Multi scale modeling of micro-coronas

1. Multi-Scale Modeling of Micro-Coronas
Fa-Gung Fan
fagung@gmaill.com
January 27, 2009
The research content outlined here constitutes part of a graduate student’s ongoing
thesis work1
at University of Wisconsin - Milwaukee. Please treat the content as
unpublished material.
(This research is in its early stage. This report represents our present thinking.
Corrections and suggestions are welcome.)
1. Introduction
Coronas are luminescent weakly ionized cold plasmas. It is called “cold plasma” because
the temperature of the electrons in the plasma and that of the surrounding gas molecules
are not in thermal equilibrium. The temperature of the electrons can be two orders of
magnitude higher than that of the gas molecules, while the latter stays near the
environment temperature. This type of plasmas is also known as non-equilibrium
plasmas, nonthermal plasmas, or low-temperature plasmas. One of the main
characteristics in plasma modeling is that different time and spatial scales have to be
resolved self-consistently. The range of time scales vary from picoseconds for the
collision cascade, to the nanosecond response time of electrons, to milliseconds for
positive ions, to tens/hundreds of milliseconds for heavy species chemistry, to hours and
days for the duration of device operation. The variation of length scale is also extremely
wide. The dimensions of commonly used plasma devices such as xerographic chargers,
plasma reactors in semi-conductor fabrication, plasma display panel (PDP) cells, and the
processing zones in plasma surface treatment equipment are in macroscopic scale. The
underlying physics that generates the charges (i.e., electrons and ions) via electron-impact
reactions occurs at the molecular scale. The electrons and ions generated through these
microscopic processes in turn determine the macroscopic space charge distribution.
One popular way of modeling plasma is to consider it as a continuum, and apply
conservation laws on each of the charge species. This method is efficient. However,
under certain conditions (e.g., in low pressure reactors), the continuum assumption breaks
down and more fundamental description of plasma is required. In some localized
phenomena, such as the ionization fronts of filament discharge streamers where steep
electron density gradients exist in the leading edge, the corona discharges from
nanostructures (e.g., nanotubes and nanowires) where Knudsen number (Kn) is large, and
maybe the extremely thin sheath in high-density plasmas where the electric field is very
high, the macroscopic model also breaks down and a microscopic view is required to
1
supervised by Professor Junhong Chen of Mechanical Engineering Department.

2. describe the physics involved. However, since a macroscopic model is computationally
much more efficient than a microscopic model, for these cases it is desirable to use the
microscopic model locally only at the locations where it is needed and leave the rest of
the region described by a macroscopic model.
This report outlines the idea of a type of multi-scale modeling technique --- domain
decomposition with different models for the micro- and macro-scale regions. There are
numerous other multi-scale modeling techniques for different applications. The reader is
referred to [1] for an overview. Here, only domain decomposition with continuum-
particle coupling multi-scale is considered. This kind of problems belongs to Type A
multi-scale modeling of [1]. The goal of this research is to develop a corona plasma
modeling tool that can span from micro- to macro-scale. The modeling/simulation
capability of plasmas in complex geometries would be useful in the development of
miniaturized charging devices, study of electrical breakdown in toner piles, and many
others. It can also find applications in other technology areas: plasma processes for
depositing or etching of micro-features, plasma growth of nanotubes/nanowires, etc.
There are three types of model that have been used to model cold plasmas. They are fluid
model, particle model, and hybrid model. For a general review of these methods as well
as their strengths and limitations when applied to various industrial problems, the reader
is referred to [2]. In this report, the aspects relevant to our specific application are
reviewed. The report is organized as the follows. In Section 2, a review of fluid model is
provided, including the governing equations for fluid model and the methods for
evaluating the necessary coefficients. Section 3 gives a review of particle model.
Among the various particle methods, particle-in-cell Monte-Carlo (PIC-MCC) method is
described in details. The direct simulation Monte Carlo (DSMC) method is briefly
outlined. Section 4 skims over the hybrid methods which are combined uses of fluid and
particle methods. Section 5 reviews the result of a literature survey on the developments
and current status of multi-scale modeling. The challenges and methods of bridging
micro-scales to macro-scales are emphasized. Section 6 is about the present research.
The uniqueness of this research is pointed out. Several particular challenges involved are
also commented. Section 7 summarizes the report.
2. Fluid Model
Fluid model describes the plasma based on continuity equations for electron, positive
ions, and negative ions coupled with Poisson’s equation.
eee
e
nun
t
n
�
�
����
�
�
)( Continuity of electron density (1)
where )( eeeeee DnEnun ����
��
� is the electron flux.
iii
i
nun
t
n
�
�
����
�
�
)( Continuity of ion densities (2)
where )( iiiiii DnEnun ����
��
� is the ion flux.

3. (The sign of the first term on the right-hand side is defined as: � for positive ions and –
for negative ions.)
)()( ei
i
i nnZe ������ ��� Poisson’s equation (3)
In Eqs.(1)-(3), en is the electron number density, and in is the number density for ion
species i . eu
�
is electron mean velocity, and iu
�
is the mean velocity for ion species i .
e� and i� are, respectively, the mobilities for electron and ion species i ; eD and iD are
the diffusivities for electron and the corresponding ion species. � is the electrical
potential, � is the electric permittivity, E
�
is the electric field, and e is elementary
charge ( 19
10602.1 �
� coulomb). eZi is the charge of ion species ( e , e2 , e3 , … for
positive ions, and e� , e2� , e3� , … for negative ions). en� denotes the source of
electron production or consumption due to ionization or recombination; in� represents the
rate of change of the corresponding ion species by electron impact collisions or chemical
reactions.
It should be emphasized that the drift-diffusion approximation for fluxes has been used in
the above formulation. This assumption is valid for high-pressure (atmospheric)
environments, but may not be adequate for low-pressure situations (like that in
conventional low pressure plasma reactors) where the effect of ion inertia is not
negligible. In that case, the ion flux is evaluated from the momentum equation
iiiiNiiiiiiiiiii unmkTnEenZuunmunm
t
�����
��������
�
�
)()()( (4)
where im is the ion mass, k is the Boltzmann constant, iT is ion temperature, and iN� is
the ion momentum transfer collision frequency. The first term on the right-hand side of
Eq.(4) is the electrostatic force, the second term describes the pressure gradient (ideal gas
assumed), the third term represents the collisions of dissimilar particles (electrons, ions)
that cause momentum change.
Electron temperature eT which is needed in some calculations can be obtained from the
electron energy equation
� � )()()(
2
5
)(
2
3
er
r
reeeeeeeee TkNnPTnDTnTn
t
���������
�
�
�� (5)
where P is power density absorbed by the electron. The second term on the right-hand
side is cooling due to collisional electron energy loss.
The coupled fluid equations can be solved using one of the finite difference, finite
volume, or finite element methods. Specifically, the drift terms may be handled with the
flux corrected transport (FCT) technique [3-5] and the diffusion terms by central
difference. For time derivatives, a Crank-Nicolson scheme may be used to treat the
diffusion terms implicitly, while the drift terms are done explicitly with FCT.
Alternatively, the drift-diffusion terms can be discretized by Scharfetter-Gummel
exponential scheme [6], and a fully implicit Euler backward method (or a Crank-

4. Nicolson-like scheme) for the time derivatives. The Poisson’s equation can be
discretized with the central difference.
Within the framework of fluid model, there are several ways to obtain the transport
coefficients (mobility e� and diffusivity eD ) and the rate coefficients of electron-
induced chemical reactions ( ion� for ionization, att� for attachment, rec� for
recombination, dis� for dissociation, etc.) The first method is based on local field
approximation. Under this approximation, the transport coefficients and the reaction rate
coefficients are functions of reduced electric field NE / only. Here, E is electric field
magnitude and N is neutral gas number density. Empirical equations for these
properties as functions of NE / can be developed from swarm experimental data, and be
used in the fluid model.
Another method of evaluating transport coefficients and reaction rate coefficients is
based on the electron energy distribution function (EEDF), f . A brief outline of the
method is given here. Since a cold plasma is non-equilibrium, the (equilibrium)
Maxwellian energy distribution
)/exp(
)(
2
)( 2/32/1 e
e
kT
kT
f ��
�
� �� (6)
does not apply. Instead, f has to be obtained by solving the Boltzmann transport
equation. The Boltzmann equation, however, is 6-dimensional (with 3 dimensions in
physical space and 3 in velocity). One way to simplify the equation is to use two-term
spherical harmonic expansion. This approach results in a four-dimensional Fokker-
Planck type of equation [7]
S
f
DfV
f
fD
f
fD
t
f
rr ��
�
�
�
�
�
�
�
�
�
�
�
�
�
���
�
�
�
�
�
�
�
�����
�
�
��
�
�
�
�
�
�
�
�
�
�
�
�
�
������
�
�
�
�
�
���
���
� ��
0
0
0
0
0
0
0 1
(7)
with three dimensions in physical space and one in energy. In Eq.(7), 0f is the
approximate electron energy distribution function, evme 2/2
�� is electron kinetic
energy (in eV), v is electron velocity, �3/2
vDr � is the electron diffusion coefficient in
physical space, � is the electron transport collision frequency, �D and �V are electron
diffusion coefficient and drift velocity along the energy axis � caused by quasi-elastic
collisions, and S is the source term describing inelastic collisions. In some limiting
cases, Eq.(7) can be further simplified (e.g., steady-state, uniform electric field, spatial or
energy predominant, etc.)
With 0f obtained from Eq.(7), the electron mobility and diffusivity are given by
�
�
��� d
f
D
n
r
e
e
�
�
� �
�
0
0
)(
1
(8)
���� dfD
n
D r
e
e )()(
1
0
0
�
�
� (9)

5. and the reactions rate coefficients can be evaluated from
������� dfkk )()( 0
0
�
�
� (10)
where k� is the collision cross-section for k -type electron-induced reaction (i.e.,
ionization, attachment, recombination, dissociation, etc.)
3. Particle Model
Particle model takes advantage of the collective behavior of charged particles in plasmas
to model the kinetics of various species. There are several particle methods that have
been used for plasma simulations, including particle-in-cell Monte-Carlo collision (PIC-
MCC) method, direct simulation Monte Carlo (DSMC) method, particle-in-cell direct
simulation Monte Carlo (PIC-DSMC) method. In this section, the PIC-MCC method
which is used in the present work is described in details, DSMC method is briefly
outlined, while PIC-DSMC is not reviewed.
The particle-in-cell Monte Carlo collision (PIC-MCC) method consists of integrating the
Newton-Lorentz equation of motion for a number of superparticles (or pseudoparticles)
coupled with Poisson’s equation for the self-consistent calculation of electric field,
j
j
v
dt
xd �
�
� (11)
)( j
j
j xF
dt
vd
M
��
�
� (12)
���� jqF
�
(13)
�
�
� ���2
(14)
Every superparticle is composed of a large number of real charged particles moving with
the same velocity. Typically, electrons and ions are simulated and background neutrals
are assumed to be uniformly distributed in space.
In Eqs.(11)-(14), )(txx jj
��
� and )(tvv jj
��
� are, respectively, the position and velocity of
the j-th (electron or ion) superparticle at time t ; jq and jM are its charge and mass, and
� is the charge density. In the PIC method, the field quantities such as potential ),( tx
�
�
and charge density ),( tx
�
� are numerically evaluated only at spatial grid points, whereas
particles are allowed to occupy any position within the computational domain. The right-
hand side term of Eq.(13) is therefore evaluated from the interpolation of values at
neighboring grid points (weighting), whereas the value of in the right-hand side of Eq.(14)
at each grid point is evaluated from charges of neighboring particles (another weighting).
For these weighting processes to be accurate, a sufficient number of simulation particles
must be present in each grid cell --- thus the name "particle-in-cell." [8] A flow diagram
of an explicit PIC simulation is shown in Fig. 1. The dashed line denotes the flow path
for simulating collisionless plasmas.

6. Fig. 1 Flow diagram of an explicit PIC simulation. The dashed line denotes the flow path when there is no
collision (i.e., PIC only).
To model collisional corona plasmas and electron-impact reactions, the Monte Carlo
collision (MCC) algorithm with null collision method [9, 10] is used. For each time step,
the kinetic properties of particles (velocity, position, and kinetic energy) are available
from the PIC solver. With the known kinetic energies and the relative velocities to target
particles, the collision frequency can be calculated. This is the first step. The second
step is to determine whether the particle is to be scattered in the current time step. The
next step is to determine which scattering process occurs, using statistics and cross
sections for each reaction. The last step is to determine the new (scattered) velocity and
new particle velocities if ionization occurs. Ionizations create an ion-electron pair and
end up with new electron velocities. Fig. 2 shows the Monte Carlo collision method as
implemented.
Fig 2. Flow diagram of the Monte Carlo collision method
PIC-MCC methods are attractive since fundamental equations are solved with few
approximations, and kinetics of the simulated species can be accurately modeled,
including both local and non-local effects. The simulations provide detailed kinetic
information about the discharge characteristics (electrons and ions velocity distribution).
Therefore, PIC-MCC simulations are well suited for low-pressure discharges where non-
local effects can be very important. The major disadvantages of kinetic simulations are
they are computationally more demanding than fluid simulations.

7. Direct simulation Monte Carlo (DSMC) [11] is a well-established algorithm for
computing rarefied gas dynamics. Using the simulated particles, the DSMC method
samples the time evolution of the distribution function of a given system. The results are
computed in the form of averages over the samples obtained. The simulation domain is
discretized into cells for the purposes of collision sampling and calculation of mean flow
properties. During each time step, particles are first moved without interaction --- free
flight. After all particles have moved, a given number are randomly selected for
collisions. Rather than exactly calculating successive collisions, as in molecular
dynamics (MD) simulations, the DSMC method generates collisions stochastically with
scattering rates and post-collision velocity distributions determined from the kinetic
theory of a dilute gas. The method has been shown to provide accurate solutions to the
Boltzmann equation at the limit of large particle number and small cell size and time step.
The method has also been developed to simulate charged and neutral particle mixture and
plasma [12]. DSMC and MD simulations were used extensively in the development of
multi-scale modeling technology which will be discussed in Section 5.
4. Hybrid Model
Hybrid models are a combination of continuum simulations and kinetic simulations. The
idea behind hybrid models is to reduce the computational cost associated with kinetic
simulations but still retain some of their advantages. Depending on the physics/chemistry
to be modeled, various hybrid models can be developed. Electrons can be treated as fluid
whereas ions are kinetically simulated [12]. Alternatively, ions can be modeled as a fluid
while electrons are treated with a kinetic method. Or, only high-energy electrons are
tracked with a kinetic simulation while ions and low-energy electrons are treated as fluids.
Another hybrid procedure is to obtain the transport properties and reaction rate
coefficients by solving the Boltzmann equation with DSMC method, and then use the
obtained coefficients in a fluid model. Hybrid models are not discussed further here
because it is not going to be used in this work according to current thinking.
5. Multi-Scale Modeling
As mentioned above, one of the central issues in modeling the behavior of plasma
processes is the disparity in length and time scales. This is the same problem that is faced
when physics occurring at different time/length scales are included in the same model.
One approach to overcome this difficulty is to use the smallest time step required for
either of the processes. This approach captures every detail of the plasma but the
simulation time maybe prohibitively long.
Because of the strong interaction and coupling of the phenomena at different scales, a
sequential one-way coupling is not adequate. For the accuracy and efficiency of the
simulation, different methods are preferable for different scales. The way to resolve the
problem is to break it down into smaller pieces to separate the length and time scales.

8. Apply microscopic simulations only where it is required, use macroscopic methods for
the rest of the domain, and devise a communication scheme for the multiple regions
which are solved by completely different simulation approaches (for example, fluid and
particles). This is a type of multi-scale modeling [1]. It is emphasized that the interfaces
between the multiples regions (i.e., the particle and the fluid regions) have to locate at the
locations where both descriptions are valid.
The critical step in implementing a multi-scale model is the communication between the
macroscopic (fluid) and the microscopic (particle) parts of the model. Time step
requirements are in general quite different in the two simulations. At the communication
interface, the amounts of information from the two sides are very different as well. On
one side is the continuum region, and there are nN computational nodes, each of them
holds one degree of freedom (for the case of computing electron or ion number density).
On the other side of the interface is the particle region, and there are pN particles, each
of them holds six degrees of freedom (3 in position and 3 in velocity). Generally, pN is
much larger than nN for the statistics to be meaningful. The information transfer from
the particle region to the continuum is less problematic. An averaging procedure can be
applied to reduce the information from pN6 to nN degrees of freedom. However, in the
transfer from continuum to particle regions, pN6 degrees-of-freedom information has to
be derived based on nN degrees of freedom available. Fluctuation and randomness have
to be introduced consistently.
Literature survey has been conducted to understand the techniques of multi-scale
modeling and the state-of-the-art of continuum-particle coupling. The other purpose of
the survey was to find out what cases people had been doing, and whether there were
similar examples that could be learned or borrow from. In this section, the result of the
literature review is summarized.
The early works of continuum-particle coupling for fluids were mostly done in the field
of gas dynamics. Wadsworth and Erwin [13] studied one-dimensional (1-d) normal
shock in a monatomic gas flow. They used DSMC simulation in the large gradient region
in the hypersonic flow field where the flow was nonequilibrium. A continuum Navier-
Stokes equation solver was used in other region of the flow. The bridging of the two was
through a coupled iteration via a flux boundary condition.
The review shows there are generally two types of coupling techniques being used in
continuum-particle multi-scale modeling of fluids. One technique aims at matching the
fluxes of mass, momentum, and energy at the interface [13-21], and the other one is
based on the Schwarz alternating method (which is also used in domain decomposition
computing) over an overlap domain where the mass density, the mean velocity, and the
temperature are imposed [22-27]. Both the flux-based and Schwarz-based coupling
schemes have been used for gas and liquid systems. Some of the more representative
ones are summarized in the following:

9. Garcia et al. [15] developed a bridging solution that used Adaptive Mesh and Algorithm
Refinement (AMAR). They considered DSMC coupled to Navier-Stokes or Euler
equations for rarefied gas flow. A particle method (DSMC) was embedded within a
continuum method at the finest level of an adaptive mesh refinement (AMR) hierarchy.
The embedding of DSMC in continuum grid made the particle method appear just like
any level of refinement in the purely continuum case. The continuum and atomistic
representations were coupled by matching fluxes at the continuum-atomistic interfaces
and by proper averaging and interpolation of data between scales. Wijesinghe et al. [19]
generalized this method to include two-species gas and moving continuum-particle
interface. They demonstrated the capability by tracking a moving shock wave.
Flekkoy et al. [16,17] developed a flux-based coupling scheme for a liquid system. They
used it to couple a molecular dynamics (MD) simulation to a Navier-Stokes solver. The
MD used the Lennard-Jones potential. The approach was explicitly based on direct flux
exchange, and the relevant conservation laws were complied.
Hadjiconstantinou and Patera [22] and Hadjiconstantinou [23] introduced Schwarz
alternating scheme for multi-scale coupling. The Schwarz technique was inherently a
steady-state solution method. However, transient problems could be treated
quasistatically --- Schwarz iterations could be applied at each time step.
Hadjiconstantinou [24] considered a liquid system (dense fluid), and coupled MD
simulation to a Stokes equation solver. The MD used Lennard–Jones potential. Length
scale decoupling was achieved through the use of an overlap region across which the
continuum and molecular subdomains exchanged information.
Aktas and Aluru’s paper [25] considered coupling of DSMC with time-independent
Stokes equation continuum model using an overlapped decomposition. The multiscale
approach used was an overlapped Schwarz method with Dirichlet–Dirichlet type
boundary conditions. They used it to study the gas flow in a microfluidic filter.
In summary, the literature survey shows that models described in open literature have
considered DSMC-continuum or MD-continuum coupling exclusively. Most of the
studies considered simple gas or simple liquid (e.g., liquid argon). One paper considered
bridging DSMC to Euler equation for two-species gas. However, the gas is charge
neutral. The result implies that the field of multi-scale modeling is still at its infancy, and
a lot of further developments need to be done before it can become a matured technique.
Furthermore, the study also shows that, between the two methods that have been
developed for domain coupling, it seems that Schwarz method is easier to implement.
6. Uniqueness and Challenges of the Present Research
There are several unique features of the present research and each of them presents a
challenge that has to be overcome:
1. Most of the studies in open literature have considered coupling continuum and
particle regions for simple gas or simple liquid. In our case, we intend to couple a

10. region of multiple species of charged particles (electrons, positive and negative
ions) to its corresponding continuum fluid model.
2. Most examples available in literature have shown studies of coupling between
DSMC or MD simulations to continuum models. In our case, we intend to couple
PIC-MCC simulation to the corresponding plasma fluid model.
3. The long range electrostatic forces between the charged particles can complicate
the dynamics in the coupling buffer or the overlapping cells.
4. And, sure there will be many unforeseen challenges and obstacles.
7. Summary
This report describes a research project on developing multi-scale modeling/simulation
capability for micro-coronas. The research focuses on a type of multi-scale modeling
technique referred to as domain decomposition. The goal of the research is a corona
simulation model that spans from micro- to macro-scales.
As background information, different methods commonly used in plasma modeling are
reviewed. These include fluid model, particle model, and hybrid model. Both the fluid
and the particle models will be used in the present study. Specifically, the plan is to
apply the PIC-MCC simulation to the microscopic region and a fluid model to the
macroscopic region, and bridge up the two regions using a multi-scale technique at a
location where both methods are valid. The critical part for this kind of bridging is the
communication scheme between the two different regions of the same model that are
simulated with completely different methods.
A literature survey on the subject of multi-scale modeling has been conducted. The result
shows that two fundamental techniques have been developed for coupling a microscopic
description to a macroscopic continuum description. One method is based on matching
the fluxes at the interface of the two regions, and the other is based on Schwarz
alternating method and overlapping cells. The survey also shows that there are very
limited examples both on the simulation methods being coupled and the systems being
studied. There have to be a lot more studies and developments before multi-scale
modeling can become a matured technology.
As the research is at its early stage, this report represents our current thinking. It is meant
to be updated and extended as more information becomes available.
References
1. Weinan E, Xiantao Li, and Eric Vanden-Eijnden, “Some recent progress in
multiscale modeling,” in Multiscale Modelling and Simulation (S. Attinger and P.
Koumoutsakos eds.), LNCSE 39, pp. 3-22 (2004).

11. 2. H C Kim, F Iza, S S Yang, M Radmilovic-Radjenovic and J K Lee, “Particle and
fluid simulations of low-temperature plasma discharges: benchmarks and kinetic
effects,” J. Phys. D: Appl. Phys. 38, pp. R283-R301 (2005).
3. Jay P. Boris and David L. Book., “Flux-corrected transport: I. SHASTA, a fluid
transport algorithm that works,” J. Comput. Phys., 11, pp. 38-69 (1973).
4. D. L. Book, J. P. Boris, and K. Hain, “Flux-correct transport: II. Generalizations
of the method,” J. Comput. Phys., 18, pp. 248-283 (1975).
5. J. P. Boris and D. L. Book, "Flux-corrected transport: III. Minimal-error FCT
algorithms", J. Comput. Phys., 20, pp. 397-431 (1976).
6. D. Scharfetter and H. Gummel, “Large-signal analysis of a silicon read diode
oscillator,” IEEE Trans. Electron Devices, 16, pp. 64-77 (1969).
7. V. I. Kolobov and R. R. Arslanbekov, “Simulation of electron kinetics in gas
discharges,” IEEE Trans. Plasma Science, 34(3), pp. 895- 909 (2006).
8. J. G. Shaw, Xerox Particle Simulation Environment (XPSE), Xerox Corporation.
9. C. K. Birdsall, "Particle-in-cell charged-particle simulations, plus Monte-Carlo
collisions with neutral atoms, Pic-Mcc," IEEE Trans. on Plasma Science, 19(2),
pp. 65-85 (1991).
10. V. Vahedi and M. Surendra, "A Monte Carlo collision model for the particle-in-
cell method: applications to argon and oxygen discharges," Computer Physics
Communications, 87, pp.179-198 (1995).
11. A. Bird, Molecular Gas Dynamics (Oxford University Press, Oxford, 1976).
12. D. J. Economou, T. J. Bartel, R. S. Wise, and D. P. Lymberopoulos, “Two-
dimensional direct simulation Monte Carlo (DSMC) of reactive neutral and ion
flow in a high density plasma reactor,” IEEE Trans. Plasma Science, 23(4), pp.
581-590 (1995).
13. D. C. Wadsworth and D. A. Erwin, “One-dimensional hybrid continuum-particle
simulation approach for rarefied hypersonic flows,” AIAA-90-1690 (1990).
14. S. T. O’Connell and P. A. Thompson, “Molecular dynamics-continuum hybrid
computations: A tool for studying complex fluid flow,” Phys. Rev. E, 52, pp.
5792–5795 (1995).
15. A. L. Garcia, J. B. Bell, W. Y. Crutchfield, and B. J. Alder, “Adaptive mesh and
algorithm refinement using direct simulation Monte Carlo,” J. Comput. Phys. 154,
pp. 134-155 (1999).
16. E. G. Flekkoy, G. Wagner, J. Feder, “Hybrid model for combined particle and
continuum dynamics,” Europhys. Lett. 52, pp. 271–276 (2000).
17. E. G. Flekkoy, R. Delgado-Buscalioni, and P. V.Coveney, “Flux boundary
conditions in particle simulations,” Phys. Rev. E, 72, p. 26703 (2000).
18. R. Delgado-Buscalioni and P. V. Coveney, “Continuum-particle hybrid coupling
for mass, momentum, and energy transfers in unsteady fluid flow,” Phys. Rev. E,
67, p. 046704 (2003)
19. H. S. Wijesinghe, R. D. Hornung, A. L. Garcia, N. G. Hadjiconstantinou, “Three-
dimensional hybrid continuum-atomistic simulations for multiscale
hydrodynamics,” J. of Fluids Engineering, 126, pp. 768-777 (2004).
20. G. De Fabritiis, R. Delgado-Buscalioni, and P. V. Coveney, “Multiscale modeling
of liquids with molecular specificity,” Phys. Rev. Lett., 97, p. 134501 (2006).

12. 21. V. I. Kolobov, S. A. Bayyuk, R. R. Arslanbekov, V. V. Aristov, A. A. Frolova,
and S. A. Zabelok, “Construction of a unified continuum/kinetic solver for
aerodynamic problems,” J. of Spacecraft and Rockets, 42(4), pp. 598-606 (2005).
22. N. G Hadjiconstantinou and A. T. Patera, “Heterogeneous atomistic-continuum
representations for dense fluid systems,” Int. J. Mod. Phy. C, 8(4), pp. 967–76
(1997).
23. N. G. Hadjiconstantinou, “Combining atomistic and continuum simulations of
contact line motion,” Phys. Rev. E, 59(2), pp. 2475–78 (1999).
24. N. G. Hadjiconstantinou, “Hybrid atomistic-continuum formulations and the
moving contact-line problem,” J. Comput. Phys. 154, pp. 245–265 (1999).
25. O. Aktas and N. R. Aluru, “A combined continuum/DSMC technique for
multiscale analysis of microfluidic filters,” J. of Comput. Phys., 178, pp. 342–372
(2002).
26. X. B. Nie, S. Y.Chen, W. N. E. Robbins, and M. O. Robbins, “A continuum and
molecular dynamics hybrid method for micro- and nano-fluid flow,” J. Fluid
Mech., 55, pp. 55–64 (2004).
27. E. M. Kotsalis, J. H. Walther, and P. Koumoutsakos, “Control of density
fluctuations in atomistic-continuum simulations of dense liquids,” Phys. Rev. E,
76, p. 016709 (2007).

13. This document was created with the Win2PDF “print to PDF” printer available at
http://www.win2pdf.com
This version of Win2PDF 10 is for evaluation and non-commercial use only.
This page will not be added after purchasing Win2PDF.
http://www.win2pdf.com/purchase/