## Abstract

The boundary element method (BEM) is a computational method particularly suited to solution of linear partial differential equations (PDEs), including the Laplace and Stokes equations, in complex geometries. The PDEs are formulated as boundary integral equations over bounding surfaces, which can be discretized for numerical solution. This manuscript reviews application of the BEM for simulation of the dynamics of “active” colloids that can self-propel through liquid solution. We introduce basic concepts and model equations for both catalytically active colloids and the “squirmer” model of a ciliated biological microswimmer. We review the foundations of the BEM for both the Laplace and Stokes equations, including the application to confined geometries, and the extension of the method to include thermal fluctuations of the colloid. Finally, we discuss recent and potential applications to research problems concerning active colloids. The aim of this review is to facilitate development and adoption of boundary element models that capture the interplay of deterministic and stochastic effects in the dynamics of active colloids.

### Keywords

- active colloids
- Brownian dynamics
- boundary element method

## 1. Introduction

Over the past 15 years, significant effort has been invested in the development of synthetic micro- and nano-sized colloids capable of self-propulsion in liquid solution [1, 2, 3]. These “active colloids” have myriad potential applications in drug delivery [4, 5], sensing [6], microsurgery [7], and programmable materials assembly [8]. Furthermore, they provide well-controlled model systems for study of materials systems maintained out of thermal equilibrium by continuous dissipation of free energy. In this context, and in comparison with driven systems (e.g., sheared suspensions), a unique aspect of active colloids is that energy is injected into the system at the *microscopic* scale of a single particle, instead of through macroscopic external fields or at the boundaries of the system. As a consequence of this, novel collective behaviors are possible, including motility-induced phase separation [9], mesoscopic “active turbulence” [10], and formation of dynamic “living crystals” and clusters [11, 12]. Furthermore, since living systems can be regarded as self-organized non-equilibrium materials systems, study of active colloids could yield insight into fundamental principles of living systems, and open a path towards development of biomimetic “dissipative materials” capable of homeostasis [13], self-repair [14], goal-directed behavior [15, 16], and other aspects of life.

Paradigmatic examples of synthetic active colloids include bimetallic Janus rods [17] and Janus spheres consisting of a spherical core with a hemispherical coating of a catalytic material [18]. In both cases, self-propulsion is driven by catalytic decomposition of a chemical “fuel” available in the liquid solution. For instance, for gold/platinum Janus rods, both ends of the rod are involved in the electrochemical decomposition of hydrogen peroxide into water and oxygen: hydrogen peroxide is oxidized at the platinum anode and reduced at the gold cathode. In this reaction process, a hydrogen ion gradient is established between the anode and cathode. The resulting gradient in electrical charge creates an electric field in the vicinity of the rod. The electric field exerts a force on the diffuse layer of ions surrounding the colloid surface, resulting in motion of the suspending fluid relative to the colloid surface. Viewed in a stationary reference frame, the final result is “self-electrophoretic” motion of the colloid in direction of the platinum end. For Janus spheres (e.g., platinum on silica or platinum on polystyrene), the mechanism of motion is still a subject of debate. Since the core material is inert and insulating, it was originally thought that these particles move by neutral self-diffusiophoresis in a self-generated oxygen gradient. Diffusiophoresis is similar to electrophoresis in that motion is driven by interfacial molecular forces. Briefly, in diffusiophoresis, the colloid surface and solute molecules interact through some molecular potential. This interaction potential, in conjunction with a gradient of solute concentration along the surface of the colloid, leads to the pressure gradient in a thin film surrounding the colloid, and therefore fluid flow within the film relative to the colloid surface. Following initial studies on chemically active Janus spheres, subsequent studies revealed a dependence of the Janus particle speed on the concentration of added salt [19], suggesting that a self-electrophoretic mechanism may be implicated in motion of the colloid. Golestanian and co-workers proposed that dependence of the rate of catalysis on thickness of the deposited catalyst can lead to different regions of the catalyst acting as anode and cathode [20]. More recently, it was proposed that if one of the redox reactions is reaction-limited and the other is diffusion-limited, the anodic or cathodic character of a point on the catalytic surface will depend on the local curvature of the surface [21]. Regardless of the detailed molecular mechanism of motion, a key point is that *interfacial flows* drive self-propulsion of chemically active colloids. A second key point is that particles need to have an intrinsic asymmetry (e.g., from the Janus character of their material composition) in order to exhibit directed motion.

These findings have motivated development of theoretical and numerical concepts for modeling the interfacially driven self-propulsion of active colloids. Motivated by classical work on phoresis in thermodynamic gradients [22, 23], an influential continuum framework for modeling neutral self-diffusiophoresis was established in Ref. 24, and will be reviewed below. This basic framework can be modified or extended to account for electrochemical effects [25], multicomponent diffusion [26], reactions in the bulk solution [27], and confinement [28, 29, 30, 31, 32, 33, 34]. An emerging area of study within this framework is autonomous navigation and “taxis” of chemically active colloids in ambient fields and complex geometries, including chemotaxis in chemical gradients [35] and rheotaxis in confined flows [15, 36]. Theoretical research on synthetic active colloids has also found common ground with an older strand of research on locomotion of *biological* microswimmers. Here, an important point of contact is again the idea of interfacial flow [37]. For a quasi-spherical microswimmer that is “carpeted” with a layer of cilia, the effect of the periodic, time-dependent, metachronal motion of the cilia can be modeled as a period-averaged interfacial flow. This “squirmer” model of locomotion was introduced by Lighthill [38] and refined by Blake [39]. More recent work has explored collective motion of suspensions of squirmers [40] and squirmer motion in confined geometries [41].

These theoretical frameworks are deterministic, and do not directly address the role of thermal fluctuations. For instance, for the model of a chemically active colloid in Ref. 42, diffusion of the chemical reaction product (i.e., the solute) into the surrounding solution is modeled with the Laplace equation, which has a smooth and unique solution for a given set of boundary conditions describing surface catalysis. Implicit in the use of the Laplace equation are the assumptions that, on the timescale of Janus particle motion, the solute diffuses very fast, and that fluctuations of the solute distribution average out to be negligible. Likewise, fluctuations of the surrounding fluid are neglected, i.e., the deterministic Stokes equation is used to model the fluid in lieu of the fluctuating Stokes equation. On the other hand, micron-sized active Janus particles are observed in experiments to exhibit “enhanced diffusion”: directed motion on short timescales

Moreover, as part of the general drive towards miniaturization, recent experimental efforts have sought to fabricate and characterize nano-sized chemically active colloids [45, 46, 47]. On the theoretical side, new questions arise when the size of the colloid becomes comparable to the size of the various molecules participating in the catalytic reaction. These questions include: When is using a continuum model appropriate [48]? Can a catalytic particle still display (time- and ensemble-averaged) directed motion when the particle and the surrounding chemical field are fluctuating on similar timescales? Relatedly, can a spherical colloid with a catalytic surface of *uniform* composition exhibit enhanced diffusion when nano-sized [49]? Can a fluctuating, nano-sized Janus particle effectively follow an ambient chemical gradient, i.e., exhibit chemotaxis [35]? These questions also connect with the burgeoning literature on chemotaxis of biological enzymes [50].

In this chapter, we review the boundary element approach to modeling the motion of active colloids. This is a “hydrodynamic” approach that resolves the detailed geometry and surface chemistry of the colloids, the velocity of the surrounding solution, and the distribution of chemical species within the solution [30, 40, 51, 52, 53, 54, 55, 56, 57]. The advantage of such an approach—in comparison with, for instance, the active Brownian particle model—is that it can resolve the detailed microscopic physics of how a colloid couples to ambient fields and other features of the surrounding micro-environment. In addition, we discuss how thermal fluctuations can be included within the approach. The aim of this review is to facilitate development and adoption of models that capture the interplay of deterministic and stochastic effects within an integrated framework.

## 2. Theory

As a starting point, we review the basic deterministic theoretical framework for understanding the motion of active colloids [24]. This is a continuum approach that coarse-grains the interfacial flow that drives colloid motion, discussed above, as a “slip velocity” boundary condition for the velocity of the suspending fluid.

We consider a suspension of

where

and the boundary condition

where

In order to close this system of equations, we require

where the integrals are performed over the surface

where the pressure

Practitioners of Stokesian Dynamics may notice some similarity between Eq. 3 and the boundary condition for an inert or passive sphere in an ambient flow field. If

### 2.1 The squirmer model: prescribed surface slip

The “squirmer” model was originally introduced by Lighthill to describe the time-averaged motion of ciliated quasi-spherical micro-organisms [38]. Lighthill’s formulation was subsequently corrected and extended by Blake [39]. The basic motivating idea of the squirmer model is that the periodic, metachronal motion of the carpet of cilia on the surface of the micro-organism drives, over the course of one period and in the vicinity of the microswimmer surface, net flow from the “forward” or “leading” pole of the micro-organism to the “rear” pole (see Figure 1, left). This interfacial flow drives flow in the surrounding bulk fluid, leading to directed motion of the micro-organism towards the forward end. The squirmer model captures some essential features of the self-propulsion of micro-organisms, including the hydrodynamic interactions between micro-organisms, and between an individual micro-organism and confining surfaces.

The slip velocity on the surface of a spherical squirmer

where

The unit vector

The squirming mode amplitudes *a priori* and do not depend on the configuration of the suspension. The set of amplitudes determine the detailed form of the flow field in vicinity of the particle. Furthermore, the lowest order squirming mode

### 2.2 Chemically active colloids: diffusiophoretic slip from chemical gradients

For chemically active colloids, the slip velocity on the surface of a colloid is driven by interfacial molecular forces. The molecular physics of phoresis and self-phoresis is reviewed in detail elsewhere [2, 23, 58]; here, we provide a brief summary. Consider a “Janus” colloid with a surface composed of two different materials. In the presence of molecular “fuel” diffusing in the surrounding solution, one of the two Janus particle materials catalyzes the decomposition of the fuel into molecular reaction products. A paradigmatic example of this reaction is the decomposition of hydrogen peroxide by platinum into water and oxygen:

(This equation is a severe simplification of the actual reaction scheme, which most likely involves charged and complex intermediates [20, 27]; nevertheless, proceeding from it, we can capture some essential features of self-phoresis.) If the reaction is reaction-limited—i.e., hydrogen peroxide is plentifully available in solution, and diffuses quickly relatively to the reaction rate—then we can approximate the production of oxygen with zero order kinetics:

where

Finally, we assume that

where

Accordingly, each Janus particle will be surrounded by an anisotropic “cloud” of oxygen molecules (“solute”), with the oxygen concentration highest near the catalytic cap (see Figure 1, right). Now we suppose that the oxygen molecules interact with the surface of the colloid through some molecular interaction potential with range *locally*, in the direction locally normal to the colloid surface, relax to a Boltzmann (i.e., equilibrium) distribution governed by the molecular interaction potential

These notions can be made mathematically rigorous through the theory of matched asymptotics. However, for the purpose of this discussion, the essential idea is that the bulk concentration

Here, the surface gradient operator is defined as

### 2.3 Lorentz reciprocal theorem

The Lorentz reciprocal theorem relates the fluid stresses

where

This theorem can be used to simplify the problem specified above for the velocities of

Additionally, the flow field vanishes far away from the particles, i.e.,

For problem

It can be shown that the right hand side of this equation vanishes. Consider the term involving

but the integral is simply the force

but the integral is the torque

Rearranging the left hand side of Eq. 17, we obtain a set of

These

where

The advantage of the reciprocal theorem approach is that if we solve the “primed” problem for a given set of particle positions *directly* solving for the

#### 2.3.1 Proof of Lorentz reciprocal theorem

We provide a short proof of Eq. 15, following the lines of Ref. 59 because some intermediate results will be useful later in the chapter. We recall that the rate of strain tensor

and that, in index notation, the stress tensor is

We consider the quantity

where we have used

We can also manipulate

Swapping the two indices in the last term,

But

so that

If there are no point forces applied to the fluid in determination of

Integrating both sides over the volume

### 2.4 Boundary integral formulation of the Laplace equation

Even with the aid of the Lorentz reciprocal theorem, it is necessary to solve the Stokes and (for self-phoretic particles) Laplace equations in a fluid domain containing the active particles as interior boundaries. For most configurations of the suspension, an analytical solution is intractable, and a numerical approach is required. Many numerical methods (e.g., the Finite Element Method) discretize and solve the governing partial differential equations in the three-dimensional fluid domain. This can be computationally intensive. Moreover, if the domain is unbounded in one or more dimensions, the computational domain must be truncated. Typically, the computational domain must be large in order to accurately approximate an unbounded solution, and significant computational effort must be expended on calculating the flow, pressure, and concentration fields far away from the particles.

An alternative approach proceeds from the following insight: a linear boundary value problems can be reformulated as a boundary integral equation (BIE) on the domain boundaries [51, 60]. Furthermore, the boundary integral equation can be discretized for numerical solution, yielding a dense linear system of coupled *boundary element* equations in the form of *boundary element method* is that it requires discretization of only bounding surfaces; for instance, to represent an unbounded system of

In order to obtain the BIE for the Laplace equation, we first consider the divergence theorem:

where the volume integral on the left hand side is carried out over the entire solution domain *into* the solution domain (see Figure 3). If

We can also write Green’s first identity for

Subtracting Eq. 34 from Eq. 33, we obtain Green’s second identity:

Now, we let

We obtain:

We have not yet specified the location of the pole

Using the divergence theorem, can show that

We recall from electrostatics that

As

By inspection, the Green’s function obeys the symmetry property

Interestingly, we have obtained an expression for *a priori*. Eq. 41 has an interesting physical interpretation: *plus* a distribution of point dipoles (i.e., infinitesimally separated pairs of mass sources and sinks) with strength

We still have two other options for where to place

Placing

This is a *boundary integral equation* (BIE) because the left hand side is the concentration

In the *boundary element method*, the boundary integral equation is discretized for numerical solution. Here, we briefly summarize the method, and direct the reader to consult the useful and comprehensive book of Pozrikidis for further information [51]. Each particle is represented as a meshed, closed surface. The meshing only needs to be done once; for a dynamical simulation, no remeshing during the simulation is required, even if the particles move relative to each other. The concentration

Choosing

The

where

and

Given either a specification of either

A certain difficulty becomes apparent when we consider the element

As a further note, issues with singular integrals have motivated development of *regularized boundary element methods*, which use a regularized Green’s function, i.e., a Green’s function with the singularity “smeared out” over a finite size

### 2.5 Boundary integral formulation of the Stokes equation

A similar approach can be taken for the Stokes equation [51, 59]. Recall that the Stokes equation is:

We can define a Green’s function

or

It can be shown that the Green’s function is

where

The stress in the fluid is given by

Now we wish to apply Eq. 30. We specify the “primed” fields

We integrate both sides over the domain

Now we apply the divergence theorem:

where the negative sign appears because of our convention that

If we choose to place

As with Eq. 41, the boundary integral representation for the flow field has an interesting physical interpretation. The first term on the right hand side of Eq. 60 can be regarded as a “single layer potential” due to a distribution of point forces with strength

If we place

Finally, if we place

For rigid body motion, *including* the *into*

For rigid body motion, there is no shear stress and the pressure is uniform, i.e.,

Examining Eq. 62, we note that

In order to obtain a single-layer boundary integral equation for

This single layer boundary integral equation can be discretized and solved numerically in a similar manner as the Laplace equation; Ref. 51 provides a comprehensive account.

### 2.6 Active suspensions in confined geometries

In the preceding, we considered a suspension of

A second, “mesh-free” approach is suitable for confining geometries with high symmetry, such as an infinite planar wall [39], an interface between two fluids with different viscosities [62], a fluid domain bounded by a solid wall and a free interface [63], or even two infinite planar walls. Additionally, it can be suitable if the domain is periodic in two or three dimensions. In this approach, the Green’s functions for the Laplace and Stokes equations are replaced with Green’s functions that obey the desired boundary conditions on the bounding surfaces. The Green’s function in the confined geometry can often be constructed by the method of images.

### 2.7 Thermal fluctuations

So far, we have considered the deterministic contributions to the

One approach to include Brownian forces on an active particle, the *hybrid boundary element/Brownian dynamics method*, simply calculates them separately and superposes them with the deterministic contributions. Using the Itô convention for stochastic differential equations, this superposition is expressed by the overdamped Langevin equation for the generalized,

where *deterministic* contribution of activity to the generalized velocity

where

The update of the orientation of each particle

The stochastic drift term in Eq. 68 can present some difficulty for numerical calculations [66]. For some simple situations, such as a single spherical colloid near a planar wall [34, 42], solutions for the configuration dependence of the mobility tensor are available in the literature [68, 69]. Alternatively, Eq. 67 can be discretized and solved via Fixman’s midpoint method to avoid calculation of the drift term [70].

This approach assumes that that the colloid and the fluid are not fluctuating on the same timescale, i.e., the fluid velocity is integrated out as a fast variable. Additionally, for self-phoretic particles, this approach necessarily neglects fluctuations of the chemical field

A recently developed variation of the boundary element method for Stokes flow, the *fluctuating boundary element method*, does not make this *post hoc* superposition of deterministic and Brownian contributions to particle motion. Rather, fluctuations are directly incorporated into boundary integral equation via a random velocity field on the boundary

## 3. Discussion and conclusions

The boundary element method is emerging as a powerful and important method for numerical simulation in the field of synthetic active colloids [30, 52, 54, 55, 56, 57]. This new area of application follows many years of fruitful application to modeling biological microswimmers, including with the squirmer model [40, 53]. For active colloids, a major advantage of the boundary element approach is that it can resolve the microscopic details of phoretic self-propulsion, including the chemical and flow fields generated by an active colloid, the surface chemistry and shape of the colloid, and the microscopic physics of how the colloid can couple to ambient fields and confining surfaces.

A few examples serve to illustrate the utility of the approach. Ref. [30] considers the dynamics of a spherical active Janus colloid near a planar wall. The colloid can “sense” and respond to the wall through self-generated chemical and hydrodynamic fields. Specifically, the wall provides a no-flux boundary condition for the solute concentration, and a no-slip boundary condition for the flow field. By confining the solute, the wall enriches the concentration of solute in the space between the particle and the wall, breaking the axial symmetry of the concentration field. Concerning the flow, the flow created by the particle scatters off the wall and back to the particle. These effects are captured by the boundary element method, including their dependence on the size of the catalytic cap and the spatial variation in the surface mobility *b* over the surface of the particle. As another example, Ref. [43] considers the dynamics of a *photo-active* spherical Janus colloid. The catalytic cap of the colloid is only active when exposed to incident light. This self-shadowing effect, in conjunction with the spatial variation of

However, some caveats are in order. For the hybrid boundary element/Brownian dynamics method discussed in this work, neither the fluctuations of the suspending fluid nor of the chemical field(s) are explicitly resolved. For self-phoretic particles in the ångstrom to nanometer size range, the particles, the solute, and the solvent fluctuate on similar timescales. Additionally, the validity of the continuum description of the surrounding solution is questionable. Molecular and mesoscopic simulation methods that resolve discrete solute and solvent particles may be more appropriate in this size range [48]. As a second caveat, boundary element methods are most suited to solution of linear governing PDEs, such as the Laplace and Stokes equations. Introducing nonlinearity in the governing equations (e.g., for a solution with nonlinear rheology or nonlinear bulk reaction kinetics) leads to the appearance of volume integrals in the boundary integral formulation. Thirdly and relatedly, the boundary element method is not as easily extensible as other methods (e.g., the finite element method) for inclusion of more complicated multiphysics. Finally, there is a caveat specific to active colloids. Much remains unknown about the reaction kinetics for self-phoretic particles. The boundary element method can have many free microscopic parameters (e.g., the values of the surface mobility

As a potential direction of research, we suggest developing a hybrid computational method combining the advantages of BEM and Stokesian Dynamics (SD). Stokesian dynamics is a method for simulating the dynamics of colloidal suspensions [73, 74, 75, 76]. Far-field hydrodynamic interactions are included in SD, truncated at the level of the stresslet (i.e., the first moment of the stress on the surface of a particle, which produces a hydrodynamic disturbance decaying as

As a second potential research direction, one could consider *deformable* active particles using the BEM. The boundary element method for Stokes flow has been coupled to methods to model particle elasticity, including the finite element method, in order to study the deformation of fluid-filled capsules [77] and elastic particles in shear flow [78], as well as the deformation of blood cells squeezing through constrictions [79].

The boundary element method could also be used to investigate questions touching upon fundamental nonequilibrium statistical mechanics. For instance, do nonequilibrium steady states of squirmers or self-phoretic particles (e.g., stable clusters of catalytic particles [57]) minimize the rate of entropy production [80]? When do hydrodynamic interactions suppress or enhance motility-induced phase separation and other nonequilibrium phase transitions? Does the pressure of an active suspension on a boundary obey an equation of state when hydrodynamic and phoretic interactions with the boundary are considered [76, 81]?

In any case, we anticipate that the boundary element method will continue to find successful application in the microswimmers field. A few potential problems include: modeling the collision dynamics and scattering of two or more non-spherical active colloids [72, 82]; the interaction of an active colloid and a passive colloid, possibly including the formation of dimeric bound states for cargo transport; and further exploration of motion near bounding surfaces and interfaces, especially fluid/fluid interfaces.

Consider an inert (non-active) sphere of radius

Additionally,

Now we recall the definitions of the (symmetric) rate of strain tensor

and the (anti-symmetric) vorticity tensor

The vorticity tensor can be related to the vorticity *vector*

Here, **ε** is the Levi-Civita tensor. The first derivative in Eq. 70 can be decomposed into symmetric and anti-symmetric contributions:

Using the Lorentz reciprocal theorem, one can obtain Faxén’s law for the drag force on the sphere (see Ref. [59] for details):

(In our shorthand notation, the Laplacian is first applied to

where the angular velocity of the fluid

where

So far we have only presented standard results, but now we raise the following question. Consider an *active* sphere with a slip velocity *linear* flow field

with

As our starting point, we write the Taylor expansion of

To obtain

We identify

Using Eq. 75, we obtain

If we consider a force-free swimmer,

This equation is one of the major results obtained in Ref. 37 by use of the Lorentz reciprocal theorem. However, our rederivation and interpretation in terms of an effective ambient flow field

The first integral on the right hand side of Eq. 85 vanishes. For the second integral on the right hand side, we use the identity

We obtain:

so that

Using Eq. 76, we obtain:

For a torque-free swimmer,

Finally, we consider how to obtain the stresslet

The first integral on the right hand vanishes, giving

Swapping the indices

Adding these two equations and dividing by two, we obtain

Accordingly,

Using the Faxén Law in Eq. 77, we obtain:

This is the major result obtained in Ref. 84 via the Lorentz reciprocal theorem. As before, this manuscript provides a novel alternative derivation and interpretation of Eq. 97 in terms of an effective ambient flow field. (Note that, due to the linearity of the Stokes equation, our approach is easily extended to model active particles in a real ambient flow field.)