Heat conduction and relaxation in liquids of high viscosity

Physica A 162 (199(I) 377-404 North-Holhmd

HEAT CONDUCTION AND RELAXATION IN LIQUIDS OF HIGH VISCOSITY Josef JACKLE Fakultfit fur Phv~ik. Unircrsitia Kon.~tanz, D-7750 Konsmnz, Fed. Rep. Germany

Received 12 June 1989 Revised verqon received 22 September 1989

The generalized hydrodynamic description of heat conductton in liquids of high vt,,co,,ity is presented, following the phenomenological theory of thermoviscocla,,ticitv The thc~ry apphes in particular to undercooled one-component liquids near the gla,,,, tran,,itlon, in ~hlch structural relaxation causes the specific heat to be frequency dependent. For ,i ~tmpic model in which a separate channel for the flow of configurational energy independent of phonon,, exists, it is shown that a frequency dependence of the thermal conductiv,ty goe~ together with a wa~exector dependence of the specific heat. The spectrum of thermal entropy fluctuatmns and the quasi-elastic light scattering spectrum, which dr, play the mterl,'rcncc of the heat conductmn mode wflh structural relaxation, are dcrwed from the phcnomcnolognc,d cqudtions.

I. Introduction Compared with the frequency dependence of mechanical or electrical properties, in the past observations of a frequency-dependent specific heat have received relatively little attention. In fact. a frequency dependence cf the specific heat at constant volume and pressure was first observed indirectly by measuring the dispersion of sound waves in polyatomic gases [1.2]. The first direct observation of a time dependence of the specific heat was made for liquid glycerol close to the glass transition [3]. Only recently a systematic study of the frequency dependence of the specific heat of viscoelastic liquids near their g~a~ - " -- transition was carried out l"-~. r, -, -'~,i-,,,, On the theoretical side. a frequency dependence of the specific heat is accounted for in the formulation of Coleman and Noll of the thermodynamics of materials with fading memory [8.9]. This theory has been applied to heat conduction by Coleman and Gurtin [10]. Gurtin and Pipkin [11 ]. Nunziato [12] and others. Apparently. this generalized theory of heat conduction has not been used much in condensed matter physics, probably because the physical conditions for its validity had not become sufficiently clear. The present paper 0378-4371/90/$03 50 © Elsevier Soence Publishers B.V. (North-Holland)

378

J. Jiickle I Heat conduction and relaxation in high-viscosttv hquids

deals with the application of generalized heat conduction equations to viscous glass-forming liquids close to the glass transition [13]. Such liquids are ideal for the study of the frequency dependence of specific heats, since the frequency dependence is large in magnitude and occurs on a macroscopic time scale. Moreover, these liquids are Newtonian and rheologically simple. Apart from exploring their physical content, I examine the reasons and conditions for the applicability of the generalized heat conduction equation (sections 2 and 3). The question concerning a possible frequency dependence of the coefficient of thermal conductivity is clarified in section 4 using a simple model for the heat transport in a viscous liquid near the glass transition. The combined effect of heat conduction and relaxation on the spectra of entropy and density fluctuations is described in sections 6 and 7. The superposition of the Rayleigh line due to heat conduction and the Mountain line arising from structural relaxation in the quasielastic light scattering spectrum [36-41] is of particular interest (section 7). Section 5 and part of section 6 are largely pedagogical. Section 5 deals with the difference between the generalized heat conduction equation and the generalized diffusion equation, and with the dependence of the solutions of initial value problems for these equations on the history prior to the initial time. In section 6 we also comment on the way in which hydrodynamic transport coefficients and thermodynamic derivatives are replaced by frequency-dependent quantities in different expressions for the correlation functions of gencralized hydrodynamics. It is shown that Oxtoby's identification I14] of a frequency- and wavevector-depcndent thermal conductivity in a simple model for heat conduction and rclaxation is not supported by the ph~nomenological theory.

2. Assumptions and basic equations The theory of thermoviscoelasticity is a macroscopic description of viscous fluids. It therefore applies only if the spatial variation of a process is characterized by wavelengths much longer than the average intermolecular distance. Unlike the hydrodynamic description, however, the theory is not restricted to processes which are slow compared with the longest relaxation times of the fluid. The thermoviscoelastic theory is based on the assumption that the relaxation times of the fluid are split in two well separated groups. It is valid for all frequencies which are low compared to the group of the short relaxation rates. The restriction in frequency arises from the requirement that at least the modes of molecular motion which have the shortest relaxation times should be in a state of local equilibrium. The assumption is fulfilled for very viscous liquids, such as undercooled liquids near a glass transition, in which the

J. Jiickle / Heat conduction and relaxation in high-viscosity liquids

379

molecular motion may be decomposed in the fast vibrational and the slow diffusive part, the latter giving rise to structural relaxation. The fast relaxing vibrational modes transport heat and determine the temperature measured by a thermometer immersed in the liquid. It is necessary that the frequency of the macroscopic motion is sufficiently low for these modes to be in local equilibrium. Otherwise the relation between the heat current and the temperature would not be well defined. An average relaxation time for the slow diffusive motion is given by the relaxation time r, for shear stress relaxation, which is obtained from the shear viscosity r/ and the high-frequency shear modulus G(:c) as 7, = T I / G ( x ) .

(2.1)

As the viscosity of an undercooled liquid increases by many orders of magnitude when it is cooled towards its glass transition, a very broad gap develops between the relaxation times for the fast and the slow molecular motions. Very close to the glass transition (for T < 1.2 Tg, say) it is sometimes observed that a contribution of so-called secondary relaxations is split off the low-frequency part of the relaxation spectrum [15, 16]. The secondary relaxation is faster than the primary one, which follows eq. (2.1). In macroscopic measurements it is observed most clearly below the glass transition. The theory of thcrmoviscoelasticity is also valid in the frequency rangc of such secondary relaxation processes. We are using only thc linear form of the theory which applies to the case of small perturbations. As is well known, the time-dependent correlation functions for thermal equilibrium fluctuations can be derived from thc hncar theory, We are mainly interested in the response to longitudinal perturbations. The equations of motion of a thermoviscoelastic fluid derive from the linearized forms of the conservation laws for particle number, momentum and energy [17]. Sincc we are interested here only in slow processes of structural relaxation and heat conduction, the second of these equations reduces to the condition of mechanical equilibrium, which reads V, er,, - n,,V V~,, = O.

(2.2)

t . ~ . stress tensor, ~;vm~.lt t u ~ S U m ~,l ,, is ,h,. "'':~'~ is .L~ _,- a n :l~utl~Jlal~......... :~ p a r t ~ i"v c n u..t,y .t.~tHc negative pressure ( - P) and the shear stress o-*'1, which has trace zero. V~,, is an external potential (per particle). The other two equations are the continuity equations for the particle number density n and the entropy per mass s,

ti + n. div v = O, mn.T,~

+ div q = q..,.,t ,

(2.3) (2.4)

J. Jiickle I Heat conduction and retaxation in high-viscosity liquids

380

where q is the heat current density and the source term t~¢,, denotes the heat supply per time and volume from an external heat source. The material properties of a particular liquid enter through the constitutive equations [18] for pressure, shear stress, entropy (or temperature) and heat current density in terms of the present and past values of the particle nm,nber density n, the drift velocity v, the temperature T (or entropy s) and the temperature gradient VT. The constitutive equations derive from the behaviour of a material after sudden step-like changes of external parameters. Experimentally, e.g., the volume or enthalpy relaxation following a sudden change of the temperature or pressure can be measured [34, 35]. It should be emphasized that temperature here means the temperature pertaining to the fast vibrational molecular motions which, on the time scale of the thermoviscoelastic description, come to equilibrium instantaneously. On the level of a molecular description this temperature can be derived from the mean kinetic energy using the classical law of equipartition. As a result of the coupling to the slow processes of structural relaxation, this temperature relaxes towards the thermodynamic equilibrium temperature. The temperature T of the vibrational degrees of freedom is not a mere theoretical construct. Since the heat contact between the liquid and an external heat bath is established by the molecular vibrations (as by the phonons in the case of a crystalline solid), this temperature is actually measured by a thermometer immersed in the liquid. We first describe the response to an externally applied step-like perturbation. Choosing, for example, n and s as independent variables, we write for the reaction of the temperature following small changes ~n and 8s at time t' aT

[ ( ~), - ( ~),,]@v.,,(t- t')} an(,') os /-~

~, as /oJ

r.'(t-

(2.5)

Here aT~On and aT~as are the partial derivatives of T as a function of n and s. The high-frequency limit ( a x / a y ) ~ determines the instantaneous response, whereas ( a x / a y ) , is the thermodynamic equilibrium derivative. The derivatives of the temperature with respect to the entropy are given by

(Or

r,,

-~-s ) . - q , % )

_ ( as I -

r,, c,

"

(2.6)

where cv(O ) and cv(x) are the low- and high-frequency specific heats at constant volume, for which Cv(0) > c,,(~:)

(2.7)

J. Jilckle / Heat conduction and relaxation in high-viscosity liquids

381

holds. That cr(0 ) is appreciably larger than cv(~c) for many supercooled liquids of high viscosity may be inferred from the compilation of Gupta and Moynihan [31] of data for the discontinuities of the specific heat at constant pressure c e, the isothermal compressibiitiy Kr and the thermal expansion coefficient a at the glass transition: For the good glass formers B:O 3, (Ca(NO3)2) o ~.(KNO3) o, and poly(vinylacetate) the relative difference [cv(0) - cv(~)] Icy(oo) amounts to 0.38, 0.47 and 0.25, respectively. It has been pointed out by Zwanzig [32] that relaxation effects in heat conduction also arise from sheer viscoelasticity, where cv(0) = cr(~c) and a retarded constitutive equation like eq. (2.5) exists only for pressure and shear stress. However, the viscoelastic model treated by Zwanzig, which is contained in the present theory as a special case [cf. eq. (3.7a)], does not describe the behaviour of some typical glass-forming liquids, and probably does not apply to the materials used in the recent ac heat conduction experiments [4-7, 30]. Given the fact that the two specific heats cv(0 ) and cv(~) are different, the entropy term on the r.h.s, of eq. (2.5) describes the retarded response of the temperature after a sudden increase of the entropy from an initial rise by T o ?~s/cv(~ c) to the final increase given by T, 8s/cv(0). This behaviour is explained by the splitting of the internal degrees of freedom into the fast and slow modes mentioned above. The amount of heat energy per mass T~ bs is first supplied exclusively to the system of fast (vibrational) modes, from which part of the heat energy leaks to the slow modes. In the course of this relaxation process the measurable temperature T, which is determined by the fast modes, decreases. For the response of the entropy to a change of n and T and for the reaction of the pressure af/~er a change of n and s, or, alternatively, n and T, expressions similar to (2.5) are obtained. The expression for the shear stress o't~(r, t) produced by displacements u(r, t') at time t' is given by { 0u,(r, t') ~,u,(r, t') 2 } ~'~" t) = G(~c)qbc;,(t t') + 8,, div u(r, t') . o',~ tr, Ox~ Ox, 3 (2.8) The relation between the displacement field u(r, t') and the local density change ~n(r, t') follows from the iinearized form of the continuity equation and reads ~n(r, t') = -n,, div u(r, t ' ) .

'~'.9)

The relaxation from the instantaneous response to the response in equilibrium is described by the relaxation functions ~.,.(r), which are normalized by the condition ~ ( r = 0 ) = 1. These functions decay monotonically to zero at long times. As the viscosity of an undercooled liquid increases with cooling, the

382

J. Jiickh' I Heat conduction and rehlxation in high-vLwosity liquids

decay becomes markedly non-exponential. For most of the decay a relaxation function can usuaUy bc described by the empirical Kohlrausch formula

• (r) =exp[-(r/rq,(7"))~l,

O < 13 <.

!,

( 2. H~)

with a temperature-dependent relaxation time rq!(T) and a fractional exponent /3. The relaxation functions ~,,,,!., !.! ~ q~, and ~r.,, are related to shear and bulk viscosity rl and ~"by [cf. eq. (2.1)]

7qtG(~) =

r,

=

(2.11)

f dr ~,.(r), II

dT I!

The asymptotic behaviour at long times of both these functions therefore exhibits a long-time tail proportional to t -~ -' [19]. For high viscosity, however. these long-time tails arc very small, except for a contribution of the heat conduction mode to ~,,,(r). which is proportional to (eric v - l) z and therefore probably small, too. From t.q. (2.5) the general form of the retarded constitutive equations is obtained using Boltzmann's supcrposition principle for linear media. The expression for ST(t) as a functional of 8n(t') and 8s(t'), for example, reads

aT -- (r,,

+ [(r,,aT ).

aT

f

,, !

(2.13) The constitutive equation for the shear stress is obtained from ( _.8) ~ ' a~ t

£

cru'""(r, t) = G(:c) J ~ , ( t x

av,(r, t')

ax,

t')

+

av,(r, t')

ax,

2 } .~ ¢~,, div v(r. t' ) d r .

( 2.1 4)

where v = ti is the local drift velocity. The memory terms naturally have the form of Stieltjes integrals. The kernels of the Stieltjcs integrals arc monotoni-

J, ]ackl¢ : t l c a t ¢¢mdm'lttm and r~4axattc,m tn htgh.vt~vo~t~, hqutd~

~/~

cally decaying functions of the time argument. For h;lrmomc pcrturb:~t,on,, ~-c '"~ the con,,titutivc equation,, dcline complex frcqucncvodcp~:ndent J c ~ ~ lives like

t-.t~)

= where

q'(,o) = i dr e'"~Olr) denotes the Laplace transform of a relaxation function. In the |oilowmg wc u,,c the slightly shorter notation T,(m) for (JT/Os)~.,. The input parameters of generalized hydrodynam1~ arc the low- and high-frequency limits of the partial derivatives of P and T with respect to n and s and the high-frequenc)" shear modulus G(:~) together with the corrL.~pondmg relaxation [unctions. An additional input parameter is the c~ctficicnt ~ thermal conductivity h. If A is frequency independent, the heat current dcn,~it~ is given by (2.17

q = -AVT.

which is Fouricr's law. We argue that in a ~cr ~, x~.~cou,, hquld heat i,~ transported mostly by the diffusion of vibrational cncrg), c~cn ~h~-ugh a ma~or fraction of the specific heat is oi ,,.tructurad tmgm. Thereat, re. t~., a geod approximation, the thermal conductivity should have the short rc|axation, time of the fast vibrational degrees of freedom and be independent of frcqucnc} in the domain of validity of generalized hydrod)qaamics. The situation is similar to the case of glasses at low temperature, where the heat is predominantly carried by the phonons, although most of the specific heat is due to the nonpropagating tunneling excitations [20. 28];. In addition, the question arises whether a frequency dependence of the thermal conductivity would be at all consistent with retarded constitutive equations for temperature and entropy like eq. (?.!~). accordino~, to our interpretation of e~a.. (_,.I~). the heat suppl~_ to a portion of a liquid first goes into the system of the fast modes. This seem,, to require that the exchange of heat between different parts of a liquid. ~.c. heat conduction, must be due entirely to the fast (vibrational) modes. As stated above, in this case the thermal conductivity would be independent of frequency in the range of validity of the thermoviscoelastic equations. A modcI for heat conduction in very viscous liquids, to be described in section 4. shows. however, how retarded constitutive equations for entropy and temperature like .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

J. Jiickle I Heat conduction and relaxation in high-viscosity liquids

384

eq. (2.13) and a frequency dependence of the thermal conductivity can both be taken into account consistently in a generalized heat conduction equation. The validity of the argument for a frequency-independent thermal conductivity needs to be tested by comparison with experimental data. However, the question is not yet completely settled experimentally. Both the dc measurements of Sandberg et al. [27] for glycerol and the ac data of Birge [5] for glycerol and propylene glycol show features which may indicate the existence of a small frequency-dependent contribution to A. In the ac data of Dixon and Nagel [33] for o-terphenyi mixtures a frequency dependence of A is not apparent, but a small frequency-dependent portion of up to 5% cannot be excluded in view of the limited accuracy. Since the available experimental data do not rule out a weak frequency of the thermal conductivity, we also formulate the equations for heat conduction for this case. The retarded constitutive equation for the heat current density q is written by analogy with the retarded equations of state for the thermodynamic quantities. The general expression reads !

q(r,t)=-W{A(O)T(r.t)+[A(~c)-A(O)] f

~,('-

,')7"(r, t ' ) d , ' } . (2.18)

It contains the thermal conductivities A(0) and A(~) at low and high frequency and a relaxation function q~A(T). Again we write the memory term in the form of a Stieltjes integral. For ST(r, t')-e -'~' eq. (2.18) defines the frequencydependent thermal conductivity as A(w) = A(O) + [A(x) - A(O)l(-ito)q~(w).

(2.19)

We note that ip, the Coleman and Noll thermodynamic theory of materials with fading memory the heat current density is assumed to depend only on the current value of the temperature gradient, leading to a constant thermal conductivity in the linear limit. A retarded equation of state for the heat current density, in which a possible frequency dependence of the thermal conductivity is taken into account, was put forward before bv several authors FIN L*'J-

1")1 t~ i .

3. Generalized heat conduction equation The generalized heat conduction equation derives from the continuity equation (2.4) for the entropy per mass s and the condition of mechanical equilibrium (2.2), combined with the constitutive equations for s and P in

J. Jtickle / Heat conduction and relaxation in high-viscosity liquids

385

terms of T and n, for the shear stress tensor o"~'~ [eq. (2.14)] and for the heat current density q. We first consider the case of a frequency-independent thermal conductivity [eq. (2.17)]. As a result of thermal expansion an inhomogeneous distribution of the temperature creates a displacement field, which has to be calculated from the condition of mechanical equilibrium. In general, the displacement field depends on the boundary conditions. Here we treat only the case of an infinite medium and of a semi-infinite medium bounded by a plane, with the additional constraint that the equilibrium temperature To is maintained at infinity. We also assume that no external forces act on the liquid (V~ t = 0). The generalized heat conduction equation is derived in the following way. First we take the Fourier transform with respect to the time of the constitutive equation for the entropy change 8s(r, t) to obtain

s(r, to) = sir(to ) T(r, w) + Sl,,(~o)n(r, to),

(3.1)

where the frequency-depcqde:~t partial derivati-'es sir(to ) and st.(to ) arc defined like Tit(to ) in eq. (2.15). :Slmitarly. the condition of mechanical equilibrium yields

V,{Pj,,(w)n(r. to) + Pir(to)T(r, to)} + G ( z c ) ( - i t o ) ~ , ( w ) {Au,(r, co) + ~V div u(r. to)} : 0.

(3.2)

where dP(,(w) is the Laplace transforn, of the shear stress relaxation function [cf. eq. (2.16)]. (Note that P),,(¢o) here means the derivative of P at constant 7", not at constant s.) Using the relation between div u and 8n [eq. (2.9)] the last equation can be written in the form C, .r(to) grad div u(r. w) - G(zc)(-ito)~.(to) rot rot u(r. to) (3.3)

= Pit(to) grad T(r. to)

with the frequency-dependent isothermal longitudinal elastic constant defined by c,

:

+

.

3.4)

For the two types of boundary conditions mentioned above the solution of eq. (3.3) reads

PIT(w)

div u(r, to) - C~.r(to) T(r, ~o),

rot u(r, to) = 0.

(3.5)

386

J. Jiickle I Heat conduction and relaxation bz high-viscosity liquids

Inserting the expression for div u into (3.1), we find the relation between s(r, w) and T(r, w). The ratio between these quantities defines an effective

frequency-dependent specific heat for heat cnna,:r.:'m c,,(w), which is given by

cA,o)= T,,{siT(,o)- st"('°)&~('°)~/--~0}"

(3.6)

Using the relations c,,(,,,)= cv(~o)

sl.(w)P Ir(W) -

(3.7)

T.

pl.(W)

'

cv(,o) = T,,siT(,o),

(3.8)

and the definition K r ( w ) = noPin(w )

(3.9)

of a frequency-dependent isothermal bulk modulus, c,,(w) can be expressed as Kr(to) c,,(~,) = c, (,o) + [ c A , o ) - c~.(~,)] c,.T(,o) .

(3.10)

We note the viscoelastic limit of eq. (3.7), in which only the frequency dependence of K r is kept: ~,~¢o~,.. , Kr(O) cp (to) = c v + [Cp(0) - cv] Kr(w ) .

(3.7a)

This is the case proposed by Zwanzig [32] for the interpretation of ac heat conduction experiments near the glass transition. For the low- and highfrequency limits of eq. (3.10) we obtain the known results for a liquid,

(3.11)

c.(O)=c,(O). and for an isotropic solid [21],

c.(~) = c~.(~) + [cA~) - c~(~)] < c,.(0).

Ki(~:) Kr(x ) + ~ G(:c) (3.12)

Formula (3.10) can be viewed as an interpolation between these two limiting

J. Jiickle

/ Heat

conduction

and

relaxation

in high-viscosity

iiquid.~

387

forms of behaviour. The occurrence of the high-frequency shear modulus in exprcssion (3.12) is due to the shear componcnt of the displaccmcnt field caused by the thermal expansion. Going back to the time representation, we can write the total entropy variation ~s(r. t) including the effect of thermal expansion in the form of a retarded constitutive equation. We define a relaxation function q~,.r.,,(r) by its Fourier transform, writing in analogy to eq. (2.15) c,(o))

= cp(O)

+

(3.13)

-

It can bc shown that ~,.r.,,(o)) is an analytic function of ~0 except for poles on the negative imaginary axis. Therefore q~,.r.,,(r) is causal, that is zero for negative times r. We expect that this function is similar to the relaxation functions occurring in the retarded constitutive equations, which decrease monotonically from the initial value of one to zero at infinity. Inserting the expression t

T,, gs(t) = c,,(O) g T ( t ) + [c,~(~c) - ce(0)] f

q ) . , . . ( t - t')7"(t') dt'

(3.14)

x

into eq. (2.4), we arrive at the generalized heat conduction equation

I . ( t - t')-](r, t ' ) d t

Dr({})

(3.15)

mn.c,~(x) "

where we have defined a frequency-dependent thermal diffusivity by A D r ( ° ) ) - mn,c,r(o)) "

(3.16)

Note that the memory term of (3.15) contains the time derivative of q~, r ,r(r) • We finally turn to the case of a dispersive thermal conductivity. Inserting o" "vVn r o~e' ~ ° s i. n. .n. t") X . . . . .I R'~Y f. t ~ . r. . t .h o. . . h. o.a r. . .c .' t ~. r. r o. n. l . . de_n_sily . . . . . . . .instead of Fourier's law leG. . . (2.17)]. the generalized heat conduction equation is obtained as Ot

c,~(x)

mn,,c,,(~c)

1

qb~.r.,T(t-- t') l'(r, t') dr'

f

A(o) f

mn,,c,r(zc )

t)

q~a(t - t') A 7"(r. t') dt' = mn,,c,~(x) "

(3.i7)

388

J. Jiickle I Heat conduction and relaxation in high.viscosity liquids

A similar equation, in which the second memory term is expressed as a Riemann integral over ~T(r, t') rather than a Sticltjes integral, has been proposed by Nunziato [12]. We shall derive an equation of the same mathematical form from a model in the next section.

4. A simple model for a frequency-dependent thermal conductivity The model is shown schematically in fig. 1. It is analogous to a two-state model for the diffusion of particles [29]. It is assumed that heat is carried mostly by the vibrational modes (or "'phonons") with diffusivity D t , which also couple to external heat baths or thermometers. The diffusivity D, of the heat stored in the configurational degrces of freedom, on the other hand, is expected to be relatively small for a viscous liquid near the glass transition (see section 2). The heat exchange between phonons and "configurational modes", characterized by a relaxation rate 3', allows the configurational energy to participate in heat conduction even if the diffusivity D 2 is zero. In this latter case the model is similar to a model proposed by Oxtoby [14], in which the configurational modes are described as localized two-state systems, as are observed in glasses at low temperatures. In terms of the same model, a non-zero D~ could arise from interactions, perhaps of elastic origin, between neighbouring two-state systems [28]. If the diffusivity D, of the configurational energy is not zero, a generalized heat conduction equation similar to eq. (3.17) correspondiog to a frequency-dependent thcrmal conductivity is obtained. In addition, however, the equation to be derived contains a term arising from a wavevector dependence of the specific heat. This result is derived under the condition that the spatial temperature variation is of sufficiently long wavelength. For shorter wavelengths additional terms containing spatial derivativcs of fourth and higher order occur in the generalized heat conduction cquation. If T~(r, t) and T~_(r, t) denote thc temperature of the phonons and of the

external heat bath "off'{ or I "or1" i

phonons (c,,DI)

T~ (£,t)

configu.r.ahonal modes (c2. D2 )

T2 (~' t )

Fig. 1. Schcmatic representation of the model for heat transport m a ver~ xiscou~ hquid. 3' denotes the relaxation rate for the heat exchange between the configuratnonal and ~lbrational modes The couphng of the latter to an external heat bath may bc switched on or off.

J. Jackle / Heat conduction and relaxation in high-viscostty liquids

389

configurational modes, the equation describing the heat exchange between the two subsystems reads 7"2- DzAT2 = -3'(T2-

(4.1)

T,).

With specific heats per v o l u m e c t and c 2 and heat current densities qt = - c l D t V T , ,

(4.2)

q2 = - c 2D 2~ TT 2

the continuity equation for the entropy (2.4) becomes ct(7" u - D t ATI) + c,(T 2 - D, AT2) : ilcx,/mn,,.

(4.3)

We first calculate the solution of (4.1) and (4.3) for plane-wave like pcrturbation t ~ x , ~ e "~'r-''~, to obtain the expression for the frequency-dependent specific heat c(to) and thermal conductivity A(to). Eliminating T 2 we obtain the following relation between the amplitudes T~ and t~c~t:

-ioJTl(cl

+ c2 - i w

+ 3' + D2 k2

)

+ kz~'t

(

_)

CtO1 + c2D~' -ito + 3' + D-,k 2

q~2'~ t

(4.4)

mtz~

For arbitrary k values eq. (4.4) must be interpreted in tcrm', of a frequencya n d wavevector-dependent specific heat and thermal conductivity. Neglecting D z k 2 compared with y in the denominators in the limit k---> 0, we identify a wavevector-independent but frequency-dependent specific heat c(to) and thermal conductivity A(to) as

Y

(4.5)

c(to) = c l + c 2 - i t o + y

and

3/

A(to)=ciO t + c 2 D 2 - i o 9 + 3"

(4.6)

The low- and high-frequency limits of A(to) are given by A(0) = c t D t

+ c2D 2 ,

A(°°) = c t D I •

(4.7)

If the configurational modes do not carry any heat themselves ( 0 2 -----0), the

390

J. J~ickle I Heat conduction and relaxation in high-viscosity hqaids

thermal conductivity is not affected by these modes and only due to the phonons. In the time representation, the expressions for the local variation of the entropy and the heat current density which correspond to (4.5) and (4.6) follow eqs. (3.14) and (2.18). In both cases the same relaxation functions is obtained, which is given by ~ . r ( T ) = q~a(z) = e x p ( - y r ) ,

z I>0.

(4.8)

In eq. (4.4), however, we cannot neglect the term D2k 2 in the denominators altogether. Expansion of the wavevector-dependent part of the specific heat term to lowest order in (D2k2/y) yields a term , it.-/)']/ 7'1 c2D2k" ( - i w + 3,) 2 '

(4.9)

which differs from the contribution of the frequency-dependent part of A to eq. (4.4) only by a factor of i w / ( - i w + y), which need not be small compared to unity. Taking the frequency dependence of the thermal conductivity into account, therefore, we must not neglect the wavevector dependence of the specific heat. In other words, in deriving a generalized heat conduction equation for our model, a non-local relation between entropy and temperature needs to be used. Expanding the first term on the l.h.s, of eq. 1,4.4) to ~(D2k2/'y), we obtain for the non-local part of the specific heat c ° " ( k , ,o) = - c 2

D~k Zy

(4.10)

( - i w + y)2 •

The expression for the non-local contribution to the change of the entropy ~s(r, t) reads t

8s" I (r, t)

= c 2 ( D 2 A/y){~Tl(r,

t) -

f o,t_t,T, r t, t}

(4.11)

with ~,(r) = ( 1 + yr) e -v" .

(4.12)

The corresponding contribution to the l.h.s, of the generalized heat conduction equation is given by t

g" ' (r, t) /c I = -(c2D2/cl y ) A f (b(t -- t')7"l(r, t ' ) d t ' ,

(4.13)

J. Jdckle 1 Heat conduction and relaxation in high-viscosity liquids

391

where

(4.14)

6(r) = - y % e-".

Apart from the contribution (4.13), the generalized heat conduction equation for the phonon temperature, which derives from eq. (4.4), has the form of eq. (3.17). The complete equation reads

7",(r, t) -

(

D t + D , --:"

aTl(r,

c,/

I

t) + --= y

Cl

C I

e-'"-"'J',(r,

t')dt'

_:~

t

+ D , - C, - f e- -re-,") ( l + y ( t - ct

t' ))AT',(r, t' ) d t ' - - -

q,,~T

rnnoc~

(4.15)

The associated expression for the heat current density, which follows from (4.6). is given by f

q(r, t) = - ( c , D ,

+ c2D2)VTl(r, t) + c , D 2 V f e - ' " - " ' 7 " , ( r , t') dt' . -x

(4.16)

It should be noted that the last term on the l.h.s, of eq. (4.15) consists of two different contributions: one comes from the frequency-dependent part of the thermal conductivity [cf. eq. (4.16)], thc other is duc to the non-local part of the entropy change |cf. eq. (4.13)]. The second term is missing in the phenomenological equation (3.17). For an ac heat conduction experiment with T~(r, t) = Re{ Tt(r, co) e -''°t } the generalized heat conduction equation can be written m the form [-io~ + Dr(o., ) AlT,(r, oJ) = c)c,,(r, ~o)/,nnoc(o))

(4.17)

with an apparent thermal diffusivity (4.18)

b,(~o) = ~(o,)/c(,o) related to an apparent frequency-dependent thermal conductivity A(¢o) = cID 1 + c2O 2 -io) + 3'

"

(4.19)

The true thermal conductivity A(ca). which determines the ac heat current

J. Jgtckle ,t Heat conduction and relaxation in high-viscosity liquids

392

density

q(r, to) = - A(to)VTt(r, o ) ,

(4.20)

is given by (4.6). Eq. (4.14) can be derived from eqs. (4.1) and (4.3) directly without using Fourier transforms in the following way. We assume that the spatial variation of T~ and T 2 is slow so that it is sufficient to treat the term D 2 AT 2 in eq. (4.1) to first order in a perturbation expansion. This leads to the following expression for T2(r, t) in terms of Tl(r, t): t

T2(r. t) = y f dt' e-*"-C)T~(r, t') + y D 2 A f dt' (t - t') e-~'~'-"~Tt (r, t') + O(D 2 A/y)2TI . Transforming both terms in (4.21) to Stieltjes integrals and inserting them into eq. (4.3), one directly obtains eq. (4.15). The perturbation expansion in powers of (D 2 A / y ) can be carried to higher order. To ~ ( D 2 A/y)" one obtains the following additional terms on the l.h.s. of eq. (4.15):

(c2/c I y )(O 2 A)2{ - T,(r, t) + i e--~"'-c'(1 + y ( , - / ' ) +

~y2(t-t')2)l",(r,t')dt'}.

(4.22)

5. Properties of the generalized heat conduction equation and comparison with the generalized diffusion equation The ordinary heat conduction equation is mathematically equivalent to the diffusion equation. Surperisingly , .t. ,. ~, for viscoelastic t,,c. . .~. c. .,.~ a . ,:__., L c u equations _. vanu liquids are not. Here we consider the generalized diffusion equation for the entropy, which is mathematically equivalent to that for the concentration in binary liquid mixtures [22-24]. We der;ve the generalized diffusion equation for the entropy in the same way as the heat conduction equation. For the infinite or semi-infinite medium in which the perturbation vanishes at infinity we can eliminate the strain field as before. We only need to solve eq. (3.14) for the temperature ~Tin terms of ~s.

J. Jiickle / Heat conduction and relaxation in high-viscosity liquids

393

Defining a relaxation function ~r,s.~ by writing

1

1

1]

[1

%(0) (-i°~) ~r":*(~°) '

(5.1)

we obtain t

= ce(O----~ + %(~)

%(0)

q~r.~.,~(t-t')~(t')dt'.

(5.2)

For a constant thermal conductivity a this leads to the following generalized diffusion equation: t g~

( ~ t - Or(0)A)s(r, , ) - - [ O r ( ~ ) - Or(0) ] J q~r..,.,,(t-t')A.qr, t') dt' ext

-

(5.3)

mno----~o.

The frequency-dependent thermal diffusivity Dr(tO ) is defined in eq. (3.16). We note that partial differential equations for the temperature and the entropy can be derived from the generalized heat conduction and diffusion equation if the relaxation functions ~r.s:,(~') and q0~.r.,(r) are exponentials [231, q~r.,.~ (r)= e-'" ,

q~,r,~(r) = e ~'-

(5.4)

The ratio of the relaxation rates is, according to cqs. (3.13) and (5.1), given by

(5.5) For both temperature and entropy the same differential equation

Ls(r, t) = 0 ,

LT(r, t) = 0

(5.6)

holds with a partial differential operator L which reads L=

O2 3 ~ + ( y - Or(~ ) A) -~ - yOr(0 ) A. 0t"

(5.7)

It is assumed in eqs. (5.6) that the coupling to external heat baths is switched off. For both equations initial values aT(r, 0) and 7"(r, 0) or ~s(r, 0) and g(r, 0) uniquely deteJmine the solution for times t t>0. However, the connection

J. Jiickle / Heat conduction and relaxation in high-viscosity liquids

394

between these initial values and the history of temperature or entropy prior to t = 0 is different. The connection is given by the eqs. (3.15) or (5.3) takcn at t = 0. In this way the difference between the two types of equations reappears. We now comment on the difference between the generalized heat conduction and diffusion equation and illustrate its physical origin and consequences. The memory terms in both equations (3.15) and (5.3) contain a factor [ D r ( ~ ) Dr(0)], which is a measure of the dispersion of the thermal diffusivity, or the specific heat. The Laplacian in the memory term of (5.3) indicates that the memory is coupled to the heat diffusion mode. In eq. (3.15), on the other hand, the memory may be conceived as arising from coupling to the relaxation modes, since the memory term includes the time derivative of the relaxation function qbs.r:,~('r) and is therefore proportional to an average relaxation rate. The absence of the Laplacian in the memory term of (3.15) also implies that the temperature can relax even under spatially homogeneous conditions, which is not possible for the entropy. The reason is that the temperature T, which is the temperature of the thermally excited vibration modes, can relax towards the temperature of the structural degrees of freedom, whereas s, which denotes the sum of vibrational and structural entropy per mass, is conserved in the linear approximation. As shown by eq. (3.17), the case of a frequencydependent thermal conductivity leads to a hybrid form of the generalized heat conduction equation for the temperature, in which both types of memory terms occur simultaneously. The solution of the generalized heat conduction and diffusion equation in the absence of the source term (q~t = 0) is unique only for given boundary and/or initial conditions. In the case of an initial value problem one has to bear in mind that the retarded equations of state (3.14) and (5.2) are valid at all times, although eqs. (3.15) and (5.3) without source term do not hold at times t < t o (we assume t o = 0 further on) during which the initial condition is prepared by coupling to external heat baths. Therefore the memory expressed by the integrals in these equations extends to times t' < 0, even though the equations only hold for t > 0 . As a consequence, the equations obtained for t > 0 generally depend on the way in which the initial condition is prepared. Only when the initial entropy or temperature perturbation is prepared adiabatically (i.e. infinitely slowly), the memory for times prior to t = 0 does not contribute since 41

f

qb(t - t ' ) A ( t ' ) dt'----~0

for x(t) = x(O) e"'

with r/---> + 0 .

(5.8)

We remark that this last result holds because the memory terms have the form of Stieltjes integrals. If one uses, for the generalized diffusion equation for the

J. Jtickle / Heat conduction and relaxation in high-viscosity liquids

395

entropy for example, the equivalent form /

r

\

L

t - O:(~.) a ! s ( r , t) - [ O F ( = ) \Ot .

-

-

OT(0~ l

cbr..,.~(t - t') As(r, t') dt'

il¢xt(r, t) mn~ T~,

(5.9)

which follows from (5.3) by partial integration, the preparation period t ' < 0 contributes to the memory integral even for an adiabatically prepared initial condition, since qP ¢-

J +(t-t')x(r)dt'---~-x(O)dp(t)

forx(t)=x(O)e"

with 7/--->+0.

(5.1o) The difference between the two generalized heat conduction equations (3.15) and (5.3) can be illustrated by considering a pure initial value problem with plane-wave-like initial condition (5.11)

~s(r, t = O) ~ e '~'r ,

and similarly for ~T. For the special case of exponential relaxation functions (5.4) the solution of these initial value problems can be obtained for arbitrary preparation of the initial condition using the solutions of the analogous equations for the diffusion of particles in a binary liquid mixture [23]. In this case the concentration and the chemical potential are the analogues of the entropy and the temperature in the heat conduction problem. In the limit of very slow structural relaxation and for adiabatically prepared initial conditions, for example, the solutions for ~T(r, t) and ~s(r, t) in leading order are given by (5.12)

~T(r, t) = 8T(r, O) e -°'(~k'' , {Or(O______~) -Dr(~)~'-, [ ~s(r, t) = ~s(r, O) DT(~¢) e + 1

r.,,(t)}

(5.13)

These expressions are even valid for non-exponential relaxation functions. The difference between the solutions (5.12) and (5.13) for bT and ~s reflects the fact that under the conditions assumed the imprint of the initial condition in the structure relaxes very slowly in proportion to the relaxation function ~ , r , , ( t ) as described by the second term in (5.13), whereas the initial modulation of the temperature of the thermal vibrations decays relatively fast via the heat conduction process.

J. Jiickle / Heat conduction and relaxation in high-viscosity liquids

396

Let us now consider the alternative case of a pure boundary value problem with no initial condition for finite times, which describes a situation in which heat baths (or heaters ) are coupled only to the boundary of the liquid. Such a situation occurs in the ac heat conduction experiments [4-7]. In this case eqs. (3.15) and (5.3) hold at all times - : ¢ < t < +:~. It is easy to show that under such conditions the two equations are, in fact, equivalent. Taking the Fourier transform of eq. (5.3) with respect to time one obtains, using the definitions (3.16) and (5.1), (-iw -

Dr(to ) A)s(r.

to) = O.

(5.14)

Eq. (3.15) yields the equivalent equation Dr(~)[

A]T(r, to)=0

-ito

for the temporal Fourier transform

(5.15)

T(r, w)

of

T(r, t).

6. The correlation function for the entropy fluctuations

We turn now to the spectrum S,.,(k, w) of the entropy fluctuations in thermal equilibrium. We first consider the case of a constant thermal conductivity A. The spectrum of the entropy fluctuations can be derived from the generalized diffusion equation (5.3) by solving the initial value problem (5.11) for adiabatic preparation of an initial sinusoidal entropy modulation of wavevector k. Taking the Laplace transform of (5.3) and averaging over the initial values of sk(O), one obtains the Laplace transform of the correlation function for the entropy fluctuations as 7J¢

S~.s(k, to) = f dt e'°"( s~(t)s k(O))

(Is, l")

(6 I~

Dr(0)k"

-iw + 1 + k2[OT(

) -- D (0)lqS ,

where



=

ks%(O)/mn o

(6.2)

is the long-wavelength limit of the intensity of entropy fluctuations of wavector

J. Jiickle / Heat conducuon and relaxation in high-viscosity liquids

397

k. The fluctuation spectrum S,.,(k, to) is given by

Ss.~(k, to)= ERe S~.~(k, to) .

(6.3)

It is interesting to see that the memory term of eq. (5.3) leads to the replacement of the hydrodynamic thermal diffusivity Dr(O ) by a peculiar toand k-dependent expression, rather than by the frequency-dependent thermal diffusivity Dr(to ) defined in eq. (3.16). The expression replacing Dr(O) reads Dr(O)

(6.4)

1 + k " [ D r ( ~ ) - Dr(O)l$, ..... (to)

From a purely formal point of view it might be tempting to interpret the new expression (6.4) as a generalized frequency- and wavevector-dependent thermal diffusivity. However, there is no physical reason for such an interpretation. We mention that ill a paper on generalized hydrodynamics and specific-heat spectroscopy near glass transitions, Oxtoby [14] interpreted a related result in terms of a to- and k-dependent thermal conductivity A(k, co). For a model of a liquid which contains two-state systems relaxing at a rate rR 1. Oxtoby's A(k, co) stands for the following expression:

( h "~)2ctkZ

A(k, to) = A* +

c~

,

(6.5)

A+c~k "-

where A" is the hydrodynamic thermal conductivity, and c v and c~ denote the total specific heat of the liquid (at constant volume) and the specific heat of the two-state systems, respectively. The high-frequency specific heat is given by the difference

cv(x)=Cv-C,.

(6.6)

It is easily verified that the r.h.s, of (6.5) can also be written in the form of expression (6.4) as A*

1 + k2[Dr(~)- Dr(O)l~r.,.,,(o~) with

(6.7)

398

J. Jiickle / Heat conduction attd relaxation in high-viscosiO' liquids

Dr(0 ) = ~tnnoC V . =

1

Dr(z¢) = mnocv(X) . -/'

Cv

"

(6.8)

(6.9)

Therefore, Oxtoby's result fits perfectly well in the phenomenological framework of the generalized diffusion equation that derives from a frequencydependent specific heat and a constant thermal conductivity. The . -mcept of a frequency- and wavevector-dependent thermal conductivity is not unreasonable in principle (cf. the model presented in section 4). However, consistency would require that other quantities (like the specific heats or the viscosities) are frequency and wavevector dependent as well. The phenomenologicai theory of thermoviscoelasticity presented here is consistent in the sense that all second derivatives of thermodynamic potentials are treated in the same way as frequency-dependent quantities. The problem of identification in expressions of generalized hydrodynamics is peculiar to the Laplace transform of time-dependent correlation functions. In the dynamical susceptibilities Xx.,.(k, z), on the other hand, the replacement of hydrodynamic transport coefficients and thermodynamic derivatives by frequency-dependent quantities is more straightforward. The dynamical susceptibilities are related to the Laplace transforms of the time-dependent correlation functions by X~.y(k, to) - X~.~ (k, O) = ito/3g,.; (k, ~o) ,

(6.10)

where the static susceptibility is given by the thermodynamic fluctuation formula X,,.,.(k, O) =/3(x~,y_k)

(6.11)

and /3 = (ksTo)-'. Indeed, in the expression for the dynamical susceptibility Xs.s(k, to) for the entropy per mass. which is obtained from (6.1) using (6.10) and t6 11~ ~,-,a which reads (h/m'-n~To)k ~ Xs.~(k, to)= -io~ + DT(to)k 2 "

(6.12)

the hydrodynamic thermal diffusivity is simply replaced by the frequencydependent one. Using the dynamiral. .suv-',-,-v,,~.,,-~ ~--,;~,a;t,,.~ is also |Itc simplest way to include the

J. Jackle / Heat conduction and relaxation in high-viscosity liquids

399

effect of a frequency dependence of the thermal conductivity on the spectrum S,.s(k. ~o) of the entropy fluctuations. We simply have to replace the constant ,~ by the frequency-dependent one of eq. (2.19), and Dr(~O ) by lt(oJ)/mnoc,,(~).

7. Quasielastic light scattering Contrary to the fluctuations of the entropy, the spectrum S,.,,(k, to) of the density fluctuations can be measured by scattering experiments. For the long wavelengths to which our theory applies, photon correlation spectroscopy is the appropriate technique [36, 39-41]. The spectrum of thermal density fluctuations in a viscous liquid has been described previously using the concept of an internal relaxing variable [37, 38]. We choose to calculatc the dynamic susceptibility X,,,(k. co) for the particle number density, from which the spectrum of density fluctuations is obtained as

S,,.,,(k, to) = (2keT/to) Im X,,.,,(k, to).

(7.1)

The expression for the dynamic susceptibility X,..,(k. to) is found by solving eqs. (2.2). (2.4), (2.9) together with the constitutivc cquations (2.13) and (2.17) for the variation Gn(r. t) of the density in the presence of a plane-wavelike perturbation Vcxt(r. t) ~ e '(k'r-"'~ in an infinite medium. The thermal source term q~,t in eq. (2.4) is dropped. For simplicity, we consider only the case of a constant thermal conductivity. According to the theory of linear response [25]. X,,.,(k. to) is given by

Gn(r,t) X,.,,(k, to)= - Ve,t(r. t ) .

(7.2)

The result of this calculation can be written as "1

n,, K,(to) - Kr(to) "

X""(k" to)= C, .r( to) - ito

(7.3)

,n

- i w + D.(w)k-

Here D0(to ) is a frequency-dependent thermal diffusivity in which the effect of thermal expansion on heat conduction is not taken into account. It is defined by

Do(w ) - mnocv(to ) •

(7.4)

The frequency-dependent isothermal longitudinal modulus C~.r(~O) is defined

400

1. J~ckle I Heat conduction and relaxation in high-viscosity liquids

in eq. (3.4). The difference between the frequency-dependent adiabatic and isothermal bulk modulus can be calculated from the constitutive equations and is given by

Ks(,,,)- KT(.,)=-no

(7.5)

sl (,o )

Comparison with eq. (3.7) yields the relation = cp(o,)/cv(

o)

(7.6)

,

which is the extension of a well-known thermodynamic formula to frequencydependent quantities. Apparently, all thermodynamic relations between second derivatives of thermodynamic potentials are also valid for the corresponding frequency-dependent quantities [18]. According to (7.6), the factor (Ks(to) - Kr(tO)), which determines the coupling of the heat conduction mode to the density fluctuations, is proportional to (ce(to) - c v ( t o ) ) , and therefore proportional to the square of the frequency-dependent coefficient of thermal expansion t~(o.,). At this point it should be mentioned that Nunziato [26] has considered previously the frequency dependence of Griineisen parameters for viscoelastic polymers. Using (7.6) and eqs. (3.10) and (3.16) for G(to) and Dr(to ) , expression (7.3) can be brought in the alternative form

Xn.n(k, oJ) - C,--~o)

c,,(to) +

1

G(to )

-ito + D r ( t o ) k 2 "

(7.7)

For a given wavevector k, expressions (7.3) and (7.7) are only valid at trequencies much lower than the frequency of longitudinal sound waves of wavevector k, since mechanical equilibrium was assumed in their derivation. Since the frequency of the longitudinal sound waves is determined by the frequency-dependent adiabatic longitudinal sound velocity G.~(to), this condition can be written as

Itol c, s(o )k.

(7.8)

Consequently, the spectrum of density fluctuations which follows from expressions (7.3) or (7.7) only contains the contribution of the slow processes of heat conduction and relaxation~ but not the Briilouin lines due to the longitudinal sound waves. An obvious mathematical consequence of this limitation is that expression (7.3) does not fulfil the Kramers-Kronig relations. The complete result for the dynamic susceptibility valid for all frcqucncics would be obtained

J. Jack& I Heat conduction and relaxation in high-viscosity liqutds

401

by multiplying numerator and denominator of expression (7.3) by a factor (kZfmno) and then adding a term (-to") to the latter. The qualitative features of the spectrum of density fluctuations are made clear by considering three limiting cases for the magnitude of the average relaxation rate "~ relative to the sound wave frequency c~.s(to)k and the frequency Dr(to)k 2 of the heat conduction mode. The first case (7.9) corresponds to the hydrodynamic limit, in which the contribution of the relaxation processes is negligible. The heat conduction mode contributes the Rayleigh line, of Lorentzian iineshape and relative intensity (1 - cv(O)/ce(O)), to the density fluctuation spectrum. The expression for the Rayleigh line in this case reads

tR) 2kaTn2o [ Sn.n(k, to)= Kr(O ) 1

cv(O)] Dr(O)k 2 cp(O) to 2 + (Dr(O)k2) 2 "

(7.10)

In the intermediate case

Dr(O)k 2 ~ ~ ~ c,.,(~c)k ,

(7.11)

the quasielastic spectrum consists of the superposition of thc Raylcigh line described by (7.10) and a broader Mountain line due to relaxation, for which the expression S,,.n(k, (M) to)

-

-

2k - - nBT o l m ,(

1

,o

c,.[(,o)

)

(7.12)

follows from (7.3). C,.,(o) is the adiabatic longitudinal modulus obtained from (3.4) by replacing the isothermal by the adiabatic bulk modulus. As in this case the relative intensity of the Briilouin lines is reduced from Kr(O)/Ks(O ) to KT(O)/C~,,(~ ), the relative intensity of the relaxational contribution is enhanced correspondingly, and given by

(1

KT(O) K,(O)

1 )

C,,[(~)

"

(7.13)

For very viscous liquids the relaxation is markedly non-exponential. Consequently the lineshape of the Mountain line is strongly non-Lorentzian. In the third case of slow relaxation "~<~ Dr(~)k 2 ,

(7.14)

402

J. liickle / Heat conduction and relaxation in high-viscosity liquids

a narrow non-Lorentzian Mountain line is superimposed on a broader Lorentzian Rayleigh line. The expression for the Rayleigh line in this "glassy limit" reads tR) 2kaTno[1 S,,.,(k, to) = ~

Cv(~) ] to2 Dr(~)k2 ~ + (Or(~)k2)2 •

(7.15)

As a result of the inequalities c,,(~) < ce(O) and Dr(~ ) > Dr(0 ), the Rayleigh line now is broader than in the other two cases. At the same time it is also considerably weaker, since the coefficient of thermal expansion at high frequencies is that of an elastic solid, and therefore much smaller than at low frequencies. The relative intensity of the Rayleigh line in the glassy limit (7.14) amounts to Kr(0) [

1

Cv(~)] < 1

Cv(:~)

cA )"

(7.16)

Using again the data quoted by Gupta and Moynihan [31], we can calculate the upper limit given by the r.h.s, of (7.16) for liquid BzO 3, (Ca(NO3)2) 0 4" (KNO3)0 6 and poly(vinylacetate) at the glass transition. We find the values 0.005, 0.037 and 0.05, respectively. For comparison, the values of the hydrodynamic intensity ratio (1 - cv(O )/ce(O)) are 0.06, 0.10 and 0.14. Therefore in the glassy limit the quasielastic part of the spectrum is by far dominated by the Mountain line [39, 41], which is described by eq. (7.12) with C~.~ replaced by C~.r. The relative intensity of the Mountain line is given by [18]

1 - Kr(O)/C,.r(~ ) .

(7.17)

For frequencies intermediate between the last two cases (7.11) and (7.14) the density fluctuations due to relaxation and heat conduction interfere in a complicated manner. 8. Concluding remarks In conclusion, we wish to point out two results of our phenomenological theory which call for new experiments. The first concerns the frequency dependence of the coefficient of thermal conductivity. In section 4 a simple model was described which combines a frequency dependence of the thermal conductivity with the constitutive equations of thermoviscoelasticity. The thermal conductivity is frequency dependent if a mechanism for direct transport of heat by the configurational modes exists which does not rely on the phonons

J. Jiickle / Heat conducuon and relaxation in high-viscosity liquids

403

(see fig. 1). However, together with a frequency-dependent part of the thermal conductivity a frequency- and wavevector-dependent part of the specific heat appears in the model. Both parts combine to give an apparent frequencydependent thermal conductivity in the generalized heat conduction equation. Although the frequency-dependent part of the thermal conductivity is expected to be relatively small [5, 27, 3], it may not be altogether negligible (of the order of a few percent). Experimental efforts to determine the actual magnitude of the effect or more accurate upper limits of its magnitude should be encouraged. It should be possible to determine the frequency-dependent specific heat c,(ta) and true thermal conductivity A(to) separately by a combination of different ac heat conduction experiments. For example, the apparent thermal diffusivity b~ (to) = A(oJ)/mn,c,,(to) can be obtained by measuring the propagation of a temperature oscillation in planar geometry across the liquid. The product A"(to)-c,r(to)/,~(to ) can be measured using the method of NageL and Birge [4, 5]. The second suggestion concerns the spectrum of quasielastic light scattering (section 7). It would be interesting to observe the hybridization of the Lorentzian Rayleigh line due to the heat conduction mode with the non-Lorentzian Mountain line arising from the structural relaxation modes, which occurs in very viscous liquids under conditions intermediate between case (7.11) and case (7.14).

Acknowledgement I wish to acknowledge the hospitality of the Rutherford-Appleton Laboratory, England, in summer 1988, when part of this paper was written.

References [1] G.W. Pierce, Proc. Am. Acad. Boston 60 (1925) 271. [2] H.O. Kneser, Handbuch der Physik, vol. 11/1, S. Flugge, ed. (Springer. Berlin, 1961), p. 129. [3] P.G. Oblad and R.F. Newton, J, Am. Chem. Soc. 59 (1937) 2495. [4] N O. Birge and S.R. Nagel, Phys. Rev. Lett. 54 (1985) 2674. [5] N O. Birge, Phys. Rev. B 34 (1986) 1631. [61 N.O. Birge, Y.H. Jeor'g and S.R Nagel, Ann. NY Acad Sct. 484 (1985) 10l. [7] T. Christensen, J. Phys. (Paris) 46 (1985) C8-635. [8l B.D. Coleman and W. Noil, Arch. Rat. Mech. Anal. 13 (1963) 167. [9] B.D. Coleman, Arch. Rat. Mech Anal. 17 (1964) 1. [10] B.D. Coleman and M.E. Gurtin, Z. Angew. Math. Phys. 18 (1967) 199. [11] M.E. Gurtin and A.C. Pipkin, Arch. Rat. Mech. Anal. 31 (1968) 113. [12] J.W. Nunziato, Quart. Appl. Math. 29 (1971) 187. [13] J. Jackle, Rep. Progr. Phys. 49 (1986) 171.

404

J. Jiickle t Heat conduction and relaxation in high-viscosity liqutds

[141 D.W. Oxtoby, J. Chem. Phys. 85 (1986) 1549. [15] G.P. Johari and M. Goldstein, J. Chem. Phys. 53 (1970) 2372; 55 (1971) 4245. [16] J. Wong and C.A. Angell, Glass Structure by Spectroscopy (Marcel Dekker, New York, 1976). [17] L.D. Landau and E.M. Lfschitz, Fluid Mechanics (Pergamon, Oxford, 1959), ch. V. [18] J. J~ickle, Z. Physik B 64 (1986) 41. [19] M.H. Ernst, E.H. Hauge and J.M,J. van Leeuwen, Phys. Lett. A 34 (1971) 419. [20] M.P. Zaitlin and A.C. Anderson, Phys. Rev. B 12 (1975) 4475. [21] L.D. Landau and E.M. Lifschitz, Theory of Elasticity (Pergamon, Oxford, 1970), p. 151. [22] J. J/iekle and H.L. Frisch, J. Polym. Sci. Polym. Phys. Ed. 23 (1985) 675. [23] J. J~ickle and H.L. Frisch, J. Chem. Phys. 85 (1986) 1621. [24] J. J/ickle and M. Pieroth, J. Phys.: Cond. Matt., submitted. [25] L.P. Kadanoff and P.C. Martin, Ann. Phys. (NY) 24 (1963) 419. [26] J.W. Nunziato and E.K. Walsh. J. Appl. Phys. 44 (1973) 1207. [27] O. Sandberg, P. Andersson and G. Backstrom, J. Phys. E: Sci. lnstrum. 10 (1977) 474. [28] M.A. Continentino, Solid State Commun. 32 (1979) 1193. [29] J.W. Haus and K.W. Kehr, Phys. Rep. 150 (1987) 263, sec. 3.4. [30] B. Biichner and P. Korpiun, Appl. Phys. B 43 (1987) 29. [31] P.K. Gupta and C.T. Moynihan, J. Chem. Phys. 65 (1976) 4136. [32] R. Zwanzig, J. Chem. Phys. 88 (1988) 5831. [33] P.K. Dixon and S.R. Nagel, Phys. Rev. Lett. 61 (1988) 341. [34] J.D. Ferry, Viscoelastic Properties of Polymers, 3rd Ed. (Wiley, New York, 1980). [35] C.T. Moynihan, P.B. Macedo, C.J. Montrose, P.K. Gupta, M.A. De Bolt, J.F. Dill, B.E. Dom, P.W. Drake, A.J. Easteal, P,B. Elterman. R.P. MoeUer, H. Sasabe and J.A. Wilder, Ann. NY Acad. Sci. 279 (1976) 15. [36] C. Demoulin, C.J. Montrose and N. Ostrowsky, Phys. Rev. A 9 (1974) 1740. [37] C. Allain-Demoulin, P. Lallemand and N. Ostrowsky, Mol. Ph~,s. 31 (1976) 581. [38] C. Ailain and P. Lallemand, J. Phys. (Paris) 40 (1q79) (,79. [39] C. Allain and P. Lallemand, J. Phys. (Paris) 40 (1979) 693. [40] G.D. Patterson, Adv. Polym. Sci. 48 (1983) 125. [41 i D.L. Sidebottom and C.M. Sorensen, Phys. Rev B 40 (1989) 461.