NBSIR 75-812

THE KAPITZA CONDUCTANCE OF THE [100] SURFACE OF

COPPER AND POLYCRYSTALLINE INDIUM AND NIOBIUM

N. S. Snyder

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

June 1975

Final Report Prepared for:

Office of Naval F^esearch

Arlington, Virginia 22217

NBSIR 75-812

THE KAPITZA CONDUCTANCE OF THE (100] SURFACE OF

COPPER AND POLYCRYSTALLINE INDIUM AND NIOBIUM

N. S. Snyder

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

June 1975

Final Report Prepared for:

Office of Naval Research

Arlington, Virginia 22217

Covers April 15, 1971 to October 14, 1974

Contract Nos. NAonr-20-71, NAonr-8-72, NAonr-13-73, NAonr-22-74.

Approved for public release; distribution unlimited.

Reproduction in whole or in part is permitted for any purpose of the

United States Government.

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretary

NATIONAL BUREAU OF STANDARDS Richard W Roberts Director
DISCLAIMER

Certain commercial materials are identified in this report in order to

adequately specify the experimental procedure. In no case does such identification imply recommendation or endorsement by the National Bureau of Standards,

nor does it imply that the material identified is necessarily the best available for the purpose.
CONTENTS

Page

1. INTRODUCTION

1.1. Background. The Acoustic Mismatch Theory ]_ 1.2. Previous Experimental Work 2 1.3. Recent Theoretical Work 5 1.3.1. Modification of the Acoustic Mismatch Theory. ... 5 1.3.2. Non-Continuum Approaches 7 1.4. Objectives of These Experiments g

2 . DESCRIPTION OF APPARATUS ..... 9 2.1. The Ultra-High Vacuum System 9 2.2. The Cryostat -^^ 2.3. Electrical Measurements .... u

3. EXPERIMENTAL PROCEDURES AND SOURCES OF ERROR I7

3.1. Preliminary Sample Preparation u

3.2. Sample Preparation in the UHV System ........ 13

3.3. Cryogenic Procedures 19

3.3.1. "Storage" Vacuum ^.9 3.3.2. Thermometry 3^9 3.3.3. Parallel Heat Flow 20

3.4. Analyses of Sample Properties 22

3.4.1. Surface Area 22

3.4.2. Analysis by X-Ray Techniques of Crystal Perfection. 22

3.4.3. Purity 24

4. RESULTS OF MEASUREMENTS 24

4.1. Corrections for Parallel Heat Flow 24

4.2. Copper Single Crystal Sample 1 25

4.3. Copper Single Crystal Sample 2. ....... . 25

5. CONCLUSIONS 30

APPENDIX A. MEASUREMENTS OF THE CONDUCTANCE OF POLYCRYSTALLINE

INDIUM_ 32

APPENDIX B. MEASUREMENT OF THE CONDUCTANCE OF POLYCRYSTALLINE

NIOBIUM 36

REFERENCES 39

DISTRIBUTION LIST , 43

iii
LIST OF FIGURES

Page

Figure 1. Experimental results and theoretical curves for

a copper-He II interface 3 Figure 2. Schematic diagram of the vacuum system during

sample preparation. ... 10

Figure 3. , Schematic diagram of the cryostat in which Kapitza

conductance measurements are made 11

Figure 4. Photograph of apparatus 12

Figure 5a. Detailed view of sample chamber 14

Figure 5b. Sample chamber in cryogenic mode 15

Figure 6. Detailed schematic of the heat flow between the elements

of the Kapitza conductance measurement system 21

Figure 7. Kapitza conductance measurements of single crystal

sample 1 26

Figure 8. Kapitza conductance measurements of single crystal

sample 2 27

Figure 9. Comparison of measurements on annealed samples IC

and 2B 29

Figure 10. Apparatus for Kapitza conductance measurements of

indium 33

Figure 11. Kapitza conductance results for two polycrystalline

indium samples, 3 and 4 34

Figure 12. Comparison of Kapitza conductance measurements on indium with theory and with previous work 35

Figure 13. Kapitza conductance of a polycrystalline niobium

sample 37

Figure 14. Comparison of the observed Kapitza conductance of

niobium with theory and with previous work 38

iv
ABSTRACT

Measurements of the Kapitza conductance to liquid helium II across the (100) surface of single crystals of copper are presented.

The temperature range of these measurements was 1.6 to 2.1 K. The sample surfaces were subjected to several different treatments. Some surfaces were cleaned by low energy argon ion bombardment, annealed in an ultrahigh vacuum system, and preserved under

vacuum until purified liquid helium was admitted. Other surfaces

were intentionally damaged by machining and/or exposure to the

atmosphere. The conductance after these latter treatments was found

to be about a factor of three higher than that of the more ideally

cleaned and annealed surfaces, and a significant increase in the

temperature dependence of the conductance was also observed. Indications are that these effects on the conductance are produced

primarily by the last few atomic layers of the solid, so extensive

cold-working, which impairs the thermal conductivity of the bulk,

need not be undertaken to improve surface heat transfer. This

finding has important implications for the design of practical heat

transfer surfaces. Good reproducibility was found for the conductances of similarly treated sufaces and good correlation with

studies of damage carried out by x-ray diffraction. The relationship of these results to the numerous current theories of the

Kapitza conductance is discussed. Conductance measurements of

polycrystalline niobium and indium are presented in the appendices.

Key words: Heat transfer to helium II; interfacial thermal resistance; Kapitza conductance; Kapitza conductance of copper; Kapitza

conductance of indium; Kapitza conductance of niobium; (100) sur- face of copper; x-ray diffraction examination of single crystal

lattice damage

.

V

THE KAPITZA CONDUCTANCE OF THE (100) SURFACE OF COPPER AND

POLYCRYSTALLINE INDIUM AND NIOBIUM

N, S. Snyder

1. INTRODUCTION

1.1. Background. The Acoustic Mismatch Theory

An analysis of the heat transfer between a solid body and liquid helium

II shows that the thermal impedance may be separated into two parts. One

thermal impedance is located precisely at the solid-liquid interface, and is

described by the heat transfer coefficient, h^, often called the Kapitza

IS. conductance. Thus, the heat flow, Q, through the interface, is given by

Q = hj^AAT, (1)

where A is the surface area of heat transfer, and AT is the temperature difference between the solid surface and the liquid. To restrict consideration to

the interface proper and eliminate processes in the bulk liquid, one can think

of h„ as the limiting value reached as AT becomes very small, i.e.

is. \ " ^^"^ aIt • ^ AT-vO ^

The second part of the thermal impedance occurs in the bulk liquid, helium II,

and is negligible compared to the interfacial impedance under most conditions.

In the temperature region near 1.9 K, the Kapitza temperature drop across the

interface is usually larger than the temperature drop across many meters of

fluid, because of the very high thermal conductivity of helium II. For heat

2 inputs of 1 W/cm , experimentally measured temperature differences across an interface range from about 0.3 K to 10 K. The Kapitza conductance is an even more important parameter at low tem3 peratures because it varies with temperature approximately as T , where T is

the absolute temperature. Thus, the conductance is so large above liquid

helium temperatures that the interfacial thermal barrier is usually not noticeable. On the other hand, at temperatures in the millikelvin region, the con- ductance may be so small that it is the limiting factor in the design of devices.

But even at temperatures near the lambda point, the smallness of the conductance

is quite noticeable, compared with the exceptionally high thermal conductivity

of helium II. It was this factor which led to the original discovery of the

interfacial thermal barrier by Kapitza."*" In fact, it was thought at first that

the interfacial resistance was related to the superfluid properties of helium

3 II, but conductances of similar magnitude were later found for solid- He mter2 faces, and these early theories were subsequently abandoned.

Later, two possible reasons for the existence of interfacial thermal

barriers at low temperatures were recognized. First, at temperatures far below

the Debye temperature of a solid, only a relatively small number of phonons of

high energy are available for heat transmission across an interface where

electronic transmission is unimportant. (Because the magnitude of the Kapitza

2 conductance is similar for metals and insulators, it is currently thought that

electronic wave function "leakage" is relatively small, but see recent work of

Wagner, et al."^) This effect, sometimes known as the phonon radiation limit,

is demonstrated in a simple calculation that is found in reviews of the sub2 4 ]ect. ' The conductance is
PRL 2tt k (3N/4^) 2/3

(3)

where 6 is the Debye temperature and N the number of atoms per unit volume

3 The T temperature dependence then results simply from the tempera

dence of the lattice portion of the specific heat at low temperatu

this is a measure of the available energy.

A second reason for the interfacial thermal barrier is the la

mismatch between liquid helium and most solids, which is expected

reduce the phonon transmission. Since the distribution of momenta

in the two media is so different, it is difficult to satisfy the 1 servation of energy and momentum for arbitrary angles of incidence

calculation was first carried out for liquid helium-solid interfac

Khalatnikov, ^ ' ^ and for solid-solid interfaces by Little.^ The re form which may be compared with eq. (3) is

ture depenres, since

rge acoustic

to further

of phonons

aws of con- This

es by

suit in a

AM o

4, 2

2tt k 3N

4lT

2/3

3n'

2ri + 1

2/3

(F^ ^ ^2^

p, . c, . ^ '^liq liq ^3

sol t

(4)

where n is the ratio of the velocities of longitudinal and transverse waves,

p refers to the density,

^ij_q ^° the velocity of sound in the liquid, and c^

to the transverse velocity of sound in the solid. is a measure of the heat

flow due to phonon transmission within the allowed angle, and is a measure

of the heat flow due to the excitation of surface modes. (It is assumed that

energy is effectively transferred from the surface modes to the bulk by some dissipative mechanism. ) F^ and F2 are of approximately the same magnitude

for most common solids, and the entire quantity between the square brackets

is of order unity. Hence, a comparison of eq. (4) with (3) indicates that the

main difference is in the appearance of the ratio of the acoustic impedance,

-3

pc, in the two media. This ratio is often about 10 , which reduces the transmission significantly from the phonon radiation limit. However, for solid-solid

interfaces, the mismatch is much smaller.

1.2. Previous Experimental Work

Quite a number of solids have been studied, usually in the polycrystalline

state. Results are available for the more common structural metals, including

aluminum, nickel, copper, and iron, as well as for the softer elements, such as

tin, lead, and indium. Several dielectrics have been studied. The measurements

2 up to about 1970 have been tabulated, and more recent results on aluminum and

8 9 niobium, and iron, cobalt, and nickel are available. (The emphasis in this

discussion will be on work in the temperature range near the lambda point; the

present situation at lower temperatures is that relatively few experiments

have been done, and the work has been carried out at a small number of laboratories, so there has not been much comparison of data by different workers.)

The major conclusion of the higher temperature work has been that for both

dielectrics and metals the observed conductances are far in excess of the

2 10 Khalatnikov acoustic mismatch model, often by two orders of magnitude. ' An example of the situation is shown in figure 1,^"*" which reproduces results

for copper on polycrystalline samples. The results fall roughly in between

the phonon radiation limit, eq. (3), and the acoustic mismatch theory, eq. (4).
40, "1 r

PHONOH RADIATION LIMIT

I r

0.5

„....-A CHALLIS, ORANSFELD, WILKS THEORY

0.1 -

KHALATNIKOV THEORY

0.04

1.3 1.4 1.5 1.6 1.7 I.e

TEMPERATURE, K

1.9 2.

I a

u o

(U -H U U

C eg m

•H EU (U M O C M 14-1 -H C ^

0) M

X -a

I (U 0)^ CO D. • cd Cu U T3 O d) a 13 0) J=

to fi JJ to

l-l JJ O ?N CO

li-i j3 x:

a) Qj -o > cx il

>-i ca 4-1 3 a. n! o o

rH J3 TJ

cfl JJ c O -H

•H C

JJ -H C • (u o a> "O -H d

O C -U -H

a) 3 ca -H

s: o V

4-1 «-l 1-1 T3

-a Q)

cs u E

i-i3 CO CO (U

0) CJ

S-i c

0)

S-i ca 0)

j-i y-i c a)

a) erf E

T-i

l-l .

o- u

!< ta

U

3

00

X5 iH

O

CO

ca XI >

ca -a

i-i

(1) 3

TJ o

ca x: E CO

3
Furthermore, results by different workers are often in considerable disagreement. There seems to be a tendency for results on surfaces which were chemically

etched before the measurements (solid lines) to exhibit a higher conductance

than those which were simply machined (dashed lines) . However, there are no results in which chemical etching or annealing were carried out on a sample

which had been measured after it was machined, so it is difficult to assess

the effect of cold work. Since the present work was started, some work along

12 these lines has been done at very low temperatures by Folinsbee and Anderson.

This will be discussed more fully below, but the applicability of these results

to this temperature range is uncertain. The dependence on the Debye temperature and liquid density predicted by eq. (4) is also not observed experimentally . ^ The characterization of the surfaces studied to date, in terms of the

current knowledge of surfaces that has developed during the past decade, can only be described as extremely sketchy. Many of the early workers in the field

(before about I960) did not provide any description of surface treatment. More

recently, typical metal samples have been machined from polycrystalline stock,

mechanically polished, and sometimes etched or electropolished . Only a few

samples have been annealed to remove damage. Exposure to the atmosphere has

always occurred prior to the Kapitza conductance measurements of metals, ex13 . cept for work done by Challis and Cheeke m which ion bombarded samples of

-4 polycrystalline lead were kept in a vacuum of about 10 torr. Such a vacuum

would not prevent an oxide layer from reforming on the surface. Also, polycrystalline copper samples were milled under liquid helium by Johnson and

14 Little and then measured, but this procedure did not permit healing of the

machining damage. Surface conditions have been, in general, better specified

and more ideal for the few dielectrics, usually single crystals, that have

been measured. There has been a measurement on a crystal of KCl cleaved under

helium,"*"^ and studies of silicon with differing amounts of surface damage have

14 been carried out. The above studies, and others up to about 1970, are described more fully in table I of reference 2. More recently, cold-worked and/j^2 or annealed polycrystalline copper exposed to the atmosphere has been studied ' in the temperature range below 0.5 K. Pumping on the sample surface for an extended period of time at room temperature has recently been found to affect

the conductance subsequently measured at low temperatures."'"^

It is unfortunate that the surfaces studied have not been better characterized, or more nearly ideal. The theory essentially is constructed for an ideal solid with a plane surface that is a "simple termination of the bulk",

although a real interface has surface roughness and includes foreign atoms,

dislocations, and other defects. Recently, some new theoretical effort related

to non-ideal surfaces has been occurring, and this will be discussed in the

next section. Before these modifications to the acoustic mismatch theory are considered, however, it should be mentioned that the theory apparently works

for solid-solid interfaces at low temperatures, according to some work carried

1

8

out in the past few years, chiefly by Peterson and Anderson and by Cheeke,

et al.

19 However, the applicability of the theory to solid-solid...helium interfaces is still controversial, as data by Mezhov-Deglin obtained on a single

crystal of ^He has a very different temperature dependence ^'^^ than does data

22 obtained by Folinsbee and Anderson. Thus the data obtained by the first

worker appear to be in reasonable agreement with theory at higher temperatures

(0.5 to 2 K) , whereas the latter data do not.

It is appropriate to discuss several recent experimental studies of phonon

transmission coefficients at this point, although a more detailed discussion

will be deferred until the treatment of more recent theoretical results in the

4

i i
next section. These experiments involve efforts to measure, in principle, the

transmission and reflection at an interface of phonons that are well-defined

in terms of frequency, angle of incidence, and mode. Successive refinements

in defining these parameters of transmitted phonons have recently occurred.

Of course in heat transmission, these parameters - are , in effect, averaged over

frequency, mode, and angle by the phonon distribution for a particular solid

2 3 at a gxven temperature. Guo and Maris measured the reflection of heat pulses

from a dielectric crystal interface with liquid and gaseous helium and with

solid neon, using a superconducting bolometer as a detector. The transmission

found was miuch higher than predicted by the acoustic mismatch theory, and in

fact was similar for solid neon and liquid helium, although the acoustic mismatch transmission factors are quite different. In experiments with helium

gas adsorbed at the interface, transmission rose to the bulk liquid value

when the film thickness was equal to about three layers of helium atoms. A

24 " 25 sim.ilar finding was reported by Long, et al. Kinder and Dietsche used

monochromatic phonon pulses of variable frequency, generated and detected by

superconducting tunnel junctions. No structure at the maximum phonon and

minimum roton frequencies is found in their results , though this v/ould be

expected on theoretical grounds if phonons were directly transmitted into

26 " excitations of the bulk liquid. Thus, the m.icrostructure of the first

few layers of helium should be relevant to an explanation of the miechanism

of the energy transfer at the interface, at least for temperatures

above '^1 K. This finding lends added interest to the present work, which is

motivated by an attempt to characterize surface details on the atomic scale.

The phonon transmission experiments, however, are apparently open to some uncertainty in that the transmission of energy directly into the liguid is not

23

proven. Also, there has not been much concern in the transmission v,'ork for

layers of gases other than helium absorbed on the surface, although this may

affect the results profoundly, because theoretical work at higher temperatures

has shown that the properties of the adsorbate and its bond, rather than the

27 bulk, control the energy and momentum exchange with incident gas particles.

These experiments have not been carried out at temperatures below the lambda

2 8 point, where phonon energies are lower. An experiment by Sherlock, et al.

which measured the angular distribution of the phonons radiated into the

liquid from the (10 0) face of a sodium fluoride crystal indicated the presence

of a sharp peak in the angular distribution. This would tend to substantiate

the acoustic mismatch theory because the requirement of conservation of

momentum and energy leads to emission in a critical cone, but phonons reflected

at larger angles were also observed, indicating a possible parallel mode of

energy transmission.

1.3. Recent Theoretical Work

Much of the recent work in the field of Kapitza conductance has been directed towards modifications of the theory to account for the possible non- ideal behavior of the actual interfaces used, rather than attempting the more

difficult job of refining the experiments so that the surfaces employed would

be more ideal and better defined. Hence, the available new theories suffer

from the problem that direct quantitative comparison with experiment has been

impossible, as will be seen in the detailed discussion below. A dichotomy of

approach has occurred, with one school working on modifying the acoustic mismatch theory while still using the continuum approach (expected to be most

applicable at temperatures less than 0.1 K) , and others attempting to discard

the old theory entirely and to construct a new theory upon the basis of atomic

interactions

.

1.3.1. Modification of the Acoustic Mismatch Theory

The acoustic mismatch theory hypothesizes that heat transfer at the interface occurs only through phonons, ana that most phonons incident on the interface will be reflected because of the large acoustic mismatch between liquid

5
helium and most solids. But, if dissipative mechanisms such as scattering

centers exist, the momenta and energy of some scattered phonons may be modified so that boundary conditions can be met, and the transmission will be

29 . increased. Khalatnikov actually took account of one type of dissipation,

since his final result included the conversion of energy into the surface or Rayleigh modes, and it was assumed that all this energy was somehow absorbed

into the bulk,"^*^ although the mechanism was not discussed. Early work on the

interaction of conduction electrons with phonons by Andreev"^''" provided a possible mechanism. However, the Rayleigh wave mechanism increases the magnitude

of the theoretical conductance only by a factor of about three. The theoretical

curve shown in figure 1 includes this factor, which is not sufficient to bring

about agreement with experiment.

In more recent work, phonons incident on the solid from the liquid outside

the critical cone of acoustic-mismatch transmission are considered to create

a non-propagating disturbance with an amplitude which decreases exponentially

with distance from the surface much more rapidly than the Rayleigh waves do.

When attenuation due to dislocations or other causes exists, energy can be

absorbed from the surface disturbances into the bulk material. This mechanism

32 33 has been examined by Khalatnikov and Adamenko, Peterson and Anderson, and

34 Haug and Weiss. The calculations employed perturbation theory and predicted

that transmission would increase above the acoustic-mismatch model in proportion

to the factor p = ^p^i^^ph' ^^^^^

''^ph phonon wavelength, and i^-^ is the

phonon mean free path. There has been no independent way of determining ^..v^ / so it has been used as an adjustable parameter to fit the theory to the

35 . experiments. As Vuorio points out in his critique of these calculations, an assumption that p is temperature-independent, or nearly temperature independent,

has to be made (though ^pj^(T) varies with the type of lattice defect causing

the scattering) , and the probable variation of p between samples is greater than

the factor of 10-100 that is the range of the different experimental results

3 6 for each metal. Challis felt that the smallness of required to explain

the data was excessive, but this criticism has apparently been answered by

37 33 Sluckin, et al. and Peterson and Anderson who point out that the depth of

penetration of the surface disturbances is relatively small (for example,

50 8 at 0.1 K), so that the more familiar concepts of bulk attenuation are

less relevant than surface attenuation, which could be very high. Experiments

12 at low temperatures (<0.1 K) showed qualitative evidence for the theory, although lack of reproducibility was a problem with the experiments. Whether

the mechanism can explain the lack of agreement between the original acousticmismatch theory and experiment in the higher region of temperature that is of

interest here is uncertain. Peterson and Anderson explicitly state that their

calculations are expected to be valid only for T < 0.1 K, but Khalatnikov and

Adamenko indicate their theory should also be valid up to the lambda point.

9 Cheeke, et al. failed to observe an increase in conductance m this higher

temperature range after machining a sample that had been measured in the annealed

state. In these experiments, mechanical polishing of the samples was followed

by an anneal and a light electropolish . It is possible that defects and impurities from the mechanical polish could not be removed by the anneal, so that

a sufficient number of defects were present in both cases. It is also possible

that surface oxidation or physically adsorbed gases, which were present in both

cases, were the determining factors in establishing the conductance. The data

11 38 for lead seem to show the opposite effect, ' and the previous results on copper seemed to also."'"'^ However, as noted above, the situation was unclear

because comparisions had to be made between different samples measured by

different workers.

6
Another possible dissipative effect results from surface roughness. Little

first pointed out that when the mean free path of the phonons is comparable to

the interface roughness, the effective area will differ for phonons of different frequency: For long wavelength phonons the area is the plane area, but for

short wavelength phonons the effective area will be the microscopic area. This can cause an increase in the exponent of temperature dependence, so that

39 it is greater than three. Little concluded that transmission would not be

enhanced for the case of microscopic roughness of the size of the phonon wave40 length. Adamenko and Fuks used different methods of approximation in their

calculation of surface roughness, and concluded that when the phonon wavelength

is comparable with the dimensions of the roughness, the scattering at the

surface becomes nearly diffuse, so that surface waves can be transformed into

volume waves, and transmission outside the critical cone can be increased.

1.3.2. Non-Continuum Approaches

Because it appears that heat transfer across solid-solid interfaces does

follow an acoustic mismatch relationship, a theoretical approach which exploits

3 4 some unique property of He and He, perhaps in both the liquid and solid

state, might be fruitful. The acoustic theory assumes that both materials are continuous media with properties which change discontinuously at the interface

to their bulk values. However, neglect of Van der Waals forces in a relatively

open system such as liquid helium may be unjustified. A number of calculations

which attempt to take such factors into account have appeared recently. There

41 was also an early attempt by Challis, et al. to consider the improvement in

acoustic matching due to the increased helium density near the interface, which

results from consideration of Van der Waals forces between the helium atoms and

the solid. The effect was not large enough to account for the experimental

results (see fig. 1). More recent efforts included dynamic effects of the

Van der Waals forces as well as static effects in a calculation patterned after

42 early calculations of the gas accommodation coefficient. Again, however, the

effect found was not large enough to bring about agreement with experiment

because the contribution to the conductance does not start to be important at

4 3 a low enough temperature. Vuorio considered that improved acoustic matching

in the right temperature range might occur if the interfacial layer contained

defects or impurity structures such as oxides that had vibrational modes of

frequencies different from those of an ideal interface, perhaps as low as 100 GHz. The simplified model for the calculation consists of a layer of solid

atoms separated from the bulk solid by a layer of liquid atoms. An enhanced

transmission of about a factor of 10 above the acoustic mismatch model is found.

This factor of 10 would not be large enough to explain many of the experimental

results, but this could be the result of using a simplified model. Transmission

from the liquid to the solid does not occur outside the critical cone, which

44 45 avoids some theoretical paradoxes. Long and Weiss have recently undertaken

calculations of energy transfer by desorption of helium atoms and obtained some qualitative agreement with the coefficient of reflection found from the transmission experiments. The weak temperature dependence may be explained by such

calculations

.

4 6 Opsal and Pollack have blended the two approaches discussed above

attenuation in the solid at low temperatures and impedance matching at higher

temperatures due to the increased density of helium near the surface. Their

theory has an adjustable parameter for each of these situations, and can be

made to fit some of the previously published data. However, there have been

no measurements on a single sample over the entire temperature range from the

millikelvin region to the lambda point, and there is considerable variation

47 now being found in the magnitude of results at millikelvin temperatures, so whether this approach has general applicability is unclear. For example, as

7
mentioned earlier, studies in the transition region from high to low temperatures, *^'22,48 Yia-ve given results for the temperature dependence that varied

4 2 from approximately T to T . 49 . . . . Saslow has raised the question of dissipation in the liquid as a possible source of increased energy transmission. He has pointed out that- the

dominant frequency of the thermal phonons, even at helium temperatures, is too

high for the application of the usual two-fluid model of the hydrodynamics of

liquid helium. (In the two-fluid theory, the helium below the lambda point is

considered to be a mixture of two components: normal fluid, which carries

entropy and has a normal viscosity; and superfluid, which carries no entropy

and has zero viscosity. The collision time, t, between excitations in the twofluid model is too long for this type of liquid to respond to the thermal

vibrations of the solid, so this model is not applicable to the Kapitza con- ductance problem. ) An intermediate region would exist for frequencies slightly larger than l/x , in which the behavior of the fluid may be described as viscoelastic in an analogy with ordinary fluids such as glycerine. As the

frequency is raised, however, an atomic description becomes necessary.

Saslow assumes that at the thermal frequencies of helium II, a viscoelastic

atomic description is appropriate, and carries out calculations on this model

which blend features of both the viscoelastic and atomic regime. He finds

that viscous dissipation is an important source of energy available for

transfer from the liquid to the solid, but does not obtain an exact solution

for the proportion of this energy which is transmitted and contributes to

the Kapitza conductance. To do this would require a solution of the Boltz- mann equation for atoms rather than phonons, which is a difficult theoretical

problem. Because of the short distances over which energy is dissipated, the

dynamics of atomic collisions probably determine the energy transfer. This

4 3 approach should give similar results for He and He, in accordance with the

experimental results that conductances to both liquids are similar.

1.4. Objectives of These Experiments

The above discussion illustrates the variety of present theoretical approaches. The fact that Kapitza conductance experiments, especially on metals,

have not really provided a test of the Khalatnikov theory or other theories

because the surfaces were not sufficiently well-defined has been the motivating

factor for the experiments described here. The objective of these studies was

to do experiments under controlled conditions in an effort to indicate whether

acoustic mismatch or an atomic interaction approach is most applicable. Attempts

to improve the Kapitza conductances of practical devices operating below the

lambda point are likely to be tedious and unproductive until more basic knowledge

of the problem is available. The difficulty with past experiments has been that

too many possible variables were poorly defined, and where attempts were made

to vary parameters, more than one was varied simultaneously, so results were

inconclusive. The experiments described here constitute a step forward towards

the goal of measuring the Kapitza conductance of an atomically clean, structurally

undamaged surface. The techniques used for sample preparation were those that

have been developed in surface studies using low energy electron diffraction

(LEED) , although the present apparatus does not incorporate such devices for

in situ surface characterization. A single crystal sample was prepared in an ultrahigh vacuum (UHV) environment by using low energy argon ion bombardment

to clean the surface and high temperature annealing to reduce damage. The

sample was then sealed against an indium 0-ring. A high vacuum was maintained

at the sample surface while the apparatus was converted to a cryostat for the

measurements of the Kapitza conductance to helium II. Measurements of the

Kapitza conductance were made over the temperature range of 1.6 to 2.1 K. LEED and Auger spectroscopy studies have shown, over approximately the last

decade, that many metals, insulators, and ionic crystals have a clean surface

structure that is essentially a "simple termination of the bulk", in which the

lattice spacing for the final few layers of atoms is larger than the bulk

spacing by only a few percent. However, the surface atoms may have a much larger

8
amplitude of vibration than the bulk atoms do. In contrast, semiconductors

with smaller coordination numbers for surface atoms and a more open lattice

exhibit rearrangements of the surface that have fewer "dangling bonds" than the

idealized structure.^"*" Such clean surfaces are frequently prepared in a UHV

environment with the pressure below 5 x 10 '^^ torr, using cycles of low energy

argon ion bombardment followed by annealing to remove embedded argon atoms and

superficial damage resulting from bombardment. The annealing is desirable be- cause ion bombardment does not necessarily leave a smooth surface. It is usu- ally necessary to use a low index face of a single crystal to get an ideal

surface. Although contamination of the order of 10% of a monolayer of foreign

atoms is difficult to detect with LEED, the identity of surface contaminants

can be determined with Auger spectroscopy, in which the energy of electrons

from atoms in the surface is detected. It was not feasible to use either of

these two techniques in this experiment, since to do so would mean a considerable increase in the complexity and expense of the apparatus. It seemed

fruitful, however, to do an experiment in which the procedures for obtaining

clean surfaces that were established by LEED and Auger studies would be used,

and the sample maintained in a UHV environment until purified liquid helium

was admitted to make the cryogenic measurements. This procedure should result

in the first Kapitza measurements on a metal surface that has a low dislocation

density, a fairly planar surface (so that the microscopic area is close to the

macroscopic area) , and little contamination (perhaps several monolayers at most)

of foreign atoms. Since a monolayer can be deposited on a fresh surface in only

one second in a vacuum of 10 ^ torr, preservation of such a well-prepared surface

in a much better vacuum is essential if heavy contamination is to be avoided.

A shiny metal surface prepared by a chemical etch and immediately installed in

a cryostat will still have an oxide layer some tens of angstroms deep, as well

as possible impurities from the etch and some physisorbed atoms, in spite of

its bright appearance. Also, to study the effect of damage in the layers next

to the surface, it is desirable to have the capability to remove this damage

by first bombarding the surface to remove foreign atoms, so that subsequent

annealing does not simply diffuse impurities into the surface where they may

interfere with recovery. Because of the strong probability that atomic collision calculations will be necessary to resolve the theoretical problems, it seems highly desirable at this point to make further observations of Kapitza

conductance only on well-defined single crystal surfaces.

In brief, the present studies with single crystal samples of copper showed

that operations which tended to make the surface more ideal decreased the con- ductance toward the original Khalatnikov acoustic-mismatch theory. Introducing

damage increased the conductance substantially, as did an oxide layer. An

attempt was made to vary only one parameter at a time in the course of the measurements. The state of the last few atomic layers of the solid seemed to have

the predominant influence on the size of the conductance.

2. DESCRIPTION OF APPARATUS

2.1. The Ultrahigh Vacuum System

The ultrahigh vacuum system is constructed of welded stainless steel, with

commercial flanges which are sealed by copper gaskets. All metal valves with

metal bellows and soft metal seats are used. A well-trapped diffusion pump is

supplemented by a titanium sublimation chamber. The system is bakeable.

Equipment for ion bombarding and annealing the sample is included. Figure 2 gives a schematic diagram of the system, while the size and relationship of the

cryostat and other components are shown in the photograph, figure 4. A commercial

titanium sublimation cartridge. A, is mounted in a chamber, B, in which the walls

are cooled by liquid nitrogen. This increases the efficiency of hydrogen pump52 ing which is important because hydrogen is expected to be a ma:]or constituent

of the gas load. The insulating vacuum space outside the wall enclosing the

9
TO DIFFUSION PUMP TO

and TRAP PUMP

t I

Figure 2. Schematic diagram of the vacuum system during sample preparation.

A, titanium sublimation cartridge; B, sublimation chamber; C, sample

chamber; Di, D2, 50 mm gold-sealed valves; E, tubing to which sample

is sealed after preparation; F, indium 0-ring groove; G, continuation

of tubing to which sample is sealed after preparation; H, pumped helium

bath; J, ion gun; K, sample manipulator; L, sample; M, radiation shield;

N, vacuum gauge; S, well; W, level indicator of tantalum wire; X, . helium inlet valve; Y, sample chamber flange.

10
TO DIFFUSION

and TRAP PUMP TO PUMP

Figure 3. Schematic diagram of the cryostat in which Kapitza conductance measurements are made. Di and D2, 50 mm gold-sealed valves; F, indium

0-ring groove; H, pumped helium bath; L, sample; N, vacuum gauge; 0,

harness which seals sample; P, split rings to which thermometers are

attached; R, cone on which heater is wound; S, well for germanium re- sistance thermometer; T, radiation shield at liquid helium temperature;

U, radiation shield at liquid nitrogen temperature; V, outer wall of

cryostat; W, level indicator (tantalum wire) ; X, helium inlet valve;

Y, sample chamber.

11
Figure 4. Photograph of apparatus. The sample chamber is at the lower right,

the cryostat is at the upper right, the titanium sublim.ation pump is on the left, and valves and connections to the diffusion pump are at

the top.

12
nitrogen is separate from the UHV system. The sublimation chamber is connected

to the sample chamber, C, by an elbow which is 150 mm in diameter. A baffle (not

shown) is a further precaution against titanium entering the sample chamber.

The sublimation pump is backed by a 100 mm oil diffusion pump. A cold cap helps

reduce backstreaming , and a commercial high conductance trap with a constant

cold zone at liquid nitrogen temperature prevents migration of any oil molecules

into the system. The trap was kept filled with liquid nitrogen by an automatic

system for the duration of the experiments. The forepump is equipped with a metal mesh trap. The entire vacuum system, except for the trap and pump, is

mounted in an oven with removable walls, so that the system can be baked to

400°C. The sublimation pump chamber is baked by operating the cartridge at

high power. The argon and helium gas lines leading to the system can be

evacuated and heated. The argon gas is further purified as it passes through

the sublimation chamber. Two high conductance gold seal valves, and Ti^t allow the sample chamber to be pumped via two paths: by the higher speed

titanium sublimation pump during sample processing, or by the diffusion pump

alone. One nude ionization gauge (not shown) measures pressure in the sample

chamber. Another ionization gauge is indicated by N.

The sample chamber contains an inner tube, E, with an elbow on which is

located a groove, F, for an indium 0-ring. Tubing E is connected by a com- mercial flange to tubing G which passes through the cryostat helium bath, H,

and on to valve D2 • The apparatus is designed to accept cylindrical samples

25 mm in diameter and 19 to 38 mm in length. The outer bellows on the sample

manipulator, K, is used to position the sample, L, for ion bombardment. The

dashed outline in figure 2 indicates the position for ion bombardment. An

ion gun, denoted by J, is mounted on a bellows that allows it to be moved

through an angle of a few degrees to cover the entire surface of the sample.

A second, inside bellows on the sample manipulator allows the sample to be

raised and lowered. Electrical contact to the radiation heater which is used

to anneal the sample is made through a feedthrough that is also located on

the sample manipulator. The heater is not shown in the schematic figure 2,

but is shown in more detail in figure 5a. It consists of tungsten wire which

passes through ceramic spacers which surround the upper two-thirds of the

sample. The heater assembly is enclosed within a molybdenum shield. The

heater is capable of raising the sample temperature to about 700°C in about

an hour. A thermocouple placed above the sample gives an approximate measure

of the temperature to within about 25°C, but avoids contamination of the sample

from surface diffusion at high temperatures.

Details of the sample sealing arrangement are shown in figure 5, since

the seal involved some development work. It was considered undesirable to

machine the sample because of the damage that would be introduced, although

this would have simplified the design of a reliable seal. Care is needed to

line up the sample manipulator with the 0-ring groove. The flange which mates

with flange Y is rotatable, as is the smaller flange welded to tubing E (fig.

2) . The sample chamber is aligned with a substitute sample when it is bolted

to the cryostat before each measurement. A beryllium copper contact ring with

many "fingers" that form an inverted cone helps to position the sample as it

is lowered to the 0-ring. The beryllium copper contact ring is located on a split ring support so that it can be removed after the sample is sealed. However, as the sealing force is applied, some rotation is allowed for at the

bearing-nut junction. Also, there is flexibility in the sample support because

it has been made partly of thin walled stainless tubing to allow enough

thermal isolation of the sample for annealing. Thus, during the application

of sealing force, the sample surface can align itself parallel to the surface

in which the 0-ring groove is cut. Some preliminary experiments on seals

showed that this provision was essential for a plane sealing surface. A

spring outside the vacuum system provides the sealing force of about 300 N. An indium 0-ring is used to take up the sealing force and minimize strain

damage to the sample. The sealing force is about 20 MPa if it is assumed

that all force is concentrated on the inner edge of the groove, which is

0.25 mm wide, and 3 MPa if the force is distributed over the entire groove

13
Figure 5a. Detailed view of sample chamber. A, beryllium copper spring; B, nut

which applied sealing force; C, inner bellows; D, outer bellows; E,

electrical feed through (connection to heater not shown); F, supports

of thin walled tubing; G, sample holder; H, sample heater; J, molybdenum radiation shield for heater; L, sample; M, radiation shield;

N, contact ring guide; Q, indium 0-ring groove; R, thin-walled stainless steel support.

14
Figure 5b. Sample chamber in cryogenic mode. Oi, beryllium copper rods; R, cone

on which heater is wound; O2, harness arrangement which seals the

sample; P, split rings of copper to which the carbon resistance thermometers are attached; S, well in which germanium resistance thermometer is inserted.

15
area. Observations have been that the indium adheres quite strongly to a position on the sample which corresponds to the inner edge, so the higher

figure may be more correct. Samples were examined by x-ray diffraction (Sec.

3.4.2) after sealing, to check for damage.

2.2. The Cryostat

Most of the cryostat is constructed of welded stainless steel, except for

the radiation shields which are copper. All welds were checked individually

on a sensitive leak detector during the course of construction and rewelded if

necessary. Repeated thermal shocking to liquid nitrogen temperatures was part

of the testing procedure. Consequently, there were no problems encountered

with superleaks in the welded or gasketed joints despite the large number of

thermal cycles from about 500 K to 2 K during the course of the measurements.

Part of the cryostat remains in place during the bakeout of the UHV system

and the processing of the sample, as may be seen by comparing figure 3 with

figure 2. After the sample is sealed, valve is closed, but the sample

surface is still pumped upon by the path leading through valve D2 • Air is

admitted to the system and the sample chamber access flange (not shown) is

removed. The sealing force is then transferred from the sample manipulator

to a harness arrangement, 0, indicated schematically in figure 3 and in detail

in figure 5b. Two beryllium copper rods (0^ in fig. 5b) now provide the spring

force as on the harness is tightened slightly. The three supporting pillars

connected to the sample manipulator are positioned so that the cone, R, which

transmits the force to the sample, and the beryllium copper rods can fit

between them. These supports are then detached from the sample holder and

raised by the sample manipulator. The sample chamber is next unbolted from

the flange Y and withdrawn.

The guide for the sample sealing operation is then removed, and the

thermometers, which are mounted on split copper rings, P, are attached to the

sample. Carbon resistors are used for thermometers on the sample. One lead

of each thermometer is soldered to the clamp to improve thermal contact. The

heater is located on the cone, R, which is already in place. A germanium

resistance thermometer is installed in the well, S, of the sample chamber by

using vacuum grease for thermal contact. (This, of course, has to be thoroughly cleaned before the next UHV operation.) A germanium thermometer is

used because it is desirable to know the absolute temperature of the helium

bath at the sample surface, without the complication of opening this clean

system to an oil manometer, since oil migration in a cold trap is difficult

to avoid. A liquid helium level detector, W, consists of a 70 ym diameter

tantalum wire which becomes superconducting in the liquid but not in the gas

if sufficient current is supplied to it. This detector, which is wound upon a glass cylinder, is bakeable and compatable with the UHV system. It remains

in place throughout the experiment. Because the large 20 cm flange Y that

remains attached to the helium bath did not cool down as well as expected (it

is welded to tubing of small diameter) , copper screws coated with indium on the ends are threaded through the bolt holes to touch the bath wall, as shown

by Z. Radiation baffles for the sample helium bath are already in place in

the neck of the sample chamber as shown in figure 3. They are designed not

to reduce the pumping speed appreciably.

After the sensors are in place, the rest of the cryostat is assembled.

The helium temperature radiation shield, T, is bolted to the cryostat bath,

H. A coat of indium on the contacting surface of the shield of annealed copper

improves thermal contact. The thermometer leads are tempered to this shield

with a polyimide varnish that has been baked on and withstands thermal cycling

well. A nitrogen- temperature shield, U, also made of copper, is bolted to the

bottom plate of the nitrogen bath. Again, lead wires have been tempered to this

shield. The outer chamber of the cryostat, V, can be raised by a pulley arrangement and bolted on to the top flange of the cryostat. A smaller flange at the

bottom contains feedthroughs for the electrical leads. Both plates are sealed

by rubber 0-rings. The cryostat is pumped out by a separate diffusion pumping

16
system to a vacuum in the 10 torr range. A sensor resistance thermometer and

a heater for temperature control are placed in the bath, H. A helium level detector mounted on the end of a thin tube is also placed in this bath. The

cryostat is precooled by admitting liquid nitrogen to space H and waiting several

hours for the large mass of shields, cryostat walls, and flanges to cool. Then

the cryostat is pressurized to blow out the nitrogen, helium is transferred to

chamber H, and helium gas from a storage cylinder is admitted to the sample

bath via tubing that passes first through a liquid nitrogen trap and then through

coiled tubing in bath H. The helium input lines are heated during each bakeout

to remove impurities. Further impurities should be removed in the tubulation

at helium temperature. (Coiling is not shown in the schematic diagram for simplicity. ) Valve X is provided in the event it is necessary to warm up the system

again to repair some cryogenic problem. Then most trapped impurities will be

prevented from migrating to the sample chamber.

The pumped chamber of the cryostat holds about seven liters of liquid.

With a loss rate of about 0.5 liters/hr., there are about eight hours for data

accumulation after a helium transfer and pumpdown. Since it was necessary to

make wall thicknesses relatively large to prevent permeation into the vacuum

system and to support the heavy vacuiam flanges, this loss rate was expected.

The sample chamber helium bath is about 0.05 K warmer than the pumped bath,

and this was expected. Design here was obviously a compromise because the

vertical tubing to the bath had to be made large enough to allow a good pumping speed at the sample after sealing, yet a large diameter here allows more

superfluid helium film flow and more radiation and conduction heat leaks.

2.3. Electrical Measurements

An electronic temperature controller stabilizes the temperature of the

pumped bath, which in turn stabilizes the sample bath. At each temperature,

zero power readings are obtained from the two carbon thermometers and the

germanium thermometer. Then the change in thermometer resistance is measured

for several power inputs to the heater. The germanium thermometer was calibrated

separately and rechecked during the course of the measurements. There is a three wire connection to each carbon thermometer, with the current being re- turned through the common ground. Current is supplied to each thermometer in

turn, only as it is read. Power levels to the thermometers were less than a nanowatt. Measurements are made on a digital voltmeter with a dc system in

which the current is reversed to cancel extraneous emfs. Heating power measurements were obtained by measuring the voltage across the heater, using a separate set of voltage leads, and the current into the heater resistance.

3. EXPERIMENTAL PROCEDURES AND SOURCES OF ERROR

3.1. Preliminary Sample Preparation

The procedure used in cutting a sample to size is important if excessive

damage is to be avoided. Conventional methods of cutting can introduce dislocations up to a depth of several mm in copper, ^"^ and even spark cutting

54 results in some damage. Sample 1 was cut by the supplier. Semi-elements, Inc.,

by an abrasive string saw and etching procedure. Sample 2 was spark cut,

according to the supplier, Aremco Products, Inc. The heat transfer face of

sample 2 was rough cut in this laboratory by spark cutting, followed by heavy

etching and electropolishing . About 0 . 5 mm was removed. The specified orientation of the heat transfer face was (100) to within 2'^ for both samples.

VJhile this was not checked precisely, analysis of the perfection of the crystal,

with the x-ray dif f ractometer (described below) indicated that there could not

have been any significant deviation from this orientation. The purity of

sample 1 was specified as 99.995%, and sample 2 was specified as 99.999% pure.

A rough measure of the thermal conductivity of the samples was obtained as a by-product of the Kapitza conductance measurement, and this indicated that the

conductivity of sample 2 was higher by about a factor of two.

Further smoothing of the sample surfaces was accomplished by electropolishing, except for measurement lA. This surface was prepared by an automatic

17
mechanical polisher using diamond abrasives of 30 ym and 15 ym size. Diamond

abrasives are thought to cut away high regions readily and thus cause less damage

than soft abrasives. Final polishing was done with 0.3 ym alumina abrasive.

This mechanical polishing was followed by chemical polishing intended to remove

some of the damage and imbedded impurities from the polishing. The solution

contained 22% HNO^, 54% H^PO^, and 24% acetic acid, and was used at a temperature between 60 and 70°C. The total amount removed was 40 ym. Some pitting occurred with a pit size of several microns. This was seen only in the

single crystal sample and not in the polycrystalline sample holder, and so may have been due to impurity inclusions in the sample.

The other surfaces investigated, IB through IF, and 2A to 2C, were prepared

55 to a mirror finish by electropolishing . The amounts removed were 0 . 1 to

0.2 mm each time. The electrolyte consisted of a 60% solution of phosphoric

acid with CuO added at the concentration of about 15 kg/m"^ . The cupric ions

suppress the liberation of hydrogen gas at the cathode, so that the cathodic

overvoltage does not change with time, and a stable polishing condition is

obtained. A cell voltage of about 1.10 V was set by measurement at the sample

surface with a copper reference electrode. The sample is placed in a jig which

serves as a guard to prevent rounding of the edges. The bottom of the beaker

which is used for the electropolishing was ground flat, because the flatness

of the surface obtained on the sample is similar to the uniformity of the

electropolishing cell. After the electropolish , a rinsing solution of 5% B.^'PO^ is used, followed by deionized water. A final, very brief (about 20 seconds)

chemical polish was carried out just before the sample was installed in the

apparatus. This was followed by rinses in deionized water, and in highly

filtered methyl ethyl ketone, after which the sample was blown dry in a stream

of nitrogen gas.

3.2. Sample Preparation in the UHV System

The sample chamber and its components, including the indium 0-ring, are cleaned carefully before each run. The usual cleaning procedure consists of

vapor degreasing in hot trichloroethylene , agitation in a detergent solution

in an ultrasonic bath, numerous rinses with hot water followed by deionized

water, and finally, a rinse with highly filtered methyl ethyl ketone followed

by drying in a stream of nitrogen gas. The last procedure was necessary be- cause the apparatus was not located in a clean room, although a clean bench

could be used for preliminary assembly work. All work in the sample chamber

was carried out with degreased tools and nylon gloves.

— 8 After assembly, the system is evacuated to a pressure in the 10 torr

range by the diffusion pump, and baked out for about 24 hours. The maximum

temperature in the oven was about 190°C, but the temperature of the indium,

which is located near the bottom of the oven, remains below the melting point

of 156°C because there is no forced air circulation in the oven. Bakeout

is followed by outgassing of the filaments and heater, and bakeout of the

titanium sublimation pump. After operation of the sublimation pump, the

pressure in the sample chamber falls to the 10 ^ torr range as measured by

the nude ionization gauge mounted in the sample chamber. The upper glass

gauge reads a slightly higher pressure. (Gauge pressure was not corrected

for the residual gas composition, which was not determined. This would not

change the pressure by more than a factor of 2.) After a good vacuum is

achieved, research grade argon (99.9995% purity) is admitted, and ion bombardment is started in the tilted position. Typical ion bombardment parameters

were: 300 V, 8 yA, pressure 2 x 10 torr, for a time of 20 minutes, followed

by an anneal at 650°C for 1 to 2 hours. This procedure is usually carried out

twice. Specific information for each sample is shown later in the table of

section 4. According to published figures for yields, an argon ion bombardment

2 current of 8 yA/cm for 1/2 hour should remove a thickness of approximately

0.6 ym from a smooth, electropolished surface. During the anneal, the sample

18
is returned to the vertical position so that the radiation shield for the

indium will be effective. The purpose of the anneal is to remove argon gas

that may have diffused into the sample, and to heal the moderate damage to the

surface layer that may have occurred during the low energy bombardment. As

noted earlier, a thermocouple was placed just above the sample to minimize

contamination. Visual examination of the glowing sample gave an approximate

verification of the readings. Annealing temperatures for each measurement

are noted in the table.

3.3. Cryogenic Procedures

3.3.1. "Storage" Vacuum

After the final anneal, some hours must elapse before the sample is cool

enough to seal because thermal conductances from the sample to other parts of

the chamber are small. During this time, the vacuum is in the low 10 '^^ torr

range. This is the range that, from LEED studies, is usually found to be

adequate for preservation of a clean surface for a period of hours. The

minimum monolayer time at 10 ^ torr is 3.6 hours, if all incident molecules

stick to the surface. However, after the sample was sealed, the vacuum at the

heat transfer surface was not as good as initial tests had indicated. This

_ g reading, indicated in the table in section 4, was usually in the 10 or upper

-9

10 torr range. There is some possibility that this reading may have been

more a characteristic of the gauge tubulation than of the pressure seen by

the sample, since the pressure, measured at room temperature, did not decrease

as the cryostat was cooled to liquid helium temperatures. The gauge is con- nected by 18 mm I.D. tubing to the 50 mm tubing that connects the sample to

the pumping system. After the seal is made, 6 to 8 hours are normally spent

in transferring the sealing force, removing the sample chamber, attaching

thermometers, and assembling and pumping out the cryostat, during which the

-9

sample is exposed to a vacuum in the 10 torr range. Liquid nitrogen is then

transferred into the pumped helium chamber. After overnight cooling at liquid

nitrogen temperatures, followed by several hours of helium gas cooling of the

rather massive (20 kg) system, liquid helium was transferred into the pumped

chamber, and then the gaseous helium was condensed into the sample chamber

through the system of traps described earlier. Thus the sample surface was in

a low temperature environment for 20 to 24 hours before measurements were

actually underway. In some cases, cryogenic problems arose during the cooldown

and an additional storage time occurred. This is indicated in the table. It

-9

is not known with certainty whether the vacuum was better than 10 torr during

this period. It is felt that this cooling probably reduced the pressure at

the sample surface, though, as noted above, this was not reflected at the

room temperature gauge. Molecules from the rest of the system should be

trapped out to some extent before reaching the sample surface, and local

outgassing from the walls surrounding the sample should be reduced. These

effects are counterbalanced by observations that surfaces held at low tempera57 tures generally seem to deteriorate faster, presumably because of the higher

5 8 sticking probability at low temperatures. On the other hand, Rhodin has

observed that oxidation of copper surfaces is retarded at low temperatures,

59 and Lee and Farnsworth felt that true oxidation of a cleaned surface did

not occur except in the presence of sizeable amounts of impurities, such as water molecules.

3.3.2. Thermometry

Before the cryostat was given a final cleaning and attached to the UHV

system, several measurements were made to verify that the cryostat would

perform as desired to make Kapitza conductance measurements. For these tests,

the sample bath was connected to a sensitive oil manometer that could measure

temperatures below the lambda point to within approximately 0.001 K. It was

found that the germanium resistance thermometer, mounted in a well extending

19
into the sample chamber as described in section 2.2, followed the sample bath

temperature quite closely as heat was put into the sample heater, or as the

pumped bath temperature was changed. The temperature measured with the germanium thermometer is offset from the sample bath temperature by an amount

which varies slightly from run to run, probably depending upon the thermal

contact achieved with the vacuum grease. The offset varies with bath temperature during a given run, from about 0.035 K at 2 . 1 K to about 0.045 K at

1.5 K. This change is not expected to seriously affect the in situ calibration

of the carbon resistance thermometers at zero power, since the slope dR/dT of

the calibration curve does not change significantly over 0.-010 K, and it is

this quantity which is involved in determining the differences in temperature

that occur when heat is applied. In two of the measurements the leads to the

thermometer developed faults at low temperatures, and so the bath vapor

pressure was measured using a temporary connection to an oil manometer with

a liquid nitrogen trap. No significant differences were found in the temperatures determined this way, as compared to those determined with the

germanium thermometer.

Another question about the thermometry is whether an annular ring on the

outside of the sample can accurately measure the temperature distribution on

a plane through the crystal when the length-to-diameter ratio is close to one. Because of the smallness of the phonon mean free paths and high conductivity,

this is thought to be the case for the present system, for in principle,

trouble would only be encountered for certain alloys with a small electronic

thermal conductivity, yet high mean free path for phonon collisions with

6 0 electrons. Also, a check was made by measuring the second sample, 2, of

twice the length-to-diameter ratio of ID-IF. The distance of the sensors

from the heater was also increased on this sample. Good reproducibility was

found, as discussed later in section 4.

The thermometer voltages and currents were read on a digital voltmeter

that has a quoted instrumental accuracy of ±0.3 yV. Noise and variations in

time of the thermal emfs resulted in a total estimated inaccuracy of about

1.5 yV in each primary voltage measurement. After a number of operations on these primary numbers, the resulting error in the measured temperature difference at the interface ranged from about 1 to 15%, depending on the conductance

of a given sample and the power level applied to it.

3.3.3. Parallel Heat Flow

Figure 6 is a schematic diagram of the various heat flow paths discussed

below. The heat flow from the heater through the supports which furnish the

sealing force was measured and found to be negligible: It was less than 0.5%

of the heat flow through the sample surface. The heat loss through the thick

part of the stainless steel tubing in which the indium groove is cut amounts

to about 2% of that passing through the sample area exposed to helium. This

9 correction is calculated using the recent measurements of Cheeke, et al. as

6

1

an estimate for the conductance, and using fin calculations. Calculation

shows heat leaks along the lead wires of 0.08 mm manganin to be negligible.

Since these leads were coated with a thin superconducting layer of lead-tin

solder. Joule heating would also be expected to be negligible.

The greatest problem with parallel heat leaks occurred because the indium

from the seal tended to extrude along the sample to a much greater extent than

had been anticipated. Separate measurements, without UHV sample preparation,

were made with a soldered joint instead of an indium joint for a comparison,

and these confirmed the importance of the effect. The Kapitza conductance of

two samples of the polycrystalline indium used in the seal (99.99% purity,

from the Indium Corporation of America) was therefore measured in separate

experiments. These experiments indicated that the correction from the indium

could be as much as 20%. Details of these measurements are given in appendix

A. In making the correction to the data for the heat flow through the indium,

uncertainties arise from two factors: the difficulty in measuring the area

of the extruded indium exactly, and the uncertainty in the value of -the Kapitza

20
n) s o <u

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4-1 iH 1 C H to 01

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•fH 4-1

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aj CO

u

01 0 4-1 0 4-1 C CJ •H 3 CU 3 ,H o c o rH c

4-1 T3 t4-l o

n) CO

m CJ <u -C 4J -C en CO CC) ^

to CU aj o a x: CU u 3

4J

u 4J <

(U

an

nj

O CJ

u 4-1 n) •H C o B M-1 (U

•r-l <u •H cn 4J 4-1 0) C CU cn 4J 00 u g >i c •1-1 a.

1-1 cn

au o 4-1 Td CO Ul c H cn

(U l-( CU -a g O C c

<u (11 CO CU •H iH u 1 cn •H 3 XI CU

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m •H !-i c

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2]
conductance of the indium. The latter uncertainty is by far the most important.

The area of the indium was determined by a photographic technique and was

found to be nearly the same in all cases. The expected uncertainty due to the

indium conductance is determined from the results on the two samples measured,

and is included in the error bars shown on the plots of the results. However,

since the indium used is from the same roll of wire (except for run 2A where a different roll was inadvertently substituted) and undergoes the same. treatment

in each measurement, results for different measurements can be compared in a relative sense. One aim of the experiments was to see whether better agreement

could be found with the ideal acoustic-mismatch theory by better surface preparation, and since previous experiments were in disagreement by an order of

magnitude or more with this theory, an absolute uncertainty of 10 to 20% was

not of great concern. The effects of different surface preparation were far

larger than this.

3.4. Analyses of Sample Properties

3.4.1. Surface Area

The area of the exposed copper in the sample was taken to be the geometrical area. Absorption studies have indicated that for similar methods of

preparation of metal surfaces, the enhancement over the geometrical area is

57 5 8 not large. ' This factor would not, therefore, be sufficient to reduce the

conductance more than 10 to 20%, which would not result in any significant

change in agreement with theory. Sample 1 had more surface structure, of

the order of micrometers, than did sample 2 after polishing. Pits and small

hillocks, of about 10 to 50 ym size, developed at scattered points on the surface, covering perhaps 2 to 4% of the total area. These structures did not

appear on the polycrystalline sample holders that were used in the polishing

processes to prevent rounding of the edges, and they were also almost

entirely absent on sample 2, so it is believed that they might have been

nucleated by impurities or defects in sample 1. Sample 2 appeared to be without

structure when examined under a metallurgical microscope; that is, on a scale

of the order of 5000 S.

3.4.2. Analysis by X-Ray Techniques of Crystal Perfection

After the first few measurements, the sample was examined by Laue back

reflection photography. Although the spots were sharp, indicating a good sur- face layer, the results were only qualitative, and numerical comparisions

were difficult. After some of the later measurements, it was possible to

examine the sample on an x-ray diffTactometer . The radiation used was the

cobalt Ka line, which has a penetration depth in copper of about 14 ym.

Significant differences in the linewidths of the (200) reflection were found,

which seemed to be well-correlated with surface treatment. The results apply

only to a spot about 3 mm in diameter, approximately in the middle of the heat

transfer surface. After measurement IC, on a very well-annealed sample, the,

width at half height, in terms of 6 (the angle of reflection) was 0.032° for

the electropolished and ion bombarded surface, and 0.060° for the opposite

surface, which was untreated and subject to pressure from the heater cone.

It was not possible to directly examine the spark-cut surface of sample 2A

due to scheduling problems, but the matching surface of the small piece that

was cut away from this sample had a width of 0.12°. The actual surface 2A was probably in somewhat better shape than this because it was heavily

electropolished. However, these values give some indication of the improvement of the surface IC over a spark-cut surface. The sample was examined

after run 2B and found to have a width of 0.035°, which shows an interesting

agreement with IC, since these two surfaces were prepared as identically as possible. Finally, the intentionally damaged sample ID was examined and found

to have a width of 0.12°. These results are included in the table. It should

be noted that these linewidths have a relative significance, but not an absolute

meaning, because there was no absolute calibration in terms of beam width, etc.

Also, of course, the sampling depth may be much deeper than is relevant for the

actual mechanism of energy transfer, since it is about 300 times larger than

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23
the phonon wavelength in the copper at 2 K, and as indicated earlier, the relevant depth may be considerably less than even this. Nevertheless, the results

of the analysis are mentioned because of the good correlation with the measurements, as discussed below.

3.4.3. Purity

6 2 The eddy current decay method was used to obtain the ratio of the

specimen resistance at 273 K to its resistance at 4 K (RRR) . This non- destructive method is particularly useful for large, relatively pure samples.

The high temperature resistivity is characteristic of the base metal, but the

low temperature resistivity reflects the impurities and physical imperfections

of the individual sample. Both samples had been annealed before examination

to remove as many physical defects as possible. Of course, this method is

sensitive to only impurities which act as electron scatterers, and thus impurities such as iron and the transition metals affect the RRR most strongly.

The stated purity of sample 1 was 99.995%, according to the supplier, and its

RRR was 70. The RRR of sample 2 was 380, which is significantly higher, and

its stated purity was 99.999%. As will be seen in the next section, this

difference in the RRR did not appear to affect the Kapitza conductance.

4. RESULTS AND MEASUREMENTS

Before examining the results in detail, the procedure for going from the raw data to the information presented in the figures will be discussed. Several

corrections to the data must be calculated.

4.1. Corrections for Parallel Heat Flow

The heat transfer through the indium is subtracted from the power input,

using an average value for the indium conductance based on the results given

in appendix A. The much smaller parallel flow through the stainless support

tubing is also subtracted. For each temperature, a plot of the interfacial

temperature difference against power per unit area of copper is prepared.

(This quantity is corrected for the 3/2(AT/T) effect when this factor is

2 appreciable. ) It is the slope determined from these points that is plotted

in the figures as h . No effect of high power analogous to that reported by

63 Cheeke, et al. was seen in the AT plots, but the inaccuracy in AT may have

been large enough to obscure this. Also, power densities in this work ranged

2 from 5 to 4 0 mW/cm , whereas the anomaly was observed by comparing the slope

2 below about 5 mW/cm with the slope at higher power levels. If undetected in

this data, the anomaly could result in an underestimation of hj^ by about 25%.

In the present studies, only equilibrium measurements were made, in contrast

to the system employed by Cheeke, et al. in which power slowly increased with

time as a ramp function. For some very high power inputs, equilibrium could

not be obtained because of bath temperature drift, and so these data were dis2 carded. A high power effect for densities of about 40 mW/cm was observed in

the measurements on indium, where the geometry was more like that of Cheeke '

s

group, but it was eliminated by simple precautions (see appendix A)

.

The errors shown in the figures are due to the inaccuracies in temperature

measurement, plus the uncertainty in the indium conductance as estimated from

the two samples measured. Each effect contributed about half of the error.

Of course the indium uncertainty, unlike the temperature jump uncertainty, is not random, and would have to lie in the same direction for each

temperature point of a given set of measurements. In evaluating the random

scatter of the data for a given sample, then, the errors should be mentally

divided in half. To allow for the uncertainty of the indium conductance, however, the whole curve should be shifted up or down by about half of the error. Another error is not shown in the figures because it is roughly similar for

almost all points. This is the uncertainty of about 0.02 K in the temperature

readings of one set of measurements relative to another, due to the somewhat

indirect method of deriving the bath temperature. However, temperatures for

24
samples IC and ID were obtained from the vapor pressure of the sample bath,

and so should be more accurate. For these measurements, the system was opened

to the manometer only after the liquid helium had been condensed into the

sample chamber. Since the slopes of IC and ID are quite different, lack of

accuracy in temperature measurement clearly is not responsible for this effect.

4.2. Copper Single Crystal Sample 1 Figure 7 shows the data obtained with sample 1, after various types of

surface preparation. The abcissa of this and the following figures is the

average temperature of the bath ana solid. The data separate into two groups

of high and low conductances, which also exhibit high and low exponents of

temperature dependence, except for sample IE. The difference in conductances

between the two groups is a factor of 2 or 3. In the higher grouping, measurement lA was obtained after the sample had been mechanically polished, as gently

as possible, and then chemically polished and ion bombarded and annealed in vacuum. However, the annealing temperature after the final ion bombardment was

480°C, which was not as high as the temperatures of over 600°C for the other

anneals. Measurements IB and IC, which showed a much lower conductance, were

made more than a year later, so some healing may have occurred at room temperature in that time interval. A similar phenomenon may have been seen by Anderson

1

6

and Johnson at low temperatures. Measurement ID, which closely reproduces

curve lA, was made after severe mechanical damage had taken place. An ion bombardment was carried out to remove surface impurities, but the sample was not

annealed after this. The sample surface had been exposed to air in situ for

several days at room temperature before measurement IF was made. It also lies

in the upper group. However, except for this exposure, conditions were identical

to measurement IE, on an annealed surface, which preceded IF, and which lies in

the lower group. The fact that nothing was changed between these two measurements, except that the sample surface was exposed to the air, tends to eliminate

various spurious causes of the pronounced differences in magnitude between the

two groups, such as unintentional variations in sample sealing forces. The

sample surface was pumped on at room temperature for approximately a 24-hour

period before measurement IF to remove the more loosely adhering molecules from

the surface.

Annealing preceded all the measurements in the lower group of conductances.

Final anneal temperatures ranged from about 625 to 675°C. Also, measurements

IB and IC were made after the sample had been heavily electropolished (0.025 cm removed each time) , so it would be expected that most mechanical damage introduced into the surface by the mechanical polishing had finally been removed.

The effects of damage and annealing are seen to be approximately reversible:

The lowest conductance obtained for an annealed sample, measurement IC, was followed by the highest, ID, after the sample had been heavily damaged. Then,

after measurement ID, further electropolishing and annealing returned the

sample to a conductance that was lower by a factor of two, although it does

not have the small slope of IC. This may have occurred because all the

dislocations introduced could not be removed. It should also be noted that

before sample ID was machined heavily, about 6 mm were inadvertantly spark

planed off the sample, yet subsequent measurements approximately reproduced

the curve for lA, This, and the subsequent reversibility shown by IE, tends

to diminish the possibility that a different impurity composition of the sample

in its interior could account for the differences in conductance, because the

results do not seem to be a function of depth.

4.3. Copper Single Crystal Sample 2 Figure 8 supports similar conclusions for sample 2, though the measurements on this sample were less extensive. Measurement 2A was made on a spark

cut surface, though much of the damage was undoubtedly removed by the heavy

electropolishing that was necessary to reduce the somewhat rough cut to a smooth

surface. Also, this sample had a layer of oxide, plus whatever impurities re- mained after the chemical polish and final wash, although it was placed under vacuum within about an hour after this procedure. An oxide layer of about

20 8 would be expected to form^'^ during this time. Thus, this sample is similar

25
26
Figure 8. Kapitza conductance measurements of single crystal sample 2. Symbols

are identified in the text and the table.

27
to IF except for probable damage. The sample received further electropolishing

and the usual UHV treatment before measurement 2B. The conductance was lowered

by a factor of two at 1.9 K, and again the exponent of temperature dependence

also decreased. These results are similar to those for sample 1. Good agreement is found between measurements IC and 2B, as would be expected, since the

surface preparation conditions for these two measurements were as identical as possible. The two measurements are compared in figure 9, which shows the good

reproducibility. The small lack of agreement in the slopes may indicate

only the uncertainty in measuring that quantity in the present setup, due to

both the use of a thermometer well and to the lack of a suitable number of

points over a wide temperature range.

The table shows that the x-ray linewidths, where available, correlate with

the observed conductances. This is true in spite of the fact that the linewidth

is composed partly of contributions from layers far below the surface layer

that would not be expected to affect the conductance.

It had not been foreseen that the indium sealing material would extrude so much and cause such a sizeable uncertainty in the results. From the details of

the indium measurements presented in appendix A, it may be seen that the mea- surement conditions approximate, but are not identical to, the conditions under

which it was used in the experiments. This does not affect the intercomparison

of the various results (except that the influence of the atmosphere upon the

indium conductance in IF and 2A may have been an additional variable) , but it

does affect the absolute value of the results as compared with theory. If the

indium conductance used to correct the data is too small by an order of magnitude, the magnitude of the experimental curves shown in figure 8 could decrease

considerably. However, when the sample parameters were varied, but those of

the indium were unchanged, as between measurements IC and ID, a sizeable change

is seen in the conductance before it is corrected for parallel heat flow. This

indicates that the indium conductance is not swamping the sample conductance.

A study of the results for sample 1 indicated that the effects observed

could be explained by changes in the composition and/or structure of the last

few layers of atoms on the solid, rather, than by modifications in the bulk

that were of the order of the phonon wavelength, '\^400 8, or larger. It was hypothesized that perhaps lA gave a relatively high conductance compared to

IB and IC because the anneal at 480°C after the ion bombardment was not at

quite a high enough temperature to completely remove the surface-disordering

effects resulting from the bombardment, although earlier anneals should have

removed the bulk damage. (However, LEED work with anneal temperatures in this

range has seemed to produce good surfaces, though much work has been done

59 using higher temperatures. ) Also, the high conductance observed in measurement ID could be due to the ion bombardment without anneal rather than the

deeper damage caused by machining. This ion bombardment was carried out so that foreign atoms could be removed from the surface to the same extent as

in other measurements. Furthermore, measurement IF exhibited a noticeably

higher conductance than measurement lE, yet the conditions were identical,

64 except that the IF surface probably had acquired a cohesive oxide layer that was very much less than the phonon wavelength. Consequently, measurement

2C was undertaken to test this hypothesis: The sample would be lightly electropolished (mainly to remove indium adhering from the previous run) ; given the

usual UHV treatment, including a preliminary, outgassing anneal; and given

an anneal after the first sequence of ion bombardment to remove any bulk damage

that could have been caused by handling the crystal after measurement 2B.

However, the crystal would not be annealed after the second, final ion bombardment. After this procedure had been carried out and helium condensed

into the apparatus, an unfortunate accident occurred which prevented the

completion of the conductance measurements in the usual manner. The helium

level detector in the pumped helium bath became lodged in the coiled tubing

through which the helium is condensed to the sample bath, shorted out because

of a defect in the insulation, and made a small hole above the liquid level

28
ure 9. Comparison of measurements on annealed samples IC and 2B. Previous

experimental work is represented by the shaded area. (Compare fig.

1) Theoretical curves are explained in the text.
in the pumped bath. This caused the helium in the sample chamber to be pumped

out until, presumably, only a thin film layer of helium II remained. The

measurements were attempted anyway and are denoted by the arrows in figure 8.

Since the helium film would eventually evaporate as more power was put in,

and a very low conductance would then suddenly be observed, the base of the arrow represents a qualitative lower limit, since some finite amount of power

had to be used to obtain an observable temperature difference. The measurements do seem to indicate that the conductance had increased markedly > in support of the hypothesis that the last few surface layers are determinate. If

this conclusion is correct, it could explain some of the previous disagreements

9 66"

between authors on the effects of surface damage, such as those on nickel, ' since damage to the bulk could be similar, yet differences in the last few

atomic layers near the surface could cause significant differences in conductance. If helium atoms exchanging energy with the solid "see" only the outermost layers, the effect of bulk damage could be obscured with oxide layers.

Alnaimi and van der Sluijs ^ have observed changes in conductance with the

length of time under vacuum which might be due to superficial layers of adsorbed atoms, although the state of the vacuum and surface preparation techniques were not comparable with the present work. Results for samples ID

(damaged) and IB and IC (annealed) are in agreement with previous work of

8 Mittag on polycrystalline samples, also carried out in this laboratory, in

a similar temperature range. Although there is apparent agreement between

12 this work and that of Folinsbee and Anderson at 0.0 3 and 0.3 K in that coldworking increases the conductance, a closer examination suggests some problems.

The temperature dependences observed here, if simply extrapolated to lower

temperatures, would cause curves ID and IC to cross, and thus reverse the

effect of cold work. Also, since the present results indicate that the

condition of the surface layer could be more influential than the state of

the bulk, it is difficult to assess agreement with the work at lower tempera33 tures, as deeper layers may be operative at low temperatures, and because

surface parameters in the low temperature work were not controlled in a manner

similar to this work.

5. CONCLUSIONS

In figure 9, the results for the two most ideal surfaces are compared, on

a compressed scale, with the original, acoustic mismatch theory of Khalat41 nikov, eq. (4), the early modifications by Challis, et al., and the upper

limit for phonon transmission, eq. (3). The shaded portion of the figure shows

the range of previous measurements with polycrystalline samples. Though agreement with the acoustic mismatch theory is still lacking by an order of magnitude,

the data presented in figures 7 and 8 show that making surface conditions more nearly ideal did lower the conductance by a factor of two or three, towards

better agreement with the theory. The large exponent of temperature dependence

decreased also to agree somewhat more closely with the theory. In view of the

large effect of an oxide layer formed in the atmosphere (measurement IF) , and

-9

the fact that the monolayer time in storage vacuum of 1 x 10 torr is less

than 1/2 hour, further work under improved vacuum conditions might lower the

conductance even more. However, it should be pointed out that the type of

oxide layer that would form during exposure to a complete atmosphere is con- tinuous and fairly homogeneous, whereas that formed over a period of time in

a vacuum will usually be inhomogeneous^^ and may have structural differences.^^

It would also be desirable to examine the extent of damage of the material more carefully and to have iri situ facilities, such as LEED and Auger instrumentation,

for determining the perfection of the surface. It would be expected that some damage would occur near the sealing area. Also, the possible effects of low

levels of impurities were not clarified in this study, as the number of samples

studied was too low, and time did not permit the study of an extremely pure

sample as had been originally planned. Thus, a more definitive test of the

theory must await more sophisticated techniques. In this regard, it is interesting to note that the conclusion from the present work that improving sample

30
surface conditions lowers the conductance is opposite to our earlier, tentative

2 conclusion based on an examination of the data that was available then. The

difference is presumably that there were no comparisons available earlier for one sample in which only one surface parameter at a time was varied.

There is a strong indication from the present work that the depth of the

disorder at the surface need not be comparable to the depth of the phonon wavelength in the solid to have a significant effect, but is more likely to be of

the order of a few atomic spacings. Further work in this direction with an even better method of preserving the sample surface, once prepared, would be

most desirable. This conclusion is quite important from the practical standpoint of developing good low temperature heat transfer surfaces: If only the

outermost layer of the solid need be disordered to improve heat transfer, then

there is no need to consider, for instance, extensive coldworking which could

seriously impair the thermal conductivity through the bulk, and perhaps not

improve the overall situation- From a theoretical standpoint, this conclusion

tends to favor a desorption theory of energy transmission, as do the phonon

transmission experiments. Surface roughness and adsorbed layers provide

mechanisms for differences in energy exchange with incident atoms — for ex- ample, energy exchange increases as the binding energy between adsorbed particles

27 and the lattice decreases, according to calculations at higher temperatures.

The conclusion that the last few layers are important in energy exchange may

2 also explain the observation that the temperature dependence is roughly T for

the cleaner, annealed surfaces: At low temperatures, the specific heat of the

68 2 surface layer is proportional to T , whereas that of the bulk usually goes

as T^.

One motivation for this study was the recognition of the difficulty in

improving practical heat transfer surfaces without better knowledge of the

mechanism of heat transfer at the solid-helium II interface. The wide range

of over an order of magnitude for the Kapitza conductance of copper surfaces

shown in figure 1 is an indication of the problems encountered when the surface

properties are not well-defined. From the results of the present work, it now appears that changing the macroscopic layer of material beneath the surface

may have little effect, unless the last few layers of the bulk also happen to

be altered by this manipulation. Also, the results indicate that oxide layers

could easily overshadow more subtle changes. This effect might well obscure

efforts to find out what changes in surfaces are important, if the surfaces are exposed to the atmosphere or an inadequate vacuum before the Kapitza conductance

measurement at helium temperatures.

ACKNOWLEDGEMENTS

The author thanks Dr. V. D. Arp for many useful discussions and for

considerable help in the design of the apparatus. Dr. A. Tench kindly carried

out the x-ray examinations of the samples. The assistance of G. Wallace was essential. Dr. F. Fickett supplied the RRR determinations. E. Ballinger,

W. Decker, and many other staff members were most helpful.

31
APPENDIX A, MEASUREMENTS OF THE CONDUCTANCE OF POLYCRYSTALLINE INDIUM

A sample, 25 mm in diameter, of the polycrystalline indium used in making

O-ring seals was obtained from the Indium Corporation of America. The purity

of the indium is quoted as 99.99%. A 25 mm long portion of this sample was machined off, degreased, and sealed by pressure into the apparatus. The seal

leaked slightly below the lambda point, which caused some deterioration of the

vacuum and thus an additional heat leak. This resulted in more scatter in the

results than usual. Consequently, additional measurements were made in a second, auxiliary apparatus which had previously been used in testing various

geometric configurations for the indium seal. The indium sample could be

soldered into this apparatus, using a low melting point solder. A schematic

sketch of the apparatus is shown in figure 10. Fine lead wires of 0.005 cm manganin coated with a superconducting lead-tin mixture appeared to isolate the

thermometers and heater satisfactorily, although the apparatus was small. After

some experimentation with methods of sealing the indium to the apparatus, it was

found that soldering with Wood's metal in an oven with a temperature of about

75°C would produce a seal that would withstand thermal cycling, and would not

leak below the lambda point. Because the thermal contraction of indium is quite

large compared to that of most metals, the joint was made to thin stainless

which could flex slightly. A further problem ,was encountered in finding a suitable bonding agent for the heater, since bakeable varnishes which withstand

thermal cycling well were unsuitable because of the low melting point of the

sample. The clamps on which the thermometers were mounted tended to work loose

on the sample, due to extrusion of the soft indium. The poor adhesion of the

heater and looseness of clamps sometimes resulted in data which was power dependent. This was eliminated by correcting these problems, and by enclosing

the heater with a radiation shield (see fig. 10) so that radiation from the

heater which did not thermally equilibrate with the walls of the vacuum enclose

could not strike the resistance thermometers.

The results from two measurements each of the first sample, 3, and a second,

longer sample, 4, are shown in figure 11. The results are consistent for each

sample (the two measurements with sample 3 were taken over a year apart in

different apparatus), but they differ from each other by about 45%. Because of

this consistency, it appears at present that the differences should be ascribed

to dissimilar characteristics of the surface of each sample, since there are a number of ways that this may arise during sample preparation. Furthermore,

the indium surface encountered in the measurements on copper samples using the

indium seals consists chiefly of exposed fresh surface as the indium extruded,

so that these results can be expected to give only the order of magnitude of

the conductance to be applied as a correction. Several attempts were made to

measure the conductance of indium samples whose surfaces were treated in a manner more nearly identical to the O-ring surfaces, but these attempts failed

because of the difficulty in making pressure seals with a bulk sample. Etching the surfaces with hydrochloric acid to reduce the oxide layer did not have

an effect.

Since previous measurements of indium have not been extensive, it is of

interest to compare these measurements with the theory and with previous work.

Figure 12 shows that the present measurements are in the range of previous

measurements by Gittleman and Bozowski,^^ and Neeper and Dillinger , though

they show a stronger temperature dependence of about t"^*^ and T^'"*. All ex- perimental results are well above the Khalatnikov acoustic mismatch theory,

but not as far above as some metals with higher Debye temperatures have been

9 found to be. Some recent work by Cheeke , et al. on the harder metals has

tended to confirm this trend. However, since all measurements have been of

polycrystalline samples, measurements on clean, single crystal surfaces would

be interesting. Because the samples are polycrystalline and the elastic

properties of indium are highly anisotropic, an average value of the trans- verse velocity of sound was used to obtain the line in figure 12 representing

the acoustic mismatch theory. Results for a single crystal would permit a more accurate comparison of theory and experiment, but the handling of such

a sample would be a problem.

32
Vacuum

Figure 10. Apparatus for Kapitza conductance measurements of indium. The chamber

is placed in a helium bath which is pumped upon to regulate the tem- perature . 33
10

8

34
50

Present Measurements

1.4 1.5 1.6 1.7 i.8 1.9 2.0 2.1

Figure 12. Comparison of Kapitza conducatnce measurements on indium with theory

and with previous work.

35
APPENDIX B. MEASUREMENT OF THE CONDUCTANCE OF POLYCRYSTALLINE NIOBIUM

The niobium sample was obtained from the Stanford group working on superconducting accelerators (courtesy of Dr. Michael Macashan) . The polycrystalline sample had a preparation similar to that of the superconducting cavities:

-9 chemically polished, electropolished , UHV-fired at 1800°C at 10 torr. The

measurements were made with the sample in the superconducting state. In this

case, the phonons have a mean free path that is a sizeable fraction of the distance between the thermometers on the sample and between the last thermometer

and the interface. Hence, the usual linear extrapolation of temperature to

the interface may not be accurate. Therefore, the data from these measurements

were analyzed in two ways. Using a linear extrapolation of the temperature

measured by the thermometer nearest the interface, a temperature dependence of

5 4 approximately T * was found for the Kapitza conductance. However, if the temperature at the interface is assumed to be most closely approximated by the

temperature of the nearest thermometer, because a temperature gradient is not

well-defined for distances of the order of a mean free path, the temperature

dependence of the present measurements of the Kapitza conductance is somewhat

4 7 lower, about T . Figure 13 shows the results for the case where the temperature is extrapolated to the interface. The error bar represents the uncertainty

in the indium correction, as gauged from the present measurements, and the un- certainty of the temperature difference measurement. Figure 13 also shows the same data, but treated differently so that the temperature was not extrapolated

to the interface.

A comparison of this data with the phonon radiation limit and the Khalatg nikov acoustic mismatch theory is shown in figure 14. Earlier work by Mittag

at this laboratory using a different apparatus is also shown. The curve shown

for comparison is the one for which the temperature is extrapolated, as this was the way in which Mittag 's data were treated. The sample represented by

Mittag 's highest curve had a surface preparation which was most similar to the

preparation of the sample used in the present work. He also used an indium

0-ring seal. It is expected that the disagreement may be due to choosing

different values for the indium conductance, since the value Mittag used was

not stated.

36
1.5 1.6 l.J 1,1 1,9 2,(1 2.1 2.2

T 1K|

Figure 13.Kapitza conductance of a polycrystalline niobium sample. Conductance

is calculated both by extrapolating the apparent temperature gradient

in the sample from the closest thermometer, and by not extrapolating

this temperature.

37
38
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NBS-114A (REV, 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. 75-8122. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

THE KAPITZA CONDUCTANCE OF THE (100) SURFACE OF COPPER AND

POLYCRYSTALLINE INDIUM AND NIOBIUM

5. Publication Date June 1975

6. Performing Organization Code

7. AUTHOR(S)

N. S. Snyder

8. Performing Organ. Report No. NBSIR 75-812

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2750187

11. Contract/Gra.nt No. NAonr-22-74

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Office of Naval Research

Arlington, VA 22217

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.) Measurements of the Kapitza conductance to liquid helium II across the (100)

surface of single crystals of copper are presented. The temperature range of these

measurements was 1.6 to 2.1 K. The sample surfaces were subjected to several different treatments. Some surfaces were cleaned by low energy argon ion bombardment, annealed in an ultrahigh vacuum

system, and preserved under vacuum until purified liquid helium was admitted. Other

surfaces were intentionally damaged by machining and/or exposure to the atmosphere.

The conductance after these latter treatments was found to be about a factor of three

higher than that of the more ideally cleaned and annealed surfaces, and a significant

increase in the temperature dependence of the conductance was also observed. Indications are that these effects on the conductance are produced primarily by the last

few atomic layers of the solid, so extensive cold-working, which Impairs the thermal

conductivity of the bulk, need not be undertaken to improve surface heat transfer.

This finding has important implications for the design of practical heat transfer

surfaces. Good reproducibility was found for the conductances of similarly treated

sufaces and good correlation with studies of damage carried out by x-ray diffraction.

The relationship of these results to the numerous current theories of the Kapitza

conductance is discussed. Conductance measurements of polycrystalline niobium and

indium are presented in the appendices.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) 'Re.a.t transfer to helium II; interfacial thermal resistance;

Kapitza conductance; Kapitza conductance of copper; Kapitza conductance of indium;

Kapitza conductance of niobium; (100) surface of copper; x-ray diffraction examination

of single crystal lattice damage.

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