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, 54–57]. 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.

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 [73–76]. 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 � <sup>1</sup>=r2.) Nearfield hydrodynamic interactions are typically included via lubrication forces acting between particle pairs. Due to these approximations, Stokesian dynamics is computationally much cheaper than BEM, allowing access to collective dynamics, the rheology of dense suspensions, etc. On the other hand, SD does not typically resolve the microscopic details of individual particles, such as shape or heterogeneous surface chemistry. A hybrid BEM-SD method could combine the detailed microscopic resolution of BEM for near-field interactions with the ability of SD to capture many-body phenomena driven by far-field interactions. (This hybrid approach would bear some similarity to the fast multipole method.) In the Appendix at the end of this chapter, we develop a starting point for including interfacial flows vsð Þ x

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

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

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 nonspherical 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,

Consider an inert (non-active) sphere of radius <sup>R</sup> in an ambient flow field <sup>u</sup><sup>∞</sup>ð Þ <sup>x</sup> . The sphere has translational velocity U and rotational velocity Ω. The flow field u can be written as u xð Þ¼ <sup>u</sup><sup>∞</sup>ð Þþ <sup>x</sup> <sup>u</sup><sup>D</sup>ð Þ <sup>x</sup> , where <sup>u</sup><sup>D</sup>ð Þ <sup>x</sup> is the velocity disturbance created by the presence of the sphere. The boundary condition for <sup>u</sup><sup>D</sup>ð Þ <sup>x</sup> on the

Additionally, <sup>u</sup><sup>D</sup>ð Þ! <sup>j</sup>xj ! <sup>∞</sup> 0. Taking the sphere position to be <sup>x</sup><sup>0</sup> <sup>¼</sup> 0, we can

<sup>u</sup><sup>D</sup>ð Þ¼ xs <sup>U</sup> <sup>þ</sup> <sup>Ω</sup> � ð Þ� xs � <sup>x</sup><sup>0</sup> <sup>u</sup><sup>∞</sup>ð Þ <sup>x</sup><sup>s</sup> , <sup>x</sup><sup>s</sup> <sup>∈</sup> <sup>S</sup>: (69)

within the standard SD formalism for spherical particles.

The Boundary Element Method for Fluctuating Active Colloids

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

phoretic interactions with the boundary are considered [76, 81]?

A. Faxén laws and connection to Stokesian dynamics

expand the ambient flow field around the sphere center as:

through constrictions [79].

especially fluid/fluid interfaces.

sphere surface S is

67

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 b on the surface of the colloid, leads to phototaxis (rotation of the cap towards the light) or anti-phototaxis (rotation of the cap away from the light.) Notably, this work uses the hybrid BEM/BD method to calculate the distribution of particle orientations as a function of illumination intensity and particle surface chemistry. Concerning the interaction of multiple particles, Ref. 57 uses the regularized BEM to calculate the dynamics of multiple isotropic spherical colloids. Interestingly, a group of N ¼ 5 particles can form a stable cluster with broken rotational symmetry. This broken symmetry allows propulsion of the whole cluster. Finally, concerning shape, the BEM has been used to model toroidal [54] and spherocylindrical [72] self-phoretic particles.

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 b on different surfaces); this raises the danger of overfitting to experimental results.
