of such a turbulent flow of phonons would allow to obtain experimental information concerning the parameter p(u). Recall that this parameter in our theory is not arbitrary, but that it is determined from the thermal conductivity and the relaxation time of the heat flux through eq. (4.4).

6. Conclusions A very simple way for the derivation of the Onsager reciprocal relations in a system where more than one vectorial flux exists is shown by direct application of the original Onsager procedure. The relaxation times associated to the coupling between the heat and electric conduction processes are shown to be equal, The non-equivalence of the generalized Maxwell relations and the Onsager ones is clarified. A comparison with an equation for the heat flux obtained from microscopic methods is carried out. The coefficients of EIT in this case are related to terms that can be measured. Finally, we present a formal analogy with the hydrodynamic flow of a fluid which leads to the possibility of a turbulent flow of phonons under certain physical conditions.

Acknowledgements

This work has been partially supported by the Comision Asesora de Investigacion Cientifica y Tecnica of the Spanish Government. One of us (J.E.Ll.) acknowledges the financial support of CIRIT (Comissio Interdepartamental de Recerca i d’Innovacib Tecnologica) of the Generalitat of Catalonia.

562

J.E. LLEBOT

et al.

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