Mathematical Modeling for Colloidal Dispersion Undergoing Brownian Motion

8/6/2019 Mathematical Modeling for Colloidal Dispersion Undergoing Brownian Motion

1/21

Mathematical modeling for colloidal dispersionundergoing Brownian motion

Chaocheng Huang

Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, United States

Received 1 January 2006; received in revised form 1 December 2007; accepted 14 December 2007

Available online 10 January 2008

Abstract

An averaged motion approach for modeling Brownian dynamics for suspension systems of electrically charged particles

in liquid is developed. The continuum model for the motion of particles consists of a system of integral equations coupled

with a degenerate parabolic equation. Existence and uniqueness of global solution for the coupled system are established,

and numerical results for the non-Newtonian viscosity of the mixture in terms of shear rate or Pechlet number are

obtained. The model reveals some non-Newtonian properties such as the well-known shear thinning phenomenon for

the viscosity of colloidal dispersions.

2008 Elsevier Inc. All rights reserved.

Keywords: Brownian dynamics; Colloidal dispersion; Particle suspension

1. Introduction

Understanding the role of viscosities in colloidal dispersions of particles suspended in fluids is very impor-

tant in many industrial applications [1]. In order to compute the viscosity of the particlefluid mixture, we first

need to fully understand the motion of the particles in the liquid. In typical colloidal dispersion systems, par-

ticles are often electrically charged and they interact with themselves as well as with the carrier fluid. The par-

ticles may also undergo the Brownian motion. It is well documented that, due to the dispersion effect on the

carrier fluid, the hydraulic property of the mixture is changed dramatically and consequently that the colloidal

dispersion systems usually behave as non-Newtonian fluids [1]. When the Reynolds number of the carrier fluidis not very small and the particles size is in the range 107109 m, the motion of the individual particles may

be modeled by the Brownian dynamics (BD). In BD models, the dynamics of the carrier fluid is often assumed

to be independent of the particles, whereas the motions of the particles depend on the motion of the fluid only

through Stokes drag. Since one needs to trace individual particles, a BD mode may involves a large number of

equations. For more detailed discussion of BD, we refer to [24]; we also refer to [5, Chapter 15] for numerical

simulations of BD.

0307-904X/$ - see front matter 2008 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2007.12.027

E-mail address: [email protected]

Available online at www.sciencedirect.com

Applied Mathematical Modelling 33 (2009) 978998

www.elsevier.com/locate/apm


2/21

In this paper, we introduce a continuum model for the colloidal dispersions based on a spatial averaging

argument for BD. This approach, which is also referred as to homogenization, has been used in many appli-

cations of upscaling. We assume that particles are fairly densely populated in fluids so that probability density

functions Px; t can be properly defined to describe particle positions. Thus, instead of dealing with individualparticles, we propose to analyze the dynamics of the density functions and their effects on the viscosities of the

mixtures. Towards this end, we shall first develop continuum models for interparticle forces, for hydrody-

namic forces and for the Brownian forces. We shall then use these continuum representations to determine

the governing equations for the density functions P

x; t

of particles with prescribed initial data. Since all

the forces acting on a particle at location x depend also on other particles, these force terms generally are

Nomenclature

a particle radius

A Hamaker constant

A0 upper bounds for bx;y; t

Fx; t interparticle forcekB Boltzmann constant

L size of container

m particle mass

n spatial dimension

N number of particles

Px; t particle densitype Pechlet number

T temperature

ua attractive potential

ur repulsive potential

V0 volume particles occupiedVx; t velocity of carrier fluidw Brownian motion

a ffiffiffi3p =pm2nb Stokes friction

d Dirac measure

e m=6pl0ae0 solvent dielectric

er relative dielectric

_c shear rate of carrier fluid

c rescaled shear rate

~c rescaled shear rate of carrier fluidCk Gamma functionCn; g; s;x;y; t fundamental solutionj inverse Debye constant

k time scaling parameter

l0 viscosity of carrier fluid (water in our case)

l viscosity of mixture

g0 g0a is the distance between two neighbouring particles

w0 surface potential of particle

r12 shear stress of the mixture

s12 shear strain of the mixture

m

ffiffiffiffiffiffiffiffiffiffiffi2kBTp

is diffusion coefficient

~m dimensionless diffusion coefficient

C. Huang / Applied Mathematical Modelling 33 (2009) 978998 979


3/21

functionals of the particle densities Px; t. Consequently, the continuum model consists of system of partiallydifferential equations and integral equations for Px; t. This integro-differential system will then be analyzedmathematically and numerically. Some numerical results based on our model will be presented and be com-

pared with Brownian dynamics simulations.

The paper is organized as follows. In the next section we introduce the Brownian dynamic model for a N-

particle system. In Section 3 we present the averaged motion approach to develop the continuum model forinterparticle forces. The BD model will then be upscaled to the averaged Brownian dynamics (ABD) that con-

sists of a stochastic differential system for random processes ft f1t; f2t where f1t and f2t representthe positions and the velocities of the particles, respectively. The coefficients in the stochastic differential equa-

tion depend on the averaged interparticle force F that is actually a functional of the entire family of solutions

emanating at t 0 from points n (with initial densityP0n) and prescribed initial velocityw0n. In other words,the continuum model is described by a stochastic differential equation strongly coupled with an integral equa-

tion. In Section 4 we use Itos formula to derive the degenerate parabolic equation associated with the stochastic

differential equation. We then introduce the fundamental solution of the degenerate parabolic equation assum-

ing that F is a given function. In Section 5 we formulate the deterministic model for ABD that governs the

dynamics of particles density Px; t and the averaged interparticle force Fx; t. We then set up a coupled inte-gro-differential system for

P;F

in a dimensionless form. In Sections 6 and 7 we mathematically analyze this

system and show that the system has a unique solution for all t> 0. In Section 8 we consider the case wherethe diffusion coefficient m of the Brownian motion vanishes, and show that our model reduces to the averaged

motion model studied in [6,7]. In Section 9 we return to the BD model introduced in Section 2 with represen-

tative physical constants and carry out a dimensional analysis. This yields an ABD model with representative

size parameters. This model is then studied numerically in Section 10, where we reproduce the well-known shear

thinning phenomenon for the viscosities of colloidal dispersions described in [1,2, Chapter 15, 3,4].

2. Brownian dynamics

Consider a N-particle system in a liquid. We assume that all the particles are spherical with radius a and

mass m. Let r be the variable vector in Rn and ri the center of the ith particle i 1; . . . ;N. Set

rij jri rjj; rij ri

rj

jri rjj :The Brownian dynamics is described by, for 1 6 i 6 N,

mri FPi FNi FBi Langevins equation; 2:1where FPi , F

Ni and F

Bi represent the sum of interparticle forces, the sum of hydrodynamic forces and the sum of

Brownian forces, respectively, acting on the ith particle. The hydrodynamic forces are assumed to be the drag

forces:

FNi b_r Vr; t; 2:2where Vr; t is the velocity of the fluid (assumed to be known), b 6pl0a (Stokes friction), and l0 is the vis-cosity of the fluid. The Brownian forces FB

iare stochastic in nature and satisfy [1, p. 66]

hFBi ti 0; hFBi t FBi si 2kBTbdt s 2:3where hi denotes the expected value, kB is Boltzmanns constant, Tthe temperature, and d is the Dirac mea-sure. For electrically charged particles, the interparticle forces are given by

FPi XN

j1;ji

ourijorij

rij; 2:4

where us is a potential function. In the case where the particle surface potential is low and remains constantduring interaction between particles, we assume that potential functions have the following form (see [2,3,5,

Chapter 15])

u ua ur; 2:5

980 C. Huang / Applied Mathematical Modelling 33 (2009) 978998


4/21

where ua represents the attractive force and ur is the repulsive force. The specific choices for ua and ur depend

on physical assumptions. For many applications (see [1,5, Chapter 15]), ua and ur are taken to be

ua A12

4a2

r2 4a2 4a2

r2 2 ln 1 2a

2

r2

& '; 2:6

ur 2pere0w2

0a ln1 ejs

; 2:7where r is the distance between the centers of two neighboring particles, s r 2a is the separation betweenthese two particles, A is the Hamaker constant, er is the relative dielectric, e0 is the solvent dielectric, w0 is the

surface potential of the particles, and 1=j is the Debyes screening length. The repulsive potential (2.7) is de-rived under the assumption that all particles have the same potential on their surfaces and that each particle is

surrounded by a cloud of ions forming a buffer which prevents other particles from reaching it [1, p. 120].

This implies that particles do not collide, that is, r> 2a. For technical reasons, we further assume that

r> 2a 2g0a; g0 > 0; 2:8where 2g0, the relative separation, is assumed to be independent of a. This assumption implies that ua is a

bounded smooth function. We shall use the choice (2.6) and (2.7) just as an example; all of our analysis will

be carried out under general assumptions. We also remark that in some of the analytical results as well as in

the numerical computations later on, we can actually remove the restriction g0 > 0; see Remark 7.1 (in Section7) and Section 10.

3. Averaged Brownian dynamics

In this section, we want to replace the discrete BD model for individual particles by a continuum model for

this density function. In view of (2.4)(2.8), the continuum model for the interparticle force has the

representation

Fx; t ZRnGjx x0j

x

x0

jx x0jPx0; tdx0; 3:1where Px; t is the density of particles and Gs is a function satisfying

Gs 0 if s 6 2g0; Gs is bounded for 0 6 s < 1: 3:2Let xt xt;x (x is in a probability space) be a random process describing particle positions. Then (2.1)leads to

d2x

dt2 Fx; t K dx

dt Vx; t

mdw

dt; 3:3

where m

m1 ffiffiffiffiffiffiffiffiffiffiffiffiffi2kBTb

p, w

t

is the standard Wiener process, and K

y

is a function of y in Rn. To represent

Stokes drag we should take Ky bm1y for all y. However, for technical reasons, we shall assume thatKy is truncated:

Ky bm1y if jyj 6 S0 for some large S0; and 3:4Ky is a bounded smooth function for y2 Rn: 3:5

In most applications, this is not a serious restriction since the expression for the Stokes drag is applicable

only in certain limited ranges of the relative velocity dx=dt Vx; t. In the numerical results, we do not needto use this truncation. The system of(3.1)(3.5) models the dynamics of particles, and is the stochastic descrip-

tion of average Brownian dynamics (ABD).

In the following section we shall recall the construction of the fundamental solution of the degenerate

parabolic equation associated with (3.3). This will be needed in the subsequent sections for establishing a

deterministic continuous model.

http://-/?-http://-/?-


5/21

4. Fundamental solutions

Consider the degenerate parabolic equation

ov

os g rnv bn; g; s rgv 1

2m2Dgv 0 4:1

for vn; g; s, where n; g vary in Rn

, 0 < s < 1, m is a positive constant and bn; g; s is a function in Rn

satisfying

m 6 C0; kbkL1 6 A0 4:2for some positive constants. By [8], there exists a unique fundamental solution Cn; g; s;x;y; t for (4.1), i.e.,

for any fixed x;y; t, Cn; g; s;x;y; t satisfies (4.1) in n; g; s for 0 < s < t, andCn; g; s;x;y; tjst0 dn xdg y;

where the product of the two Dirac measures is understood as the measure defined byZR2ngn; gdn xdg ydndg gx;y; for any g2 C10 :

Furthermore, for any bounded smooth function f

x;y

, the function

vn; g; s ZR2n

Cn; g; s;x;y; tfx;ydxdyis the unique bounded solution of (4.1) in 0 < s < t, satisfying

vn; g; t fn; g:Finally, for fixed n; g; s, Cn; g; s;x;y; t satisfies the adjoint differential equation

ouot

y rxu ry bx;y; tu 12m2Dyu 0

with respect to the variables x;y; t for s < t< 1.Remark 4.1. In [8], it was assumed that bx;y; t is locally Lipschitz continuous. However, the construction ofthe fundamental solution remains unchanged for any uniformly bounded b

x;y; t

, and C

n; g; s;x;y; t

is then

a weak solution of (4.1).

We now briefly review the construction ofC in order to trace how the various estimates depend on b and m.

Let

Rn; g; s;x;y; t jy gj2

2m2t s 6 n x yg

2t s

2m2t s3 : 4:3

A direct computation shows that for s < t, the function

Wn; g; s;x;y; t at s2n expRn; g; s;x;y; t; a ffiffiffi

3p

=p n

m2n 4:4

is the fundamental solution for (4.1) with b 0. In other words, W satisfies, for fixed x;y; t,oW

os g rnW 1

2m2DgW 0; 0 < s < t;

Wn; g; t 0;x;y; t dn xdg y:Furthermore, W has the following semigroup property:Z

R2nWn; g; s; n; g;sWn; g;s;x;y; tdndg Wn; g; s;x;y; t: 4:5

We shall construct Cn; g; s;x;y; t in the form

C

n; g; s;x;y; t

W

n; g; s;x;y; t

Zt

s

ds ZR2n Wn; g; s; n; g;sQn; g;s;x;y; tdndg; 4:6



6/21

where Q is the solution to the integral equation

Qn; g; s;x;y; t bn; g; st rgWn; g; s;x;y; t

bn; g; s Zts

ds

ZR2n

rgWn; g; s; n; g;sQn; g;s;x;y; tdndg: 4:7

Formally a solution of (4.7) can be written in the form

Q X1k0

Qk; 4:8

where

Q0n; g; s;x;y; t bn; g; s rgWn; g; s;x;y; tand, for kP 1,

Qk1n; g; s;x;y; t bn; g; s Zts

ds

ZR2n

rgWn; g; s; n; g;sQkn; g;s;x;y; tdndg:

We now show convergence of the series in (4.8). Note that at

n; g; s;x;y; t

jQ0j

abeR

t s2n rgR

6 5A0amt s2n12 eRR12 6 CA0amt s2n12 eR2 ; 4:9where and in what follows, C is always a universal constant. Next

jQ1j 6CA20m2

Zts

ds

ZR2n

a

s s2n12exp Rn; g; s;

n; g;s2

a

t s2n12exp R

n; g;s;x;y; t2

dndg

and, by changes of variables n 2~n; g 2~g, by (4.4) and (4.5),

jQ1

j6CA20

m2W

n

ffiffiffi2p ;g

ffiffiffi2p ; s;x

ffiffiffi2p ;y

ffiffiffi2p ; t Zt

s

1

s s1

2

1

t s1

2

ds 6CA20

m2

a

t s2n

exp

Rn; g; s;x;y; t

2 :Proceeding by induction, we can obtain

jQk1j 6Ck1Ak20 at s

k2

mk2t s2nC k2

exp R n; g; s;x;y; t 2

;

where Cs is the Gamma function. This yields the convergence in (4.8) and

jQj 6 CA0amt s2n12

exp Rn; g; s;x;y; t2

CA0t s12

m

!: 4:10

Using (4.9) and (4.10) in (4.6), we easily obtain:

Lemma 1. There exists a constant C independent of A0 and m such that

jCn; g; s;x;y; tj 6 aeR

t s2n Cae

R2

t s2n expCA0t s

12

m

! 1

!; 4:11

where R Rn; g; s;x;y; t defined in (4.3).

Remark 4.2. In the case when bx;y; t is also locally Lipschitz, the fundamental solution C can also be con-structed in the slightly different form:

C

n; g; s;x;y; t

W

n; g; s;x;y; t

Zt

s

dsZR2n Qn; g; s; n; g;sWn; g;s;x;y; tdndg; 4:12



7/21

where Q is the solution to a slightly different integral equation

Qn; g; s;x;y; t ry bx;y; tWn; g; s;x;y; t

Zts

ds

ZR2nQn; g; s; n; g;sry bx;y; tWn; g;s;x;y; tdndg: 4:13

In Section 10 we shall only need to calculate Cn; 0; 0;x;y; t, rather than Cn; g; s;x;y; t, and this is much eas-ier to do by using (4.12) and (4.13) (by taking g 0; s 0) than by using (4.6) and (4.7).

5. Deterministic formulation of ABD

Throughout the paper, we assume that the initial particle density P0n satisfies

0 6 P0n 6 C1;ZRnP0ndn < 1; 5:1

where C1 is a positive constant. We also assume that the initial particle velocity w0n satisfies

jrw0nj 6 C2; detI trw0nP h0 for all t> 0; 5:2where I is the unit n n matrix and h0 > 0. The last condition is satisfied if, for instance, the matrix rw0 isnon-negative definite. We introduce the stochastic process vector

ft f1t; f2t def xt;dxtdt

;

where f1t is the position of the particle and f2t is its velocity at time t. By (3.3), ft satisfies the stochastic

differential system

df1t f2tdt; 5:3df2t Ff1t; t Kf2t Vf1; tdt mdwt:

Consider this system for t> s, with initial condition

f1s n; f2s g 5:4and denote the solution by fn;g;st. Let Cn; g; s;x;y; t be the fundamental solution of the degenerate parabolicequation

ou

os g rnu Fn; s Kg Vn; s rgu 1

2m2Dgu 0: 5:5

Since Cn; g; s;x;y; t is the probability density offf1n;g;st x; f2n;g;st yg, it is natural to formulate the aver-aged Brownian dynamics as follows:

Problem (ABD): Find a function Px; t satisfying

Px; t ZR2n

Cn;w0n; 0;x;y; tP0ndndy; 5:6

where Cn; g; s;x;y; t is the fundamental solution of (5.5) in which the coefficient Fx; t satisfies

Fx; t ZRnGjx x0j x x

0

jx x0jPx0; tdx0: 5:7

We observe that the auxiliary function

bPx;y; t ZRn Cn;w0n; 0;x;y; tP0ndn; 5:8



8/21

satisfies the parabolic system

bPt y rxbP ry Fx; t Ky Vx; tbP 12m2DybP 0 5:9

for t> 0, and the initial condition

bPx;y; 0 P0xdy w0x: 5:10Obviously,

Px; t ZRn

bPx;y; tdy: 5:11

Remark 5.1. Problem (ABD) can be formulated for general measures P0. Consider the special case where P0 is

a finite linear combination of Dirac measures at points nj; P0 PMj1pjdn nj. Then

P

x; t

XM

j1pj ZRn Cnj;w0nj; 0;x;y; tdndy:

We can also write

Px; t XMj1pjPrf1j t x;

where Pr is the probability measure, and fjt f1j t; f2j t is the solution of the stochastic differential equa-tion (5.3) with

Fx; t

XM

j

1

< Gjx f1j tjx f1j t

jx

f

1j

t

j

>

and

f1j 0 nj; f2j 0 w0nj:

One can prove (by successive iterations, as in [9, Chapter 5, Section 5]) that this stochastic differential sys-

tem has a unique solution.

6. Some estimates

Lemma 2. If F;P is a solution of Problem (ABD) for 0 < t< T, thenZRnPx; tdx

ZRnP0xdx 6:1

and

jFx; tj 6 kP0kL1 sup jGj; 0 6 t< T; 6:2where G is defined in (3.2).

Proof. Representing the function f 1 in terms of the fundamental solution, we have

ZR2n Cn; g; 0;x;y; tdxdy 1 6:3



9/21

and (6.1) then follows from (5.6). Next, by (3.2) and (5.7),

Fx; t 6 sup jGjZ

jxx0j>2g0Px0; tdx0 sup jGj

ZRnP0xdx:

Lemma 3. If

F;P

is a solution of Problem (ABD) for 0 < t< T, then

Px; t 6 Cexp A1ffiffit

pm

for 0 < t< T; 6:4

where C andA1 are positive constants independent of T (and m).

Proof. From (5.6) and Lemma 1 we obtain

Px; t 6 Cat2n

expA1ffiffit

pm

ZR2n

exp 12Rn;w0n; 0;x;y; t

dndy; 6:5

where by (6.2), A1 is a constant independent of T. To estimate the last integral we change the variables as

n n x y w0

n2 t; y y w0n:Note that by (5.2),

deton; yon;y

det It2rw0 t

2I

rw0 I

! detI trw0P h0: 6:6

It follows that the integral in (6.5) is bounded by, after changes of variables,

1

h0

ZR2n

exp jyj2

4m2t 6j

nj2m2t3

!dndy 1

h0

ZRn

exp 6jnj2

m2t3

!dn

ZRn

exp jyj2

4m2t

!dy

1

h0 mn

t

3n

2 ZRn

exp6jzj2

dzmn

t

n

2 ZRn

exp jz

j2

4 !dz6 Cm2nt2n:Hence, by recalling the definition ofa (see (4.4)),

a

t2n

ZR2n

exp 12Rn;w0n; 0;x;y; t

dndy6 C: 6:7

Substituting this in (6.5), the assertion (6.4) follows. h

Remark 6.1. If we allow g0 0 in assumption (2.8), then, for the potentialsua, ur as in (2.6), (2.7), the functionGs in (3.1) will have a singularity like s1n at s 0. All the estimates in this section then remain unchangedexcept that in (6.4) A1 must be replaces by CkP; tkL1 .

7. Existence and uniqueness for Problem (ABD)

Theorem 1. If (5.1) and (5.2) hold, then there exists a unique solution F;P to Problem (ABD) for all t> 0.

Proof. Let T be any positive number. Denote by AT the set of all functions Fx; t defined inXT fx; t : x 2 Rn; 0 6 t< Tg

having finite norm

kFkL1 kFkL1XT 6 sup jGj:



10/21

For every F 2 AT, we define a function Px; t by (5.6), where C is the fundamental solution for (5.5) corre-sponding to this F, and a function F by

Fx; t ZRnGjx x0j x x

0

jx x0jPx0; tdx0: 7:1

From the proof ofLemma 2, we see that (6.1) is still valid and, consequently, F

x; t

still satisfies (6.2) in X

T.

Hence, F 2 AT. Thus the mapping S defined bySF F

maps AT into itself. We shall next prove that

S is a contraction mapping if T is small; 7:2this will establish existence and uniqueness for a small time interval.

To prove (7.2), we take any F1, F2 in AT and denote the corresponding P;C;Q;Qj in (5.6), (4.7), (4.8) byP1;C1;Q

1;Q1j and P2;C2;Q2;Q2j , respectively. We shall first estimate

C1n; g; s;x;y; t C2n; g; s;x;y; t Zt

s

dsZR2n Wn; g; s;n; g;sQ1 Q2n; g;s;x;y; tdndg: 7:3

As in Section 4, we have the estimates

jQ10 Q20j 6 kF1 F2kL1 jryWj 6 kF1 F2kL1Ca

t s2n12e

R2

and

jQ1k1 Q2k1j 6 kF1 F2kL1Zts

ZR2n

rgWQ1k C Zt

s

ds

ZR2n

rgWQ1k Q2k :

Proceeding by induction and using the estimates for Qk in Section 4, we deduce that

jQ1

k1 Q2

k1j6

kF1 F2kL1Ck2a

t

s

k2e

R2

mk2t s2nCk12 where C is a constant independent of T and m. It follows that

jQ1 Q2j 6X1k0

jQ1k1 Q2k1j 6 kF1 F2kL1Ca

mt s2n12exp

Ct s12m

!and then, from (7.3) (as in the derivation of (4.11)),

jC1 C2j 6 kF1 F2kL1Ca

t s2n expCffiffiffiffiffiffiffiffiffiffit spm

1

e

R2:

Using (6.7) we then obtain

ZR2n

jC1 C2n;w0n; 0;x;y; tjdndy6 Cexp Cffiffitpm

1 kF1 F2kL1and consequently, from the representation (5.6) for P1 and P2,

jP1x; t P2x; tj 6ZR2n

jC1 C2n;w0n; s;x;y; tjP0ndndy6 Cexp Cffiffit

pm

1kF1 F2kL1 ;

where in the last inequality we used the first part of the assumption (5.1). By definition of the mapping S, it

then follows that

kSF1 SF2kL1 6CffiffiffiffiT

p

mkF1 F2kL1 ; 0 < t< T

so that S is a contraction mapping if T is small enough, as asserted in (7.2).



11/21

To extend the solution uniquely to all t> 0, it suffices to extend it to 0 6 t6 T, where T is any positivenumber. Suppose that the solution F;P exists for all 0 6 t6 t0 where 0 < t0 6 T. We shall prove that it canthen be extended uniquely to 0 6 t6 t0 T, where Tis independent oft0 (although it may depend on T); thisclearly will complete the proof of the theorem.

From the semigroup property (4.5) for Wand from (4.6), one can easily derive the semigroup property for C :

ZR2n

Cn; g; s; ~n; ~g;sC~n; ~g;s;x;y; td~nd~g Cn; g; s;x;y; t:

This property can then be used to deduce from (5.8) the relation

bPx;y; t ZR2n

Cn; g; s;x;y; tbPn; g; sdndg:By (5.11) we then have, for t> t0,

Px; t ZR3n

Cn; g; t0;x;y; tbPn; g; t0dndgdy; 7:4where bPn; g; t0 is given by (5.8).Using the relation (7.5) for P1 and P2 and proceeding as in the proof of (7.2), we derive, for t> t0, theestimates

jP1x; t P2x; tj 6ZR3n

jC1 C2n; g; t0;x;y; tjbPn; g; t0dndgdy6 kF1 F2kL1

ZR3n

Ca

t t02nexp

Cffiffiffiffiffiffiffiffiffiffiffit t0pm

1

exp R

2

bPn; g; t0dndgdy;where R Rn; g; t0;x;y; t, and as in the proof of Lemma 3,Z

R2n

Ca

t

t0

2n

exp Rn; g; t0;x;y; t2

dndy6 C:

Hence

jP1x; t P2x; tj 6 CkF1 F2kL1 expCffiffiffiffiffiffiffiffiffiffiffit t0pm

1

ZRn

supn

bPn; g; t0dgand, by (5.8), (4.11) and (6.7), the last integral is bounded by a constant independent oft0. It follows that Sis a

contraction ift t0 < T for some small T> 0 independent oft0. This completes the proof ofTheorem 1 h.

Remark 7.1. It can be shown, by a standard potential theory argument, that the function Cn; g; s;x;y; t isHolder continuous in t and, by means of(5.6), that Px; t is Holder continuous in t. From (5.7) it then followsthat Fx; t is Holder continuous in both x and t. Consequently the fundamental solution Cn; g; s;x;y; t sat-isfies the parabolic equation in the classical sense (cf. Remark 4.1).

We conclude this section by deriving a bound on Px; t for jxj ! 1. From (5.6) and Lemma 1, we have

Px; t 6 C0eCt

t2n

ZRnP0ndn

ZRn

exp Rn;w0n; 0;x;y; t

2

dy; 7:5

where the constant C0 may depend on m. Setting

X x n; Y y w0n

2:

We can write

1

2R

n;w0

n

; 0;x;y; t

3

m2t3

t2

3 jY

w

0

j2

jX

j2

2X

Yt

jY

j2t2& 'P

3

m2t3

h

4 jX

j2

e

jY

j2t2

Cht

2& '



12/21

for any 0 < h < 1, where Ch and e are positive numbers depending on h. It follows thatZRn

exp Rn;w0n; 0;x;y; t

2

dy6 exp 3h

4m2t3jXj2 3Ch

m2t

ZRn

exp 3em2t

jYj2

dy

6 C0 exp

3h

4m2

t

3

jX

j2

3Ch

m2

t ffiffitp

:

Substituting this into (7.5), we obtain

Px; t 6 C0t2n

12

exp Ct 3Chm2t

ZRn

exp 3h4m2t3

jx nj2

P0ndn: 7:6

This yields a decay of Px; t as jxj ! 1 (for any fixed t> 0). In particular, we have the following:Theorem 2. IfP0n is compactly supported in Bd, the ball of radius d centered at the origin, then, for all t> 0,

Px; t 6 C0t2n

12

exp Ct C0t

exp C1jxj d

2

t3

!if jxjP d; 7:7

where C0 andC1 are constants depending only on m and initial data.

8. Case m 0

In [6] the authors introduced a model of averaged motion of charged particles:

d2wx; tdt2

Hwx; t; t; 8:1

wx; 0 0; dwx; 0dt

w0x; 8:2

where Hx; t is the averaged electric force given byHx; t rux; t; 8:3

u is the solution of

Dux; t Px; t in RnnP 3; 8:4ux; t ! 0 if jxj ! 1;

Px; t is the density of particles at time t and, by conservation of mass,Px; t P0w1x; tJw1wx; t; 8:5

where P0x is the initial density and J is the Jacobian determinant.It was proved in [6] that system (8.1)(8.5) has a unique classical solution for small time t> 0, and that a

global solution does not exist in general.

Theorem 3. The model of averaged Brownian dynamics formally reduces, in the case m 0 andKx 0, to themodel (8.1), (8.2), and (8.5) with force Hx; t Fx; t.

Proof. Setting f w w1;w2, we have, by (3.3),dw1 w2; dw2 Fw1; t; 8:6

also,

w1x; 0 x; w2x; 0 w0

x:



13/21

We can rewrite (8.6) in the form

dw

dt Mw; t; Mx;y; t y

Fx; t

: 8:7

With

bP defined as in (5.8), we formally have, from (5.9) with m 0,d

dtbPw1;w2; t obP

ot rxbP ow1

ot rybP ow2

ot obPot

w2 rxbP F rybP 0so that, by (5.10),bPw1;w2; t bPx;y; 0 P0xdy w0xor bPx;y; t P0w11dw12 w0w11; 8:8where w1 w11; w12. For any continuous function gx with compact support, we have

ZRn Px; tgxdx ZR2nbPx;y; tgxdydx:By changing variables n; g w1x;y; t and using (8.8), we find that the right-hand side is equal toZ

R2nP0ndg w0ngw1n; g; tJwn; g; tdndg:

Since, by (8.7) and direct computation,

d

dtJw JwtracerM 0;

we obtain

Jwn; g; t Jwn; g; 0 1:We thus conclude thatZ

RnPx; tgxdx

ZR2nP0ndg w0ngw1n; g; tdndg

ZRnP0ngw1n;w0n; tdn: 8:9

Changing variables x w1n;w0n; t, we find that the right-hand side of (8.9) is equal toZRnP0w11 Jw11 gxdx

where w11 is the inverse of the mapping n#w1n;w0n; t for fixed t. Since g is arbitrary, (8.5) follows. h

Remark 8.1. If we denote by

bPm the density

bPx;y; t corresponding to m > 0, then, by Itos formula,

d

dtbPmw1;w2; t m22 DybPmw1;w2; t 0;

where w1;w2 is a solution of(8.6) (assuming Kx 0). Repeating the proof ofTheorem 3, one can show thatformally, if bPmx;y; t ! bPx;y; t as m ! 0, then Px; t satisfies (8.5).9. Dimensional analysis

In this section we non-dimensionalize the ABD model in R3 using representative numbers for various phys-

ical constants. The related dimensionless coefficients and potentials in the ABD model developed in Section 5

will then be calculated explicitly in preparation for numerical simulations of the next section.



14/21

We take the fluid flow as a shear flow with velocity

V _cx2; 0; 0T

where _c > 0 is the shear rate. The shear stress r12 of the fluid of the particleliquid system is related to theshear rate _c by

r12 l _c;where l is the shear viscosity. For Newtonian fluids, l is independent of _c. Since colloidal dispersions behave

as non-Newtonian fluids, we expect the shear viscosity l to depend on _c, and thus we shall write l l _c. Theshear stress r12 consists of two parts. The first part is due to the carrier (Newtonian) fluid with a constant vis-

cosity l0 > 0. The second part, s12, is caused by the particle motion and thus depends on the particle density P.Therefore

r12 l0 _c s12 or l l0 s12

_c: 9:1

For isolated particles

s12 12V0

XNi;j1;ji

rij;x1rij;x2rij

ouij

orij:

where rij;xk is the kth component of ri rj and V0 is the volume that the particlefluid system occupies; see[2,3]. From (9.1) it follows that

l l0 1

2 _cV0

XNi;j1;ji

rij;x1rij;x2rij

ouij

orij: 9:2

In applications, one is interested in understanding how the viscosity l depends on the shear rate _c.

We recall that the discrete interparticle force FPi is defined by (2.4) and (2.5) where, by (2.6) and (2.7),

dua

dr 2Aa

2

3

r

r2 4a22 1

r3 1r3 2a2r

!; 9:3

dur

dr 2pere0w20ja

1

ejr2a 1 : 9:4

Consider a volume element DV in R3 with a point x0 2 DV, and denote by P the density of particles. Thenumber of particles contained in DV is given by

NDV 34pa3

jDVjPx0; t; jDVj volume of DV:

Therefore, for any smooth function f

x

,X

xj2DVfxj % 3

4pa3fx0jDVjPx0; t % 3

4pa3

ZDV

fxPx; tdx:

Let L be the characteristic length scale of the fluid region (for instance, the size of a container). Assuming that

fx fx1;x2 is independent ofx3, it follows that:XNj1fxj % 3

4pa3

ZD3L

fxPx; tdx0 dx3 3L2pa3

ZD2L

fxPx; tdx0; x0 x1;x2;

where DkL L;Lk is the fluid region for k 3. Since the velocity is independent of x3, it is reasonable toassume that particle motion takes place only in x0-plane and that P

x; t

is also independent of x3. Hence,

by (9.3), for small a, the attractive force



15/21

XNj1ji

dua

drjxi xjj x

i xjjxi xjj

can be approximated by

LApa

ZD2L

ri

r2i 4a22 1r3i

1:r3i 2a2ri

! xi xjxi xjPx; tdx; 9:5

where ri jxi xj. Analogously by (9.4), the repulsive force

Xj1ji

dur

drjxi xjj x

i xjjxi xjj

can be approximated by

ere0w20ja3L

a3 ZD2L

1

ejri2a 1xi

x

jxi xjPx; tdx: 9:6

Therefore, in (5.7) the potential G is given by Gs Gasv Grsv, where

Gas LApam

s

s2 4a22 1

s3 1s3 2a2s

!;

Grs ere0w20ja3L

a3m

1

ejs2a 1and v is the characteristic function ofD2L n D22a1g0.

Let e be the inertial time scale, and k the real time for computation, scaled on e [1, p.162]. By [1, p. 442],

e mb

m6pl0a

: 9:7

We scale the phase variables and the time respectively, by

x L~x; n L~n; y N~y; g N~g; t ke~t; s ke~s; 9:8where

N Lke

9:9

is the scale for the velocity. In the new variables, ABD model (5.5)(5.7) becomes

eP~x;~t ZR2R2

eC~n; ~w0~n; 0; ~x; ~y;~tP0~nd~nd~y; 9:10eF~x;~t Z

R2

eGj~x ~x0j ~x ~x0j~x ~x0j eP~x0;~td~x0; 9:11where eC~n; ~g;~s; ~x; ~y;~t is the fundamental solution of

o~u

o~s ~g r~n

eF~n;~s

eK~g

eV~n r~g~u 1

2~m2D~g~u 0; 9:12

http://-/?-http://-/?-


16/21

where

e_c ke_c LN

_c;

eK~n keb

m~n k~n for j~nj < L1S0;

eV~n e_c~n20

!since g Vn N~g eV~n;

~w0~n 1N

w0n;eGs eGas~v eGrs~v

and

eGas keL

2

NGaLs A1 ss2 4a2L22

1

s3 1s3 2a2L2s

!;

eGrs keL2N

GrLs U1j1 1ejLs2aL1 1 ;

~v is the characteristic function ofD21 n D22aL11g0, and

~m2 kem2

N2 2kekBTb

m2N2; 9:13

A1 kAepaNm

; 9:14

U1 ere0w203keL3

a3mN; 9:15

j1

ja

particle radius

double layer thickness 9:16

are dimensionless constants. The relationship between the scaled and unscaled quantities for P, F, C areeP~x;~t Px; t;

eF~x;~t keNFx; t;eC~n; ~g;~s; ~x; ~y;~t L2N2Cn; g; s;x;y; t:

Physically, a2me2

is the inertial energy, A is the dispersion energy, ere0w20a is the electrostatic energy, and 2kBT is

the thermal energy (see [1, p. 465]). We choose the initial density such that

ZD31 eP0xdx /0jD31j 8/0/0 is volume fraction: 9:17The relation (9.2) becomes

l l0 m

2 _cV0

3L

2pa3

ZD4L

Gjx x0j x1 x01x2 x02

jx x0j Px; tPx0; tdxdx0

or in the dimensionless form,

l

l0 1 l1

ZD4

1

eGjx x0j x1 x01x2 x02jx x0j ePx; tePx0; tdxdx0; 9:18where, since V0

2L

3,



17/21

l1 ke

2l0e_c8L3 3L2pa3 m keL

2

N

1L5 9:19

is also dimensionless.

We now compute the above constants by using representative physical constants. In the SI unit system,

these constants are chosen and or calculated as follows:

Symbol Name Expression Value k 103 Referencel0 Viscosity of water 8:91 104 [1, p. 507]a Radius of particle 107 m Mass of particle 8pa3=3 8:38 1021 kB Boltzmanns constant 1:38 1023 [1, p. xvii]T Temperature 300

A Hamakers constant 24kBT 9:94 1020 [1, p. 260]j1 Debyes length a 107 [1, p. 214]ere0w

20a Electrostatic 150kBT 6:21 1019 [1, p. 472]

b Stokes friction 6pl0a 1:68

109 (??)

e Inertial time m=b 5 1012 (9.7)L Length scale 106 N Velocity scale L=ke 2:0042 102 (9.9)m Brownian

m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2bkBT

p=m 4:45 105 (3.3)

Using the above quantities, we obtain

Symbol Expression Value (k 103) Reference~m2 2:46 1023k3L2 2:46 102 (9.13)A1 9:40 1016k2L1 9:40 104 (9.14)U1 5:53 10

7

k2

L2

5:53 101

(9.15)j1 ja 1 (9.16)

l1 5:63 1013L2k1e_c1 5:63 102e_c1 (9.19)Remark 9.1. We observe that, by the change of variables x aL1y,

ZD2

1

eGajxjdx A1La

ZD2L=a

nB22g0

jyjjyj2 42

1

jyj3 1

jyj3 2jyj

!dy

9:4 103 ZD2L=a

nB22g0jyjjyj2 42 1jyj3 1jyj3 2jyj !dy

9:4 103ZLa1

2103

jyjjyj2 42

1

jyj3 1

jyj3 2jyj

!dy;Z

D21

eGrjxjdx U1j1a2L2

ZD2L=a

nB22g0

1

expjyj 2 1 dy

5:53 101ZD2L=a

nB22g0

1

expjyj 2 1 dy

2p

5:53

101 Z

L=a

2104

1

expjyj 2 1dy:

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


18/21

The two integrands on the right-hand sides are integrable away from r 2. The ratio between the factors ofthe attractive and the repulsive forces is of order 102. The first integral depends on the relative separation g0.

Some numerical values for supFa and supFr, for eP0 2:4~v and several choices ofg0, are shown in the follow-ing table:

g0 5 1 101 102 103 104

sup eFa 8:2 104 1:2 102 0.2 1.7 17.8 177.1sup eFr 5:3 102 2.4 5.3 5.8 5.8 13.9As in [1, p. 69], we introduce the diffusion coefficientD0 of an isolated spherical particle of radius a in a liquid

with viscosity l0 by D0 kBT=6pl0a, and the Pechlet number [1, p. 464] (before scaling) Pe a2 _c=D0. ThePechlet number quantifies the relative importance of the Brownian force to the shear forces. IfPe( 1, thenBrownian forces dominate. IfPe) 1, then the shear forces dominate. By the scaling in (9.8), we obtain

Pe

a2

e_c

D0ke

27pl20a

e_c

2kBTk

:

9:20

The range of interests for (dimensionless) e_c is the range corresponding to the Pechlet number in the interval

101 < Pe < 10. Since, by (9.20),

e_c 2kBTk27pl20a

Pe 1:23 106Pe;

this range corresponds to

1:23 107 < e_c < 1:23 105: 9:21In the next section we demonstrate an example ofg

e_c for

e_c in the range (9.21).

10. A simple numerical example

In this section, we compute a simple example to illustrate our model is consistent with particle method.

Extensive numerical schemes and numerical analysis for our ABD model will be discussed in separate papers.

I choose the initial data

w0 0;P0x 8/0 for x 2 D1=2;0 otherwise;

&10:1

where /0 is the volume fraction of particles, and compute the viscosity l as a function of _c. This initial data

represent the situation that particles stay away from the boundary. The choice of the constant for P0 is con-

sistent with (9.17). By Theorem 2, one sees that the density Px; t decays exponentially for large jxj. It followsthat, in the dimensionless form (9.10)(9.12), the corresponding density eP is very small for jxj > 1 when L isreasonably large. Therefore, we shall assume that Px; t 0 for jxj > 1. Since we assumed that the velocityVx; t Vx1;x2; t as well as all the quantities we need to compute are independent ofx3. The simulation willactually take place only in the domainD21 1; 12. We shall first compute the density eP of the dimensionlessABD model. For convenience, we shall drop all the tildes in (9.10)(9.12). By (4.12) and (4.13), formula (9.10)

(without tildes) can be rewritten as

Px; t ZD4

1

Cn; 0; 0;x;y; tP0ndndy bP0x; t Z11

Z11Hx;y1;y2; tdy1 dy2; 10:2

where

bP0x; t Z1

1 Z1

1 Z1

1 Z1

1W

n; n; 0; 0;x;y1;y2; t

P0

n

dn1 dn2 dy1 dy2;

10:3



19/21

Wn; g; s;x;y; t is defined in (4.4), and Hx;y; t defined by

Hx;y; t Zt

0

ds

Z11

Z11

Z11

Z11

Z11

Z11Qn; 0; 0; n; g;sWn; g;s;x;y; tP0ndndgdn;

where Q is defined by Eq. (4.13). By (4.13) and integration by parts, we find that H

x;y; t

satisfies

10-1

100

101

Pe

0

50

100

150

A = 24

= 0.30

0

Fig. 1. Hamaker number A = 24kBT, 30% volume fraction of particles, time step = 0:01.

10-1

100

101

Pe

0

50

100

150

A = 50

0 = 0.3




20/21

Hx;y; t Zt

0

ds

Z11

Z11

Z11

Z11

bn; g;s rgWn; g;s;x;y; tHn; g;sdndg 10:4

Zt

0

ds

Z11

Z11

Z11

Z11bn; g;s rgWn; g;s;x;y; t

Z11

Z11Wn; 0; 0; n; g;sP0ndndndg;

where

bx;y; t Fx; t ky Vx: 10:5

10-1

100

101

Pe

0

50

100

150

200

250

A = 24

= 0.38

0

0

0.2


10-1

100

101

Pe

0

50

100

150

200

250

A = 50

= 0.380

0.2




21/21

The ABD model then reduces to (9.11) (without tildes) and Eqs. (10.2)(10.5).

We shall use an explicit iteration scheme to solve the system of integral equations as follows. We start with

Px; 0 P0x defined in (10.1). The force term Fx; t is evaluated by (9.11) in terms of Px; t. We thenadvance time by the Euler forward scheme to compute Hx;y; t Dt from the integral equation (10.4), andPx; t Dt from (10.2). All the spatial integrations are evaluated numerically using the 5 points Gauss quad-rature in [-1,1]. The viscosity

l

l0 is finally calculated by (9.18). The results with the time stepD

t 0:01 for/0 30% and 38% are presented in Figs. 1 and 3. Figs. 2 and 4 give similar results when the Hamaker con-stant is chosen as 50kBT (instead of 24kBT). The four graphs show the shear thinning phenomenon in the range

of Pechlet number 101 < Pe < 10: the viscosity first decreases rapidly and then decreases more gradually untilit levels off. The graphs are similar to those in [5, p. 166] by the particle method.

11. Conclusions

Within the microscopic range where Brownian dynamics is valid, colloidal dispersions can be described by

the ABD model presented in Section 5. This continuous model has a unique solution for all time. The model

reduces to the system of integral equation (10.2)(10.5) where Fis defined by (9.11) (without tildes). The com-

putational time for solving this system could be much less than required for the particle method, especially

when particles are concentrated in a compact subdomain. A numerical example obtained by the ABD schemeexhibit the same shear thinning phenomenon as computed by the particle method (as in [1,2, Chapter 15, 3,4]).

References

[1] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge, England, 1989.

[2] H.A. Barnes, M.F. Edwards, L.V. Woodcock, Applications of computer simulations to dense suspension rheology, Chem Eng. Sci. 42

(1987) 591608.

[3] D.M. Heyes, Rheology of molecular liquids and concentrated suspensions by microscopic dynamical simulations, J. Non-Newtonian

Fluid Mech. 22 (1988) 4785.

[4] D.M. Heyes, J.R. Mclrose, Brownian dynamics simulations of models of hard sphere suspensions, J. Non-Newtonian Fluid Mech. 46

(1993) 128.

[5] A. Friedman, Mathematics in Industrial Problems, Part 5, IMA, vol. 49, Springer-Verlag, 1992.

[6] A. Friedman, C. Huang, Averaged motion of charged particles under their self-induced electric field, Indiana Univ. Math. J. 43 (1994)

11671225.

[7] A. Friedman, C. Huang, Averaged motion of charged particles in a curved strip, SIAM J. Math. Anal. 57 (6) (1997) 15571587.

[8] M. Weber, The fundamental solution of a degenerate partial differential equation of parabolic type, Trans. AMS Math. Soc. 71 (1951)

2437.

[9] A. Friedman, Stochastic Differential Equation and Applications, vol. 1, Academic Press, New York, 1975.

[10] W.B. Russel, Dynamics of concentrated colloidal dispersions. Statistical mechanical approach, in: M.C. Roco (Ed.), Particulate Two-

phase Flow, Butterworths, 1991.

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

Documents

Mathematical Modeling for Colloidal Dispersion Undergoing Brownian Motion