The heat pulse transport in single phase ceramics

Solid State Communications, Vol. 83, No. 5, pp. 365-370, 1992. Printed in Great Britain.

0038-1098/92 $5.00 + .00 Pergamon Press Ltd

THE HEAT PULSE TRANSPORT IN SINGLE PHASE CERAMICS S.N. Ivanov, A.G. Kororezov, E.N. Khazanov and A.V. Taranov Institute of Radioengineering and Electronics of Russian Academy of Sciences, GSP-3, Marx av. 18, Moscow, Russia (Received l0 February 1992 by V.M. Agranovich) The phonon transport through the single phase ceramic materials at low temperatures is studied. Simple transport equation is derived under the assumption that phonons move ballistically within the grains. The phonon effective diffusion coefficient is shown to depend crucially upon how the phonons pass the contact between neighbouring grains. The heat pulse signal maximum arrival time can either rise or fall as a function of the generator's temperature provided the excitation level is low. The low temperature part of heat conductivity should deviate in such ceramics from T 3 law. The experimental results on heat pulse propagation prove both the aforementioned possibilities. THE ELASTIC and thermal properties of nonuniform materials are known to be quite different from that of uniform medium [1,2]. In ceramics the porosity, grain boundaries, the vibrational properties at the contact between neighbouring grains and the distribution in grain size are the factors which can affect both the elastic and thermal properties. The aim of this paper is to present the results of theoretical and experimental study on the heat pulse transport and heat conductivity in single phase ceramics. In typical ceramics the grain size R is of the order of few micrometres R i> 1 #m and in the temperatures range of interest T i> 1 K the characteristic phonon wavelength ~[ is at least by two orders of magnitude less than R. We can, therefore, attribute the heat transport to the same vibrational excitations, for example, phonons as in the grain materials continuous medium. The vibrational density of states and specific heat are easily expressed in terms of the corresponding quantifies for the continuous medium with the account of the ceramics porosity. However, in ceramics the phonon transport is completely different and one of the main questions here is how do phonons pass through the contact between the neighbouring grains. First we discuss the simple qualitative model of the phonon transport in ceramics. Let zp be the phonon lifetime with respect to any inelastic scattering process. The characteristic time for a given phonon to pass the path from one boundary of the grain to the other is either t0 = R/vs, or t~ " R~[D~ dependent upon whether phonon moves ballistically or diffusively (% is the sound velocity, D1 is the diffusion coefficients

in the grain material). At low temperatures T ~< 4.2 K) both values t b and tR are much smaller than Tp. In what follows we consider only the case of low excitation level. That means AT[T ,~ 1, where AT is the excess in phonon generator temperature over the temperture of heat bath T. Typical value of cp for the anharmonic three-phonon processes at T = 4.2 K is more than 10-3s. The next characteristic time to describes the phonon transition from a given grain to the neighbouring one. This value obviously depends upon the properties of the Contact between grains. If to < Zp, then to is the characteristic phonon lifetime within a grain. First, we assume this inequality to be fulfilled. Let tr be the mean contact area, ~ > 1 the mean number of contacting grains and Y. = tSa the total contact area. If either Y. is small with respect to the grain surface area S, or the rate of phonon transmission through the contact is somehow restricted, then to ~> tb, or to >> tR. In that case phonon transport looks like hopping conductivity, to being the excitation lifetime at a given site and R the mean hopping distance. For the time scale cp >> t >> to such a process is described as a diffusion with a diffusion coefficient D ,~ R2[to. We estimate Sl RSl if phonons move ballistically within the grain. Herefo, stands for the phonon transmission probability at frequency o~ through the contact area. Now the signal at the bolometer reaches its maximum at L2 L2 s t~ )

365

~

__

D

~

_

_

%Rfo, E '

THE HEAT PULSE T R A N S P O R T IN SINGLE PHASE CERAMICS

366

where L is the sample length. If on the other hand phonons move diffusively within each grain, then

p(v ~ v') = p ( V ~ v). Introducing

V(r~) = ~ p ( v ~ v')a~,

to "~ t . ~

1E -=

and signal maximum arrival time is (L)2R2 s l

L 2

,,~

ON(rv, t)

-

Y'. [ p ( v -~ ~,

(1)

Here p(v ~ v') is the transition probability per unit time for an incident phonon to pass contact area from vth grain to the v'th ncighbouring one. The summation in equation (1) is over all the contacting grains. If to take into account the diffusive reflection ofphonons at grain boundaries, one can rewrite the right-hand side in equation (1) as if phonons are transmitting from one half space to another provided no and n~, are phonon densities at each half space respectively and both half spaces are conncected only by a single contact o. Proceeding in such a way we arrive at 0" V s

and

=

(3) Let's designate (A(r~)) the statistical average of a function A(r~) over the distribution of the grains in ceramics. Let D -= (D(r)) and d(r~) = D(r~) - D, then aN(rv, 0t

t)

l

DAN(r~, t) =

I(V(r~)V) + d(r~)A

+ ½ E [ ( ~ ' v ) 2 - - ~3 a , , ~ , A ] ¢),

p(v

~

v')

] N(r~, t). 1

(4) Taking the statistical average in equation (4) we arrive at DAn(r~,t) =

(£(r)N(r,t)),

£(r~) -

[V(r~)V] + d(rv)~<

+½Et(a~'v) v)'

~ - ~3 :a,~,A]p(v ~ v').

Inserting into the equation (5) the plane driving source of nonequilibrium phonons at z = 0, injecting phonons at t = 0 after the Fourier transform and the decoupling of the term (£(r)N(r)) we find n(k, m) = x

a

(27~)2

6(kx)6(k') --/co + D/~ - (2.) 3 f dq G(q + k, m)Lq2+k,k(q)'

(6) where G(k, co) is the diffusion equation Green function and Lq~,q2(q) is the Fourier transform of the function L~.~(r - r') - (Lql(r)Lq2(r'))where

Lq(r) -= i q V ( r ) - q2d(r)

- ~ [a~.VN(r~, t) + ½(a,.,.V)~ N(r., t)]p(v ~ v'),

(5)

where n(r, t) - (N(r, t)) and

p(v ~ v') = &o.

Assuming N(r~, t) to be slow varying function in space we expand N(r~,, t) in power series and ON(r~, t)

1

= ~ ~, [a~,V) 2 - aXa2,~,Alp(v ~ v')N(r~, t).

an(r~, t) 0t

v')N(r~, t)

- p(v' ~ v)N(r,,, t)].

p(v ~ v') " -~-~ fo

v')~,,

6 o, p ( v --,

[V(rv)Vl N(r~, t) - D(r~)AN(ro, t)

Ot

=

D(r~)

we rewrite equation (2)

L2Sl

This result has a simple physical meaning. The effective diffusion coefficient in porous materials turns out to be smaller than that of a continuous medium by the factor (Z[S)fo). This factor takes into account the reduction in the phonon spatial displacement rate as compared to the continuous medium (compare [3]). The passage through the contact from one grain to another is a kind of a bottleneck effect. Let N(r~, t) be phonon density at frequency co in a given grain at r~ and t. For simplicity we omit further the variable ca. We assume phonons to move ballistically within grains and their reflection at grain boundaries to be diffusive. Since (S/Y,)fo, ~ 1 an arbitrary initial phonon distribution within a particular grain becomes uniform quite rapidly (in a time of the order of tb). N(r~, t) obviously must obey the continuity equation which we write in the following form ON(rv, t) at

and

C

s 1

.(2)

Vol. 83,'No. 5

(2)'

where a~, = r e , - r~ and it was also supposed

-- ½ ~ [(a,,~,q)2 -- XaaL,q2lp(v ~ v'). The dominant contribution to the integral SdqgG(q +

Vol. 83, No. 5

THE HEAT PULSE TRANSPORT IN SINGLE PHASE CERAMICS

k,~n)Lq+~.k(q) comes from the integration over q, where q ,~. k ... I/L. Expanding the intergrand in power series under the assumption klq ,~ I and keeping only the major terms, we find

n(k, o,)

a 6(k~)6(ky) (2703 --iw + k2[D +

=

'

(7)

that ~(v,/R)to >t 1, where ~ ,~ I is the parameter to account for the rare inelastic phonon scattering events at the grain boundaries and (%/R)to >> 1, the number of phonon collisions with grain boundaries during its lifetime within a given grain. Phonon distribution within each grain is, therefore, locally equilibrium, i.e.

n(r,,, t) =

where 1

So dq V02(q)

(2~)2 So dq q2 V02(q) is the correlation length and V02 - ~ d q V 2 ( q ) -

> 1 the fluctuations scale is given bye/6. The first to second term ratio in D~ is of the order of (l/a) (Ro/R)2 and can be either less, or more than 1 dependent upon the ration Ro/R. In slightly nonuniform ceramics with small dispersion in grain size Ro -~ R and D~ - D. On the other hand if the dispersion in grain size is big, i.e. the small grains fill the space between large ones, the rati o Ro/R can be sufficiently large. For instance, for each small grain, contacting with large one the vector quantity V is directed towards the large grain, correlations length being of the order of large grain size. The structural term Vo21~/D in the expression for D~frhas a simple meaning. Consider the plane Z = const, at which there is a contact between small and big grains. If at this plane the small grains of radius r are contacting almost the same size grains at Z + r, the phonon transition rate between those grains is determined by the small difference in phonon densities in neighbouring grains. For a contact with a large grain such a difference is much more pronounced because it is determined by the mean phonon density between the planes Z = const, and Z = const. + R. Note that the phonon density within each grain independent on the distribution of incoming phonons sources rapidly becomes uniform due to the diffusive phonon scattering at grain boundaries. The bigger difference in phonon densities, i.e. N(r¢) - N(ro,) at the smalllarge grains contact causes more fast phonon transport in such a ceramics as compared with that of slightly nonuniform ceramics with the same mean grain size. The signal maximum arrival time is tm ~L2/D~ and if D >> Vo2/~/D, i.e. for slightly nonuniform ceramics, t m = L2/D ~ L2/tpR 2 that coincides with the qualitative arguments. Now consider the other limiting case. We suggest

367

1

exp ~noJ/ks T(r~, t)]"

Instead of equation (1) now we have a continuity equation for phonon energy within each grain Er~, t)

OE(ro, t) = at

J'o do, x ho,[n(o~; r~, t) -

-.

n(r~, t)]

(8)

and again/3o,(v --, v') ~ (a/S)(v,/R)f,. Assuming an n((w); r~, t) -- n(o~; ro,, t) ~ - ~ [ T ( r e . ) - T(r~)] an

-- b-~ [Ca., V) + ½(a.. V~:)~'(r~, t) and aE aE aT C d__.T at = aT at = ~at' where Co is the ceramics specific heat we have finally aT(r~, t) at =

~/~(v --+ v')[(a~.V) -4-½(a~.V)2]T(r,,t) 17'

~.

(9)

which differsfrom equation (2) by/~(v ~ v'), i.e.the averaged over phonon distribution transition probability. The further analysis of equation (9) is similar to equation (2) and results in the obvious generalization of the previously discussed results. We discuss now the temperature dependence of t,,. The only frequency or temperature [if ~(v~/ R)to >i 1] dependent quantitites in D~ are the transition probabilities p or ~, De~ being proportional to p or/~. There are few possible cases. (1) Acoustic mismatch model. If this is the case, then the transition probability does not depend upon the phonon frequency or nonequilibrium phonon temperature and t,, = const. (2) Phonon diffraction. These affects the phonon transport if the linear size of the contact area r0 is less than the characteristic phonon wavelength ~[. To estimate the effect we take the known result for the wave diffraction on the round hole on the plain screen. If >> to, then p ~ l / ~ 4 according to Rayleigh law. Since ~[ is getting smaller when the temperature increases, then arm~aT < 0.

368

THE HEAT PULSE TRANSPORT IN SINGLE PHASE CERAMICS

Vol. 83, No. 5

i

I IO

91 !

i

o~

i

5

, 0

/

I0o

i

E 92

i

tlmkc)

Fig. 1. The detected signals in CTLL, L = 120/~-n. (]) r = 3.82K; (2) r = 3.4K; (3) r = 3.0] K; (4) T = 2.2K.

I I

I I

I 2

I 4 T(K)

Fig. 3. The t=(T) dependencies in (l) CTLL ceramics, L = 120~; (2) fused silica L - - 1 6 0 # m , Pz < lO-IWt/mm2;(3) tM T 5 line. "~

Fig. 2. Electron microscope (EM) micrograph of the cleaved CTLL sample surface. (3) Defect or glassy contact. If r0 >> ~[, then the dependence of transition probability upon ~ might occur due to a large amount of defects near the contact, the same is true if the contact area is amorphous. The incident phonon then has a considerable probability to be scattered back. If ~[decreases (T increases) phonon penetration rate gets smaller and Ot=/OT > O. (4) Fractal contact. If the contact has fractal structure and the characteristic phonon frequencies in the heat pulse correspond the fracton part of vibrational states in contact area, then the heat pulse transport through the contact might be due to hopping o f localized excitations resulting in Otm/OT < O. The main conclusion is, therefore, that for different ceramics at low temperatures if phonons move ballistically within grains one can observe any type of behavior: t=(T) = const., Otm/OT < O, at~/OT > O. The unusual behavior with Ot=/OT < 0 might occur due to the effect which we could not take into

account accurately. We consider this possibility qualitatively. Let the phonon motion within the grain have the diffusive component. In that case D = D ( T ) . The structural term in D~ still must be present. Assuming D ( T ) to be a decreasing function of temperature we conclude that ~tm/OT changes sign at a certain temperature To, defined to fulfil the equation D(To) = VoRo. It is easy to show that if T < To, then atm/ OT > 0 and if T > To, then ~tm/~T < O. To discuss the low temperature heat conductivity we taken qualitatively ~ ,,0 C~Daf. We can write K(T) ~ 7". At low temperature Co ~ T 3 • In acoustic mismatch model Dca(T) = const, and n = 3. For Otm/OT < 0, Dar gets bigger if Tincreases and n > 3. For &,JOT > 0 we have n < 3. We have studied experimentally the heat pulse transport in two different ceramic materials: zirconium-lead-lanthanum (CTLL) and Ruby 22HS. The samples were the thin polished to optical precision fiat parallel slabs with thickness 100-200#m. Thin ,,-1000A thermally sputtered Au film on one side of the slab was used as heat pulse generator, while superconducting In or Sn film at the sample's back was used as a bolometer. The dissipated power PH in Au film was less than 10-~Wt/mm2. The excess in generator temperature with respect to heat bath was

•

V o l . 83, N o . 5

THE HEAT PULSE TRANSPORT IN SINGLE PHASE CERAMICS

5

369

3.0

I

2.5 4

2.0

2 1.0 05

I

I

I

I

tlmkc)

(a)

h

t(mkc)

(a)

(b) (b)

Fig. 4(a). The detected signal in Ruby 22HS ceramics. L = 140/an, P~ < 10-'Wt/mm2; (1) T = 3.8K; (2) T = 3.65K; (3) T = 3.37K; (4) T - - 3 . 1 3 K ; (5) T = 2.44K. (b) The micrograph for (a) sample structure before mechanical treatment.

Fig. 5(a). The detected signal in Ruby 22HS ceramics. L - - 140/an, PH < 10-'Wt/mm~; (1) T - - 3.78K; (2) T = 3.49K; (3) T - - 3 . 1 4 K ; (4) T - - 2 . 4 6 K . (b) The EM micrograph for (a) sample structure before mechanical treatment.

carefully controlled and did not exceed 0.2 K in the whole temperature range of the experiment from 2.2 up to 3.8K [4]. Figure I shows the detected signal S(t) in CTLL. The curves as expected have typical for the diffusion process shape with a distinctive maximum. The cleaved surface EM micrograph of this sample is shown on Fig. 2. The ceramics is comparatively dense packed and grain mean size is 3-5 #m. The ceramics is transparent and has no detectable by optical and acoustical microscope defects. The measured dependence t , ( T ) N T" with n I> 5 excludes the assumption about the possible effect of Rayleigh type scattering of phonons at contacts. It is very close to what was observed under the same experimental conditions in glasses [5]. Figure 3 shows tin(T) curves for CTLL ceramics and fused silica. For both cases tin(L) ~ L 2. The weakening of tin(T) dependencies at fixed PH is a result of AT

Fig. 6. Part of Fig. 4(b) under higher resolution.

370

THE HEAT PULSE TRANSPORT IN SINGLE PHASE CERAMICS

increase while T gets smaller and can be easily accounted for [4]. The observed tin(T) behaviour is thus consistent with the assumption of the formation of amorphous area near the grain contact. Figures 4(a) and 5(a) shows the detected signals in 22HS Ruby ceramics; the EM micrographs of those samples are given in Figs. 4(b) and 5(b) at the same resolution before polishing. In agreement with theory the more grain size and the more nonuniform ceramics are the faster is the signal maximum. Contrast to CTLL is the temperature behavior of tm for 22HS Ruby ceramics: tm gets bigger if the temperature increases. Figure 6 shows EM micrograph of the part of the sample at Fig. 4(b) under higher resolution. The grain boundaries are sharp. The contact structure, however, is not resolved. Therefore, one might think that there are fine continuous blocks at the contact area and phonon transmission is affected

Vol. 83,' No. 5 .

by the diffraction. On the other hand there are also the broad but smooth contact areas at which fractal structure can be of importance. The observed tin(T) dependence with Otm/OT < 0 might be due to both reasons. REFERENCES 1. 2. 3. 4. 5.

W. Mason, Physical Acoustics, Vol. 3, pp. 62121, Moscow, Mir (1968). K. Okazaki, Ceramic Engineering for Dielectrics, 336 pp, Moscow, Energia, (1976). M.P. Zaitlin & A.C. Anderson, Phys. Rev. BI2, 4475 (1975). S.N. Ivanov, E.N. Khazanov & A.V. Taranov, FTT 29, 672 (1987). S.N. lvanov, E.N. Khazanov & A.V. Taranov, in Proc. )(IV All Union Conf. on Acoustoelectronics and Physical Acoustics in Solid State, Vol. I, pp. 229-231, Kishinev (1989).