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 N active colloids in an unbounded liquid solution. The position of each colloid α, with α∈ f g 1; 2; …; N , is described in a stationary reference frame by a vector xα. The solution is modeled as an incompressible Newtonian fluid with dynamic viscosity μ. The solution is governed by the Stokes equation,

$$-\nabla P + \mu \nabla^2 \mathbf{u} = \mathbf{0},\tag{1}$$

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

The slip velocity on the surface of a spherical squirmer α is specified by fiat and

2 ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>x</sup><sup>2</sup> <sup>p</sup> n nð Þ þ 1

squirmer has an axis of symmetry, and a propulsion direction d^<sup>α</sup> oriented along this axis. We define the body-frame polar angle θp,<sup>α</sup> at a point x<sup>s</sup> on the surface of squirmer <sup>α</sup> as the angle between <sup>d</sup>^<sup>α</sup> and a vector <sup>x</sup><sup>s</sup> � <sup>x</sup><sup>α</sup> from the center of the

The two types of microswimmer considered in this chapter. In the spherical "squirmer" model (left), the slip velocity on the surface of the particle is specified by fiat and fixed (in a frame co-moving and co-rotating with the particle) for all time. For an axisymmetric distribution of surface slip, the particle moves in the direction of the green arrow, i.e., opposite to the (surface-averaged) direction of the slip velocity. For a Janus particle (right), a fraction of the particle surface (black) catalyzes a reaction involving various molecular species diffusing in the surrounding solution. The resulting anisotropic distribution of product molecules or "solute" (green spheres) drives a phoretic slip velocity (purple) in an interfacial layer surrounding the particle. For a repulsive interaction between the solute and the particle surface, the slip is towards high concentration of solute,

Bn,αVnð Þ cos θ e

dPnð Þ x

^<sup>θ</sup>p,<sup>α</sup> is defined in following manner (see Figure 1, left). The

^<sup>θ</sup>p,<sup>α</sup> , (8)

dx : (9)

^<sup>θ</sup>p,<sup>α</sup> is oriented in the direction of increasing θp,α,

does not depend on the configuration of the suspension. It is given as [41]:

n

vs,αð Þ¼ x<sup>s</sup> ∑

Vnð Þ� x

i.e., locally tangent to the squirmer surface along a longitudinal line.

an individual micro-organism and confining surfaces.

The Boundary Element Method for Fluctuating Active Colloids

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

where

Figure 1.

53

The unit vector e

squirmer x<sup>α</sup> to xs. The unit vector e

and the particle moves in the direction of the green arrow.

where Pð Þ x is the pressure at a position x in the solution, and u xð Þ is the velocity of the solution. The velocity obeys the incompressibility condition,

$$\nabla \cdot \mathbf{u} = \mathbf{0},\tag{2}$$

and the boundary condition

$$\mathbf{u}(\mathbf{x}\_t) = \mathbf{U}\_a + \mathbf{\Omega}\_a \times (\mathbf{x}\_t - \mathbf{x}\_a) + \mathbf{v}\_{\mathfrak{s},a}(\mathbf{x}\_t), \ \mathbf{x}\_t \in \mathcal{S}\_{ab} \tag{3}$$

where S<sup>α</sup> is the surface of colloid α, x<sup>s</sup> ∈ S<sup>α</sup> is a position on Sα, and U<sup>α</sup> and Ω<sup>α</sup> are the translational and rotational velocities, respectively, of colloid α. The quantity vs,αð Þ x<sup>s</sup> is the slip velocity on the surface of colloid α, which is either prescribed (for a squirmer) or determined by the distribution of chemical species in solution (for a chemically active colloid). The form of vs,αð Þ x<sup>s</sup> for the two types of particles will be discussed in detail below. Additionally, far away from the N particles, the fluid velocity vanishes:

$$\mathbf{u}(|\mathbf{x}| \to \infty) = \mathbf{0}.\tag{4}$$

In order to close this system of equations, we require 6N more equations, corresponding to the 6N unknown components of U<sup>α</sup> and Ωα. The net force and torque on each colloid vanishes:

$$\int\_{S\_d} \boldsymbol{\sigma} \cdot \hat{\mathbf{n}} \ \mathrm{d}S + \mathbf{F}\_{\mathrm{ext},a} = \mathbf{0},\tag{5}$$

$$\int\_{S\_a} (\mathbf{x}\_t - \mathbf{x}\_a) \times \boldsymbol{\sigma} \cdot \hat{\mathbf{n}} \ \mathrm{d}S + \mathbf{T}\_{\mathrm{ext},a} = \mathbf{0},\tag{6}$$

where the integrals are performed over the surface S<sup>α</sup> of each colloid α, and Fext,<sup>α</sup> and Text,<sup>α</sup> are, respectively, the net external force and net external torque on the colloid. The stress tensor is given by

$$\boldsymbol{\sigma} = -\boldsymbol{P}\mathbf{I} + \mu \left(\boldsymbol{\nabla}\mathbf{u} + \left(\boldsymbol{\nabla}\mathbf{u}\right)^{T}\right),\tag{7}$$

where the pressure Pð Þ x is determined by the incompressibility condition.

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 vs,αð Þ x<sup>s</sup> could be expressed as an effective ambient flow field at the position of particle α, the tools of Stokesian dynamics could be straightforwardly applied to simulation of active suspensions. This analogy will be developed in the Appendix.

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

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 α is specified by fiat and does not depend on the configuration of the suspension. It is given as [41]:

$$\mathbf{v}\_{s,a}(\mathbf{x}\_s) = \sum\_{n} B\_{n,a} \, V\_n(\cos \theta) \, \hat{\mathbf{e}}\_{\theta\_{p,a}, \mathbf{o}} \, \tag{8}$$

where

coarse-grains the interfacial flow that drives colloid motion, discussed above, as a "slip velocity" boundary condition for the velocity of the suspending fluid.

with dynamic viscosity μ. The solution is governed by the Stokes equation,

of the solution. The velocity obeys the incompressibility condition,

and the boundary condition

Non-Equilibrium Particle Dynamics

torque on each colloid vanishes:

�∇<sup>P</sup> <sup>þ</sup> <sup>μ</sup>∇<sup>2</sup>

We consider a suspension of N active colloids in an unbounded liquid solution. The position of each colloid α, with α∈ f g 1; 2; …; N , is described in a stationary reference frame by a vector xα. The solution is modeled as an incompressible Newtonian fluid

where Pð Þ x is the pressure at a position x in the solution, and u xð Þ is the velocity

where S<sup>α</sup> is the surface of colloid α, x<sup>s</sup> ∈ S<sup>α</sup> is a position on Sα, and U<sup>α</sup> and Ω<sup>α</sup> are the translational and rotational velocities, respectively, of colloid α. The quantity vs,αð Þ x<sup>s</sup> is the slip velocity on the surface of colloid α, which is either prescribed (for a squirmer) or determined by the distribution of chemical species in solution (for a chemically active colloid). The form of vs,αð Þ x<sup>s</sup> for the two types of particles will be discussed in detail below. Additionally, far away from the N particles, the fluid velocity vanishes:

In order to close this system of equations, we require 6N more equations, corresponding to the 6N unknown components of U<sup>α</sup> and Ωα. The net force and

where the integrals are performed over the surface S<sup>α</sup> of each colloid α, and Fext,<sup>α</sup> and Text,<sup>α</sup> are, respectively, the net external force and net external torque on

σ ¼ �PI þ μ ∇u þ ð Þ ∇u

where the pressure Pð Þ x is determined by the incompressibility condition. 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 vs,αð Þ x<sup>s</sup> could be expressed as an effective ambient flow field at the position of particle α, the tools of Stokesian dynamics could be straightforwardly applied to simulation of active suspensions. This analogy will be developed in the Appendix.

The "squirmer" model was originally introduced by Lighthill to describe the time-averaged motion of ciliated quasi-spherical micro-organisms [38]. Lighthill's

<sup>T</sup> � �

ð Sα

ð Sα

2.1 The squirmer model: prescribed surface slip

52

the colloid. The stress tensor is given by

u xð Þ¼ <sup>s</sup> U<sup>α</sup> þ Ω<sup>α</sup> � ð Þþ x<sup>s</sup> � x<sup>α</sup> vs,αð Þ x<sup>s</sup> , x<sup>s</sup> ∈ Sα, (3)

u ¼ 0, (1)

∇ � u ¼ 0, (2)

uð Þ¼ jxj ! ∞ 0: (4)

σ � n^ dS þ Fext,<sup>α</sup> ¼ 0, (5)

, (7)

ð Þ� x<sup>s</sup> � x<sup>α</sup> σ � n^ dS þ Text,<sup>α</sup> ¼ 0, (6)

$$V\_n(\mathbf{x}) \equiv \frac{2\sqrt{\mathbf{1} - \mathbf{x}^2}}{n(n+1)} \frac{dP\_n(\mathbf{x})}{d\mathbf{x}}.\tag{9}$$

The unit vector e ^<sup>θ</sup>p,<sup>α</sup> is defined in following manner (see Figure 1, left). The squirmer has an axis of symmetry, and a propulsion direction d^<sup>α</sup> oriented along this axis. We define the body-frame polar angle θp,<sup>α</sup> at a point x<sup>s</sup> on the surface of squirmer <sup>α</sup> as the angle between <sup>d</sup>^<sup>α</sup> and a vector <sup>x</sup><sup>s</sup> � <sup>x</sup><sup>α</sup> from the center of the squirmer x<sup>α</sup> to xs. The unit vector e ^<sup>θ</sup>p,<sup>α</sup> is oriented in the direction of increasing θp,α, i.e., locally tangent to the squirmer surface along a longitudinal line.

### Figure 1.

The two types of microswimmer considered in this chapter. In the spherical "squirmer" model (left), the slip velocity on the surface of the particle is specified by fiat and fixed (in a frame co-moving and co-rotating with the particle) for all time. For an axisymmetric distribution of surface slip, the particle moves in the direction of the green arrow, i.e., opposite to the (surface-averaged) direction of the slip velocity. For a Janus particle (right), a fraction of the particle surface (black) catalyzes a reaction involving various molecular species diffusing in the surrounding solution. The resulting anisotropic distribution of product molecules or "solute" (green spheres) drives a phoretic slip velocity (purple) in an interfacial layer surrounding the particle. For a repulsive interaction between the solute and the particle surface, the slip is towards high concentration of solute, and the particle moves in the direction of the green arrow.

The squirming mode amplitudes Bn,α, which can potentially vary from squirmer to squirmer, are fixed 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 B<sup>1</sup> determines the velocity of an isolated squirmer in unbounded solution: Ufs,<sup>α</sup> ¼ ð Þ 2=3 B1,α. According to our definition of d<sup>α</sup> and θp,α, we require that B1,α>0. Simulations of squirmers typically truncate Eq. 8 to n ≤2 or n≤ 3. The justification for this is that the contributions of the higher order squirming modes to the flow around the squirmer decay rapidly with distance from the squirmer.

## 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:

$$\text{H}\_2\text{O}\_2 \xrightarrow[\text{Pt}]{} \text{H}\_2\text{O} + \frac{1}{2}\text{O}\_2.\tag{10}$$

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 δ ≪ R [23]. Each colloid is surrounded by an interfacial layer of thickness � δ in which the molecular interaction of the solute and the colloid is significant. Outside of this layer, the solute freely diffuses in the solution. We can regard cð Þ x as the bulk concentration, i.e., the concentration outside the interfacial layer. Near a location x<sup>s</sup> on the surface of the colloid, the interfacial layer concentration is enhanced or depleted, according to the attractive or repulsive character of the

� �. Here, the plus sign emphasizes that c x<sup>þ</sup>

vs,αð Þ¼� x<sup>s</sup> bð Þ x<sup>s</sup> ∇kc: (14)

u<sup>0</sup> � σ � n^ dS, (15)

� �, where η is a coordinate defined at xs

s � � is

s

evaluated outside the interfacial layer. Moreover, since δ ≪ R, the interfacial layer concentration can locally, in the direction locally normal to the colloid surface, relax to a Boltzmann (i.e., equilibrium) distribution governed by the molecular interaction potential Φ. (The timescale for this local relaxation is much faster than the characteristic timescale for colloid motion R=U0.) Accordingly, the local pressure

s

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 cð Þ x determines the pressure in the interfacial layer in the vicinity of a point x<sup>s</sup> on the colloid surface. Moreover, cð Þ x varies over the length scale R of the colloid. Accordingly, within the interfacial layer, the pressure varies over the size of the colloid, driving flow within the interfacial layer. From the perspective of the outer solution for the flow field, this interfacial flow looks like a

Here, the surface gradient operator is defined as ∇<sup>k</sup> � ð Þ� I � n^n^ ∇. The material-dependent parameter bð Þ x<sup>s</sup> encapsulates the details of the molecular interaction between the solute and the surface material, and can be calculated from the molecular potential Φ [23]. Since the surface of the Janus colloid comprises different materials, b depends on the location on the colloid surface. In fact, a spatial variation of b over the surface of colloid is a necessary condition to obtain phoretic rotation of a colloid near a wall [30] or chemotactic alignment with a gradient of

The Lorentz reciprocal theorem relates the fluid stresses σ; σ<sup>0</sup> ð Þ and velocity fields u; u<sup>0</sup> ð Þ of two solutions to the Stokes equation within the same domain V:

> ð S

where S is the boundary of V. For the N active particles in unbounded solution,

This theorem can be used to simplify the problem specified above for the velocities of N active particles. We designate that problem as the "unprimed" problem. Additionally, we specify that Fext,<sup>α</sup> ¼ 0 and τext,<sup>α</sup> ¼ 0 for all α. (Since the Stokes equation is linear, the contributions of the external forces and torques to the

u � σ<sup>0</sup> � n^ dS ¼

molecular interaction, relative to c x<sup>þ</sup>

<sup>s</sup> ; <sup>η</sup> � � can be calculated from <sup>Φ</sup> and <sup>c</sup> <sup>x</sup><sup>þ</sup>

The Boundary Element Method for Fluctuating Active Colloids

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

that is locally normal to the colloid surface.

P x�

slip velocity:

"fuel" molecules [35].

<sup>S</sup> <sup>¼</sup> <sup>∪</sup><sup>N</sup>

55

<sup>α</sup>¼<sup>1</sup>Sα.

2.3 Lorentz reciprocal theorem

ð S

(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:

$$-D[\nabla c \cdot \hat{\mathbf{n}}]|\_{\mathbf{x}=\mathbf{x}\_{\mathbf{t}}} = \kappa(\mathbf{x}\_{\mathbf{t}}),\tag{11}$$

where D is the diffusion coefficient of oxygen, cð Þ x is the number density of oxygen, and κð Þ x<sup>s</sup> is the rate of oxygen production on the surface of the particle. (The validity of assumption of reaction-limited kinetics is quantified by the Damköhler number Da ¼ κ0R=D, where κ<sup>0</sup> is a characteristic reaction rate; we assume Da ≪ 1.) Furthermore, we assume that the Péclet number Pe � U0R=D is very small, where U<sup>0</sup> is a characteristic particle velocity and R is the particle radius. Accordingly, we can make a quasi-steady approximation for the diffusion of oxygen in the solution:

$$
\nabla^2 \mathfrak{c} = \mathbf{0}.\tag{12}
$$

Finally, we assume that

$$\mathcal{c}(|\mathbf{x}| \to \infty|) = \mathfrak{c}\_{\infty} \tag{13}$$

where c<sup>∞</sup> is a constant. Eqs. 11, 12, and 13 specify a boundary value problem (BVP) for the distribution of oxygen in the fluid domain containing the N active particles. This problem can be solved numerically, e.g., by the boundary element method, as will be outlined in a later section.
