A UNITED STATES DEPARTMENT OF COMMERCE

PUBLICATION

NBS TECHNICAL NOTE 385

Thermal Conductance at the Interface

of a Solid and Helium II

(Kapitza Conductance)
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NBS TECHNICAL NOTE 385

ISSUED DECEMBER 1969

Nat. Bur. Stand. (U.S.), Tech. Note 385, 90 pages (Dec. 1969)

CODEN: NBTNA

Thermal Conductance at the Interface

of a Solid and Helium II (Kapitza Conductance)*

N. S. Snyder

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

This work was carried out at NBS under the sponsorship of the U.S. Atomic Energy Commission, Contract Agreement AT (49-2) - 1 165.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide

a means for making available scientific data that are

of transient or limited interest. Technical Notes may

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For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C13.46:385), Price $1.00

TABLE OF CONTENTS

Page

LIST OF FIGURES vi

LISTOF TABLES. vii

1. INTRODUCTION 1

2. PHONON RADIATION LIMIT .

o 4

3. EXPERIMENTAL DATA FOR AT << T 8

3.1. Comparison with the Phonon Radiation Limit „ . . 8

3. 2. Reproducibility of Experiments 10

3. 3. Empirical Analysis of Data 13

a. Surface condition 13

b. Correlation with molecular weight and elastic

properties 19

c. Metals -effect of electrons ....«,.....„ 24

4. THEORIES OF KAPITZA CONDUCTANCE . . 30

4. 1. Acoustic Mismatch o m 31

4.2. Electron Mechanisms and Surface Disturbances. . 38

4. 3. Thermal Accommodation 40

5. INTERFACE CONDUCTANCE WHEN AT ~ T 46

5. 1. Kapitza Region o 4o

5. 2. Pre-Film Boiling Region 47

in

5. INTERFACE CONDUCTANCE WHEN AT ~ T (continued)

5. 3. q* - Maximum Heat Flux. , 51

5. 4. Film Boiling 56

6. SUMMARY 59

7. REFERENCES 82

v
LIST OF FIGURES

Page

Figure 1. Kapitza conductance at 1. 5 K - largest values

for each solid are plotted 20

Figure 2. Kapitza conductance at 1.9 K for cleaned surfaces . . 21

Figure 3. Kapitza conductance at 1. 9 K for polished surfaces . . 22

Figure 4. Kapitza conductance at 1. 9 for unpolished, oxided

surfaces 23

Figure 5. Kapitza conductance at 1.9 K for chemically clean,

unstrained surfaces plotted against l/pc « .34

Figure 6. Kapitza conductance of Ag for large AT 48

Figure 7. Maximum values of the peak heat flux density for

horizontal cylinders. ...... . '50

Figure 8. Reduced peak heat flux as a function of temperature. . 53

Figure 9. Pressure dependence of the maximum peak heat flux . 55

Figure 10. Peak temperature difference for several surfaces. . . 57

VI
LIST OF TABLES

Page

Table I. Compilation of Kapitza Conductance Data . . 62

Table II. Comparison with Phonon Radiation Limit -

Metals (1.9 K) 76

Table III. Comparison with Phonon Radiation Limit --

Non-Metals (1. 9 K) 77

Table IV. Measurements of Kapitza Conductance for Copper

[1.9K] . 78

Table V. Measurements of Kapitza Conductance for Lead

in Normal State ( 1. 9 K) 80

VII
ABSTRACT

A review is presented of the experimental and theoretical work

on Kapitza conductance, including a compilation of the available data

on conductance to helium II. A short derivation of the phonon radiation

3

limit indicates the reason for the approximate T temperature dependence

and the small size of the Kapitza conductance for most solids. Considerable qualitative and quantitative correspondence of the data with

this limit is found. From the limited evidence available, the role of

surface conditions, and of bulk parameters such as the Debye temperature in determining the conductance are considered empirically.

Theoretical knowledge of the Kapitza conductance is seen to give an

inadequate explanation of the data. In addition, the phenomena which

occur when the heat flux is high enough that properties of the bulk

liquid are also involved in the measured conductance are described

briefly.

Key Words: Heat transfer; helium II; Kapitza conductance.

Vlll
THERMAL CONDUCTANCE AT THE INTERFACE OF A SOLID AND

HELIUM II (KAPITZA CONDUCTANCE)

by

N. S. Snyder

1. INTRODUCTION

When a heat flow per unit area, Q, occurs across a solid-liquid

interface, it is found to be limited by a finite conductance, h , which is

well-defined and independent of heat transfer processes in the liquid as

AT, the temperature difference between the solid and liquid, approaches 0.

V iim "w (1 > AT-0

In the limit of AT « T. , where T. is the liquid temperature, h, is

expected to be the same for heat flow in either direction. When AT is

large, however, convection and other processes in the liquid become

important and reversibility is not expected. A discussion of these latter

effects is postponed to Section 5, since the emphasis in the present

paper is on interface properties. High AT experiments are also considered more fully in a recent review paper of Frederking (1968).
To measure h by the conventional steady state method, at

least two temperature sensors are required in the solid to give an extrapolated temperature value at the interface. By choosing helium II

for the liquid, the temperature at the liquid side of the interface is

well-defined, and in fact the first measurements of solid-liquid interface

conductance were made by Kapitza (1941) using helium II. Kapitza and

3

later investigators found that h varies approximately as T , where

T is the absolute temperature, which indicates why interface thermal

resistances were not observed first at higher temperatures. Magnitudes

of the interface conductance at 1.9 K have been found to range between

2

1. to 50 kW/m K for the 19 solids investigated thus far. Because of

these magnitudes and the temperature dependence, the interface conductance is found to be the limiting factor in many cases of heat generation for which helium II cooling is necessary. Interface conductance

assumes even more importance at temperatures in the millikelvin range

3

at which He dilution refrigerators operate.

A qualitative understanding of the temperature dependence and

variation of h with solid properties can be obtained from thermodynamic considerations and a simple model of a solid at low temperatures.

This is discussed in Section 2. The experimental work on interface

conductance to date has been fairly limited, in terms of both the range

of solids investigated and the characterization of surface conditions at
the interface. Considerable disagreement exists between results for

samples of the same solid. Table I presents the available data for all

solids. The experimental results and some empirical correlations that

may be derived from the measurements are treated in Section 3. Theoretical work has been based mostly upon the acoustic mismatch theory

of Khalatnikov (1952), (1965), but experimental results do not agree

qualitatively or quantitatively with the theories. For this reason, the

treatment of theories in Section 4 is brief. An extensive discussion is

available in a review by Pollack (19 69).

In addition to the presentation of a complete tabulation of the

experimental results, the present discussion differs from the previous

reviews of Frederking ( 1968) and Pollack ( 19 69) in several ways. The

treatment of the interface conduction problem (low AT) is more detailed

than that presented by Frederking, who emphasizes the high heat current

region. The present discussion differs from Pollack's review in that it

is oriented primarily toward presentation of design information.

There are also several differences between this discussion and that of

Pollack regarding the interpretation of the data (See Sections 3. 3 and 5).

Perhaps the most important aspect of the present study is the correlation noted between the phonon radiation limit and some of the data

(Sections 2 and 3.1). The phonon radiation limit is not mentioned by

Pollack, and its applicability to non-metals, and clean metal surfaces

is not discussed by Frederking.
2. PHONON RADIATION LIMIT

The small size of the Kapitza conductance at low temperatures

follows simply from an application of thermodynamics to well-established

models of a solid. Heat can be transported through the bulk solid both

by conduction electrons and by vibrations of the atoms, but since heat

transfer at an interface probably involves chiefly the lattice vibrations,

the case of an insulator will be considered first. When the vibrations of

the atoms are treated by the methods of quantum mechanics, the solution

in terms of traveling waves is restricted to certain normal modes with

discrete energies, each of which obeys the classical equations of motion

of a harmonic oscillator. (See, e. g. , Klemens, 1958. ) These quantized

modes of vibration are referred to as phonons, by analogy with photons,

which are the quantized electromagnetic waves. The change in the atomic

vibrations with temperature is described in terms of an increase with

temperature in the number of phonons excited for each discrete energy

level. The heat conduction process in an insulator may be understood

by considering the phonons to be particles of a "gas" with a mean free
path between collisions of the phonons with each other and with defects

and boundaries of the solid. A higher density of phonons at the hot end

of a solid then results in a net flow to the cold end. Because the lattice

contribution to the specific heat of most solids at low temperatures has

3 a T dependence, the energy density of phonons in this temperature

region is given by

U = a T (2)

where a is a factor which depends upon properties of the solid. This

equation has the same form as the blackbody radiation law for the electromagnetic energy radiated by photons, which also have some of the

kinetic properties of a gas.

Now, if the interface resistance is considered to be localized to

within a distance of one phonon mean free path on both sides of the boundary surface, the reason for a thermal barrier can be understood by considering the analogous problem of heat conduction in a low pressure gas.

In this case, for distances less than the mean free path, the heat flow is

limited by the number of molecules, or the pressure, and for a solid,

j the heat flow will be limited by the available number of phonons, which

decreases rapidly as the temperature decreases. According to the usual

kinetic result, the heat flux across a plane is

1-1-4

q =TUc = -acT (3)
where c is the average velocity of the phonons and the arrow indicates

flow in one direction. Then the net heat flux across a layer with one

side at temperature T and the other at T + AT, where AT « T, is

- 3

Q = a c T AT = h AT. (4) k

The constant a for the solid will be the limiting factor in (4) because

phonons are more easily excited in helium II. In deriving this expression

the assumption of perfect transmission at the interface was implicitly

made; hence (4) gives the upper phonon radiation limit for the conductance that may occur across a real interface. If the mechanisms that may

cause less than perfect transmission are not strongly temperature de3

pendent, the T dependence should be a good approximation for the real

interface. The actual mechanism of energy transport across the interface is still unspecified in (4), which only states the thermodynamic

basis for the thermal barrier. Although for non-superconducting metals

the electrons also participate in heat transport, unless surface waves are

important (Little, I960), or unless the electron wavefunctions "leak"

across the boundary (Johnson and Little, 1963), the electrons must give

up their thermal energy to phonons before it is transmitted across the

interface, so the upper limit expressed by (4) remains valid.
The factor a, which is different for each solid, can be obtained

in theory from an integration involving the dispersion relation for phonon

energy versus wave -vector. This dispersion relation can be obtained

either from neutron and x-ray scattering data or from a theoretical

calculation based upon the forces between neighboring atoms. At low

temperatures, however, the integral over the detailed spectrum can be

characterized to a good approximation by a single parameter, the Debye

temperature, . For example, for temperatures below about © /50,

3 3

the specific heat is proportional to T /©„ . The Debye approximation

and the evaluation of © from specific heat or elastic constant measurements is discussed in many texts (Kittel, 1966). Roughly speaking,

a high © corresponds to a lattice with strong interatomic forces and

light atoms. In this approximation the phonon radiation limit is

h = Sllii (3N/4.)

2/3 T

3

, (5)

5 6*1,

where N is the number of atoms per unit volume and the other symbols

have their usual meaning. Evaluating (5) for copper, which has a Debye

3 3 2 4

temperature of about 343 K, gives a conductance of 4. 45 X 10 T W/m K .

Lead has a considerably lower © of about 100 K, which gives a con4 3 2 4

ductance of 2. 83 X 10 T W/m K . Since few solids of practical interest have such a low ©„, it is illuminating to note that the approximate
upper limit of the interface conductance of lead at 1.9 K, which is

5 2

1.9 X 10 W/m K, is comparable with the conductance through 175 km

of a tube filled with superfluid helium at 1.9 K„ (This comparison is

valid for laminar flow conditions: for example, heat flux less than about

2 2 2x10 W/m through a 10 mm diameter tube. A rp; 19*70). This shows

why Kapitza conductance is the limiting factor in many situations of heat

transfer to superfluid helium. For heat transfer with higher fluxes which

cause non-laminar flow, the conductance through the liquid becomes more

comparable to the Kapitza conductance.

3. EXPERIMENTAL DATA FOR AT«T

3. 1. Comparison with Phonon Radiation Limit

Table I is a compilation of all experimental measurements of the

Kapitza conductance of various solids. (Some indirect measurements

of low accuracy have been excluded.) To date (Sept. 1969), 19 solids

have been investigated; prominent examples of solids of technical and

theoretical interest for which no data are available include Al, Be,

Ge, Fe, brass, stainless steel and high field magnet alloys, glass

and plastics. In Table I, the range of temperatures over which measurements were made is indicated in each case, but the conductance is

quoted only at 1.9K since this is usually an optimum temperature for

heat transfer processes with helium II. (See Arp, 1970). If the authors

reported a temperature dependence, this is also presented. Table I

8
3

shows that the T law is approximately obeyed for a variety of solids,

although the actual exponent of T ranges from 2. 6 to 4. 2. There is

some indication that below about 0.1 K, conductances for solid-solid

3

interfaces follow a T dependence more exactly (Syomi, Anderson, and

3 Holmstrom, 1968). The conductance between copper and both liquid He

4 3

and He also has been found to vary more nearly as T at very low

temperatures (Anderson, Connolly and Wheatley, 1964). This behavior

is expected because the derivation of (5) is valid as T approaches K.

Mercury has a very low Debye temperature, so its specific heat does

3

not follow a T law in the temperature range of the measurements.

However, the ratio for mercury of specific heat to conductance is

approximately constant with temperature (Neeper, Pearce and Wasilik,

1967).

In Tables II and III, the highest reported values of the Kapitza

conductance at 1.9 K for metals and non-metals are compared with the

phonon radiation limit of (5). For metals (Table II), the most interesting comparisons are those for copper and lead, because these two metals

have been most extensively investigated, and data is available for

chemically clean, unstrained surfaces which should approximate properties of the bulk solid. The steady state result for copper is about a

factor of 4 below the limit and the highest conductance reported for

lead is about a factor of 6 lower than the limit. The ac conductance

9
observed for a copper foil in a second sound resonator by Brow and

Osborne (1958) is about a factor of 1.2 below the limit. Because of an

error of a factor of 10 in evaluating the phonon radiation limit for copper,

Frederking (1968) apparently did not realize the extent of agreement of

this limit with experiment. It should be emphasized that precise agreement with the phonon radiation limit is not expected, because perfect

transmission at the interface may not occur, and the one-parameter

Debye temperature description of the solid is an approximation. Other

metals, with the exception of tin, fall below the phonon radiation limit

by larger factors. However, (5) does give a rough idea of the relative

magnitudes of the conductance for the different metals.

The non-metallic samples (Table III) are all single crystals. Data

for carbon films (Hesser, Chapman, Chang, and Frederking, 1969) are

not included because of the difficulty of defining a Debye temperature.

Although only five crystals have been investigated, it is striking that

the conductances are much closer to the phonon radiation limit than is

the case for metals. The largest disagreement is a factor of 3 for KC1.

The ranking of crystals by the magnitude of expected conductance from

(5) is confirmed by experiment, which again indicates the usefulness of

this expression for qualitative guidance.

3. 2 Reproducibility of Experiments

The chief problem a user of Table I will encounter is the wide

variation of the measured conductance between samples of the same

solid. For example, the steady state conductances reported for copper

10
2

at 1.9 K vary from 1. 2 to 7. 5 kW/m K. The conductance of even a

single sample has been found to change after a period of time (Neeper

and Dillinger, 1964). The differences in conductance do not seem to be

related to the purity of the bulk solid. It is also unlikely that they can

be traced to experimental errors. In almost all cases, precautions

have been taken to eliminate parallel paths of heat flow, so that this

problem cannot affect the results by more than 1 - 2%. If the experiments are performed with liquid helium II, there will not be any significant uncertainty about the liquid temperature at the interface. However,

for the solid, the interface temperature is usually extrapolated from the

gradient measured between two sensors located a finite distance from

the boundary. In the case of a dielectric with a long phonon mean free

path, one can see that all phonons impinging upon the boundary surface

may not come from regions of identical temperature. Also the thermal

gradient along a crystal surface may not be uniform near the ends of

the crystal. An investigation of these effects in sapphire by Neeper and

Dillinger (1964) indicated that they were small. For a metal sample,

since part of the heat is carried by electrons, similar trouble could

arise if there were simultaneously a mean free path between electron

collisions comparable to sample dimensions and a large thermal resistivity. But disagreements exist even for conductance measurements on

metal samples of high purity with negligible thermal gradient effects.

11
If such a metal is in the superconducting state, at a given temperature,

the conductivity is considerably reduced, and in one case it was necessary

to make a theoretical calculation of the extrapolated thermal gradient

(Challis, 1962). Also, if the interface region of a superconductor is

strained from the differential thermal contraction of the mounting, the

effect on the phonon conductivity may be sufficient to cause a significant

error (Challis and Sherlock, 1969). This problem does not affect most

of the results in Table I. It will be discussed further in Section 3. 3. c.

Because the experimental errors of the conductance measurements are small in comparison with the discrepancies observed between

measurements of the same solid, it seems most probable that these

variations are due to differences in the condition of the sample surface.

For many measurements, the state of the surface with regard to possible

cold-working, chemical purity, surface asperity, and similar parameters

has not been well-defined. Sizeable changes in the interface conductance

of an individual sample have been brought about by varying the surface

treatment (Challis, 1962). In some cases the state of the surface layer

may affect the extrapolation of the temperature gradient to the interface.

For example, an excess thermal resistance could result from poor contact of an oxide layer covering the surface with the bulk metal below it.

Dislocation damage from cold-working can extend for a distance of the

order of mm into a metal unless very careful techniques are used. (See

12
eg., Young and Wilson, 1961). Since the resistivity due to a high

dislocation density can be significant at low temperatures for a pure

metal, extrapolation of the bulk temperature gradient to the surface may

result in a spurious extra thermal resistance which is indistinguishable

experimentally from a true interface resistance. Not enough material

preparation details have been given, in most cases, to evaluate this

possibility.

Aside from such spurious effects, the effect of surface parameters

upon the actual heat transfer processes at the interface is highly dependent

upon the mechanism postulated for this process. The theories of these

mechanisms will be discussed in Section 4. But because none of the

theories has been substantiated by the experimental evidence, it is felt

that the following empirical analysis of the data with regard to surface

and other parameters may be more useful in furnishing estimates of

conductance for design purposes.

3. 3. Empirical Analysis of Data

a. Surface Condition

In Table IV, the results of boundary conductance measurements

for copper are ranked by the magnitude of the conductance at 1.9 K. With

the exception of measurements 4 and 6, the seven highest conductances observed refer to surfaces which had had at least some of the cold-worked

layer formed during machining and polishing removed chemically. Fur13
thermore, the highest two conductances are for surfaces which had been

partially machined while under liquid helium and then allowed to recrystallize at room temperature for about a week before measurement. Recrystallization should have been considerably facilitated by such a procedure since the impurities that would have become imbedded in the surface

during room temperature machining were not present. By contrast, all

the surfaces of measurements 8 through 13 (below the dashed line) were

subjected to cold-working through machining and polishing. For the lowest

three conductances reported, it is probable that considerable oxidation of

the surfaces had occurred over long periods of time since no precautions

or cleaning procedures were noted by the authors. Oxide layers for the

other surfaces with higher conductances, were probably lighter. In

some cases, the sample was immersed in helium only a few minutes

after etching or polishing, and in other cases, the period of exposure to

the atmosphere was limited to one week. The conclusion is that a clean,

unstrained copper surface should have a boundary conductance of ap2 proximately 8 to 5 W/m K at 1.9 K, whereas for a cold-worked, "dirty"

2

Surface, allowance for values as low as 1 to 2 W/m K should be made

for design purposes. Unless stringent measures are taken to protect a

clean, etched surface from oxidation, its surface conductance might decrease somewhat with time if it was exposed to the atmosphere between

uses at cryogenic temperatures. The relatively large conductance ob14
served recently by Goodling and Irey (1969) for a machined surface is a

puzzling exception to the above regularities. Although the geometry

employed by these authors is different from the other workers, calculations showed that end effects and temperature imhomogeneities were

negligible.

The only other solid for which measurements with a variety of

surface conditions are available is lead; a similar ranking for the metal

in the normal state is given in Table V. Samples which were freshly

machined or had had surface layers removed by electropolishing, ion

bombardment, or etching all fall above the horizontal dashed line.

Samples machined under liquid helium with recrystallization permitted

at higher temperatures also fall in this upper region, with the exception

of some measurements by Cheeke (1969), which are on the border

between the two regions. Samples which did not have special treatment

fall below this line and can have conductances that are an order of

magnitude less than the highest values. From measurements on the

same sample, the deleterious effect of an increasing oxide layer is

evident (see Challis; 1962), Wey-Yen (1962) carried out an interesting

series of experiments on a superconducting lead sample which was

machined under liquid helium, then held at room temperature for one

week, remachined under liquid helium and finally kept at room

temperature for four months (see Table I). Machining decreased

15
the conductance while room temperature recrystallization increased it.

These results have been confirmed by recent work of Cheeke (1969).

The similarity of behavior noted here for the conductance of

lead and copper with variation of surface conditions contrasts with comments of Pollack (1969), that the highest conductances are observed for

the cleanest surfaces of lead, but that the reverse is true for copper.

This involves the problem of defining a "clean" surface. In comparing

data of Challis, Dransfeld, and Wilks ( 1 96 1) for a polished and an etched

copper surface, Pollack described the mechanically polished surface

with the lower conductivity, as the cleaner one. However, such a surface will be work-hardened, and may have chemical changes at the

surface from the polishing agent. Such a surface is probably a poorer

approximation to a simple termination of the bulk than an etched surface

is. Other data on copper by Challis (1962) showed a small decrease in

the conductance of a mechanically deformed sample after it was annealed,

But because the surface was polished for the measurements both before and after the annealing, Challis stated that the rather small conductance difference might be ascribed to variations in the polishing

procedure. Hence this example does not necessarily indicate that

cold working increases the conductance of copper. With regard to the

16
presence of an oxide layer, the experiments of Johnson and Little (1963)

seem to show, at first glance, that an oxide layer improves the conductance (See Tables I and IV). But since the state of recrystallization of

the surface layer is also a variable here, no definite conclusion can be

drawn from these experiments alone.

When one examines the Johnson and Little data in conjunction with

the data of other experimenters, though, the trends stated in the first

paragraph of this section become apparent; that is, the highest conductances

are found for copper surfaces which are a closer approximation to a simple

termination of the bulk because of chemical removal of cold-worked, chemically foreign layers. It would clearly be more useful, however, if results from the same experimental apparatus, preferably with the same

sample, could be compared with respect to the variation of surface conditions. If experiments were carried out which avoided the problems

mentioned above of varying two or more surface conditions simultaneously,

the problem of whether lead and copper behave similarly could probably

be resolved.

Unfortunately, lead and copper are the only solids for which

enough measurements have been made at present to permit some rough

conclusions as to the size of conductance to be expected with a given

surface treatment. An examination of Table I indicates that from the

limited data available, ranking of silicon and platinum would follow the

17
same general principles observed above, though the effect of etching on

the non-metallic silicon, for example, is very small (Johnson and Little,

1963). However for nickel, indium and tin, Table I shows some cases

where recrystallization or removal of a surface layer by electropolishing

had little effect or actually resulted in some decrease in conductance.

Since electropolishing may cause chemical changes at the surface, the

situation here is unclear.

18
b. Correlation with Molecular Weight and Elastic Properties

Figure 1, due to Challis (1968), shows values of the conductances

at 1. 5 K for various solids plotted against the reciprocal of the Debye

temperature © on a logarithmic scale. The highest values reported

were used. Although the surface conditions of the solids may vary

considerably, there is evidence for some degree of correlation.

Figures 2 through 4 give the same type of plot for the three cases of

chemically clean and unstrained surfaces, polished metal surfaces,

and untreated, oxided surfaces. The bars indicate the range of results

for surface conditions that are approximately the same. The extremely

low conductance for electropolished Ni is not shown in figure 2 (Wey-Yen,

1962). With the possible exception of figure 4, a correlation is again

apparent, but the dependence of h on (1/0 ) goes as n~ 0. 8 in

figure 2 and n ~ 0. 6 in figure 3. The lesser amount of correlation

with the Debye temperature for the untreated surfaces of figure 4 demonstrates the need for careful surface preparation to insure a predictable

value of Kapitza conductance. However, even for surfaces of the types

represented in figures 2 and 3, the predictive value of the plots is

limited; at most the order -of -magnitude conductance for an untested

solid could be estimated if the Debye temperature were known. Corre2/3 -2

lations of conductance with molecular weight, with (p/m) (from

(5)), and with other parameters were also examined and found not to add

19
IO

3 0n (K"1 )

10

Figure 1. Kapitza conductance at 1.5 K —largest values observed for

each solid are plotted. From Challis (1968). Ag data from

Frederking (1968).

20
Figure 2. Kapitza conductance at 1.9 K for cleaned surfaces. (Surface

etched or otherwise treated to remove oxided and cold-worked

layers.) Hg (normal state) (Neeper, Pearce and Wasilik, 1967).

Pb (normal state) (Challis, Dransfeld, andWilks, 1961; Challis,

1962; Challis and Cheeke, 1965). Sn (Wey-Yen, 1962). Pt (Kapitza,

1941). KC1 (Johnson, 1964). Cu(Challis, Dransfeld and Wilks,

1961; Johnson and Little, 1963); Si (Johnson and Little, 1963).

21
IO

3

0d (K

Figure 3. Kapitza conductance at 1.9 K for polished surfaces. In, Sn

(Gittleman and Bozowski, 1962). Ag (Clement and Frederking, 1966).

SiC-2 (Challis, Dransfeld, andWilks, 1961 ; Wey- Yen, 1962). Cu

(Challis, Dransfeld, andWilks, 1961; Challis, 1962). Si (Johnson

and Little, 1963).

22
I.sj

1.0

.8

I Au

^ .6 M

E Sn Pb

*.4

i O

Ni

W In

.2 Pt

Cu

8 10

IO

3

D

~' (K"

1 )

Figure 4. Kapitza conductance at 1.9 K for unpolished, oxided surfaces.

Pb (normal state) (Challis, 1962: Gittleman and Bozowski, 1962;

Challis and Cheeke, 1965). In (Neeper and Dillinger, 1964). Au

(Johnson and Little, 1963). Sn, Pt (Wey-Yen, 1962). Cu (White,

Gonzales, and Johnston, 1953; Wey-Yen, 1962). W (Johnson and

Little, 1964). Ni (Wey-Yen, 1962).

23
anything to the presentation of data in figures 1-4. Measurements for

other solids with clean surfaces would be helpful in developing correlations. Also the anomalous behavior of Sn and Pt indicated in figures 2

and 3, and the sharp decrease in conductance of Ni after electropolishing merit further investigation before such correlations are used to

support new theories. A correlation recently reported for relatively

clean surfaces (Fig. 1 of Hesser, Chapman, Chang, and Frederking,

19o9) includes an error of a factor of 10 in the Pt data; hence the

agreement with the data is actually poorer than indicated on the figure.

c. Metals -- Effect of Electrons

Experiments with metals can help answer questions about the

way in which electrons participate in the heat transport processes at an

interface. Since electron participation would involve magnetoresistive

effects, these questions are of importance in high field applications of

liquid helium cooling, as well as of theoretical interest. It is possible

for the electrons in the metal to interact directly with the phonons of

the liquid in two ways. The quantum-mechanical electron wave function

has a finite probability density in the liquid, which dies away exponentially. This concept is also known as tunneling. Thus one mechanism

is an interaction between tunneling electrons and phonons of the fluid.

The second mechanism arises because the phonons of the liquid also

penetrate somewhat into the metal, and can cause a density change of

the charge near the surface and thus a modulation of the potential in

24
which the electrons move. (Little, 1962 discusses these mechanisms

in greater detail.) There have been several experiments related to the

first mechanism. In an experiment by Gittleman and Bozowski (1962),

an ac electric field was applied at a platinum-helium interface by placing

a high voltage probe in the liquid a short distance from the surface. If

the electron wave functions from the metal were "leaking" into the liquid

and coupling with the phonons of the liquid, ac modulation of the Kapitza

conductance might have been observed. No modulation was seen, but

the high work function of the platinum or a surface layer of adsorbed

helium atoms might have inhibited the effect. In a study by Johnson and

Little (1963), a copper surface was milled under liquid helium to see if

this would increase the penetration of electron wave functions into the

liquid. It was concluded that the effect was either very small or masked

by factors that were not understood. In addition, a calculation of such

an effect by Bloch (see Johnson and Little, 1963) gives a temperature

5 3 dependence of T instead of the approximate T dependence which is

normally observed. Negative results were also obtained by Cheeke (1969)

on a bare lead surface.

The second mechanism of electron-phonon coupling has been investigated by comparing the interface conductance of a sample in the superconducting and normal state. At temperatures well below the superconducting transition, according to current energy gap theories, most

25
electrons cannot be easily excited and therefore do not interact with

phonons or act as entropy carriers. If electron heat transport dominates

phonon transport, as is often the case for a fairly pure metal, the

thermal conductivity will be lower in the superconducting than in the

normal state. Experiments with lead indicated a significant increase in

Kapitza conductance when a superconducting sample was made normal

by the application of a magnetic field (Challis, 1962; Gittleman and

Bozowski, 1962 ; Barnes and Dillinger, 1963; Challis and Cheeke, 1965;

Hesser, Chapman, Chang and Frederking, 1969). The ratio of normal to

superconducting boundary conductance varied from about 3.0 to 1. 35

for different samples. Recently, even higher ratios of 10 and 15 were

reported for lead (Cheeke, 1969). Lower ratios were found for other

metals: 1. 3 for mercury (Neeper, Pearce, and Wasilik, 1967), 1. 1 for

tin, and 1.06 for indium (Gittleman and Bozowski, 1962). The last two

metals have a relatively small electron-phonon coupling. It is also

possible that the tin and indium measurements were influenced by the

superconducting transition in the solder which was used to mount the

specimens (Neeper, Pearce and Wasilik, 1967). No change in the Kapitza

conductance of tin within 2% was reported by Wey-Yen (1962). Decreases

in conductance in the superconducting state at solid- solid interfaces have

also been observed (see Frederking (1968) for further references).

All of the superconducting-normal measurements discussed

above were made by steady state methods on relatively thick samples,

26
where the possibility of strain from differential thermal contraction in

the mountings exists. Strain near the interface could cause a large

decrease of the thermal conductance in the superconducting state

(Rosenberg, 1963), which would result in a spuriously lower boundary

conductance. Since the surfaces of the samples studied by Cheeke (1969)

were machined under liquid helium,this mechanism was probably

responsible for the very small conductances observed in the superconducting state. To avoid the problem of extrapolating thermal

gradients in the bulk to the surface, measurements on the attenuation

of second sound by a thin foil of lead were made more recently by Challis

and Sherlock (1969), and these indicated no difference in the Kapitza

conductance of the two states. Second sound consists of very low frequency thermal oscillations which propagate in helium II (see, e.g.,

Wilks, 1967), so this type of experiment actually measures the ac thermal

conductance. When second sound measurements were made for copper

by Brow and Osborne (1958), the observed conductance was significantly

higher than that of any of the numerous steady state measurements,

although no special surface preparation was carried out. The interpretation of the second sound measurements is somewhat complicated by

the relationship of the phonon mean free path to the foil thickness (Challis

and Sherlock, 1969). Thus the correspondence between ac and steady

state measurements is not clear.

27
The superconducting studies of the Kapitza conductance seem to

indicate that electron participation in the interface heat transport is

small. However, some experiments with magnetic fields have shown

sizeable effects. Field strengths of up to 2 tesla (20 kOe) have been

employed. The first experiments with transverse fields on lead (Challis,

19ol) showed an increase of about 1% in the conductance per 0. 1 T

change in magnetic field up to 0.4 T, but the size of the experimental

errors was about the same as the size of the effect. More such data

were presented by the same author in a later paper (Challis, 1962) in

which an end effect correction to the bulk thermal resistivity was applied

and the measurements were extended to 0. 7 T. Between 0. 1 and 0. 7 T

there was a fairly constant increase of about 1% per 0. 1 T in the Kapitza

conductance. This is approximately in agreement with the increase

observed with ac methods up to 2 T for a lead foil (Challis and Sherlock,

19d9). However, a considerable decrease in the effect was observed

when the lead surface was cleaned of its oxide layer (Challis, 1964).

For mercury, the change in conductance with increasing magnetic field

was positive or negative for different samples (Neeper, Pearce and

Wasilik, 1967), and was always less than 6% over 0. 35 T. An 0. 4 T

field had no observable effect on the Kapitza conductance of copper

(Challis, Dransfeld and Wilks, 1961). One explanation advanced for

these effects of magnetic field upon conductance is that phonon scatter28
ing is reduced when the electron energies are crowded into Landau levels,

and further experiments on single crystal foils by Challis may validate

this explanation (Challis and Sherlock, 1969). Since the Kapitza conductance could increase by as much as 100% in a 10 T field if the slope observed at low fields remains constant, further measurements at higher

fields would also be of interest.

29
4. THEORIES OF KAPITZA CONDUCTANCE

At present, there are no theories of the mechanism of thermal

energy transfer at a solid-liquid interface which agree quantitatively

with experiment. Nor have any theories been advanced which are useful

in making qualitative predictions. The phonon radiation limit (5) is as

good a guide to the temperature dependence and the relative magnitude

of Kapitza conductance for a particular solid as are any of the theories

to be discussed below. The following survey of theoretical work is

presented primarily to give a background which may be of assistance in

evaluating future theoretical papers.

Historically, because the existence of a solid-liquid interface

resistance was discovered by Kapitza using helium II, the first

theories hypothesized that superfluid properties were responsible for

the observations. However, the fact that the resistance is appreciable

only at low temperatures is explained by the thermodynamic considerations of Section 2. In fact, resistances of similar size have been

3 observed with He (e. g. , Lee and Fairbank, 1959), and also between

two solids (see Syomi, Anderson and Holmstrom, 1968) in the same

temperature range. There are no observable changes in the boundary

conductance at 0. 6 K where the spectrum of excitations in helium II

exhibits a drastic change (Fairbank and Wilks, 1955). Neutron scattering experiments have shown that the short-wavelength thermal phonons

30
of the liquid are not affected by the X -point transition (Woods, 1965).

4. 1 Acoustic Mismatch

Khalatnikov (1952, 1965) has considered the various possible

interactions of the phonons in the solid with the excitations in helium II.

He concluded that above 2 K, collision of helium phonons and rotons

(see Wilks, 1967) with the solid would account for about 40% of the heat

transfer, but that between 1 and 2 K the dominant effect would be the

radiation of phonons by the oscillating solid surface. Assuming that the

solid is essentially vibrating in free space because the density of liquid

helium is so low, and that energy is rapidly removed from the surface,

the calculation gives for the conductance for AT « T,

lbrrl Jl_> %• Cli

q-

[ F (Tl) + F (T\)] T

3

(6) k 15 3 3 p , L V ' Z

K n K ' h c sol.

where p.. is the density and c. is the velocity of sound in the

liq. liq.

liquid, p , is the density and c is the velocity of transverse waves

sol. t

in the solid, and T) is the ratio of the velocities of longitudinal and

transverse waves. F and F are of order unity, and have about the

same value for most solids. F, refers to the allowed phonon trans

1 mission, and F to the surface waves on the free surface of the solid.

Energy transfer by F should be negligible for a dielectric containing no

impurities, or for a superconductor far below its transition temperature,

31
3

The T dependence arises because the low temperature Debye approximation for the phonon energy density is used, as in the derivation of the

phonon radiation limit. For comparison with this limit, equation (5) of

Section 2, the Khalatnikov result may be expressed as

2/3

k" 5

e

^U. 31_3 .1

(F

1 +F2

» liq. liq. 3

sol. t

Since T| is about 2 for most solids, the factor in brackets has the order

of magnitude unity. The factor containing the ratio of acoustic impedances pc of the liquid and the solid reduces the predicted Khalatnikov

conductance to two or more orders of magnitude below the radiation

limit. Since the velocities of sound are so different, the momenta of

phonons in the solid and liquid at the same temperature is very different,

and few phonons will be transmitted because it is difficult to satisfy

simultaneously the laws of conservation of momentum and energy for

arbitrary angles of incidence.

Quantitative agreement of this theory with experiment is very

poor, since for most solids the highest observed boundary conductances

are less than one order of magnitude below the radiation limit (Tables

-3

II and III). Qualitatively, a dependence on © should be observed

when boundary conductances of different solids are compared at a given

temperature. However, figures 1, 2, and 3 indicate that a dependence

32
upon © is more probable. Figure 5, a logarithmic plot of conductance

3

at 1.9 K for clean, unstrained surfaces against 1/P c , indicates that

3 X 0. 38

the observed conductances vary more nearly as (1/P c )

Since the Khalatnikov theory applies to an "ideal" interface, it is

necessary to consider how it might be changed by the real interfaces for

which data are available. Wey-Yen (1962) pointed out that according to

an acoustic mismatch theory a layer of adsorbed gas or oxide on the

surface would have little or no effect since the thickness of these layers

is typically less than the wavelength of acoustic phonons. However, the

thickness of the amorphous surface layer formed by the cold-working of

a metal is comparable to the phonon wavelength of the solid. The elastic

constants of such a layer are unknown, but Wey-Yen found that his data

for cold-worked surfaces showed better agreement with the Khalatnikov

theory if he assumed that c was approximately constant. In this case,

the conductance at a given temperature for cold-worked metals should

vary as 1/P , but this relation is not confirmed when data of Wey-Yen

are combined with data of other observers. With the exception of one

case, Ni (Wey-Yen, 1962), removal of the cold-worked surface layer

has always resulted in values of the conductance that are similar to or

larger than the previous values, which makes agreement with the

Khalatnikov theory worse.

The effect of surface roughness should also be considered. The

effective area for acoustic transmission could be increased by only about

33
ooCDO^ « -° o £ X ~ U 1

fc mH , X d S en 8CO ined s d

Was

Challi 1941). Si

(J

ctiC.

(0 . vO

N vO

fl <u °^ .ti o d u " ft -1 O h (ti o

clean,

r,

Pea

Wilks Pt

(K

Little ^^ >~ <u v . 'V o o>

for

chemicall 1

state)

(Neep

Dransfeld

an

y-Yen,

1962)

1:

Johnson

an CD ro O o

CO

roi +- .„ (ti . « vO O w g .2 £ s — -: 2 ^ S * 1 C rj CO "J

O Q»

C\] 00 Jl HOC. . ^

I)

w "-U1 > O anc tate 196 TM .nd 

+->on en <ti % <? "-a G ^ H 4) m £ CD co o If n -c a

O

00 pitza agains Pb

(n.

and C ,

Dra

to 2

5.

Ka

plotted ,

1967).

Challis

(Challis

v}"

ud

00

C\J

fe o oo co

(>i 2 ^/M) "H^OI

34
a factor of two by roughness (Challis, Dransfeld, and Wilks, 1961),

which is not enough to improve the agreement significantly. Little

(1961b) has shown that "microscopic" surface roughness which is less

than the phonon mean free path will decrease the predicted Khalatnikov

conductance by a small amount. If the roughness is greater than the

phonon mean free path (Little, 1959), the temperature dependence of the

3

conductance will be different from T because of the change with temperature of the wavelengths of excited phonons.

According to (6), it should be possible to change the conductance

by a substantial percentage by pressurizing the liquid helium. However,

several experiments showed that the change was much less than expected

(Challis, Dransfeld, and Wilks, 1 961 ; Wey-Yen, 1962; Ander son,

Connolly and Wheatley, 1964). If the density of helium atoms near the

solid surface is much higher than in the bulk liquid because of the van

der Waals forces, the actual density at the interface does not vary as

strongly with the applied pressure as does the density of the bulk liquid,

and in addition, this denser layer should improve the acoustic matching

and therefore the phonon transmission. A theory of these effects was

developed by Challis, Dransfeld, and Wilks (1961). Although their

procedure improves agreement with data, it still predicts conductance

values that are order s of magnitude smaller than observed. Furthermore,

4. 2 a temperature dependence of T is predicted although most observations indicate that the exponent should be less than 3. 5. The difference

35
in temperature dependence is due to the fact that the acoustic matching

will depend upon phonon wavelength, and the number of phonons excited

with a given wavelength varies with temperature. To evaluate the Challis,

Dransfeld, and Wilks modification further, Johnson and Little (1963)

calculated van der Waals interaction energies between helium atoms

and a number of solids, making allowance for oxide layers and monolayers of adsorbed gases. They then found that even the predicted relative magnitudes for different solids failed to agree with their data, and

that in some cases the disagreement was worse than for the unmodified

Khalatnikov theory. There is also a question at present about whether

the layer of atoms at the interface has a density near that of solid helium

(Steele, 1956; Manchester, 1967). A calculation by Singh and Band ( 19 55)

showed that the repulsive energy between such densely packed helium

atoms would be so much greater than the van der Waals attractive energy

that the difference in free energy would not be made up by any reasonable

value for the entropy of the film. Manchester has reviewed the experimental evidence and shown that it does not necessarily lead to a densely

packed, immobile surface layer of atoms.

An attempt to consider a somewhat different transition region

between solid and liquid was made by Abbe (1968). In this case, penetration of atoms from the liquid into the solid is postulated (Abbe, 1967).

Using kinetic theory and the Debye approximation for the specific heat

36
3 Abbe obtained a conductance varying as T with an adjustable parameter

in the expression. With this adjustable parameter, Abbe was able to fit

some experimental results almost exactly, but at present it is not possible to obtain a priori values of the parameter, or to apply the theory to

explain the different results for the same solid.

Experimental attempts to improve impedance matching were made

by Whelan and Osborne (1969). Films of barium stearate of various

thicknesses which were deposited on gold, copper, and sapphire improved

the conductance. According to the numerical results, which were quoted

only for gold, even the highest conductance obtained in this series of

experiments is slightly less than that obtained earlier by Johnson and

Little (196 3) for a bare gold surface. Whelan and Osborne point out

that there are several uncertainties in the interpretation of these results.

The conclusion is that acoustic mismatch theories do not succeed

in interpreting the experimental observations, although the reason for

this failure is unclear. It is sometimes stated that the acoustic mismatch

theory as extended by Little (1959) has been reasonably successful in

explaining the interface conductances between two solids (Frederking,

1968; Pollack, 19o9). Actually, the acoustic mismatch between two solids

is relatively small, so the theory predicts conductances that are typically

about one-half of the phonon radiation limit. Because of experimental

problems in making a good thermal contact between two solids, it is dif37
ficult to decide whether this or an acoustic mismatch causes the measured

conductances to fall slightly below the phonon radiation limit (Von Gutfeld,

Nethercot and Armstrong, 1966).

4.2. Electron Mechanisms and Surface Disturbances

Electron transport accounts for most of the thermal conduction in

a pure metal at low temperatures, so the Khalatnikov theory, which is

based upon phonon transmission at an interface, actually implies that a

further resistance will arise from the necessity of transferring the electron

energy to the phonons at the interface. Direct energy exchange between

the electrons and the vibrating surface is not considered. However,

Little ( 196 la) has pointed out that the large number of phonons from the

liquid reflected at the interface may produce a spatially varying disturbance of the lattice ions which would extend approximately l/10th of a

wavelength into the bulk solid. Then energy may be rescattered from the

surface layer of ions by imperfections in the solid. For single crystal

silicon, however, no significant difference in Kapitza conductance was

found between a highly perfect crystal and one with a dislocation density

one hundred times greater (Johnson and Little, 1963).

In a metal, the surface disturbance changes the ion charge density

which provides a mechanism of electron-phonon interaction which could

increase the interface heat transport beyond that due to phonon transmission. Calculations of this effect have been made by Little (196la)

38
and by Andreev (1962a, 1962b), using a free electron model of the metal.

A discussion and comparison of the two calculations is given by Challis

and Cheeke (1968). The most important feature of the calculations is

that both result in an expression similar to (6) of the Khalatnikov theory,

except that F and F are replaced by expressions depending upon

electron parameters. (In the limit of strong coupling of the electrons

to the surface waves, the Andreev calculation equals F ) These new

expressions are still of order of magnitude unity, so this approach does

not increase the theoretical conductance to a point where agreement with

experiment is possible. Although approximations were used in carrying

out these calculations, it seems unlikely that orders of magnitude would

be changed in a more exact calculation. Little (196 la) originally stated

that the lack of agreement with experiment might be due to these approximations, but in a later paper (Johnson and Little, 1963) he concluded that

the mechanism was inadequate as an explanation of the data. Qualitatively,

the theory predicts a value of about 2. 4 for the maximum ratio of conductances in the normal and superconducting state. This is in accord

with the early results for lead, and was regarded as a success of the

electron-mechanism theory until recent work cast doubt upon the lead

experiments (Challis and Sherlock, 1969). The theories did not make

any quantitative predictions of the effects of large magnetic fields, though

the possibility of a de Haas-van Alphen type of oscillatory behavior (for

39
single crystals) was discussed (Little, 196la).

4. 3. Thermal Accommodation

Although it is often asserted that Kapitza discovered interface

resistance in 1941, the existence of a discontinuity of temperature between a solid surface and a gas was experimentally demonstrated in 1898,

by Smoluchowski (1898). This gas-solid temperature discontinuity is

ascribed to incomplete transfer of energy by the collisions of gas atoms

at temperature T with atoms of the solid at a temperature T . If the

g

s energy with which the gas molecules rebound from the wall is considered

to correspond to an intermediate temperature T

1 , the effect can be defined

by the thermal accommodation coefficient

T 1 - T

a = T _ /

. (8)

s

" g

A more rigorous definition of (X may be given in terms of distribution

functions of molecular velocities (Hurlbut, 1966). By definition, CL is

independent of pressure, and this independence has been verified by

careful experiments in which the solid surface composition does not vary

with pressure (Wachman, 1962). For cases of limited heat input in which

certain assumptions of kinetic theory are valid (Harris, 1967), the temperature jump between the solid and gas is given by

1/2

O nM

2 a AT=T

sT

g =p(^^) "ST (9)

40
where P is the pressure, M the molecular weight of the gas, and

O = - K(dT/dr), with K the gas thermal conductivity.

For gases, the accommodation coefficient concept has been so

influential that nearly all analyses of gas- solid surface interactions have

been expressed in terms of a and similar parameters. At very low

pressures and/or high temperatures, of course, the mean free path of

the gas molecules is too large to permit the propagation of phonon waves

from the solid, or equivalently, the acoustic mismatch is very great.

Whether the acoustic mismatch approach or the energy accommodation

approach is most relevant for a liquid-solid interface seems unclear at

present. Johnson and Little (1963) suggested that, in view of the failures

of the various modifications of the acoustic mismatch theory, an entirely

different mechanism of adsorption and desorption of helium atoms on the

solid surface should perhaps be considered. Such a mechanism is related to thermal accommodation processes, although the exact equivalence

may be questioned (Wachman, 1962). In a recent experiment by Mate

and Sawyer (1968), the thermal resistance between gaseous helium and

copper was measured in the range of 1. 5 to 4. 2 K„ At these temperatures, several layers of helium atoms will be adsorbed on the solid surface, and the authors interpreted their results as due to an energy accommodation resistance between the bulk gas and the adsorbed atoms in series

with a Kapitza resistance between the adsorbed atoms and the copper.

41
Equation (9) was used to separate the accommodation term from the total

measured resistance by selecting a value of cc which would remove the

pressure dependence for the remaining data. The temperature dependence and magnitude of the Kapitza resistance thus derived was similar to

that measured by other workers between liquid helium and an untreated

copper surface. Since most of the accommodation coefficients in the

literature were measured for solids with one or more layers of adsorbed

gas, it is difficult in general to make this separation of the data.

In later, more extensive experiments with an improved apparatus

(Mate and Sawyer, 1969), the Kapitza resistances of both the liquid and

3 4

vapor phases of He and He were measured between 1. 5 and 4. 5 K.

After the calculated contribution from the accommodation term is

subtracted for the gas phases, the resistances for gaseous and liquid

3 4

He and He above the lambda point are similar. Below the lambda

transition, the measured Kapitza resistances are anomalously low.

The accommodation term becomes negligible at the higher pressures

used in these experiments, since the authors use an cc of order unity

in (9).

42
If experimental results for the accommodation coefficient of

helium gas on various surfaces were available in the low temperature

range, it would be interesting to compare the data with Kapitza conductances. However, such data do not exist. One conclusion that may be

drawn from data at higher temperatures, most of it for gases other than

helium, is that ct approaches 1, that is, energy exchange improves, as

the surface becomes more contaminated with foreign gases. This

seems opposite to the effect of surface contamination observed for Kapitza

conductance, but actually the same type of contamination is probably not

being observed, for this consists of a few monolayers of gas in the accommodation coefficient measurements in contrast to oxide or other chemically

different layers in the Kapitza measurements. Gas monolayers are generally present in the latter measurements, but their effect has not been

systematically investigated. A result of a simple classical analysis is

that cc should be inversely proportional to the molecular weight of the

atoms in the solid for light gases. This relation appears to hold approximately for helium, but careful investigations have only been made for

two solids (Thomas, 1966). The Kapitza conductance, however, seems

to increase as M increases.

43
It is also interesting to compare the temperature dependence of

ot to that of the Kapitza conductance. Early experiments (Roberts, 19 32)

indicated that a for helium on a tungsten surface decreased markedly

between room temperature and 80 K, and a one -dimensional theory

(perpendicular incidence of molecule) based on the Debye approximation

for the solid and a Morse potential for the helium-solid interaction could

be fitted to these experimental points (Devonshire, 1937). More recent

measurements (Thomas and Schofield, 1955; Thomas, 1966) on cleaner

tungsten surfaces gave a smaller value for oi

t and showed that it decreased less rapidly in this temperature range. Below 50 K, oi started

to increase rather rapidly. This increase is in sharp constrast with the

3

earlier theory of Devonshire, in which oi decreased roughly as T at

low temperatures, as the lattice vibrations and the Kapitza conductance

do. A rigorous, general theory of accommodation coefficients does not

exist at present (Hurlbut, 1966; Goodman, 1968). A recent semi-empirical

expression of Goodman and Wachman (1967) which fits the newer data for

clean surfaces is

1/2 2

or(T) - 1 - exp(-T /T) + (2. 4n/( 1+H)^ tanhf

7

' 1(MT) -

b(1+^ } ] exp(-T /T). o l m © -I o

(10)

44
The mass of the gas atoms is M, that of the solid atoms is m,

|i = M/m, © is the Debye temperature, and b is a Morse potential

parameter. T is a measure of the temperature at which 01 begins

o

to depart from unity. It is higher for gases with higher heats of adsorption. For helium, a very low value of T of a few tenths of a degree

brings (10) into agreement with the new data for a tungsten surface

mentioned above.

Because of the lack of both a general theory and relevant experimental data, an explanation of thermal resistance at a liquid-solid interface in terms of energy accommodation does not seem profitable at

present. Nevertheless, in interpreting Kapitza conductance data, it may be useful to keep such concepts in mind. For example, Johnson

and Little (1963) carried out an experiment in which the Kapitza conductance was measured before and after a copper surface was milled under

liquid helium. Contrary to expectations based upon the Challis, Dransfeld, and Wilks modification of the acoustic mismatch theory, the conductance was reduced after machining. Presumably the surface before

milling had one or more layers of N~, O and H_,0 physically adsorbed,

which could be expected to improve the energy accomodation of impinging

molecules according to accommodation coefficient experiments. While

there were other important factors in this experiment, such as room

temperature recrystallization of the work-hardened surface, this example

45
does indicate how the predicted effects of surface treatment can depend

upon the model assumed for energy exchange.

5. INTERFACE CONDUCTANCE WHEN AT ~ T

When the interface temperature difference AT rises above the

range where AT « T, properties of the bulk liquid increasingly determine the heat transfer. Three fairly distinct regions of behavior may

be described which depend upon both the magnitude of the heat input and

properties of the solid surface. Initially, the data obeys a Kapitza

relation such as (4) expanded to include higher powers of AT/T. When

the surface temperature becomes approximately equal to T, , deviations

occur and the slope of dh/d(AT) decreases slightly (fig. 6). At a surface

temperature approximately a degree higher than T, , a maximum heat

transfer rate q is reached, after which the liquid adjacent to the wall

is vaporized, and behavior similar to film boiling for ordinary liquids

* . 2

is observed, q is about 10 kW/m for a flat surface.

5. 1. Kapitza Region

In deriving (4) for the net energy flux across the interface, only

4 4

terms to first order in AT/T were kept in the expansion ( T+AT) - T .

Thus as AT increases, initially one would expect the conductivity to

follow the form:

h' = h [l + (3/Z)(AT/T) + (AT/T)

2 + (

/4)( AT/T) 3

] (11)

.K.

46
where h is the function of T defined in (4). In figure 6, which shows

data of Clement and Frederking (1965) for a polished silver surface, the

dashed lines correspond to (11). The data begin to deviate for this equation because the surface reaches the ^-transition temperature, before

the highest order terms become effective. Similar behavior for h

1 was

found with a Ni-Fe film heater (Holdredge and McFadden, 1966; Madsen

and McFadden, 1968). In the Kapitza region, the effect of increasing

the pressure on the bath up to at least 1 atm. is not noticeable (Madsen

and McFadden, 1968). There are also no observable effects of the depth

of immersion of the heater upon the heat transfer (Irey, McFadden, and

Madsen, 1965). The total heat flux across a surface of a given geometryis simply proportional to its surface area, as was the case in the limit

AT ~* 0. The only effect of the bath temperature, Tn

, is to determine

b

at what AT deviations from (11) will begin to occur, since this AT is

approximately equal to T, - T . Hence, in this region, the observed

effects are essentially still determined by only the properties of the

solid surface.

5. 2. Pre -Film Boiling Region

After the temperature at the solid surface becomes approximately

equal to T, , the observed behavior is determined by a combination of

liquid and solid properties. Geometrical effects, such as the depth of

immersion of a test specimen compared to the diameter of the specimen,

47
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CM

COO

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48
and to the diameter of the dewar, become important. Pressure and

temperature of the bath now have noticeable effects on the heat transferred.

Studies with an interferometer by Peshkov (1958) show that the density of

a layer of liquid adjacent to the surface is different from the bulk value,

and that the boundary between the regions of different fluid density is

sharp. A peak heat flux, q , is reached before vaporization of the layer

adjacent to the surface occurs. The available data seem to indicate that

q is entirely independent of the solid surface properties (Frederking,

1968). However, the corresponding temperature discontinuity, AT ,

is related to the properties of the solid, as will be discussed further in

Section 5. 3.

Not much data are available on the effects related to the liquid for

AT less than At" . Madsen and McFadden (1968) found that the heat

transfer was roughly doubled if the pressure over the bath was increased

from the saturated value to approximately 1 atm. Andronikashvili and Mirskaia (1956) reported that the heat transfer at 8 atm.

was increased by a factor of 10 to 20 over the saturated vapor pressure

values. Increasing the depth of the liquid above the heated surface also

increased the heat transfer (Holdredge and McFadden, 1966). The heat

flux per unit area for two cylinders was observed to be greater for the

cylinder of smaller diameter in this AT region, but these observed

fluxes were not reproducible from run to run (Holdredge and McFadden,

1966).

49
E

o

Q

COu

S•1-1

i—

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SI

•rH

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CD

CDa,

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5. 3. q - Maximum Heat Flux

Considerable information is available about the parameters which

affect the magnitude of q , the maximum heat flux. This quantity varies

2

roughly from 10 to 150 kW/m . The effects of geometrical configuration

bath temperature, and pressure on q will be considered first, and then

the effect of the solid surface on AT will be discussed. Frederking

(1968) presents a more detailed discussion. Although q for flow in a

capillary tube has been investigated by Rorschach and Romberg (1957),

most studies have concerned heat dissipation from a horizontal cylinder

placed in a helium bath. The diameters have ranged in size from thin

wires to infinity, that is, a flat surface. Figure 7, reproduced from the

review paper by Frederking (1968), shows that the highest q values

are obtained for fine wires, and that a limiting value of about 10 kW/m

would be expected for a flat surface at about 1. 9 K. This is in agreement with experimental results for flat surfaces (Clement and Frederking,

1965; Lyon, 1965), and the Kutateladze correlation followed by ordinary

fluids (Brentari, Giarratano and Smith, 1965). The scatter of the points

in figure 7 and the extrapolations and corrections used to get the points

are discussed by Frederking (1968).

Since the data presented in figure 7 represents different solids,

the fact that almost all of the data fall on the same curve would seem to

indicate that the properties of the solid are of lesser importance. The

51
extent of influence of these properties could be better gauged from

studies in which the surface characteristics of a sample were varied,

while other parameters were kept constant. Lyon (1969) has observed

significant effects upon the peak heat flux density when a platinum

surface is coated with a thin layer of organic material or water.

As noted above, the test dewar diameter and depth of immersion

of the heater also affect q . The effect of immersion depth has been

investigated for wire heaters of 17 to 81M- diameter in an infinitely large

pool between T, and 2 o 00 K by Frederking and Haben (1968). Lemieux

A. and Leonard (1968) have reported data on the effect of immersion depth

upon heat transfer for a fine wire over a larger temperature range.

The effect of immersion depth decreased in importance as T~> K and

T

-* T, . A few other scattered measurements of the immersion depth

effect are given in the literature (Frederking, 1961; Irey, McFadden

and Madsen, 1965), but not enough data are available to derive an analytic form for q as a function of the geometric parameters.

The experimentally observed variation of q with temperature

below the X point is shown in figure 8. A maximum occurs at about

1.9 K. Other types of experiments also show a maximum in heat transport at about 1.9 K, and it appears that this behavior is explicable entirely

52
9<xi

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u

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X

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rsj

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o

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o

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<v

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XI

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h

53
on the basis of liquid properties. If the two-fluid model of helium II

superfluidity is invoked to explain this temperature dependence, the

Gorter-Mellink model for mutual friction between the normal and superfluid components of helium II gives

q = p ST(S/Ap ) 1/3 (VT) 1/3. (12)

Equation (12) is discussed further by A rp (1970). It has a maximum at

T— 1.9 K, as illustrated in figure 3 of Arp (1970). Another expression

with similar temperature dependence has also been proposed (Clement

and Frederking, 1965; Frederking, 1968).

The dependence of q on pressures up to about one atm. is

shown in figure 9, for the case of horizontal and vertical thin wires.

The solid lines in the figure are correlations obtained for ordinary

liquids. These expressions are exceeded for helium II, at least for fine

wires.

In contrast with q , which seems to be essentially hydrodynamically limited, AT appears to be correlated with the observed Kapitza resistance, from the small amount of evidence available so far. Asa functio

54
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a

QC CDp. ITO a 1- a

•» x Q. rt

•» hi UJ

3

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Ll)

cr G Q.

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<D -H

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u r

3

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2wo/m • xrnd 1V3H wnwixviAi

55
of temperature, AT appears to have about the same dependence as q .

Both these trends are indicated in figure 10, taken from Frederking (1968),

Since the Ni-Fe alloy had a higher Kapitza conductance than the silver,

and the observed peak heat flux q was about the same for both,

AT = q /h is expected to be smaller for the alloy. The information

presented earlier in this section can be used to estimate q for a particular geometry, pressure and temperature. (Equation (12) may be

scaled to give the temperature dependence.) If h(T) can be estimated

using (11) and low AT measurements from information in Section 2,

AT can be found to a first approximation. Where solid-liquid interface

data are not available, upper and lower bounds for AT can be obtained

from simple considerations suggested by Frederking (1968). These give

(T

x -T

b

) * AT" * AT"

He l + (T

x -T

b

) , (13)

where T, is the bath temperature, and AT the peak nucleate

b He I boiling temperature difference, which appears to be less than about 2 K

from available He I experimental data.

5. 4. Film Boiling

After T is reached, the heated surface is surrounded by a film

of vapor, and the conductance across the boundary decreases by a factor

of about a hundred. Since Kapitza conductance is proportional to the

third power of the temperature, it rapidly ceases to be a limiting factor

56
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57
in this AT range. Hence this region is outside the scope of the paper,

and will be treated very sketchily with references to fuller discussions

elsewhere. A good brief discussion is found in Frederking (1968).

At very high AT, film boiling correlations for ordinary liquids

were in approximate agreement with the experimental data of Holdredge

and McFadden (1966). Frederking (1961) has also investigated the extent

to which He II film boiling data can be described by conventional correlations. He presents data on film boiling for thin wires which indicates

the dependence of the heat transfer upon wire diameter, pressure, and

depth of immersion. No discontinuity in film heat conductance h at the

X point was observed, but dh/d(AT) is discontinuous because of these

effects. Recently, a theory which applies to lower AT values has

appeared (Rivers and McFadden, 1966); at present the validity of the

model hypothesized in the calculations has not been fully checked experimentally, though agreement over a small range of experimental

parameters is satisfactory. It should also be noted that the presence

or absence of noise associated with the breakup of a smooth gas film

over the surface affects the heat transfer in this AT range considerably

(Frederking, 1968; Lemieux and Leonard, 1968).

58
6. SUMMARY

The thermal barrier at the interface between a solid and liquid

helium II will be much larger than the thermal impedance of the bulk

liquid in most cases. Hence a better understanding of Kapitza conductance is essential to the further development of the technology of heat

transfer to helium II. The phonon radiation limit, based on thermodynamic considerations, is closely approached by the non-metallic crystals

which have been studied, and by the two metals, copper and lead, with

which some work has been done to bring surface conditions closer to

the ideal of a simple termination of the bulk. This possibility of approaching the theoretical limit merits further investigation, but the parameters

which determine the conductance of practical interfaces also remain to

be identified. These parameters are unknown at present because the

specific mechanism of energy exchange at the surface has not been established. The phonon transmission process on which the acoustic mismatch

theory is based has been most used in attempts to explain the experimental

data, but both the original theory of Khalatnikov and its various modifications predict conductances that are one or more orders of magnitude below those actually observed, A possible alternative process is energy

exchange between the atoms of the fluid and those of the solid by adsorptiondesorption. This process is related to the thermal accommodation of gases

on solid surfaces, but accommodation coefficient calculations also have not

59
achieved agreement with experiment at present. The question of whether

the conduction electrons in metals exchange energy directly with the

phonons of the liquid is also unclear because of conflicting experimental

evidence and because the theories developed for such processes are in

order of magnitude disagreement with experiment.

A theoretical solution of the interface conductance problem would

be of great intrinsic interest as well as of considerable practical assistance in the development of helium II technology. For present design

problems, an order -of -magnitude estimate for a chemically cleaned or

polished metal surface can be made from figures 1 through 3 if the

Kapitza conductance of the metal has not been measured. If the surface

is not cleaned or polished, allowance for a conductance of as much as a

factor of 10 less should be made. Judging from the limited information

now available on non-metals, the phonon radiation limit, (5), can give

an approximation to within a factor of about 3 of the conductance if the

Debye temperature is known or can be estimated. (See Blackman, 1955

for further Debye temperature information. ) Even if the conductance of

the material under consideration has been previously measured, considerable caution should be exercised in applying the data from Table I,

since it has been shown that conductances of different samples can vary

by about an order of magnitude. The condition of the surface for which the

measurements were made should be noted. If heat inputs are large enough

60
that the AT « T limit is exceeded, heat transfer mechanisms in the

bulk liquid are also important, and the considerations of Section 5 must

also be taken into account.

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75
Table II. Comparison with Phonon Radiation Limit — Metals (1.9 K)

Solid

Phonon Radiation Highest Observed

eD Limit for hK ( 1.9 K) hK (1.9K)

(k W/m2 K)

30

32

(k W/m2 K)

Hg 72 440

Pb 100 190

In 111 171

Au 162 155

Pt 221 62

Ag 226 55

Sn 195 54

Cu 343 30

Ni 44 19

W 405 18

11

Reference

Neeper, Pearce

& Wasilik[ 1967]

Challis & Che eke

[1965]

Neeper & Dillinger [ 1964]

8.8 Johnson and

Little [1963]

lot Lyon [1968]

6 Clement & Frederking [ 1965]

12. 5 Gittleman &

Bo^owski [ 1962]

7. 5 Johnson & Little

[1963]

4. Wey-Yen [ 1962]

2. 5 Johnson & Little

[1963]

t There is a large uncertainty in this value because of the scatter of

the data for small AT,

76
Table III. Comparison with Phonon Radiation Limit — Non-Metals ( 1 . 9 K)

Phonon Radiation Highest Observed

Solid GD Limit for hK( 1.9 K) hK (1.9K) Reference

(k W/ms K) (k W/mS K)

KC1 Z30 22 6.9 Johnson [ 1964]

Si0 2 290 19 5.7 Wey-Yen [ 1962]

(quartz)

Si 636 6.4 4.2 Johnson & Little

[1963]

LiF 7 50 5. 1 4. 5 Johnson & Little

[1963]

A12 3 1000 1.5 1.6 Gittleman &

Bozowski [ 1962]

77
Table IV. Measurements of Kapitza Conductance for Copper [1.9 K)

Conductance

at 1.9 K

(kW/m2 K)

7. 5

7. 3

6.7

6.0

5.4

Condition

of Surface

Machined; cleaned chemically and rinsed in water

& ethanol; 8 5 mm of

154 mm machined under

liquid helium; exposed to

atmosphere 1 week at room

temperature.

Same as preceeding except

47 mm'

chined.

Temperature

Dependence

.,2.6

of 1 6 1 mm ma

Machined; cleaned chemically & rinsed in water &

ethanol; 47 mm of 1 61

mm machined under

liquid helium.

,3.1

-,3.1

Machined

Etched with dilute HN0 3 for

3 min. ; exposed to atmosphere a few min.

T

2.7

Reference

Johnson &t Little [1963]

6.3 Polished; washed in NaOH

& CCU.

T Kapitza [ 194

6. 2 Machined; cleaned chem- T

3.0 Johnson &

ically and rinsed in water & Little [ 1963]

? 2 ethanol; 85 mm of 154 mm

machined under liquid

helium

Goodling &

Irey [ 1969]

Challis, Drans.

feld & Wilks

4.2

3.5

Severely deformed by hammering; mechanically polished;

Same as preceeding; then

cooled to liquid helium temperatures; annealed for 4

hrs. at 625°C; lightly repolished mechanically.

Challis [1962]

78
Table IV. Measurements of Kapitza Conductance for Copper (1.9 K)

(continued)

Conductance

at 1.9 K

(kW/mS K)

2. 7

2. 6

2.3

1. 2

Condition

of Surface

Polished with jewelers

rouge; exposed to atmosphere a few min.

Temperature

Dependence

-,2.5

,2 6

,2.6

R eference

Challis, Dransfeld, & Wilks

[1961]

Wey-Yen [ 1962]

Zinoveva [ 1953]

White, Gonzales,

and Johnson [ 1953]

79
Table V. Measurements of Kapitza Conductance for Lead

in the Normal State (1.9 K)

Conductance

at 1. 9 K

(kW/m 2 K)

32

29

25

23

22

20.0

19

16.7

15. 1

Condition

of Surface

Annealed; machined;

lightly electropolished;

heavily bombarded by

ions.

Annealed; machined;

lightly electropolished;

heavily bombarded by

ions.

Freshly machined.

Machined; heavily electropolished and immediately placed in a 10~

bar vacuum.

Machined; exposed to

atmosphere 8 days; lightly

bombarded by ions; 2 00

M-m removed by electropolishing; heavily bombarded by ions.

Annealed a few weeks at

300 K; machined under

liquid helium; annealed

2-3 days at 77 K.

Machined; exposed to

atmosphere 8 days; lightly

bombarded by ions.

Annealed a few weeks at

300 K; Machined under

liquid helium.

Annealed at 300 K for 2

months; machined under

liquid helium.

Temperature

Dependence

n 3.5

-,3.5

2.t> 7

T

3.00

Reference

Challis &

Cheeke [ 1965]

Cheeke 1 1969]

Challis &

Cheeke [ 1965]

Cheeke ] 1969]

80
Table V. Measurements of Kapitza Conductance for Lead

in the Normal State (1.9 K) (continued)

Conductance

at 1 . 9 K

(kW/m 2 K)

15

Condition

of Surface

Etched; exposed to at- mosphere a few min.

Temperature

Dependence

3.4

Reference

Challis, Drans.

feld, & Wilks

[1961].

14

14

13

6.3

5.3

4.7

2. 53

Annealed a few weeks at 300 K;

machined under liquid helium;

annealed 2-3 days at 77 K;

annealed 5 days at 300 K.

Machined; exposed to atmosphere 1-8 days.

Machined; annealed in

vacuum for 16 hrs. at 270°C.

Machined; annealed in vacuum for 16 hrs. at 270°C

Machined; exposed to atmosphere a total of 18 months.

Annealed a few weeks at 300 K;

machined under liquid helium.

Oxidized

T

3.5

1.58

>2.4

Cheeke [ 1969]

Challis &

Cheeke [1965]

Challis [1962]

Cheeke [ 1969]

Hesser, Chapman,

Chang & Frederkin^

[1969]

81
7. REFERENCES

Abbe, W. J. (1967), Simple model for the Kapitza thermal resistance,

Bull. Am. Phys. Soc. 12, No. 5, 708.

Abbe, W. J. (I968), A note on the Kapitza resistance, Nuovo Cimento

56B , No. 1, 187-189.

Anderson, A. C. , J. I. Connolly, and J. C. Wheatley ( 1964 ), Thermal

boundary resistance between solids and helium below 1 K, Phys.

Rev. L3_5, No. 4A, A910-A921.

Andreev, A. F. (1962a), Anomalous reflection of sound from the surface

of a metal at low temperatures, Soviet Phys. - JETP 16, No. 1,

257-258, Transl. from Zh. Eksperim. i. Teor. Fiz. 43, No. 1,

358-360.

Andreev, A. F. (1962b), The effect of conducting electrons on the Kapitza

temperature jump, Soviet Phys. - JE TP J_6 , No. 4, 1084-1088,

Transl. from Zh. Eksperim. i. Teor. Fiz. 43, No. 4, 1535-1542.

Andronikashvili, E. L. , and G. G. Mirskaia (1955), The behavior of

helium II in the neighborhood of a heat radiating surface, Soviet

Phys - JETP 2, No. 3, 406-409, Transl. from Zh. Eksperim. i. Teor. Fiz. 29_, No. 4, 490-494.

Arp, V. D. (19 70), Heat transport through helium II, Cryogenics

(to be published).

Barnes, L. J., and J. R. Dillinger ( 1963 ), Thermal boundary resistance

between some superconducting and normal metals, Phys. Rev.

Letters 1_0, No. 7, 287-289.

Blackman, M. (1955), The specific heat of solids, Hand, der Phys. Vol.

7, Part 1, 325-382 (Springer-Verlag, Berlin).

82
Brentari, E. G. , P. J. Giarratano, and R. V. Smith (1965), Boiling

heat transfer for oxygen, nitrogen, hydrogen, and helium, NBS

Tech. Note No. 317, National Bureau of Standards, Boulder, Colo.

Brow, N. J. , and D. V. Osborne (19 58), Heat transfer between copper

and liquid helium II, Phil. Mag. 3_, No. 36, 1463-1466.

Challis, L. J. (1961), The heat flow between metals and liquid helium II, Proc. 7th Intern. Conf. on Low Temperature Physics, 476-480,

(University of Toronto Press, Toronto).

Challis, L. J. (1962), Some experiments on the Kapitza resistance between metals and liquid helium II, Proc. Phys. Soc. 80, Part 3,

759-767.

Challis, L. J. (1964), On surface superconductivity in lead, Phys. Letters

L3, No. 1, 20-21.

Challis, L. J. (1968), Experimental evidence for a dependence of the

Kapitza conductance on the Debye temperature of a solid, Phys.

Letters 26A, No. 3, 105-106.

Challis, L. J. , and J. D. N. Cheeke (1965), The Kapitza conductance in

lead, Prog, in Refrig. Sci. and Tech., Vol. I, 227 -230 (Pergamon

Press, London).

Challis, L. J. , and J. D. N. Cheeke (1968), Some comments on the theory

of the Kapitza conductance between metals and liquid helium II, Proc.

Roy. Soc. A304 , No. 1479, 479-486.

Challis, L. J. , K. Dransfeld, and J. Wilks (1961), Heat transfer between

solids and liquid helium II, Proc. Roy. Soc. A260, No. 1300, 31-46.

Challis, L. J., and R. A. Sherlock ( 1969), Second sound measurements

of the Kapitza boundary conductance to lead, Proc. 11th Intern.

Conf. on Low Temperature Physics Vol. I, 571-574 (University

of St. Andrews Printing Dept. , St. Andrews, Scotland),

83
Cheeke, J. D. N„ (1969), The Kapitza conductance of lead, Proc. 11th

Intern. Conf. on Low Temperature Physics, Vol I, 567-570

(University of St. Andrews Printing Dept. , St. Andrews, Scotland).

Clement, B. W. , and T. H. K. Frederking ( 1965), Thermal boundary

resistance and related peak flux during supercricial heat transport

from a horizontal surface through a short tube to a saturated bath

of liquid helium II, Liquid Helium Technology, Proc. Intern. Inst.

Refrig. Commission I, Boulder, Colo., 1966, 49-59 (Pergamon

Press, Oxford).

Devonshire, A. F. (1937), The interaction of atoms and molecules with

solid surfaces. VIII - The exchange of energy between a gas and

a solid, Proc. Roy. Soc. A158 , No. 894, 269-279.

Fairbank, H. A. , and J. Wilks (1955), Heat transfer in liquid helium

below 1 K, Proc. Roy. Soc. A231 , No. 1187, 545-554.

Frederking, T. H. K. (1961), Warmeubergang bei der Verdampfung der

verfliissigten Gase Helium und Stickstoff (Heat transfer with vaporization of liquid helium and nitrogen), Forschung 27, No. 1, 17-30

and No.2, 58-62.

Frederking, T. H. K. (1968), Thermal transport phenomena at liquid

helium II temperatures, Chem. Eng. Prog. Symp. Series 64, No.

87, 21-55.

Frederking, T. H. K. , and R. L. Haben(1968), Maximum low temperature dissipation rates of single horizontal cylinders in liquid helium

II, Cryogenics 8_, No. 1, 32-35(1968).

Gittleman, J. I. , and S. Bozowski (1962), Some aspects of the Kapitza

resistance, Phys. Rev. 128, No. 2, 646-649.

84
Glattli, H. (1968), Kapitza resistance of cerium ethyl sulphate, Can. J. Phys. 46, No. 2, 103-110.

Goodling, J. S. , and R. K. Irey (1969), Non-boiling and film boiling

heat transfer to a saturated bath of liquid helium, Advances in

Cryogenic Engineering 14, 159-169 (Plenum Press, New York).

Goodman, F. O. (1968), Classical perturbation theory of the thermal

accommodation coefficient in n dimensions, Surface Sci. 11, No.

2, 283-316.

Goodman, F. O. , and H. Y. Wachman (1967), Formula for thermal

accommodation coefficients, J. Chem. Phys. 46, No. 6, 2376-

2386.

Harris, R. E. (1967), On the determination of thermal accommodation

coefficients in the temperature jump region, J. Chem. Phys. 46 , No. 8, 3217-3220.

Hesser, R. J., R. C. Chapman, Y. W. Chang, T. H. K. Frederking

(1969), Thermal boundary conductance between solids with impure

surface and superfluid liquid helium II, A.I.Ch.E. Preprint 10,

Eleventh National Heat Transfer Conference (American Institute

of Chemical Engineers, New York).

Holdredge, R. M. , and P. W. McFadden ( 1966), Boiling heat transfer

from cylinders in a saturated liquid helium II bath, Advances in

Cryogenic Engineering 11 , 507-515 (Plenum Press, New York).

Hurlbut, F. C. (1966), Current developments in the study of gas-surface

interactions, Proc. 5th Intern. Symp. Rarefied Gas Dynamics,

Vol. I, 1 -34 (Academic Press, New York).

Irey, R. K. , P. W. McFadden, andR. A. Madsen ( 1965), Heat transfer

to a saturated bath of liquid helium II, Advances in Cryogenic

Engineering 10, 361-370 (Plenum Press, New York).

85
Johnson, R. C. (1964), Kapitza resistance of a crystal cleaved while

immersed in liquid helium, Bull. Am. Phys. Soc. 9, No. 7, 713.

Johnson, R. C. , and W. A. Little (1963), Experiments on the Kapitza

resistance, Phys. Rev. 130 , No. 2, 596-604.

Kapitza, P. L. (1941), The study of heat transfer in helium II, J. Phys.

(Moscow) 4, No. 3, 181-210.

Khalatnikov, I. M. (1952), Heat exchange between a solid body and

helium II, Zh. Eksperim. i. Teor. Fiz. 2_2, No. 6, 687-704.

Khalatnikov, I. M. (1965), Introduction to the Theory of Superfluidity,

Chap. 23 (W. A. Benjamin, Inc. , New York).

Kittel, C. (1966), Introduction to Solid State Physics, Chapters 5 and 6,

third ed. (Wiley and Sons, New York).

Klemens, P. G. (1958), Lattice thermal conductivity, Solid State Physics

7, 1-98 (Academic Press, New York).

3 Lee, D. M. , and H. A. Fairbank (1959), Heat transport in liquid He , Phys. Rev. 116 , No. 6, 1359-1364.

Lemieux, G. P., and A. C. Leonard (19 68), Maximum and minimum heat

flux in helium II for a 76. 2 M- diameter horizontal wire at depths of

immersion up to 70 cm, Advances in Cryogenic Engineering 13,

624-631 (Plenum Press, New York).

Little, W. A. (1959), The transport of heat between dissimilar solids at

low temperatures, Can. J. Phys. 37, No. 3, 334-349.

Little, W. A. (I960), The Kapitza resistance, Proc. 7th Intern. Conf.

on Low Temperature Physics, 482-483 (University of Toronto

Press, Toronto).

86
Little, W. A. (196la), Kapitza resistance between helium and metals

in the normal and superconducting states, Phys. Rev. 123, No. 2,

435-441.

Little, W. A. (1961b), Unimportance of surface roughness upon the Kapitza

resistance, Phys. Rev. 123 , No. 6, 1909-1911.

Little, W. A. (1962), The Kapitza resistance of metals in the normal and

superconducting states, I. B.M. J. Res. Dev. 6, No. 1, 31-33.

Lyon, D. N. (1965), Boiling heat transfer and peak nucleate boiling fluxes

in saturated liquid helium between the ^ and critical temperatures,

Advances in Cryogenic Engineering 10, 371-379, (Plenum Press,

New York).

Lyon, D. N. (1968), Pool boiling of cryogenic liquids, Chem. Eng.

Prog. Symp. Series 64, No. 87, 82-92.

Madsen, R. A. , and P. W. McFadden (I968), Heat transfer to an unsaturated bath of liquid helium II, Advances in Cryogenic Engineering

13 , 617-623 (Plenum Press, New York).

Manchester, F. D. (1967), Solid layers in the adsorbed helium film and

mobility of the film at low coverages, Rev. Mod. Phys. 39 , No. 2,

383-394.

Mate, C. F. , and S. P. Sawyer (1968), Kapitza effect in gaseous helium,

Phys. Rev. Letters _2_0, No. 16, 834-837.

Mate, C F., & S P. Sawyer (1969), The Kapitza effect in the liquid

and vapor phases of He3 and He 4

, Proc. 11th Intern, Conf„ on

Low Temperature Physics, Vol I, 579-581 (University of St.

Andrews Printing Dept„ , St„ Andrews, Scotland),

87
Neeper, D. A. , and J. R. Dillinger (1964), Thermal resistance at indiumsapphire boundaries between 1. 1 and 2. 1 K, Phys. Rev. 135, No.

4A , A1028-A1033.

Neeper, D. A., D. C. Pearce, and R. M. Wasilik ( 1967), Kapitza resistance of mercury between 1. 1 and 2. 1 K, Phys. Rev. 156, No. 3,

764-768.

Peshkov, V. P. (1958), Observations of the X-transition in helium in the

presence of a thermal current through the phase boundary, Soviet

Phys.-JETP8, No. 6, 943-946, Trasl. from Zh. Eksperim. i. Teor.

Fiz. 3_5, No. 6, 1350-1354.

Pollack, G. L. (1969), Kapitza resistance, Rev. Mod. Phys. 41_, No. 1,

48-81.

Rivers, W. J. , and P. W. McFadden (1966), Film free convection in

helium II, J. Heat Trans. , Trans, of the ASME Ser. C 88, No. 4,

343-350.

Roberts, J. K. (1932), The exchange of energy between gas atoms and

solid surfaces. II - The temperature variation of the accommodation

coefficient of helium, Proc. Roy. Soc. A135, No. 826, 192-205.

Rorschach, H. E. , and F. A. Romberg (1957), Pressure gradients produced by heat flow in liquid helium, Proc. 5th Intern. Conf. on

Low Temperature Physics and Chemistry, 35-38 (University of

Wisconsin Press, Madison).

Rosenberg, H. M. (1963), Low Temperature Solid State Physics, Section

5. 13 (Clarendon Press, Oxford).

Singh, R. P., and W. Band (1955), The anomalous monolayer adsorption

of helium, J. Phys. Chem. 59, No. 7, 663-665.

88
Smoluchowski, M. (1898), Conduction of heat by rarefied gases, Phil.

Mag. 46, No. 279, 192-206.

Steele, W. A. (1956), Concerning a theory of multilayer adsorption,

with particular reference to adsorbed helium, J. Chem. Phys. 25,

No. 5, 819-823.

Suomi, M , A. C. Anderson, and B. Holmstrom (1968), Heat transfer

below 0. 2 K, Physica 38_, No. 1, 67-80.

Thomas, L. B. (1966), A collection of some controlled surface thermal

accommodation coefficient measurements, Proc. 5th Intern. Symp.

Rarefied Gas Dynamics, Vol. I, 155-162 (Academic Press, New

York).

Thomas, L. B. , and E. B. Schofield ( 1955), Thermal accommodation

coefficient of Helium on a bare tungsten surface, J. Chem. Phys.

23 , No. 5, 861-866.

Von Gutfeld, R. J., A. H Nethercot, Jr., and J. A„ Armstrong ( 1966),

Transport of heat from metals to insulators at low temperatures,

Phys. Rev. 142 , No. 2, 436-441.

Wachman, H. Y. (1962), The thermal accommodation coefficient: A

critical survey, Am. Rocket Soc. J. 32, No. 1, 2-12

Wey-Yen, K. (1962), Investigation of the temperature discontinuity at

the boundary between a solid and superfluid helium, Soviet Phys, - JETP JL5, No. 4, 635-645, Transl. from Zh. Eksperim. i. Teor.

Fiz. 42, No. 4, 921-935.

Whelan, M. F„ , and D. V. Osborne (1969), The modification of the

Kapitza resistance due to a surface film, Proc. 11th Intern.

Conf. on Low Temperature Physics, Vol. I, 575-578 (University

of St. Andrews Printing Dept. , St. Andrews, Scotland).

89
White, D , O. D. Gonzales, and H. L. Johnston (1953), Heat conduction

of the boundary layer in liquid helium II, Phys„ Rev. 89, No. 3,

593-594.

Wilks, J„ (1967), The Properties of Liquid and Solid Helium, (Clarendon

Press, Oxford).

Woods, A. D. B. (1965), Neutron inelastic scattering from liquid helium

at small momentum transfers, Phys. Rev. Letters 14, No. 10,

355-356.

Young, F. W. Jr., and T. R. Wilson ( 1961), Acid cutting and acid

polishing of copper crystals, Rev. Sci. Instrum. 32, No. 5,

559-562.

Zinoveva, K. N. (1953), Zh. Eksperim. i. Theor. Fiz. 25, 235.

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