## Abstract

We study the motion of two colloidal (‘probe’) particles translating along their line of centres with fixed, equal velocities in an otherwise quiescent colloidal dispersion. The moving probes drive the microstructure of the dispersion out of equilibrium; resisting this is the Brownian diffusion of the dispersion ‘bath’ particles. As a result of the microstructural deformation, the dispersion exerts an entropic, or thermal, force on the probes. The non-equilibrium microstructure and entropic forces are computed to first order in the volume fraction of bath particles, as a function of the probe separation (** d**) and the Péclet number (

*Pe*), neglecting hydrodynamic interactions. Here,

*Pe*is the dimensionless velocity of the probes, which sets the degree of microstructural distortion. For

*Pe*≪1—the linear-response regime—the microstructural deformation is proportional to

*Pe*. In this limit, for sufficiently large

**, the deformation is nearly fore-aft symmetric about each probe; consequently, the entropic forces on them (which are opposite the direction of motion) are almost equal. However, for sufficiently small**

*d***, the symmetry is broken and, rather unexpectedly, the entropic force on the trailing probe is now in the direction of motion, whereas the force on the leading probe remains opposite to the direction of motion. Away from equilibrium,**

*d**Pe*>1 (and for all

**), the leading probe acts as a ‘bulldozer’, accumulating bath particles in a thin boundary layer on its upstream side, while leaving a wake of bath-particle free suspending fluid downstream, in which the trailing probe travels. In this (nonlinear) regime, the entropic forces are once more both opposite the direction of motion; however, the force on the leading probe is greater (in magnitude) than that on the trailing probe. Finally, far from equilibrium (**

*d**Pe*≫1), the entropic force on the trailing probe vanishes, whereas the force on the leading probe approaches a limiting value, equal to that for a single probe moving through the dispersion.

## 1. Introduction

Colloidal dispersions, comprising (sub-) micrometre-sized particles in a suspending fluid, occur in a variety of natural and man-made settings: inks, aerosols, foodstuffs, paints and biological materials are a few everyday examples. From an academic viewpoint, the length and time-scales relevant to colloidal dispersions place them in a perhaps unique position at the intersection of fluid dynamics, statistical mechanics and macromolecular chemistry. A central goal of colloid science is the calculation of the macroscopic, or effective, properties of dispersion (e.g. diffusion coefficients, viscosity, conductivity, etc.) from interactions at the microscopic level or microscale. In these materials, it is the interplay of hydrodynamic, interparticle and Brownian (or thermal) forces that determines the microscale configuration, or microstructure, of the colloidal particles. In equilibrium, the balance is between interparticle and Brownian forces, and the microstructure is given by the familiar Boltzmann distribution. The action of external agents, such as ambient flow-fields or body forces, drives the microstructure out of the equilibrium, where hydrodynamic forces now enter the description.

In this work, we study the non-equilibrium microstructure created by two colloidal (‘probe’) particles translating with equal, fixed velocities through an otherwise undisturbed colloidal dispersion. The driven motion of the probes pushes the dispersion's microstructure out of equilibrium; counteracting this is the Brownian diffusion of the colloidal ‘bath’ particles, which acts to heal this microstructural wound. As a consequence of the microstructural deformation, the dispersion exerts an entropic, or thermal, force on each of the probes, which is a function of the separation between the probes, ** d**, and the dimensionless velocity of the probes or Péclet number (

*Pe*). Our aim is to calculate the entropic forces on the probes over the entire range of

**and**

*d**Pe*, for which one must, of course, determine the non-equilibrium microstructure.

In equilibrium, it is well known that entropic forces between colloidal particles are often produced by the addition of macromolecular entities to the suspending fluid. These entities may be other (usually smaller) colloidal particles (Crocker *et al*. 1999), polymers (Verma *et al*. 1998) or stiff rods (Helden *et al*. 2003). The generated forces have important physical consequences and applications, including crystallization (Anderson & Lekkerkerker 2002) and self-assembly (Yodh *et al*. 2001). A classic example, first noted by Asakura & Oosawa (1958), is the so-called ‘depletion attraction’, where two colloidal particles in a dilute bath of smaller colloids experience an attractive (depletion) force when the excluded-volume surfaces of the large particles overlap, owing to an increase in volume available to the bath particles. The depletion force can lead to depletion flocculation (Jenkins & Snowden 1996) and has been seen to promote phase separation in binary colloid mixtures (Kaplan *et al*. 1994). The diluteness of bath particles is crucial; for more concentrated systems, interactions between the bath particles themselves can lead to a repulsive force between the two large colloids (Crocker *et al*. 1999; Tehver *et al*. 1999).

Away from equilibrium, the depletion interaction between colloidal particles must compete with external driving mechanisms. This can lead to pattern formation on macroscopic (i.e. much greater the particle size) length-scales, e.g. ‘lane’ formation in binary mixtures of oppositely driven colloids (Dzubiella *et al*. 2002; Chakrabarti *et al*. 2004). To our knowledge, the only theoretical study of depletion forces out of equilibrium is that of Dzubiella *et al*. (2003), who computed the forces on two fixed colloidal particles in a drifting bath of smaller Brownian particles. As expected, for non-zero drift velocities, the forces on the fixed particles are not equal, which Dzubiella *et al*. (2003) interpreted as a violation of Newton's third law. Their analytical approach (Brownian dynamics (BD) simulations were also performed) consisted of superimposing the microstructural deformations induced by a single fixed particle in the drifting bath, at the (two) locations relevant to the two-particle configuration. This is valid only if the fixed particles are widely separated and the (appropriately non-dimensionalized) drift velocity is small; somewhat surprisingly, the theoretical results are in good agreement with the BD simulations even when these two conditions are not met. In this work, subject to the assumptions listed below, we are able to avoid such approximations and restrictions in computing the microstructural deformation caused by the moving probes.

In the present study, we make four important assumptions. First, for simplicity, we take the probes and bath particles to be spherical and of equal size. However, note that the theory developed in §2 for the non-equilibrium microstructure may be easily extended to different-sized probe and bath particles. Hence, we could, if desired, recover the small bath particle limit of Dzubiella *et al*. (2003). Second, in analogy to classic theories of depletion forces in equilibrium, we take the volume fraction of bath particles to be small so that interactions between bath particles may be neglected. Furthermore, to facilitate an analytical treatment, we assume that the microstructure is determined by interactions of the probes with a single bath particle. Third, hydrodynamic interactions mediated by the suspending fluid are neglected. While seemingly over-restrictive, this condition can be realized for particles whose excluded-volume (or hard-sphere) radii are much greater than their physical (or hydrodynamic) radii, such as present in sterically or charge-stabilized dispersions. Fourth, we assume that the particles move along their line of centres, ** d**. Therefore, the microstructural deformation is axisymmetric about

**, and the (entropic) forces on the probes are directed along**

*d***. With these conditions in place, we are able to derive a closed equation for the non-equilibrium microstructure in terms of a ‘conditional pair-distribution function’ (CPDF), which gives the probability of finding a bath particle at a particular location, given the two-probe configuration.**

*d*The rest of the paper is organized as follows. In §2*a*, we present the three-body Smoluchowski equation governing the spatio-temporal evolution of the non-equilibrium microstructure, and the entropic forces on the probes are derived in §2*b*. The Smoluchowski equation must be solved numerically and, as discussed in §3, different approaches are used for overlapping and non-overlapping probe excluded-volumes. Results are presented in §4 and concluding remarks offered in §5.

## 2. Governing equations

### (a) Non-equilibrium microstructure

Consider a collection of *N* colloidal particles of radii *a* suspended in a Newtonian fluid of density *ρ* and viscosity *η*. The *N*-particle probability density function for finding the particles in a given spatial configuration at time *t* is *P*_{N}(*x*_{1},*x*_{2},*x*_{3},…,*x*_{N},*t*), where the labels 1 and 2 refer to the trailing and leading probes, respectively, and 3→*N* are the *N*−2 bath particles. The probability density satisfies an *N*-particle Smoluchowski equationwhere *j*_{i}=*U*_{i}*P*_{N} is the probability flux carried by particle *i*. Neglecting hydrodynamic interactions, the configuration-specific, or instantaneous, velocity of particle *i*, *U*_{i}, is given by(2.1)where *kT* is the thermal energy; *F*_{i} is the external force acting on particle *i*; and −*kT**∇*_{i}ln *P*_{N} is the thermal, or Brownian, force on particle *i*, owing to random thermal fluctuations of the solvent molecules. We proceed by integrating the *N*-particle Smoluchowski equation over the configurational degrees of freedom of *N*−3 bath particles, neglecting interactions between bath particles. The neglect of such higher-order couplings restricts our theory to low bath particle volume fractions, *ϕ*=4*πna*^{3}/3≪1 (*n* being the number density of bath particles), for which only one bath particle interacts with the probes. The three-particle distribution function, *P*_{3}(*x*_{1},*x*_{2},*x*_{3},*t*), defined as , satisfies a three-body Smoluchowski equationwhere 〈⋯〉_{3} denotes a conditional average with the trailing probe, leading probe and bath particle at *x*_{1}, *x*_{2} and *x*_{3}, respectively. For a statistically homogeneous suspension, we adopt a coordinate system relative to the trailing probe; defining *r*_{1i}=*x*_{i}−*x*_{1}, we have(2.2)

The absolute position of the trailing probe does not matter; hence, derivatives with respect to *x*_{1} are zero. The probability fluxes in the relative coordinate system are given bywhere *D*_{3}=*kT*/6*πηa* is the Stokes–Einstein–Sutherland diffusivity of the bath particle. Thus, equation (2.2) becomes(2.3)

In this study, we assume that the probes translate with equal velocities; hence, the second term in equation (2.3) vanishes. The three-particle distribution function may be written as(2.4)where *P*_{1/2}(*x*_{3}|*x*_{1},*x*_{2}) is the conditional probability of finding the bath particle at *x*_{3}, given the leading and trailing probes at *x*_{1} and *x*_{2}, respectively. Substituting this into equation (2.3) gives(2.5)

Furthermore, *P*_{1/2} may be written as *P*_{1/2}(*x*_{3}|*x*_{1},*x*_{2})=*ng*(*r*_{13}|*r*_{12}), where *g*(*r*_{13}|*r*_{12}) is the CPDF, which gives the probability of locating the bath particle at a separation *r*_{13} (henceforth ** r**) from the trailing probe, given a (fixed and finite) separation

*r*_{12}(henceforth

**) between the leading and trailing probes. Thus, equation (2.5) becomes**

*d*We make quantities dimensionless by scaling aswith *U*=|*U*_{1}|. In this study, we consider time-independent microstructures, for which the scaled three-body Smoluchowski equation reads(2.6)where ; all quantities are dimensionless and the subscripts on and *∇*_{13} have been dropped. Physically, equation (2.6) expresses a balance between advection by the moving probes, which drives the microstructure out of equilibrium, and Brownian diffusion of the bath particles, which acts to restore equilibrium. The degree to which the microstructure is distorted is given by the Péclet number, *Pe*=*Ua*/*D*_{3}, which emerges naturally from scaling as a ratio of advective (*U*) to diffusive (*D*_{3}/*a*) ‘velocities’. We leave the subscript on *D*_{3} to emphasize that only the bath particle undergoes Brownian diffusion, whereas the motion of the probes is deterministic.

To fully determine the microstructure, the Smoluchowski equation must be accompanied with suitable boundary conditions. At large separations, it is assumed that the dispersion has no long-range order, which implies(2.7)where and . Henceforth, we drop the tildes on and ; they are to be understood. The rigidity of the particles requires that the normal component of the relative flux vanishes when the bath particle is contacting either of the probes(2.8)where *n*_{i3} is the outward unit normal from probe *i* to the bath particle, and *S*_{i3} is the excluded-volume surface between probe *i* and the bath particle.

### (b) Forces on the probes

From the CPDF, it is possible to compute a variety of microstructurally averaged properties. The most interesting in the present context is the average external force on each of the probes. The average force is computed as an integral of the configuration-specific force *F*_{i} on the probe (equation (2.1)) weighted by the admissible positions of a bath particle, given the (fixed) locations of the probes. Formally, we havewhere 〈⋯〉_{2} denotes a conditional average with the trailing and leading probes at *x*_{1} and *x*_{2}, respectively. In the relative three-particle coordinate system, the difference in the forces is then(2.9)where the first term on the right-hand side of equation (2.9) is the difference in the Stokes drags, which vanishes if the probes move with equal velocities (as assumed henceforth). Using equation (2.4), we may write equation (2.9) as

Applying the divergence theorem to the volume integral in the above giveswhere d*S*_{i3} is the differential area element of the excluded-volume surface *S*_{i3}. In terms of the CPDF, *g*(** r**|

**), we have(2.10)**

*d*The two integrals in equation (2.10) represent the effect of a third (bath) particle, while the 2*kT*∇_{12}ln *P*_{2} term is the isolated two-probe contribution. We remove this two-probe contribution by defining an ‘entropic’ force, 〈Δ* F*〉

_{2}=〈Δ

**F**_{l}〉

_{2}−〈Δ

**F**_{t}〉

_{2}, owing solely to the presence of the bath particles, where(2.11)are the entropic forces on each probe. These forces are exerted

*by*the dispersion

*on*the probes as a result of the microstructural deformation; they are simply the integral over the surface of probe of the ‘osmotic’ pressure exerted by the bath particles. Thus, to ensure that the probes move with constant velocities, one must adjust the external forces acting on them by an amount equal in magnitude but opposite in direction to their respective entropic forces. Above, and henceforth, we replace the probe labels 1 and 2 with

*t*(trailing) and

*l*(leading), respectively.

## 3. Solution of the Smoluchowski equation

Computation of the dispersion microstructure is technically challenging: one has to solve the Smoluchowski (advection–diffusion) equation (2.6) subject to the far-field constraint (2.7) and the no-flux boundary condition (2.8) on the two excluded-volume surfaces, *S*_{t} and *S*_{l}. We assume that the probes move along their line of centres; consequently, the microstructural deformation is axisymmetric about the direction of motion, which somewhat simplifies the problem. Furthermore, the axisymmetry of *g*(** r**|

**) implies that the entropic forces are directed along**

*d***.**

*d*The shape of the excluded-volume surfaces, *S*_{t} and *S*_{l}, and hence the microstructural deformation, is crucially dependent on the probe spacing, ** d**. For

*d*=|

**|>4 (recall,**

*d**d*is made dimensionless with the probe radius,

*a*),

*S*

_{t}and

*S*

_{l}are the non-intersecting spheres of radii 2 (see figure 1). In contrast, for

*d*<4, both

*S*

_{t}and

*S*

_{l}are spheres (again, of radii 2), which intersect to form a closed dumb-bell (see figure 3). Physically, for

*d*>4, a bath particle is able to pass between the probes; for

*d*<4, it is not. Mathematically, the two scenarios require different coordinate systems to solve the Smoluchowksi equation: for

*d*>4, we employ bispherical coordinates, and for

*d*<4, we use toroidal coordinates (Morse & Feshbach 1953). Note that the case

*d*=4, for which

*S*

_{t}and

*S*

_{l}are the touching spheres, may be treated using tangent-sphere coordinates (Moon & Spencer 1961); however, we do not study this special value here.

Before discussing the cases *d*>4 and *d*<4 in more detail, we simplify the mathematical complexity of the problem by following Squires & Brady (2005) in writing the CPDF aswhere *κ*=*Pe*/2. Substituting this into the Smoluchowski equation (2.6) yields(3.1)i.e. a modified Helmholtz equation for *f*(** r**|

**), as compared to the advection–diffusion equation for**

*d**g*(

**|**

*r***). The far-field boundary condition on**

*d**f*(

**|**

*r***) isand the no-flux conditions (for**

*d**i*=

*t*,

*l*) are(3.2)

### (a) Non-intersecting excluded volumes: bispherical coordinates

We consider the probes to be moving along their line of centres, , which is taken as the *z*-axis of a two-dimensional Cartesian [*x*, *z*] coordinate system, whose origin is at the midpoint of the probes (see figure 1). Thus, the leading probe is at [0, *d*/2] and the trailing probe is at [0, −*d*/2]. Let us introduce the bispherical coordinates *μ* and *η* defined bywhere *c* is a scale factor, 0≤*η*≤*π* and −∞<*μ*<∞. The surface *μ*=*μ*_{0} is a sphere of radius *c*/|sinh *μ*_{0}| centred at [0, *c* coth *μ*_{0}]. Thus, we find

In bispherical coordinates, we can further simplify the Helmholtz equation for *f*(** r**|

**)=**

*d**f*(

*μ*,

*η*;

*d*) via the substitution(3.3)

The function *h*(*μ*,*η*;*d*) satisfies(3.4)subject to the boundary conditions

The bispherical coordinate system maps the two-probe configuration onto the rectangle {0≤*η*≤*π*, −*μ*_{0}≤*μ*≤*μ*_{0}}. However, we gain this rectilinear geometry at the cost of losing separability in the Helmholtz equation (3.1). Therefore, to compute *h*(*μ*,*η*;*d*), we approximate equation (3.4) by a finite difference equation (central differences are used for all derivatives) on a uniform grid and solve the resulting linear system of equations by a simple Jacobi iteration. The method closely resembles that of Khair & Brady (2006), who calculated the microstructural deformation around a single forced probe. However, the bispherical geometry raises a couple of issues that warrant comment. First, to ensure axisymmetry, one must impose the boundary condition ∂*h*/∂*η*=0 at *η*=0 and *π*. Second, it is desirable to have a high density of grid points near the excluded-volume surfaces; to this end, we use the transformation(3.5)where −1≤*λ*≤1 and *α*>0. Increasing *α* places a larger number of grid points near the excluded-volume surfaces (i.e. near *μ*=±*μ*_{0}). A typical grid discretization is shown in figure 2.

Finally, from equation (2.11), the entropic forces on the probes, which are in the *z*-direction, are given by

### (b) Intersecting excluded volumes: toroidal coordinates

When the probes are sufficiently close, *d*<4, a bath particle is not able to pass between them. In this case, the excluded-volume surfaces intersect at an angle *ψ*=arccos[(*d*^{2}/8)−1] (see figure 3) to form a dumb-bell shape. To solve the Smoluchowski equation in this geometry, it is appropriate to use toroidal coordinates; we take the same Cartesian frame as before and define the toroidal coordinates *μ* and *η* viawhere *c* is a scale factor, −*π*≤*η*≤*π* and 0<*μ*<∞. The surface *η*=*η*_{0} (*η*_{0}>0) is that part of a sphere of radius *c*/sin *η*_{0}, centred at [0, *c* cot *η*_{0}], which is above the *x*–*y* plane (i.e. for which *z*>0). Its mirror image about the *x*–*y* plane is the surface *η*=−*η*_{0}. Hence, it is easy to show

Using the substitution (3.3), in toroidal coordinates, the Helmholtz equation for *f*(*μ*,*η*;*d*) transforms intowhere *h*(*μ*,*η*;*d*) must also satisfy

In toroidal coordinates, we again lose separability of the Helmholtz equation for *f*(*μ*,*η*;*d*); therefore, *h*(*μ*,*η*;*d*) is calculated using finite differences. Axisymmetry of the microstructure about the *z*-axis requires ∂*h*/∂*μ*=0 at *μ*=0. The excluded-volume surfaces intersect at *μ*=∞, and *h*(*μ*,*η*;*d*) should be continuous at this point, which requires ∂*h*/∂*μ*=0 at *μ*=∞. In practice, this is difficult to implement owing to the semi-infinite range of *μ*. Thus, we move the condition to *μ*=*μ*_{max} and increase *μ*_{max} until convergence of the entropic forces is achieved. As with bispherical coordinates, it is desirable to place a large number of grid points near the excluded-volume surfaces. Furthermore, in toroidal coordinates, there is a natural clustering of grid points near the intersection point of the surfaces, *μ*=∞, as shown in figure 4. Therefore, we transform *η* according to equation (3.5) and *μ* via *μ*=exp(*βξ*)−1, where 0≤*ξ*≤1 and *β*=ln(*μ*_{max}+1), which places a greater density (as compared to the ‘naive’ toroidal discretization) of grid points near the upstream half of the leading probe [*μ*=0, *η*=*η*_{0}] and the downstream half of the trailing probe [*μ*=0, *η*=−*η*_{0}].

From equation (2.11), the entropic forces on the probes are given by

## 4. Results

Before presenting our results, we comment briefly on numerical details. As *Pe* is increased, the demand for grid points (and hence the number of iterations) increases to capture accurately the boundary layers on the upstream side of the leading and, depending on the value of *d*, trailing probes (see figures 5 and 6). Thus, the computational cost of the finite difference scheme also increases with *Pe*, the largest value for which we obtained a convergent solution being *Pe*=10. Typically, in both the toroidal and bispherical geometries, 300×300 grid points were used and the accuracy was tested by comparing the resulting entropic forces, and , with those computed using a 350×350 grid.

In figure 5, we plot the microstructural deformation, *g*(** r**|

**)−1, as a function of**

*d**Pe*for

*d*=5. For

*Pe*≪1, where Brownian diffusion dominates advection, the deformation is proportional to

*Pe*, and the probes act as a pair of diffusive dipoles, with an accumulation of particles on their upstream sides and a deficit on their downstream sides. For sufficiently large

*d*, the two dipoles do not interact, and the microstructure is fore-aft symmetric about each probe. However, for smaller

*d*(such as

*d*=5), the probes sense each other's presence; consequently, the fore-aft symmetry is broken. Moving to

*Pe*≈1, advection now comes into play, and the deformation around the probes exhibits the beginnings of a classic boundary layer and wake structure. Physically, in a frame fixed on the probes, the advective flux of bath particles (moving with velocity ) leads to an accumulation on the upstream sides of the (impenetrable) probes, which act as obstacles that the bath particles must navigate. The mechanism for passing around the probes is via Brownian diffusion. Thus, the boundary layers signify a balance between advection, in transporting bath particles towards the probes, and diffusion. On the downstream sides of the probes, advection carries bath particles away; a region of low particle density, or wake, is formed. As

*Pe*is increased further, the stronger advective flux results in thinner boundary layers and longer wakes. Note that the boundary layer on the trailing probe is significantly less particle-rich as compared to the leading probe (which has an

*O*(

*Pe*) accumulation), as the trailing probe travels in the (low particle density) wake created by the leading probe, while the leading probe moves through the undisturbed dispersion. Of course, for

*d*much larger than the characteristic wake length (which grows as

*Pe*), one expects the boundary layers on the trailing and leading probes to be almost identical. Conversely, for

*d*much smaller than the wake length, the boundary layer on the trailing probe disappears; essentially, it moves through a particle-free tunnel of suspending fluid.

In figure 6, we plot the microstructural deformation as a function of *Pe* for *d*=3. In this case, and for all *d*<4, the excluded-volume surfaces, *S*_{t} and *S*_{l}, of the probes join and, from the bath particle's viewpoint, form a single dumb-bell shaped obstacle. At small *Pe*, this dumb-bell acts as a diffusive dipole, which is approximately fore-aft symmetric about the intersection point of the two excluded-volume surfaces (save for a small accumulation of particles on the upstream side of the trailing probe and a small deficit on the downstream side of the leading probe). For larger *Pe*, we again see the formation of a boundary layer on the upstream side of the leading probe. As bath particles cannot reside in the overlap of *S*_{t} and *S*_{l}, there is neither a wake on the downstream side of the leading probe nor a boundary layer on the upstream side of the trailing probe. There is, however, a wake on the downstream side of the trailing probe, as bath particles are transported away from the rear of the dumb-bell by advection.

Having discussed the dispersion microstructure, we move to the entropic forces on the probes. Figure 7 plots the scaled difference in entopic forces, , as a function of *Pe* for various *d*. For *Pe*≪1 and large *d* (take the data for *d*=25, say), the difference in entropic forces is small [*O*(*Pe*^{2})]; this scaling arises because to *O*(*Pe*), the probes act as identical diffusive dipoles; thus, the *O*(*Pe*) contributions to their respective entropic forces are equal. However, as mentioned above and shown in figure 8, for smaller *d* the diffusive dipoles interact, and the difference in entropic forces grows (i.e. it is no longer *O*(*Pe*^{2})). In fact, for *d*<4, the entropic forces on the probes have opposite signs; and are directed along ** d** and −

**, respectively, and both are**

*d**O*(1) in magnitude, resulting in an

*O*(1) difference. By construction, the velocity of each probe is fixed at

**. The microstructure around the leading probe consists of an accumulation of particles on its upstream side and a deficit on what is left of its downstream portion (before the intersection of**

*U**S*

_{t}and

*S*

_{l}). Thus, the leading probe is retarded by the net accumulation of particles on its excluded-volume surface; the dispersion exerts an

*O*(1) entropic force opposite to the direction of motion. Hence, the additional external force on the leading probe (which is equal in magnitude to the entropic force on it) is in the direction of motion, thereby maintaining the probe velocity at

**. Conversely, the deformation about the trailing probe is deficit of particles on its downstream side and an accumulation on what is left of its upstream side. Hence, there is a net deficit of bath particles around the trailing probe, and the dispersion exerts an entropic force in the direction of motion. To ensure that the trailing probe velocity remains equal to**

*U***, the additional external force applied to it must be opposite to the direction of motion.**

*U*At large *Pe*, for all values of *d*, the differences in entropic forces approach a common asymptote, , as *Pe*→∞. As shown by Khair & Brady (2006), this is nothing but the entropic force on a single probe at high *Pe*, the dominant contribution to which is from the large *O*(*Pe*) build-up of bath particles in the thin *O*(*Pe*^{−1}) upstream boundary layer. For two probes at high *Pe*, there is still a boundary layer on the upstream side of the leading probe (almost identical to that for a single probe); however, the trailing probe moves through the wake of the leading probe. Moreover, the trailing probe generates its own wake; effectively, then, it moves through a tunnel of suspending fluid, and hence there is a much smaller (as compared to the leading probe) entropic force exerted on it. In figure 9, we plot the individual entropic forces as a function of *Pe* for *d*=3.5, where the behaviour described above is clearly seen. Furthermore, as shown in figure 10, for a given value of *Pe*(>1), the magnitude of the entropic force on the trailing probe monotonically decreases with decreasing *d*; the force on the leading probe is almost constant over the entire range of *d*. This is readily understood as the force on the leading probe is determined by the boundary layer structure, which is relatively insensitive to *d*, whereas decreasing *d* places the trailing probe deeper in the wake of the leading probe, thus decreasing the magnitude of the entropic force on it.

## 5. Discussion

We have calculated the entropic forces acting on two colloidal (probe) particles moving with fixed, equal velocities through a dispersion of colloidal bath particles. It is well known that the presence of such bath, or contaminant, particles can generate forces between colloidal particles; a prime example is the attractive depletion force between a pair of particles immersed in a dilute suspension of (smaller) particles at equilibrium. Like the forces calculated in the present work, depletion forces are entropic in nature, scaling with the thermal energy, *kT*. Furthermore, near equilibrium (*Pe*≪1) and for small separations (*d*<4), the entropic forces on the trailing and leading probes point upstream and downstream, respectively, which can be interpreted as the dispersion exerting a net attractive force between the probes—somewhat akin to depletion attraction. However, there is an important difference; in our study, the forces are generated as a consequence of the translating probes driving the microstructure of the dispersion out of equilibrium, whereas the equilibrium depletion forces do not rely on imposed motion of the particles, i.e. they are purely thermodynamic (excluded-volume) effects. Moreover, in our case, the dominant contribution to the (total) force on each probe is from its Stokes drag; the entropic force represents a small [*O*(*ϕ*)] correction.

Recently, Dzubiella *et al*. (2003) have attempted to generalize the concept of depletion forces to non-equilibrium by computing the entropic forces on two stationary probes in a stream of smaller bath particles. When the bath particles flow along the line of centres of the probes, for moderate stream velocities, they find (their fig. 3*b*) that the magnitude of the force on the trailing probe is less than that on the leading probe, in agreement with our results for *Pe*>1 (figure 10). However, unlike us, they do not report a change in the direction of the entropic force on the trailing probe with increasing *Pe* for *d*<4 (figure 9). Although Dzubiella *et al*. (2003) investigated more general (non-axisymmetric) probe configurations, their analytical methods (as discussed in §1) are restricted to widely separated particles and small stream velocities. On the other hand, we have computed the non-equilibrium microstructure and entropic forces for all (axisymmetric) separations and probe velocities.

In this study, we neglected hydrodynamic interactions between particles. Although, as mentioned in §1, there exist experimental systems for which this is a reasonable assumption, one should pause and consider their effect. For example, in the absence of hydrodynamic interactions, at large *Pe*, the entropic force on the trailing probe vanishes, while the entropic force on the leading probe approaches the single probe limit (figure 7). However, with hydrodynamic interactions this may not be so. As discussed by Khair & Brady (2006), in the limit of infinite *Pe*, the microstructure around a single moving probe attains spherical symmetry, i.e. the upstream boundary layer wraps over the entire (excluded volume) surface of the probe and the wake disappears. Physically, this occurs as bath particles from upstream ‘stick’ to the probe (owing to hydrodynamic lubrication forces), as they are advected around it. Consequently, the force on the probe has equal contributions from the pushing of bath particles upstream of it and pulling of particles downstream of it. (Note that at infinite *Pe*, the force exerted by the dispersion on the probe is hydrodynamic, not entropic, in origin.) Now, imagine two probes in the presence of hydrodynamic interactions at large *Pe*. The leading probe no longer creates a wake for the trailing probe to move through; likewise, the trailing probe does not create a wake of its own. We expect this qualitative change in the microstructure at large *Pe* owing to hydrodynamic interactions to be reflected in the computed forces. However, the inclusion of hydrodynamic interactions is beyond the scope of analytical theory, and one must resort to computer simulations such as Stokesian dynamics.

The theoretical framework developed in §2 may be used to study other related problems. A natural extension is to consider two probes moving perpendicular to their line of centres, for which the microstructural deformation is no longer axisymmetric about their separation, ** d**. Consequently, we expect the entropic forces on the (top and bottom) probes to have components parallel and perpendicular to

**. For large**

*d**d*=|

**|, the parallel components for each probe vanish, while the perpendicular components are, by symmetry, identical and equal to that of a single moving probe. For smaller**

*d**d*(close to, but greater than 4), a bath particle is just able to squeeze between the probes. The perpendicular components for each probe remain identical, but are now not equal to that for a single probe. In contrast, the parallel components should be in opposite directions; the parallel forces on the top and bottom probes point upward and downward, respectively, which may be viewed as the dispersion exerting a net repulsive force between the probes. Indeed, this behaviour was seen by Dzubiella

*et al*. (2003) (their fig. 3

*a*). Furthermore, as mentioned in §1, our theory can be easily extended to study the small bath particle limit of Dzubiella

*et al*. (2003), which is somewhat more representative of experimental studies on depletion forces. Lastly, the Smoluchowski equation (2.3) applies when there is a relative motion between the probes. Hence, for example, it can be used to study the motion of a probe approaching a stationary wall at fixed velocity (which is, of course, just a probe of infinite radius): a simple problem, which could provide a starting point for investigating the patterning of surfaces using driven colloidal particles.

On a complementary note to the present study, one could fix the external forces on the probes and calculate their average velocities. Although qualitatively similar, there are important (and subtle) differences between the fixed-force and fixed-velocity cases; indeed, the formulation of the three-body Smoluchowski equation for the fixed-force problem is more involved. Nevertheless, one can arrive at an equation for the CPDF analogous to equation (2.6) in the limit, where the bath particles are much more mobile (i.e. they have a far higher diffusivity) than the probes, which, in turn, requires the bath particles to be much smaller than the probes. As a first study, it would be wise to consider two (large) probes forced along their line of centres, ** d**, for which their average velocities will be directed along

**also. For**

*d**Pe*>1 (here,

*Pe*is defined as the dimensionless external force on the probes) and

*d*>4, the trailing probe would move through the wake created by the leading probe, which is practically devoid of bath particles. Furthermore, the trailing probe generates its own wake; hence, as per the fixed-velocity problem, it travels through a tunnel of suspending fluid. In contrast, the leading probe is retarded by the need to push bath particles out of its path. As such, one expects, in general, that the trailing probe moves faster than the leading probe, and it may eventually ‘catch-up’ and contact the leading probe. Continuing this process, one may add a second trailing probe (this is, unfortunately, beyond the realm of analytical theory; one must employ computer simulations), which will catch-up to the first trailing and leading probes. In this way, by adding yet more probes, one may form a ‘train’ of probes moving through the dispersion, which could provide a route for pattern formation in colloidal dispersions and perhaps other complex materials. However, we shall leave this fascinating world of multi-particle interactions for future studies.

## Acknowledgments

The authors wish to thank Jim Swan for his fruitful discussions and Justin Bois for his aid in preparing figures 5 and 6.

## Footnotes

- Received May 13, 2006.
- Accepted July 27, 2006.

- © 2006 The Royal Society