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An extension of Khalatnikov's theory of Kapitsa thermal resistance.

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Ann. Phys. (Berlin) 523, No. 3, 208 – 225 (2011) / DOI 10.1002/andp.201000111
An extension of Khalatnikov’s theory
of Kapitsa thermal resistance
Bair V. Budaev∗ and David B. Bogy∗∗
University of California, Etcheverry Hall, MC 1740, Berkeley, CA 94720-1740, USA
Received 19 August 2010, revised 27 October 2010, accepted 28 October 2010 by U. Eckern
Published online 15 December 2010
This paper develops a novel approach to Kapitsa thermal resistance which extends Khalatnikov’s acoustic
mismatch theory by inclusion of a previously excluded broad class of acoustic waves that violate the Sommerfeld radiation condition, which is essential for the theory of sound but is not relevant for the analysis of
heat transport. This extension not only preserves the main ideas of Khalatnikov’s theory but we show it also
provides reasonable estimates of the Kapitsa resistance.
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
In the 1930s, W. H. Keesom and A. P. Keesom studied thermal conductivity of liquid helium II in capillary tubes and discovered that it was so high that special care had to be taken “to eliminate the thermal
resistance at the limiting surface between copper and liquid helium II, and which appeared to be relatively
very considerable” [1]. Later, P. L. Kapitsa demonstrated that the abnormally high thermal conductivity of
helium II in capillary tubes was caused by super-fluidity which was the main, but not the only, discovery
of his study [2, 3]. Trying to understand why helium in a narrow tube conducts heat much better than in a
larger volume, P. L. Kapitsa analyzed the temperature distribution near a heated copper block suspended in
liquid helium and found the thermal resistance was clearly coming from the interface.
The interface thermal resistance, which is widely known as “Kapitsa resistance”, has attracted much
attention for about seventy years but it still remains surprisingly poorly understood [4–6]. It was found that
the Kapitsa resistance decreases with increasing temperatures approximately as T −3 and that it was highly
sensitive to the structural properties of the copper and helium as well as to the quality of the interface
between them. It also became clear that the Kapitsa resistance was not related to the specifics of helium
or copper, and that it could be observed between virtually any pair of different materials. However, the
mechanism of this resistance remained a mystery until I. M. Khalatnikov proposed an acoustic mismatch
model [7] based on the concept that heat was carried through the interface by acoustic waves propagating
by the same laws as describe the transmission of sound from one medium to another.
Khalatnikov’s theory provides a plausible description of the principle cause of the interface thermal resistance, but in some cases its predictions are almost two orders of magnitude higher than observed values.
Such noticeable disagreement with experiments has caused much controversy because, normally, it would
be sufficient to refute any theory, but on the other hand, no flaws have been found in Khalatnikov’s reasoning that the principal part of the interface thermal resistance is caused by the acoustic mismatch. Attempts
to resolve the mystery inevitably led to the conjectures that either thermal vibrations “are able to cross the
interface substantivally more easily than expected” [6], or “that some other, more important mechanism is
also present” [4]. Since these ideas are not contradictory it is not surprising that both of them have been
∗
∗∗
Corresponding author E-mail: bair@berkeley.edu
E-mail: dbogy@berkeley.edu
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Ann. Phys. (Berlin) 523, No. 3 (2011)
209
intensively explored separately and together. Thus, the diffuse mismatch model [8] preserves the idea that a
major part of the heat flow through the interface is carried by acoustic waves, which, however, interact with
the interface not by the laws of acoustics but by some phenomenological probabilistic rules. This model,
however, often predicts thermal resistance as much as two orders below experimental observations. The
search for additional heat carriers across the interface was even more intensive than the efforts to improve
the acoustic mismatch model. Much of the work in these directions is reviewed in surveys [4, 6, 8], all of
which come to a similar conclusion that “the reason for this so-called ‘anomalous transmission’ at these
low temperatures has still not been established” [6].
Here we present a new approach to Kapitsa resistance which extends Khalatnikov’s theory by inclusion
into consideration a broad class of acoustic waves violating the Sommerfeld radiation condition which, as
discussed in [9], is essential for the theory of sound but is not relevant for the analysis of heat transport.
It is shown that this extension not only preserves the main ideas of Khalatnikov’s theory but also provides
reasonable agreement with experimentally measured values of the Kapitsa resistance.
To illustrate the difference between the proposed and the conventional approaches to the Kapitsa resistance it suffices to analyze vibrations of the one-dimensional space −∞ < x < ∞ subdivided into the
half-lines x < 0 and x > 0 separated by the interface x = 0. It is easy to see that an arbitrary acoustic field
in this space has the structure
⎧
⎨a eik− x−iωt + b e−ik− x−iωt ,
x < 0,
−
−
(1.1)
φ=
⎩a+ eik+ x−iωt + b+ e−ik+ x−iωt ,
x > 0,
where the pairs of coefficients are coupled by an equation (a+ , b+ ) = T (a− , b− ) with the operator T
determined by the interface conditions. Therefore, either pair of the coefficients (a− , b− ) or (a+ , b+ ) determines an acoustic field in the entire space, and our approach indiscriminately takes into account all such
fields. On the contrary, the conventional acoustic mismatch theory takes into account only fields of one of
the following types
⎧
⎨a eik− x−iωt + b e−ik− x−iωt ,
x < 0,
−
−
φ=
(1.2)
⎩a+ eik+ x−iωt ,
x > 0,
or
⎧
⎨b e−ik− x−iωt ,
−
φ=
⎩b+ e−ik+ x−iωt + a+ eik+ x−iωt ,
x < 0,
(1.3)
x > 0,
which are particular cases of (1.1) with b+ = 0 or a− = 0, respectively. It is easy to see that our approach incorporates both of the conjectures that the acoustic waves may have additional easy ways to cross
the interface and that heat may be conducted by carriers ignored in the conventional acoustic mismatch
theory. Indeed, interface crossing is determined by coupling of waves propagating on different sides of
the interface, and our approach allows four-way couplings instead of three-way couplings permitted in the
conventional theory. Also, acoustic fields of the type (1.1) which do not fit particular classes (1.2) and
(1.3) may be considered as additional heat carriers ignored by conventional theories but included in our
approach.
The next three sections of the paper provide background material about ensembles of acoustic vibrations
of the entire homogeneous space with and without a heat flux. Section 2 explicitly describes an equilibrium
ensemble of thermal waves at temperature T . In Sect. 3, we apply an analog of the Doppler transformation
and generalize the formulas representing equilibrium ensembles to formulas describing ensembles of vi Then, in Sect. 4 the results of the previous sections are re-arranged to a
brations with a steady heat flux Q.
more convenient form for our purpose. In Sects. 5 and 6 the focus shifts to the analysis of acoustic fields in
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B. V. Budaev and D. B. Bogy: Khalatnikov’s theory of Kapitsa thermal resistance
joined half-spaces occupied by different materials. First we formulate the coupling conditions, and then we
describe the structure of an equilibrium ensemble of vibrations of joined half-spaces considered as a single
bi-material medium. Finally, in Sects. 7 and 8 we estimate the Kapitsa resistance and discuss the obtained
numerical results and the directions of the future developments of the proposed theory.
2 Ensemble of acoustic vibrations in thermodynamical equilibrium
It is convenient to describe vibration of an acoustic medium in terms of the pressure p(x, t) related to the
particle displacements ξ and to the average energy density E by the expressions
2
p
∂ 2 ξ
ρ 2 = −∇p,
E=
,
(2.1)
∂t
ρc2
where ρ and c are the material density and sound speed, respectively.
Let p(x, k) be the pressure at the point x of a monochromatic acoustic wave
ρc
p(x, k) =
F (k)eik·x−iωt ,
(2.2)
3
8π
where ω is the frequency of vibrations and ρc/8π 3 is a scaling factor introduced for future convenience,
F (k) is a function of the wave vector k related to the frequency ω by the dispersion equation
c|k| = ω.
(2.3)
From (2.1) it follows that the average density of the acoustical energy E(ω) of the ensemble of waves (2.2)
with different wave vectors corresponding to the frequency ω has the value
1
|F (k)|2 dS(k),
(2.4)
E(ω) =
3
8π c |k|=ω/c
where dS(k) is the area element on the sphere of radius |k|, so that
|
k|=ω/c
2
ω
dS(k) = 4π 2 .
c
(2.5)
Formula (2.4) presents the energy spectrum of vibrations (2.2) considered as acoustic waves. However,
these vibrations can also be considered as independent harmonic oscillators at frequency ω. Then, the
quantum hypothesis implies that if this ensemble is thermodynamically balanced at temperature T its
average energy density has the value
E(ω) = P 2 (ω, T )D(ω) +
1
ω D(ω),
2
(2.6)
where
P 2 (ω, T ) =
ω
eω/κT − 1
(2.7)
is the average thermal energy of a single harmonic oscillator, 12 ω is its ground-state (zero-point) energy
which is unrelated to thermal processes, and D(ω) is the “density of states” defined in such a way that
D(ω)dω is the number of vibrational modes of a unit volume of the medium with frequencies from the
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Berlin) 523, No. 3 (2011)
211
interval [ω, ω + dω). As discussed in [10], the density of states D(ω) of an isotropic acoustic media has
the value D(ω) = V (ω)/(2π)3 , where V (ω) is the volume of the sphere |k| < ω/c, so that
4π ω 3
4πω 2
1
V (ω) =
,
V (ω) =
=
dS(k),
(2.8)
3 c
c3
c |k|=ω/c
and
D(ω) =
1
8π 3 c
dS(k).
(2.9)
|
k|=ω/c
Now we have two different representations (2.4) and (2.6), (2.9) for the energy spectrum E(ω). Equalizing
these representations we conclude that
ω
|F (k)|2 = P 2 (ω, T ) +
,
2
ω=
|k|
.
c
(2.10)
Then, assuming that the ground-state vibrations are uncorrelated with the thermal vibrations we represent
the amplitudes of the ensemble (2.2) as
ω iα0
|k|
e + P (ω, T )eiα ,
,
(2.11)
ω=
F (k) =
2
c
where α0 and α are independent uniformly distributed random phase shifts. To verify that F (k) from (2.11)
satisfies (2.10) we obtain the expression
ω
ω
2
2
+ P (ω, T ) + 2
P (ω, T ) Re{ei(α0 −α) },
|F (k)| =
(2.12)
2
2
where the average of the last term vanishes because of the uniform distributions of α0 and α.
Substitution of (2.11) into (2.2) leads to the decomposition p = pT + p0 where the first term describes
the pressure fluctuation due to thermal motion while the second term describes the ground state vibrations
which are not related to thermal processes. Taking into consideration the asymptotic structure of P (ω, T )
in the domain ω/κT 1 it is possible to demonstrate that the ground state vibrations do not affect
the thermal properties of liquid helium and that, therefore, for our purposes it suffices to assume that the
equilibrium ensemble of the vibrations of the acoustic medium can be modeled by a set of waves
ρc
P (ω, T )eik·x−iωt+iα ,
ω = c|k|,
(2.13)
p(x, k) =
8π 3
where α is a random phase shift and P (ω, T ) is the Planck function (2.7)
The above discussion is limited to an isotropic acoustic medium supporting propagation of only one type
of waves which may have an arbitrary frequency. On the other hand, real materials may be anisotropic, and
even isotropic solids usually support propagation of longitudinal and shear elastic waves whose frequency
can not exceed a certain threshold determined by the material’s structure. Comparison of these observations suggests that a straightforward application of the results that follow may not adequately describe
thermal transport in realistic solids. However, it is well known that the thermal properties of solids can be
sufficiently well described by the Debye approximation briefly described below.
In the Debye theory any solid is modeled by an isotropic acoustic medium supporting propagation of
three different kinds of waves all of which have the same sound speed
3vs3 vp3
3
3
≡ 3 3
,
(2.14)
c=
3
3
1/vp + 2/vs
vs + 2vp3
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B. V. Budaev and D. B. Bogy: Khalatnikov’s theory of Kapitsa thermal resistance
where vp and vs are the speeds of longitudinal and shear waves in the real solid, respectively. The density
of vibrational states in this model is determined by the formula
⎧
⎨3ω 2 /8π 3 c3 , if ω ≤ Ω
max ,
D(ω) =
(2.15)
⎩0,
if ω > Ωmax ,
which differs from (2.9) by the presence of the factor “3” introduced to account for the three different kinds
of waves, and by the presence of the cut-off frequency Ωmax determined from the condition that the total
number of vibrational modes in the real material coincides with the total number of vibrational modes in a
model with the density of states (2.15). Finally, it is worth mentioning that the frequency Ωmax is usually
represented by
Ωmax =
κ
TD ,
(2.16)
where TD is the Debye temperature considered as a material parameter, and that the density of thermal
energy in the Debye model has the value
3
E=
2π 2 c3
κTD /
0
3κ4 T 4
ω 3 dω
=
2π 2 c3 3
eω/κT − 1
TD /T
0
ξ 3 dξ
,
eξ − 1
(2.17)
which is obtained by the substitution of P 2 (ω, T ) and D(ω) from (2.7) and (2.15) into the the integral
∞
E = 0 P 2 (ω, T )D(ω)dω.
3 Ensemble of acoustic vibrations with a heat flux
If the ensemble of vibrations has a heat flux then it cannot be described by the formula (2.13) derived from
the assumption of thermodynamical equilibrium. However, the method which led to the expression (2.13)
can be adjusted to handle non-equilibrium systems with a steady heat flux.
it suffices to find the moving frame in which there is
To consider vibrations with a steady heat flux Q
no heat flux. Let v be the speed of the moving frame, which means that the observer’s positions x and x1
in the reference and the moving frames are related by the formula
x1 = x − v t,
(3.1)
1 in the moving frame has the value
illustrated in Fig. 1. Then, the heat flux Q
1 = Q
− v E + o(v 2 ),
Q
(3.2)
is the heat flux in the reference frame and E is the density of thermal energy from (2.17). If
where Q
v = Q/E then the ensemble of vibrations in the moving frame has no heat flux and can, therefore, be
described by the methods of equilibrium thermodynamics.
A medium with the density of thermal energy E
0
Q
vt
0
1 = Q
− vE
Q
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
x
(reference frame)
x1 = x − vt
(moving frame)
Fig. 1 Heat flux in the reference and moving coordinate frames.
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Ann. Phys. (Berlin) 523, No. 3 (2011)
213
Assume for definiteness that the heat flux is parallel to the x-axis which means that
= Qex ,
Q
|ex | = 1,
(3.3)
where ex is the unit vector parallel to the x-axis. Then, taking into account (3.1) we find that in terms of
x1 the plane wave (2.2) can be represented as
p(x1 , ω1 ) =
ρc
F (k)eik·x1 −iω1 t+iα ,
8π 3
(3.4)
where
ω1 = ωq(k) ≡ c|k|q(k),
(3.5)
k · ex
q(k) = 1 − Q
cE
(3.6)
and
is the Doppler factor.
Since the ensemble (3.4) is balanced its energy density has the Planck spectrum
E1 (ω1 , T, Q) = P 2 (ω1 , T )D1 (ω1 , Q),
(3.7)
with the density of states
D1 (ω1 , Q) =
1 dV1 (ω1 , Q)
,
(2π)3
dω1
(3.8)
where
V1 (ω1 , Q) =
ω1 /cq(
k)
2
k dk
0
|
e|=1
1
dS(e) =
3
ω13 dS(e)
,
c3 q 3 (e)
|
e|=1
e =
k
,
|k|
(3.9)
is the volume of the domain |k| < ω1 /cq(k). From (3.7)–(3.9) we obtain an explicit expression for
E1 (ω1 , T, Q) and observing that E(ω)dω = E1 (ω1 )dω1 converts it to the formula
1
E(ω, T, Q) =
8π 3 c
|
e|=1
ω2 2
1
P (ωq(k)) dS(e) =
c2
8π 3 c
P 2 (ωq(e) dS(k),
(3.10)
|
k|=ω/c
which represents the energy spectrum in terms of the frequency ω in the reference frame.
The last expression represents the spectrum of the ensemble (2.2) computed from the thermodynamical
point of view. This spectrum, however, can also be computed from the acoustical formula (2.17). Equalizing
both expressions we conclude that F (k) = P (ωq(k), T ) and that an ensemble (2.2) with the energy flux
consists of the waves
Q
p(x, k) =
ρc
P (ωq(k), T )eik·x−iωt+iα ,
8π 3
ω = c|k|,
(3.11)
which reduce to (2.13) in the equilibrium case when q(k) = 1.
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B. V. Budaev and D. B. Bogy: Khalatnikov’s theory of Kapitsa thermal resistance
4 Ensembles of pairs of acoustic waves
Although any acoustic field can be decomposed into plane waves (2.2), a set of plane waves does not
adequately describe the vibrations in a domain other than the entire space. In particular, vibrations in a
half-space do not consist of individual plane waves but consist of the pairs of waves
p(x, t) = Aeik(ex x+ey y+ez z)−iωt + Beik(−ex x+ey y+ez z)−iωt ,
k=
ω
,
c
ex ≥ 0.
(4.1)
Therefore, to study thermal transport carried by acoustic vibrations between half-spaces we need to analyze
the ensembles of acoustic fields of the type (4.1).
To analyze the energy characteristics of the fields (4.1), it is convenient to re-arrange them by representing the two complex-valued coefficients A and B in the “polar” form
A = Λ cos ηeiβ+iα ,
B = Λ sin ηeiβ−iα ,
(4.2)
with four real-valued parameters, including the absolute and relative phase shifts
1
1
[arg(A) − arg(B)] ,
β = [arg(A) + arg(B)] ,
(4.3)
2
2
the amplitude Λ = |A|2 + |B|2 and the angle η = arctan (|B|/|A|), which will be referred to as the
“composition index”. In terms of these new parameters the field (4.1) appears as
(4.4)
p(x, t) = Λ cos ηeikx cos θ+iα + sin ηe−ikx cos θ−iα eik(ey y+ez z)+iβ−iωt ,
α=
where
(4.5)
θ = arccos(ex ) > 0
is the angle between the x-axis and directions of propagation of the waves from the field (4.1).
Consider a thermodynamically balanced ensemble of acoustic fields defined by the formula (4.4) where
all parameters are independent random numbers. Since this ensemble consists of the same set of plane
waves as included in the ensemble (2.13), it is natural to expect that the parameters of (4.4) admit simple
representations in terms of the Planck function P (ω, T ) from (2.7). To obtain this representation we observe that the average energy densities Er and El carried by the ensemble (4.4) to the right and to the left
have the values
2 2 Λ
Λ
cos2 η
cos2 η
ω2
ω2
and El = 2π 2 ·
.
(4.6)
Er = 2π 2 ·
c
ρc2
c
ρc2
On the other hand, the equilibrium ensemble (2.13) carries the energy density
E∗ = 2π
ω 2 P 2 (ω, T )
·
c2
8π 3 c
(4.7)
in each of the directions x → +∞ and x → −∞. Therefore, in order for the ensemble (4.4) to be in
thermodynamical equilibrium, its parameters must satisfy the conditions
2
ρc 2
Λ =
P (ω, T ),
4π 3
1
cos2 η = sin2 η = ,
2
(4.8)
which follow from the substitution of (4.6) and (4.7) into the identities E∗ = Er = El . Then, combining
(4.8) with (4.4) we conclude that an equilibrium ensemble of vibrations at temperature T can be modeled
by a set of acoustic fields
ρc
ikx cos θ+iα
−ikx cos θ−iα
eik(ey y+ez z)+iβ−iωt , η ∈ L(0) (4.9)
P
(ω,
T
)
cos
ηe
+
sin
ηe
4π 3
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Berlin) 523, No. 3 (2011)
215
where α and β are arbitrary random numbers, while η = η(ω) is a random number uniformly distributed
over some domain L(0) which guarantees the identity
(4.10)
cos 2η(ω) |L(0) = 0.
It is obvious that this requirement is satisfied by any domain on the (ω, η)-plane which is symmetric with
respect to the horizontal line η = π/4. However, for our purposes it suffices to assume that L(0) has a
simple one-piece structure and is defined as a curvilinear strip
π
π
(4.11)
− δω ,
+ δω ,
L(0) :
η(ω) ∈
4
4
where δω is as of now an indefinite function.
The representation (4.9)–(4.11) of the ensemble of vibrations in thermodynamical equilibrium admits
simple generalization to the case with a steady heat flux. Thus, following the method employed in the
previous section we first find that the ensemble of vibrations with a heat flux Q along the x-axis consists
of the waves
ρc
P (ω + τ ωQ, T ) cos η(ω + τ ωQ)eikx cos θ+iα
4π 3
+ P (ω − τ ωQ, T ) sin η(ω − τ ωQ)e
−ikx cos θ−iα
eik(ey y+ez z)+iβ−iωt ,
(4.12)
where
τ=
cos θ
,
cE
(4.13)
is the coefficient from the representation of the Doppler factor from (3.6) in the form q = 1 − τ Q, and
ω ± τ ωQ are the arguments of the function η(·) whose values are uniformly distributed in the domain L(0)
satisfying (4.10). Then, defining PQ (ω, T ) from the equation
PQ2 (ω, T ) = P 2 (ω + τ ωQ, T ) cos2 η(ω + τ ωQ) + P 2 (ω − τ ωQ, T ) sin2 η(ω − τ ωQ), (4.14)
we straightforwardly convert (4.12) to the form
ρc
ikx cos θ+iα
−ikx cos θ−iα
eik(ey y+ez z)+iβ−iωt ,
P
(ω,
T
)
cos
η
e
+
sin
η
e
Q
4π 3
(4.15)
which is similar to (4.9) except that here the composition index η is no longer uniformly distributed in the
symmetric domain L(0) but is defined by the formula
P (ω − τ ωQ, T ) sin η(ω − τ ωQ)
η ≡ η(ηω ) = arctan
·
) ,
(4.16)
P (ω + τ ωQ, T ) cos η(ω + τ ωQ
where η(ω) is distributed over the domain L(0) satisfying (4.10).
It is important to note that the composition indices η and η of the equilibrium and non-equilibrium
ensembles (4.1) and (4.15) are spread over domains of different shapes. Thus, the equilibrium ensemble η
is spread over a symmetric domain L(0) from (4.11) which is shaded in Fig. 2. In an ensemble (4.15) with
a small heat flux Q the composition index η is distributed over the asymmetric domain
L(Q) = η L(0) ,
(4.17)
where the domain L(Q) is dotted in Fig. 2.
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c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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B. V. Budaev and D. B. Bogy: Khalatnikov’s theory of Kapitsa thermal resistance
π
2
L(Q)
π
4
L(0)
0
Fig. 2 (online colour at: www.annphys.org) Distributions of the composition index of ensembles with a heat flux.
ω
5 Acoustic fields in joined half-spaces
Now we switch to a discussion of acoustic fields in joined half-spaces x > 0 and x < 0 occupied by
different acoustic media in perfect contact along the plane x = 0.
Let ρ+ and c+ be the material density and the sound speed in the domain x > 0, and let ρ− and c− be
these parameters in the domain x < 0. Then, any monochromatic acoustical vibration of this bi-material
space can be decomposed into the pressure fields
⎧
⎨p (x, t), if x < 0,
−
p(x, t) =
(5.1)
⎩p+ (x, t), if x > 0,
where p± (x, t) are the superpositions of plane waves
p− (x, t) = A− ei(ex x+ey y+ez z)ω/c− −iωt + B− ei(−ex x+ey y+ez z)ω/c− −iωt ,
ex ≥ 0,
p+ (x, t) = A+ ei(dx x+dy y+dz z)ω/c+ −iωt + B+ ei(−dx x+dy y+dz z)ω/c+ −iωt ,
dx ≥ 0,
(5.2)
coupled by the interface conditions
p+ (0, k+ ) = p− (0, k+ ),
1 ∂p+ 1 ∂p− =
.
ρ+ ∂x x=0
ρ− ∂x x=0
(5.3)
Although each of the fields p− (x, t) and p+ (x, t) consists of several waves, it is convenient to associate
them with the vectors
k+ = ω d,
k− = ω e,
(5.4)
c−
c+
defined in terms of unit vectors e = (ex , ey , ez ) and d = (dx , dy , dz ) coupled by the formulas
e2y + e2z = sin θ− ,
d2y + d2z = sin θ+ , dx = 1 − sin2 θ+ ,
ex = cos θ− ,
(5.5)
where θ± are the incidence angles shown in Fig. 3. Straightforward substitution of (5.2) into the interface
conditions (5.3) leads to the formulas
2
c+
dy
dz
c+
ex = cos θ− ,
dx = 1 −
sin2 θ− ,
=
=
,
(5.6)
c−
ey
ez
c−
which reduce the interface conditions (5.3) to a system of linear algebraic equations
1 1 + μg, 1 − μg
A−
A+
·
=
2 1 − μg, 1 + μg
B+
B−
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(5.7)
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Ann. Phys. (Berlin) 523, No. 3 (2011)
217
where
g ≡ g(γ, θ− ) =
1 − sin2 θ−
,
1 − γ 2 sin2 θ−
γ=
c+
,
c−
μ=
ρ + c+
,
ρ − c−
(5.8)
are the characteristics of the contrast between the media and the incidence angle.
B− ei(−ex x+ey y+ez z)ω/c− −iωt
ρ− , c −
ρ+ , c +
A+ ei(dx x+dy y+dz z)ω/c+ −iωt
θ+
θ−
A− ei(ex x+ey y+ez z)ω/c− −iωt
B+ ei(−dx x+dy y+dz z)ω/c+ −iωt
Fig. 3 (online colour at: www.ann-phys.org) Coupling of waves at an interface.
It is important to emphasize that the vibrations in the two joined half-spaces can not be independent
of each other and that each acoustic field p− in the half-space x < 0 has a unique continuation p+ to the
domain x > 0 defined in such a way that (5.1) determines a continuous acoustic field in the entire space.
The importance of this observation comes from Debye’s fundamental concept that the part of the heat
which is stored in lattice vibrations is stored in eigen-vibrations of the entire body. If Debye’s concept is
adopted, then in order to compute the energy of lattice vibrations of the system of two joined half-spaces it
is necessary to study the eigen-vibrations of the entire composite space which are described by the formula
(5.1) with the components p− and p+ related to each other by the one-to-one correspondence determined
by the interface conditions, as discussed above.
It is also worth noticing that the existence of a one-to-one correspondence between the fields on different sides of the interface plays a crucial role in the phenomenon of the boundary thermal resistance.
T− )
Indeed,the results of the previous section make it possible to describe the statistical ensembles F− (Q,
and F+ (Q, T+ ) of acoustic fields in the half spaces x < 0 and x > 0 which have a constant heat flux Q and
temperatures T− and T+ , respectively. Then the interface conditions establish a one-to-one correspondence
between these ensembles
T
T− )} ←→
T+ )},
{F+ (k; Q,
{F− (k; Q,
(5.9)
which can be considered as a single “equation” connecting the two temperatures T− and T+ with the heat
After this equation is solved, the thermal resistance between the half spaces is computed as the
flux Q.
ratio R = (T− − T+ )/Q.
6 Joined half-spaces in thermal equilibrium
Let the ensembles of vibrations in two joined acoustic half-spaces x > 0 and x < 0 be in equilibrium at
the temperature T . Then, as shown in Sect. 4, these ensembles consist of the fields
√
p− = C ρ− c− eiβ− P (ω, T ) cos η− eiωx cos θ− /c− +iα− + sin η− e−iωx cos θ− /c− −iα− , x < 0,
√
p+ = C ρ+ c+ eiβ+ P (ω, T ) cos η+ eiωx cos θ+ /c+ +iα+ + sin η+ e−iωx cos θ+ /c+ −iα+ , x > 0,
(6.1)
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B. V. Budaev and D. B. Bogy: Khalatnikov’s theory of Kapitsa thermal resistance
where
C=
ei(ey y+ez z)ω/c− −iωt
√
,
4π 3
(6.2)
is a common factor taken out for brevity, θ− is the incident angle of the waves in the domain x < 0,
and η± = η± (ω) are random numbers uniformly distributed in as yet unspecified domains L± (0) that
guarantee the ensembles (6.1) have no heat flux.
Formulas (6.1) and (6.2) take into account the thermodynamics of the vibrations in the half-spaces x < 0
and x > 0, but ignore the requirement that the vibrations in these half-spaces are coupled. To take into
account the coupling we first observe that the frequencies of the acoustic fields p± from (6.1) are limited to
±
−
+
the bands ω < κTD
/, where TD
and TD
are the Debye temperatures of the materials in the half-spaces
x < 0 and x > 0, respectively. Therefore, real coupling between the fields p± happens only in the band
ω<Ω≡
− +
κ
min TD
, TD ,
(6.3)
where both of them exist. As for the vibrations with frequencies ω > Ω, they exist only in one of the
half-spaces and therefore can not contribute to heat transport across the interface.
Assume that ω < Ω and substitute (6.1) into (5.7) which leads to the system of equations
√ i(β+ −β− ) cos η+ e+iα+
1 + μg, 1 − μg
cos η− e+iα−
2 μe
=
·
.
(6.4)
1 − μg, 1 + μg
sin η+ e−iα+
sin η− e−iα−
Then, eliminating η+ we get a single equation
sin 2η− cos 2α− = r,
r=
1 + μ2 g 2 (γ, θ− ) − 2μ
,
μ2 g 2 (γ, θ− ) − 1
which implies that η− = η− (ω) may take any value in the domain
1
π π
ω < Ω,
R− (0) :
η− − ≤ − arcsin(r),
4
4
2
(6.5)
(6.6)
where | sin 2η− | ≥ r, so that (6.5) can be enforced by the right selection of the phase shift α− .
The above analysis shows that if the vibrations (6.1) obey the interface conditions (5.3) then the composition index η− must be located in the domain R− (0) of the type (6.6). At the same time, since this
ensemble has a vanishing average heat flux, the composition index η− has to be uniformly distributed over
a domain L− (0) loosely defined by (4.10) as a domain symmetric with respect to the axis η = π4 . Therefore, observing that the strip (6.6) also has this symmetry, we conclude that L− (0) = R− (0) is the largest
domain that contains admissible values of η− .
It is appropriate to view the width of the strip (6.6) as a measure of the “number” of members in an
equilibrium ensemble of vibrations in a bi-material medium. Therefore, to understand the structure of this
ensemble, it is important to study the dependence on the strip’s width on the parameters μ, ν and θ− which
determine the right-hand sides of (6.5) and (6.6).
Consider first the degenerate case when both half-spaces have identical sound speeds c+ = c− . In this
case γ = c+ /c− = 1 and (6.6) reduces to the inequality
π
μ−1
1
π R− (0) :
.
(6.7)
η− − ≤ − arcsin
4
2
2
μ+1
Since |(μ − 1)/(μ + 1)| ≤ 1, the strip (6.7) is always well-defined and, as shown in Fig. 4, it expands when
μ → 1 and contracts towards the axis η = π4 when μ → 0 or μ → ∞. These observations imply that an
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Berlin) 523, No. 3 (2011)
219
η
The width of the strip R is determined by the acoustic mismatch.
π
2
If the materials are sharply dissimilar
(μ 1, or 1 μ), then R− is narrow
(shaded strip).
π
4
If the materials are similar (μ ≈ 1),
then R− is wider (hatched strip)
0
Ω
ω
Fig. 4 (online colour at: www.ann-phys.org) Dependence of the domain R− (0, 0) on the impedance mismatch µ.
ensemble of vibrations in a space made of very dissimilar materials, as μ → 0 or μ → ∞, contains only
standing waves which do not carry energy.
The general case, when c+ = c− is not radically different from the case of equal sound speeds, but the
results are more complex because the width of the strip (6.6) depends not only on the parameter μ, but also
on the ratio of the sound speeds γ = c+ /c− and the incidence angle θ− . In particular, for an arbitrary γ the
existence of a non-empty strip (6.6) can only be guaranteed if the incidence angle θ− ∈ A(μ, γ), where
A(μ, γ) :
|μ2 g 2 (γ, θ− ) − 1| ≤ |1 + μ2 g 2 (γ, θ− ) − 2μ|,
(6.8)
is the set of angles θ− which expands to 0 ≤ θ− ≤ π2 in the case γ = 1 and which is not empty because it
always contains θ− = 0.
It is worth noting that the restriction θ− ∈ A does not mean that the considered bi-material structure
does not support the propagation of wave fields with an incident angle θ− ∈
/ A. Such waves can propagate,
but they are not contained in an equilibrium ensemble of vibrations.
7 Interface thermal resistance
Let Q be a small heat flux between two acoustic half-spaces x > 0 and x < 0 maintained at the temperatures T− = T and T+ = T + ΔT , respectively. Then, the results of Sect. 5 imply that the ensembles of
thermal vibrations in these media consist of the waves
√
p− = C ρ− c− eiβ− PQ (ω, T− ) cos η− eiωx cos θ− /c− +iα− + sin η− e−iωx cos θ− /c− −iα− ,
(7.1)
√
iβ+
iωx cos θ+ /c+ +iα+
−iωx cos θ+ /c+ −iα+
,
+ sin η+ e
p+ = C ρ+ c+ e PQ (ω, T+ ) cos η+ e
which are similar to (6.1) except that here η± are not distributed in symmetric domains L± (0) from (4.11)
but are distributed in asymmetric domains L± (Q) defined by (4.17).
To simplify the calculations we limit ourselves here to the analysis of the interface between sharply
contrasting materials for which the domains L± (0) are narrow strips around the line η = π/4, which
makes it possible to simplify the expressions for the function PQ (ω, T ) and of the domains L± (Q).
Thus, elementary calculations based on Taylor expansions show that PQ (ω, T ) admits the approximation
PQ (ω, T ) ≈ P (ω, T ), and L− (Q) can be approximated by the domain
P (ω − τ ωQ, T )
L− (Q) :
tan η ,
η ∈ L− (0),
η− (ω) = arctan
(7.2)
P (ω + τ ωQ, T )
which has the shape of the dotted curvilinear strip in Fig. 2.
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B. V. Budaev and D. B. Bogy: Khalatnikov’s theory of Kapitsa thermal resistance
To formulate the conditions which guarantee that the fields (7.1) are coupled at the interface we follow
the method employed in Sect. 6 and find that η− must obey the equation
(7.3)
sin 2
η− cos 2α− = rω (ΔT ),
where the right-hand side
rω (ΔT ) =
1 + μ2 g 2 (γ, θ− ) − 2μνω (ΔT )
μ2 g 2 (γ, θ− ) − 1
(7.4)
is similar to (6.5) but involves an additional factor
νω (ΔT ) =
eω/κ(T +ΔT ) − 1
P 2 (ω, T + ΔT )
,
≡
2
P (ω, T )
eω/κT − 1
(7.5)
which depends on ΔT and ω, as well as on the base temperature T considered as a constant parameter. To
solve Eq. (7.3) we select η− as any point of the domain
R− (ΔT ) :
1
π π
ηω − < − arcsin rω (ΔT ) ,
4
4
2
(7.6)
where | sin 2
η− | ≥ |rω |, and then enforce (7.3) by an appropriate choice of α− . This important domain
is sketched in Fig. 5 for two different cases: the hatched curvilinear area corresponds to ΔT > 0 and the
shaded strip corresponds to ΔT = 0 for which (7.6) reduces to (6.6).
η
π
2
Interface conditions imply that the com-
R− (ΔT ) position index η− must belong to the doπ
4
R− (0)
0
main R which degenerates to a strip if
ΔT = 0 and expands to a more complex
shape when ΔT > 0.
ω
Fig. 5 (online colour at: www.ann-phys.org) Dependence of the domain R = R− (ΔT ) on the temperature differential.
Now we can easily explain and estimate the interface thermal resistance from the observation that the
domains L− (Q) and R− (ΔT ) are subordinated as
L− (Q) ⊆ R− (ΔT ),
(7.7)
because L− (Q) is the domain over which the composition index η is spread in the case when the ensemble
(7.1) has the heat flux Q, and R− (ΔT ) is where η must be located in order to maintain the temperature
differential ΔT .
Consider first the equilibrium state illustrated in the left diagram of Fig. 6 below. As discussed above,
if ΔT = 0 and Q = 0 both of the domains R− (0) and L− (0) coincide and appear as a strip (6.6), which
is simultaneously shaded and dotted in order to emphasize its dual role. Next, assume that ΔT = 0 but
Q > 0. In this case, illustrated in the right diagram of Fig. 6, the shaded domain R− (0) remains unchanged
but the dotted domain L− (Q) moves up and takes an asymmetric shape which does not fit into R− (0) and,
therefore, makes the inclusion in (7.7) impossible.
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Berlin) 523, No. 3 (2011)
η
221
η
L− (0) = R− (0)
π
2
π
2
π
4
L− (Q) ⊆ R− (0)
π
4
0
0
ω
ω
Fig. 6 (online colour at: www.ann-phys.org) Arrangement of the domains R− (0) and L− (Q).
Next we assume that the heat flux Q > 0 remains constant and try to satisfy (7.7) by increasing ΔT
from its equilibrium level ΔT = 0. This process is illustrated in Fig. 7, where the constant domain L− (Q)
is dotted while the domains R− (ΔT ) are shaded. In the beginning, when ΔT remains very small R− (ΔT )
remains close to R− (0) and does not cover L− (Q), as shown in the left diagram of Fig. 7. However, as
ΔT increases it reaches the threshold ΔT ∗ (Q) at which R− (ΔT ∗ ) covers L− (0) as shown in the right
diagram of Fig. 7.
η
η
L− (Q) ⊆ R− (ΔT )
π
2
L− (Q) ⊆ R− (ΔT ∗ )
π
2
R− (ΔT )
π
4
π
4
R− (ΔT ∗ )
ΔT<ΔT ∗
0
ω
If ΔT < ΔT ∗ then the shaded domain
does not cover the dotted domain
0
ω
If ΔT ≥ ΔT ∗ then the shaded domain
covers the dotted domain
Fig. 7 (online colour at: www.ann-phys.org) Estimation of the interface thermal resistance.
Now, recall that the domain R− (ΔT ) from (7.6) depends on the incident angle θ− belonging to the
set A(μ, γ) defined by (6.8) in terms of the material parameters μ and γ. Therefore, the threshold ΔT ∗ =
ΔT ∗ (Q, θ− ) depends not only on the heat flux Q but also on θ− . Therefore, to satisfy the inclusion (7.7)
for all waves of the ensemble (7.1) the temperature differential ΔT has to exceed all of the thresholds
ΔT ∗ (Q, θ− ) for all admissible incidence angles θ− ∈ A(μ, γ), which means that the following inequality
must hold
ΔT ≥ max{ΔT ∗ (Q, θ− )}θ− ∈A(μ,γ) .
(7.8)
Then, applying the principle of minimum entropy production [11], which says that the steady state of an
irreversible process is characterized by a minimum value of the rate of entropy production, we conclude
that
ΔT = sup{ΔT ∗ (Q, θ− )}θ− ∈A(μ,γ) .
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(7.9)
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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B. V. Budaev and D. B. Bogy: Khalatnikov’s theory of Kapitsa thermal resistance
which is the smallest value of the temperature differential ΔT compatible with (7.9). As a result, we
conclude that the interface thermal resistance is determined by the formula
ΔT ∗ (Q, θ− )
RK = sup
,
(7.10)
Q
θ− ∈A(μ,γ)
which can be easily calculated on a computer.
8 Numerical example and discussion
To examine the predictions of the developments presented above we estimate the Kapitsa resistance between copper and liquid helium for the temperatures from 0.05 K◦ to 5 K◦ which are low enough to justify
the approximation of lattice vibrations by acoustic waves with the dominating wavelength considerably
exceeding the characteristic dimensions of the lattice. The parameters were taken from [5, Table 1] which
provide the values ρ− = 8960 kg/m3 and ρ+ = 145 kg/m3 for the densities of copper and helium, respectively; the values of cp,− = 4910 m/s and cs,− = 2500 m/s for the longitudinal and transverse sound
speeds in copper, and the value c+ = 238 m/s for the sound speed in liquid helium. The base temperature
is defined as the temperature in helium, and the temperature differential varies in the range 0.001–0.01 K◦
in order to make sure that the thermal resistance linearly depends on the temperature differential.
The computations were made using a simple algorithm straightforwardly mimicking the procedure as
described above. First we set ΔT = 0 and compute the boundaries of the domain L− (0) = R− (0) which
are located on the plane (ω, η− ) between the horizontal lines η− = f1 and η− = f2 , where
f1◦ =
1
arcsin(r),
2
f2◦ =
1
π
− arcsin(r),
2
2
(8.1)
and r is determined by the formula (6.5) with the parameters μ = 0.0014 and γ = 0.085 corresponding
to the selected pair of materials. As for the incidence angle it was varied between θ− = 0◦ and θ− = 90◦
with 15◦ increments. Then we assume that Q = 0 and compute the curvilinear lines η− = f1 (ω; Q) and
η− = f2 (ω, Q), where
P (ω − τ ωQ, T )
fi (ω; Q) = arctan
tan fi◦ , i = 1, 2,
(8.2)
P (ω + τ ωQ, T )
P (ω, T ) is the Planck function (2.7), and τ is obtained from (4.13). These curves surround the area L− (Q)
which must be completely covered by the domain R− (ΔT ) between the curves η− = h1 (ω; ΔT ) and
η− = h2 (ω; ΔT ), where
h1 (ω; ΔT ) =
1
arcsin (rω (ΔT )) ,
2
h2 (ω; ΔT ) =
1
π
− arcsin (rω (ΔT )) ,
2
2
(8.3)
and rω (ΔT ) is defined by (7.4). In other words, for a given Q = 0, we need to find ΔT which guarantees
satisfaction of the inequalities
h1 (ω; ΔT ) ≤ f1 (ω; Q) < f2 (ω, Q) ≤ h2 (ω; ΔT )
(8.4)
for all frequencies ω below the threshold Ω ≈ 3.5 · 1012 Hz determined from (6.3).
Since all of the functions in (8.4) are explicitly defined, the algorithm for determining ΔT reduces to a
simple step-by-step process repeated in a double loop enumerated by the integers m and n which determine
the incidence angle and the temperature differential by the formulas θ− = 15◦ m and ΔT = n10−4 . For
any fixed pair of these indices, starting from m = 0 and n = 0, we verify the inequalities (8.4) on a
sufficiently dense mesh on the frequency interval 0 < ω < Ω. If any of the inequalities is violated, then
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Berlin) 523, No. 3 (2011)
223
we increase the index n by one and repeat the process for a new temperature differential. After several
iterations we get an approximate value of the threshold ΔT ∗ (θ− ) corresponding to the current incident
angle θ− = 15◦ m. Repeating this process for every incident angle, and we find ΔT ∗ = max{ΔT ∗ (θ− )
and then the Kapitsa resistance is determined as the ratio RK = ΔT ∗ /Q. To get results that are comparable
with available experimental data we performed the computations for several values of Q corresponding to
temperature differentials in the range ΔT ≈ 0.001 K◦ – 0.01 K◦ , used in Kapitsa’s original experiments.
The results of our computations are plotted in Fig. 8 which also includes experimental and other theoretical data recompiled from [5, Fig. 1]. To suppress the strong dependence of the Kapitsa resistance RK
on the temperature T , this figure plots the product RK T 3 against the temperature T on a log-log scale.
The curves (a), (b), (c), and (d) correspond to the measurements from [12–15], respectively, and the curve
(e) shows results of the original Kapitsa experiments [3]. The horizontal lines (f) and (g) correspond to
Khalatnikov’s acoustic mismatch theory [7] and to the diffuse mismatch theory [8], respectively. Finally,
the thick solid line with additional circular dots is computed by the approach presented above.
.1
Scaled Kapitsa resistance RK T 3
4 2 K m /W
.05
(f) Acoustic mismatch theory
.02
(a)
.01
(b)
.005
(d)
.002
(e)
.001
(c)
.0005
(g) Diffuse mismatch theory
RK T 3 = 0.00028 K4 m2 /W
.0002
.0001
.05
.1
.2
.5
1
2
5
Temperature (◦ K)
Fig. 8 (online colour at: www.ann-phys.org) Measured and computed Kapitsa resistance using three models: Acoustic mismatch theory of Khalatnikov; Diffuse mismatch theory; the present extension of Khalatnikov’s theory.
www.ann-phys.org
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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B. V. Budaev and D. B. Bogy: Khalatnikov’s theory of Kapitsa thermal resistance
It is easy to see that the experimental results are not very consistent. Moreover, as mentioned in [5, Fig. I]
the experiments are “typical irreproducibility from run to run, even on the same sample”. Nevertheless,
the natural spread of the measured values of the Kapitsa resistance is not sufficient to cover the almost two
orders of magnitude of discrepancies between the experimental data and the predictions of the acoustic and
diffuse mismatch theories. At the same time, the approach presented here gives very reasonable order-ofmagnitude estimates of the Kapitsa resistance.
9 Summary and conclusions
In an attempt to clarify the role of acoustic waves in heat transport across a material interface we re-examine
the foundations of the acoustic mismatch model of the Kapitsa thermal resistance and generalize the conventional theory by inclusion into consideration some additional details which do not alter the fundamental
ideas of the model but may significantly affect its predictions. First we observe that the conventional theory
limits the analysis to waves radiated by a certain source of vibrations that satisfy the Sommerfeld radiation
condition. This assumption is natural for the theory of sound which is supposed to propagate away from
its source, but it is not relevant to the analysis of spontaneously generated thermal vibrations. Obviously,
inclusion of waves which are not restrained by any kind of radiation conditions significantly increases the
pool of potential heat carriers. Next we observe that the conventional theory assumes that the ensembles
of thermal vibrations on each side of the interface have Planck-like statistical distributions which are valid
in equilibrium systems, but, strictly speaking, are not valid in any system with a non-vanishing heat flux,
including those where an interface thermal resistance arises. To eliminate this inconsistency we derive a
slight modification of the Planck formula which describes the ensemble of thermal vibrations with a steady
heat flux. As is obvious from Fig. 8, the implementation of the above mentioned extensions of Khalatnikov’s theory radically changes the predictions of Kapitsa resistance and brings them to the reasonably
accurate “order-of-magnitude” agreement with experimental data.
It is certainly true that Kapitsa thermal resistance is a complex phenomenon caused by many different
mechanisms acting simultaneously and affecting each other. Although acoustic waves are believed to be
the dominant heat carriers across the interface between materials, at least one of which is dielectric, it is
impossible to completely exclude other factors such as the thermal motion of electrons and the quality
of contact at the interface. Even the acoustics-only model may be improved to include the consideration
of anisotropy of the materials, existence of waves with different polarizations, existence of surface waves
and local non-homogeneity which smoothes the interface. Moreover, the basic model considered here
also has room for improvements which may lead to more accurate predictions of the Kapitsa thermal
resistance and especially as regards its dependence on the temperature. Most notably, this model may be
improved by the inclusion into consideration the ground-state vibrations ignored in (2.13), and by a more
accurate computation of the domain L( Q) defined by the expression (4.17) but evaluated by a simplified
formula (7.2) which may become inaccurate at higher frequencies when the ratio sin η(ω − τ ωQ) and
cos η(ω − τ ωQ) can not be replaced by sin η(ω) and cos η(ω), respectively.
All of the above suggests that the approach to the Kapitsa thermal resistance developed here may be
a good starting point for further studies which may result in a full theory capable of delivering not only
order-of-magnitude estimates but accurate values of the interface thermal resistance between any materials.
Acknowledgements
This work was supported by the William S. Floyd, Jr., Distinguished Chair, held by D. Bogy.
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Ann. Phys. (Berlin) 523, No. 3 (2011)
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