2.1 Concentration field of product solute

A simple form for the activity of the colloidal particles is taken which consists of a constant flux production of solutes uniformly distributed on the catalytic surface. For small solute in infinite dilution limit, the non-dimensional concentration field of the solute product in the comoving frame can be written as (Khair Reference Khair2013; Michelin & Lauga Reference Michelin and Lauga2014):

(2.1) $$\begin{eqnarray}\displaystyle Pe\left(\frac{\partial n}{\partial t}+\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}n\right)={\rm\nabla}^{2}n, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{v}$ is the fluid velocity profile, $t$ is time and the Péclet number, $Pe$ , appears as a direct consequence of non-dimensionalization identifying the relative importance of the time scale for solute advection ( $a/U_{c}$ ) to the time scale for diffusion of product solute ( $a^{2}/D_{B}$ ). At small Péclet numbers ( $Pe\ll 1$ ), the left-hand side of the above equation can be neglected and hence we have the diffusion equation which can be written as,

(2.2) $$\begin{eqnarray}\displaystyle {\rm\nabla}^{2}n=0, & & \displaystyle\end{eqnarray}$$

with the following boundary conditions,

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{n}^{j}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}n=-{\it\vartheta}_{j}u_{j}(\boldsymbol{r}),\quad \boldsymbol{r}\in S_{c}^{j},(j=1,2), & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle n\rightarrow 0,\quad (\left|\boldsymbol{r}\right|\rightarrow \infty ), & \displaystyle\end{eqnarray}$$

where $\boldsymbol{e}_{n}^{j}$ is the unit normal vector pointing outward to the surface of the $j$ th colloid, $n$ is the non-dimensional concentration of the product solute, $\boldsymbol{r}$ is the non-dimensional position vector with respect to the $j$ th colloid centre, $S_{c}^{j}$ accounts for the surface domain of the $j$ th colloid, $u_{j}$ is the coverage function of colloid $j$ which is one on its active section and zero elsewhere and ${\it\vartheta}_{j}$ represents the scaled solute flux production for colloid $j$ defined as,

(2.5) $$\begin{eqnarray}\displaystyle {\it\vartheta}_{j}=\frac{N_{j}}{N_{1}}, & & \displaystyle\end{eqnarray}$$

where $N_{j}$ denotes the dimensional solute flux production for colloid $j$ . All variables mentioned above were non-dimensionalized according to the characteristic concentration of the solute $n_{c}\sim N_{1}a/D_{B}$ and particle radius $a$ . To satisfy the boundary conditions at both colloid surfaces, we utilize a bispherical coordinate $({\it\alpha},{\it\beta},{\it\phi})$ . Bispherical coordinates are related to cylindrical coordinates ( ${\it\rho},{\it\phi},z$ ) with the following transformation:

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}=\frac{c\;\sin {\it\alpha}\;}{\cosh {\it\beta}-\cos {\it\alpha}}, & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle z=\frac{c\;\sinh {\it\beta}}{\cosh {\it\beta}-\cos {\it\alpha}}. & \displaystyle\end{eqnarray}$$

Note that the $z$ -axis is always aligned in the direction of the centre-to-centre line of the particles. The range of coordinates are limited to $0\leqslant {\it\alpha}\leqslant {\rm\pi}$ , ${\it\beta}_{2}<{\it\beta}<{\it\beta}_{1}$ and $0\leqslant {\it\phi}\leqslant 2{\rm\pi}$ , for positive scale factor $c$ . Furthermore, ${\it\beta}={\it\beta}_{1}>0$ represents a sphere whose centre is located on the positive $z$ -axis at point $z=c\coth {\it\beta}_{1}$ with radius of $r_{c1}=c/\sinh {\it\beta}_{1}$ and ${\it\beta}={\it\beta}_{2}<0$ is the other sphere whose centre is located on the negative $z$ -axis at point $z=c\coth {\it\beta}_{2}$ with radius of $r_{c2}=c/|\sinh {\it\beta}_{2}|$ (see figure 1 for details). The parameter $c$ is non-dimensional and can be related to the separation distance $d$ (for two equal spheres) according to,

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\beta}_{1}=-{\it\beta}_{2}=\cosh ^{-1}\left(\frac{d}{2a}\right), & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle c=\sinh {\it\beta}_{1}. & \displaystyle\end{eqnarray}$$

The general three-dimensional solution of solute distribution in bispherical coordinates can be found via the following ansatz,

(2.10) $$\begin{eqnarray}\displaystyle n({\it\alpha},{\it\beta},{\it\phi}) & = & \displaystyle \sqrt{\cosh {\it\beta}-\cos {\it\alpha}}\mathop{\sum }_{m=0}^{\infty }\mathop{\sum }_{n=m}^{\infty }[(\tilde{A}_{nm}\cosh (n+0.5){\it\beta}\nonumber\\ \displaystyle & & \displaystyle +\,\tilde{B}_{nm}\sinh (n+0.5){\it\beta})\cos m{\it\phi}+ (\tilde{C}_{nm}\cosh (n+0.5){\it\beta}\nonumber\\ \displaystyle & & \displaystyle +\,\tilde{D}_{nm}\sinh (n+0.5){\it\beta})\sin m{\it\phi}]P_{n}^{m}(\cos {\it\alpha}),\end{eqnarray}$$

where $\tilde{A}_{nm}$ , $\tilde{B}_{nm}$ , $\tilde{C}_{nm}$ and $\tilde{D}_{nm}$ are unknown coefficients that must be determined by applying the boundary conditions on particle surfaces and $P_{n}^{m}(\cos {\it\alpha})$ is the associated Legendre function of the first kind. Having utilized the orthogonality of trigonometric functions and the associated Legendre polynomials, we have a set of recursive relationships which must be solved numerically. These relations are given in appendix A.

2.2 Hydrodynamics

The hydrodynamics in the Stokes regime in non-dimensional (see below for non-dimensionalization schemes) form can be written as

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}=0, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle -\boldsymbol{{\rm\nabla}}p+{\rm\nabla}^{2}\boldsymbol{v}=\mathbf{0}, & \displaystyle\end{eqnarray}$$

where $p$ is the pressure. The above equations are then subjected to a slip boundary condition at the surfaces of the colloids ( $j=1$ or $2$ ) and the quiescent far-field condition while considering the fact that particles are force and torque free

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}(\boldsymbol{r}\in S_{c}^{j})=\boldsymbol{v}_{s}^{j}+\boldsymbol{U}^{j}+{\it\bf\Omega}^{j}\times (\boldsymbol{r}-\boldsymbol{r}_{0}^{j}), & \displaystyle\end{eqnarray}$$

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}(\left|\boldsymbol{r}\right|\rightarrow \infty )\rightarrow \mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{F}_{T}^{j}=\unicode[STIX]{x0222F}_{S_{c}^{j}}(\boldsymbol{e}_{n}^{j}\boldsymbol{\cdot }{\bf\sigma})\,\text{d}S=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{T}_{T}^{j}=\unicode[STIX]{x0222F}_{S_{c}^{j}}(\boldsymbol{r}-\boldsymbol{r}_{0}^{j})\times (\boldsymbol{e}_{n}^{j}\boldsymbol{\cdot }{\bf\sigma})\,\text{d}S=\mathbf{0}, & \displaystyle\end{eqnarray}$$

where ${\bf\sigma}$ is the stress tensor, $\boldsymbol{r}_{0}^{j}$ is the location of the $j$ th particle centre of mass, $\boldsymbol{U}^{j}$ and ${\it\bf\Omega}^{j}$ are the non-dimensional translational and angular velocities of the $j$ th colloid yet to be determined and finally $\boldsymbol{v}_{s}^{j}$ is the non-dimensional slip velocity on the surface of the $j$ th colloidal motor, given as

(2.17) $$\begin{eqnarray}\displaystyle \boldsymbol{v}_{s}^{j}=-\boldsymbol{{\rm\nabla}}_{s}\,n(\boldsymbol{r}\in S_{c}^{j}), & & \displaystyle\end{eqnarray}$$

where the surface gradient is defined as $\boldsymbol{{\rm\nabla}}_{s}=(\boldsymbol{I}-\boldsymbol{e}_{n}\boldsymbol{e}_{n})\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}$ and all variables introduced in the above were non-dimensionalized with $U_{c}\sim bn_{c}/a$ , ${\it\Omega}_{c}\sim bn_{c}/a^{2}$ , $p$ and ${\bf\sigma}\sim {\it\mu}U_{c}/a$ , $F_{c}\sim {\it\mu}U_{c}a$ and $T_{c}\sim {\it\mu}U_{c}a^{2}$ . The mobility coefficient $b$ is defined as

(2.18) $$\begin{eqnarray}\displaystyle b=\frac{k_{B}TL^{2}}{{\it\mu}}\int _{0}^{\infty }y^{\prime }\left[\exp \left(-\frac{{\it\psi}}{k_{B}T}\right)-1\right]\text{d}y^{\prime }, & & \displaystyle\end{eqnarray}$$

where ${\it\mu}$ is the fluid viscosity, $k_{B}$ is the Boltzmann constant, $T$ is the absolute fluid temperature, $y^{\prime }$ is the distance to the particle edge and ${\it\psi}$ is the net potential interaction of the solute molecules with the particle in the presence of solvent. For a hard-sphere potential between solutes and colloids, this reduces to $b=-k_{B}TL^{2}/2{\it\mu}$ .

Figure 3.

Decomposition of velocity field into (a) surface diffusiophoretic slip with no net movement (translational or rotational) of the colloids (b) translation of colloids parallel to the line of centres ( $z$ axis) with no rotation or slip (c) rotation of colloids around the line of centres ( $z$ axis) with no slip or translation (d) translation of colloids perpendicular to the line of centres ( $x$ or $y$ axis) with no slip or rotation (e) rotation of colloids around the third axis ( $y$ or $x$ axis) with no slip or translation.

The arbitrary motion of two colloidal particles in space can be decomposed into translational motion parallel  $\Vert$  ( $z$ -axis) and perpendicular $\bot$ ( $x$ - and $y$ -axes) to the line of centres and rotational motion around an axis parallel $\Vert$ ( $z$ -axis) to the line of centres and the third axis $\otimes$ ( $y$ - and $x$ -axes), (see figure 3 for details). Furthermore, due to linearity of the Stokes flow equations and the boundary conditions, the fluid flow around two catalytically active particles can be decomposed into five subproblems (see figure 3):

1. (i) The colloidal particles are stationary, i.e. $U_{\Vert }^{j}=U_{\bot }^{j}={\it\Omega}_{\Vert }^{j}={\it\Omega}_{\otimes }^{j}=0$ and $j=1$ , $2$ , but the fluid slips according to the diffusiophoretic slip (2.17) on particles surfaces.

2. (ii) The colloidal particles are translating with velocities $U_{\Vert }^{1}$ and $U_{\Vert }^{2}$ parallel to the line of centres ( $z$ -direction) with no translations along the axis perpendicular to the line of centres ( $U_{\bot }^{1}=U_{\bot }^{2}=0$ ) or rotations around any axes ( ${\it\Omega}_{\Vert }^{j}={\it\Omega}_{\otimes }^{j}=0$ ). In this case, the colloids experience hydrodynamic forces in the direction of motion $F_{\Vert }^{j,H}$ , but the motion is axisymmetric and hence there are no hydrodynamic torques on particles.

3. (iii) The colloidal particles are rotating with angular velocities ${\it\Omega}_{\Vert }^{1}$ and ${\it\Omega}_{\Vert }^{2}$ around the axis parallel to the line of centres ( $z$ -direction) with no translations along any direction ( $U_{\Vert }^{j}=U_{\bot }^{j}=0$ ) or rotations around any other axes ( ${\it\Omega}_{\otimes }^{1}={\it\Omega}_{\otimes }^{2}=0$ ). In this case, the colloids experience hydrodynamic torques $T_{\Vert }^{j,H}$ , but their motions are axisymmetric and hence there are no hydrodynamic forces on the particles. We will show below that this contribution is identically zero in self-diffusiophoresis.

4. (iv) The colloidal particles are translating with velocities $U_{\bot }^{1}$ and $U_{\bot }^{2}$ along an axis perpendicular to the line of centres with no translations along the axis parallel to the line of centres ( $U_{\Vert }^{1}=U_{\Vert }^{2}=0$ ) or rotations around any other axes ( ${\it\Omega}_{\Vert }^{j}={\it\Omega}_{\otimes }^{j}=0$ ). In this case, due to the hydrodynamic coupling between the motions of the particles, there are hydrodynamic forces in the direction of motion and torques around an axis along the third direction.

5. (v) The colloidal particles are rotating with angular velocities ${\it\Omega}_{\otimes }^{1}$ and ${\it\Omega}_{\otimes }^{2}$ around the third axis with no translations along any axis ( $U_{\Vert }^{j}=U_{\bot }^{j}=0$ ) or rotations around the axis parallel to the line of centres ( ${\it\Omega}_{\Vert }^{1}={\it\Omega}_{\Vert }^{2}=0$ ). In this case, due to the hydrodynamic coupling between the motions of the particles, there are hydrodynamic forces in a direction perpendicular to the line of centres.

Note that in the above decomposition, each one of problems (iv) and (v) are generally two problems (one along $x$ and the other along $y$ -directions); nonetheless, the treatment for both are identical. To obtain the swimming velocities of the active colloids, the forces and torques associated with each one of the above five problems should be determined. Additionally, analytical expressions of hydrodynamic forces and torques on two particles for problems (ii)–(v) have already been derived in the literature (Jeffery Reference Jeffery1915; Stimson & Jeffery Reference Stimson and Jeffery1926; Maude Reference Maude1961; Goldman, Cox & Brenner Reference Goldman, Cox and Brenner1966; Wakiya Reference Wakiya1967; Davis Reference Davis1969; O’Neill & Majumdar Reference O’Neill and Majumdar1970a ; Spielman Reference Spielman1970) and (in a general linear form) can be written as

(2.19) $$\begin{eqnarray}\displaystyle \boldsymbol{F}^{H}=-\unicode[STIX]{x1D64D}\boldsymbol{\cdot }\boldsymbol{U}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D64D}$ (a $12\times 12$ matrix) is the grand resistance matrix, $\boldsymbol{F}^{H}$ is the $12$ -component vector including the hydrodynamic forces and torques on two particles and the $12$ -component vector $\boldsymbol{U}$ contains the translational and rotational velocities ( $\boldsymbol{U}^{j}$ and ${\it\bf\Omega}^{j}$ ) of the particles in Cartesian coordinates. Having known that both colloids are force and torque free, i.e. $\boldsymbol{F}^{H}+\boldsymbol{F}^{P}=0$ , the swimming translational and rotational velocities of the particles are linearly related to the propulsive (diffusiophoretic) forces and torques (problem (i) in the above decomposition) through the grand mobility matrix $\unicode[STIX]{x1D64D}^{-1}$ which in general form can be given as

(2.20) $$\begin{eqnarray}\displaystyle \boldsymbol{U}=\unicode[STIX]{x1D64D}^{-1}\boldsymbol{\cdot }\boldsymbol{F}^{P}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{F}^{P}$ is the $12$ -component vector including the propulsive forces and torques on two particles (force and torques on particles in figure 3 a). To obtain $\boldsymbol{U}$ and eventually the trajectory of swimmers, the only task that has to be done is to evaluate the propulsive forces and torques $\boldsymbol{F}^{P}$ . This can be achieved by utilizing the analytical solution for Stokes flow in bispherical coordinates (Lee & Leal Reference Lee and Leal1980). In this case, the velocity components are in terms of infinite series with coefficients that must be found numerically. Thereafter, the forces and torques on the particles associated with problem (i) in figure 3(a) can be numerically calculated. This method has been used in a number of studies, including ones involving the dynamics of two spherical particles, a spherical particle in the vicinity of a planar solid wall in electrophoresis (Keh & Chen Reference Keh and Chen1988; Zeng, Zinchenko & Davis Reference Zeng, Zinchenko and Davis1999) and the self-diffusiophoresis of a catalytic particle near a solid wall (Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016). This approach is straightforward, however the details of the calculation and algebra are cumbersome and hence our objective here is to employ an alternative approach to obtain expressions for swimming translational and angular velocities $\boldsymbol{U}^{j}$ and ${\it\bf\Omega}^{j}$ of the particles due to self-diffusiophoresis. Here, we use the RRT (Teubner Reference Teubner1982; Happel & Brenner Reference Happel and Brenner1983; Stone & Samuel Reference Stone and Samuel1996; Masoud & Stone Reference Masoud and Stone2014) to simplify the calculation of propulsive forces and torques on particles as explicit integrals in terms of the stress tensor of classical Stokes flow problems involving translational and rotational motion of two spherical particles. A significant advantage of RRT is that the hydrodynamics (2.11) and (2.12) does not need to be solved in order to evaluate the force and torque and therefore the calculations is significantly reduced.

Suppose ( $\boldsymbol{v}^{\prime },{\bf\sigma}^{\prime }$ ) and ( $\boldsymbol{v}^{\prime \prime },{\bf\sigma}^{\prime \prime }$ ) are the solutions of the Stokes flow of an incompressible fluid for some arbitrary fluid volume ${\it\Sigma}$ . According to RRT, one can state that:

(2.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0222F}_{\partial {\it\Sigma}}\boldsymbol{n}_{{\it\Sigma}}\boldsymbol{\cdot }{\bf\sigma}^{\prime }\boldsymbol{\cdot }\boldsymbol{v}^{\prime \prime }\,\text{d}S=\unicode[STIX]{x0222F}_{\partial {\it\Sigma}}\boldsymbol{n}_{{\it\Sigma}}\boldsymbol{\cdot }{\bf\sigma}^{\prime \prime }\boldsymbol{\cdot }\boldsymbol{v}^{\prime }\,\text{d}S, & & \displaystyle\end{eqnarray}$$

where $\partial {\it\Sigma}$ represents the boundaries enclosing the volume ${\it\Sigma}$ ; for instance, in the case of two spherical colloids, $\partial {\it\Sigma}$ accounts for a union of the boundaries including the colloids surfaces and an enclosing surface boundary far away from the two colloids, where $\boldsymbol{n}_{{\it\Sigma}}$ are the unit normals on these boundaries. Inasmuch as the velocity field of the test problem around the colloidal particles attenuates as $v\sim |\boldsymbol{r}|^{-1}$ or faster, the far-field area integral does not contribute to the calculation and can be neglected.

Having utilized RRT to evaluate the propulsive forces and torques on the particles $\boldsymbol{F}^{P}$ (see below for detail), we can substitute the propulsive forces and torques on the particle in (2.20) to construct a compact formula which express swimming translational and angular velocities of the particles according to (see Stone & Samuel Reference Stone and Samuel1996; Elfring Reference Elfring2015; Elfring & Lauga Reference Elfring and Lauga2015)

(2.22) $$\begin{eqnarray}\displaystyle \boldsymbol{U}=\unicode[STIX]{x1D64D}^{-1}\boldsymbol{\cdot }\mathop{\sum }_{j=1}^{2}\unicode[STIX]{x0222F}_{S_{c}^{j}}\boldsymbol{v}_{s}^{j}\boldsymbol{\cdot }(\boldsymbol{e}_{n}^{j}\boldsymbol{\cdot }\unicode[STIX]{x1D64F})\,\text{d}s, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D64F}$ denotes the triadic hydrodynamic stress fields (a $[3\times 3]\times 12$ matrix) associated with translations and rotations of two spherical particles and note that in the above expression we utilized the linearity of the Stokes equations by letting ${\bf\sigma}^{\prime \prime }=\unicode[STIX]{x1D64F}\boldsymbol{\cdot }\boldsymbol{U}$ . Below we evaluate the integral in (2.22) to discuss the propulsive forces and torques on particles in different directions.

2.2.1 Propulsive forces on two colloidal motors along the centre-to-centre line

To quantify the forces on the particles along the line of centres ( $z$ -direction), ( $\boldsymbol{v}^{\prime \prime }$ , ${\bf\sigma}^{\prime \prime }$ ) in (2.21) is chosen to be the solution of translational motion of two colloids along their line of centres which satisfies the following boundary conditions at the particle surfaces and far field,

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }(\boldsymbol{r}\in S_{c}^{j})=\boldsymbol{e}_{\Vert },\quad (j=1,2), & \displaystyle\end{eqnarray}$$

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }(|\boldsymbol{r}|\rightarrow \infty )=\mathbf{0}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{e}_{\Vert }$ is the unit normal vector along the line of centres. This problem was analysed in detail by Stimson & Jeffery (Reference Stimson and Jeffery1926). Having substituted ( $\boldsymbol{v}^{\prime \prime }$ , ${\bf\sigma}^{\prime \prime }$ ) in (2.21), it can be shown that

(2.25) $$\begin{eqnarray}\displaystyle F_{\Vert }^{1,P}+F_{\Vert }^{2,P}=\unicode[STIX]{x0222F}_{S_{c}^{1}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }{\bf\sigma}_{S-J}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S-\unicode[STIX]{x0222F}_{S_{c}^{2}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }{\bf\sigma}_{S-J}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S, & & \displaystyle\end{eqnarray}$$

where ${\bf\sigma}_{S-J}$ is the stress tensor corresponding to the Stimson & Jeffery (Reference Stimson and Jeffery1926) analysis. At the moment we have one equation (2.25) with two unknowns ( $F_{\Vert }^{1,P}$ and $F_{\Vert }^{2,P}$ ) and hence one more equation is required to close the problem. To obtain the second relationship, RRT can be used one more time by assuming ( $\boldsymbol{v}^{\prime \prime }$ , ${\bf\sigma}^{\prime \prime }$ ) in (2.21) is the solution of translational motion of two colloids along their line of centres and which translate in an opposite direction,

(2.26) $$\begin{eqnarray}\boldsymbol{v}^{\prime \prime }(\boldsymbol{r}\in S_{c}^{j})=\mp \boldsymbol{e}_{\Vert },(j=1,2),\end{eqnarray}$$

(2.27) $$\begin{eqnarray}\boldsymbol{v}^{\prime \prime }(|\boldsymbol{r}|\rightarrow \infty )=\mathbf{0},\end{eqnarray}$$

where the minus sign in (2.26) is for particle $1$ on top and the plus sign is for the particle $2$ on the bottom. This problem was analysed initially by Maude (Reference Maude1961) (but with some errors) and, by substitution of the solution, we have

(2.28) $$\begin{eqnarray}\displaystyle -F_{\Vert }^{1,P}+F_{\Vert }^{2,P}=\unicode[STIX]{x0222F}_{S_{c}^{1}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }{\bf\sigma}_{M}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S-\unicode[STIX]{x0222F}_{S_{c}^{2}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }{\bf\sigma}_{M}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S, & & \displaystyle\end{eqnarray}$$

where ${\bf\sigma}_{M}$ is the stress tensor corresponding to the Maude (Reference Maude1961) analysis. Having solved equations (2.25) and (2.28) simultaneously, the two unknowns $F_{z}^{1,P}$ and $F_{z}^{2,P}$ can be evaluated. Note that the above integrals were calculated numerically to find the magnitude of the propulsive forces along the line of centres ( $z$ -direction) of the two colloids.

2.2.2 Propulsive torques on two colloidal motors parallel to their line of centres

Propulsive torques on colloids around the line of centres ( $z$ -axis) can be directly evaluated with RRT by letting ( $\boldsymbol{v}^{\prime \prime }$ , ${\bf\sigma}^{\prime \prime }$ ) in (2.21) be the solution of the rotations of the two colloids around their line of centres. This analysis was performed by Jeffery (Reference Jeffery1915) and it is reiterated in appendix B. In this case, the problem is axisymmetric and the only non-zero component of the velocity field is in the azimuthal angle direction ${\it\phi}$ , and hence all non-zero components of the stress tensor associated with this problem do not have a ${\it\phi}$ dependency. On the other hand, the solute concentration gradient does depend on ${\it\phi}$ and hence the propulsive torques on colloids along the line of centres can be proven to be identically zero,

(2.29) $$\begin{eqnarray}\displaystyle T_{\Vert }^{j,P}=\mp \unicode[STIX]{x0222F}_{S_{c}^{j}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }{\bf\sigma}_{J}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\;\text{d}S=0, & & \displaystyle\end{eqnarray}$$

where ${\bf\sigma}_{J}$ is the non-dimensional stress tensor associated with Jeffery’s problem.

2.2.3 Propulsive forces and torques on two colloidal motors perpendicular to their line of centres and around the third axis

In this case, the propulsive forces and torques on colloidal particles are in a direction perpendicular to the line of centres and around the third axis respectively. These directions can be either in $x$ or $y$ (see figure 2) and the analysis for each one of the directions is similar. The translational (or rotational) motion of colloidal particles in this case is no longer axisymmetric and the colloids experience a torque (or force) in the $y$ (or $x$ ) direction.

To use RRT (2.21) for calculation of the propulsive forces and torques on the colloids, ( $\boldsymbol{v}^{\prime \prime }$ , ${\bf\sigma}^{\prime \prime }$ ) is chosen to be the solution of four subproblems:

(I) Two colloids are translating with no rotation in the same direction with a unit velocity perpendicular to the line of centres

(2.30) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }(\boldsymbol{r}\in S_{c}^{1})=\boldsymbol{e}_{\bot }, & \displaystyle\end{eqnarray}$$

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }(\boldsymbol{r}\in S_{c}^{2})=\boldsymbol{e}_{\bot }, & \displaystyle\end{eqnarray}$$

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }(|\boldsymbol{r}|\rightarrow \infty )=\mathbf{0}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{e}_{\bot }$ can be either $\boldsymbol{e}_{x}$ or $\boldsymbol{e}_{y}$ . This problem was analysed by Goldman et al. (Reference Goldman, Cox and Brenner1966). Having utilized RRT (2.21), one can show

(2.33) $$\begin{eqnarray}\displaystyle F_{\bot }^{1,P}+F_{\bot }^{2,P}=\unicode[STIX]{x0222F}_{S_{c}^{1}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }\tilde{{\bf\sigma}_{1}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S-\unicode[STIX]{x0222F}_{S_{c}^{2}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }\tilde{{\bf\sigma}_{1}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S, & & \displaystyle\end{eqnarray}$$

where $\tilde{{\bf\sigma}}_{1}$ is the non-dimensional stress tensor corresponding to analysis in Goldman et al. (Reference Goldman, Cox and Brenner1966).

(II) Two colloids are rotating with no translation in opposite directions around the third direction which can be either the $y$ - or $x$ -axis,

(2.34) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }(\boldsymbol{r}\in S_{c}^{1})=\boldsymbol{e}_{\otimes }\times (\boldsymbol{r}-\boldsymbol{r}_{0}^{1}), & \displaystyle\end{eqnarray}$$

(2.35) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }(\boldsymbol{r}\in S_{c}^{2})=-\boldsymbol{e}_{\otimes }\times (\boldsymbol{r}-\boldsymbol{r}_{0}^{2}), & \displaystyle\end{eqnarray}$$

(2.36) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }(|\boldsymbol{r}|\rightarrow \infty )=\mathbf{0}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{e}_{\otimes }$ accounts for the unit vector in the third direction, i.e. $y$ or $x$ . This problem was also analysed by Goldman et al. (Reference Goldman, Cox and Brenner1966). Having utilized RRT (2.21), one can show

(2.37) $$\begin{eqnarray}\displaystyle T_{\otimes }^{1,P}-T_{\otimes }^{2,P}=\unicode[STIX]{x0222F}_{S_{c}^{1}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }\tilde{{\bf\sigma}_{2}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S-\unicode[STIX]{x0222F}_{S_{c}^{2}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }\tilde{{\bf\sigma}_{2}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S, & & \displaystyle\end{eqnarray}$$

where $\tilde{{\bf\sigma}}_{2}$ is the non-dimensional stress tensor corresponding to the analysis performed by Goldman et al. (Reference Goldman, Cox and Brenner1966).

(III) Two colloids are translating with no rotation in an opposite direction with a unit velocity perpendicular to the line of centres,

(2.38) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }\left(\boldsymbol{r}\in S_{c}^{1}\right)=\boldsymbol{e}_{\bot }, & \displaystyle\end{eqnarray}$$

(2.39) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }\left(\boldsymbol{r}\in S_{c}^{2}\right)=-\boldsymbol{e}_{\bot }, & \displaystyle\end{eqnarray}$$

(2.40) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }\left(|\boldsymbol{r}|\rightarrow \infty \right)=\mathbf{0}. & \displaystyle\end{eqnarray}$$

This problem was analysed by O’Neill (Reference O’Neill1969). Having utilized RRT (2.21), we find

(2.41) $$\begin{eqnarray}\displaystyle F_{\bot }^{1,P}-F_{\bot }^{2,P}=\unicode[STIX]{x0222F}_{S_{c}^{1}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }\tilde{{\bf\sigma}_{3}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S-\unicode[STIX]{x0222F}_{S_{c}^{2}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }\tilde{{\bf\sigma}_{3}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S, & & \displaystyle\end{eqnarray}$$

where $\tilde{{\bf\sigma}}_{3}$ is the non-dimensional stress tensor corresponding to the analysis performed by O’Neill (Reference O’Neill1969).

(IV) Two colloids are rotating with no translation in the same direction

(2.42) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }\left(\boldsymbol{r}\in S_{c}^{1}\right)=\boldsymbol{e}_{\otimes }\times \left(\boldsymbol{r}-\boldsymbol{r}_{0}^{1}\right), & \displaystyle\end{eqnarray}$$

(2.43) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }\left(\boldsymbol{r}\in S_{c}^{2}\right)=\boldsymbol{e}_{\otimes }\times \left(\boldsymbol{r}-\boldsymbol{r}_{0}^{2}\right), & \displaystyle\end{eqnarray}$$

(2.44) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}^{\prime \prime }\left(|\boldsymbol{r}|\rightarrow \infty \right)=\mathbf{0}. & \displaystyle\end{eqnarray}$$

This problem was also analysed by O’Neill (Reference O’Neill1969). Having utilized RRT (2.21), we find

(2.45) $$\begin{eqnarray}\displaystyle T_{\otimes }^{1,P}+T_{\otimes }^{2,P}=\unicode[STIX]{x0222F}_{S_{c}^{1}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }\tilde{{\bf\sigma}_{4}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S-\unicode[STIX]{x0222F}_{S_{c}^{2}}\boldsymbol{e}_{{\it\beta}}\boldsymbol{\cdot }\tilde{{\bf\sigma}_{4}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{s}n\,\text{d}S, & & \displaystyle\end{eqnarray}$$

where $\tilde{{\bf\sigma}}_{4}$ is the non-dimensional stress tensor corresponding to the O’Neill (Reference O’Neill1969) analysis.

By combining equations (2.33) and (2.41) and (2.37) and (2.45), the forces and torques around the third axis can be evaluated respectively.

The solution to the four classical problems (Goldman et al. Reference Goldman, Cox and Brenner1966; O’Neill Reference O’Neill1969) mentioned above are reiterated in appendix B. The above relations provide forces and torques on the particles in (2.20) and the integral on the right-hand side of (2.22). The integrands were all evaluated according to the analytical relations, however, the integrations were performed numerically. In the evaluation of these quadratures, all stress tensors associated with the classical problems are in the form of infinite sums and, in our calculation, these series are truncated according to the fact that the coefficients vanish for sufficiently large number of terms. Also, more terms in the truncated series were required for small separation distances to reach numerically accurate and consistent results for both concentration field and stress tensor components. Once the propulsive forces and torques on the particles $\boldsymbol{F}^{P}$ are known, we can utilize the mobility relation given in (2.22) (or alternatively (2.20)) to obtain swimming translational and angular velocity of the particles.

3.1 Concentration field

The solute concentration field around the colloids is independent of azimuthal angle ${\it\phi}$ when the inclination angles of the colloids are either $0^{\circ }$ or $180^{\circ }$ . For the situation where the active sections of colloids are opposite to each other, i.e. ${\it\Xi}_{1}=180^{\circ }$ and ${\it\Xi}_{2}=0^{\circ }$ , the solute distribution does not change drastically, except for very high coverage, as the colloids approach each other since the solutes can freely diffuse away from them (see figure 4 a,d). By contrast, in the other case where the active caps of the colloids face towards each other, ${\it\Xi}_{1}=0^{\circ }$ and ${\it\Xi}_{2}=180^{\circ }$ , as the separation distance between the particle decreases, the concentration of solute increases precipitously in the gap region adjacent to the colloids (see figure 4 b,e). When the active section of one colloid faces the passive section of the other one, i.e. ${\it\Xi}_{1}={\it\Xi}_{2}=0^{\circ }$ , the concentration of solute in the gap increases again as the separation distance between the colloids decreases. The major difference between this situation and the other two previous axisymmetric cases is the fact that the concentration field is no longer symmetric with respect to the midplane $x$ – $y$ and hence, as we show later, the swimming velocities of the colloids are no longer equal in magnitude (see figure 4 c,f).

Figure 4.

Non-dimensional concentration field for half active Janus colloids $({\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=90^{\circ })$ in the $x$ – $z$ plane for axisymmetric orientation of colloids with ${\it\Xi}_{1}=180^{\circ }$ and ${\it\Xi}_{2}=0^{\circ }$ at two different non-dimensional centre-to-centre distances of (a)  ${\it\Delta}=4$ and (d) ${\it\Delta}=3$ , and ${\it\Xi}_{1}=0^{\circ }$ and ${\it\Xi}_{2}=180^{\circ }$ at (b) ${\it\Delta}=4$ and (e) ${\it\Delta}=3$ , and ${\it\Xi}_{1}={\it\Xi}_{2}=0^{\circ }$ at (c) ${\it\Delta}=4$ and (f) ${\it\Delta}=3$ . Note that the red caps indicate the catalytic surface of the colloids.

For all other orientations, the solution of the concentration field is three-dimensional in bispherical coordinates, $n\left({\it\alpha},{\it\beta},{\it\phi}\right)$ . Representative concentration fields in the $x$ – $z$ plane around two Janus colloids $({\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=90)$ for various relative inclination angles and a fixed separation distance ${\it\Delta}=3$ are shown in figure 5. The pair interaction effect is more pronounced when the active sections are close to each other and therefore the solute concentration is boosted in the gap. Furthermore, for all values of the inclination angles except the symmetric cases, the concentration fields around the colloids are no longer symmetric with respect to the centre-to-centre line (see figure 5) and hence results in the rotation of particles. These rotations are the direct consequence of diffusiophoresis, which is due to the asymmetric distribution of product solute; this effect will be discussed in detail presently. Furthermore, the asymmetric distribution around the colloid axis is more pronounced as the inclination angles of colloids approach ${\it\Xi}_{j}\rightarrow 90^{\circ }$ or ${\it\Xi}_{j}\rightarrow 270^{\circ }$ ( $j=1,2$ ).

Figure 5.

Non-dimensional concentration field for half active Janus colloids $({\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=90^{\circ })$ in the $x$ – $z$ plane for asymmetric orientations and non-dimensional centre-to-centre distance of ${\it\Delta}=3$ . Note that the red caps indicate the catalytic surface of the colloids.

The concentration field around colloids can be altered not only by changing the orientation of colloids but also by varying the extent of the active sections. Figure 6(a–c) demonstrates concentration fields in the $x$ – $z$ plane for small ( ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=40^{\circ }$ ) and large ( ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=140^{\circ }$ ) active coverages. Note that for ${\it\theta}_{cap}^{j}=140^{\circ }$ , there is a substantial increase in solute concentration in the gap relative to the case where ${\it\theta}_{cap}^{j}=40^{\circ }$ for similar inclination angles. Thus, the diffusiophoretic effect for colloids with large active coverage is more significant compared to the colloids with smaller coverage.

Figure 6.

Non-dimensional concentration field for active colloids at various inclination angles in the $x$ – $z$ plane for asymmetric orientation of colloids at non-dimensional centre-to-centre distance of ${\it\Delta}=3$ with (a–c) ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=40^{\circ }$ and (d–f) ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=140^{\circ }$ . Note that the red caps indicate the catalytic surface of the colloids.

3.2 Swimming velocities and trajectories

3.2.1 Axisymmetric motion of two colloidal motors swimming in the direction parallel to the centre-to-centre line

To study the hydrodynamics of colloidal self-propulsion, we first focus on the axisymmetric motion of two motors, i.e. the inclination angles of the colloids are either $0^{\circ }$ or $180^{\circ }$ . In this situation, the only non-zero component of the swimming velocity is in the $z$ -direction (see figure 2). Moreover, since the flow is axisymmetric, the colloids do not undergo a rotation around any axis. This condition simplifies the governing equations significantly and the flow field can be found by using the conventional streamfunction technique in bispherical coordinates for the Stokes flow regime (Popescu et al. Reference Popescu, Tasinkevych and Dietrich2011; Sharifi-Mood Reference Sharifi-Mood2013; Reigh & Kapral Reference Reigh and Kapral2015). Additionally, to obtain a far-field solution, we employ a method of reflections to validate our analytical solution. The detailed derivation of this solution is discussed in appendix C.

Figure 7.

Non-dimensional swimming velocities of two half active Janus colloidal motors ( ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=90^{\circ }$ ) as a function of non-dimensional separation distance ${\it\Delta}-2$ for axisymmetric motions. Open symbols represent the full solution while the solid lines are the results from the method of reflections.

Figure 7 illustrates the non-dimensional swimming velocities of two Janus colloids as a function of the centre-to-centre separation distance ${\it\Delta}$ for three relative orientations. In all cases, the colloids move along the centre-to-centre line ( $z$ -direction), however, the dependency of the swimming velocities on the separation distance are substantially different for them. The concentration field can be symmetric or asymmetric with respect to the $x$ – $y$ plane. For a situation where the active sections of the motors face opposite to each other, figure 7(a), the swimming velocities of the colloids are equal in magnitude due to the symmetric orientation of the colloids with respect to the $x$ – $y$ plane, and the magnitudes of swimming velocities of the colloids are equal to the reference swimming velocity $U_{\infty }$ when the colloids are a few radii away from each other. This magnitude decreases monotonically as a function of separation distance between the colloids. In this case, the propulsive force does not change as the particles approach each other because the concentration field around each one of them does not change drastically by the presence of the other (see figure 4 a,d) and therefore the reduction in swimming velocity is solely due to an increase in hydrodynamic resistance which in the near-field scales as ${\sim}1/({\it\Delta}-2)$ (Kim & Karrila Reference Kim and Karrila2005). By contrast, when the active sections of the colloids face towards each other, the swimming velocities of particles increase considerably as the separation distance between them becomes smaller as shown in figure 7(b). In this case, the concentration of solute in the gap between the colloids becomes larger in magnitude precipitously as the separation distance between the colloids decreases (see figure 4 b,e) and consequently the propulsive force is greatly amplified so that it overtakes the increase in lubrication resistance. When the relative orientation of the colloids is not symmetric with respect to the $x$ – $y$ plane, e.g. the active section of one colloid faces the passive section of the other colloid, the swimming velocities of the colloidal particles are no longer identical in magnitude because the concentration field around the colloids are no longer symmetric with respect to $x$ – $y$ plane. Here, for a given orientation in figure 7(c), as the separation distance decreases, the concentration gradient around the colloid on top (colloid $1$ ) increases while the concentration gradient around the one on the bottom (colloid $2$ ) decreases and hence they swim faster and slower, respectively. At very small separation distances, ${\it\Delta}-2\sim 0.3$ , the concentration build up in the thin gap becomes so strong that it ultimately forces the colloid at the bottom (colloid $2$ ) to cease and for even smaller separation distances ${\it\Delta}-2<0.3$ , the direction of translation reverses as the location where the surface concentration is maximum shifts from the centre of the active cap to the perimeter closer to the passive cap. The results found from the method of reflections (solid black lines in figures 7 and 8) can reasonably capture the far-field swimming velocities of the particles, however, for smaller separation distances, this method loses its accuracy and more reflected terms should be taken into account for precise results.

Figure 8.

Non-dimensional swimming velocities of (a) two catalytically active colloids ( ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=180^{\circ }$ ) and (b) a catalytically active colloid ( ${\it\theta}_{cap}^{1}=180^{\circ }$ ) and an inert colloid (cargo, ${\it\theta}_{cap}^{2}=0^{\circ }$ ). Solid black lines are the method of reflections solutions.

Similarly, a catalytically active colloid with a uniform reaction at its surface can be propelled due to the presence of another active (or passive) colloid. Figure 8(a) represents the non-dimensional velocities (scaled with $U_{c}$ ) of two catalytically active colloids ( ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=180^{\circ }$ ) as a function of their separation distance. When the colloids are far from each other, they do not interact as the concentration field is uniform. However, when the separation distance becomes smaller, the presence of the other colloid distorts the concentration field around the first one and it eventually yields a propulsion. Likewise, this scenario would take place for a catalytic colloid ( ${\it\theta}_{cap}^{1}=180^{\circ }$ ) and an inert colloid ( ${\it\theta}_{cap}^{2}=0^{\circ }$ ) which can be considered as a cargo, figure 8(b). Notice the swimming velocities are not equal in magnitude in this case because the solute concentration gradient around the inert particle is stronger compared to the catalytically active particle and hence it translates faster in all separation distances. The solid black lines in figure 8 represent the results found from the method of reflection analysis suggesting the far-field velocity of the catalytic colloids ${\sim}1/{\it\Delta}^{5}$ and for the inert one ${\sim}1/{\it\Delta}^{2}$ (see appendix C for details). In a similar fashion, it has been recently shown that the symmetry can be also broken not only by taking different chemical activity for two colloids but also by considering two different particle sizes (Michelin & Lauga Reference Michelin and Lauga2015).

3.2.2 Asymmetric motion of two colloidal motors

For arbitrary inclination angles of the colloidal particles, they swim in all three directions and they also undergo solid body rotation. In particular, the motions of colloidal motors are influenced not only by hydrodynamic interactions (problem (b), (d) and (e) in figure 3) but also via diffusive interaction of the solute concentration generated at the active caps of the colloids with the boundary of the other colloid which modifies the propulsive force (problem (a) in figure 3). To disentangle these effects better, we focus on situations where the orientation vectors of the particles $\boldsymbol{n}_{p}^{1}$ and $\boldsymbol{n}_{p}^{2}$ are in the same plane $x$ – $z$ and do not have a component in $y$ -direction.

As we have observed in the axisymmetric motions, the swimming velocities of colloids depend considerably on their relative orientation, e.g. it does matter in which direction two colloidal motors approach or move away from each other. In order to systematically elucidate this effect, three sets of relative orientations of the colloids are considered. The inclination angles of the colloidal particles are chosen to be complimentary, ${\it\Xi}_{1}+{\it\Xi}_{2}=180^{\circ }$ in the first two cases and ${\it\Xi}_{1}-{\it\Xi}_{2}=180^{\circ }$ for the third case. In this way, by considering equal surface coverages for the particles, the magnitudes of the swimming translational and angular velocities are identical.

Figure 9.

Non-dimensional translational and angular velocities of two Janus colloidal motors ( ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=90^{\circ }$ ) for three distinct relative orientations as a function of non-dimensional separation distance ${\it\Delta}-2$ . The translational non-dimensional velocities in the (a) $z$ -direction (parallel to the centre-to-centre line) and (b) $x$ -direction (perpendicular to the centre-to-centre line) are non-dimensionalized with the swimming velocity of a Janus colloid at infinite medium $U_{\infty }$ and angular velocity (c) around the $y$ axis is non-dimensionalized with $U_{\infty }/a$ .

In the first scenario, the active areas of the colloidal motors are on one side and adjacent to each other. Figure 9 illustrates the translational and angular velocities of two Janus colloids as a function of separation distance for three relative orientations. The non-dimensional swimming velocity component, which is along the direction of the centre-to-centre line, approaches the reference value when the colloids are far from each other, i.e. $\tilde{U} _{z}^{j}\rightarrow \text{cos}{\it\Xi}_{j}$ , however, for smaller separation distances, this value increases as the concentration of solute becomes larger in the thin gap for all given inclination angles as shown in figure 9(a). The component of non-dimensional swimming velocity in the direction perpendicular to the centre-to-centre line, $\tilde{U} _{x}$ , behaves similarly; when the motors are far away from each other, it approaches the reference value $\tilde{U} _{x}^{j}\rightarrow \text{sin}{\it\Xi}_{j}$ and when the separation distance is smaller, the colloids translate faster (figure 9 b). This behaviour is predictable based on what was discussed regarding the axisymmetric motions: for smaller separation distances, the solute concentration is strengthened in the thin gap and brings about stronger propulsions. In addition, the hydrodynamic resistance associated with problem (d) in figure 3 becomes weaker as the particles translate in the same direction (see Goldman et al. Reference Goldman, Cox and Brenner1966) and therefore particles can swim faster. The angular rotation around the $y$ axis, on the other hand, is non-monotonic. To explain this trend, we should first determine the sign of the propulsive torques (torques associated with problem (a) in figure 3) on the particles. For the complementary relative orientations given in the first scenario, the propulsive torques on the colloids are always equal and opposite, however, depending on the relative orientations, the signs can be positive (clockwise with respect to the negative $y$ axis) or negative (counterclockwise with respect to the negative $y$ axis). To resolve this complex behaviour, we have to first elucidate the sign of propulsive torques on the particles. The propulsive torques on particles are controlled by the local distribution of the force in the gap which depends on the relative orientations of the colloids and their separation distance. In figure 10(a), the directions of slip velocity $\boldsymbol{v}_{s}$ on particle surfaces are schematically shown. In this case, the high surface concentration is along the active caps and the low concentration at the inert sides. The generated surface flow due to solute gradient adjacent to the gap (shown with solid lines) brings about a counterclockwise (clockwise) torque on the particle on top (bottom) while the surface flow on the other section of the particle (shown with dotted lines) yields a clockwise (counterclockwise) torque on the particle on top (bottom). The surface flow along the solid lines are greater in magnitude relative to the dotted lines because the concentration gradient in the gap is stronger and therefore by neglecting the flow generated by the other particle, it is expected the particle on top (bottom) rotates counterclockwise (clockwise) with negative (positive) propulsive torque. However, the local shear stress (velocity gradient) in the gap region can be drastically altered by the interaction of the surface flows generated by the particles. In figure 10(b) the velocity profiles in the gap region and the regions facing opposite to the gaps are schematically demonstrated. As evident from the figure, the velocity profile inside the gap is always more uniform compared to the regions opposite to the gap. This effect can alleviate the local force in the thin gap and reverse the direction of propulsive torques on the particles and hence the competition between these two effects shown in figure 10 at different relative orientations of the particles brings about the complex trend observed in angular velocity (figure 9 c). Furthermore, it is important to note that the resultant hydrodynamic torque due to translational motion of two colloids (figure 3 d) is also involved in determining the angular velocities of the particles however this effect is weaker relative to the propulsive torque in this case.

Figure 10.

Schematic representations of (a) the surface flows generated by the solute concentration gradient along the particle surfaces in the first scenario (the dotted and solid lines indicate weak and strong surface flows respectively) and (b) the local velocity profile around the two particles due to their surface flows (phoretic effect).

In the second scenario, the passive areas of the colloids are next to each other in an inclined angle and they are both translating in the positive $x$ -direction (see figure 11). Figure 11(a) shows the $z$ component of the non-dimensional swimming velocity as a function of separation distance for four inclination angles. For three relative inclined orientations, the colloids velocities decrease as the particles swim towards each other and they eventually stops due to the lubrication force in the thin gap. However, for the motion along the positive $x$ -direction where ${\it\Xi}_{1}={\it\Xi}_{2}=90^{\circ }$ , the non-dimensional swimming velocity in the $z$ -direction is amplified by the propulsive force due to the presence of excess solute in the gap between the colloids. The swimming velocity component along $x$ and the rotational velocity around $y$ for all relative orientations (except the parallel motion ( ${\it\Xi}_{1}={\it\Xi}_{2}=90^{\circ }$ )) behaves exactly as in the previous case (compare figure 11 b,c with figure 9 b,c). In the case of parallel motion ( ${\it\Xi}_{1}={\it\Xi}_{2}=90^{\circ }$ ), the swimming velocity along the $x$ -axis increases initially as the separation distance becomes smaller. However, once the separation distance becomes smaller than ${\it\Delta}\sim 2.4$ , the swimming velocity monotonically decreases (see figure 11 b). The rotational velocity around the $y$ axis in this case is always positive and it monotonically increases as the colloids approach (open circles in figure 11 c). This complex trend is again due to the coupling of the rotational motion around the $y$ axis (positive angular velocity for the colloid on top and negative for the one on the bottom) and positive translational motion in the $x$ -direction according to the forces and torques determined from problems (a), (d) and (e) in figure 3.

Figure 11.

Non-dimensional translational and angular velocities of two Janus colloidal motors ( ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=90^{\circ }$ ) for four distinct relative orientations as a function of non-dimensional separation distance ${\it\Delta}-2$ . The translational non-dimensional velocities in (a) $z$ -direction (parallel to the centre-to-centre line) and (b) $x$ -direction (perpendicular to the centre-to-centre line) are non-dimensionalized with the swimming velocity of a Janus colloid at infinite medium $U_{\infty }$ and angular velocity (c) around the $y$ axis is non-dimensionalized with $U_{\infty }/a$ . Inset in (c) demonstrate the non-dimensional angular velocity of the relative orientations with small magnitude of rotation.

In the third scenario, the active caps face opposite each other in all the given orientations, figure 12. The $z$ component of the swimming velocity is identical to the second scenario, figure 12(a). However, the swimming velocity in the $x$ -direction and angular velocity behave differently. Figure 12(b) demonstrates the non-dimensional swimming velocity in the $x$ -direction as a function of separation distance. The speed of the colloids for all orientations decreases as they move towards each other. To realize the trend in this scenario, we allude again to the hydrodynamic forces associated with problems (d) and (e) in figure 3. For two passive colloidal particles which are translating in opposite directions under a constant force (O’Neill Reference O’Neill1969), the hydrodynamic torques on the two colloids are equal in sign and magnitude and hence the particles rotate in the same direction, e.g. if two passive colloids on the top and bottom of the $x$ – $y$ plane translate in negative and positive $x$ -directions, respectively, the hydrodynamic torque forces both colloids to rotate counterclockwise with respect to the negative $y$ -axis. Moreover, the hydrodynamic resistance to the motion of two passive colloids translating in opposite directions along the $x$ -axis increases as the separation distance decreases and therefore results in a smaller translational velocity. Figure 12(b) demonstrates the non-dimensional swimming velocity in the $x$ -direction as a function of separation distance; the speed of colloids for all of the orientations decreases as they move towards each other since the hydrodynamic resistance scales as ${\sim}\text{log}({\it\Delta}-2)$ (Kim & Karrila Reference Kim and Karrila2005). The colloids also rotate faster as they approach each other (see figure 12 c). For the parallel motion ( ${\it\Xi}_{1}=270^{\circ }$ and ${\it\Xi}_{2}=90^{\circ }$ ), the rotational velocity is initially positive and then it undergoes a transition to the negative values. The initial positive values are due to the fact that the propulsive torque in this case is always positive, i.e. the torques associated with problem (a) in figure 3 always tend to force the colloids to rotate in such a way that they bring their active sections close to each other and this effect is dominant in the far field once ${\it\Xi}_{1}=270^{\circ }$ and ${\it\Xi}_{2}=90^{\circ }$ . However, once the colloids approach each other, the hydrodynamic torque associated with problem (d) in figure 3 offsets this positive torque and ultimately compels the colloids to rotate with negative angular velocities.

Figure 12.

Non-dimensional translational and angular velocities of two Janus colloidal motors ( ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=90^{\circ }$ ) for three distinct relative orientations as a function of non-dimensional separation distance ${\it\Delta}-2$ . The translational non-dimensional velocities in (a) $z$ -direction (parallel to the centre-to-centre line) and (b) $x$ -direction (perpendicular to the centre-to-centre line) are non-dimensionalized with the swimming velocity of a Janus colloid at infinite medium $U_{\infty }$ and angular velocity (c) around the $y$ axis is non-dimensionalized with $U_{\infty }/a$ .

3.3 Pair trajectories

To investigate the pair trajectories of colloidal motors, we assume that, similar to previous section, the initial orientation vectors of the motors are both in the $x$ – $z$ plane and consequently the colloids stay in the same plane for a long time since it was shown that there are no rotations around the $z$ and $x$ axes and hence the colloids never exit the $x$ – $z$ plane over a long time period. Generally, for two colloidal particles approaching each other, there are two ultimate possible scenarios: either they (i) interact and ‘escape’ from each other or (ii) they persistently approach and come into contact to assemble. We refer to ‘assembly’ when the non-dimensional separation distance is less than ${\it\Delta}<2.01$ . Here, complementary angles for relative orientations are considered, i.e. the initial inclination angles with respect to the line of centres are either ${\it\Xi}_{1}+{\it\Xi}_{2}=180^{\circ }$ or ${\it\Xi}_{1}-{\it\Xi}_{2}=180^{\circ }$ ; the former indicates the conditions where the $x$ component of swimming velocities of both motors are in the positive direction while the latter includes situations where the $x$ component of swimming velocities are in opposite directions. To obtain particle trajectories, the swimming translational and rotational velocities of the particles were determined initially at each time step in the comoving (Lagrangian) frame and then the particles centre of mass and orientation vectors were updated by explicit integration in time with Euler’s method. The non-dimensional time interval was considered sufficiently small so that the particles do not move (or rotate) considerably in the time interval and the errors due to numerical integration are negligible. For two swimming colloids, the direction of rotation can be altered depending on two parameters: (i) relative orientations and (ii) the active coverages of the colloids.

Figure 13.

Pair trajectories of Janus motors for various initial inclination angles where ${\it\Xi}_{1}+{\it\Xi}_{2}=180^{\circ }$ .

The relative orientation indicates whether the active sections of the colloidal motors are adjacent or not. If the initial orientation of the motors is set in a fashion similar to the first scenario (see the schematic in figure 9), the build up of the excess concentration of solute in the gap between the particles gives rise to a situation where particles persistently move away from each other until they fail to interact and each one of them swims independently (see figure 13 a).

If the initial relative orientation of the motors is similar to the second scenario, as shown in figure 11, the colloids approach and rotate at the same time until they assemble. For two Janus motors, the directions of rotation at most relative orientations are in a way such that they enhance the assembly process of the particles, i.e. the particles rotate in opposite directions in a way to bring their passive sections aligned and closer. The pair trajectories of two Janus colloids with various initial relative orientations and separation distances of ${\it\Delta}=4$ are shown in figure 13(b–d). For initial inclination angles ${\it\Xi}_{1}=120^{\circ }$ (figure 13 b) and ${\it\Xi}_{1}=100^{\circ }$ (figure 13 c), the particle on top (bottom) rotates clockwise (counterclockwise) with respect to the negative $y$ -direction and this in turn enhances the assembly of the particles. For ${\it\Xi}_{1}=135^{\circ }$ (figure 13 d), the directions of rotation of the colloids are reversed, however the magnitude of the rotations are too small to allow the particles to escape.

Figure 14.

Pair trajectories of Janus motors for various relative initial inclination angles where ${\it\Xi}_{1}-{\it\Xi}_{2}=180^{\circ }$ .

In the third scenario, as shown in figure 12, the passive sections of the colloids are adjacent. The typical pair trajectories of two Janus motors for various initial inclination angles ${\it\Xi}_{1}$ are illustrated in figure 14. Here, the colloids always rotate in the same direction around the $y$ -axis. The particle tend to face their passive sections and for all initial inclination angles ${\it\Xi}_{1}>194^{\circ }$ at ${\it\Delta}=4$ , the colloids pass each other and escape. For smaller initial inclination angle ${\it\Xi}_{1}\leqslant 194^{\circ }$ , the particles block each other’s paths and hence they assemble.

Figure 15.

Pair trajectories of catalytically active motors for fixed relative initial inclination angles of ${\it\Xi}_{1}={\it\Xi}_{2}=90^{\circ }$ and (a) ${\it\theta}_{cap}=90^{\circ }$ , (b) ${\it\theta}_{cap}=40^{\circ }$ and (c) ${\it\theta}_{cap}=140^{\circ }$ .

In all three scenarios, the active coverages of the colloidal motors play an important role on the pair trajectories. To resolve this fact more clearly, in figure 15 we demonstrate the pair trajectories of active colloidal motors for a fixed relative orientation of ${\it\Xi}_{1}={\it\Xi}_{2}=90^{\circ }$ and three different cap angles of ${\it\theta}_{cap}^{j}=90^{\circ }$ , ${\it\theta}_{cap}^{j}=40^{\circ }$ and ${\it\theta}_{cap}^{j}=140^{\circ }$ . In figure 15(a), the pair trajectory of two Janus motors ${\it\theta}_{cap}^{j}=90^{\circ }$ is shown; the particles initially move away from each other because the swimming velocity along the $z$ -direction is positive (see figure 11 a). However, since the angular velocity around the $y$ axis is positive (negative) for the particle on top (bottom) (see figure 11 c), the colloids rotate to bring their passive sections to face the gap and approach in order to assemble. For ${\it\theta}_{cap}^{j}=40^{\circ }$ , the directions of rotation of the particles around the $y$ axis are reversed (see figure 15 b). This type of pair trajectory is a typical behaviour of two swimming bacteria whose dynamics is usually studied in the context of the squirmer model (Lighthill Reference Lighthill1952; Blake Reference Blake1971; Ishikawa, Simmonds & Pedley Reference Ishikawa, Simmonds and Pedley2006). For larger coverages, the directions of rotation of the particles around the $y$ -axis are also reversed. However, the presence of excess solute concentrated in the gap repels the particles to the distant regions where the particles do not interact and therefore swim independently (see figure 15 c). Moreover, for this initial separation distance ${\it\Delta}=4$ and inclination angles of ${\it\Xi}_{1}={\it\Xi}_{2}=90^{\circ }$ , there is a critical cap angle (coverage) ${\it\theta}_{cap}^{1}={\it\theta}_{cap}^{2}=63^{\circ }$ at which the magnitude of the propulsive torque becomes precisely equal in magnitude to the resultant hydrodynamic torques of problem (d) in figure 3 and hence the particles do not rotate. However, this condition is unstable because the swimming velocity component in the $z$ -direction is non-zero and hence the particles move away from each other.

Figure 16 shows the pair trajectories that correspond to the third scenario (see figure 12) for three different cap angles. In all cases, the shape of the trajectories are almost similar. However, the direction of rotations are not similar for all coverages. For ${\it\theta}_{cap}^{j}=40^{\circ }$ and ${\it\theta}_{cap}^{j}=90^{\circ }$ the particles rotate with negative angular velocities since the particles can reach smaller separation distances where the positive propulsive torques (torque associated with problem (a) in figure 3) are small compared to the negative hydrodynamic torques generated by their translational motions in the opposite direction. For high coverage ${\it\theta}_{cap}^{j}=140^{\circ }$ on the other hand, the particles are so far that the colloids have positive angular velocities. Moreover, as the particle coverage becomes smaller in size, the particles stay in the sticky lubrication zone much longer and this leads to a more curved pair trajectory (see figure 16 a).

Figure 16.

Pair trajectories of catalytically active motors for a fixed relative initial inclination angle ${\it\Xi}_{1}=200^{\circ },{\it\Xi}_{2}=20^{\circ }$ and (a) ${\it\theta}_{cap}=40^{\circ }$ , (b) ${\it\theta}_{cap}=90^{\circ }$ and (c) ${\it\theta}_{cap}=140^{\circ }$ . The numbers on (a) are for clarity on particles trajectories.

In the second scenario, particles with very high coverage initially move away to distant regions where they attain positive (particle on top) and negative (particle on bottom) rotations and translate until they reach an axisymmetric orientation, i.e. $180^{\circ }$ for the one on top and $0^{\circ }$ for the one on bottom, and then approach to assemble (see figure 17 a). Furthermore, the analysis indicates that beyond a critical coverage of ${\it\theta}_{cap}=152^{\circ }$ the particles can never be assembled. However, there is one interesting feature in the pair trajectories where the motors become completely stationary (see figure 17 b). At the stationary point, the concentration field around each one of the particles is uniform and hence the propulsive forces/torques are identically zero. Thus they come to a complete stop and then remain stationary indefinitely. This type of trajectory has also been observed for catalytic motors with high coverages in the vicinity of a solid boundary (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016).

Figure 17.

Pair trajectories of catalytically active motors for a fixed initial inclination angles of ${\it\Xi}_{1}=97^{\circ },{\it\Xi}_{2}=83^{\circ }$ and (a) ${\it\theta}_{cap}=140^{\circ }$ (b) ${\it\theta}_{cap}=160^{\circ }$ .