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IMECE2008-69152

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Proceedings of IMECE2008
2008 ASME International Mechanical Engineering Congress and Exposition
November 2-6, 2008, Boston, Massachusetts, USA
Paper No. IMECE2008-69152
INTERFACIAL THERMAL RESISTANCE IN NANOSCALE HEAT TRANSFER
Ganesh Balasubramanian
Virginia Polytechnic Institute and
State University
Blacksburg, VA, USA
Soumik Banerjee
Virginia Polytechnic Institute and
State University
Blacksburg, VA, USA
ABSTRACT
We investigate nanoscale thermal transport across a solidfluid interface using molecular dynamics simulations. Cooler
fluid argon (Ar) is placed between two heated iron (Fe) walls,
thereby imposing a temperature gradient within the system.
Fluid-fluid and solid-fluid interactions are modeled with
Lennard-Jones potential parameters, while Embedded Atom
Method (EAM) is used to describe the interactions between
solid molecules. The Fe-Ar interaction causes ordering of fluid
molecules into quasi-crystalline layers near the walls. This
causes temperature discontinuity between these solid-like Ar
molecules and the adjacent fluid. The time evolution of the
interfacial (Kapitza) thermal resistance (Rk) and Kapitza length
(Lk) are observed. The averaged Kapitza resistance (Rk,av)
varies with the initial temperature difference between the wall
and the fluid (∆Tw ) as Rk,av ∝ ∆Tw-0.82 .
INTRODUCTION
Thermal properties of nanostructures differ from the
corresponding ones at the continuum since their length scales
are comparable to those of the energy carriers at the nanoscale,
viz. phonons[1, 2]. The large surface area to volume ratio
exerts a strong influence on thermal transport at the molecular
level[3, 4]. The structural ordering that occurs due to the
attraction of liquid molecules to hydrophilic surfaces enhances
the local thermal conductivity during heat transfer across
nanoscale solid-liquid interface[5]. A finite temperature
discontinuity is observed because of the local intermolecular
interactions. Thus, the role of thermal resistance at nanoscale
interfaces is much pronounced than at larger scales[6]. P.L.
Kapitza[7] was the first to report the existence of this
interfacial contact resistance (also known as Kapitza
resistance[8], Rk) through his experiments on heat transfer
between liquid helium and copper. It is expressed as the ratio of
Ishwar K. Puri
Virginia Polytechnic Institute and
State University
Blacksburg, VA, USA
temperature drop (∆T) at the interface to the normal heat flux
across it
Rk = ∆T / q
(1)
This resistance to thermal transport is also often expressed in
terms of a length parameter (Kapitza length, Lk)[9], which
represents the width of the bulk medium over which there will
be same temperature jump as at the interface
Lk = ∆T / (dT/dx)
(2)
Previous studies have reported impact of Rk and Lk on heat
transfer across interfaces at steady state[10, 11]. We attempt to
understand the nature of temporal evolution of the resistance
and length parameters using calculation of the heat flux from
the first principles, instead of the more widely adopted
approach using continuum approximation. Further, the
dependence of Rk on initial temperature difference between the
walls and the fluid (∆Tw) provides some more insight to the
role of thermal resistance in nanoscale heat transfer.
METHODOLOGY
Molecular Dynamics (MD) is a fundamental tool applied
to the study of thermal transport at the nanoscale. These
atomistic simulations are applicable for systems both in
equilibrium, where absence of temperature gradients allow free
interactions between atoms, as well as non-equilibrium, such as
interfacial heat transfer, where temperature gradients are
imposed on the system[10].
An empirical potential function is used to represent atomic
interactions in MD simulations. We model the interactions
between solid-solid and solid-fluid atoms using Lennard-Jones
(LJ) potential parameters (values listed in Table 1)[12], which
is represented as,
1
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⎡⎛ σ
uij = 4ε ij ⎢⎜ ij
⎢⎜⎝ rij
⎣
12
⎞
⎛ σ ij
⎟⎟ − ⎜⎜
⎠
⎝ rij
⎞
⎟⎟
⎠
6
⎤
⎥
⎥
⎦
(3)
where, ‘rij’ denotes the scalar distance between sites ‘i’ and ‘j’,
‘σij’ and ‘εij’ are the LJ interaction parameters.
Interactions
ε (eV)
σ (Å)
Ar-Ar
Ar-Fe
0.0103
0.0516
3.4
3.7
Table 1: Lennard Jones interaction parameters for Fe and Ar
The interaction amongst solid atoms has been previously
modeled with pair-potential schemes like LJ as well as with
harmonic potential functions. These, however, fail to
incorporate the many-atom interactions characteristic of solids.
Hence, we represent Fe-Fe interactions with a more realistic
Embedded Atom Method (EAM) potential[13, 14], expressed
as,
⎡
1
E = ∑ ⎢ Fi ρi +
2
i ⎣
( )
⎤
Φij ( Rij ) ⎥
∑
j ( ≠i )
⎦
(4)
Here, E is the total energy of a system of atoms, the sums are
over the atoms ‘i’ and ‘j’, the embedding function ‘Fi’ is the
energy to embed an atom of type ‘i’ into the background
electron density at site ‘i’, ‘ρi.’ ‘Φij’ is a pair interaction
between atoms ‘i’ and ‘j’ whose separation is given by ‘Rij’.
The electron density of the solid is considered as a linear
superposition of the densities (ρa) of the individual atoms.
The simulation configuration as shown in Fig. 1 consists of
a 3 nm × 3 nm × 58 nm cuboid that contains two solid Fe walls,
each 3 nm × 3 nm × 1 nm in volume. The Fe blocks extending
from 0-10 nm and 29-30 nm in the x-direction restrain 3402
atoms of liquid-vapor mixture of Ar (with an initial 33% vapor
volume fraction) between them. The fluid is initialized in the
form of face centered cubic lattice that equilibrates through
simulations. We eliminate wall effects by applying periodic
boundary conditions in all three dimensions. A cut-off distance
of 10 Å is applied for both Ar-Ar and Fe-Ar interactions
modeled by LJ potential parameters.
Fig. 1: A 3-dimensional view of the MD simulation domain in
which Ar (cyan) fills a space of 28 nm between two 1 nm thick
blocks of solid Fe. Fluid atoms on both sides of the walls
indicate periodicity.
After initializing the system at a temperature of 100K,
velocity-rescaling temperature control is applied over all the
atoms for the first 300 picoseconds. An initial equilibrium state
for a uniform temperature distribution over the entire system is
achieved. For the next 700ps, temperature of the solid blocks is
controlled and the fluid is allowed to equilibrate. Subsequently,
at 1000ps, the Fe walls are provided with a step increase to a
higher temperature, which is thereafter maintained with
constant velocity rescaling for the remainder of the simulation.
Linear momentum is conserved throughout the entire
simulation. The simulations are performed with the massively
parallel LAMMPS code[15] advanced through successive timesteps of 0.1 femtosecond.
RESULTS
In order to obtain the spacial distribution of temperature
and number density of the Ar atoms when the system attains
near steady-state configuration, we divide the fluid domain into
several slabs, each 4 Å wide along the x-direction. The solid
walls are maintained at a constant temperature of 125K and the
density and temperature distribution for the fluid averaged over
uniform time intervals is shown in Fig. 2. In accordance with
previous literature [6, 10, 16], we observe the tendency of the
fluid atoms to migrate towards the heated walls. This localized
higher concentration of Ar atoms increases the interfacial
pressure so that the fluid atoms behave like a quasi-crystalline
solid. Atoms further away from the wall exhibit a vapor
behavior. The migration of Ar atoms, few molecular diameters
away from the solids, towards the Fe walls thereby creates local
vacancies. Fluid atoms that are further removed from the walls
are sparsely affected by the intermolecular interaction with the
Fe atoms, and hence remain homogeneously distributed. The
fluid being essentially a liquid-vapor mixture, phase
segregation occurs and causes this inhomogeneous density
distribution.
2
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60
128
Temperature
Density
Fe
50
124
Temperature(K)
40
30
120
Ar
20
Number of Molecules
Fe
116
10
112
0
0
50
100
150
200
250
300
Distance between Iron Walls(Angstrom)
Fig. 2: The temperature (dotted lines) and density (solid lines)
distributions of the atoms across the x-wise direction after the
system is near steady state. The solid Fe walls are located
between 0-10 nm and 29-30 nm.
temperature gradients (dT/dx) by central difference method on
the temperatures computed over the successive fluid slabs
where a drop is observed.
The temporal variation of Rk and Lk for two different cases
of wall temperature are presented in Fig. 3. Both the interfacial
thermal resistance and the Kapitza length increase with time,
though their natures of increment are different. As the system
approaches steady state configuration, the rate of flow of
thermal energy across the interface decreases, and as evident
from the simulations, this decrease is faster than the
corresponding rate of reduction of ∆T. Further, the rate of
change in the temperature gradient is steeper at the interface
than the corresponding the temperature drop, and hence the
observed nature of Lk. However, from the presented results for
the two cases, it seems that the rate of change of interfacial
temperature gradient and temperature drop remain proportional
till a critical time, after which this length shoots upwards from
a moderately constant value.
1.0E-06
40
q& sl = ( E& ls − E& sl ) / 2
(5)
The total thermal energy transferred from a high temperature
wall to the low temperature fluid adjacent to it is
q& = ∑s ∑l q& sl
Kapitza resistance(m2.K/W)
35
Lk for 125K
Lk for 120K
Kapitza length (nm)
Local equilibrium approximation is applied to obtain the
averaged temperature distribution of the fluid domain, also
shown in Fig. 2. The densely packed Ar atoms adjacent to the
walls have a temperature nearly same as that of the constant
temperature solid. The strong attraction of these fluid atoms
(extending for about four atomic layers from the solid) to the
Fe walls renders them practically motionless, thereby
increasing their effective thermal conductivity. A distinct
temperature discontinuity is observed at the interface between
these quasi-solid atoms and the neighboring vacancies. This
can be attributed to the impediment to energy transport across
the thinly scattered region of Ar vapor atoms.
The typically adopted approach to calculate the heat flux
across nanoscale interfaces is by a quasi-continuum
approximation using the material bulk thermal conductivity. We
have instead determined this flux through a more fundamental
first principles method by considering the energy transfer
rate[17, 18] from a solid molecule s to a fluid molecule l as
Rk for 125K
Rk for 120K
30
25
20
1.0E-07
15
10
5
0
1.0E-08
0.5
1
1.5
2
2.5
3
3.5
Time (nanoseconds)
Fig. 3: The temporal evolution of the interfacial thermal
resistances and Kapitza lengths for two different cases with wall
temperatures of 120 and 125 K. The interfacial temperature
drop at any instant is calculated as a time average over 100 ps.
The dependence of Rk on the wall temperature is also
examined. We calculate an averaged Kapitza resistance, Rk.av,
using averaged values for ∆T and heat flux over a period of
2000-4000 ps for each simulated case.
q& av = (1 / 2000)∑i = 2000 ps q& i
4000 ps
(6)
(7)
This enables us to determine Rk (Equation 1) explicitly from
the simulation. To compute Lk, we obtain the spacial
3
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The results presented in Fig. 4 show the variation of Rk.av with
increasing initial temperature difference between the wall and
the fluid (∆Tw).
Kapitza resistance, m2K/W
6.0E-07
4.0E-07
2.0E-07
Rk,av = (3x10-6)∆Tw-0.82
0.0E+00
5
15
25
35
45
Initial temperature difference, K
Fig. 4: Change in the Kapitza resistance with respect to varying
initial temperature difference between fluid and wall. The fluxes
and temperature drops are averaged over 2000-4000 ps.
With increasing wall temperatures, the pressure within the
fluid and hence on the ordered layers of Ar atoms increase.
This causes more pronounced intermolecular interactions. This
leads to enhanced thermal transport across these layers as a
result of which resistance to heat transfer at the interfaces
reduces. A dependence of the form Rk.av∝ ∆Tw-0.82 is obtained.
CONCLUSION
In summary, we find that the inhomogeneous density
distribution and the non-uniform temperature profile across the
fluid domain occur due to phase segregation. The temperature
discontinuity between the layers of quasi-crystalline Ar atoms
and the adjacent rarified region is caused by hindrance to
energy transfer through the sparse population of fluid atoms.
Heat flux and temperature gradient decrease at a faster rate than
the temperature drop. So, Rk and Lk exhibit an increasing trend
with time. A denser packing of the solid-like fluid atoms with
increasing wall temperatures enhances heat transfer and the
averaged interfacial thermal resistance decreases with ∆Tw
obeying a power law relation.
ACKNOWLEDGMENTS
We thank the Virginia Tech Advanced Research
Computing Facility for use of the terascale System X.
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