HAL Id: jpa-00217518
https://hal.archives-ouvertes.fr/jpa-00217518
Submitted on 1 Jan 1978
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
THE KAPITZA RESISTANCE AND PHONON
SCATTERING AT SOLID-LIQUID He INTERFACES
A. Khater
To cite this version:
A. Khater.
THE KAPITZA RESISTANCE AND PHONON SCATTERING AT
JOURNAL DE PHYSIQUE Colloque C6, supplement au n" 8, Tome 39, aout 1978, page C6-258
THE KAPITZA RESISTANCE AND PHONON SCATTERING AT SOLID-LIQUID He INTERFACES A.F. Khater
Institute de Fisica-VFF, Nitevoi-RJ-CEP 24000, Brazil
Résumé.- Nous présentons une théorie qui explique le problème de résistance thermique de Kapitza en-tre un solide et l'hélium liquide. La diffusion des phonons dans l'interface inégal est étudiée et on démontre que la dynamique de vibrations des microaspéritês dans la surface du solide est responsa-ble de la haute transmission d'énergie entre le solide et l'hélium. La théorie explique bien d'autres aspects du problème.
Abstract.- A theory is presented which explains the Kapitza resistance between a solid and liquid he-lium. Scattering of phonons in the microirregular interface is studied and the elastic flexure of surface microfeatures is shown to be responsible for the relatively high transmission of energy across. The theory accounts for a large number of experimental observations.
The Kapitza thermal boundary resistance R^ is an effect observed when a heat current Q passes across the boundary (area A) between a solid and liquid helium. It was first observed by Kapitza/1/ and has been extensively reported for a wide varie-ty of solids since. See/2/ for a recent review ar-ticle. For moderate heat currents Q a linear rela-tion can be written giving
\ = AT/{Q/A} (1)
where AT is the measured temperature drop across the boundary. There is for magnetic materials and liquid 3He a separate magnetic Kapitza resistance which has been shown to depend on dipole—dipole interactions between the electronic spin in the so-lid and the nuclear spin of 3He/3/. We do not
dis-cuss this magnetic effect here, and the subsequent discussion concerns solely the ordinary R^. This, further, is only a summarized version of a full ac-count to be published/4/.
The first theory proposed for the effect was given by Khalatnikov/5/. This author studied the heat transfer between the two media in terms of phonon reflection and transmission coefficients at a plane interface separating them. The predictions of this so called acoustic mismatch theory, however, do not compare favourably with experimental results. In particular, the theoretical values given for R^ are in general one or two orders of magnitude grea-ter than the measured ones. There has been a number of theoretical attempts to explain this discrepan-cy as well as other relevant experimental observa-tions but with seemingly no satisfactory conclu-sion/6-7-8/.
Recently the R^ problem has gained further significance in a new set of experiments in which phonon reflection and transmission at a solid-liquid helium interface was specifically investigated/9/. The results of these experiments demonstrate that
the processes of scattering of transverse (T) and longitudinal (L) phonons are quite different at such an interface. L-phonons scatter with little loss of energy whereas T-phonons lose an apprecia-ble amount of energy.
This has prompted us to investigate T-phonon scattering at an uneven solid-liquid helium inter-face since solids have uneven surinter-faces in general. Field ion microscopy and LEED show that even surfa-ces obtained by crystal cleavage are not perfectly plane/10/.
We represent in our theory the micromorpho-logy of any given arbitrary solid surface as a
dis-tribution of cylindrical microfeatures. The n feature is characterized by a radius n and height
E and its base is taken fixed to the local substra-n
te. The feature sizes are microscopic. It is conse-quently valid to argue that a feature can flex and to treat its corresponding elastic behaviour clas-sically.
Furthermore, there is evidence that the first adsorbed layer of helium atoms on surfaces of dif-ferent solids is itself a 2 dimensional helium so-lid which supports 2 dimensional phonons in the range of temperatures D(T) = [5.01 K, 2.2 K] over which R^ is observed and measured/11/. The helium atoms are pinned down in this layer by extremely high Van Der Waals forces. We suppose the layer, hence, to replicate in solid form in adsorption
equilibrium the exposed microtopography of the so- lid surface.
For a free surface, a T-phonon incident from the solid bulk can be shown to excite a new kind of localized surface mode of elastic flexure on the nth feature, with a dispersion relation
This has a cut-off frequency determined by the li- mit of the validity of assuming flexure in the Ber- noulli type of bending theory. w is the frequency of the incident T-phonon. Es, vs and p, are respec- tively Young's modulus of elasticity, Poisson's ra- tio and the density of the solid. A and I are res-
n n pectively the cross-sectional area and the moment of area about the axis of flexure of the nth micro- feature. k is a wavevector.
n
Given the strong Van Der Waals coupling bet- ween the helium atoms and the exposed surface this flexure mode can generate 2 dimensional phonons in the 2d solid helium layer. These phonons scatter into the surrounding liquid helium bulk at the ed- ge of a given microfeature and lose their energies. Their lifetimes are given approximately by
on the nth feature. v2d stands for the velocity of transverse or longitudinal 2d phonons in the solid helium layer.
The Van Der Waals force in the interface cou- ples strongly the helium atoms to the exposed solid surface. It is also significantthatw~~ varies from
<< I to <
-
1 over D(T) for thermal phonons in the solid, Furthermore, the arrival of these T-phonons is random. Under such conditions the creation and annihilation of the 2d phonons constitute a thermal bath mechanism for the transfer of energy from T- phonons in the solid to the bulk liquid helium. The solid L-phonons in contrast do not excite the fle- xure mode in equation (2), and they scatter accor- ding to the boundary conditions of the acoustic mis- match theory/5/.We derive a transfer energy function for a transverse phonon between the solid and liquid helium over a probability distribution of features on any arbitrary solid surface. For a black body spectrum of T-phonons travelling to the interface the heat flow across the boundary is derived and a theoretical expression for
R;;'
= 2 ~ ~ { g ( ~ ~ ) ) $ ~ ~ ( 2 8 i ~ + ~ 8 ~ ~ ~ (4)and governs the efficiency of the process. T in (4) is the temperature of the heat source. is the Debye temperature of the 3 dimensional helium solid whose elastic properties scale simply to those of
the 2 dimensional helium solid at the same atomic separations of helium atoms/l2/. $ is a constant for a given solid and varies little with choice of solid. M{g(Sn)) is the mean of a function g(S ) over
n the normalized probability of finding a feature with (17,,En) dimensionsl41. It turns out that
M{~(S
)l%'
n a - l ~ ~ , where a is a lattice constant in the solid and f3 is a dimensionless random quantity which sca- les r) to
5,
for any given arbitrary surface. Hen-n
zler/lO/ has measured a comparable quantity with va- lues bounded between 5 and 15. For B = 9 we ob- tain excellent agreement with experimental RK re- sults over the temperature range D(T). The locus with temperature 8 = 8 (T) is a variable too,
3d 3d
bounded by the highest and lowest possible helium densities in the 2d solid helium layer. The quanti- ties f3 and 83d are thus independent but bounded ran- dom variables for a given arbitrary surface.
The theory accounts for the miscellaneous scatter of the experimental
$
results, as well as a significant number of other observations obtainedfrom experimental data.
References
/I/ Kapitza,P.L., J. Phys. (Moscow) (1941) 181 121 Challis,L.J., J. Phys. C :
1
(1974) 481 /3/ Mills,D.L. and Beal-Monod,M.T., Phys. Rev. @(1974) 343
141 Khater,A.F., to be published
/5/ Khalatnikov,I.M., Zh. Eskp. Teor. Fiz.
22
(1952) 687161 Sheard,F.W., Bowley,R.M. and Toornbs,G.A., Phys. Rev. (1973) 3135
/ 7 / Cheeke,D. and Ettinger,H., Phys. Rev. Lett.
21
(1976) 1625/8/ Nakayama,T., J. Phys. C.
10
(1977) 3273 ./g/ Long,A.R;, Sherlock,R.A. and Wyatt,A.F.G., J. Low Temp. Phys
.
15
( 1974) 523/10/ Henzler,M., Surf. Sci.
19
(1970) 159/l]/ Elgin,R.L. and Goodstein,D.L., Phys. Rev.
2
(1974) 2657/l 2/ Stewart ,G.A., Siegel, S. and Goodstein,D.L., Proceedings of the XI11 Conf. on Low Temp. Phys.
(1972) 180
1131 Sample,H.H. and Swenson,C.A., Phys. Rev.