Novel technological applications often involve fluid flows in the Knudsen
regime in which the mean free path is comparable to the system size. We use
molecular dynamics simulations to study the transition between the dilute gas
and the dense fluid regimes as the fluid density is increased.Comment: REVTeX, 15 pages, 4 EPS figures, to appear in Physica 

Allen

Bhattacharya

Bhushan

Bitsanis

Cercignani

Cieplak

Ehrfeld

Grest

Hoover

Jayanth R. Bavanar

Joel Koplik

Kennard

Kim (Ed.)

Koplik

Loose

Marek Cieplak

Marlow (Ed.)

Maxwell

Morris

Muntz

Nose

Sugano

Tehver

Thompson

English

arXiv

Novel technological applications often involve fluid flows in the Knudsen regime in which the mean free path is comparable to the system size. We use molecular dynamics simulations to study the transition between the dilute gas and the dense fluid regimes as the fluid density is increased

Jayanth R Bavanar

CiteSeerX

Physica A 287 (2000) 153–160
www.elsevier.com/locate/physa
Molecular dynamics of ows in the
Knudsen regime
Marek Cieplaka ;∗, Joel Koplikb , Jayanth R. Bavanarc
aInstitute of Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland
bBenjamin Levich Institute and Department of Physics, City College of New York, NY 10031, USA
cDepartment of Physics and Center for Materials Physics, 104 Davey Laboratory,
The Pennsylvania State University, University Park, PA 16802, USA
Received 2 June 2000
Abstract
Novel technological applications often involve uid ows in the Knudsen regime in which
the mean free path is comparable to the system size. We use molecular dynamics simulations
to study the transition between the dilute gas and the dense uid regimes as the uid density is
increased. c© 2000 Elsevier Science B.V. All rights reserved.
PACS: 51.10.+y; 34.10.+x; 92.20.Bk
Keywords: Knudsen ows; Raried gases; Molecular dynamics; Fluid–solid interfaces
In a landmark paper [1], Maxwell applied his kinetic theory [2] to the study of the
slip length associated with uid ow close to a solid surface. He postulated a linear
combination of two kinds of behaviour when a molecule was incident on a wall – it is
either specularly reected or it undergoes enough collisions with the wall molecules so
that it loses memory of the incident velocity and emerges with a velocity characteristic
of the wall temperature. We revisit this problem with molecular dynamics simulations
– as the properties of the wall are varied, we nd both kinds of behaviour suggested
by Maxwell, but the crossover from one to the other is not as simple as envisioned by
him. In addition, rich behaviour is found in the uid ow properties on varying the
uid density from the dense liquid regime to the Knudsen regime (large mean free path
compared to the system size). Our studies of ows in the subcontinuum regime provide
a unique window on the nature of boundary conditions at a uid–solid interface and are
∗ Corresponding author.
0378-4371/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.
PII: S 0378 -4371(00)00353 -8
154 M. Cieplak et al. / Physica A 287 (2000) 153–160
relevant to the behaviour of nanoscale mechanical devices, high-altitude aerodynamics,
aerosol dynamics, isotope separation schemes, ow cooling techniques, and energy
transfer in molecular collisions [3–6].
Studies of ows at nanoscales involve questions pertaining to a sub-continuum
regime [7,8] for which experiments are often unable to provide the requisite answers.
The behaviour of materials at the nanoscale is often quite dierent from that in the
bulk [9–12]. Numerical simulations based on molecular dynamics (MD) oer a natural
tool to handle the interacting many-body nanoscale physics. A detailed understanding
of the behaviour of materials at very small scales is necessary for the fabrication of
miniature devices and for the manipulation of materials at the molecular scale. Such
microscopic studies are useful for assessing the range of validity of traditional con-
tinuum theories. In addition, atomistic simulations are an invaluable tool for deducing
constitutive relationships, which may then be fed into continuum calculations.
Here, we report on MD studies of the boundary conditions in single-uid ows
in the Knudsen regime [13] in which the mean free path, , becomes comparable
to the characteristic device dimension, L. Such ows behave very dierently from
bulk, continuum expectations. There is considerable work on the modied boundary
conditions based on either semi-heuristic kinetic theory arguments or using an analysis
of the Boltzmann equation, but inevitably severe simplications are made concerning
the dynamics of the system [14]. Furthermore, in such treatments, the solid boundaries
are not handled in a realistic manner as having atomic structure on length scales
comparable to that of the uid constitutents [15,16].
We study Lennard–Jones uids undergoing two kinds of ow, Couette and Poiseuille,
between two solid walls with an atomic structure. Three kinds of walls, two attractive
and one repulsive, are discussed. We consider a range of uid densities to study the
crossover from ows described by continuum uid mechanics to dilute gas ows in
the Knudsen regime.
In a Lennard–Jones uid, two atoms separated by a distance r interact with the
potential
VLJ(r) = 4
[( r

)−12
−
( r

)−6]
; (1)
where  is the size of the repulsive core [the potential is truncated at 2:2]. The
ows take place between two solid planar walls parallel to the xy plane with periodic
boundary conditions imposed along the x- and y-direction. Couette ow was generated
by moving the walls at constant velocity in opposite directions along the x-axis whereas
Poiseuille ow was induced by an application of uniform acceleration g on all the uid
molecules in the x-direction.
Following Thompson and Robbins [17], the wall of atoms form two [0 0 1] planes of
an fcc lattice with each wall atom tethered to a xed lattice site by a harmonic spring
of spring constant k. Our studies were done for k=400 so that the mean-square displace-
ment about the lattice sites did not exceed the Lindemann criterion for
melting but was close to the value set by this criterion. The wall–uid interactions
M. Cieplak et al. / Physica A 287 (2000) 153–160 155
were modelled by another Lennard–Jones potential
Vwf(r) = 16
[( r

)−12
− A
( r

)−6]
; (2)
with the parameter A varying between 1 and 0, corresponding to attractive and repulsive
walls, respectively. For the narrowest channel that we studied, the uid atoms were
conned to a space measuring 13:6 by 5:1 in the xy plane and 12:75 between the
walls. The “basic” walls in our studies contain 96 wall atoms each (in each wall plane,
the atomic periodicity is 0:85). We have also considered the case of walls which are
about 2.5 times denser and consist of 176 wall atoms each – we shall refer to these
as “dense” walls.
The equations of motion were integrated using a fth-order Gear predictor–corrector
algorithm [18] with a step size of 0:005, where  = (m2=)1=2 and m is the atomic
(wall and uid) mass. The uid and wall temperatures were xed at kBT= = 1:1.
We present results obtained with the use of Langevin thermostat [19] but the results
obtained with the use of the Nose–Hoover thermostat [20,21] were virtually identical.
The noise was applied in all directions during equilibration and only in the y-direction
during the data-taking phase. The averages were obtained over runs which lasted at
least 5000, and in the extreme dilution case, for up to 400 000. An initial time of at
least 400 was spent for equilibration. The spatial averaging was carried out in slabs
of width =4 along the z-axis.
In the dilute gas limit, the Knudsen number, Kn, is dened as =L, i.e., as
(
√
22L)−1, where  is the mean free path,  is the uid number density and L
is the characteristic width of the channel. A study of the density prole shows qualita-
tively dierent behaviours of the density proles for the attractive and repulsive walls.
In the former case, two adsorbed layers form near both walls whereas there is merely
a depletion zone, whose thickness is of order , alongside the repulsive walls. The
rst layer at the attractive walls is fairly periodic, as measured by the static struc-
ture factor, and the resulting local order corresponds to either a square or hexagonal
lattice depending on whether the wall is basic or dense. L is dened as an eective
channel width in which the ow takes place (in the attractive case this is the distance
between centers of the second layers). In the centre of the channel, the uid density
is essentially constant and this value is chosen as  in the denition of Kn.
Fig. 1 shows the Poiseuille ow velocity proles for three dierent values of  (or
Kn) for the attractive wall case. We have conrmed with studies of channels of three
dierent widths and the two wall densities that the crossover value of  above which
continuum uid mechanics holds, K , is somewhat less than 0.19, corresponding to
a Kn of order 0.1 for the channel of the smallest width. In this regime, the proles
are clearly parabolic and essentially with no slip at the walls. In contrast, the velocity
prole at the highest Kn that we were able to study (corresponding to just 4 particles
in the interior of the channel) is almost that of plug ow with a huge slip length.
For repulsive walls, the plug-ow regime extends to higher densities because such
walls provide little coupling to the uid and correspondingly, K becomes of order
156 M. Cieplak et al. / Physica A 287 (2000) 153–160
Fig. 1. Velocity proles of Poiseuille ow with g = 0:01=2 for the smallest width channel with basic
attractive walls. The values of the interior uid density and the corresponding Knudsen number are indicated.
The lled circles denote the highest Kn case. The inner wall layers are at the edges of the gure. Eectively,
one observes no slip boundary conditions for the two higher density cases (the slip length is around 1:7
measured with respect to the wall location, but note the presence of the adsorbed layers at the wall).
0.3. Furthermore, in the continuum regime, the velocity eld exhibits substantial slip.
A consistent behaviour is found for Couette ow with higher slip for the repulsive walls
(compared to the attractive wall case) for ¿K and essentially zero uid velocity at
the walls at the highest Kn.
We now proceed to study the eect of interactions between the uid particles un-
dergoing Poiseuille ow on increasing  within the channel. In Poiseuille ow, the
velocity eld is highest in the middle of the channel and is denoted by vmax. At the
lowest densities, one expects that vmax ought to scale as L. This is because the driving
force is eective in accelerating the uid particles only during the time spent traversing
the channel between collisions with the walls (and the associated adsorbed layers). This
time simply scales as the channel width. On the other hand, at large , corresponding
to usual uid behaviour, vmax ought to scale as L2 as given by the Navier–Stokes equa-
tion with no-slip boundary conditions. Fig. 2 shows a plot of how vmax interpolates
between the low and high Knudsen number behaviours as the uid density is varied.
Strikingly, vmax shows a maximum around K with the peak being more pronounced
for larger L’s. We have conrmed that this qualitative behaviour is essentially indepen-
dent of the nature of the walls. Physically, K corresponds to the situation when the
uid intermolecular interactions begin to dominate leading to a diminished inuence
of the wall in the interior of the uid.
M. Cieplak et al. / Physica A 287 (2000) 153–160 157
Fig. 2. Plot of the uid velocity in the middle of the channel in Poiseuille ow with g = 0:01=2 as a
function of uid density (and the Knudsen number in the inset). The plot is for three dierent channel widths.
The L0 in the gure denotes the narrowest channel. For this case, both attractive walls were considered –
the basic (hexagons) and higher (asterisks) wall densities. 2L0 and 4L0 denote channels that are eectively
twice and four times as wide as the smallest channel.
Fig. 3 shows a vivid illustration of the role played by the intermolecular interactions
in leading to an enhancement of vmax even as the uid density is moderately increased
from the high Kn limit. For a repulsive wall (not shown), the particle is reected
approximately specularly and moves out of the vicinity of the wall immediately. The
bottom panel shows qualitatively dierent behaviour in the (basic) attractive wall case.
Here, again as suggested by Maxwell, a particle either thermalizes in the vicinity of the
wall for a relatively long time and then escapes from it or occasionally still undergoes
specular reection. On increasing the uid density, one gets the situation illustrated in
the middle panel in which intermolecular collisions keep the particles longer within the
channel thus reducing the time of localization near the walls and eectively increasing
the velocity eld in the middle of the channel (Fig. 2). At still higher densities (∼ K),
the intermolecular interactions become suciently large so that liquid-like behaviour
sets in and vmax begins to decrease.
A key quantity that characterizes the uid–solid interface is the collision kernel
[14,22] – the probability density that a molecule striking the surface with a given ve-
locity reemerges with a specic velocity after a given residence time. For the repulsive
wall, we nd that Maxwell’s specular reection scenario holds exceedingly well and the
residence time is negligible. On the other hand, for the attractive walls the distribution
of residence times develops a long tail. Strikingly, the velocity distribution of a particle
158 M. Cieplak et al. / Physica A 287 (2000) 153–160
Fig. 3. Plot of the time dependence of the z-coordinate of a typical uid particle undergoing Poiseuille ow
in the narrowest channel. The horizontal dotted lines indicate the z-coordinates of the wall molecules. The
solid horizontal line refers to the centres of the adsorbed rst layers.
after the collision with the wall is substantially independent of the incoming velocity.
As an illustration, Fig. 4 shows the case of particles with unit speed incident normal
to the wall. The inset shows the probability distribution of residence times whereas
the main gure shows the distributions of the normal (z) and tangential (x) velocities
of the scattered molecules. The results are in excellent accord with Maxwell’s thermal
wall scenario in which the probability densities corresponding to vx and vz are [1,13,23]
x(vx) =
√
m
2kBT
exp
(
− mv
2
x
2kBT
)
(3)
and
z(vz) =
m
kBT
vz exp
(
− mv
2
z
2kBT
)
; (4)
respectively. As expected for a thermal wall, the residence time is found to be uncor-
related with the outgoing velocity.
How does the behaviour crossover from that of specular reection to that of a thermal
wall? Following Maxwell [1], a common hypothesis is a linear combination of both
behaviours with weights (1 − f) and f, respectively. Such an intermediate case is
realized, e.g., on considering the scattering from a wall with the wall–uid potential
given by Eq. (2) with A=1=4. The velocity distributions of the scattered particles clearly
deviate from pure thermal behaviour and the data might be interpreted as containing
a specular component. However, the resulting distributions are more complex than
the simple linear combination form [24]. 1 The behaviour of the collision kernel is
1 The velocity distributions in this case are shown (Fig. 10) in a conference paper (see Ref. [24]). The
complementary case of a repulsive wall is considered in detail in this work and the results are benchmarked
against our basic ndings presented here.
M. Cieplak et al. / Physica A 287 (2000) 153–160 159
Fig. 4. The probability density distributions for the z (normal) and x components of the outgoing velocity
of a uid molecule after collision with the wall and the associated adsorbed layer. The solid histograms
are obtained based on 3000 scattering events. The dotted lines are the analytic predictions of Maxwell,
Eqs. (3) and (4), discretized with a bin size of 0:1=0 as in the simulations. The inset shows the distribution
of residence times. The residence time is measured as the time interval between a uid molecule leaving
a force-free region (near the wall) and its reemergence. The peak of the curve (at a non-zero value of )
corresponds to a virtually instantaneous bounce back of the uid molecule and accounts for the time needed
to approach the wall and return.
further enriched on considering the role of the interactions between the uid molecules
as the Knudsen number decreases to a value of 0.27 corresponding to a density of
 = 0:066 (Fig. 10 in Ref. [24]). The inuence of other uid molecules leads to the
distributions more closely approaching the thermal ones. The nature of the boundary
conditions at a uid–solid interface is thus complex but akin to those envisioned by
Maxwell.
We are indebted to Mark Robbins and Riina Tehver for many valuable discus-
sions. This work was supported by KBN (Grant No. 2P03B-025-13), NASA, and the
Petroleum Research Fund administered by the American Chemical Society.
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