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Shang Yik
Reigh
*^{a},
Prabha
Chuphal
*^{b},
Snigdha
Thakur
*^{b} and
Raymond
Kapral
*^{c}
^{a}Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany. E-mail: reigh@is.mpg.de
^{b}Department of Physics, Indian Institute of Science Education and Research Bhopal, India. E-mail: prabhac@iiserb.ac.in; sthakur@iiserb.ac.in
^{c}Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada. E-mail: rkapral@chem.utoronto.ca

Received
29th May 2018
, Accepted 18th June 2018

First published on 25th June 2018

In the presence of a chemically active particle, a nearby chemically inert particle can respond to a concentration gradient and move by diffusiophoresis. The nature of the motion is studied for two cases: first, a fixed reactive sphere and a moving inert sphere, and second, freely moving reactive and inert spheres. The continuum reaction–diffusion and Stokes equations are solved analytically for these systems and microscopic simulations of the dynamics are carried out. Although the relative velocities of the spheres are very similar in the two systems, the local and global structures of streamlines and the flow velocity fields are found to be quite different. For freely moving spheres, when the two spheres approach each other the flow generated by the inert sphere through diffusiophoresis drags the reactive sphere towards it. This leads to a self-assembled dimer motor that is able to propel itself in solution. The fluid flow field at the moment of dimer formation changes direction. The ratio of sphere sizes in the dimer influences the characteristics of the flow fields, and this feature suggests that active self-assembly of spherical colloidal particles may be manipulated by sphere-size changes in such reactive systems.

In this article, we investigate the dynamics of a pair of small colloidal particles, one of which is chemically active and converts fuel to product, while the other is nonreactive. Further, we suppose that the interactions of the fuel and product molecules with the colloidal particles are the same for the reactive particle but different for the nonreactive particle, so that the nonreactive particle can respond to the chemical gradient produced by the catalytic particle as a result of diffusiophoresis. We consider interactions such that diffusiophoresis causes motion towards high product concentrations, and situations where the reactive particle is either fixed or free to move.

These specific choices are only a few among several other possibilities. For instance, the interaction potentials may be chosen so that either or both colloidal particles may be diffusiophoretically active with different responses to gradients.^{17} Also, either particle may be fixed or free to move, or their internuclear separation can be fixed as in a sphere-dimer motor.^{18–20} All of these situations are potentially interesting to study. A study, based on a continuum description of the fluid, of the dynamics of a pair of colloidal particles each of which could be Janus particles or active or inert is related to the work presented here.^{21,22} In order to investigate the dynamical properties of the spheres we use deterministic continuum theory as well as coarse-grain microscopic simulations. The particle-based simulations include fluctuations relevant for experimental studies of small active colloidal particles in solution,^{23} and automatically account for chemical-gradient, hydrodynamic and direct intermolecular interactions between the spheres without imposing specific boundary conditions.^{24}

The diffusiophoretic mechanism for the motion of a colloidal particle in an external concentration gradient is well known.^{5–8} By choosing the fixed reactive particle in our study to be diffusiophoretically inactive, it serves simply as reactive source that produces concentration gradients in the system.^{13–16} The nonreactive colloidal particle responds to this chemical gradient, which is analogous to an external chemical gradient, but presents some additional features as a result of pinning and reaction. We may contrast this case with that when the reactive sphere is free to move. The reactive particle again only generates concentration gradients in the system but when the two spheres closely approach we show that they form a self-assembled sphere-dimer motor that moves autonomously in solution, and we find that substantial changes in the flow fields occur at the moment of the dimer formation.

On a basic level, investigations of the mechanisms that give rise to the concentration and fluid flow fields that are responsible for the dynamics provide insight into the relative roles of chemical and hydrodynamic interactions, a topic that is important for studies of the collective dynamics of active particles.^{25–27} In this connection, recent experimental and computational studies have considered mixtures of chemically active and inactive spherical particles that exhibit interesting self-assembly and emergent dynamics.^{28–30} As in the present study, the dynamics of such mixtures will depend on both hydrodynamic and chemical, temperature, or electric fields that exist in the system.^{21,22,31–35}

In Section 2 we present continuum solutions for the reaction–diffusion and Stokes equations for this problem, and Section 3 describes the particle-based simulation method. Sections 4 and 5 discuss the physical phenomena that are observed for fixed and freely moving catalytic spheres, respectively. The conclusions of the investigation are given in Section 6.

In this circumstance the concentration gradient in the system arising from chemical activity on S_{1} will induce a body force on the noncatalytic sphere S_{2}. The diffusiophoretic mechanism will then operate and lead to a mean velocity component along the line of centers between the two spheres due to the axial symmetry of the system. In the continuum description our interest is in the value of the mean velocity that results from this mechanism, as well as the forms of the concentration and fluid velocity fields that accompany it.

The two-sphere system can be solved in a bispherical coordinate system.^{20,36–39} The bispherical coordinates are (θ, η, ϕ), where 0 ≤ θ ≤ π, −∞ ≤ η ≤ ∞, and 0 ≤ ϕ ≤ 2π as shown in Fig. 2. In Cartesian coordinates (x, y, z), the relations, x = ξsinθcosϕ/(coshη − cosθ), y = ξsinθsinϕ/(coshη − cosθ) and z = ξsinhη/(coshη − cosθ) are satisfied with a scale factor ξ (>0).^{40} The surfaces of the S_{1} and S_{2} spheres are represented by the parameters η = η_{1}(>0) and η = η_{2}(<0), respectively. Inversely, from the values of the radii of the S_{1} and S_{2} spheres, R_{1} and R_{2}, and any separation distance, L, which is greater than the sum of their radii, the bispherical coordinate parameters, ξ, η_{1} and η_{2} are found by , , and .

∇^{2}c_{A} = 0, | (1) |

(J·)_{η=η1} = _{0}c_{A}(η = η_{1}), |

(J·)_{η=η2} = 0, | (2) |

The total concentration c_{0} = c_{A} + c_{B} is conserved in the reaction–diffusion system with the boundary conditions on the surfaces of the spheres and infinity, and we can write c_{A} = c_{0} − c_{B} locally; thus, we can eliminate c_{A} and consider only c_{B}. In bispherical coordinates, the concentration of B is now given by

(3) |

2.2.1 Fixed catalytic sphere.
We suppose that the catalytic sphere S_{1} is fixed in space by external force and the noncatalytic sphere S_{2} is able to move in the solution. The concentration field around the S_{1} is asymmetric as given by eqn (3); hence, a flow is generated at the surface of the S_{2} sphere by the diffusiophoretic mechanism.^{7,8} The slip velocity is the fluid velocity at the outer edge of a boundary layer beyond which the interaction potentials vanish, and is given in the body-fixed frame of the sphere by

where I is the unit dyadic, the surface normal vector,

is the diffusiophoretic factor, with the shear viscosity, k_{B} the Boltzmann constant, and T the temperature.^{8,12}

subject to the boundary conditions in the laboratory frame of reference,

where p is the pressure, v the fluid velocity field, and V the velocity of the noncatalytic sphere.

where E^{4} = E^{2}(E^{2}) and E^{2} = (coshη − μ)/ξ^{2}[∂/∂η{(coshη − μ)∂/∂η} + (1 − μ^{2})∂/∂μ{(coshη − μ)∂/∂μ}]. This equation has an exact solution given by^{36}

where W_{n}(η) = a_{n}cosh(n − ½)η + b_{n}sinh(n − ½)η + c_{n}cosh(n + )η + d_{n}sinh(n + )η and V_{n}(μ) = P_{n−1}(μ) − P_{n+1}(μ). The unknown coefficients a_{n}, b_{n}, c_{n}, and d_{n} in eqn (9) are determined by boundary conditions at the outer edges of the boundary layers around the S_{1} and S_{2} spheres, i.e.eqn (7). In the laboratory frame where the motor moves with velocity V, these boundary conditions are given in terms of the stream function by

By writing in eqn (9), we can replace the boundary conditions, eqn (10) in terms of χ by

Here, can be expressed in a series of Legendre function P_{n}, (1 − μ^{2})P_{n} and μV_{n} are rewritten by Gegenbauer functions V_{n−1} and V_{n+1}, and (1 − μ^{2})dP_{n}/dμ is rewritten by V_{n}.^{20,36,41} Then, we may expand the right sides of eqn (11) for η = η_{2} in a series of V_{n} as

where γ_{n} = f_{n}V and f_{n} is given in Table 1 in the Appendix. The solution of the above equations for the unknown coefficients a_{n}, b_{n}, c_{n}, d_{n} is expressed by

where X = {a_{n},b_{n},c_{n},d_{n}}, Y^{(e)} = {Y^{(2)}_{n},Y^{(4)}_{n},Y^{(6)}_{n},Y^{(8)}_{n}}, and Z = {z^{(1)}_{n},z^{(2)}_{n},z^{(3)}_{n},z^{(4)}_{n}}. The elements of the vectors are given in Table 1 in the Appendix. The solution for two inactive spheres can be obtained easily by taking κ = 0, which gives X = γ_{n}Y^{(e)}/Δ_{n}. In this case, one colloidal sphere (S_{2}) with constant velocity V moves to the other sphere (S_{1}) fixed in space.

Also, the force F_{1} exerted on the fixed catalytic sphere by the fluid found here is used for the plots in Fig. 10.

v_{s} = −κ(I − )·∇c_{B}, | (4) |

(5) |

The Reynolds number is assumed to be small so that viscous forces dominate inertial forces and the fluid flow field outside of the boundary layer is found by solving the Stokes equation with the incompressibility condition,

∇p = ∇^{2}v, ∇·v = 0, | (6) |

v_{η=η1} = 0, v_{η=η2} = (V + v_{s})_{η=η2}, | (7) |

Introducing the stream function ψ, which is related to the flow velocity by v = /ρ × ∇ψ, where ρ = ξsinθ/(coshη − μ), one may replace the Stokes equation with the incompressibility condition in terms of stream functions by^{36,40}

E^{4}(ψ) = 0, | (8) |

(9) |

(10) |

(11) |

(12) |

Since both sides of eqn (12) are expanded in a series of Gegenbauer function V_{n}, we can determine the unknown coefficients of W_{n}(η) in eqn (9) from the following equations:

(13) |

(14) |

The forces (F_{1},F_{2}) on the individual spheres (S_{1},S_{2}) are given by integrating the stress on the surface of the boundary layer, (i = 1, 2), where Π_{i,z} = ẑ·Π_{i} and Π is the stress tensor. The system is symmetric around the azimuthal angle ϕ and only the force in the z-direction needs to be considered. The analytic expressions for the force exerted on the spheres by the fluid are given in Stimson and Jeffery^{36} as

(15) |

The velocity can be found from these force expressions. Since no external force is applied to the S_{2} sphere, although the S_{1} sphere is fixed in space by an external force, the total force on the S_{2} sphere at the outer edge of the boundary layer is zero, F_{2} = 0. Noting that γ_{n} = f_{n}V, one can find the following expression for velocity of the noncatalytic sphere,

(16) |

2.2.2 Freely moving catalytic sphere.
We now suppose that both spheres are free to move and construct the solutions for this force-free case. Letting the velocities of the S_{1} and S_{2} spheres be V^{(1)} and V^{(2)}, respectively, one may replace the boundary conditions in eqn (7) by

Then the boundary conditions for the stream function are

where Θ_{1} = 0, Θ_{2} = 1, and i = 1, 2.

where the upper and lower signs are taken for i = 1 and 2, respectively.

where γ^{(i)}_{n} = f_{n}V^{(i)} and the upper and lower signs correspond to i = 1 and 2, respectively.

where X = {a_{n},b_{n},c_{n},d_{n}}, Y^{(o)} = {Y^{(1)}_{n},Y^{(3)}_{n},Y^{(5)}_{n},Y^{(7)}_{n}}, Y^{(e)} = {Y^{(2)}_{n},Y^{(4)}_{n},Y^{(6)}_{n},Y^{(8)}_{n}}, and Z = {z^{(1)}_{n},z^{(2)}_{n},z^{(3)}_{n},z^{(4)}_{n}}. The elements of the vectors are given in Table 1 in the Appendix. Applying the force-free conditions on both the spheres, F_{1} = F_{2} = 0 in eqn 15, one can find the solution for the velocities of the S_{1} and S_{2} spheres as

where

v_{η=η1} = (V^{(1)})_{η=η1}, v_{η=η2} = (V^{(2)} + v_{s})_{η=η2}. | (17) |

(18) |

In this case, the boundary conditions for streamlines in eqn (18) are rewritten in terms of by

(19) |

As discussed previously, we may expand the right sides of eqn (19) in a series of Gegenbauer function V_{n} as

(20) |

Since both sides of eqn (20) are expanded in a series of V_{n}, we can determine the unknown coefficients of W_{n}(η) in eqn (9) from the following equations:

(21) |

The solution of the above equations for the unknown coefficients a_{n}, b_{n}, c_{n}, d_{n} is given by

(22) |

(23) |

(24) |

The solutions for two inactive spheres moving with constant velocities V^{(1)} and V^{(2)} along the axisymmetric direction can be obtained easily by setting κ = 0, which gives X = (γ^{(1)}_{n}Y^{(o)} + γ^{(2)}_{n}Y^{(e)})/Δ_{n}. Also, the solutions for the sphere-dimer can be obtained by setting V^{(1)} = V^{(2)} = V, which gives Δ_{n}X = γ_{n}Y − ½ξκΦ_{n}Z, where Y = Y^{(o)} + Y^{(e)} and γ_{n} = γ^{(1)}_{n} = γ^{(2)}_{n}. This expression is consistent with the formula given earlier.^{20,42}

The coarse-grain particle-based simulations do not make such assumptions. The input parameters are the intermolecular potentials and multiparticle collision parameters for the solvent.^{24} The resulting dynamics then yields all other properties such as the transport coefficients of the system, and other dimensionless numbers that characterize the system. One can show that on long distance and times scales the continuum hydrodynamic and diffusion equations are recovered,^{43} but the dynamics is not restricted to this limit. Consequently, it is of interest to examine the extent to which the continuum model can capture the active dynamics of these small particles.^{20,44}

The coarse-grain microscopic dynamics we employ combines molecular dynamics (MD) with multiparticle collision (MPC) dynamics.^{43,45} More specifically, the fluid is composed of N_{s} point particles of mass m with positions r_{i} and velocities v_{i}, where i = 1,…,N_{s}. There are no explicit intermolecular potentials among these fluid particles and their interactions are accounted for by multiparticle collisions. The dynamics consists of two alternating steps: streaming and collision. In the streaming steps of duration h, all particles in the system move by Newton's equations of motion with forces determined by the sphere–sphere and sphere–solvent intermolecular potentials. At each collision time the solvent particles are sorted into cubic cells of side length a, which is larger than the mean free path, and their relative velocities are rotated around a randomly oriented axis by a fixed angle α with respect to the center-of-mass velocities of each cell. The velocity of particle i after collision is given by v_{i}(t + h) = v_{cm}(t) + (α)(v_{i}(t) − v_{cm}(t)), where (α) is the rotation matrix, is the center-of-mass velocity of the particles in the cell to which the particle i belongs, and N_{c} is the number of particles in that cell. A random shift of the collision lattice is applied at every collision step to ensure Galilean invariance.^{46} The dynamics locally conserves mass, momentum and energy.^{24}

The spheres interact with the fluid particles through repulsive Lennard-Jones (LJ) potentials, U = 4ε[(σ/r)^{12} − (σ/r)^{6}] + ε for r < 2^{1/6}σ and U = 0 for r ≥ 2^{1/6}σ with energy ε and distance σ parameters. In addition, repulsive LJ potentials are employed to take into account excluded volume interactions between the two spheres with σ_{s} denoting the value of σ in this case. In order to make only the noncatalytic sphere hydrodynamically active, we choose the interaction energies of the A and B molecules with the S_{1} catalytic sphere to be the same (ε_{A} = ε_{B} = ε) and those with the S_{2} noncatalytic sphere to be different (ε_{B} < ε_{A} = ε). Setting ε_{B} < ε_{A}, so that the A particles are more strongly repelled from the S_{2} sphere than the B particles, causes it to move towards the S_{1} sphere; hence B plays the role of chemoattractant. An irreversible chemical reaction A → B takes place on the S_{1} sphere with intrinsic reaction rate k_{0} whenever A encounters S_{1}. Collisions of A or B particles with the S_{2} sphere do not lead to reaction. To maintain the system in a steady state, the B particles are converted to A at a distance d_{p} (= L_{b}/2) far from the spheres.

All quantities are reported in dimensionless units where length, energy, mass and time are measured in units of the MPC cell length a = σ/2, ε, the solvent mass m, and , respectively. The cubic simulation box with linear dimension L_{b} = 50 and periodic boundary conditions in all dimensions is divided into L_{b}^{3} = 50^{3} cubic cells. Multiparticle collisions are carried out in each cell by performing velocity rotations by an angle α = 120° about a randomly chosen axis every collision time h = 0.1. The average solvent number density is c_{0} = 10 and the temperature is k_{B}T = 1. The MD time step is Δt = 0.01. The energy parameters for the S_{2} sphere-fluid repulsive LJ potentials are ε_{A} = 1.0 and ε_{B} = 0.1 for A and B, respectively, while ε_{A} = ε_{B} = 1.0 for the S_{1} sphere. The size parameters are σ = 2 and σ_{s} = 4 to give effective sphere radii of R_{1} = R_{2} = 2^{1/6}σ. The sphere mass is taken to be M = 4πσ^{3}c_{0}/3 corresponding to neutral buoyancy. The intrinsic reaction rate constant for the A + S_{1} → B + S_{1} reaction can be estimated from simple collision theory so that . The transport properties of the fluid depend on h, α, and N_{c}. The fluid viscosity is = mN_{c}ν = 7.9, where ν is the kinematic viscosity, and the common A and B diffusion constant is D = 0.0611. The Schmidt number is S_{c} = ν/D = 13 > 1, which ensures that momentum transport dominates over mass transport, the Reynolds number Re = c_{0}Va/ < 0.1, implying that viscosity is dominant over inertia, and the Péclet number Pe = Va/D < 1, diffusion being dominant over fluid advection.

The parameter values given above are used as input to obtain the analytic solutions in the continuum theory. For example, the factor κ in eqn (5) is obtained from the repulsive cut-off LJ potentials with the energy parameters ε_{A} and ε_{B} given in simulations, along with the viscosity from the microscopic model. Using the analytical continuum solutions and simulations of the microscopic equations of motion, we can discuss the physics underlying dynamics of these two-sphere systems. Since the phenomena depend on whether the catalytic sphere is fixed or free to move, we discuss these two cases separately.

The velocity of the S_{2} sphere, V, is plotted in Fig. 3 as a function of the distance L separating the centers of the two spheres. The figure shows the expected increase in velocity as the S_{2} sphere approaches the S_{1} sphere until, at a short distance, it begins to decrease as the capture event takes place. The figure compares the simulation results with the exact analytical continuum theory result in eqn (16). The results are also compared with an approximate theory where the two spheres are assumed to be separated by a large distance. In this case, the concentration field may be approximated by calculating it in the absence of the S_{2} sphere^{47,48} as follows. Taking the origin of a spherical polar coordinate (r_{1},θ_{1},ϕ_{1}) at the center of the S_{1} sphere in Fig. 2, the B species concentration field may be obtained from the solution of the diffusion eqn (1) subject to the radiation boundary condition in eqn (2) as

(25) |

Fig. 3 The velocity V of the noncatalytic S_{2} sphere as a function of the separation L between the S_{2} and S_{1} catalytic spheres. The black solid line is the exact solution calculated from continuum hydrodynamic theory in eqn (16) and the black dashed line is the approximate velocity V_{a} from eqn (27) that is valid for large L. The red circles with error bars are the results of microscopic simulations. Averages were obtained from 80 realizations of the dynamics. |

The approximation to the propulsion velocity of the S_{2} sphere may be then found by averaging the slip velocity like eqn (4) at the edge of the boundary layer of the S_{2} sphere^{8,12,49} in a coordinate system (r_{2}, θ_{2}, ϕ_{2}) where the origin is at the center of the S_{2} sphere. The result is

(26) |

(27) |

In the microscopic simulations the colloidal particles undergo Brownian motion as a result of thermal fluctuations, as well as directed motion due to diffusiophoresis. Fig. 4 shows some examples of noncatalytic sphere trajectories. At large distances (L/σ > 8) the noncatalytic sphere exhibits small thermal fluctuations in its displacement which are less than its radius, as well as larger random displacements. When L/σ < 6, diffusiophoretic interactions are stronger and the deterministic component of the motion dominates. Thus, fluctuations lead to a dispersion of capture times seen in Fig. 4, and only the average in Fig. 5 can be compared to the deterministic theory.

Fig. 4 Plot of the distance between the fixed catalytic and moving noncatalytic spheres as a function of time. Five realizations of the dynamics are shown, each with an initial separation L/σ = 10. Contact occurs at approximately L/σ ∼ 2.3. The time where the distance achieves its minimum value is the capture time (see Fig. 5). |

The capture time, τ, which is defined by the time it takes the S_{2} sphere, initially at L, to reach the S_{1} sphere, i.e., the spheres are separated by a distance equal to the sum of their radii, R_{1} + R_{2}. The time τ can be calculated easily by integrating the velocity (eqn (27)) to obtain the simple expression, τ = (k_{0} + k_{D})(L^{3} − (R_{1} + R_{2})^{3})/(2κc_{0}k_{0}R_{1}). Fig. 5 shows how τ varies with L. The exact continuum solutions agree well with simulations, while here are discrepancies with the approximate theory.

The concentration and fluid velocity fields vary during the capture process, and these variations play a role in determining the details of the capture mechanism. The B species concentration fields and their gradients on the surface of the S_{2} sphere are shown in Fig. 6. The concentration field decays as 1/r at long distances^{20} but again there are discrepancies in the magnitude of the field close to the S_{2} sphere. Such discrepancies might be expected because the dynamics in the finite-size boundary layer cannot be simply represented by the continuum boundary conditions. It is interesting that the tangential gradient of this field on the surface corresponds very closely to that of the continuum model. Consequently, even though the microscopic nature of the concentration fields is manifest in the boundary layer, the gradient, which determines the propulsion, is accurately given by the continuum theory. As a result many of the other observable properties are accurately given.

The velocity fields generated by the moving S_{2} sphere present a more interesting and complex structure as a function of L. Fig. 7 shows the streamlines and flow fields in the laboratory frame of reference. The streamlines are plotted by setting ψ equal to a constant. At large separations, we see that the fluid near the head of the S_{2} sphere (portion closest to the S_{1} sphere) is pushed to the lateral directions (in the xy plane) with respect to the axisymmetric z axis, and executes broad fluid circulation near the S_{1} sphere. Fluid also flows towards the rear of the S_{2} sphere. The flow near the S_{2} sphere shows a puller-like behavior; i.e., fluid enters from the front and back and is expelled from the sides.^{32,50} (A pusher-like behavior can be also seen in our system if ε_{B} > ε_{A}.) As the two spheres approach each other (L/σ ∼ 3.5) the circulating flows between and to the sides of the spheres reduce in size and disappear, leaving a puller-like flow pattern. Near the contact distance (L/σ ∼ 2.5), the fluid is pushed from the back to the front of two spheres.

That the flow patterns are affected by the pinning of the catalytic sphere are clearly seen in the plots of the far field streamlines in Fig. 8. The flow near the spheres resembles that due to stresslet fields (similar to that for L/σ ∼ 3.5 and ∼5 in Fig. 7), but at distances far from the spheres (see Fig. 8(b) and (c)) the flow resembles a drift flow (Stokeslet).^{51} When the separation between the spheres is large (Fig. 8(d), L/σ ∼ 7.5), the flow circulation (stresslet fields) expands to occupy a larger portion of space, but a drift flow (Stokeslet) again appears when viewed at large distances from the spheres. These far-field flows are characterized quantitatively by calculating the magnitude of fluid velocity , where v = v_{θ} + v_{η},^{40} as shown in Fig. 9. For example, at L/σ = 7.5, one sees a 1/r^{2} decay, characteristic of stresslets, for distances up to approximately r/σ ∼ 20, but eventually the flow velocity decays asymptotically as 1/r. As the separation distance decreases, it is notable that the flow velocity increases, the stresslet contribution disappears, and the Stokeslet contribution increases. The asymptotic expressions are found by introducing the spherical polar coordinates (r, ϑ, ϕ) in Fig. 2, where two coordinate systems share the origin, and expanding the variables θ and η in terms of 1/r. Then one may obtain asymptotic expressions for flow velocity up to (1/r^{2}) as

(28) |

Fig. 8 Far-field streamlines for various sphere separations, (a) L/σ = 2.5 (b) L/σ = 3.5 (c) L/σ = 5 (d) L/σ = 7.5. |

Fig. 9 The magnitude of fluid velocity, , for ϑ = π/2 as a function of distance r, where the spherical polar coordinates (r, ϑ, ϕ) are taken with a common origin in Fig. 2. The black, red, green, blue, brown, magenta lines (from top to bottom) correspond to the separation distances, L/σ = 2.5, 3.5, 5, 7.5, 10, 15, respectively. |

Since the fluid between the spheres flows from the S_{1} to S_{2} spheres with a broad circulation pattern, one may expect that the force the fluid exerts on the fixed catalytic sphere is in the same direction; i.e., an attractive force. (If ε_{B} > ε_{A} then the flow directions are reversed and one has a repulsive force.) The force is given by eqn (15) and is plotted in Fig. 10, along with the simulation result. In the microscopic simulations, the force is calculated by summing the forces on the catalytic sphere due to all of the fluid particles. The continuum theory and simulations agree very well. The force is almost zero for large L, and becomes more negative (attractive) as L decreases, reaching its largest negative value at L/σ ∼ 2.5, near the contact distance, L/σ ∼ 2.25. If L decreases further, the force take positive (repulsive) values.

Fig. 10 The force on the catalytic sphere exerted by fluid. The black solid line and red circles correspond to the continuum theory and simulations, respectively. Negative values (−z direction in Fig. 2) imply the force is attractive. |

Fig. 11 Plot of the velocities V^{(1)} and V^{(2)} of the S_{1} and S_{2} spheres in a force-free system. The solid blue and red lines denote the continuum theoretical values of V^{(1)} and V^{(2)}, respectively, while the circles with error bars are the microscopic simulation results. The inset shows the velocity difference V^{(2)} − V^{(1)} (solid lines) and, for comparison, the velocity of the S_{2} sphere (dashed line) when the S_{1} sphere is fixed in space (eqn (16)). |

Note that although the velocities of the two spheres have opposite signs (− for S_{1} and + for S_{2}) as they approach, the sign of the S_{1} velocity changes so that both sphere velocities are positive (+z) as the two spheres meet to form a self-propelled sphere-dimer that moves with the S_{1} sphere at its head (see Movie 2, ESI†).^{19,20} In contrast to the sphere-dimer motors previously studied that are made from spheres with a rigid bond, this sphere-dimer motor self-assembles from isolated spheres to form a bound pair with a bond length that may fluctuate around a mean value depending on parameters used. Once the sphere dimer is formed by self-assembly it behaves like the sphere-dimer with a fixed bond length. Similar motion of two spheres was observed in a numerical study of a thermocapillary system consisting of a solid particle and a gas bubble.^{52}

The streamlines and flow field are shown in Fig. 12 (left two columns) in the laboratory frame of reference. When L is relatively large (L/σ = 5), the streamlines are roughly similar to those when the S_{1} sphere is fixed but there is no local fluid circulations at small distances from the spheres and no drift flow at large distances. The fluid flow near the S_{2} sphere exhibits a puller-like pattern and near the S_{1} sphere fluid is simply dragged to the S_{2} sphere. As discussed above, this difference is attributed to the contributions of Stokeslets in a forced system and these effects are pronounced at small L (L/σ = 2.3, 3.5). The streamlines in a force-free system do not significantly change at small separations, while those in a forced system are more distorted in the direction of the applied external force (Fig. 7). The quantitative variations of streamlines and flow fields can be seen by plotting the magnitudes of flow velocity as displayed in Fig. 13 (left panel). The flow velocity of force-free spheres decays as a r^{−2} (stresslet) in a distance r/σ ∼ 5 for various values of L, and this power-law behavior remains unchanged at long distances. However, the flow velocity in a system with sphere S_{1} fixed exhibits a r^{−2} decay for distances r/σ ∼ 5 when L/σ = 5, and it shows a r^{−1} decay (Stokeslet) for L/σ = 2.5, although the velocity in all cases eventually decays a r^{−1} at long distances (Fig. 9).

Fig. 13 The magnitude of fluid velocity, , along the side direction (ϑ = π/2) as a function of distance r for the unlinked two spheres (left) and the linked dimer (right). The spherical polar coordinate (r, ϑ, ϕ) is taken by setting the origin of the coordinate at the middle of two spheres as in Fig. 2 and 9. The black, red, and blue lines correspond to the separation distance, L/σ = 2.3, 3.5, 5, respectively. |

These puller and pusher flow patterns remain unchanged as L increases (second and third rows in Fig. 12). The magnitudes of flow velocity for the unlinked and linked spheres are compared quantitatively in Fig. 13. Both cases exhibit a r^{−2} decay in contrast to that for a fixed S_{1} sphere. For small L (L/σ < 3.5), the magnitudes of flow velocity for both linked and unlinked spheres are very similar; only the flow directions have opposite signs. The asymptotic expressions are given by eqn (28) without Ω_{1} terms since Ω_{1} is zero by the force-free condition.

The characteristics of the flow fields depend on the sphere sizes. Two separated spheres behave as a puller, regardless of their sphere size ratio, while the sphere-dimer motor that is formed can have either puller or pusher characteristics, and this does depend on the size ratio. Consequently, it should be possible to construct self-propelled dimers with either of these flow characteristics by simply manipulating the sphere sizes. This feature may be used to aid in the understanding of the collective behavior of many-sphere systems, and to provide a route to the construction of complex self-assembled structures in the laboratory.^{25–27}

The two-sphere dynamics studied in this paper may be regarded as an elementary process that contributes to the collective dynamics of mixtures of active and passive particles^{28–30} and sphere dimers with non-rigid bonds. The study provides insight into the mechanisms that could lead to dynamic clusters of various types that not only move but may also fragment and reassemble. In this connection, situations not considered in this paper could be of considerable interest to investigate further. If the interactions are such that the nonreactive sphere moves to lower product concentrations, in dilute solution the two sphere will simply avoid each other. However, in more dense colloidal suspensions they will be forced to interact and lead to different active collective states, analogous to the different collective dynamics of forward and backward moving sphere dimers.^{53}

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## Footnote |

† Electronic supplementary information (ESI) available: Two movies of two spheres motions, the catalytic sphere fixed in space (S_{1}) and both spheres freely moving (S_{2}). See DOI: 10.1039/c8sm01102h |

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