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Journal of Colloid and Interface Science 283 (2005) 251266

www.elsevier.com/locate/jcis

Comparison between theoretical values and simulation results of viscosityfor the dissipative particle dynamics method

Akira Satoh a,, Tamotsu Majima b

a Department of Machine Intelligence and System Engineering, Faculty of System Science and Technology, Akita Prefectural University, 84-4, Ebinokuchi,

Tsuchiya-aza, Honjo 015-0055, Japanb Graduate School of Science and Technology, Chiba University, 1-33, Yayoicho, Inage-ku, Chiba 263-8522, Japan

Received 10 October 2003; accepted 13 September 2004

Abstract

In order to investigate the validity of the dissipative particle dynamics method, which is a mesoscopic simulation technique, we have derived

an expression for viscosity from the equation of motion of dissipative particles. In the concrete, we have shown the FokkerPlanck equation

in phase space, and macroscopic conservation equations such as the equation of continuity and the equation of momentum conservation.

The basic equations of the single-particle and pair distribution functions have been derived using the FokkerPlanck equation. The solutions

of these distribution functions have approximately been solved by the perturbation method under the assumption of molecular chaos. The

expressions of the viscosity due to momentum and dissipative forces have been obtained using the approximate solutions of the distribution

functions. Also, we have conducted nonequilibrium dynamics simulations to investigate the influence of the parameters, which have appeared

in defining the equation of motion in the dissipative particle dynamics method. The theoretical values of the viscosity due to dissipative forces

in the HoogerbruggeKoelman theory are in good agreement with the simulation results obtained by the nonequilibrium dynamics method,

except in the range of small number densities. There are restriction conditions for taking appropriate values of the number density, numberof particles, time interval, shear rate, etc., to obtain physically reasonable results by means of dissipative particle dynamics simulations.

2004 Published by Elsevier Inc.

Keywords: Dissipative particle dynamics method; Mesoscopic approach; Nonequilibrium dynamics method; Transport coefficients; Viscosity

1. Introduction

The hydrodynamic solution for a three-particle system

has to be combined into a simulation method in order to take

into account multibody hydrodynamic interactions among

colloidal particles more precisely. However, it is highly dif-

ficult even to solve analytically the flow field for a three-

particle system [1,2], and, therefore, it seems to be almost

hopeless to obtain an analytical solution for a nonspherical

particle system such as a system composed of rodlike par-

ticles. Thus, we have the choice to take another approach

to develop a more precise simulation method for colloidal

dispersions. If both colloidal particles and solvent mole-

* Corresponding author. Fax: +81-184-27-2188.

E-mail address: [email protected] (A. Satoh).

cules are simulated, we can obtain the particle motion and

the solution of the flow field simultaneously. However, if

molecules themselves are treated in simulations, we cannot

develop an effective simulation method, since the character-

istic time for the motion of colloidal particles is significantly

different from that of solvent molecules. In other words, this

kind of molecular-dynamics-like method is unrealistic as a

technique for simulation of a colloidal dispersion from a

simulation time point of view. To circumvent this difficulty,

the concept of fluid particles seems to be promising. Hooger-

brugge and Koelman [3,4] have developed the dissipative

particle dynamics method in terms of fluid particles. In their

theory, a fluid is regarded as being composed of such virtual

particles, and the flow field is solved by simulating these par-

ticles. The fluid particles interact with each other, exchange

momentum, and should make random motion like Brownian

0021-9797/$ see front matter 2004 Published by Elsevier Inc.

doi:10.1016/j.jcis.2004.09.050
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252 A. Satoh, T. Majima / Journal of Colloid and Interface Science 283 (2005) 251266

particles. From now on, such fluid particles are called dissi-

pative particles.

For the simulation of the flow field for a colloidal dis-

persion, the motion of colloidal particles is dependent on

the interaction between colloidal particles themselves, the

interaction between colloidal particles and dissipative par-

ticles, and the interaction between colloidal particles andan applied field such as a magnetic field. Hence, in this

simulation technique, the solution of pair or three-body hy-

drodynamic interactions need not be combined with it, in

order to simulate colloidal particles. This is in high con-

trast with the ordinary simulation methods such as Stokesian

dynamics methods [511]. Multibody hydrodynamic inter-

actions among colloidal particles are automatically taken

into account from the interactions of colloidal particles with

dissipative ones. This is a significant feature in the dissi-

pative particle dynamics method compared with ordinary

simulation methods. It is clear from these discussions that

the concept of the dissipative particle dynamics method isstraightforwardly applicable to a colloidal dispersion com-

posed of nonspherical particles, and, therefore, seems to be a

highly promising technique from a simulation point of view.

As will be shown later, the equation of motion for the dis-

sipative particle dynamics, which has been presented to date

[3,4,1214], is not unique, but has considerable freedom;

for example, the equation of motion has unknown constants

which are parameters representing the strength of conserv-

ative and dissipative forces. However, whether or not the

equation of motion can yield physically reasonable solu-

tions has not been sufficiently clarified [3,4,15], although

there is uncertainty in the equation of motion for dissipa-

tive particles. Hence, in order to apply the dissipative par-

ticle dynamics method widely to the usual flow problems

and physicsengineering problems of colloidal dispersions,

it is very important to investigate the influence of such un-

certainty in the equation of motion on the solution of the

flow properties such as the flow field and transport coeffi-

cients. This may be accomplished by comparing simulation

results with theoretical solutions concerning transport coef-

ficients, which can be derived analytically from the equation

of motion.

The present study, therefore, attempts first to derive an

analytical expression for viscosity using the equation of mo-

tion, secondly to obtain the numerical data in terms of dis-sipative particle dynamics simulations, and finally to com-

pare the simulation results with the analytical solutions. In

the concrete, we show the FokkerPlanck equation in phase

space, and macroscopic conservation equations such as the

equation of continuity and the equation of momentum con-

servation. The basic equations of the single-particle and pair

distribution functions are derived using the FokkerPlanck

equation. These equations of the distribution functions are

approximately solved to obtain an expression for the viscos-

ity due to momentum and dissipative forces. Nonequilibrium

dynamics simulations have been conducted under circum-

stances of simple shear flow. Finally, these simulation results

are compared with the theoretical solutions to investigate the

influence of the model parameters, which have appeared in

definitions of the equation of motion, number density, num-

ber of particles, time interval, etc., on this transport coeffi-

cient and also the internal structure of a dissipative particle

system. The present study may stimulate the development of

a new equation of motion for the dissipative particle dynam-ics, which gives rise to more physically reasonable simula-

tion results.

2. Dynamics of dissipative particles

2.1. Kinetic equation of dissipative particles

The equation of motion for dissipative particles has to

obey some physical constraints in order for the solution

obtained by the dissipative particle dynamics method to

agree with that obtained by the NavierStokes equation. Theconservation of the total momentum of a system and the

Galilean invariance have to be satisfied by the equation of

motion as a minimum request [3,4,12,13].

We concentrate our attention on particle i and consider

the forces acting on this particle. The following three kinds

of forces may be physically reasonable as forces acting on

particle i: a repulsive conservative force FCij exerted by the

other particles, a dissipative force FDij providing a viscous

drag to the system, and a random or stochastic force FRij in-

ducing the thermal motion of particles. With these forces, the

equation of motion of particle i can be written as [1214]

(1)mdvi

dt=

j (=i)

FCij +

j (=i)FDij +

j (=i)

FRij,

in which m is the mass of particle i , vi is the velocity, and,

concerning the subscripts, for example, FCij is the force act-

ing on particle i from particle j.

Now we have to embody specific forms of the above-

mentioned forces. It may be reasonable to assume that the

conservative force FCij depends only on the relative position

rij (= ri rj), and not on the particle velocities. An ex-plicit expression for this force will be shown later. Since the

Galilean invariance has to be satisfied, the dissipative force

FDij and the random force FRij should not be dependent onthe position ri and velocity vi themselves, but should be

functions of the relative position rij and relative velocity vij(= vi vj), if necessarily. Additionally, it may be reason-able to assume that the random force FRij does not depend on

the relative velocity but the relative position rij alone. Fur-

thermore, we have to take into account the isotropy of the

particle motion and the decrease in the magnitude of forces

with particleparticle separation. The following expressions

for FDij and FRij

satisfy these physical requirements [1214],

(2)FDij = wD(rij)(eij vij)eij,

(3)FRij = wR(rij)eijij,


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A. Satoh, T. Majima / Journal of Colloid and Interface Science 283 (2005) 251266 253

in which rij = |rij|, and eij is the unit vector denoting thedirection of particle i from particle j, expressed as eij =rij/rij. Also ij is a random variable inducing the random

motion of particles and has to satisfy the following stochastic

properties:

ij(t )

= 0,(4)

ij(t)ij (t

)= (ii jj + ij j i )(t t).

This variable satisfies the characteristic of the symmetry

ij = j i , which ensures that the total momentum of thesystem is conserved. The wD(rij) and wR(rij) are weight

functions to reproduce the decrease in forces with particle

particle separation, and and are constants specifying the

magnitude of forces. These constants can be related to the

system temperature and friction coefficient, which will be

shown later. It is seen from the expression for FDij that this

force acts in such a way as to relax the relative motion of

particles i and j. On the other hand, FRij acts as if inducing

the thermal motion of particles, but this force satisfies theactionreaction formula, so that the conservation of the total

momentum of the system is ensured.

The substitution of Eqs. (2) and (3) into Eq. (1) leads to

the following equation:

mdvi

dt=

j (=i)

FCij(rij)

j (=i) wD(rij)(eij vij)eij

(5)+

j (=i) wR(rij)eijij.

If this equation is integrated with respect to time over a small

time interval from t to t+

t, then the finite difference equa-

tions governing the particle motion in simulations can be

obtained as

(6)ri = vi t ,

(7)

vi = 1m

j (=i)

FCij(rij)

j (=i) wD(rij)(eij vij)eij

t

+ 1m

j (=i)

wR(rij)eijWij,

in which

(8)Wij =

t+tt

ij d.

This Wij has to satisfy the following stochastic properties,

which arise from Eq. (4):

Wij = 0,(9)WijWij = (ii jj + ij j i )t.

If a new stochastic variable ij is introduced from the defini-

tion Wij = ij

t, the third term in Eq. (7) can be written

as

(10)1

m

j (=i) wR(rij)eijij

t ,

in which ij has to satisfy the following stochastic proper-

ties:

(11)ij = 0, ijij = (ii jj + ij j i ).It is seen from Eq. (11) that the stochastic variable ij is sam-

pled from a uniform or normal distribution with zero average

value and unit variance [3,4,1214].

The equation of motion in Eq. (5) satisfies the conserva-

tion law of the total momentum of the system, but not the

energy conservation law. The equation of motion has to be

modified to satisfy the conservation of the system energy

[16,17].

The coefficients and and the weight functions

wD(rij) and wR(rij) cannot be determined independently,

because there are some relationships among these quanti-

ties. The derivation of these relationships will be shown in

the next section.

2.2. FokkerPlanck equation

We use the notation ri for the position vector of particle i

and vi for the velocity vector. Also, for simplicity of expres-

sion, the vector vN is used for describing the velocity vectors

of all the particles (that is, vN represents v1, v2, v3, . . .), and

similarly rN is for the position vectors of all the particles

(that is, rN represents r1, r2, r3, . . .). If the probability that

a particle position and velocity are found within the range

from (rN, vN) to (rN + rN, vN + vN) is denoted byW (rN, vN, t) drN dvN, then the probability density function

W (rN, vN, t) satisfies the following stochastic formula, i.e.,

the ChapmanKolmogorov equation,

W (rN, vN, t) =

W (rN rN, vN vN, t t )

(12)

(rN rN, vN vN; rN, vN) d(rN) d(vN),

in which (rNrN, vNvN; rN, vN) is the transi-tion probability for the state (rN rN, vN vN) trans-ferring to another state (rN, vN) during a time interval t

and this does not depend on the time point t. If the veloc-

ity vN does not change appreciably during the small time

interval t, then can be written as

(rN rN, vN vN; rN, vN)

(13)

= (rN rN, vN vN; vN)(rN vNt),

in which is the transition probability and is independent

of rN. The substitution of Eq. (13) into Eq. (12) leads to

the following different form of the ChapmanKolmogorov

equation:

W (rN + vNt , vN, t+ t )=

W (rN, vN vN, t)

(14) (rN

, v

N

vN

; vN

) d(v

N

).


4/16


The FokkerPlanck equation in phase space (or the Chan-

drasekhar equation) can be derived by reforming Eq. (14) in

terms of Taylor series expansions with the finite difference

equations (6) and (7). Since the detailed derivation proce-

dure is shown in Ref. [18], we here show the final result:

Wt

+

i

vi W ri

+

i

j

(i=j )

FCij

m W

vi

=

i

j

(i=j )

eij vi

1

m wD(rij)(eij vij)W

+ 12

i

j

(i=j )

1

m22w2R(rij)

(15) eij

vi eij

vi eij

vjW.

2.3. Final expression of equation of motion and its

nondimensional form

If a system composed of dissipative particles is in equi-

librium, then the equilibrium distribution Weq becomes the

canonical distribution for an ensemble which is specified by

a given particle number N, volume V, and temperature T.

With the notation U for the potential energy, Weq is written

as

(16)Weq =1

Zexp

1

kTN

i=1

mv2i

2+ U,

in which Z is the partition function, and U is related to the

conservative force FCij by

(17)FCij = U

rij.

The equilibrium distribution Weq has to satisfy the Fokker

Planck equation in phase space in Eq. (15). Since the left-

hand side in Eq. (15) vanishes for the substitution of Weq,

the right-hand side also has to become zero. This is accom-

plished by the requirements

(18)wD(rij) = w2

R(rij), 2

= 2 k T ,in which k is Boltzmanns constant. The second equation is

the fluctuationdissipation theorem for the dissipative parti-

cle dynamics.

We now have to determine the explicit expressions for the

conservative force FCij(rij) and the weight function wR(rij).

FCij(rij) is a repulsive force preventing unphysical exces-

sive overlaps between particles, and wR(rij) has to be set

so that interparticle forces decrease with increasing particle

particle separations. These requirements are satisfied by the

expressions

(19)FCij = wR(rij)eij,

(20)wR(rij) =

1 rijrc

for rij rc,

0 for rij > rc,

in which is a constant representing the magnitude of the re-

pulsive forces. By substituting these equations into Eqs. (6)

and (7) with considering Eq. (10), the final expression for

the equation of motion of the dissipative particle dynamicscan be written as

(21)ri = vi t ,

(22)

vi =

m

j (=i)

wR(rij)eijt

m

j (=i)

w2R(rij)(eij vij)eijt

+ (2 k T )1/2

m

j (=i)

wR(rij)eijij

t ,

in which the expression for wR(rij) has already been shownin Eq. (20). The stochastic variable ij has to obey the sto-

chastic properties shown in Eq. (11) and is sampled from a

uniform or normal distribution with zero average and unit

variance, as already pointed out. The dissipative particle dy-

namics method uses Eqs. (21) and (22) to simulate the mo-

tion of dissipative particles.

Lastly, we show an exampleof the nondimensionalization

method used for actual simulations. To nondimensionalize

each quantity, the following representative values are used:

(kT/m)1/2 for velocities, rc for distances, rc(m/kT )1/2 for

time, (1/r 3c ) for number densities, etc. With these represen-

tative values, Eqs. (21) and (22) can be nondimensionalized

as

(23)ri = vi t,

(24)

vi =

j (=i)wR(r

ij)eijt

j (=i)w2R(r

ij)(eij vij)eijt

+ (2)1/2

j (=i)wR(r

ij)eijij

t,

in which

(25)wR(rij) =

1 rij for rij 1,0 for rij > 1,

(26) = rckT

, = rc(mkT )1/2

.

In these equations, the quantities with superscript * are di-

mensionless. Equations (23) and (24) clearly show that one

can start a dissipative particle dynamics simulation if appro-

priate values of the nondimensional parameters and are adopted, and also the time interval t , the number den-sity n0 (= r3c N/V , V is the system volume), and the particlenumber N are properly specified. Results obtained by the


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simulations should not depend on the nondimensional para-

meters which have been introduced in embodying the equa-

tion of motion; one has to be, therefore, careful in setting

appropriate values for these values in conducting dissipa-

tive particle dynamics simulations. For example, since time

is nondimensionalized by the representative value based on

the mean velocity v ( (kT/m)1/2) and the radius rc, themagnitude of the dimensionless time interval t may haveto be taken sufficiently smaller than unity [19].

3. Transport equations

In the present section, the dissipative particle dynamics

method, the equation of continuity, and the momentumequa-

tion of the fluid are derived using the equation of motion of

the dissipative particle dynamics.

If an arbitrary physical quantity A(rN, vN) is not de-

pendent on time explicitly, the time average A can beexpressed, using the probability density function W which

satisfies Eq. (15), as

(27)A =

AW (rN, vN, t) drN dvN,

in which

(28)

W (rN, vN, t) drN dvN = 1.

Hence, the time variation of A can be expressed, fromEq. (15), as

tA =

A

W

tdrN dvN

=

A

i

vi W ri

i

j

(i=j )

FCij

m W

vi

+

i

j

(i=j )

eij vi

1

m wD(rij)(eij vij)W

+ 12 i j

(i=j )

1

m22w2R(rij)

(29)

eij

vi

eij

vi eij

vj

W

drN dvN.

By applying integration by parts, this equation is written as

tA =

i

vi A

ri

i

j

(i=j )

FCij

m A

vi

i j

(i=j )

mwD(rij)(eij vij)eij

A

vi

+ 12

i

j

(i=j )

1

m22w2R(rij)

eij

vi

(30)

eij

vi eij

vj

A

.

Next we derive the equation of continuity and the momen-

tum equation of the fluid using Eq. (30). If A is defined by

the equation

(31)A =N

i=1m(r ri ),

then the average A, which is evaluated from Eq. (27), isequal to a local density (r). That is,

(32)A =

N

i=1

m(r ri )

= (r).

IfA is taken as

(33)A =N

i=1mvi (r ri ),

then the average A is now

(34)A = (r)u(r),in which u(r) is a macroscopic fluid velocity at position r.

The substitution of Eq. (31) into Eq. (30) leads to the fol-

lowing expression:

(35)

t+

r (u) = 0.

This is no other than the equation of continuity.

Next the substitution of Eq. (33) into Eq. (30) leads to the

following equation:

t(u) =

i

mvi ri

vi (r ri )

+

i

j

(i=j )

FCij

vi

vi (r ri )

i

j

(i=j )

wD(rij)(eij vij)

eij vi

vi (r ri )

+ 12

i

j

(i=j )

1

m2w2R(rij)

eij

vi

(36) eij

vi

vi (r ri )

vj

vj(r rj)

.

Thus, by taking into account Eq. (2), and conducting a simi-

lar reformation, Eq. (36) can be simplified as


6/16


t(u) =

r

i

mvi vi (r ri )

(37)+

i

j

(i

=j )

FCij + FDij

(r ri )

.

By taking account of the relation FDij = FDijeij, the followingequation can be obtained:

i

j

(i=j )

FDij(r ri )

= 12

i

j

(i=j )

FDij(r ri ) + FDj i (r rj)

(38)= 12 i j

(i=j )

FDijeij

(r ri ) (r rj)

.

If the following relationship concerning the Dirac delta func-

tion is taken into account,

(r rj) = (r ri + rij)

(39)

= (r ri ) + r

eij

rij0

(r ri + eij) d

,

then Eq. (38) can be reformed further as

i

j(i=j )

FDij(r

ri )

(40)

= 12

i

j

(i=j )

r

FDijeijeij

rij0

(r ri + eij) d

.

By conducting a similar reformation concerning FCij, Eq. (37)

can finally be written as

(41)

t(u) =

r (uu) +

r (K + U),

in which

K =

i

m(vi u)(vi u)(r ri )

,

(42)

U = 1

2

i

j

(i=j )

FCij + FDij

eijeij

rij0

(r ri + eij) d

.

If the equation of continuity in Eq. (35) is taken into consid-

eration, Eq. (41) reduces to the momentum equation of the

fluid,

(43)u

t +u

u

r=

r (K

+

U),

in which K and U are stress tensors due to the particle

momentum and the forces acting between particles, respec-

tively. It has now been shown that the equation of motion of

the dissipative particle dynamics gives rise to the momen-

tum equation of the fluid. The virial equation of state can be

derived by evaluating directly K and U in Eq. (42).

4. Expressions for transport coefficients

4.1. Distribution functions

The local number density n(r, t) at position r at time t

can be expressed using W (rN, vN, t ) as [18]

n(r, t) =

Ni=1

(r ri )

= i

(r ri )W (rN, vN, t ) drN dvN

= N

W (r, r2, . . . , rN, vN, t)

(44) dr2 . . . drN dvN.Similarly, the pair correlation function g(r, r, t) can be writ-ten as [18]

g(r, r, t )

= 1n20

N

iN

j(i=j )

(r ri )(r rj)

W (rN, vN, t ) drN dvN

= N(N 1)n20

(45)

W (r, r, r3, . . . , rN, vN, t) dr3 . . . drN dvN,

in which n0 is the mean number density, given by n0 =N/V, as defined before.

If the distribution function f (r, v, t) is defined as [14]

(46)f (r, v, t) = N

i=1(r ri )(v vi )

,

f (r, v, t ) can be written, using W (rN, vN, t ), as

(47)

f (r, v, t) = N

W (r, r2, . . . , rN, v, v2, . . . , vN, t )

dr2 . . . drN dv2 . . . dvN.By comparing Eq. (47) with Eq. (44), it is seen that there is

a relationship between f (r, v, t) and n(r, t ):

(48)n(r, t)

= f (r, v, t) dv.


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Similarly,

(49)n(r, t )u(r, t) =

Ni

vi (r ri )

=

vf (r, v, t) dv.

Next, if we define the pair distribution f(2)(r, r, v, v, t )by the equation [14]

f(2)(r, r, v, v, t)

=

Ni=1

Nj=1

(i=j )

(r ri )(r rj)(v vi )(v vj)

= N(N 1)

(50)

W (r, r, r3, . . . , rN, v, v, v3, . . . , vN, t)

dr3 . . . drN dv3 . . . dvN,

then thepair distribution can be related to the pair correlationfunction g(r, r, t) as

(51)n20g(r, r, t) =

f(2)(r, r, v, v, t) dv dv.

It is clear from Eq. (30) that the average of an arbi-

trary quantity A(rN, vN) can be evaluated using f (r, v, t )

and f(2)(r, r, v, v, t ) without obtaining the expression forW (rN, vN, t ). Hence, we here first show the basic equations

for f and f(2), and then solve the equations analytically to

obtain the approximate solutions.

If we set A =i

(r ri )(v vi ) in Eq. (30), and re-form the equation with the formulae of vector operations, the

following equation can be obtained:

tf (r, v, t) + v

rf (r, v, t)

= m

wD(R)RR

: v

(v v)f(2)(r, r + R, v, v, t)dR dv

+ 2

2m2

w2R(R)RR

(52)

:2

vv

f(2)(r, r

+R, v, v, t) dR dv.

It has been assumed in deriving this equation that there are

no conservative forces. In Eq. (25), R is the unit vector,

denoted by R = R/|R|. It is seen from Eq. (52) that thepair distribution function f(2)(r, r, v, v, t ) is necessary forobtaining the solution off (r, v, t), and, similarly, the multi-

body distribution functions more than the pair distribution is

required to get the solution of f(2). This clearly shows that

Eq. (52) is not closed, but a hierarchy equation. To close the

equation for f, the following assumption of molecular chaos

is introduced [14]:

(53)f

(2)

(r, r, v, v, t) f (r, v, t ) f (r, v,t).

This assumption theoretically enables us to solve the equa-

tion of the distribution function. The substitution of Eq. (53)

into Eq. (52) leads to the following equation:

tf (r, v, t) + v

rf (r, v, t)

= m

wD(R)f(r + R, v, t)RR

: v

(v v)f (r, v, t)dR dv

+ 2

2m2

w2R(R)f (r + R, v, t )RR

(54): 2

v vf (r, v, t ) dR dv.

The validity of solutions which are obtained from Eq. (54)

is strongly dependent on the physical reasonability of the

assumption of molecular chaos.

The nondimensionalform of Eq. (54) may be more under-standable to be solved analytically. According to the nondi-

mensionalization method, which has been shown in Sec-

tion 2.3, Eq. (54) is nondimensionalized as

tf(r, v, t) + v

rf(r, v, t)

= n0

wD(R)f(r + R, v , t)RR

: v

(v v )f(r, v, t)dR dv

+ n0 wD(R)f(r + R, v , t)RR

(55): 2

v vf(r, v, t) dR dv ,

in which the distribution function f has been nondimension-

alized as f = (1/n0)(kT/m)3/2f. If we set = 1/n0and assume to be much smaller than unity, the perturba-

tion method is applicable. Hence, f is expanded, with theperturbation parameter , as

f(r, v, t) = f0 (r, v, t) + f1 (r, v, t)(56)+ 2f2 (r, v, t) + .

By substituting this equation into Eq. (55) and neglecting the

higher-order terms, we can obtain the following expression:

tf0 + v

rf0

+ 2

tf1 + v

rf1

=

wD(R)

f0 (r + R, v , t)

+ f1 (r + R, v , t)

RR

: v

(v v )f0 (r, v, t)+ f1 (r, v, t)

dR dv

+ wD(R)f

0 (r

+ R, v , t)

+ f1 (r + R, v , t)RR


8/16


(57)

: 2

vv

f0 (r, v, t) + f1 (r, v, t)

dR dv .

The equation for the zeroth order of can be written, from

this equation, as

wD(R)f0 (r + R, v , t)RR

: v

(v v )f0 (r, v, t)

dR dv

+

wD(R)f0 (r

+ R, v , t)RR

(58): 2

vvf0 (r

, v, t) dR dv = 0.Similarly, the equation for the first order of can be ex-

pressed as

tf0

+v

rf0

=

wD(R)

f0 (r + R, v , t)RR

:

v(v v )f1 (r, v, t)

+ f1 (r + R, v , t)RR

: v

(v v )f0 (r, v, t)

dR dv

+

wD(R)

f0 (r + R, v , t)RR

:2

vv f1 (r, v, t)+ f1 (r + R, v , t)RR

(59): 2

vvf0 (r

, v, t)

dR dv .

Equation (58) can straightforwardly be solved as

(60)f0 (r, v, t) = n

1

2

3/2exp

1

2(v u)2

.

By reforming Eq. (59) with the formulae of vector opera-

tions, the following equation is obtained:

t

f0 + v r f0 =

n(r + R, t)wD(R)RR

(61):

v

(v u)f1 (r, v, t)

+ 2

vvf1 (r

, v, t)

dR.

If it is assumed that n(r +R, t) (= n/n0) is independentof the position and is constant, and the integration is carried

out using the polar axis coordinates of R, then Eq. (61) can

be simplified as

t f0 + v

r f0

= [w]3

v (v u)f1 (r, v, t)

(62)+ v

v

f1 (r, v, t)

,

in which [w] is defined as

(63)[w] =1

0

wD dR =

10

wD4 R2 dR.

The left-hand side of Eq. (62) can finally be reformed, by

taking account of Eq. (60), as

tf0 + v

rf0

(64)= 12

f0 UU :

ru +

rut

,

in which the notation U, defined by U

=(v

u), has

been used. Also, the superscript t means a transposed ten-sor. Now we have prepared for solving Eq. (62) to get the

solution off1 .As the first trial to find the solution of f1 , we substitute

f0 UU into f1 in Eq. (62) to get the following expression

for the right-hand side of Eq. (62):

(65)1

3[w]2UUf0 + 2If0 .

As the second trail, we substitute f0 U2I into f1 in

Eq. (62) to get the following equation for the right-hand side

of Eq. (62):

(66)13[w]2U2f0 I + 6f0 I.

As the third trial, we substitute f0 I into f1 in Eq. (62) to

obtain zero for the right-hand side of Eq. (62). Finally, if the

expression for f1 is assumed to be

(67)UUf0 1

3U2f0 I,

then the right-hand side of Eq. (62) becomes the following

expression:

(68)23[w]U

Uf0 1

3U2f0 I.

After all, from these considerations, the solution of Eq. (62)

can straightforwardlybe seen to be the following expression:

f1 = 3

2

1

n0[w]

UUf0 1

3U2f0 I

(69): 12

ru +

rut

.

If the equation of continuity, / r u = 0, is taken intoaccount, it is seen that the following relationship is satisfied:

(70)I

:

ru

+

ru

t

= 0.


9/16


We now have obtained the zeroth and first order solutions

off,

(71)f = f0 + f1 ,in which f0 and f

1 are given in Eqs. (60) and (69), respec-

tively.

4.2. Viscosity due to momentum

The viscosity due to momentum, K, can be obtainedfrom Eqs. (42) and (71) as

K = n0

UU

f0 + f1

dv

= n0I +3

2

1

2

3/2 n0[w]

UU

UU 13

U2I

exp1

2U2

dU : 1

2

ru +

rut

(72)= n0I +3

2

1

[w]

ru +

rut

,

in which the following relationship has been used:

UUUU 1

3

U2I :D exp

1

2

U2dU

(73)

= (2 )3/2

2D11 D12 + D21 D13 + D31

D12 + D21 2D22 D23 + D32D13 + D31 D23 + D32 2D33

.

In this equation, D is defined by

(74)D = 12

ru +

rut

.

Hence, by comparing Eq. (72) with the general expression

of a stress tensor for Newtonian fluids,

(75) = PI +

ru +

rut

,

the viscosity due to momentum, K, can be expressed as

(76)K = 32

1

[w] =45

4

1

.

The dimensional expression for this expression is written as

(77)K

=

3

2

mkT

r

3

c [w].

4.3. Viscosity due to dissipative forces

We derive an expression for the stress tensor D. Withthe following Taylor series expansion of the Dirac delta

function,

(r ri + eij)= (r ri ) + eij

r(r ri )

(78)+ 12!

2eijeij :

r

r(r ri ) + ,

the stress tensor due to dissipative forces can be written as

D = 1

2

i

j

(i=j )

wD(rij)(eij vij)eijrij(r ri )

+ 14

ri j

(i=j )

wD(rij)

(79) (eij vij)eijrijrij(r ri )

+ .

The second term on the right-hand side in this equation is

negligible unless there is a significant nonuniformity of the

system. Equation (79) is, therefore, rewritten as

D = 1

2n20

RwD(R)

R (v v )RR(80) f(2)(r, r + R, v, v , t) dR dv dv .

By the assumption of molecular chaos, written in Eq. (53),with Eq. (71) for f, the expression for f(2) can be writtenas

f(2)(r, r , v, v , t) = f0 (r, v, t)f0 (r , v , t)+ f0 (r, v, t)f1 (r , v , t)+ f0 (r , v , t)f1 (r, v, t)

(81)+ 2f1 (r, v, t)f1 (r , v , t).By evaluating Eq. (80) with Eq. (81), the expression for Dcan be obtained, which is now shown in the following.

We first derive the expression for the stress D0 due to thefirst term on the right-hand side in Eq. (81). From Eq. (80),

we get the following expression:

D0 = 12 n

20

RwD(R)

R (v v )RR(82) f0 (r, v, t)f0 (r , v , t) dR dv dv .

If we take into account the relationship

(83)

v v = (v u) (v u ) R r

u(r, t),

then Eq. (82) can be rewritten as

(84)D0 =1

2n20

wD(R)R2RRRR : D dR.

The integral on the right-hand side in this equation can be

carried out using the polar coordinate system to give the final


10/16


expression for D0 ,

(85)D0 =n20

15[R2w]1

2

ru +

rut

,

in which

[R2w] =

R2wD(R) dR

(86)=1

0

R2wD(R)4 R2 dR.

Next, we have to evaluate the stresses due to the second,

third, and fourth terms on the right-hand side in Eq. (81).

From a similar procedure, it can straightforwardly be shown

that all these stresses become zero.

By taking into account all these results, the expression for

D can easily be derived as

(87)D = D0 .Hence, the comparison of Eq. (85) with Eq. (75) leads to

the final expression for the viscosity D due to dissipativeforces:

(88)D = n2030

[R2w] = 21575

n20 .

The dimensional form is expressed as

(89)D = n20

30

r 3c R

2w

.

The viscosity is finally obtained as the sum ofK and D

as = K + D in the present case.It is clearly seen from the above discussion that the per-

turbation method with perturbation parameter using the

nondimensional basic equation written in Eq. (55) can be

significantly understandable in notifying the validity range

of the analytical expression of the viscosity, Eq. (88) or

Eq. (89), although the result essentially agrees with that

which was derived by Marsh et al. [14].

4.4. Relationship with HK theory

In order to compare the present results with the HK the-

ory [3,4] more straightforwardly, we rewrite Eq. (89) as

(90)D = n20

30

r2

1 rrc

2dr = n

20

30

r 2wD(r)dr.

The viscosity due to dissipative forces, DHK, in the HK

theory is written as

(91)DHK =mn20

30t

r2wHK(r) dr,

in which wHK(r) is a weight function in the HK theory

and this weight function corresponds to wD(r) in the present

theory. However, it should be noted that wHK(r) does not

satisfy the fluctuationdissipation theory [12,13], which is,

in constant, satisfied by the present theory. If we take into

account that the notation in the HK theory is related to the

notations here as = t/m, it is seen that D essentiallyagrees with DHK. This clearly shows that the expression in

the HK theory, DHK, has a similar validity range to the

present expression D, which has derived for the cases of

small values of.

5. Evaluation of viscosity by means of the

nonequilibrium dynamics method

5.1. Evaluation by GreenKubo expression

There are two methods for evaluating transport coef-

ficients by means of simulations: the first method is for

thermodynamic equilibrium and evaluates them using the

GreenKubo expression; the second one is the nonequilib-

rium dynamics method, shown in the following, in whichtransport coefficients are evaluated under circumstances of a

simple shear flow.

The theory of the nonequilibrium molecular dynamics

method [20] for a molecular system is applicable to the

present case, if a simple shear flow is considered as a flow

field. In this case, the viscosity yx is evaluated from the fol-

lowing equation in simulations [18,20],

(92)yx = 1

VJyx ne,

in which

Jyx (t) =N

i=1

mviy (t)vix (t ) + yi (t)Fix (t )

(93)=N

i=1mviy (t)vix (t ) +

Ni=1

Nj=1

(i


11/16


cases for a molecular system. According to the nonequi-

librium molecular dynamics method, the finite difference

equations of the equation of motion for the present system

are modified as [20]

xi = vix + yi t, yi = viy t,(96)zi = viz t,

vix =

Fix viy

t, viy = Fiy t,(97)viz = Fiz t,

in which Fi denotes the force acting on particle i.

5.2. Parameters for simulations

Simulations have been conducted under various condi-

tions in order to clarify the characteristics of the equation

of motion for dissipative particles dynamics by comparing

simulation results with theoretical solutions. Typical cases

such as n0 = 0.5, 3.16, and 10.0 were taken for the numberdensity, and the particle number N was 500, unless specif-

ically notified. From sufficient numbers of simulations for

a small system, the time interval t was taken as t =0.0001, and the total time steps were adopted as 5,000,000,

which ensures sufficient accuracy of average values. The er-

ror analysis [20] concerning almost all simulation results

was conducted to show the error ranges of simulation data;

the error ranges of K and D are denoted by Kerror andDerror, respectively, and these values will be shown withresults in Section 6. Also, the parameters and , whichhave appeared in the equation of motion for dissipative parti-

cles, were taken as = 10, 25 and = , /10, unlessspecifically noted.

6. Results

6.1. Influence of on viscosity

Figs. 1 and 2 show the influence of the values of onthe viscosity, which were obtained by the nonequilibrium

dynamics method. Fig. 1 is for the viscosity D due to dis-sipative forces and Fig. 2 is for the viscosity K due tomomentum.

It is seen from Fig. 1 that the results obtained by thenonequilibrium dynamics simulations are in good agreement

with the theoretical solutions shown in Eq. (88) for both

cases of = /10 and = . However, for the casesof small number densities such as n0 = 0.5, the simula-tion results have significant errors and, therefore, qualitative

properties are relatively difficult to identify. That the num-

ber density n0 is smaller than unity means that a sufficientlylarge number of particles do not exist around an arbitrary

particle within the range of the radius rc from its center. In

this situation, the accuracy of the viscosity data is strongly

dependent on whether or not there are other particles which

interact with the particle of interest. Hence such a situation

(a)

(b)

Fig. 1. Influence of on viscosity due to dissipative forces obtainedby nonequilibrium dynamics method (a) for = /10 and (b) for = (Derror = (5.2, 1.5) for n20 = (10,000, 1000), respec-tively, for n0

=10,

(1.5, 0.32) for n20

=(1000, 100), respectively,

for n0 = 3.16, and (0.049, 0.022) for n20 = (25, 2.5), respectively, forn0 = 0.5, in (a); similar errors are included in (b)).

causes significant errors of the simulation results of viscos-

ity, which has been seen for n0 = 0.5 in Fig. 1. We mayconclude from this fact that it is desirable to take the number

density as n0 1.0 in simulating a flow problem by meansof the dissipative particle dynamics method.

Fig. 2 clearly shows that the simulation data are larger

than the theoretical solutions, and this tendency becomes

more significant for larger values of . When the influenceof dissipative forces is of the same order of the momentum,

that is, when 1, the simulation results of K agreewell with the theoretical solutions. On the other hand, at

1, where dissipative forces are more dominant thanthe momentum, the discrepancy between the simulation re-

sults and the analytical solutions has a tendency to increase

with values of . As will be shown later in Fig. 8, the ra-dial distribution function does not significantly change for

values of, so that this discrepancy is presumed to be dueto the mechanism other than the assumption of molecular

chaos, adopted here. Some researchers attempted to explain

the discrepancy between simulation results and theoretical

solutions concerning the viscosity due to the momentum, in

terms of the pair collision theory [21].


12/16


(a)

(b)

Fig. 2. Influence of on viscosity due to momentum obtained by non-equilibrium dynamics method (a) for = /10 and (b) for = (Kerror = (0.57, 1.47) for 1/ = (0.01, 0.1), respectively, for n0 = 10,(0.40, 0.62) for 1/ = (0.01, 0.1), respectively, for n0 = 3.16, and(0.44, 0.12) for 1/ = (0.01, 0.1), respectively, for n0 = 0.5, in (a);similar errors are included in (b)).

6.2. Influence of and number density on viscosity

Fig. 3 shows the influence of on the viscosity due todissipative forces. It is seen from Fig. 3 that the values of

do not have significant influence on the viscosity for bothcases ofn0 = 3.16 and 10.0. As will be shown later in Fig. 9,the radial distribution function changes significantly for dif-

ferent values of. Hence, this means that the viscosity dueto dissipative forces, D, is not strongly dependent on theshape of the radial distribution function.

Fig. 4 shows the dependence of the viscosity on the num-

ber density of particles: Fig. 4a is for the viscosity D dueto dissipative forces, and Fig. 4b is for the viscosity Kdue to momentum. Since the theoretical solutions are valid

for n0 1.0, the simulation results are in good agree-

ment with the analytical solutions for the values of n0 whichsatisfy the condition n0

1.0 for a given value of .This agreement becomes worse with decreasing values ofn0smaller than unity; especially, the results for = 10 and = 1 significantly deviate from the theoretical values withlarge error ranges. As already pointed out, this is presumed

to be due to the situation in which the ambient particles of an

Fig. 3. Influence of on viscosity due to dissipative forces (Derror =(1.2, 3.1) for = (100, 1), respectively, in the case of n0 = 10 and = 25, and (0.17, 0.32) for = (40, 1), respectively, in the case ofn0 = 3.16 and = 10).

(a)

(b)

Fig. 4. Influence of number density on viscosity: (a) for viscosity due

to dissipative forces and (b) for viscosity due to momentum (Derror =(2.7, 0.12) for n0 = (10, 1), respectively, in the case of = 25 and = 2.5, and (0.78, 0.022) for n0 = (10, 0.5), respectively, in the caseof = 10 and = 1, in (a); Kerror = (1.1, 0.24) for n0 = (10, 1),respectively, in the case of = 25 and = 2.5, and (1.0, 0.12) forn0 = (10, 0.5), respectively, in the case of = 10, and = 1, in (b)).

arbitrary particle come to be less for decreasing the number

density smaller than unity, which leads to the large fluctu-

ation in the contribution of dissipative particles to physical


13/16


quantities such as transport coefficients with time. Although

the viscosity K due to momentum shown in Fig. 4b has alarge error range, it is clearly seen that there is a significant

discrepancy between the simulation and theoretical values,

and this tendency does not strongly depend on the values

ofn0. As already pointed out in Fig. 2, since the radial distri-bution function does not significantly deviate from g(r) = 1(shown later in Fig. 10a), it is presumed that this discrepancy

for = 10 and = 1 is induced by the mechanism otherthan the invalidity of the molecular chaos assumption.

6.3. Influence of time interval on mean velocity of particles

and viscosity

Fig. 5 shows the influence of the time interval h (=t), which is used in conducting simulations based on thefinite difference equations, on the mean velocity of particles

and the viscosity. It is seen from Fig. 5a that the viscos-

ity D does not strongly depend on the time interval forh 0.1. On the other hand, as shown in Fig. 5b, the vis-cosity due to momentum, K, is significantly dependenton the values of h. Especially, this dependence comes toappear from h 0.001, and values of K significantlyincrease with increasing values of the time interval. A sim-

ilar tendency can be observed in Fig. 5c, in which the mean

velocity of particles significantly increases from h 0.01with the values of h. Since the mean velocity v is nondi-mensionalized in such a way of being v = 1, this conditionhas to be satisfied in actual simulations. We may, therefore,

conclude that the simulations have to be conducted using a

sufficiently small time interval such as h = 0.0001, adoptedhere.

6.4. Influence of shear rate and number of particles on

viscosity

Fig. 6 shows the influence of the shear rate of a simple

shear flow and the number of particles on the viscosity. It

is seen that the simulation results come to have significant

errors with decreasing shear rate, since just a weak simple

shear flow is generated for such situations. This character-

istic is observed for both cases of D in Fig. 6a and K

in Fig. 6b, especially in the results of K. It is seen fromFig. 6c that the mean velocity v significantly increases withthe shear rate after 1. Hence, there is an appropriaterange for the shear rate , which is sufficiently small forsatisfying the condition of v = 1, and sufficiently large forgenerating a simple shear flow with an appropriate shear

strength. By taking account of these constraints, we here

adopted = 0.01 in conducting simulations.Fig. 7 shows the influence of the number of particles on

the viscosity D due to dissipative forces. It is seen fromFig. 7 that a larger system is required for a smaller num-

ber density to remove the dependence of the results on the

number of particles. This means that the interactions of the

(a)

(b)

(c)

Fig. 5. Influence of time interval on (a) viscosity due to dissipative forces,

(b) viscosity due to momentum, and (c) average velocity of particles

(Derror = (0.43, 2.7) for h = (0.01, 0.0001), respectively, in the caseof n0 = 10, = 25, and = 2.5 (Case A), and (0.051, 0.31) forh = (0.1, 0.0001), respectively, in the case of n0 = 3.16, = 10, and = 1 (Case B), in (a); Kerror = (0.34, 1.1) for h = (0.01, 0.0001),respectively, in Case A, and (1.1, 0.62) for h = (0.1, 0.0001), respec-tively, in Case B, in (b)).

particle of interest with the ambient particles come to de-

pend on the dimensions of the system, unless a sufficiently

large system is used for the cases of small number densities.

As already pointed out, the number density has to be taken

as sufficiently large, and a large system such as N = 500,adopted here, at least has to be adopted to remove the de-

pendence of results on the system size.


14/16


(a)

(b)

(c)

Fig. 6. Influence of shear rate on (a) viscosity due to dissipative forces,

(b) viscosity due to momentum, and (c) average velocity of particles

(Derror = (0.055, 2.7) for = (0.5, 0.01), respectively, in Case A,and (0.0064, 0.32) for = (0.5, 0.01), respectively, in Case B, in (a);Kerror = (0.028, 1.1) for = (0.5, 0.01), respectively, in Case A, and

(0.017, 0.62) for

=(0.5, 0.01), respectively, in Case B, in (b)).

6.5. Influence of, , and number density on radialdistribution function

Finally, we show the results of the radial distribution

function [18] in Figs. 810 in order to clarify the quantitative

dependence of the internal structure on , , and the num-ber density of particles. Figs. 8, 9, and 10 show the influence

of, , and the number density, respectively.It is seen from Fig. 8 that the radial distribution function

is almost independent of the values of , and the curve for

= 10 agrees well with that for = 100 for both cases

Fig. 7. Influence of number of particles on viscosity due to dissipative forces

(Derror = (1.7, 6.1) for N = (1372, 108), respectively, in Case A, and(0.19, 0.68) for N= (1372, 108), respectively, in Case B).

(a)

(b)

Fig. 8. Dependence of radial distribution function on for = 10 (a) forn0 = 3.16 and (b) for n0 = 10.

ofn0 = 3.16 and 10.0. Although the theoretical solutions ofthe viscosity have been derived under circumstances of ne-

glecting the internal structure of the system, that is, under

the molecular chaos assumption, it is very interesting that

the theoretical solutions agree well with the simulation re-

sults (Fig. 1), irrespective of the system being in a gas-like

or liquid-like internal structure, as shown in Fig. 8.

Fig. 9 clearly shows that the radial distribution function

strongly depends on the values of . A more significant

repulsive force acts between particles with increasing the


15/16


(a)

(b)

Fig. 9. Dependence of radial distribution function on (a) for n0 = 3.16and = 10, and (b) for n0 = 10 and = 25.

values of , so that the radial distribution function shows aliquid-like structure for = 25 and = 40. That the valueof is much larger than that ofmeans that the influenceof dissipative and random forces is much smaller than that

of repulsions due to conservative forces. This means that the

merits of the dissipative particle dynamics method cannot

function effectively under such circumstances. It is, there-

fore, presumed that has to be taken as < to makethe dissipative particle dynamics method effective. It is seen

from Fig. 9a that the molecular chaos assumption is satisfied

to a certain degree for = 0.4. To the contrary, the resultsfor = 4 and 25 cannot sufficiently satisfy this assumption.Nevertheless, as already shown in Fig. 3, the viscosity Ddue to dissipative forces is almost independent of the values

of . We may conclude form these results that dissipativeforces do not strongly depend on the internal structure of the

system.

It is seen from Fig. 10 that the radial distribution function

is not strongly dependent on the number density. However,

the curve for the case of n0 = 6.0, in which the particle ofinterest can interact with a sufficient number of ambient par-

ticles, slightly deviates from the results for n0 = 0.2 and 1.0,in which there are not sufficient particles around the particle

of interest.

(a)

(b)

Fig. 10. Dependence of radial distribution function on number density of

particles (a) for = 10 and = 1, and (b) for = 25 and = 2.5.

Fig. 11. Results of radial distribution function obtained by Espanol et al.

[22].

Finally, we compare the present results of the radial distri-

bution function with other simulation results obtained by Es-

panol et al. [22]. They investigated the relationship between

the dissipative particle dynamics and the smoothed particle

dynamics methods. In the concrete, the behavior of the clus-

ters of particles, which correspond to dissipative particles

in the present study, was investigated by means of conduct-

ing equilibrium molecular dynamics simulations of a system

composed of particles with repulsive interactions. Fig. 11


16/16


shows one of the results concerning the radial distribution

function, which were obtained by Espanol et al. [22]. From

the potential curves of the mean force in their paper [22], it

is seen that R in their notation roughly corresponds to 3rin the present one, and that lower curves correspond to re-

sults for a larger value of in the present study. We clearlysee from comparing Fig. 9 (also Fig. 10) with Fig. 11 that theshape of the curves are quite similar between the present and

their results, in which a small increase at very short distances

can be observed more significantly with decreasing values

of. A system with larger values of approaches a usualsystem such as a molecular one, so that dissipative parti-

cles cannot penetrate each other due to a high-energy barrier.

This leads to the fact that the curves of the radial distribution

function comes to have an ordinary shape for a liquid or gas,

which has no small increase within a short distance range.

7. Conclusions

In order to investigate the validity of the dissipative par-

ticle dynamics method, which is a mesoscopic simulation

technique, we have derived expressions for transport coef-

ficients such as viscosity, from the equation of motion of

the dissipative particles. In the concrete, we have shown the

FokkerPlanck equation in phase space, and macroscopic

conservation equations such as the equation of continuity

and the equation of momentum conservation. The basic

equations of the single-particle and pair distribution func-

tions have been derived using the FokkerPlanck equation.

The solutions of these distribution functions have approx-

imately been solved by the perturbation method under the

assumption of molecular chaos. The expressions of the vis-

cosity due to momentum and dissipative forces have been

obtained using the approximate solutions of the distribution

functions. Also, we have conducted nonequilibrium dynam-

ics simulations to investigate the influence of the parame-

ters, which have appeared in defining the equation of motion

in the dissipative particle dynamics method. The theoreti-

cal values of the viscosity due to dissipative forces in the

HK theory are in good agreement with the simulation re-

sults obtained by the nonequilibrium dynamics method, ex-

cept in the range of small numberdensities. There are restric-

tion conditions for taking appropriate values of the numberdensity, number of particles, time interval, shear rate, etc., to

obtain physically reasonable results by means of dissipative

particle dynamics simulations.

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dpd simulation

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