2.7 Thermal fluctuations

equation as follows [59]. Consider extending the volume filled by "fluid" to Vp. Within Vp, the flow field u is simply the flow field uRBM for rigid body motion, since it must obey no-slip on the surface S. Now we apply Eq. 59 for the field uRBM inside

ij ð Þþ <sup>x</sup> <sup>Σ</sup>ijkð Þ <sup>x</sup>0; <sup>x</sup> <sup>u</sup>RBM

Examining Eq. 62, we note that u x<sup>ð</sup> ), i.e., the flow velocity in <sup>V</sup>, is equal to <sup>u</sup>RBM

In order to obtain a single-layer boundary integral equation for x<sup>0</sup> ∈S, note that the jump discontinuity responsible for the factor of 1=2 in Eq. 62 is strictly from the double-layer potential [59]. The contribution of the single layer potential to the

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

In the preceding, we considered a suspension of N particles in an unbounded three-dimensional geometry. However, the presence of confining boundaries can have a significant effect on the dynamics of a suspension. It is possible to include a solid surface by explicitly meshing it and including it as a "fixed" or immobile particle in the calculations [53]. This approach is typically necessary for solid surfaces with corners or complex topography. One disadvantage of this approach is that an infinite surface (e.g., an infinite planar wall) must be truncated and

included as a finite size object. Care must be taken that the mesh is sufficiently fine near the particles, so that, for instance, the concentration and flow fields do not "leak" through a solid wall, but also that the mesh is sufficiently coarse far away

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

<sup>i</sup> ð Þ x

<sup>G</sup>ikð Þ <sup>x</sup>0; <sup>x</sup> <sup>σ</sup>ijð Þ <sup>x</sup> � �nj dS, <sup>x</sup><sup>0</sup> <sup>∈</sup>V: (65)

<sup>G</sup>ikð Þ <sup>x</sup>0; <sup>x</sup> <sup>σ</sup>ijð Þ <sup>x</sup> � �nj dS, <sup>x</sup><sup>0</sup> <sup>∈</sup>S: (66)

. The first term is simply the integral of Gikð Þ� x0; x n^<sup>0</sup> over

n0

<sup>j</sup> dS ¼ 0: (63)

<sup>j</sup> dS ¼ 0: (64)

V<sup>p</sup> and x<sup>0</sup> ∈V, noting that we must use a normal n^<sup>0</sup> ¼ �n^ pointing into Vp:

h i

the surface S for x<sup>0</sup> ∈V, which vanishes identically by incompressibility. This

<sup>Σ</sup>ijkð Þ <sup>x</sup>0; <sup>x</sup> <sup>u</sup>RBM <sup>i</sup> ð Þ <sup>x</sup> � �n<sup>0</sup>

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

� ð S

Non-Equilibrium Particle Dynamics

ij ð Þ x n^<sup>0</sup> ¼ �p0n^<sup>0</sup>

on S. Therefore, we conclude:

prehensive account.

64

i.e., σRBM

leaves:

<sup>G</sup>ikð Þ <sup>x</sup>0; <sup>x</sup> <sup>σ</sup>RBM

� ð S

ukð Þ¼� x<sup>0</sup>

velocity field is continuous across S. We obtain:

ukð Þ¼� x<sup>0</sup>

2.6 Active suspensions in confined geometries

from the particles, so that computation time is tractable.

ð S

ð S

So far, we have considered the deterministic contributions to the 6N components of velocity for a suspension of N particles. However, as outlined in the Introduction, the interplay of these deterministic contributions and the stochastic Brownian forces on the particles is important—and in some problems, such as the long-time behavior of an active colloid, it is absolutely essential.

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, 6N-component coordinate q:

$$\frac{d\mathbf{q}}{dt} = \mathbf{V} + k\_B T(\nabla \cdot \mathbf{M}) + \sqrt{2k\_B T} \mathbf{B} \cdot \mathbf{W},\tag{67}$$

where V is the deterministic contribution of activity to the generalized velocity ð Þ <sup>U</sup>α; <sup>Ω</sup><sup>α</sup> <sup>T</sup>, i.e., the solution to the problem outlined above; <sup>M</sup> is the grand mobility matrix <sup>M</sup> <sup>¼</sup> <sup>R</sup>�<sup>1</sup> ; <sup>B</sup> satisfies <sup>B</sup> � <sup>B</sup><sup>T</sup> <sup>¼</sup> <sup>M</sup>; and <sup>W</sup> is a collection of independent Wiener processes. Discretizing time in steps of Δt, one can write a generalized displacement Δq as the following Euler-Maruyama equation [34, 64]:

$$
\Delta \mathbf{q} = \mathbf{V} \Delta t + k\_B T (\nabla \cdot \mathbf{M}) \Delta t + \sqrt{2k\_B T} \mathbf{B} \cdot \Delta \mathbf{w}, \tag{68}
$$

where Δw is a stochastic variable with h i Δw ¼ 0 and Δwi Δwj � � <sup>¼</sup> <sup>δ</sup>ijΔt. The stochastic drift term ð Þ ∇ �M is a consequence of having a configuration-dependent mobility tensor in the framework of the Itô interpretation.

The update of the orientation of each particle α should respect the constraint that ∣dα∣ ¼ 1 and avoid any errors arising from application of (non-commuting) rotation matrices in arbitrary order to dα. There are robust algorithms for rigid body motion that represent the orientations of the particles with quaternions [65], Euler angles [66], or rotation matrices that transform between body-fixed and global reference frames [67].

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 cð Þ x in the fluid domain V.

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 S [71].
