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An Atomistic-to-Continuum Coupling Method for Heat Transfer in Solids
An Atomistic-to-Continuum Coupling Method
for Heat Transfer in Solids
G.J. Wagner
a,
, R.E. Jones
a
, J.A. Templeton
a
, M.L. Parks
b
a
Sandia National Laboratories, Livermore, CA 94551
b
Sandia National Laboratories, Albuquerque, NM 87123
Abstract
In this work, we present a seamless, energy-conserving method to couple atomistic
and continuum representations of a temperature eld in a material. This technique
allows a molecular dynamics simulation to be used in localized regions of the compu-
tational domain, surrounded and overlaid by a continuum nite element representa-
tion. Thermal energy can pass between the two regions in either direction, making
larger simulations of nanoscale thermal processes possible. We discuss theoretical
developments and numerical implementation details. In addition, we present and
analyze a set of representative simulations.
Key words: atomistic-to-continuum coupling, heat transfer, nite elements,
multi-scale simulations
1
Introduction
As technological advances allow the engineering of devices at ever decreasing
length scales, and as ever increasing delity is demanded in the computational
simulation of these devices, it has become clear that traditional material mod-
els based on continuum descriptions of solids can be inadequate at the micro-
and nano-scales. Surface eects, grain boundaries, defects, and other devia-
tions from a perfect continuum can have a large eect on material behavior
at these scales, and simulation techniques based on descriptions at the atom
scale, such as molecular dynamics (MD), have become an important part of
Corresponding author.
Email addresses: gjwagne@sandia.gov (G.J. Wagner), rjones@sandia.gov
(R.E. Jones), jatempl@sandia.gov (J.A. Templeton), mlparks@sandia.gov
(M.L. Parks).
Preprint submitted to Elsevier Science
1 February 2008 the computational toolbox. However, molecular dynamics simulations on even
the largest supercomputers are currently limited to systems on the order of
a billion atoms [1], large enough for the study of some nano-scale phenom-
ena but still far too small to resolve the micro-to-macroscale interactions that
must be captured in the simulation of any real device. The recognition of this
limitation on MD has led to the development of several methods for the cou-
pling of atomistic and continuum material descriptions in a single simulation;
see [2,3] for reviews of these methods. The goal of these methods is to allow
the use of a continuum-based technique such as nite elements (FE) in parts
of the domain where such a description is valid, while using MD near defects
or in other regions in which the continuum description breaks down.
To date, most of these atomistic-to-continuum coupling methods have been
based on the coupling of the momentum equation (or in the case of quasi-
static problems, the equilibrium equation) in the continuum to the equations
of motion for the atoms, usually by combining the Hamiltonians of the two
systems [4] or by ensuring that internal forces are properly balanced [5]. Most
often, these methods assume that the temperature of the continuum region is
in eect zero, and quite a bit of attention has been paid to reducing unwanted
internal reections of waves in the MD lattice at the MD-continuum interface.
However, a much more typical scenario for real devices is a temperature that
is far above absolute zero. In this case, it is more accurate to recognize lattice
waves as energy-carrying phonons, and to think of the surrounding continuum
as a thermal bath that maintains the correct balance of incoming and outgoing
phonons at the interface at the local temperature.
Some attempts have been made previously to accurately account for the eects
of non-zero temperature. Dupuy et al. [6] have developed a nite-temperature
version of the Quasicontinuum Method that uses a local-harmonic approxima-
tion, at a constant temperature, to account for thermal uctuations of atoms.
Rudd and Broughton [7] have developed the coarse-grained molecular dynam-
ics (CGMD) technique for simulations of anharmonic solids at nite temper-
atures. The bridging scale decomposition method of Wagner and Liu [8] has
been extended to nite temperatures by Park et al. [9]. However, to our knowl-
edge, no technique exists to couple the thermal uctuations in the MD region
with an energy equation in the continuum to eect true two-way temperature
coupling between the MD and continuum regions. In this work, we present a
technique for such a coupling, allowing the simulation of nonequilibrium heat
transfer between MD and continuum regions of a domain.
Two-way temperature coupling implies that thermal information can pass out
of the MD region into the continuum, and that the temperature of the con-
tinuum aects the thermal uctuations of the MD region. The rst direction
of information ow, from MD to continuum, is important in applications in
which phenomena at the atom scale lead to what would be measured in the
2 laboratory as changes in macroscale temperature. Examples of such phenom-
ena include friction [10], laser heating [11], fracture [12], and plastic failure
[13], all of which have been studied using MD or even coupled MD-continuum
simulations but without a complete treatment of macroscale temperature in-
teractions. By coupling a continuum energy equation to the atom dynamics,
we can simulate temperature changes in the continuum, possibly over large
distances, that are caused by these atom-scale phenomena.
At the same time, in a coupled simulation any temperature eld that is im-
posed on the continuum should have an eect on the thermal uctuations of
atoms in the MD region. For example, a macroscale temperature gradient on
the continuum should lead to a heat ux through an MD region embedded
within it. It is important to capture this behavior correctly in a simulation
method, because it is known that structures at the atomic scale such as inclu-
sions or grain boundaries can have a large eect on the thermal conductivity of
the material [14]. The ability to do two-way temperature coupling allows the
nanostructure of the MD region to have the proper eect on the continuum
temperature eld.
Several previous authors have coupled MD simulations to a continuum energy
equation. Ivanov et al. [11] have used a two-temperature model to incorporate
the eects of the electron temperature on the dynamics of the atomic nuclei
in simulations of laser heating of metal lms. Schall et al. [15] employed a
thermostat acting on the atoms in an MD simulation to enforce the correct
thermal conductivity in simulations of metals; this conductivity is otherwise
underpredicted by classical MD. Padgett and Brenner [16] used a similar tech-
nique to capture the eects of Joule heating in metal nanowires. The principal
innovation of the current work is the ability to couple an MD simulation to
the temperature eld of a continuum that overlaps and surrounds it, such that
the two-way coupling of energy between two dierent domains is eected.
In this work we will use nite elements to solve the heat equation in the con-
tinuum. We begin in Section 2 by dening the basic problem to be solved
and stating the assumptions used. In Sections 3 and 4, respectively, we derive
the forms of the energy equations to be solved in the MD and FE domains;
the coupling between the domains follows naturally from our derivation. Time
ltering is introduced in Section 5 to reduce uctuations in the temperature
eld, and in Section 6 we present some of the details of the numerical imple-
mentation of the method. Example problems are presented in Section 7, and
we conclude with a discussion in Section 8.
3 2
Problem Denition
Consider the problem geometry shown in Figure 1. A domain is discretized
with a nite element mesh; the outer boundary of the domain is denoted
with outward normal vector n. At the same time, an internal portion of
the domain
md
is lled by a set of atoms A. The remaining portion of the
domain in which there are no atoms but only nite elements is denoted
f em
,
so that
md

f em
= and
md

f em
= . The boundary between the two
subdomains is given by
md
, with normal vector n
md
oriented into the MD
region. Note that the entire domain, including
md
, is discretized with nite
elements, so that the atomistic and nite element descriptions co-exist in
md
.
In the following, the vector X represents the reference coordinates of a given
point in .
We are concerned with heat transfer problems in which we can assume t