Thermophoresis in Rarefied Gas Flows
D. J. Rader, M. A. Gallis, and J. R. Torczynski
Engineering Sciences Center; Sandia National Laboratories; Albuquerque, New Mexico 87185-0834 USA
Abstract. Numerical calculations are presented for the thermophoretic force acting on a motionless, spherical particle
suspended in a quiescent, rarefied gas between parallel plates of unequal temperature. The rarefied-gas heat flux and
temperature profiles are calculated with the Direct Simulation Monte Carlo (DSMC) method, which provides a timeaveraged, spatially resolved approximation to the molecular velocity distribution. A force Green’s function is used to
calculate the spatial and pressure dependence of the thermophoretic force directly from the computed velocity distribution.
INTRODUCTION
Particle transport in rarefied gas flow is of interest in applications such as MEMS devices (small length scales) and
semiconductor processing (low pressures). Of particular interest is the thermophoretic force, which arises when a small
particle is placed in a gas with a temperature gradient. The thermophoretic force will induce particle motion even in
the absence of gas flow, causing the particle to move from warmer towards cooler gas regions. Many authors have
recognized the possibility of using thermophoresis to protect critical surfaces from particle contamination by keeping
the surface warmer than the surrounding gas [1-2], but questions have emerged about the effectiveness of
thermophoresis at low pressures. This concern is well founded, as in the limit of a perfect vacuum the thermophoretic
force must vanish along with the gas molecules that cause it. Recently, thermophoresis has been found to play a key
role in the transport of small dust particles generated in low-pressure plasma discharges [3]. Because dust particles are
known to accumulate at the plasma-sheath boundary near chamber walls [4-6], there has been renewed interest in the
problem of near-wall thermophoresis in low-pressure plasma systems [7-10].
This paper presents calculations of the thermophoretic force on a spherical, free-molecular particle suspended in a
stationary, rarefied gas between two infinite parallel plates held at unequal temperature (Figure 1). For the present
discussion, the temperatures of the hot plate, T h , and the cold plate, T c , do not differ by much (i.e., T h – T c « T c ).
The particle concentration is dilute, so that the thermophoretic force calculation has two aspects: simulating the
rarefied gas and computing the force acting on an isolated particle suspended in this gas. The rarefied gas is described
with the well-established Direct Simulation Monte Carlo (DSMC) method [11-12]. The thermophoretic force acting
on a suspended particle is calculated directly from DSMC calculations of the local molecular velocity distribution
using the force Green’s function method [13]. Gas-particle interactions are treated as diffuse with complete thermal
accommodation; gas-plate interactions can be specular or diffuse. These calculations are intended to elucidate the
behavior of the thermophoretic force in a rarefied gas, with particular attention given to near-wall effects.
GENERAL FORCE ON A FREE-MOLECULAR PARTICLE
Under the following assumptions, expressions for the force on a particle in a rarefied gas with an arbitrary
molecular velocity distribution function can be found [13]. The particle concentration is dilute, the gas is monatomic,
and the gas is strictly free-molecular on the particle scale (but not necessarily on the system scale). The particle is
spherical, with uniform temperature (not necessarily the gas temperature) and uniform surface properties. All gasparticle interactions are assumed here to be diffuse and fully accommodated (isothermal at the particle’s temperature),
although other interaction models have been considered [13]. Brownian motion is neglected here.
The force F on a spherical particle is given by an integral of the molecular velocity distribution function f [ u ]
multiplied by the force Green’s functions F δ (which represents the force on a spherical particle from a delta-function
molecular velocity distribution) [13]:
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
234
Th
z=L
heat flux q
force F
z=0
Tc
FIGURE 1. Thermophoresis geometry: gas and particle are motionless (particle size is greatly exaggerated).
F [ f ] = ∫ F δ [ u – u p ] f [ u ] du ,
2
F δ [ c ] = mn ( πR p )c{ c + ( π ⁄ 6 )c p } where c p = ( 8k B T p ⁄ ( πm ) )
∫ f [ u ] du
(1)
1⁄2
= 1.
,
(2)
(3)
Here, u and u p are molecule and particle velocities, c = u – u p is their difference, m is the molecule mass, n is the
molecule number density, R p is the particle radius, T p is the particle temperature, and k B is Boltzmann’s constant.
An analogous heat-transfer Green’s function has also been derived [13] but may be neglected in the present case of a
motionless gas and particle for which the heat flux vanishes and the gas and particle temperatures are equal.
The above expressions are implemented in a computational framework by providing a numerical representation of
the molecular velocity distribution function. In this implementation, the calculated velocity distribution function is
used to compute the moments 〈 u – u p〉 and 〈 ( u – u p ) u – u p 〉 , which are computed like moments for any other
macroscopic quantities [12]. This approach has been implemented in the DSMC codes DSMC1 [12] and Icarus [14]
in the manner described in [13]. The particles do not collide with each other and the computational molecules are not
affected by collisions with a particle. Since the force is exactly proportional to the cross-sectional area of the particle
in the locally free-molecular limit considered here, the force per unit cross-sectional area is reported.
For the present case of a motionless particle suspended in a motionless gas, the force Green’s function method
directly yields the thermophoretic force, which arises from the asymmetry of the molecular velocity distribution. For
a motionless particle and gas, the particle-temperature dependent term in Equation (2) vanishes and the force Green’s
2
function becomes F δ = mn ( πR p )u u .
ANALYTICAL EXPRESSIONS FOR THE THERMOPHORETIC FORCE
Theoretical expressions for the thermophoretic force and heat flux in the present parallel-plate geometry are
available in both the free-molecular ( Kn → ∞ ) and continuum ( Kn → 0 ) limits [13, 15]. The system Knudsen
number, Kn, is defined as the ratio of the gas mean free path, λ = 2µ ⁄ mnc , to the inter-plate separation, L , where
µ is the gas viscosity and c = 8k B T ⁄ πm is the mean molecular speed of an equilibrium gas at temperature T .
In the free-molecular (collisionless) limit, molecules travel back and forth between the plates without colliding with
each other. For diffuse (fully accommodated) gas-surface interactions, the molecules reflected from each plate possess
a half-range Maxwellian velocity distribution at the plate temperature, and the space between the plates is
235
characterized by two half-range Maxwellians. When the temperature difference between the plates is small, the freemolecular heat flux [12] and the thermophoretic force [16] are given by
Th – Tc
1
(4)
q FM = – --- Pc ------------------ ,
2
T
3
2 q FM
F FM = --- ( πR p ) ----------- ,
c
4
(5)
where P is the gas pressure and T = T h T c is the gas temperature. The free-molecular thermophoretic force is
directly proportional to pressure and is constant across the domain; in particular, there is no wall effect.
In the continuum limit with small temperature differences, the heat flux and the thermophoretic force are
∂T
q C = – k ------- ,
∂z
(6)
32
2 qC
F C = --------- ( πR p ) ------ ,
c
15π
(7)
where k = 15µk B ⁄ 4 m is the gas thermal conductivity. The thermophoretic force expression given by Equation (7)
was first derived by Waldmann [17] for a free-molecular particle suspended in a first-approximation Chapman-Enskog
gas. Gallis and coworkers [15,18] have shown that higher-order approximations to the Chapman-Enskog molecular
velocity distribution lead to expressions of the same form as Equation (7) wherein the numerical coefficient depends
on the choice of molecular collision model. Interestingly, the coefficient given by the collision-model-independent
first approximation, 32 ⁄ 15π , turns out to be the exact, high-order value for the Maxwell collision model.
A comparison of Equations (5) and (7) reveals a striking result first realized by Vestner [19] and later exploited by
Gallis and coworkers [13,15,18]: the thermophoretic force is related to the local heat flux, the particle cross-sectional
area, and the molecular thermal speed scale in a particularly simple fashion:
q
F = ξ ( πR 2p ) --- ,
c
(8)
where the constant of proportionality, ξ , is termed here the thermophoresis parameter. The thermophoresis parameter
provides a single, dimensionless number that can be used to describe the variation of the thermophoretic force with
molecular collision model. The value of ξ depends on the local molecular velocity distribution but in a surprisingly
weak way. Although the free-molecular and Chapman-Enskog velocity distributions described previously are
substantially different, the thermophoresis parameter for the free-molecular limit ( ξ FM = 3 ⁄ 4 = 0.75 ) differs by
only 10% from the continuum Chapman-Enskog (Maxwell molecule) value ( ξ C = 32 ⁄ 15π = 0.67906 ).
As the Knudsen number increases from zero, the continuum approximation begins to break down. Rarefaction
effects first become apparent at the walls in the form of temperature discontinuities (jumps). For near-continuum flow
in the present geometry, both the Chapman-Enskog approximation and Equation (7) will not apply near the plates
although both still may apply in the interior. The behavior of the thermophoretic force near walls has been the subject
of several theoretical investigations [7-10, 20]. All of these studies assume that a Chapman-Enskog distribution is
obtained far from the wall, but they disagree on the magnitude of the wall effect. For example, Havnes et al. [7]
predicted a substantial (50%) decrease, Williams [20] predicted a slight (2-5%) increase, while Chen [8-10] predicted
a substantial (up to 40%) increase in the near-wall thermophoretic force compared to the asymptotic value given by
Equation (7). Recently, Gallis et al. [21] derived an expression for the near-wall thermophoretic force based on a
heuristic model originally proposed by Chen [8-10]. Gallis’ derivation approximates the near-wall velocity distribution
as the sum of a half-range Maxwell Chapman-Enskog distribution streaming toward the wall and a reflected half-range
distribution streaming away from it. The reflected molecules are described by a distribution in which a fraction ε w are
specularly reflected and the remainder ( 1 – ε w ) are diffusely reflected. For small temperature differences between the
incoming gas and the wall, the near-wall (Knudsen layer) thermophoretic force becomes [21]
236
1
2 qC
F KN = --- [ ξ C ( 1 – ε w ) + ξ FM ( 1 + ε w ) ] ( πR p ) ------ ,
2
c
(9)
where ξ FM and ξ C are the free-molecular and Maxwell Chapman-Enskog values given above. Interestingly, for a
fully diffuse wall, the near-wall thermophoresis parameter lies exactly half-way between these two theoretical limits.
Although both the heat flux and the thermophoretic force vanish for a fully specular wall, the thermophoresis
parameter has a well-defined limit: ξ KN ( ε w → 1 ) = ξ FM , the free-molecular value. Thus, Equation (9) indicates that
the wall effect increases the thermophoretic force by less than 10% compared to that experienced by a particle in an
unbounded Maxwell Chapman-Enskog gas with the same heat flux.
TRANSITION-REGIME INTERPOLATION FORMULAS
A simple interpolation formula suggested by Sherman [22] approximates the heat flux in the transition regime as
q FM
q TR ≈ ----------------------------------- ,
1 + ( q FM ⁄ q C )
(10)
where the free-molecular and continuum heat fluxes are given by Equations (4) and (6), respectively. Springer [23]
noted that Equation (10) agrees well with the limited experimental data. Recently, Gallis et al. [15] proposed an
interpolation formula for the transition regime thermophoretic force based on Equations (8) and (10):
q TR
F TR ≈ ξ C ( πR 2p ) --------- .
c
(11)
The use of ξ C in Equation (11) is exact in the continuum limit for a Maxwell Chapman-Enskog distribution but would
under-predict the free-molecular thermophoretic force by 10%. For argon, helium, and nitrogen, Gallis et al. [15]
showed that Equation (11) always agreed with DSMC/Green’s function calculations of the thermophoretic force to
better than 10%; for Kn < 1.5 the agreement was better than 4%. The interpolation formula of Equation (11) provides
no information about spatial variations and is intended to provide a spatially-averaged value of the thermophoretic
force between the plates.
NUMERICAL SIMULATIONS OF THE THERMOPHORETIC FORCE
Numerical simulations using the DSMC/Green’s function method are used to explore the spatial and pressure
dependence of the thermophoretic force in the parallel-plate geometry. The DSMC calculations are performed with
Bird’s one-dimensional code DSMC1 using the Variable Hard Sphere (VHS) collision model [12]. To facilitate
comparison with theory, VHS parameters are chosen to simulate a Maxwell collision model with argon-like properties:
– 26
–5
m = 6.63 ×10 kg , µ ref = 2.117 ×10 Pa ⋅ s at T ref = 273 K , and a viscosity temperature exponent of ω = 1 .
– 10
The corresponding molecular diameter [12] is d ref = 4.59 ×10 m . A one-dimensional 1-mm domain with 500
uniform cells is used. The time step employed in the simulations ( ∆t = 2 ns ) is chosen to be small enough to ensure
that molecules travel not more than about 1/3 of the cell size in a time step. The cold and warm plates are held at 263
and 283 K, respectively. Each cell has 30 computational molecules on average. Calculations are performed over a
pressure range from 0.1 to 1000 mtorr (0.01333-133.3 Pa). The gas is initially at 273 K and the selected pressure. To
obtain steady-state results, 1 million transient steps are typically followed by 100 million or more steady-state time
steps (a total time of at least 0.2 s).
Figure 2 shows temperature profiles from DSMC simulations for an argon-like Maxwell gas at five system
Knudsen numbers: 0.0475, 0.158, 0.475, 1.58, and 4.75 (corresponding to pressures of 1000, 300, 100, 30, and 10
mtorr, respectively). A temperature profile from a DSMC simulation for the free-molecular regime is also shown, in
237
283
um
inu torr
C
0m
00
r
1
=
tor
P
0m
0
3
rr
mto
100
torr
30 m
rr
10 mto
281
t
on
VHS Maxwell, diffuse walls
263-283 K, 1 mm
279
Temperature (K)
277
275
273
.75
271
Kn = 4
1.58
269
75
0.4
58
5
47 um
0.0 tinu
n
Co
0.1
267
265
263
Free-Molecular
Free-Molecular
0
0.2
0.4
0.6
0.8
1
z/L
FIGURE 2. DSMC-calculated temperature profiles for an argon-like Maxwell gas.
which collisions are suppressed. The theoretical continuum temperature profile is also included for reference. All
calculations assumed fully diffuse walls ( ε w = 0 ). The calculated temperature profile for the free-molecular case is
nearly uniform at the geometric mean temperature of 272.8 K, while the theoretical continuum temperature profile
matches the wall temperatures and varies linearly in between. For intermediate Knudsen numbers, the calculated
temperature profiles exhibit temperature jumps at the walls and varying degrees of curvature over a distance extending
a few mean free paths from the walls. For the smallest Knudsen number (0.0475), temperature jumps of ~1 K are
observed at the walls, while the temperature profile is reasonably linear over the domain interior. The smoothness of
the profiles is a result of the large number of time steps used in the DSMC calculations.
Figure 3 shows plots of the spatial variation of the thermophoresis parameter for the same DSMC calculations
shown in Figure 2. The thermophoresis parameter in each cell is calculated from the cell-based moments for the heat
flux, the thermophoretic force, and the temperature. The results show the striking progression from free-molecular
flow (for which ξ is constant) to near-continuum flow (for which ξ is constant over the interior but increases toward
the walls inside the Knudsen layers). The thermophoresis parameter in the free-molecular limit is in excellent
agreement with the theoretical value of 0.75, and the calculations clearly demonstrate that there are no wall effects in
the free-molecular regime. As Kn decreases, the profiles are increasingly displaced below the free-molecular limit and
increasingly exhibit a concave-upward curvature.
In the near-continuum regime ( Kn = 0.0475 ), the profile of ξ shows a flat central region surrounded by Knudsen
layers wherein the thermophoresis parameter increases toward the walls. The lower solid line plotted in Figure 3 shows
the theoretical prediction for the thermophoresis parameter for the Maxwell Chapman-Enskog velocity distribution,
ξ C = 0.679061 . An average of the thermophoresis parameter over the uniform central region ( 0.3 < z ⁄ L < 0.7 ) gives
an average value of 0.6792 that is in excellent agreement (0.02%) with the theoretical value. Consequently, these
results suggest that the DSMC method is calculating a velocity distribution that very closely approximates the Maxwell
Chapman-Enskog distribution within the central part of the domain. The behavior of ξ near the walls clearly illustrates
the structure of the Knudsen layer for the near-continuum calculation. The Knudsen layers are fairly symmetric about
the centerline and extend into the domain about 5 mean free paths from each wall. The magnitude of the wall effect is
shown to be fairly small, however, with ξ at the cold and warm walls increasing by 4.0% and 4.7%, respectively,
compared to the interior value. The prediction of ξ KN based on Equation (9) for a fully diffuse wall is that ξ near the
238
0.76
FREE-MOLECULAR
Thermophoresis Parameter
0.75
0.74
Kn = 4.75
0.73
1.58
0.72
0.71
0.475
0.70
0.158
0.69
0.68
CHAPMAN-ENSKOG
0.0475
VHS Maxwell, diffuse walls
263-283 K, 1 mm
0.67
0.66
0
0.2
0.4
0.6
0.8
1
z/L
FIGURE 3. Thermophoresis-parameter profiles: symbols, DSMC simulations; solid lines, theory.
wall should be 5.2% larger than the value in the interior. The agreement between DSMC calculations and Equation (9)
is quite good. Note that for increasing Kn the wall effect becomes relatively less evident until it vanishes for freemolecular flow. For all cases, the calculated values of the thermophoresis parameter are bounded in a narrow range:
ξ C ≤ ξ ≤ ξ FM . Moreover, for cases in which a Chapman-Enskog region is obtained in the interior of the domain, an
even tighter bound on the thermophoresis parameter is observed: ξ C ≤ ξ ≤ ξ KN .
To further explore near-wall thermophoresis, DSMC simulations are performed for several specular reflection
values ( ε w = 0, 0.25, 0.50, and 0.75) at Kn = 0.0475 . The results are shown in Figure 4, where DSMC calculations
of the thermophoretic force ratio (TFR) are plotted against wall specularity. Here, the TFR is defined to be the ratio of
the Knudsen and the continuum forces, TFR = F KN ⁄ F C . For the DSMC calculations, the near-wall force is taken as
the average of the 5 adjacent near-wall cells (to reduce statistical scatter), and the continuum force is taken as the
average over the uniform central region. One trend is clear: the magnitude of the thermophoresis parameter at the wall
increases with increasing specular fraction. In the diffuse-wall limit, the DSMC calculations indicate near-wall TFR
values of 1.04 at both the cold and warm walls. For ε w = 0.75 , the force ratio increases slightly 1.08. Theoretical
values using the present analysis, Equations (9) and (7), are seen to be in good agreement with DSMC calculations.
The most recent theory of Chen and Xu [10], however, shows the wrong qualitative behavior.
The spatially-averaged thermophoretic force is shown in Figure 5 for a more realistic argon model (VSS,
ω = 0.81 , α = 1.4 [12]) with a 1-cm gap [15]. As expected, the theoretical continuum and free-molecular limits are
closely approached for the highest and lowest pressures, respectively. The proposed interpolation strategy for
predicting the transition-regime thermophoretic force, Equations (10) and (11), is shown to be in good agreement
(within ~10%) with the DSMC calculations over the entire range; for Kn < 1.5 , the agreement is better than 4%.
CONCLUSIONS
DSMC/Green’s function calculations are used to explore the spatial and pressure variations of the thermophoretic
force in rarefied gas between parallel plates. A highly refined mesh, long convergence times, and high heat fluxes are
used to provide results of high accuracy so that near-wall effects can be clearly observed. Calculations of the spatial
239
1.40
VHS Maxwell, Kn = 0.0475
263-283 K, 1 mm
Thermophoretic Force Ratio, FKN/FC
1.35
DSMC Cold Wall
DSMC Warm Wall
1.30
1.25
Chen and Xu
1.20
1.15
This work - Equations (9) ÷ (7)
1.10
1.05
1.00
0
0.25
0.5
Specular Fraction, εw
0.75
1
FIGURE 4. Thermophoretic force ratio: symbols, DSMC simulations; solid lines, theory.
100
VSS Argon, diffuse walls
263-283 K, 1 cm
F
Force/Area (N/m2)
10-1
e
re
M
o
u
lec
lar
Continuum
10-2
DSMC/Green’s
Eqns. (10)-(11)
10-3
10-4 -1
10
100
101
Pressure (mTorr)
102
103
FIGURE 5. Pressure dependence of the thermophoretic force (per unit particle cross-sectional area).
240
profile of the thermophoresis parameter show a striking progression from free-molecular flow (for which ξ is
constant) to near-continuum flow (for which ξ is constant over the interior but increases toward the walls inside the
Knudsen layer). The calculated free-molecular thermophoresis parameter is in excellent agreement with theory. As the
Knudsen number is decreased, the profiles are increasingly displaced below the free-molecular limit and increasingly
exhibit a concave-upward curvature. For the smallest Knudsen number (0.0475) simulated, the profile for ξ shows a
flat central region surrounded by Knudsen layers wherein the thermophoresis parameter increases toward the walls.
For all conditions examined, the DSMC-calculated values of the thermophoresis parameter are bounded in the narrow
range between the free-molecular and Maxwell Chapman-Enskog values, which differ by only 10%. Moreover, when
a Chapman-Enskog region is obtained in the domain interior, an even tighter bound on ξ is observed. A new
approximate model for the near-wall thermophoretic force agrees well with the present calculations.
At small Knudsen numbers, exceptional agreement is found in the central part of the domain between DSMC
calculations of and Maxwell Chapman-Enskog theory predictions for ξ (to within 0.02%). Consequently, the
calculations suggest that the DSMC method is producing a velocity distribution that closely approximates a ChapmanEnskog distribution within the central part of the domain. Finally, a transition-regime interpolation strategy for
estimating the thermophoretic force compares well with the present numerical simulations.
Acknowledgments - This work was performed at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation,
a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.
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