Theory of thermal conduction in low density materials

] O U R N A L OF

Journal of Non-Crystalline Solids 145 (1992) 191-195 North-Holland

NON-CRYSTALLINE SOLIDS

Theory of thermal conduction in low density materials P.G. K l e m e n s Department of Physics and Institute of Materials Science, University of Connecticut, Storrs, CT 06269-3046, USA

In rigid low density materials, there must be a continuous path for the transmission of force, and thus also a continuous path for heat flow. T h e elastic stress field due to an applied force arranges itself to minimize the strain energy. Heat currents and local temperature gradients similarly arrange themselves to minimize the rate of entropy production. In open structures, both strain energy and entropy production are concentrated in the regions of contact between the structural units which make up the material. This results in the same geometrical reductions of the effective thermal conductivity and the effective bulk modulus from the values for the fully dense solid. Thus, the thermal conductivity, Ae, relative to that of the ,2 ,2 where p is the density, and L, is the sound velocity, either in the low bulk material, A0, should be A e / / ~ 0 = PeCe/PObo, density material (suffix e) or the bulk solid (suffix 0). If the particles are very small or contain internal substructure, A0 would be appropriately reduced. Observations indicate that Ae / A 0 is larger at low density. Contacts between particles may have a larger effective area for heat conduction than for the transmission of force.

1. Introduction

The conduction of heat in low density insulations has been frequently discussed. It has components due to conduction in the pore gas, radiative conduction and conduction through the solid. It is often assumed that these components are additive. In the case of radiative conduction, this seems well satisfied; the role of the solid is to limit the photon mean free path by scattering and absorption. In the case of gas conduction and solid conduction, the additivity is not so obvious: gas conduction acts not only parallel to solid conduction, but can also form bridges between solid particles, so that these components act partially in parallel and partially in series. Another complication is that the atomic mean free path is often influenced by the geometry of the empty regions between the particles (Knudsen effect). In the case of a low density assembly of fibers, it is not obvious whether the limiting mean free path is determined by the dimensions of the voids or by the radius of the fibers. This paper is concerned with the effective conductivity through the solid particles, disregarding

gas conduction and the radiative component, such as would be the case at ordinary temperatures in evacuated insulations. The effective conductivity is then governed by the contacts between particles and by the bulk thermal conductivity of the solid itself. The main question to be discussed is the relation between the effective thermal conductivity and the effective elastic modulus as determined by the velocity of ultrasonic waves through the same particulate structure. An analogy will be drawn between the rate of entropy production due to heat conduction in a temperature gradient, and the strain energy due to a compressive strain. The original motivation was to find a justification for the semi-empirical relation A e / A 0 = C e U e / C o v 0 = PeVe/PoL; 0

(1)

proposed by the Wfirzburg group [1]. Here Ao, Co, v o and Po are the thermal conductivity, the specific heat per unit volume, the sound velocity and the density of the bulk material making up the particles, while the symbols with suffix e denote the corresponding effective properties of the particulate structure. It is assumed that C e is

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192

P.G. Klemens / Thermal conduction in low density materials

proportional to Pc, which should be true except at extremely low temperatures, where particles vibrate as a unit. It is also assumed that A0 is unaffected by the particle size. This is discussed below.

For a uniform elongation, e, in one direction, without elongation or contraction in the two perpendicular directions, as in the case of a longitudinal elastic wave,

W(r) = ½ ( K + 4 / x / 3 ) e 2.

(5a)

For a uniaxial strain without transverse stress, 2. M i n i m u m principles

W(r) = ~rYe 2 = ½9K/.L(3K -}- Ida) le2,

H e a t currents resulting from t e m p e r a t u r e gradients give rise to entropy production. In a small element which has conductivity, A, the rate of entropy production per unit volume due to a t e m p e r a t u r e gradient, VT, is given by

T 2 dS/dt = AVT 2,

(2)

where T is the absolute temperature. In an inhomogeneous medium, this equation can be used to define the effective thermal conductivity in terms of dS/dt and the average t e m p e r a t u r e gradient [2]. Further, dS/dt can be shown to be a minim u m with respect to small changes of the local t e m p e r a t u r e field from the actual t e m p e r a t u r e field. The gradient of the actual t e m p e r a t u r e field will, of course, depart from the average t e m p e r a t u r e gradient, due to variations in the conductivity. The condition that dS/dt be an extremum yields the same equation as the heat conduction equation. The fact that the extremum is a minimum can be established from simple cases. There is a second condition which must be satisfied: the heat current density J = - A r T must be continuous, or divJ = 0. In the case of an elastic strain field, the stresses and strains adjust themselves so as to minimize the total strain energy. If W(r) is the strain energy density as function of position r,

W= fW(r) dr

(3)

is a minimum, subject to the given boundary stresses. Also, in an isotropic medium W(r)

= gKA 1 2

+ ~1 e i ,

2

(4)

where K is the bulk modulus, A the dilatation, the shear modulus and e i one of the three components of shear strain.

(5b)

where Y is Young's modulus. In a composite structure, where K and /, vary with position, stresses and strains adjust themselves so as to minimize eq. (3), and again there is the additional requirement that the total forces over a cross-sectional area be conserved.

3. Structural model The material consists of solid particles, making contact with each other at contact 'points', each of which actually has a finite cross-sectional area, a c. The material forms a rigid structure, i.e., each particle has at least three contacts, but on average more. Also, the material can support a load, so that there is a continuous path for transmitted force. The individual particles are made of material with bulk and shear moduli K 0 and /*0, respectively, and with thermal conductivity, A0. In some cases, this may be smaller than the bulk thermal conductivity, either because of a small particle size, or because the particles have internal structure, as in the case of aerogels. Let P0 be the bulk density of the particle, and Pe the overall density of the material, which is less because of the spaces between the particles. Space can be divided up into rectangular blocks, so that each block, of length L and crosssectional area A0, contains one particle. The particle itself has length L, and a maximum cross-section A, which is less than A 0. On average, the volume of each particle is (pe/Po)Ao L. The contact 'points' between particles have a cross-sectional area, a¢, on average. In the presence of a t e m p e r a t u r e gradient, the t e m p e r a t u r e drop in each particle is concentrated

P.G. Klemens / Thermal conduction in low density materials

193

in the immediate vicinity of each contact point, _3/2 within a volume Vc ~ u~ . In the presence of an overall stress, the force between particles is transmitted at the contact points, so that the stresses are similarly concentrated in a volume, Vc, near the contact points. Continuity of heat flow and continuity of force require that the maximum current density and the maximum stress at the contacts are increased above their average value by a factor of order Ao/a~.

pend on pressure, but also on the previous history of the sample. Nilsson et al. [3] found a variation of the solid conductivity of an aerogel on compression a s 101"2. This density variation is much more rapid than predicted by eq. (10) and would have to be ascribed to a change in structure on compression. An increase in a c with density, almost as rapid as t) 2, does not seem very probable. Other structural changes, such as a change in the average number of contacts, cannot be ruled out.

4. Effective thermal conductivity

5. Effective elastic moduli

We use eq. (2) to define the effective thermal conductivity, Ae. The entropy generation occurs mainly at the contact points, where the average value of the t e m p e r a t u r e gradient is

Analogous but slightly different considerations can be used for the elastic moduli. It is here assumed that each particle consists of an elastically isotropic material, characterized by two moduli, such as the bulk modulus, K0, and the shear modulus, /x 0. The strain energy is concentrated in the vicinity of the contact points. Using the same notation as in the previous section, the strain energy per unit volume is

(gT)loc = cos O V T ( A o L ) l / 3 / a l c / 2 ,

(6)

where 0 is the angle between the contact normal and the direction of VT. If there are on average n c o n t a c t points on each particle, since on average cos20 = 1/3, and since each particle occupies an average volume A o L , the entropy generation per unit volume is T 2 dS/dt

= nAo(VT)~o~Vc/(AoL ) = (n/3)Ao(VT)2alc/2/(AoL)l/3

'

(7) so that the reduction in thermal conductivity becomes A e / A 0 --

( n / 3 ) a l c / 2 ( A o L ) -1/3

(8)

Now the fractional reduction in density is approximately Pe/PO = ( 7 / 2 4 ) A L / A o L ,

(9)

1

2

3/2

W = 7Yoelocna c

(AoL)

- 1

,

(11)

where I10 is the Young's modulus of the solid and eloc is the local value of the strain. The Young's modulus is chosen because at the contact point the local stress, Cqoc, is uniaxial, and O-loc = Y0eloc. If the medium is subjected to a hydrostatic stress, or, the effective bulk modulus is defined by 1

2

(12)

W = 7o" / K e.

Also the continuity of force requires a~OSoo = aoYoe~o ¢ = o - ( A o L ) 2/3.

(13)

Equating espressions (11) and (12) for W, and using eq. (13) to express qo~ in terms of o-, one finds

so that ge/g

Ae/A o ~ ( n / 3 ) ( a c / A ) l / 2 ( p e / P o ) l / 3 .

0 = alc/2/q,

l(Aog)-1/3

(14)

(10)

A p a r t from a form factor, the fractional reduction thus varies as the one-third power of the density. The form factor, varying as the square root of the average contact area, may well de-

The reduction in bulk modulus can now be compared with the reduction in thermal conductivity, given by the analogous eq. (8) he/a 0 = (Ke/K0) n2Ko/3Yo .

(15)

P.G. Klemens / Thermal conduction in low density materials

194

One should note that if one compares materials of different density due to differences in packing, n can be expected to increase with density. However, if one compares changes in thermal conductivity and bulk modulus during hydrostatic compaction of the same specimen, one may expect n to be unchanged or possibly to increase. Nilsson et al. [3] compared the changes in thermal conductivity with density with changes in Young's modulus. Well above the critical density which is the minimum for a continuous structure, oz they found that Ye CI Pe, Ae (X Pe, where r = 1.5c~. This would be in accord with eq. (15), if the effective shear modulus decreases more rapidly with decreasing density than the bulk modulus, since Ye-----3/Xe if /*e
6. Thermal conductivity and sound velocity

mal conductivity of low density materials, even though there is a tendency for Ae to be above the values given by eq. (17) at lowest densities [3]. The present considerations, leading to eq. (15), suggest that Ae/A ° = B K e / K o = BPeV e/polYo, 2 2

(18)

where B = n2Ko/3Yo ~ 1 depends on the structure. Equation (18) makes Ae lower than given by eq. (17) by a factor BVe/V o, so that eq. (17) should hold reasonably well at higher densities: when p~ approaches P0, t~ should approach v 0. It is only at lowest densities that one would expect systematic departures from eq. (17). This seems borne out for two powders and for two aerogel samples [3], but the departures are in the opposite sense to eq. (18). In order to account for this departure on the present model, one would have to postulate that the effective contact area, ac, is larger for heat conduction than for the transmission of force. This may be due to bridging by gas conduction, by residual moisture or by radiation. Wittwer [5] has pointed out that wherever the space between two solid surfaces is smaller than about 1 txm, radiation can 'tunnel' from one particle to the other. Depending on the curvature of the particles near the contact point, this could result in heat transfer over an enhanced contact area.

The lattice thermal conductivity can be expressed in the form

A = (1/3)Ct,l,

(16)

where C is the specific heat per unit volume, v the phonon velocity and l the phonon mean free path. From this, it had been suggested [1] that Ae/A 0 =

PeUe/POVO,

(17)

where v e is the sound velocity of the low density material and t~0 is that of the bulk material. This assumes that the phonon m e a n free path is the same in both cases, and that C is proportional to the amount of material per unit volume. While one can identify v 0 with the phonon velocity of the bulk material, the phonon velocity in each solid particle is also v 0, not re, so that eq. (17) does not really follow from eq. (16). Nevertheless, eq. (17) is quite successful in predicting the ther-

7. Particle size and phonon mean free path So far it has been assumed that the thermal conductivity in the immediate vicinity of the contact points is equal to A0, the thermal conductivity of the bulk material. This implies, of course, that the phonon mean free path, l, is well below the smallest linear dimension of the region which contributes to the overall thermal resistance. Thus, l must be not only smaller than the particle, but also smaller than the diameter of the contacts, i.e., l should be smaller than arc/2. Application of this criterion is made difficult not only by an imperfect knowledge of a c, but also by the fact that the phonon mean free path of crystalline solids varies with phonon frequency,

P.G. Klemens / Thermal conduction in low density materials

f, as I cxf-2, SO that the low frequency phonons make a relatively large contribution to A0 [4]. Thus, for fine oxide powders of 0.05 p~m diameter, as studied by Nilsson et al. [3], this must be considered. Using an equation (eq. (17) of ref. [4], the reduction in thermal conductivity within each grain at 300 K is roughly 30%. Since a c is probably an order of magnitude smaller, the reduction could be even larger, although that particular equation could then no longer be used. When the basic material is amorphous, such as in aerogels, the phonon mean free path is much shorter, of the order of 1 nm, and also almost independent of phonon frequency, except below about 30 K. One thus expects the thermal conductivity to be that of the bulk material, except that the particles have a substructure which could reduce the phonon mean free path, possibly by a factor 2. In any case, there is so much uncertainty in the theory giving Ae/A 0 that a reduction by even a factor 2 becomes a minor issue.

8. Summary In principle, eq. (10) allows an estimate of the effective thermal conductivity. It is based on the assumption that the thermal resistance arises at

195

the contact points between the particles. On this model, it must then also be sensitive to residual gas conduction as well as moisture at the contacts. Since the true nature of the contacts is usually not well known, this equation has limited usefulness. More promising is the relation between thermal conductivity and elastic wave velocity. The semi-empirical relation (17) obtained by the Wiirzburg group serves surprisingly well, except at lowest densities. The present attempt to relate thermal conductivity to the effective bulk velocity and thus to the elastic wave velocity may be of some use, but needs to be tested. It too was based on the idea that both thermal resistance and strain energy reside mainly at the contacts. This assumption needs further investigation.

References [1] O. Nilsson, G. Riischenp6hler, J. Grosz and J. Fricke, High Temp.-High Pressures 21 (1989) 267. [2] P.G. Klemens, Int. J. Thermophys. 10 (1989) 1213. [3] O. Nilsson, X. Lu and J. Fricke, in: Thermal Conductivity, Vol. 21 (Plenum, New York, 1990) p. 359. [4] P.G. Klemens, in: Thermal Conductivity,Vol. 21 (Plenum, New York, 1990) p. 373. [5] V. Wittwer, Fraunhofer Institut fiir Solare Energiesysteme, Freiburg, Germany, private communication.