Lattice-Boltzmann modeling of phonon hydrodynamics
Wen-Shu Jiaung
*
and Jeng-Rong Ho
†
Department of Mechanical Engineering, National Chung Cheng University, Ming-Hsiung, Chia-Yi 621 Taiwan, Republic of China
Received 2 November 2007; published 23 June 2008
Based on the phonon Boltzmann equation, a lattice-Boltzmann model for phonon hydrodynamics is devel-
oped. Both transverse and longitudinal polarized phonons that interact through normal and umklapp processes
are considered in the model. The collision term is approximated by the relaxation time model where normal
and umklapp processes tend to relax distributions of phonons to their corresponding equilibrium distribution
functions—the displaced Planck distribution and the Planck distribution, respectively. A macroscopic phonon
thermal wave equation PTWE, valid for the second-sound mode, is derived through the technique of
Chapman-Enskog expansion. Compared to the dual-phase-lag DPL -based thermal wave equation, the PTWE
has an additional fourth-ordered spatial derivative term. The fundamental difference between the two models is
discussed through examining a propagating thermal pulse in a single-phased medium and the transient and
steady-state transport phenomena on a two-layered structure subjected to different temperatures at boundaries.
Results show that transport phenomena are significantly different between the two models. The behavior
exhibited by the DPL model, as thermal wave behavior goes over to diffusive behavior,
T
→
q
is incompatible
with any microscopic phonon propagating mode. Unlike the DPL model, in which
T
only has an effect on the
transient phenomena, in the PTWE model
T
shows effects on phenomena at both transient and steady state.
With the intrinsic compatibility to the microscopic state, discontinuous quantities, such as a jump of tempera-
ture at a boundary or at an interface, can be calculated naturally and straightforwardly with the present
lattice-Boltzmann method.
DOI: 10.1103/PhysRevE.77.066710 PACS numbers: 05.10.-a, 44.90.+c, 63.20.-e
I. INTRODUCTION
The transport of heat described by Fourier’s law has been
demonstrated to give rise to unreasonable results in several
situations. The anomaly is due to the assumption that the
heat flux vector and the temperature gradient occur at the
same instant of time; this leads to an infinite speed of heat
propagation 1. Cattaneo 2 and Vernotte 3 proposed a
relaxation model to resolve this dilemma that results in the
wave-based hyperbolic heat conduction equation. A compre-
hensive literature survey of thermal waves can be found in
the review papers by Joseph and Preziosi 4,5 and by Ozisik
and Tzou 6. Although the CVW wave model proposed by
Cattaneo and Vernotte remedies the paradox of the instanta-
neous response of thermal disturbance, it also introduces
some unusual solutions 7,8.
Instead of the precedence assumption of the lead of the
temperature gradient to the heat flux, Tzou proposed a dual-
phase-lag DPL model that allows either the temperature
gradient to precede the heat flux or the heat flux to precede
the temperature gradient 9–11. The heat conduction equa-
tion based on the DPL model is given by
ˆ
q
2
ˆ
t
ˆ
2
+
ˆ
t
ˆ
= ˆ
2
ˆ
x ˆ
2
+ ˆ
ˆ
T
3
ˆ
x ˆ
2
t
ˆ
, 1
where ˆ
q
represents the phase-lag time between the tempera-
ture gradient and the commencement of heat flow while ˆ
T
is
the lag of the temperature gradient to the heat flow. This
model can reduce to diffusion, CVW, the phonon-electron
interaction, and the pure phonon scattering models under
suitable values of ˆ
q
and ˆ
T
. It thus can cover a wide range of
physical responses from microscopic to macroscopic scales
in both space and time. More recent research interests based
on the DPL model are summarized in our previous work of
lattice-Boltzmann modeling for the DPL-based heat conduc-
tion equation 12.
The above review quickly portrays models for heat trans-
port from the macroscopic approach. Microscopically, trans-
port of energy in a dielectric solid is accomplished through
atomic vibrations that travel within the solid as waves. The
interatomic coupling present in a solid allows many different
vibrational modes that correspond to waves with different
frequencies. Just as the energy of an electromagnetic wave is
quantized, the energy of the waves can be quantized as
phonons and the solid medium can be treated as phonon gas.
The Boltzmann equation Boltzmann-Peierls equation is one
of the tools that is often used in the description of phonon
interactions, such as modeling for lattice thermal conductiv-
ity, phenomena of the second sound, and the Poiseuille flow
13–19. Depending on the relative strength of the normal
processes and resistive processes, the phonon propagation
can be classified as the following modes: ballistics, diffusion,
second sound, and heat conduction. These modes can be fur-
ther separated into two groups as the individual propagation
modes that include the ballistics and diffusion modes as well
as the collective propagation modes which cover the second
sound, and heat conduction modes 20.
Thermal wave is the collective or hydrodynamic behav-
ior of the interacting phonon system that fits in the second-
*
Researcher, Mechanical & Systems Research Laboratories, In-
dustrial Technology Research Institute, Hsin-Chu, Taiwan, Republic
of China.
†
Author to whom correspondence should be addressed; FAX:
+886-5-2720589; imerjrho@ccu.edu.tw
PHYSICAL REVIEW E 77, 066710 2008
1539-3755/2008/776/06671013 ©2008 The American Physical Society 066710-1