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 [1–3]. 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 selforganized 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 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

The Boundary Element Method for Fluctuating Active Colloids

DOI: http://dx.doi.org/10.5772/intechopen.86738

"enhanced diffusion": directed motion on short timescales t < τ<sup>r</sup> and random walk behavior on long timescales t ≫ τr. For the latter, the effective diffusion coefficient Deff is enhanced relative to the "bare" diffusion coefficient D<sup>0</sup> of an inactive colloid, i.e., Deff ≫ D0. The reason for this behavior is that the orientation of the particle is free to fluctuate, and the particle changes its direction of motion by rotational

cient of the particle [18]. Therefore, thermal fluctuations qualitatively affect the motion of even a micron-sized catalytic Janus particle in unbounded, uniform solution. For a catalytic Janus particle in an ambient field or in confinement, thermal fluctuations affect whether and for how long the particle can align with the ambient field [42, 43] or stay near confining surfaces [34, 44]. Overall, a full theoretical understanding of the behavior of micron-sized active colloids requires

Moreover, as part of the general drive towards miniaturization, recent experimental efforts have sought to fabricate and characterize nano-sized chemically active colloids [45–47]. 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

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, 51–57]. 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

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

burgeoning literature on chemotaxis of biological enzymes [50].

stochastic effects within an integrated framework.

2. Theory

51

<sup>r</sup> , where Dr is the rotational diffusion coeffi-

diffusion over the timescale <sup>τ</sup><sup>r</sup> <sup>¼</sup> <sup>D</sup>�<sup>1</sup>

modeling thermal fluctuations.

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 [28–34]. 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 quasispherical 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].

The Boundary Element Method for Fluctuating Active Colloids DOI: http://dx.doi.org/10.5772/intechopen.86738

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

Non-Equilibrium Particle Dynamics

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.

fined geometries [41].

50

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 [28–34]. 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 quasispherical 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 con-

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 t < τ<sup>r</sup> and random walk behavior on long timescales t ≫ τr. For the latter, the effective diffusion coefficient Deff is enhanced relative to the "bare" diffusion coefficient D<sup>0</sup> of an inactive colloid, i.e., Deff ≫ D0. The reason for this behavior is that the orientation of the particle is free to fluctuate, and the particle changes its direction of motion by rotational diffusion over the timescale <sup>τ</sup><sup>r</sup> <sup>¼</sup> <sup>D</sup>�<sup>1</sup> <sup>r</sup> , where Dr is the rotational diffusion coefficient of the particle [18]. Therefore, thermal fluctuations qualitatively affect the motion of even a micron-sized catalytic Janus particle in unbounded, uniform solution. For a catalytic Janus particle in an ambient field or in confinement, thermal fluctuations affect whether and for how long the particle can align with the ambient field [42, 43] or stay near confining surfaces [34, 44]. Overall, a full theoretical understanding of the behavior of micron-sized active colloids requires modeling thermal fluctuations.

Moreover, as part of the general drive towards miniaturization, recent experimental efforts have sought to fabricate and characterize nano-sized chemically active colloids [45–47]. 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 ensembleaveraged) 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, 51–57]. 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.
