flm1000160a - Zia2010p11494J_Fluid

J. Fluid Mech.

(2010),

vol.

658,

pp.

188–210.

Cambridge University Press 2010

doi:10.1017/S0022112010001606

Single-particle motion in colloids:

force-induced diffusion

ROSEANNA N. ZIA

†

AND

JOHN F. BRADY

Department of Mechanical Engineering, California Institute of Technology, Pasadena, CA 91125, USA

Department of Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA

(Received

7 July 2009; revised 26 March 2010; accepted 29 March 2010;

first published online 9 June 2010)

We study the fluctuating motion of a Brownian-sized probe particle as it is dragged

by a constant external force through a colloidal dispersion. In this nonlinear-

microrheology problem, collisions between the probe and the background bath

particles, in addition to thermal fluctuations of the solvent, drive a long-time diffusive

spread of the probe’s trajectory. The influence of the former is determined by the

spatial configuration of the bath particles and the force with which the probe perturbs

it. With no external forcing the probe and bath particles form an equilibrium

microstructure that fluctuates thermally with the solvent. Probe motion through

the dispersion distorts the microstructure; the character of this deformation, and

hence its influence on the probe’s motion, depends on the strength with which the

probe is forced,

ext

, compared to thermal forces,

kT/b

,definingaP

eclet number,

ext

(

kT /b

), where

is the thermal energy and

the bath particle size. It is

shown that the long-time mean-square fluctuational motion of the probe is diffusive

and the effective diffusivity of the forced probe is determined for the full range of P

eclet

number. At small

Brownian motion dominates and the diffusive behaviour of the

probe characteristic of passive microrheology is recovered, but with an incremental

flow-induced ‘microdiffusivity’ that scales as

micro

∼

, where

is the volume

fraction of bath particles and

is the self-diffusivity of an isolated probe. At the

other extreme of high P

eclet number the fluctuational motion is still diffusive, and the

diffusivity becomes primarily force induced, scaling as (

ext

/η

)

, where

is the vis-

cosity of the solvent. The force-induced microdiffusivity is anisotropic, with diffusion

longitudinal to the direction of forcing larger in both limits compared to transverse

diffusion, but more strongly so in the high-

limit. The diffusivity is computed for

all

for a probe of size

in a bath of colloidal particles, all of size

, for arbitrary

size ratio

a/b

, neglecting hydrodynamic interactions. The results are compared with

the force-induced diffusion measured by Brownian dynamics simulation. The theory

is also compared to the analogous shear-induced diffusion of macrorheology, as well

as to experimental results for macroscopic falling-ball rheometry. The results of this

analysis may also be applied to the diffusive motion of self-propelled particles.

Key words:

colloids, particle/fluid flow, rheology, Stokesian dynamics, suspensions

1. Introduction

The motion of active microscale particles driven through complex fluids is a physical

process central to many current scientific problems: vesicle trafficking in cells, artificial

†

Email address for correspondence: roseanna@caltech.edu

Single-particle motion in colloids: force-induced diffusion

189

nano-motors and nanotherapeutic drug delivery devices are a few important examples

(Janke

et al.

2005; Shirai

et al.

2005; Heath, Davis & Hood 2009). The increased

demand for knowledge of this small-scale behaviour has made microrheology a key

step in the understanding, use and design of such systems. Among the collection

of techniques known as microrheology, most involve tracking the movement of a

colloidal particle (or a set of particles) in order to determine the properties of the

surrounding environment (MacKintosh & Schmidt 1999). There are two main types of

particle tracking microrheology:

passive

– tracking the random motion due to thermal

fluctuations – and

active

– applying a constant or oscillatory force to the particles,

for example by using optical tweezers or magnetic fields. A detailed comparison can

be found in Khair & Brady (2006). Most microrheological work to date has focused

on passive microrheology, to obtain linear viscoelastic properties by correlating the

random thermally driven displacements of tracers to the complex modulus through a

generalized Stokes–Einstein–Sutherland relation – a process which is well understood

but limited in its scope to equilibrium systems. Yet as noted above, many systems

of practical interest are driven out of equilibrium and display (indeed, rely upon)

nonlinear behaviours. Recently a body of work has emerged focusing on this active

nonlinear microrheology regime (Habdas

et al.

2004; Meyer

et al.

2005; Squires &

Brady 2005; Khair & Brady 2006; Wilson

et al.

2009). In such a system, tracer

particles undergo displacements not only due to random thermal fluctuations, but

also due to the application of an external force applied directly to the tracer, or

‘probe’. The dispersion is driven out of equilibrium, and as with macrorheology,

dynamic responses such as viscosity can be measured. Since the tracer interrogates

the material at its own (micro)scale, much smaller samples are required compared

to traditional macrorheology, and localized material heterogeneity can be explored.

This is a particular benefit for rare biological materials and small systems such

as cells. Khair & Brady (2006) recently established the theory that predicts the

microviscosity of dilute systems of colloids, and defined the relationship between

micro- and macroviscosity – a critical step in the development of microrheology as

an experimental tool. Recent experiments confirm the theory (and raise additional

questions) (Meyer

et al.

2005; Squires 2008; Wilson

et al.

2009).

But in both theory and experiment, the focus thus far has been on the mean

response of the material – the viscosity – and far less work has been devoted to

particle fluctuations that occur due to collisions between the probe and bath particles.

As the probe moves through the dispersion it must push neighbouring particles

out of its way; these collisions induce fluctuations in the probe’s velocity, scattering

it from its mean path. Is this scattering diffusive? Is it isotropic? How important

is the scattering compared to the Brownian diffusion the probe simultaneously

undergoes? The answers to these questions are fundamental to understanding the

motion of an active microscale particle – important for both scientific and technology

considerations. Little work has been published on this topic (Habdas

et al.

2004), even

though it has major implications for a wide range of technologies beyond material

interrogation.

Previous study of particle fluctuations in colloidal macrorheology shows that

imposing a shearing flow on a suspension enhances particle diffusivity (Leighton &

Acrivos 1987; Morris & Brady 1996; Brady & Morris 1997; Breedveld

et al.

1998).

The imposed shear flow drives the microstructure from equilibrium, giving rise

to mechanisms of diffusion not present in a quiescent suspension: a deformed

microstructure and interparticle collisions. A forced microrheological probe also

imposes a flow that drives the suspension from equilibrium, again leading to a

190

R. N. Zia and J. F. Brady

(

)

–

(

)

F

Figure 1.

Microstructural deformation under (

) macrorheological shear flow and (

)micror-

heological forcing. In simple shearing motion (

), bath particles accumulate along the

compressional axis and deplete along the extensional axis. A tracer at the origin (the dark

sphere) experiences both in-plane and out-of-plane bath particle gradients. In (

), the (dark)

tracer itself deforms the microstructure, accumulating particles on its upstream face and leaving

a wake of depletion behind, creating an axisymmetric structure with only longitudinal and

transverse components.

deformed microstructure and interparticle collisions – and hence to an analogous

force-induced diffusion, or ‘microdiffusivity’. It is the primary objective of this study

to extend the theoretical model of active nonlinear microrheology to one that is

explicit in the fluctuations of the microstructure, and thereby develop expressions for

predicting the resultant force-induced diffusion.

It is also useful to ask whether the qualitative agreement between micro- and

macroviscosity can be extended to the micro- and macrodiffusivity. Both the shear-

and force-induced diffusions grow out of fundamentally similar mechanisms: external

forcing causes the tracer to scatter off of the microstructure, rather than wander

passively through it. But the directionality and magnitude of the scattering depend

on the shape of the deformed microstructure, and this asymmetry is distinct for the

two cases, as illustrated in figure 1. The idea of whether a direct correlation between

macro shear-induced diffusion and micro force-induced diffusion is possible (or even

necessary) will be explored in this study, and a comparison is sought between the two.

To build up a physical model, we follow the example of Squires & Brady (2005)

and consider the motion of a Brownian probe driven by an externally applied force

through a dispersion of neutrally buoyant-force- and torque-free colloidal particles.

The size ratio of probe to bath particle is arbitrary. As the probe particle moves

through the suspension it must push neighbouring particles out of its way; a build-

up of background particle concentration forms in front of the advancing probe

and a deficit or wake trails it. The bath particles (including the probe) undergo

Brownian diffusion due to thermal fluctuations of the solvent, which acts to restore

the deformed microstructure to an equilibrium configuration. The ratio of advective

forcing to entropic restoring force is the P

eclet number,

ext

(

kT /b

), where

is the thermal energy and

the bath particle size. In the linear-response regime

(

→

0), diffusion of the bath particles dominates. As the probe forcing increases

and

grows beyond

∼

(1) – the active nonlinear-response regime – advection

plays an increasingly important role in the shape of the distorted microstructure.

Since the details of this shape govern the strength and likelihood of collisions between

probe and bath particles, its determination is critical to understanding the resultant

force-induced diffusive spread of the probe’s trajectory.

Single-particle motion in colloids: force-induced diffusion

191

The spatiotemporal distribution of bath particles obeys a Smoluchowski equation.

Prior treatments of active microrheology formulated the expression for the steady

microstructure moving relative to a fixed probe. We extend this approach by

considering both the steady microstructure deformation and fluctuations in the

microstructure responsible for diffusion of the probe. The resulting Smoluchowski

equations are solved for all

by a combination of perturbation methods and

numerical computation to obtain both the steady microstructure along with a new

quantity, the probability-weighted collisional displacements of the probe relative to

the bath particles.

Scaling arguments are useful for predicting the behaviour for extreme values of the

eclet number. For random-walk processes, the diffusivity scales as

∼

(1.1)

where

is the size of a probe step and

is the decorrelation time. For very weak

forcing,

1, Brownian diffusion dominates the motion; so the time scale is

∼

(

)

, where

is the self-diffusivity of an isolated probe particle of size

In this linear-response regime,

∼

(

). The number of diffusive steps depends

on the number of bath particle collisions; thus, for very small

, the microdiffusivity

should scale quadratically in the forcing and linearly in volume fraction of bath

particles

micro

∼

(1.2)

For large forcing,

1, the time scale is now advective,

∼

(

)

,theprobe

can move

∼

(

) in that time and the force-induced microdiffusivity should scale

linearly with the P

eclet number and in volume fraction of bath particles:

micro

∼

(1.3)

In the remaining sections of this paper, we propose and examine an extended

model for active nonlinear microrheology that is explicit in the probe fluctuations, and

explore the resultant force-induced diffusion. In

2, we formulate the Smoluchowski

equation that governs the evolution of the microstructure in physical space, along

with kinematic expressions for the probe flux. The latter comprises advective and

interparticle contributions, and from these the diffusive flux of the probe is extracted

and separated into Brownian and flow-induced components. To make analytical

progress an assumption of diluteness is made. In

3 the Smoluchowski equation is

separated into steady and fluctuating components, completing the formulation of the

problem. In

4.1 the case of asymptotically weak probe forcing,

1, is studied.

Regular perturbation expansions are sufficient to obtain the

(

) correction to the

long-time self-diffusivity, which corresponds to passive diffusion:

∼

−

where

is the (dilute) volume fraction of bath particles and

is the diffusivity

of an isolated probe (Batchelor 1976). Higher orders in

are required in order

to find the first effect of the forcing on particle self-diffusion and we proceed to

(

) whereupon the problem becomes singular, which requires the use of matched

asymptotic expansions. In

4.2 we shift focus to the opposite extreme of very strong

probe forcing, or

1, and the nonlinear response of the microstructure is exposed.

This limit is also singular with a boundary layer at the probe surface. To solve for

probe fluctuations at arbitrary values of the P

eclet number, a numerical solution

is required, and a finite difference scheme is employed to this end in

4.3. In

we present an alternative solution of the problem based upon measurements of

192

R. N. Zia and J. F. Brady

the probe’s displacements obtained by Brownian dynamics simulation, presented

juxtaposed to the Smoluchowski results. Section 6 is devoted to a comparison of

microdiffusivity to the macrodiffusivity (theory and experiments). Throughout, we

consider a simplified model that neglects hydrodynamic interactions between particles;

this simplification affords insight into the basic physics of diffusive behaviour arising

from interparticle forcing, but could be extended to include other interparticle forces

such as hydrodynamic interactions; the approach also offers a direct solution to related

problems such as self-propelled objects. The study is concluded with a discussion in

7, including a brief discussion of hydrodynamic interactions, self-propulsion and

non-spherical shapes.

2. Microdiffusivity

The theoretical framework of active microrheology begins with a simple model: a

single Brownian probe particle of radius

is dragged by a constant external force

ext

through a dispersion of colloidal particles, all of size

, which are immersed in

a solvent of density

and viscosity

. The importance of fluid inertia relative to the

viscous shearing forces is characterized by the Reynolds number,

ρUa/η

,where

is the characteristic velocity of the moving probe, and for micrometre-sized probes

1, so that the fluid mechanics are governed by Stokes flow. The advective forcing

of the probe acts to deform the microstructure of the bath, while the Brownian motion

of the bath particles counteracts it in an attempt to restore equilibrium. This interplay

drives fluctuations in the probe’s velocity that give rise to diffusive behaviour. The

primary goals of this section are to examine the dispersive contributions to the probe’s

flux relative to the bath, formulate the expression for the force-induced component

and show that the force-induced dispersive motion is indeed diffusive.

We begin by defining the hard-sphere model for the interactive potential

(

)

between a particle of size

and a particle of size

located at positions

and

respectively; they are separated by

−

. Thus defined, the particles exert no

force on each other until their surfaces touch,

, at which point an infinite

repulsive potential is exerted to prevent their overlap:

(

{

∞

,r<a

,r>a

(2.1)

In general, the radii

and

at which particles exert the hard-sphere force are not the

same as their hydrodynamic radii,

and

, the surface at which the no-slip boundary

condition is obeyed. Various physical conditions of the colloids or the solvent can

extend the effective size of the particle beyond the hydrodynamic radius, e.g. steric

hindrance or an ionic screening layer; two particles may then experience a hard-sphere

repulsive force at overlap of their increased effective or ‘thermodynamic’ radii.

Following the excluded annulus model of Morris & Brady (1996) and Bergenholtz

et al.

(2002), the ratios

λ

a/a

and

λ

b/b

give the relative importance

of hydrodynamic interactions between particles (figure 2). When

λ

λ

∼

(1),

hydrodynamic interactions strongly affect the particle configuration; as

λ

λ

→

the particles get close enough that short-range lubrication forces become important.

We shall assume a system of particles for which

λ

λ

1, where hydrodynamic

interactions are negligible compared to interparticle and thermal forces. This model

captures the essential features of the dispersive process while keeping the analyses

Single-particle motion in colloids: force-induced diffusion

193

F

ext

Figure 2.

Pair interaction for the probe and bath particle.

as simple as possible; the effect of hydrodynamic interactions are discussed briefly

We consider the probe amid a dispersion of

−

1 bath particles in a configuration

. The distribution of particles is given by an

-particle probability density,

(

,...,

;

), which obeys a Smoluchowski equation

∂P

∂t

∑

∇

(2.2)

where the sum is over all particles in the dispersion, and the flux of particle

is given

(

;

)

−

∑

·∇

(ln

(

;

/kT

)

(

;

)

(2.3)

Here,

is the thermal energy,

is the relative Brownian diffusivity

between particles

and

is the mobility tensor relating the velocity of

particle

to the force exerted on particle

. The first term on the right-hand side of

(2.3) is the advective flux of particle

with velocity

, the second term is the flux

due to entropic gradients in the microstructure and

(

∇

/kT

)

gives the flux

of particle

due to the forcing of interactive potential with particle

In the dilute limit, only pair interactions are important, and the probability

(2.2) and (2.3) reduces to the pair probability of finding the probe at position

and

a bath particle at position

. To analyse the relative flux between probe and bath, it

is convenient to change to a frame of reference moving with the probe, placing the

probe at

, and a bath particle at

−

. The pair Smoluchowski equation

becomes

∂P

(

;

)

∂t

∇

(

−

)=0

(2.4)

Here, the subscripts

and

refer to the probe and the bath particle, respectively. We

are interested in the flux of the probe relative to any other particle, and so we integrate

(

;

) over all possible positions of a bath particle, to obtain the single-particle

Smoluchowski equation for the probe particle. Applying the divergence theorem and

194

R. N. Zia and J. F. Brady

noting that relative flux is zero at contact and at infinity, we have

∂P

(

;

)

∂t

∇

(2.5)

where

≡

∫

. The unbounded domain of the probe makes a transformation

to Fourier space convenient. Taking the Fourier transform, denoted by

,ofthe

Smoluchowski equation (2.5) yields

∂

(

;

)

∂t

(2.6)

where the average flux

from (2.3) has also been transformed to Fourier space:

(

−

)

(

;

∫

∇

(

;

(2.7)

Here,

ext

π

ηa

is the probe velocity due to the imposed constant external

force,

≡

the probe self-diffusivity, i the imaginary unit and the angle brackets

denote an ensemble average over all possible suspension configurations.

(

;

)

and

(

;

) are the Fourier transforms of

(

;

)and

(

;

), respectively.

The last term in (2.7) explicitly preserves the effect of bath particles on the probe.

In order to determine the average probe flux, the distribution of the bath relative to

the probe must be determined. To this end we define the structure function

(

;

)

≡

(

;

)

(

;

)

(2.8)

where

is the undisturbed number density of bath particles far from the probe.

Expression (2.8) is similar to the familiar definition of the pair-distribution function

in physical space,

(

;

(

;

)(since

is the origin, it is typically omitted).

But

(

;

) is not simply the Fourier transform of

(

;

). Rather, we have

defined in Fourier space the microstructure

(

;

) where the

-dependence explicitly

preserves fluctuations of the probe relative to the origin.

Also note that

(

) is not to be confused with the structure factor,

(

[

(

)], corresponding to a Fourier transform with respect to the

separation vector

between the probe and the bath particle. Indeed, we solve for

the distribution of bath particles relative to the probe in physical (real) space

.The

Fourier transform variable

is with respect to the absolute position of the probe,

. To determine the diffusive motion of the probe one imagines a concentration

gradient of a dilute collection of probes; these physical-space gradients correspond to

algebraic multiples of the wave vector,

, in Fourier space. The probe flux down this

concentration gradient is influenced by the interaction with bath particles distributed

according to

(

), a distribution that must be determined for all

Combining (2.7) and (2.8) we obtain the following for the steady average probe

flux:

[

−

∫

∇

(

]

(

)

(2.9)

For the long-time self-diffusion of the probe, we consider the short wave vector (long

length scale) limit and expand

(

)forsmall

, corresponding to a weak gradient

in the ‘concentration of probes’:

(

)+i

(

···

(2.10)

Single-particle motion in colloids: force-induced diffusion

195

which immediately yields the two terms governing the scattering of the probe: the

steady microstructure,

(

), and the probability-weighted displacement of the probe,

(

) – i.e. the likelihood of a collision and the strength and direction of the probe

displacement upon collision, respectively. Substituting this into the expression for

probe flux yields

([

∫

∇

(

]

−

[

−

∫

∇

(

])

(

)

(2.11)

where

is the identity tensor. Examination of (2

11) reveals the effect of the bath

particles on both the mean (

(1)) and fluctuating (

(

)) response of the overall

suspension. The first bracketed term gives the probe’s average speed through the fluid:

is reduced by the entropic reactive force of the microstructure, as given by the

integral term. This reduction in probe speed due to the suspended particles was used by

Squires & Brady (2005) to define the microviscosity. Recalling that i

terms represent

diffusion, the second bracketed group gives the effective diffusivity of the probe; the

third term its free Brownian diffusion, plus an increment due to interactions with the

bath. This increment corresponds to hard-sphere interactions between the probe and

bath that scatter the probe’s mean path. From a phenomenological perspective, the

effect of the bath particles is to reduce the mean velocity of the probe and increase

the diffusive spread of its trajectory – the effective diffusivity. Defining the second

bracketed group as the effective diffusivity of the probe and integrating by parts we

obtain

eff

≡

[

−

∮

]

(2.12)

where

is the unit surface normal pointing outwards from the probe.

In the limit

→

0, the entropically hindered diffusion of a particle in a

dilute suspension without hydrodynamic interactions must be recovered,

eff

∞

−

) (Batchelor 1976), for

a/b

= 1. Motivated by this fact, we denote the

corresponding displacement field for the unforced probe

entropic

≡

(

= 0). Hence,

we express the total displacement field as a sum of entropic and mechanical

contributions:

entropic

(2.13)

where

is the probe fluctuation over and above that for hindered passive diffusion.

As we show below,

entropic

contributes

−

to the effective diffusivity; so we write

eff

(

−

)

micro

(2.14)

where we have defined

micro

≡

∮

(2.15)

The effective diffusivity of a tracer particle is its bare diffusivity,

, minus the

entropic hindrance of the bath, 2

, plus an enhancement due to mechanical

scattering by the other bath particles,

micro

. The force-induced microdiffusivity is

proportional to the number density of bath particles, the isolated probe self-diffusivity,

and to the first moment of the hard-sphere deflections.

It remains only to obtain the steady microstructure

(

) and the first fluctuation

correction,

(

196

R. N. Zia and J. F. Brady

= 0.01

= 1

= 5

= 20

Figure 3.

(Colour online) Theoretical predictions for the deformed microstructure around a

moving probe particle in the absence of hydrodynamic interactions at the pair level. The test

particle is moving to the right and there is a build-up of background particle density in front

(red) of the probe and a deficit (dark blue) in the trailing wake (Squires & Brady 2005).

3. Non-equilibrium microstructure

The goal of this section is to formulate an expression governing the evolution of

the microstructure

(

). The Smoluchowski equation governing the pair probability

in Fourier space gives the evolution of the fluctuating microstructure:

∂

∂t

∇

[

−

∇

]

·∇

(3.1)

where we have defined the relative Brownian diffusivity between the probe and the

bath particle,

≡

and

≡

−

. With the definition of

(

) in (2.8)

and substituting (2.6) into (3.1) we have leading order in diluteness, at steady state:

∇

[

−

∇

]

·∇

(3.2)

[

−

∇

]=0 at

(3.3)

∼

1as

→∞

(3.4)

in which conservation requires a no-flux boundary condition at contact and there is

no long-range order. The equations are made dimensionless by scaling quantities as

∼

b, U

∼

ext

π

ηa, D

∼

π

(

)

(3.5)

and together with of the expansion (2.10) of

(

), the steady microstructure obeys

∇

−

·∇

(3.6)

[

∇

−

]=0 at

(3.7)

∼

1as

→∞

(3.8)

where

is the unit vector parallel to probe forcing. Squires & Brady (2005) have

solved this problem analytically for all

ext

(

kT /b

). A contour plot in figure 3

shows the perturbed steady microstructure

for a range of

The expression governing the probability-weighted displacement also forms an

advection–diffusion equation, but is forced by gradients in the steady microstructure:

∇

−

·∇

∇

(3.9)

(

∇

−

(3.10)

∼

0as

→∞

(3.11)

Single-particle motion in colloids: force-induced diffusion

197

where

≡

(1 +

a/b

). Hence, the displacement field

is coupled to

. In the next

section the coupled system is solved analytically in the limit of small and large

and numerically for arbitrary values of the P

eclet number.

4. Results

4.1.

Low-Pe limit

For small P

eclet number, Brownian diffusion of the bath particles easily repairs the

deformation of the microstructure caused by the probe’s motion. Since the bath is

hardly displaced from equilibrium, we approach the solution with a perturbation

expansion in small

. Recalling (3.6)–(3.8), however, it is apparent that the problem

is singular: at some distance

∼

from the probe, advection is as important as

diffusion. The domain is divided into two regions, and matched asymptotic expansions

yield the expression for the steady microstructure

(

;

)=1

−

(

−

[

−

]

−

)

(4.1)

which agrees with the solution that Squires & Brady obtained to

(

), which we

have extended here to

(

A similar method is applied to (3.9)–(3.11) to obtain the fluctuation field. The

term of the expansion of

yields the solution

entropic

≡

(0)

−

(4.2)

which gives

eff

(

=0)=

[

−

(

a/b

)

]

(4.3)

which, for equal probe and bath particle size, recovers the long-time self-diffusivity

of an isolated sphere in a quiescent solvent reported by Batchelor (1976),

eff

(

=0)=

∞

−

). Since there is no flow at

=0,this

(

) correction

is due to the entropic hindrance of the bath.

It is interesting to note that the

(1) solution for

is the same as the

(

)

solution for

. In fact the problems for

and

are identical in the limit

→

In the linear-response regime, whether the forcing is by external means,

,orby

thermal fluctuation,

, the resulting mobility reduction or diffusivity –

times the

mobility – is the same.

At the next order in

, we resolve the vector

into scalar components parallel

and transverse to the direction of the probe’s velocity,

⊥

and

⊥

.The

(

) fluctuation makes no contribution to the microdiffusivity –

in keeping with scaling predictions, since

(1)

∼

nnn

and the displacement of a

sphere has no coupling to odd tensors.

Proceeding to the next order in

, the singular nature of the problem becomes

evident, as the solution by regular perturbation expansion fails to decay to zero

far from the probe. An asymptotic expansion in the inner region is matched to

the solution in the outer region, giving the proper far-field condition for the inner

solution. The first correction to the fluctuation that gives rise to microdiffusive

198

R. N. Zia and J. F. Brady

behaviour is then

(2)

[

−

(

360

−

480

[

−

rrr

]

)

]

−

[

144

(

[

−

rrr

]

−

(

−

rrr

))

]

(4.4)

(2)

⊥

{

−

480

[

−

]

−

]

}

(4.5)

which yields for the microdiffusivity in the limit of

1 (plotted in figure 6):

micro

180

(

)

(4.6)

micro

⊥

(

)

(4.7)

When forced very weakly through a dilute suspension, a probe particle diffuses with

its bare diffusivity

minus an entropic hindrance due to the presence of the bath

particles that scales as

plus an enhancement due to hard-sphere collisions with

the bath particles – characteristic of the Taylor dispersion for particles in a bulk

flow. As predicted by scaling arguments in

1, in the low-

limit, the force-induced

enhancement to the diffusion is quadratic in the forcing; it is also linear in the

volume fraction of bath particles,

, and anisotropic, preferentially diffusing along

the direction of forcing (as compared to the transverse direction) by a factor of 2

39.

4.2.

High-Pe limit

For very large P

eclet number, the shape of the microstructure in front of the probe

is deformed into two distinct regions: an outer region in which advection dominates

diffusion and the microstructure is undisturbed, and an inner region – a 1

-thin

boundary layer that forms on the upstream face of the probe – where diffusion

balances advection. A

-long wake of particle deficit forms behind the probe, where

probability for a probe/bath particle collision is small. The particles that reside

inside the boundary layer provide the most probability for a strong hard-sphere

deflection of the probe (cf. figure 3). Inside the boundary layer, a coordinate rescaling

−

∼

(1) preserves the diffusive term, properly reflecting the physics of the

inner region and allowing satisfaction of the no-flux condition at contact. A singular

perturbation expansion in powers of

−

then obtains the deflection field in the

boundary layer on the upstream face of the probe,

π

π

−

(

)

(

1+(

)

(

−

(1)

(4.8)

⊥

−

(

)(

(

−

(1)

(4.9)

where

is the angle between

and the normal

and

is a unit vector in

the direction of

. As expected from earlier scaling arguments, the microdiffusivity is

linear in the forcing when

micro

(

)

(

ln 2

−

)

(1)

(4.10)

micro

⊥

(

)

(1)

(4.11)

Single-particle motion in colloids: force-induced diffusion

199

= 20

= 5

= 1

= 0.01

Figure 4.

(Colour online) The fluctuation field longitudinal to probe forcing. Blue areas

indicate regions of weak or no deflection; red areas indicate probability of strong deflection.

= 20

= 5

= 1

= 0.01

Figure 5.

(Colour online) The fluctuation field transverse to probe forcing. Blue areas

indicate regions of weak or no deflection; red areas indicate probability of strong deflection.

As in the low-

limit, the large

microdiffusivity is also transversely anisotropic,

with a longitudinal-to-transverse preference of approximately 7

2. The effect of the

hard-sphere collisions is a

-large diffusive scattering of the probe’s trajectory.

The high-

analytical results are shown in figure 6 alongside those for small

When scaled with the volume fraction of bath particles and the probe’s bare diffusivity

, the asymptotic limits of

1and

1 form a framework to guide the analysis

for intermediate values of the P

eclet number, which is developed in the next section.

4.3.

Numerical solution for arbitrary Pe

To obtain the fluctuating microstructure over the full range of

, a numerical solution

of the full Smoluchowski equations (3.6)–(3.8) and (3.9)–(3.11) is required. The radial

coordinate is rescaled with

−

to obtain the stretched coordinate

(

−

1).

Because the flow is axisymmetric about the line of external forcing, derivatives of

and

in the azimuthal angle are zero.

A central difference scheme is used to discretize gradients over the two-dimensional

domain. Once a boundary layer forms, i.e. beyond

(1), the radial gradients in

the microstructure are almost entirely confined to the boundary layer. As

continues

to grow and the boundary layer thins, a grid point concentration function that varies

with

increases the density of grid points close to contact, yet retains sufficient

resolution far from the probe to capture the physics throughout the upstream domain.

The difference coefficients and operators for both radial and angular directions are

compactly arranged into first- and second-order sparse matrices (Swaroop 2004). The

solutions for the steady and fluctuating pair-distribution function are then obtained

Matlab

using a

LaPack

iterative banded solver.

The steady microstructure is solved first (figure 3), and the gradients

∇

used

to drive the fluctuation field

. Contour plots for the deflection field are shown in

figures 4 and 5; the anisotropy is evident. The longitudinal fluctuations show that at

200

R. N. Zia and J. F. Brady

–2

–1

(

micro

–2

–1

ext

(ln 2 – )

Figure 6.

The force-induced diffusivity

micro

, scaled with the probe bare diffusivity

and

the volume fraction of bath particles

. Analytical solutions for asymptotically small and large

forcing are shown by solid lines (longitudinal) and dotted lines (transverse). Curved asymptotes

exclude the entropic contribution (dashed for longitudinal; dash-dotted for transverse). Open

symbols represent the numerical solution of the full Smoluchowski equation (circles for the

longitudinal microdiffusivity, squares for the transverse microdiffusivity).

very small

(similar to

01 in the figure), the fluctuations form a dipole about

the probe, with highest probability of a strong kick at the upstream face of the probe,

and decaying as

∼

.AstheP

eclet number is increased, the boundary layer thins

(similar to

= 20 in figure 4), and strong kicks to the probe result from particles

swept into the boundary layer on the front of the probe. Particles diffuse and weakly

advect around the probe, the boundary layer detaches and a wake forms behind it –

resulting in strong fore–aft asymmetry of probe fluctuations.

The transverse probe fluctuation field is shown in figure 5; for very weak forcing

(similar to

01 in the figure), the region of highest probability for a strong

deflection is at

π

2, with the distribution mirrored across the axis of symmetry.

The resulting probe deflection is perpendicular to its mean motion. As

is increased,

the probability of a lateral deflection is confined to the boundary layer.

The first moment of the fluctuation is numerically integrated over the surface of

contact between the probe and the bath to obtain the microdiffusivity for a range of

1000, as shown in figure 6. For very weak and very strong forcing, the

numerical solution matches the analytical asymptotes.

Note that two sets of asymptotes are shown for

1 in figure 6. The straight

asymptotes correspond to the high-

microdiffusivity as given by (4.10) and (4.11).

Recall that for the large-

analytical solution we found

, the total fluctuation

field, rather than that due to force-induced collisions only,

(cf. (2.13)). This is

asymptotically correct for

micro

→∞

.But

micro

is defined in terms of

and,

therefore, to be consistent, we should use

rather than

– which gives the curved

asymptote. For large values of the P

eclet number the two coincide.

Single-particle motion in colloids: force-induced diffusion

201

5. Solution via Brownian dynamics simulation

The dynamics of probe and bath particle motion are governed by the Langevin

equation, a force balance which includes Brownian, external, hydrodynamic and other

interparticle forces. In the present case this equation reads

ext

(5.1)

where the left-hand side is zero because inertia is not important in colloidal

dispersions, and

ext

for all particles except the probe. A probe of size

placed among a randomly distributed bath of particles of size

. The external force

is prescribed, and the other forces in (5.1) are given by

(0)

(

)=2

π

ηa

)

(

)

(5.2)

(5.3)

Here, the overbar denotes a time average and

(

) is the Dirac delta function;

≡

for the probe and

≡

for a bath particle. At each time step in the simulation the

particle positions are updated with a Brownian step and in the case of the probe, an

externally forced step. The hard-sphere displacement due to a collision between probe

and bath particle is added next; since the hard-sphere force is singular – non-zero at

contact only – special treatment is needed. To this end we use a modified ‘potential-

free’ algorithm (Heyes & Melrose 1993; Carpen & Brady 2005), in which overlaps

resulting from the external and Brownian steps are corrected along the line of centres

of the two particles, for a hard-sphere step

. For a complete description of

Brownian dynamics of active microrheology, see Carpen & Brady (2005).

A dilute bath can be achieved in two ways: first, a single bath particle and a

single probe can be placed in the simulation cell and many, many simulations run in

order to obtain a statistically large number of interparticle collisions. Alternatively,

many ‘ideal-gas’ bath particles can be placed in the cell with one probe – i.e. only

probe–bath particle collisions occur, and the bath particles simply pass through each

other. Thus, an individual time step contains only one deterministic and one Brownian

step – for each particle in the cell – but it could contain zero, one or several hard-

sphere displacements, depending on the number of bath particles within one step of

contacting the probe. Since the bath particles do not directly see each other, they have

no size except when they encounter the probe. It is their number concentration

the contact length scale (

) and the Brownian diffusivities

and

that govern

the system dynamics. Varying the value of

thus provides a means to compress the

time required to obtain a sufficient number of collisions for statistical analysis, and

should have no effect on probe diffusivity (although this turns out not to be precisely

the case, as we show below). To this end, volume fractions of bath particles from

9 were tested.

Simulations were conducted with values of the P

eclet number ranging from 0.1

to 100, volume fractions

7and0

9, and

a/b

= 1. On average, a

hard-sphere collision occurs during approximately 2 % of the total number of time

steps. Hence, to obtain sufficient resolution of the data, a set of 480 simulations of

time steps each was run for each

and for each

. The overall displacement of

the probe,

(

), was recorded at each time step.

The effective diffusivity of the probe,

eff

, is obtained from the time rate of change

of the mean-square displacement of the probe according to

eff

(

)

(

)

(5.4)

202

R. N. Zia and J. F. Brady

200

180

160

140

120

100

〈

′

〉

= 20

= 5

= 1

= 0

Figure 7.

Longitudinal mean-square displacement of the probe as a function of time

via Brownian dynamics simulation. Volume fraction of bath particles shown is

Displacements are made dimensionless as (

); time is scaled with the correlation time

where

∼

for

1and

∼

a/U

for

1. Each curve is an ensemble average over

480 simulations.

where

≡

(

)

−

(

)

and the angle brackets

denote an ensemble average over

time and over all simulations. A plot of the probe’s average mean-square displacement

versus time is shown in figure 7, where it can be seen that at long times

grows

linearly in time – confirming that the force-induced dispersion of the probe is indeed

diffusive, with a constant diffusivity.

To determine the effect of external forcing on the probe’s diffusivity, the effective

diffusivity in the absence of flow, i.e. at

≡

0, is subtracted from

eff

to yield the

force-induced diffusion

micro

eff

−

eff

(

=0)

(5.5)

which corresponds to (2.14) defined in text. Results are plotted in figure 8, where

micro

is made dimensionless with probe self-diffusivity

and scaled with the volume

fraction of bath particles

For

1, the Brownian dynamics data match the theoretical solution over the

full range of

, but the data for other values of

do not collapse together as

expected. The data follow the same qualitative trend, but for

1 lie beneath

the numerical solution, indicating a dependence on volume fraction – even though

the bath is modelled as an ideal gas. This surprising result can be understood by

noting that the bath particles’ motion is correlated via their interactions with the

probe. Although the bath particles do not interact directly, a probe–bath particle

collision changes the position of the probe relative to the other nearby particles,

giving rise to a correlation between the bath particles. That this correlation must

exist can be seen from the dilute pairwise expression for the equilibrium long-time

self-diffusivity of a probe equal in size to the bath particles,

∞

=(1

−

)

.This

result predicts a negative diffusion coefficient for

5 if only pairwise interaction