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
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
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
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
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
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
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
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
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
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
We still have two other options for where to place
Placing
This is a
In the
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
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,
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
where
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
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
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
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
Here,
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
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.)
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