1

BRIDGING DIVERSE PHYSICAL SCALES WITH THE DISCRETE-

PARTICLE PARADIGM IN MODELLING COLLOIDAL DYNAMICS WITH MESOSCOPIC FEATURES

Witold Dzwinel1, David A.Yuen2 , Krzysztof Boryczko1,2,

1Institute of Computer Science, AGH University of Technology, Mickiewicza 30, 30-059 Kraków, Poland

dzwinel@uci.agh.edu.pl, boryczko@uci.agh.edu.pl 2Minnesota Supercomputing Institute, University of Minnesota, MN 55455, Minneapolis, USA

davey@msi.umn.edu

(submitted for publication to Chemical Eng. Sci. January 2003)

Abstract Mesoscopic features embedded within macroscopic phenomena in polymers, when coupled together with microstructural dynamics and boundary singularities, produce complex multiresolution patterns, which are difficult to capture with the continuum model using partial differential equations, for example the Navier-Stokes equation and the Cahn-Hillard equation. The continuum model must be augmented with discretized atomistic microscopic models, such as molecular dynamics (MD) in order to provide an effective solver across the diverse scales with different physics. The high degree of spatial and temporal disparities of this approach makes it a computationally demanding task. We can view the cross-scale endeavor characteristic of a multiresolution homogeneous particle model, as a manifestation of the interactions present in the discrete particle model, which allow them to produce the microscopic and macroscopic modes in the mesoscopic scale. First, we describe a d iscrete-particle model in which the following spatio -temporal scales are obtained by subsequent coarse-graining of hierarchical systems consisting of atoms, molecules, fluid particles and moving mesh nodes. We then show some examples of 2D and 3D modeling of phase separation, colloidal arrays and colloidal dynamics in the mesoscale by using fluid particles as the exemplary discretized model. The modeled multiresolution patterns look amazingly similar to those found in laboratory experiments and can mimic a single micelle, colloidal crystals, large-scale colloidal aggregates up to scales of hydrodynamic instabilities and the macroscopic phenomenon involving the clustering of red blood cells in capillaries. We can summarize the computationally homogeneous discrete particle model in the following hierarchical scheme: non-equilibrium molecular dynamics (NEMD), dissipative particle dynamics (DPD), fluid particle model (FPM), smoothed particle hydrodynamics (SPH) and thermodynamically consistent DPD coupled together with off-lattice techniques, such as lattice-Boltzmann gas (LBG). A powerful toolkit over the GRID can be formed from these discrete particle schemes to model successfully multiple-scale phenomena in biological vascular and mesoscopic porous-media systems. Keywords: discrete-particle methods, non-equilibrium molecular dynamics, dissipative particle dynamics, fluid particle model, smoothed particle hydrodynamics, colloidal dynamics, blood flow.

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1 Introduction The successes enjoyed by nanosciences in many fields [Nakano et al. 1998, 2001, Goddard, 1999, Chelikowsky, 2000, Grom et al., 2000, Banfield and Navrotsky , 2001, Rustad et al., 2001, Davis, 2002] have resulted in a need for adequate theory and large-scale numerical simulations in order to understand what the various roles played by surface effects, edge effects or bulk effects in nanomaterials. The dynamics of colloidal particle transport also call for not only passive transport, but additional processes such as agglomeration/dissociation, driven interfaces, adsorption to pore wall grains, and biofilm interactions [Albert et. al., 1998, Benson, 1998, Kechagia, 2001, Knutson and Travis, 2002, Davis, 2002]. In many cases, there is a dire need to investigate these multiscale structures, ranging from nanometers to micrometers, in complex geometries, such as in capillary or porous systems [Davis and Lobo, 1992, Li et al., 1999, Davis, 2002]. Cur rently the most cost-effective numerical approach, which is able to couple some of these processes and to capture their feed-back effects on the macroscropic flow field is the continuum multiscale multi-grid approaches [Hou and Wu, 1997, Moulton et., al., 1998], the lattice Boltzmann gas (LBG) (see e.g., [Travis et. al., 1993, Ladd and Verberg, 2001]) and particle/continuum hybrid algorithms [Hadjiconstantinou and Patera 1997, Nakano et al., 1998 ].

Capillary systems and porous media in the mesoscale can be viewed as two mutually interacting, complex constituents, namely the colloidal suspension and the porous material. The complexity of these components is connected mainly with the development of multiple spatio-temporal scales involved in a proper description of their physical, chemical and geometrical properties. In the case of colloidal suspension the multiple scales come from: 1. the spatial factor - the size of colloidal bead differing by a few orders of magnitude from the

sizes of solvent molecules, 2. dynamical factor - chemical reactions and thermal fluctuations occurring over time scales,

which are several orders of magnitude smaller than those associated with hydrodynamical flows.

The spatial character of the multiple scales dominates in capillary systems and porous material from their intermolecular attractive forces. The diameters of capillaries and micro-pores can vary from as small as nanomters to tens of micrometers [Davis, 2002]. The pores can be classified from micropores, of the order of nanomters, mesopores, from 2 nm to 60 nm and those above 60 nm are called macropores [Davis, 2002]. Another type of complexity is associated with the complex geometry of these systems consisting of bifurcating and branching capillaries with vastly different shape.

Because the phenomena occurring in porous material due to the flow of the colloidal suspension involve multiple spatio-temporal scales, porous media are extremely intricate physical systems impossible to model within a single numerical paradigm. Physical phenomena occurring in complex materials are usually described within hierarchical, multi- level numerical models. In these models each level is responsible for different spatio -temporal behavior and passes out the averaged parameters to the level, which is next in the hierarchy. In realistic cross-scale simulations, communication between the levels must be bidirectional. Critical phenomena occurring due to hydrodynamic instabilities and mixing (e.g., combustion, cement hardening) [Werner, et., al., 2001, Dzwinel et., al., 2000, Dzwinel and Yuen, 2002, Devaney, 2002] or fracture dynamics (crack propagation [Mathur et.,al., 1996, Holian and Ravelo, 1995, Abraham et. al., 1998, Vashishta and Nakano, 1998, Nakano et al., 2001] are well-documented examples of such complex cross-scale behavior.

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In Fig.1 we show a schematic diagram depicting the spatio-temporal, hierarchical nature of hybrid numerical model, which can be used effectively in the cross-scale modeling of flows of colloidal suspensions in porous media. In spite of its conceptual correctness, the methodological and computational disadvantage involved in this grandiose scheme is quite obvious. The hybridized model composed of heterogeneous mathematical and numerical concepts, such as molecular dynamics (MD) and finite elements (FEM) or finite differences (FDM) techniques, involve different discretized primitives (particles vs. finite elements) and different numerical approaches (summation of intermolecular forces and integration of ordinary differential equations vs. solving multi-dimensional and non-linear partial differential equations). Moreover, a large computa tional gap opens between these two approaches [Dzwinel et.,al., 1999]. The size of a sample in which microscopic effects can be averaged out by using continuum hydrodynamics is about hundreds of nanometers in the transition region between quantum mechanics and classical mechanics. For modeling such the system employing MD in three dimensions more than 109 particles have to be simulated in 105-106 timesteps [Beazley et. al., 1996, Vashishta and Nakano, 1999]. Matching continuum and MD models requires hundred of iterations per timestep. This means a lot of floating point operations per timestep, at least 1012.

Molecular Dynamics (MD)

FINER GRID RESOLUTION INCONSISTENCY?

LARGER NUMBER OF PARTICLES

SPATIAL SCALE [m]

TIME [s]

10-14

10-12

10-6

1

60

10-3

10- 9 10- 6 10-3 1

atoms ⇒ Schwarz procedure ⇒ grid

Finite Differences (FDM) & Finite Elements (FEM)

• chem

ical reactions •

microscopic rheological properties

• mesoscopic flows of colloidal suspension in a capillary

• mesoscopic rheological properties,

• viscoelasticity

• permeability • phase diagram • thermal

conductivity • other global

parameters

ab initio

Molecular Dynamics

• potentials • parameters of chem. reactions

Fig.1 Schematic diagram showing the applicability of various kinds of numerical models to study the diverse spatial and temporal scales associated with the multitudinous phenomena spanning over multiple scales in the modelling of colloidal dynamics and flows in porous media.

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Another computational task for modeling complex flows in porous media covering both the nanoscale, dissipative particle dynamics (DPD) [Hoogerbrugge and Koelman, 1992] and fluid particle method (FPM) [Espanol, 1998] can be treated within a common framework. This framework in the mesoscale consists in the successive coarse-graining of the underlying molecular dynamics system. As shown in Fig.2, both lattice Boltzmann gas (LBG) [Rothman, 1988, Chopard and Droz, 1998], direct simulation Mone-Carlo (DSMC) [Bird, 1994], and gridless particle methods such as dissipative particle dynamics (DPD) [Hoogerbrugge and Koelman, 1992] and fluid particle method (FPM) [Espanol, 1998] can be treated within a common methodological framework. We can list the components as : 1. common primitives- particles - defined by a set of attributes whose physical mining is

different depending which spatio -temporal scale is currently being considered, 2. local collision operator Ω - defined as a sum of short-ranged additive forces or collisions

between particles, 3. the rules of particle system evolution, which can be treated as a component of particle

dynamics. As shown in Fig.2, the methods can capture different scales from this general philosophy of successive coarse graining. Because the direct simulation Monte Carlo is used for modeling flows of rare gases and the systems defined by large Knudsen number, macroscopic scales can be reached by using relatively small number of particles. In the modeling of mesoscopic flows great success has been enjoyed by lattice gas and lattice Boltzmann gas (LBG), which were used among others in modeling colloidal suspensions [Swift et al, 1996; Stockman, 1990, 1997; Baudet et al 1989; Ladd, 2001; Flekkoy, 1993] and porous media (e.g., [Rothman, 1988; Chen et al 1991; Lutsko et al 1992; Chen and Lu, 1992; Travis et al, 1993; Martys and Chen, 1996; Manwart, et., al., 2002]). This success comes about from the computational simplicity of the method, which comes from coarse-grained discretization of both the space and time and drastic simplification of collision rules between particles. Today one can simulate by using LBG the scales up to milimeters by using grids with 107 collision nodes. The gridless particle methods, such as DPD and FPM, can capture easily mesoscopic scales of hundreds of micrometers employing up to 107 fluid particles currently, i.e., the scales in which temperature fluctuations and depletion forces interact with mesoscopic flows.

Therefore, gridless particle methods can mimick the complex dynamics of fluid particles in the mesoscale more realistically than LBG. The fluid particle methods save also computational time taken by molecular dynamics for calculating thermal noise. Instead, in dissipative particle dynamics and fluid particle method we introduce the random Brownian force.

Here we would recommend that one should use multi- level gridless particle approach for simulating mesoscopic phenomena, such as colloidal dynamics in complex geometries. These particles can be treated as fluid packets of the size reflecting with great fidelity the spatial scale of simulation. The particles interact with each other and their dynamics is governed by the Newtonian equations of motion. We show that by combining fluid particles with particles modelling solid colloid al beads we can couple together macroscopic hydrodynamical modes with microstructures emerging due to interactions between the beads and between the beads and fluid particles. Thus, unlike in MD-FEM and LBG approaches, we can also control more precisely both the micro-structural parameters and interactions between microscopic and macroscopic modes.

In the first section we present briefly the best candidates, which can be combined into a homogeneous, particle based, cross-scale numerical solver. We then discuss the equations describing interactions between particles and the issues for bridging different particle models

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both to the atomic scales and the macroscopic scales. We also demonstrate the results from DPD and FPM simulations of colloidal dynamics in the mesoscale involving the flow of red blood cells dynamics in capillary vessels, obtained by using dissipative particle dynamics and fluid particle model. In the last sectio n we discuss the possibility of extending the gridless particle model to larger scales, which will involve a hybridized scheme connecting it with the lattice Boltzmann gas method.

Fig.2. Methodological framework for discrete particle methods with a listing of the spatial scales able to be captured today and number of particles simulated on reasonably large shared-memory computer systems. 2 Discrete particle techniques From the standpoint of traditional fluid dynamics, a general problem in modeling of colloidal dispersion comes from a basic difficulty in defining physically consistent models, which can couple together continuum and discrete approaches. Continuum models, such as the Navier-Stokes equations for fluid mechanics or the Cahn-Hillard equation for metallurgical phase separation (e.g. [Pego, 1989]) and crystallization processes [Kloucek and Melara, 2002], which are usually based on simple conservation laws, represent "top-down" way of chemical-physical description of matter, i.e., from large to small length scales, and can be used successfully for simple Newtonian fluids and constrained crystals. For complex fluids with complicated physics such as two-phase flow and compaction [e.g. Mc Kenzie, 1984, Bercovici et., al., 2001],

6

however, equivalent phenomenological representations are usually unavailable and must be approximated by empirically derived constitutive relations and other empirical parameters, such as surface tension [Bercovici et., al., 2001], obtained from computationally complex direct numerical simulations (DNS) [Glowinski et., al, 2000] or heterogeneous models combining both continuum and discrete particle models (e.g., fluid particle dynamics (FPD) by [Tanaka and Araki, 2000]).

Conversely, the modeling approach can be based on the microscopic description working from the smallest scales upward along the general lines of the program for statistical mechanics. Discrete-particles techniques represent such an approach. The system of discrete particles is defined by a set of boundary and initial conditions and by interactions between particles represented by a collision operator Ω ij. The particle system evolves according to Newtonian equations of motion, which can be described in a discrete form as follows:

tm

tn

ini

nij

N

jij

ni ∆=∆∆⋅=∆ ∑ P

reOP , (1)

where r is the position of particle, P is its momentum and ∆t the integration timestep, N is the number of particles in the interaction range .

Non-equilibrium molecular dynamics (NEMD) simulations in which collision operator Ω ij is defined as a central, mostly short-ranged, conservative force have been used extensively in the past few years to study microscopic fluid instabilities, such as the Rayleigh-Taylor instability shown in Fig.3, and the rheology of fluids represented by models of varying complexity [Kroger, 1995, Vashishta and Nakano, 1999, Dzwinel et. al., 2000a]. Because large-scale NEMD simulation can bridge time scales dictated by fast modes of motion together with the slower modes, which determine the viscosity, it can capture the effects of varying molecular topology on fluid rheology resulting, e.g., from chemical reactions (see e.g. [Alda et. al., 2000]) or mixing [Pierrehumbert and Yang, 1993, Ten et al., 1997] with complicated velocity fields. We need billions or more of particles modelled in millions of timesteps in order to capture spatio -temporal scales occurring in porous systems filled with colloidal suspension and defined by the conditions of: 1. colloidal beads of sizes larger than 10 nm, 2. the material structures with pores or capillary diameters ranging from 1µm to millimeters, billions of particles simulated in millions of time-steps are required.

Fig.3 The snapshots from 2-D NEMD simulation of the Rayleigh-Taylor hydrodynamics instability using one million Lennard -Jones particles. The heavy fluid is shaded black the lighter fluid is white.

Mesoscopic regimes involving scales of the porous system exceeding 1 ns and 1 µm entail the fast modes of motion to be eliminated in favor of a coarse grain representation. At this level, the particles will represent clusters of atoms or molecules, so-called, dissipative particles [Flekkoy

7

and Coveney, 1999]. Flekkoy and Coveney (1999) have shown the feasibility to link and pass the averaged properties of molecular ensemble onto dissipative particles by using the bottom-up approach from molecular dynamics by means of a systematic coarse-graining procedure. The dissipative particles are represented by cells defined on the Voronoi lattice [see e.g., Braun and Sambridge, 1995, Augenbaum and Peskin, 1985] with variable masses and volumes (see Fig.4). The notion of the Voronoi cells allows for a very clear statement of the problem of coupling continuum equations and molecular dynamics. This is important when the continuum description breaks down due to complex molecular details in certain regions such as the contact line between two fluids and a solid, or the singularity of the tip in propagating fracture [Mathur et. al., 1996]. Entire representation of all the MD particles can be achieved in a general way by introducing a sampling function:

( ) ( )( )∑ −−

=−

ll

kkkf

rrrr

rrθ

θ (2)

where the positions rk and rl define the centers of dissipative particles, r is an arbitrary position, and θ(r) is the Gaussian function. The mass, momentum, and internal energy Ek of the kth dissipative particle are then defined as:

( ) ( )∑∑ ==i

iikki

ikk mffM vrPr (3)

( ) ∑∑ ∑ ≡

+=+

≠iik

i ijijMD

iikk

kk frm

fEUM

εφ )(21

2)(

2

22

rv

r (4)

where vi is the velocity of ith MD particle having identical masses m, Pk is the momentum of the kth dissipative particle, and φMD (rij) is the potential energy of the MD particle pair i,j separated by a distance rij. The particle energy ε i contains both the kinetic term and a potential term. In order to derive the equations of motion for dissipative particles the time derivatives of Eqs.(4) must be resolved [Flekkoy and Coveney, 1999]. Finally, after averaging over the velocities, masses and interactions on the Voronoi lattice, we can obtain the following:

( )[ ] ∑∑∑ +

⋅++−

++=

lkl

lklklklkl

klkl

klkl

l

lkklk

k

rp

LMMdt

dFeeUUe

UUg

P ~22

η& (5)

where Uk(l) - velocity of dissipative particles k,l; pkl

- a pressure term between k and l dissipative particles resulting from conservative MD interactions; Lkl - a parameter of the Voronoi lattice; η - the dynamic viscosity of the MD ensemble and the last summation symbolizes the fluctuations of the coarse-grained representation. This procedure links all the forces between the dissipative particles to a hydrodynamic description of the underlying molecular dynamics atoms. The method may be used to deal with situations such as this shown in Fig.3 in which several different dynamical length scales are simultaneously present. To increase the computational efficiency, the Voronoi cells can be approximated by spheres (see Fig.4). Using additional simplifications, such as the unification of dissipative particle sizes, we can arrive at a model which converges to the dissipative particle dynamics (DPD).

In dissipative particle dynamics [Hoogerbrugge and Koelman,1992] the two-body interactions between two fluid particles i and j are assumed to be central and short-ranged. This can be defined as a sum of a conservative force FC, dissipative component FD and the Brownian

8

force FB. The Brownian factor represents the thermal fluctuations averaged out due to coarse graining procedure. The equations below show the basic formula describing the interparticle forces.

( ) ( ) ( ) ( )

( )

( )( )( ) cutijijij

cutijBDCijij

ij

cut

ij

cut

ij

ijijij

BijijijijDijijC

rrr

rrr

densityparticlenumber- random n,-

r

r

rnr

rt

rMr

>=Ω

<++=Ω

−∈

−

⋅=

⋅⋅∆

⋅=⋅⋅⋅⋅=⋅⋅=

for 0,

for ,

11

1 3

, ,

2

2

p

FFFp

eFeveFeF

θ

πω

ωθσ

ωγωπ o

(6)

The cut-off radius rcut is defined arbitrarily and reveals the range of interaction between the fluid particles. DPD model with longer cut-off radius reproduces better dynamical properties of realistic fluids expressed in terms of velocity correlation function [Espanol and Serrano, 1999]. Simultaneously, for a shorter cut-off radius, the efficiency of DPD codes increases as O(1/rcut

3), which allows for more precise computation of thermodynamic properties of the particle system from statistical mechanics point of view. A strong background drawn from statistical mechanics has been provided to DPD [Marsh, et. al, 1997, Espanol, 1998, Espanol and Serrano, 1999] from which explicit formulas for transport coefficients in terms of the particle interactions can be derived. The kinetic theory for standard hydrodynamic behavior in the DPD model was developed by Marsh et al. [Marsh, et. al., 1997] for the low friction (small value of γ in Eq.6), low density case and vanishing conservative interactions FC. In this weak scattering theory, the interactions between the dissipative particles produce only small deflections.

The strong scattering theory, where the friction between the DPD particles is large (but still for π→0 in Eq.6), was considered by Masters and Warren [Masters and Warren, 1999]. In that paper, the Fokker–Planck–Boltzmann equation has been replaced by the Boltzmann equation with a finite scattering cross section. For large friction, in the collective regime, the dynamics is controlled by mode coupling effects by including an internal energy variable such that total energy becomes a conserved quantity. Comparing Fig.5 and Fig.3, one can conclude that the DPD simulations display clearly the collective effects reflected by phenomena such as droplets formation [Cohen et., al., 1999] and fingering instabilities [Chen and Ortoleva, 1990, Dzwinel and Yuen, 2001].

There exist several other methods, which generalize the dissipative particle technique, such as:

1. the fluid particle model (FPM), 2. the bottom-up atomistic approach – coarse graining procedure replacing atoms interacting

with conservative and central forces with clusters of atoms interacting with dissipative forces [Serrano and Español, 2002],

3. the top-down thermodynamically consistent approach – the fragments of continuum medium is approximated by fluid particles.

Fluid particle model [Español, 1998] finds itself situated between classical dissipative particle dynamics and new formulations of DPD: bottom-up approach from the tiny to the large length

9

scales, by Flekkoy and Coveney [Flekkoy and Coveney, 1999] and the top-down method from the large to small scales [Serrano and Español, 2002].

colloidal bead

dissipative particle

BOTTOM-UP APPROACH

MD – particles creating Voronoi clusters

Fig.4 The schematics of atomistic bottom-up coarse-graining paradigm by [Flekkoy and Coveney, 1999]. The Voronoi cell contains particles which are closest to the cell center. The centers of all the Voronoi cells correspond to the centers of coarse-grained representation of the system - dissipative particles - which can be approximated by spheres One drawback of DPD is the absence of a drag force between the central particle and the second one orbiting about the first particle. To eliminate this side effect a greater number of neighbors can be considered in calculating forces, which decreases the computational efficiency. Unlike the classical formulation of dissipative particle dynamics method, fluid particles interact via additional non-central forces, producing a drag between circumventing particles. This saves computational time spent for searching of the neighboring particles. Fluid particle model (FPM) represents generalized smoothed particle hydrodynamics (SPH) method for which, unlike in SPH, angular momentum is conserved exactly. The fluid particles are represented by their centers of mass, which posses several attributes, such as the mass mi, position ri, translational and angular velocities and a type. These “droplets” interact with each other by two-body forces dependent on the type of particles. This type of interaction involves a sum of: 1. the repulsive conservative force FC. 2. the frictional forces FT and FR (translational and rotational), proportional to the relative

velocities of the particles, 3. the Brownian force F~ representing the microscopic degrees of freedom smaller than the

mesoscopic scales, which have been previously eliminated in the coarse grained mesoscopic model,

The equations for forces are becoming more complex than for the classical DPD technique and are as follows:

10

ijR

ijTij

Cijij FFFFF

~+++=

( ) ijijC

ij rV eF ⋅−=

ijijTij m vTF •⋅−= γ

( )

+ו⋅−= jiijij

Rij m ??rTF

21

γ

( ) ( ) ( ) [ ] ( ) ijA

ijijijS

ijijBij drCdtrD

rBdrAmTkdt eW1WWF •

++⋅=

~1~~2

~ 2/1γ

( ) ( ) ijijijijij rBrA ee1T += (7)

where ω is the angular velocity and W are random matrices consisted of independent Wiener increments weighted by scalar weighting functions A(rij), B(rij), Ã(rij), B(rij), C(rij) defined in [Español, 1998]. In a similar fashion as in DPD the particle system yields the Gibbs distribution as the steady-state solution to the Fokker-Planck-Boltzmann equation. The fluid particle model satisfies the fluctuation dissipation theorem [Marsh et., al., 1997], which defines the relationship between the normalized weight functions under the condition of detailed balance, i.e.,

mTkB ⋅= γσ 22 (8)

where: T – is the temperature of particle system, kB – the Boltzmann constant and σ,γ are the scaling factors of both the random and dissipative forces. The temporal evolution of the FPM particle i is described by the Newtonian laws of motion:

( )∑<

=cutij rrj

iiiiji;

,, ωvrFP& , ii vr =& , ( )∑<

=cutij rrj

iiiiji

i I ;

,,1 ωω vrN& , ijijij FrN ×−=21 (9)

Because the particles have nonzero angular momenta, one can investigate the positive feedback effects of elastic interactions such as those exerted by red blood cells on plasma flow displayed in Fig.6 [Dzwinel et. al., 2003, Boryczko et., al., 2003].

In the macroscale the particles can be also considered as moving mesh nodes, as it is so conceived in smoothed particle hydrodynamics (SPH) [Libersky et. al., 1994] and moving particles semi- implicit method (MPS) [Koshizuka and Ikeda, 1999]. The entire system is defined by mass mj distribution in space, where fluid density in r is given by the following approximation formula:

( ) ( )∑=

−=iN

jjijji hWm

1

,rrrρ (10)

where W(rij, h) is Gaussian weighting function and h defines a cut-off radius rcut for which if |rij|>rcut ⇒W()=0.

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Fig.5 Droplets formation accompanying the fingering instability (falling sheet) and the Rayleigh-Taylor mixing for two superimposed immiscible DPD fluids (right). Both snapshots come from 2D dissipative particle dynamics simulations with 105 particles.

Fig.6 A snapshot from FPM simulation of red blood cells flowing through a choking point. The elastic blood cells were made of particles interacting by harmonic forces [Dzwinel et., al., 2003, Boryczko et., al., 2003]. Periodic boundary conditions were applied for the particle ensemble consisting of 4 million particles. Only the fastest plasma particles are displayed. The zoom-in fragment of the picture shows the difference in length scale between the red blood cell and fluid particles

From the principles of approximation [Libersky et. al., 1994] and statistical mechanics we can show that for a continuum function f() the spatial average of this function and its gradient are defined respectively as:

12

( ) ( ) ( ) ( )i

ii

N

ii

i

ii

N

ii

mhWff

mhWff

ρρ,)( ;,)(

11

rrrrrrrr −∇⋅=∇−⋅= ∑∑==

(11)

This spatial averaging over different length-scales transforms the partial differential equations of continuum hydrodynamics into a set of non-linear ordinary differential equations for the particle velocity v and internal energy ε :

( ) ( ) ( ) , ,,1

21

22hWm

pdt

dhW

ppm

dtd

ji

N

jjij

i

iiji

N

j j

j

i

ij

iii

rrvvrrv

−∇−−=−∇⋅

+−= ∑∑

−−

oρ

ερρ

(12)

where Ni is the number of particles in O(ri, rcut) sphere. The first equation can be interpreted as the equation of motion for a set of nodes i and j in which transient pressures are pi and pj,,

respectively and which “interactions” are derived in a canonical manner from the force laws of continuum hydrodynamics. The evolution of the particle system is then described by the Newtonian equations of motion. Therefore, one can employ the same algorithms and codes as in molecular dynamics in modeling fluids over macroscopic scales by SPH. In Fig.7 we display the snapshots from modeling of the Rayleigh-Taylor hydrodynamic instabilities using smoothed particle hydrodynamics in which 105 smoothed particles were modeled in about 3000 timesteps.

Although originally designed for astrophysical problems [Monaghan, 1992], as shown in [Ellero et al, 2002], SPH can also be used for modeling polymers in the macroscale. However, smoothed particle hydrodynamics does not include thermal fluctuations in the form of a random stress tensor and heat flux as in the Landau and Lifshitz theory of hydrodynamic fluctuations. Therefore, the validity of SPH to the study of complex fluids is problema tic at scales where thermal fluctuations are important. In order to bridge mesoscopic scales with the scales described by SPH technique, Serrano and Espanol [Serrano and Espanol, 2002] propose a new version of dissipative particle dynamics - the thermodynamically consistent DPD (TC-DPD). The TC-DPD method at present represents a superset for the classical dissipative particle dynamics, fluid particle model and smoothed particle dynamics models.

The main features of this new approach in comparison to the classical DPD are that: 1. apart from position ri and velocity vi, the volume ν i (or density ρi), temperature Ti and

entropy Si of the particle Xi are the relevant dynamical variable thus ( )iiiiii STX ,,,, νvr≡ and

( ) ( ) ijijiji

ij

iji rWrW ee ′=== ∑ ~ ,1

,ρ

νρ

where W() is a Gaussian weighting function. 2. the forces are given in terms of discrete versions of the gradient of the stress tensor.

This approach involves finite volume Lagrangian discretization of the continuum equations of hydrodynamics through the Voronoi tessellation technique shown in Fig.7.

Finally, we can describe the evolution of the fluid particle system by the reversible (SPH)

∑

+−=

j i

i

j

jiji

pp22

~ ρρ

eP& (13)

and irreversible terms for momentum P* and entropy S*

13

( )( ) ( ) ( )∑∑∑ =−+⋅−=j

ijijjiiiij

ijijijijj

jiiji TTfSSdtd

TTT evFeevP ~,,ˆ,,~

,~~~,

ˆ1 ** νγν

& (14)

whereν - average volume of i and j particles, γij – viscosity, S~

,~F - are random factors for the force and entropy. The temperature Ti, the chemical potential and the pressure for each of fluid particles can be computed as corresponding partial derivatives of internal energy function ε(S*,N,V) which form can be approximated by the fundamental equation of ideal gas (as it is in [Serrano and Espanol, 2002]) or derived experimentally for a realistic fluid.

The formalism of thermodynamically consistent dissipative particle dynamics (TC-DPD) represents a consistent discrete model for the Lagrangian fluctuating hydrodynamics. Equations (13-14) conserve the mass, momentum, energy and volume. The irreversible (produced) entropy S* is a strictly increasing function of time in the absence of fluctuations. Thermal fluctuations represented by S

~,~F are consistently included, which lead to an increase of the entropy and to

correct for the Einstein distribution function [Serrano and Espanol, 2002]. Viscous forces between a pair of fluid particles depend not only on the velocities of this

pair but also on the velocities of the nearest neighbors. Therefore, we need to retain a great deal of information about the fluid state. Moreover, unlike in classical dissipative particle dynamics, thermodynamically consistent DPD method adaptatively controls the spatio -temporal scale of the model. The transport properties of thermodynamically consistent DPD fluid are known a priori. The size of the thermal fluctuations is given by the typical size of the volumes of the particles, scaling as the square root of this volume. The need to incorporate thermal fluctuations in a particular system will be determined by the external length scales that need to be resolved. In the case of submicron colloidal particles, we require to resolve the size of this particle with fluid particles of size an order of magnitude or two smaller than the diameter of the colloidal particle. For these small volumes, fluctuations are important and lead to the Brownian motion. The grains, one order of magnitude greater, require fluid particles to be much larger, in which case thermal fluctuations are small or negligible.

Fig.7 The snapshots from a simulation using smoothed particle hydrodynamics of the Rayleigh-Taylor instability. The size of the box is about 1 cm. The dark grey system represents fluid which is 10 times heavier than the lighter one placed in the bottom (invisible in this figure). About 105 SPH particles have been modeled.

14

This thermodynamically consistent DPD represents a truly multiscale discrete-particle model, which shows its true ability by the effect of switching off or on thermal fluctuations depending on the size of the fluid particles. This phenomenon is demonstrated in Fig.9 for phase separation process. By switching the fluctuation off, the lamellar structures are generated. Otherwise, only irregular patterns are produced.

Colloidal bead

Dissipative particle

TOP-DOWN APPROACH

Finite Volume - contiuum description.

Fig.8 The schematics of coarse-graining idea realized by the top-down thermodynamically consistent DPD model. The Voronoi cell represents a fragment of continuum fluid. This fragment can be treated as a dissipative particle with variable mass, volume, temperature and entropy. The thermodynamically consistent DPD particles interact with forces dependent not only on their positions and velocities but on the current thermodynamic states of interacting particles as well. This hierarchy of gridless particle methods is presented in Fig.10. The methods establish a sound foundation for cross-scale computations, ranging from nanometers to centimeters. They can provide a framework to study the interactions between microstruc tures and large-scale flow, which are of value in blood flow [Dzwinel, et. al. 2002b, Boryczko et. al., 2002] and other applications in polymeric and blood dynamics.

We should emphasize strongly here that molecular dynamics forms the centerpiece from which the other techniques are derived and are employed for attaining greater length scales. Therefore, the numerical models of the particle methods having similar framework are very interesting for modeling multiscale phenomena. As shown in [Dzwinel et. al., 1999, 2000b, Boryczko et. al., 2002], by generalization of the well known numerical MD models onto mesoscopic scales, one can solve many technical problems by correctly implementing other discrete particle methods.

Back 4 years ago [Dzwinel et. al, 1999] we have already outlined a computational paradigm of a hierarchical multiscale model employing discrete-particles and in [Boryczko et. al., 2002] we describe concrete parallel implementation of the fluid particle model for simulating complex with many constituents fluids in the mesoscale. Within this computational framework, the colloidal mixture can be made of fluid particles of uniform or various types and more than two kinds of particles can be studied. The particles can be perfectly mixed initially, or separated by a sharp interface from solid particles representing colloidal beads with arbitrary shapes. In the

15

following section we demonstrate some interesting results in colloidal dynamics obtained by using this particle code.

Fig.9 Phase separation realized by a thermodynamically consistent DPD model designed especially for binary systems with thermal fluctuations switched on a) and off b) (cross-section is shown to display lamellar structures )

GGGRRRIIIDDDLLLEEESSSSSS PPPAAARRRTTTIIICCCLLLEEE MMMEEETTTHHHOOODDDSSS

conservative interactions

MMDD

DDPPDD + dissipative and Brownian forces

FFPPMM + particles rotation, non-central forces

SSPPHH Regularized tensor interactions

AT

OM

S or

CL

UST

ER

S OF A

TO

MS

FL

UID

PA

RT

ICL

ES-V

OL

UM

EL

ESS

MO

VIN

G M

ESH

NO

DE

S

+ variable mass and particle volume non-isothermal model

TTCC-- DDPPDD FLU

ID PA

RT

ICL

ES O

N V

OR

ON

OI L

AT

TIC

E

10-9 10-6 10-3 10-1 Spatial scale (meters)

Fig.10 Schematic diagram illustrating the various hierarchies of the gridless particle methods and their resolved length-scales, spanning from nanometers to macroscopic dimensions

16

3 The role played by fluid particles in modelling colloidal dynamics DPD and FPM models have both attracted a great deal of attention mainly from the chemical and engineering modeling community. As demonstrated already in [Coveney and Novik, 1996, Groot and Warren, 1997, Clark et. al., 2000, Dzwinel and Yuen, 1999, 2000a,b,c, Rustad et. al., 2001, Dzwinel et. al. 2002a] by changing just the nature of the conservative interactions between the fluid particles and by introducing also larger solid particles, one can easily model the different dynamics of colloids, micelles, colloidal crystals and aggregates.

In Fig.11 we show explicitly the way the solid particles (colloidal beads) interact with dissipative particles (solvent) results in creation of various micellar structures. We define the collision operator as follows:

( ) ( ) ( ) ( )

−

∆

+⋅⋅−⋅

=Ω particle colloidal or if 2

24

particlessolvent and if ~

126

121

lkrrr

lkrt

rMr

nij

kln

ij

kln

ij

kl

nij

ijklnij

nij

nijikl

nijkl

nij σσε

ωθδ

ωγωπ ve o

(15)

Depending on the ratio ? of the depths of the potential wells εsolvent-solvent and εsolvent-colloid, one can observe the emergence of lamellar, hydrophobic the hydrophilic colloidal arrays or coexistence of the two phases (Fig.11a). For other physical parameters, the micelles produce fractal- like colloidal aggregates with distinct multi- resolutional structure (Fig.11b).

The appearance of similar multiscalar structures can be observed in Fig.11a along two phases interface due to small-scale mixing. This nucleation results in rapid changes in fragmentation speed. As shown in Figs.12b,c and in Fig.13 in different type of solid-liquid flows (characterized with different kind of particle-particle interactions εSP-DP) we can easily recognize the characteristic dispersion structures caused by the microstructural dynamics including phenomena such as rupture and erosion (Figs.12b,13), shatter (Fig.12c) and agglomeration [Dzwinel and Yuen, 2002]. The colloidal particles create long streaks, which shrinks for less vigorous flow. As shown in Fig.13, the characteristic streaks are formed also in 3D particle systems due to accelerated shear flow. All of these dispersion phenomena would be very difficult to model, using partial differential equations such as the Cahn-Hillard equation.

We can model flows in the presence of larger, elastic microstructures besides systems consisted of solvent and colloidal particles of similar sizes. They can be built on many particles linked together with harmonic interactions. As shown in [Dzwinel, et., al., 2002b, Boryczko et., al., 2002c], red blood cells (RBC) flowing in FPM fluid can be modeled in such a way. Due to spatio-temporal scale, which can be captured by DPD, the modeling can be performed for a single capillary (see Fig.13) of different shape with fixed, elastic and moving walls such as we have employed in simulating blood flow in capillary vessels [Dzwinel, et, al 2002b, Boryczko et al, 2002c]. We can study many clotting factors in blood flow such as aggregation of RBCs due to hydrodynamic and depletion forces and different geometry of capillary channels. Larger systems consisting of many capillaries can also be studied within the discrete-particle model in which the very notion of particle must be redefined, such as defined in thermodynamically consistent DPD model [Serrano and Espanol, 2002].

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Fig.11 Colloidal arrays and colloidal aggregates made up of micelles. Micelles and arrays can self-organize and emerge spontaneously from the random mixture of solid and fluid particles . In Fig a) we display the arrays produced by the particle system for which ?=εsolvent -solvent/εsolvent-colloid = 4.5. Conversely, in Fig b) ?=16

Extracting multiresolution features from the results generated by continuum methods involves modern feature-extraction techniques based on second-generation wavelets (see [Erlebacher et. al., 2001, Yuen et. al., 2002, Thompson et al., 2002]). The raw data produced by particle codes comprise positions and velocities of particles. The detection and visualization of multiresolution patterns created due to flow are crucial tasks for understanding complex flows in vascular and porous media. Fast algorithms and codes for the analysis and detection of microscopic coherent structures such as aggregates, clusters, droplets, etc. have to be constructed. In the case of out-of-core data mining, we propose to combine parallel clustering procedures - similar to these described in [Faber, 1994, Boryczko et al, 2002b] - with wavelet codes. The goal of cluster extraction is to collect statistical knowledge about micro-structural properties of complex systems at various spatial resolutions. This knowledge could be also used for bridging scales in subsequent coarse graining procedures such as: NEMD-DPD-SPH. As shown in Fig.14, we have used modern visualization packages such as Amira [www.amiravis.com, 2000] together with large display visualization systems (Power Wall) at the University of Minnesota (htpp://www.lcse.umn.edu).

18

Fig.12 Snapshots from simulations of dispersion of colloidal slab made of solid particles in DPD and FPM fluids in 2D (a,b) and 3D (c). The g ravitational acceleration is directed downward. We can observe crystallization process along the mixing front (a) for a properly defined interparticle force.

19

-G

G

DPD fluid

Cubic slab made of colloidal

particles

DPD fluid

x

y z

Fig.13 The particle system, which models the dispersion of a cubic colloidal slab positioned at the interface of counter flow sketched in a). The flows are accelerated in both directions. The colloidal slab is made of colloidal particles (about 105 particles) and fluid consists of 106 DPD particles (invisible). The color of colloidal particle represents particle velocity. The red color indicates the largest velocity of particles flowing to the right, the blue color corresponds to the largest velocity of particles flowing to the left. Figures b) and c) represent the projection of the colloidal particles after 2000 timesteps on x-y and x-z plains, respectively. In figure d) the break-up instant (after 3000 timesteps) is displayed.

20

Fig.14 Clot formation in blood flow simulated by using discrete-particles in bifurcating capillary vessels

Fig.15 Blood flow in a curved capillary visualized on the Power-Wall at the University of Minnesota (http://www.lcse.umn.edu).

21

4 Concluding remarks In recent years, new discrete particle methods have been developed for modeling physical and chemical phenomena occurring in the mesoscale. The most popular are grid-type techniques such as: cellular automata (CA), lattice gas (LG), lattice-Boltzmann gas (LBG), diffusion and reaction limited aggregation [Chopard and Droz, 1998] and stochastic gridless methods, e.g., direct simulation Monte-Carlo (DSMC) used for modeling systems characterized by a large Knudsen number [Bird, 1994] and stochastic rotation dynamics (SRD) [Ihle and Kroll, 2003]. Unlike direct simulation Monte-Carlo, in SRD collisions are modeled by simultaneous stochastic rotation of the relative velocities of every particle in each cell.

Gridless discrete-particle methods have a few significant advantages over grid techniques. These advantages can be enumerated as follows:

1. In contrast to methods based on spatial discretization of partial differential operators, the dynamics of fluid particles develop over continuum space in real time, thus allowing for realistic visualization and statistical analysis, such as clustering.

2. Within the context of cross-scaling systems , they are homogeneous with both microscopic molecular dynamics and macroscopic smoothed particle hydrodynamics techniques. From a numerical standpoint we do not need to switch over from particles to a static grid with control parameters, such as the Knudsen number.

3. The methods employing fluid particles are also homogeneous for solid-liquid simulations in which both solid and liquid can be represented by particles.

4. Many types of complex boundary conditions such as a free surface and complex porous structure can be easily implemented.

5. Particle methods are homogeneous from implementation point of view. Well-known sequential and parallel algorithms from MD and FPM simulations can be employed directly without any complications.

On the other hand , the LBG method can capture both mesoscopic and macroscopic scales even larger than those that can be modeled by discrete particle methods. This advantage is due to computational simplicity of the method, which comes from coarse-grained discretization of both the space and time and drastic simplification of collision rules between particles. We can look at the validity of these simplifications by comparing them with more realistic discrete-particles simulation. We regard both DPD and LBG as being complementary computational tools for modeling the slow dynamics in porous media over wide spatio-temporal scales.

The aggregated DPD/LBG model has many other powerful features, which are lacking in the two models, when taken separately.

1. MD, DPD and LBG methods together can capture both microscopic and macroscopic scales. 2. The common mesoscopic scale of confrontation of the two methods allows for more precise

scales bridging and adjust more precisely the rheological parameters for both systems. 3. Both DPD and LBG methods are homogeneous within the context of solid- liquid simulations,

i.e., both solid and liquid can be represented by particles. 4. They are also homogeneous in terms of implementing on a parallel computing, i.e., similar

parallel algorithms based on geometrical decomposition and load-balancing schemes can be employed.

Large scale simulations must be carried out to model multiresolution structures ranging from millimeters to micrometers emerging in vascular systems and porous media. To obtain

22

satisfactory resolutions we have to use at least 10 million DPD particles and huge LBG meshes with 108-109 sites. These large-scale modeling requires not only high-performance multiprocessor systems and fast parallel codes. These calculations efforts several TBytes of data, which must be stored, co- and post-processed, analyzed and visualized. Additionally, to realize a complete cross-scale computational system, which combines large-scale computations, mass storage, data processing and visualization, simultaneously making it user friendly and remotely accessible, a new system-user interface and data flow organization has to be implemented, such as in a GRID system (see e.g., [Foster and Kesselman, 1998, Fox, 2002]). Grid computing - and the software that enables it – is absolutely essential to building a cyberinfrastructure that will help to improve the usefulness of mesoscopic modelling.

Therefore, in the not-too-distant future with grid computing coming to the fore, realistic large-scale and cross-scale simulations will be really affordable and intricate tasks from both conceptual and computational points of view. These two factors cannot be considered separately for hybridized cross-scale models. Conceptual structure of these models has to be easily mapped into a modern computational environment, which is currently based on distributed and shared-memory computational resources, object oriented and component way of programming. The discrete-particle approach meets all of these requirements under the aegis of GRID computing.

Acknowledgments

We thank discussion with Dr. Gordon Erlebacher, Jim Rustad and Geoffrey C. Fox concerning computational environments. Support for this work has come from the Polish Committee for Scientific Research (KBN) project 4T11F02022, the Complex Fluid Program of U.S. Department of Energy and from AGH Institute of Computer Science internal funds. References 1. Abraham, F., Broughton, J., Q., Bernstein, N., Kaxiras, E., Spanning the Length Scales in

Dynamic Simulation“, Computers in Physics, 12/6, 538-546, 1998. 2. Albert, R., Barabasi, A-L, Carle, N., Dougherty A., Driven Interfaces in Disordered Media:

Determination of Universality Classes from Experimental Data, Phys.Rev.Let , 81, 14, 2926-29, 1998

3. Alda, W., Yuen, D.A., Luthi, H.P., Rustad, J.,R., Exothermic and Endothermic Chemical Reactions Modeled With Molecular Dynamics, Physica D, 146, 261-274, 2000

4. Amira v. 2.3 - Advanced 3D Visualization and Volume Modeling, http://www.amiravis.com. 5. Augenbaum, J.M. and Peskin , C.S., On the construction of the Voronoi mesh on a sphere,

J. Comput. Phys. 14, 177-198, 1985. 6. Banfield ., J.F. and Navrotsky, A., Nanoparticles and the Environment, Rev. Mineral.

Geochem. 44, Mineral. Soc. Am., Washington,D.C., 349, 2001. 7. Baudet, C., Hulin, J.P., Lallemand, P., d’Humieres, D., Lattice-gas automata: a model for the

simulations of dispersed phenomena, Phys. Fluids A, 1, 507-512, 1989 8. Beazley, D.,M., Lomdahl, P.,S., Gronbech-Jansen, N., Giles, R., Tomayo, P., Parallel

Algorithms for Short Range Molecular Dynamics, in World Scientific’s Annual Reviews of Computational Physics III, World Scientific, 119-175, 1996

9. Benson, D.A. The Fractional Advection-Dispersion Equation: Development and Application. PhD dissertation, U. Nevada-Reno, 1998.

23

10. Bercovici, D., Ricard, Y. and G. Schubert, A two-phase model of compaction and damage. I. general theory, J. Geophys. Res., 101(B5) 8887-8906, 2001.

11. Bird G A., Molecular Dynamics and the Direct Simulation of Gas Flow, Oxford Science Publications: Oxford, 1994

12. Boryczko, K., Dzwinel, W., Yuen D., A., Parallel Implementation of the Fluid Particle Model for Simulating Complex Fluids in the Mesoscale, Concurrency and Computation: Practice and Experience, 14, 137-161, 2002a

13. Boryczko, K., Dzwinel, W., Yuen D., A., Clustering revealed in high-resolution simulations and visualization of multi-resolution features in fluid-particle model, Concurrency and Computation: Practice and Experience, 14, 16 pages 2002b (in press).

14. Boryczko, K., Dzwine l, W., Yuen D.,A., Dynamical clustering of red blood cells in capillary vessels, Journal of Molecular Modeling, 2003 (in press)

15. Braun, J. and Sambridge M., A numerical method for solving partial differential equations on highly evolving grids, Nature, 376, 655 - 660, 1995.

16. Brezany, P., Bubak, M., Malawski, M., Zajac, K., Large-Scale Scientific Irregular Computing on Clusters and Grids, in: Sloot, P.M.A. Kenneth Tan, C.J., Dongara, J.J., Hoekstra, A.G. (Eds.), Proc. Int. Conf. Computational Science - ICCS, Amsterdam, April 21-24, vol. I, Lecture Notes in Computer Science, no. 2330, Springer, 2002, pp. 484-493.

17. Chelikowsky, J. R., The pseudopotential-density functional method (PDFM) applied to nanostructures, J. Phys. D, 33, R33-R50, 2000.

18. Chen, W. and Ortoleva, P., Reaction front fingering in carbonate- cemented sandstones, Earth Science Reviews, 29, 183-198, 1990.

19. Chopard B. and Droz, M., Cellular Automata Modelling of Physical System, Cambridge University Press, Cambridge, 1998

20. Clark, A.T., Lal, M., Ruddock, J.,N., Warren, P.,B., Mesoscopic Simulation of Drops in Gravitational and Shear Fields, Langmuir, 16, 6342-6350, 2000

21. Cohen, M.P., Brenner, J. Eggers, and Nagel, S.R., Two fluid drop snap-off problem: experiments and theory, Phys. Rev. Lett., 83, 1147-1150, 1999

22. Coveney, P.V., Novik, K.E., Computer simulations of domain growth and phase separation in two-dimensional binary immiscible fluids using dissipative particle dynamics, Physical Review E, 54(5), 5134-5141, 1996

23. Davis, M.E., Ordered porous materials for emerging applications, Nature, 417, 813-821, 2002. 24. Davis, M.E. and R.F. Lobo, Zeolite and molecular sieve synthesis, Chem.. Mater., 4, 756-768,

1992. 25. Devaney, J., Visualization of High Performance Concrete,

http://math.nist.gov/mcsd/savg/vis/concrete/index.html, 2002 26. Dutka, L., Kitowski, J., Application of Component-Expert Technology for Selection od Data-

Handlers in CrossGrid, in: Kranzlmüller, D., Kascuk, P., Dongarra, J., Volkert, J. (Eds.), Recent Advances in Parallel Virtual Machine and Message Passing Interface - 9th European PVM/MPI Users' Group Meeting Linz, Austria, September 29-October 2, 2002, Proceedings, no. 2474, Lecture Notes in Computer Science, Springer, 2002, pp. 25-32.

27. Dzwinel, W., Alda, W., Yuen, D.A., Cross-Scale Numerical Simulations Using Discrete Particle Models, Molecular Simulation, 22, 397-418, 1999

28. Dzwinel, W., Yuen, D.A., Dissipative particle dynamics of the thin- film evolution in mesoscale, Molecular Simulation, 22, 369-395, 1999

29. Dzwinel, W., Yuen D.A., A Multi-level Discrete Particle Model in Simulating Ordered Colloidal Structures, Journal of Colloid and Interface Science, 225,179-190, 2000a

24

30. Dzwinel, W., Yuen, D.A., A two-level, discrete particle approach for large-scale simulation of colloidal aggregates, Int. J. Modern Phys.C, 11/5, 1037-1061, 2000b

31. Dzwinel, W., Yuen, D.A., Matching Macroscopic Properties of Binary Fluid to the Interactions of Dissipative Particle Dynamics, International Journal of Modern Physics C, 11/1, 1-25, 2000c

32. Dzwinel, W., Alda, W., Pogoda, M., Yuen, D.A., Turbulent mixing in the microscale, Physica D, 137, 157-171, 2000a

33. Dzwinel, W., Alda W., Kitowski, J., Yuen, D.A., Using discrete-particles as a natural solver in simulating multiple-scale phenomena, Molecular Simulation, 25, 6, 361-384, 2000b

34. Dzwinel, W., Yuen, D.A., Rayleigh-Taylor Instability in the Mesoscale Modelled by Dissipative Particle Dynamics, Int. J. Modern Phys.C, 12/1, 91-118, 2001.

35. Dzwinel, W., Yuen, D.A., Mesoscopic dispersion of colloidal agglomerate in complex fluid modeled by a hybrid fluid particle model, Journal of Colloid and Interface Science, 217, 463-480, 2002

36. Dzwinel, W., Yuen D.A., Boryczko, K., Mesoscopic Dynamics of Colloids Simulated with Dissipative Particle Dynamics and Fluid Particle Model, Journal of Molecular Modeling, 8, 33-43, 2002a.

37. Dzwinel, W., Boryczko, K., Yuen D., A., A Discrete-Particle Model of Blood Dynamics in Capillary Vessels, Journal of Colloid and Interface Science, in press, 2003

38. Ellero, M., Kröger, M, Hess, S., Viscoelastic flows studied by smoothed particle dynamics, J. Non-Newtonian Fluid Mech. 105, 35–51, 2002

39. Erlebacher, G., Yuen, D.A., and F. Dubuffet, Current trends and demands in visualization in the geosciences, Electronic Geosciences, 6, 2001. http://link.springerny.com/link/service/journals/10069/free/technic/erlebach/index.htm

40. Español, P., Fluid particle model, Physical Review E, 57/3, 2930-2948, 1998 41. Español, P., Serrano, M., Dynamical regimes in DPD, Phys. Rev. E, 59/6, 6340-7, 1999 42. Flekkøy EG, Coveney PV, Foundations of dissipative particle dynamics, Phys. Rev. Lett.

83:1775-1778, 1999 43. Flekkøy EG, Lattice Bhatnagar-Gross-Krook models for miscible fluids, Phys.Rev. E, 52,

4952-62, 1993 44. Faber V, Clustering and the Continuous k-Means Algorithm, Los Alamos Sci., 22, 138-149,

1994 45. Foster, I., Kesselman, C., The Grid: Bluepr int for a New Computing Infrastructure, Morgan

Kaufmann Pbls., 1998 (www.mkp.com/grids) 46. Fox G, Presentation on Web Services and Peer-to-Peer Technologies for the Grid ICCS

Amsterdam April 24 2002 URL: (http://grids.ucs.indiana.edu/ptliupages/publications/presentations/iccsapril02.ppt)

47. Fox G, internal Presentation on Collaboration and Web Services April 4 2002 URL: (http://grids.ucs.indiana.edu/ptliupages/publications/presentations/collabwsncsaapril02.ppt)

48. Glowinski, R., Pan, T.-W., Hesla, T. I., Joseph D. D. and Periaux J., A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate flow, J. Comput. Phys., 169 363-427, 2001

49. Goddard III, W., A., Computational materials chemistry at the nanoscale, J. Nanoparticle Research, 1, 51-69, 1999

50. Groom, G.F., Lockwood, D.J., Ordering and self organization in nanocrystalline silicon, Nature, 407, 358-361, 2000.

25

51. Groot, R.,D., Warren, P.,B., Dissipative Particle Dynamics: Bridging the gap between atomistic and mesoscopic simulation, J. Chem . Phys, 107:4423-4435, 1997

52. Hadjiconstantinou, N.G., Patera, A.T., Heterogeneous atomistic-continuum representation for dense fluid systems , Int. Journal of Modern Physics C, 8, 4, 967-976, 1997

53. Hochella, M.F. , Jr., Nanoscience and technology: the next revolution in the Earth sciences, Earth Planetary and Sci. Lett., 203, 593-605, 2002

54. Holian, B., L., Ravelo, R., Fracture Simulation Using Large -Scale Molecular Dynamics, Phys. Rev. B., 51, 17, 11275-11285, 1995

55. Hoogerbrugge, PJ, Koelman, JMVA, Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics, Europhysics Letters, 19, 3, 155-160, 1992.

56. Hou, T.Y. and Wu, X-H, A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media, J. Comp. Physics, 134, 169-189, 1997.

57. Ihle, T., Kroll, D.M., Stochastic Rotation Dynamics I: Formalism, Galilean invariance, and Green-Kubo relations, Phys Rev E, in press, 2003.

58. Kechagia, P.E., I. Tsimpanogiannis, Y.C. Yortsos, and P.C. Lichtner, On the upscaling of reaction-transport processes in porous media with fast kinetics. Submitted for publication to Chem. Eng. 2001

59. Kloucek, P., and Melara. L., A., The Computational Modelling of Branching Fine Structures in Constrained Crystals, J. Computational Physics, 183, 623-651, 2002.

60. Knutson, C. E., and B. J. Travis, A Pore Scale Study of Permeability Reduction Caused by Biofilm Growth, Eos Trans. AGU, 83(47), Fall Meet. Suppl., Abstract H71B-0797, 2002.

61. Koshizuka Seiichi, Ikeda Hirokazu, MPS – moving particles semi- implicit method, http://www.tokai.t.u-tokyo.ac. jp/usr/rohonbu/ikeda/mps/mps.html, 1999

62. Kröger, M., NEMD computer simulation of polymer melt rheology. Appl. Rheol. 5, 66-71, 1995

63. Ladd, AJC and Verberg, R., Lattice-Boltzmann Simulations of Particle-Fluid Suspensions, Journal of Statistical Physics, 104/ 5/6, 1191-1251, September 2001

64. Lutsko, J.F., Boon, J.P., and Somers, J.A., Lattice gas automata simulations of viscous fingering in a porous medium, in T.M.M. Verheggen Numerical Methods for the Simulation of Multi-Phase and Complex Flow, Springer Verlag, Berlin, 124-135, 1992

65. Li, H., Eddaoudi, M., O'Keefe, M. and Yaghi, O.M., Design and synthesis of an exceptionally stable and highly porous metal-organic framework, Nature, 402, 276-279, 1999.

66. Libersky L.D., Petschek, A.G. Carney, T.C. Hipp, J.R..Allahdadi, F.A, High Strain Lagrangian Hydrodynamics, J. Comp. Phys., 109/1, 67-73, 1993

67. Manwart, C., Aaltosalmi, U., Koponen, A., Hilfer R., Timonen, J., Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media, Phys.Rev. E, 66, 2002, in press

68. Marsh, C., Backx, G., Ernst, M.H., Static and dynamic properties of dissipative particle dynamics, Physical Review E, 56, 1976, 1997

69. Martys, N., Chen, H., Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method, Phys.Rev E., 53, 743-750, 1996

70. Masters A. J. and Warren, P. B., Europhys. Lett., 48:1, 1999 71. Mathur, K., Needleman, A. and Tvergaard V., Three dimensional analysis of dynamic ductile

crack growth in a thin plate, J. Mech. Phys. Solids, 44, 439-464, 1996 72. McKenzie, D.P., The generation and compaction of partial melts, J. Petrol., 25, 713-765,

1984.

26

73. Monaghan, J., J., Smoothed Particle Hydrodynamics, Annu. Rev. Astron. Astrophys., 30, 543, 1992.

74. Moulton, J. D., Dendy, J. E. Jr., and Hyman J. M., The Black Box Multigrid Numerical Homogenization Algorithm, J. Comp. Physics, 142, 80-108, 1998

75. Nakano, A., Bachlechner, M., Campbell, T., Kalia, R. Omeltchenko, A., Tsuruta, K., Vashishta, P., Ogata, S., Ebbsjo, I. and Madhukar,A., Atomistic simulation of nanostructured materials, IEEE Computational Science and Engineering, Vol. 5 , No. 4, 68-78, 1998

76. Nakano, A., Bachlechner, M.E., Kalia, R.K., Lidorikis, E.,Vashishta, P., Multiscale simulation of nanosystems, Computing in Sc ience and Engineering, 3/4, 42-55, 2001

77. Pierrehumbert, R., T., Yang, H., Global Chaotic Mixing on Isentropic Surfaces, Journal of the Atmospheric Sciences, 50/15, 2462-2479, 1993

78. Pego, R.L., Front migration in the nonlinear Cahn-Hillard equation, Proc. Roy. Soc. London, Ser. A, 422, 261-278, 1989

79. Rothman, D.,H., Cellular Automaton Fluids: a model for flow in porous media, Geophysics, 53, 509-518, 1988

80. Rustad, J., R., Dzwinel, W., Yuen, D.A., Computational Approaches to Nanomineralogy, Rev. Mineralogy and Geochemistry, 44, 191-216, 2001

81. Stockman, H.W., Stockman, C. T. and Carrigan C.R., Modelling viscous segregation in immiscible fluids using lattice-gas automate, Nature, 348, 523-525, 1990.

82. Stockman, H.W., Li, C. and J. L. Wilson, A lattice-gas and lattice Boltzmann study of mixing at continuous fracture junctions: Importance of boundary conditions, Geophys. Res. Lett. 24/12, 1515-1518, 1997.

83. Serrano, M., Español, P., Thermodynamically consistent mesoscopic fluid particle model, Phys.Rev. E, 64, 2002, in press

84. Swift, M.,R., Orlandini, E., Osbors, W.,R., Yeomans, J., Lattice Boltzmann Simulations of Liquid-Gas and Binary-Fluid systems, Phys.,Rev., E., 54/5, 5041-5052, 1996

85. Tanaka, H. and Araki, T., Simulation method of colloidal suspensions with hydrodynamic interactions: Fluid Particle dynamics, Phys. Rev. Lett., 85/6, 1338-1341, 2000

86. Thompson D. S., Machiraju R., Dusi V. S., Jiang M., Nair J., Thampy S., Craciun G., Physics-based Feature Mining of Computational Fluid Dynamics Data Sets, Special Isssue of IEEE Computing in Science and Engineering on Data Mining, pp 22-30, July/August, 2002.

87. Ten, A., Yuen, D.A., Podladchikov, Yu.Yu., Larsen, T.B., Pachepsky, E. and A.V. Malvesky, Fractal features in mixing of non-Newtonian and Newtonian mantle convection, Earth Planet. Sci. Lett., 146, 401-414, 1997.

88. Travis, B.J., K. Eggert, D. Grunau, S.Y. Chen, and G. Doolen, Lattice Boltzmann Models for Flow in Porous Media, pp. 42-47, in Computing at the Leading Edge: Research in the Energy Sciences, eds. A. A. Mirin, and P. T. Van Dyke, National Energy Research Super-computer Center, Lawrence Livermore National Laboratory Rep. UCRL-TB-111084,1993

89. Theodoris S, Koutroumbas K, Pattern Recognition, Academic Press, San Diego, London, Boston, 1998

90. Werner, A., Echtle, H., Wierse, M., High Performance Simulation of Internal Combustion Engines, http://www.supercomp.org/sc98/TechPapers/sc98_FullAbstracts/Werner772/index.htm, 2001

91. Vashishta, P., Nakano, A., Dynamic fracture analysis, Computing in Science and Engineering, Sept/October, 20-23, 1999

27

92. Yuen D.A., Vincent, A.,P., Kido, M., Vecsey, L., Geophysical Applications of Multidimensional Filtering with Wavelets, Pure and Applied Geophysics, 159, 2285-2309, 2002.