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P.
Malgaretti
*^{ab},
M. N.
Popescu
^{ab} and
S.
Dietrich
^{ab}
^{a}Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, D-70569 Stuttgart, Germany. E-mail: malgaretti@is.mpg.de
^{b}IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

Received
29th November 2017
, Accepted 18th January 2018

First published on 18th January 2018

The influence of a fluid–fluid interface on self-phoresis of chemically active, axially symmetric, spherical colloids is analyzed. Distinct from the studies of self-phoresis for colloids trapped at fluid interfaces or in the vicinity of hard walls, here we focus on the issue of self-phoresis close to a fluid–fluid interface. In order to provide physically intuitive results highlighting the role played by the interface, the analysis is carried out for the case that the symmetry axis of the colloid is normal to the interface; moreover, thermal fluctuations are not taken into account. Similarly to what has been observed near hard walls, we find that such colloids can be set into motion even if their whole surface is homogeneously active. This is due to the anisotropy along the direction normal to the interface owing to the partitioning by diffusion, among the coexisting fluid phases, of the product of the chemical reaction taking place at the colloid surface. Different from results corresponding to hard walls, in the case of a fluid interface the direction of motion, i.e., towards the interface or away from it, can be controlled by tuning the physical properties of one of the two fluid phases. This effect is analyzed qualitatively and quantitatively, both by resorting to a far-field approximation and via an exact, analytical calculation which provides the means for a critical assessment of the approximate analysis.

In this context, here we address self-phoresis of a catalytic particle in the vicinity of a fluid–fluid interface. First, in order to emphasize the influence of the interface, we study the case of a particle which is homogeneously catalytic. In order to keep the system as simple as possible, in the following we assume that the concentration of reactants is kept constant in space and time. In such a situation the particle releases the products of the catalytic reaction isotropically. Therefore, even though the system is kept out of equilibrium, due to the isotropic coverage of the catalyst in the bulk there is no symmetry breaking† and hence no net displacement. However, the presence of an interface breaks the homogeneity of the transport coefficients (i.e., the diffusivities of the reaction products in the two fluid phases). This leads to an inhomogeneous distribution of the concentration of the reaction products along the interface normal. Accordingly, such an inhomogeneous density profile leads to an interface-induced phoresis, the direction of which is normal to the interface, similarly to what has already been reported for the case of a hard wall.^{30,32} Interestingly, in the present case the sign of the resulting velocity depends not only on the surface properties of the particle, as it is the case near a hard wall, but it depends on both the contrast between the diffusivities and the distinct solvability of the catalysis products in the two fluid phases.

If the catalyst is not homogeneously distributed along the surface of the particle, a net motion will arise due to self-diffusiophoresis even in a homogeneous fluid. Accordingly, if such particles are close to a fluid–fluid interface, a competition arises between the intrinsic motility and the one induced by the interface. Since the motility induced by the interface is directed solely along its normal, in order to grasp the interplay between the self-phoresis and the interface-induced phoresis we focus on the case of an asymmetrically coated colloid (ideal Janus particle) the axis of symmetry of which is parallel to the normal of the interface.

The paper is organized as follows. In Section 2 we formulate the model describing the dynamics of active colloids close to a fluid interface. In Section 3 we study the velocity of these active colloids by using an approximate far-field expansion as well as an exact solution. While the approximate far-field expansion allows us to straightforwardly grasp the phenomenology emerging from the dynamics of active colloids close to fluid interfaces, the exact solution, by providing quantitatively reliable results, allows us to critically asses the strengths, as well as the shortcomings, of the approximate far-field approach. Finally in Section 4 we provide concluding remarks. The details of the necessary calculations beyond the ones presented in the main text are included in the Appendices A–D.

We restrict our study to the case in which the diffusion of the solute in the two fluids is sufficiently fast such that the transport of solute by advection due to induced hydrodynamic flow is negligible compared to the one by diffusion. Accordingly, the solute number density distribution is determined solely by diffusion. Furthermore, generically in experiments with chemically active particles, the liquid media are aqueous solutions and the flows induced by self-phoresis correspond to very small Reynolds numbers. Therefore, in the following we shall describe the corresponding hydrodynamics by the Stokes equation for an incompressible Newtonian fluid.

Concerning the solute we assume that the role of the interface is simply to provide distinct diffusion coefficients and distinct solvation energies in the two fluid phases. This amounts to a partitioning effect, which is associated with the Donnan potential (see p. 75 in ref. 52). The transport by advection is negligible relative to that by diffusion (see the discussion above), so that at steady state the number density c(r) of solute is the solution of the Laplace equation

∇^{2}c(r) = 0, | (1) |

(a) constant values in each half-space far from the particle:

(2) |

(b) discontinuity at the interface solely due to the different solvability in the two media (Donnan potential):

λc(r)|_{z→0+} = c(r)|_{z→0−}, | (3) |

λc^{∞}_{1} = c^{∞}_{2}; | (4) |

(c) no transport parallel to the interface, i.e., all the solute flux leaving medium “1” enters into medium “2”:

n_{1}·(−D_{1}∇c)|_{z→0+} = −n_{2}·(−D_{2}∇c)|_{z→0−}, | (5) |

(d) at the catalyst covered part of the surface Σ_{p} of the particle the molecular flux of solute (i.e., the number density current along the outward normal n of the particle surface Σ_{p}) equals the “production” rate of solute by the catalytic reaction, while on the rest of the surface it vanishes (i.e., the particle is impermeable):

n·(−D_{1}∇c)|_{Σp} = Qf(θ), | (6) |

(7) |

Due to the asymmetry introduced via the partial coverage by catalyst as well as due to the presence of the interface, the distribution of solute around the colloidal particle is spatially inhomogeneous. Since the solute molecules interact with the colloid differently than the solvent ones, this inhomogeneity induces gradients in the local osmotic pressure along the surface of the particle. As a result, flow of the fluid and motion of the particle emerge.^{10,53} If the range of these molecular interactions is much smaller than the size of the particle, which is a plausible assumption for most of the experimental realizations, the effects of the osmotic pressure gradient are captured via a so-called phoretic slip velocity v_{p}(r_{p}) of the fluid relative to the surface of the particle. The phoretic slip is related to the local gradient of the number density of solute via^{10,53}

v_{p}(r_{p}) = −b∇_{‖}c(r_{p}), | (8) |

(9) |

Once the slip velocity is provided one can set out to solve the Stokes equation under the proper boundary conditions at the fluid interface. Alternatively, the velocity of the particle in the lab reference frame can be obtained via the reciprocal theorem.^{14,54–56} The reciprocal theorem states that in the absence of volume forces any two incompressible flow fields u(r) and û(r), which are distinct solutions of the Stokes equations within the same domain , i.e., solutions subject to different boundary conditions but on the very same boundaries ∂, obey the relation

(10) |

However, the present system is somewhat different, in that the phase-separated condition divides the fluid domain into two sub-domains (on each side of the interface the flow velocity along the interface normal vanishes), but the flows in the two domains _{1} and _{2} are connected due to the requirement of continuity of the tangential stress and of the tangential velocity at the interface. In spite of these complications, it has been shown that for an infinitely large fluid domain with a planar interface the reciprocal theorem takes the exact same form as in the case of a particle immersed in a single fluid medium,^{57}i.e.,

(11) |

In the following we restrict the discussion to cases in which the system exhibits axial symmetry, i.e., the symmetry axis of the particle coincides with the z-direction, which in turn is the normal of the interface, and we neglect the effects of thermal fluctuations, in particular the rotational diffusion of the axis of the active colloid. Accordingly, concerning the description of the present system there is a single unknown quantity, which is the translational velocity of the particle along the direction normal to the interface. We therefore select as the dual problem the Stokes problem of the translation of a chemically inert particle (located at z = L) with velocity Û(L) along the normal of the interface, due to a force acting on it (located at z = L) with a no-slip boundary condition at its surface, i.e.,

û(r_{p}) = Û. | (12) |

u(r_{p}) = U + v_{p}(r_{p}), | (13) |

(14) |

We discuss separately two cases, first the one in which the whole surface of the particle is chemically active, and second the case of a Janus colloid for which only a spherical cap is chemically active. The first case allows one to highlight those effects which are solely due to the way in which the solute partitions between the two fluid phases. The second case provides insight into the interplay between the above mechanism and the asymmetry in the distribution of the reaction sites across the surface of the particle.

We start our analysis by decomposing the stress tensor of the auxiliary problem:

= _{0} + _{im}, | (15) |

(16) |

_{im}(x, y, z) = _{im}(0, 0, L) + δ_{im}(x, y, z). | (17) |

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

As expected from the behavior reported for similar active particles near a hard wall or near another particle,^{30,32,63} in the vicinity of the interface the particle exhibits motion along the direction normal to the interface. Upon leaving the interface the magnitude of the velocity of the particle decays as (R/L)^{2}. Near the interface the velocity can reach values of the same order of magnitude as the maximum velocity V_{o}/4 of a Janus particle in an unbounded fluid^{15} (see Fig. 2). We note that in the limit α → 0 eqn (25) takes the form corresponding to a hard wall,^{30,32} whereas for α → ∞ it is expected that it will recover the form corresponding to a direct calculation for a fluid–gas interface.

Fig. 2 Rescaled velocity U_{z}/|V_{o}| (see eqn (26)) as a function of the distance L/R of the center of the particle from the interface. The whole surface of the particle is chemically active, i.e., θ_{0} = π. The symbols (blue triangles for λα = 0.2 and red circles for λα = 5) show the exact results calculated by using bi-polar coordinates (see Appendix A), while the dashed lines show the results from the far-field approximation (eqn (25)), for λα = 0.2 (blue) and λα = 5 (red) (see eqn (3) and note that ), and V_{o} < 0 (eqn (26)). The inset shows the ratio of the exact velocity U^{bp}_{z} obtained by using bi-polar coordinates and the velocity U^{ff}_{z} calculated within the far-field approximation. |

Apart from the parameters included in V_{o} (eqn (26)), the velocity depends on the ratio λ of the solvabilities (eqn (3)) and on the ratio α (eqn (20)) of the diffusion constants of the solute in the two fluid phases 1 and 2 (or equivalently, due to the Stokes–Einstein relation, on the inverse ratio of the viscosities of the two fluids). These additional dependences are particularly interesting because they imply that, in contrast to the behavior near a hard wall, also the direction of the motion depends on λα and not solely on the sign of the phoretic mobility b, which enters via V_{o} (eqn (26)).

Taking, for example, b < 0, which implies V_{o} < 0, and λ = 1 (the discussion can be straightforwardly extended to the cases in which λ ≠ 1 or b > 0), one infers from eqn (25) that for α < 1, i.e., if the particle is suspended in the less viscous fluid phase, U_{z} is positive and thus the particle moves away from the interface. For α > 1, i.e., if the particle is suspended in the more viscous fluid phase, U_{z} turns negative and thus the particle moves towards the interface. Furthermore, eqn (25) exhibits the symmetry relation . Since replacing λα by amounts to interchanging media 1 and 2, one concludes that, if one would perform an experiment in which the active particle is placed at z = L and another one with an identical particle now placed at z = −L the outcome will be the following. If in the first experiment the particle will move towards the interface, then in the second experiment it will move away from it, the speed of the motions being the same in the two cases. Vice versa, if in the first experiment the particle will move away from the interface, in the second it will move towards the interface, with the same speed of motion.

As shown in Fig. 2, the predictions of the far-field approximation (eqn (25), dashed lines) accurately capture the qualitative behavior of U_{z}, in particular the most important feature of the above noted change of sign of the velocity for λα ≷ 1. Fig. 2 also shows that the magnitude of the velocity changes upon the replacement λα → 1/(λα) (see the asymmetry of the blue and red symbols); thus the symmetry predicted by eqn (25) is an artifact of the truncation in the far-field analysis.

Quantitative discrepancies are noticeable, as highlighted by the inset of Fig. 2: the inset shows that, even in the limit L/R → ∞, the amplitude of the far-field result deviates from that of the exact solution by ca. 50%. As it is apparent from the main panel of Fig. 2, the far-field calculation correctly captures the asymptotic result of a vanishing velocity. The fact that the ratio of the far-field approximated velocity and the exact one happens in this case to be a constant, rather than a function of the distance from the interface, is somewhat peculiar. However, it merely translates into the difference between the approximated value and the exact one being 1/2 of the exact value. Since the latter decays to zero with increasing distance from the interface, the deviation of the approximation from the exact value also vanishes with increasing distance, as it should.

We have identified two causes of these discrepancies. First, close to the interface the mismatch between the far-field predictions and the exact solution is due to the fact that we have kept only the lowest order singularities in the far-field approximation, while in such situations higher order terms are clearly not negligible. Second, in the far-field approximation we have accounted solely for those images which are needed to enforce the boundary conditions at the fluid interface, while the changes induced by those images in the boundary conditions at the surface of the particle have been disregarded, irrespectively of the distance from the interface. In this case, this kind of issue, which demonstrates the need for critical comparisons with exact solutions, leads to discrepancies between the far-field approximation and the exact result even in the limit L → ∞ (see the inset in Fig. 2 and Appendix C).

(27) |

(28) |

(29) |

(30) |

(31) |

The result in eqn (31) deserves further discussion. Similarly to the symmetry discussed in the context of eqn (25), a straightforward calculation shows that upon simultaneously replacing λα → 1/(λα) and “cap up” → “cap down”, U_{z} changes sign but its magnitude stays the same. Thus an experiment with a particle at z = L and oriented “cap up” (away from the interface) and one in which an identical particle is immersed at z = −L in the other fluid but oriented “cap-down” (towards the interface) will show in one case the particle approaching the interface and in the other moving away from the interface, in both cases with precisely the same speed. Furthermore, it can be inferred that there are cases in which the two terms within the square brackets on the rhs of eqn (31) have opposite signs. To this end we consider the situation of a “cap-up” particle, i.e., in eqn (31) for the second term the “−” sign holds so that for λα < 1 the first term is positive, while the second is negative. In this case it is straightforward to show that if

(32) |

(33) |

Fig. 3 shows the dependence of U_{z}/|V_{o}| for three values of the size of the catalytic cap and for V_{o} < 0. In particular, Fig. 3(a) indeed confirms that for sufficiently large coverages, such as θ_{0} = 3π/4, active particles can be trapped at a finite distance (i.e., U_{z}(L_{0}) = 0 for cap down and λα = 5), similar to what has been reported for active particles close to hard walls.^{32} Actually, in Fig. 3(a) the cap-down configuration with zero velocity is an unstable configuration in that, if the particle deviates from this position L_{0}, it does not return there, as can be inferred by inspection of the sign of the velocity to the left and to the right of the zero-crossing. As discussed above, such states do occur only if the coverage is sufficiently large, and therefore they are not observed in Fig. 3(b) and (c). As in the previous case of a homogeneously active particle, the far-field approximation (dashed lines in Fig. 3) misses to quantitatively capture the magnitude of the velocity if the active Janus particle is close to the interface – even though it captures the asymptotic values of U_{z} at large distances L. As in the case of homogeneously covered particles, the symmetry properties inferred from eqn (31) are not confirmed by the exact solution, but at a qualitative level the predicted change in sign of the velocity upon the simultaneous change λα → 1/(λα) and “cap up” → “cap down” indeed holds for large L/R. Finally, we remark that in the case λα = 1 there is no distortion of the number density profile due to the presence of the interface. Therefore in this case the variation of the velocity as function of L (green symbols in Fig. 3) is of purely hydrodynamic origin. It stems from the boundary conditions imposed on the velocity profile by the fluid–fluid interface.

Fig. 3 Rescaled velocity U_{z}/|V_{o}| calculated by using bi-polar coordinates (full and open symbols), as a function of the distance L/R of the center of mass of the particle from the interface for θ_{0} = 3π/4 (panel (a)), θ_{0} = π/2 (panel (b)), and θ_{0} = π/4 (panel (c)) with the chemically active cap pointing upwards (full symbols) or downwards (open symbols) for λα = 0.2 (blue), λα = 1 (green), and λα = 5 (red). The results shown correspond to the choice V_{o} < 0 (b < 0). The predictions of the far-field approximation are reported as dashed lines using the same color scheme as for the results obtained by employing bi-polar coordinates. As expected, the comparison between Fig. 2 and 3 shows that U_{z}(L → ∞, θ_{0} = π) = 0 whereas U_{z}(L → ∞, θ_{0} < π) ≠ 0. |

In order to characterize the dynamics of catalytic active particles close to a fluid–fluid interface we have developed a truncated far-field expansion of the diffusion equation for the product molecules of the catalysis and of the Stokes equation. As well, an exact solution, in terms of bi-polar coordinates, has been constructed based on the results available in ref. 46. We have found that both the magnitude and the sign of the velocity of the particle can be controlled by tuning the ratio of the diffusivities and solvabilities of the catalysis products in the two coexisting fluid phases. In particular, both the exact solution and the far-field approximation show that the sign of the velocity is controlled by the product λα of the ratio α of the diffusion coefficients of the catalysis products in the two fluid phases, and the ratio λ of the equilibrium solvabilities. Indeed, for a positive phoretic mobility b > 0 an active particle with λα > 1 moves towards the interface, whereas it moves away for λα < 1; for b < 0, the directions are reversed. This effect is strongest close to the interface; the interface-induced velocity decays algebraically (∼L^{−2}) with the distance L from the interface.

If the coverage of the particle is not homogeneous, there is a velocity (eqn (31)) already in a homogeneous fluid. Therefore, if such particles are close to a fluid–fluid interface the interface induced velocity (eqn (31)) sums up with v_{0}, leading to rich scenarios. For example, a particle with sufficiently large coverage θ_{0} (eqn (32)) experiences a stagnation point close to the interface (eqn (33)) where its velocity vanishes (Fig. 3(a)). If stable, these states are the equivalent of the steady-state hovering near a planar wall. Clearly, for this kind of particles adsorption at the interface is hindered due to the interface-induced component of the velocity. On the contrary, for coverages below the critical value given by eqn (32) such a tug-of-war scenario between the eigen-velocity of the particle and the interface-induced velocity does not occur and the sign of the velocity is always determined by the eigen-velocity v_{0}.

A.1 System of bi-polar coordinates.
The bi-polar coordinates (ξ,η) with −∞ < ξ < ∞ and 0 ≤ η ≤ π are defined such that the vertical coordinate z and the radial distance r from the z-axis are given by^{55,64}

where with ξ_{0} = arccosh(L/R) is chosen such that the manifold ξ = ξ_{0} corresponds to the spherical surface of radius R centered at z = L (which is the surface of the particle). The plane z = 0 of the interface corresponds to ξ = 0. In order to simplify the notations we introduce the quantity ω := cosη.

i.e., the edge is the intersection of the manifolds ξ = ξ_{0} and η = η_{0}. Therefore, for the “cap up” setup the catalyst covered area corresponds to (ξ_{0}, 0 ≤ η ≤ η_{0}), while for a “cap down” setup the catalyst covered area corresponds to (ξ_{0}, η_{0} ≤ η ≤ π).

(34) |

Here we focus on the case in which the catalytic cap, characterized by the opening angle θ_{0} (which is expressed in terms of the spherical coordinates attached to the particle), is oriented away from the interface, as drawn in Fig. 1. (The opposite case, i.e., the cap facing the interface, follows from a minor change in the equations determining the coefficients in the expansion of the solute number density; this change will be pointed out at the corresponding step in the derivation.) In this case, the point (ξ = ξ_{0}, η = 0) farthest from the interface (the “north” pole) belongs to the catalytic cap, while the point (ξ = ξ_{0}, η = π) closest to the interface (the “south” pole) belongs to the chemically inert part. The boundary between the catalytic and the inert caps (the “edge”) is a circle parallel to the plane z = 0. The points P_{e} = (r_{e}, z_{e}) on the edge have the spherical coordinates r_{e} = Rsinθ_{0} and z_{e} = L + Rcosθ_{0} (see Fig. 1); since the edge is part of the sphere ξ = ξ_{0}, i.e., r_{e} and z_{e} satisfy eqn (34) with ξ = ξ_{0}, the points on the edge have the same coordinate η_{0} given by

(35) |

A.2 Solution of the diffusion problem.
Since all details of the corresponding calculations are provided by the openly accessible supplementary material of ref. 46, the brief outline given below for obtaining the solution of the diffusion problem and of the auxiliary stream function is considered to be sufficient.

in fluid 1, with a similar expression but with different coefficients , Â_{n}, and _{n} in fluid 2 (ξ < 0). The prefactor QR/D_{1} has the units of a number density, so that the coefficients A_{n}, B_{n}, Â_{n}, and _{n} are dimensionless. (We note that the same prefactor QR/D_{1} is used for both ξ > 0 and ξ < 0.) We focus on the solution in fluid 1 (ξ > 0) because only that one enters into the expression for the phoretic slip at the surface of the colloid.

inserting them into the boundary conditions at the interface (eqn (3) and (5)) in combination with the requirement that the density is bounded everywhere leads to

with α = D_{2}/D_{1} (as defined in eqn (20) in the main text). (Note that for a constant flux boundary condition on the particle surface, which is time- and position-independent over the catalyst part, the velocity of the particle turns out to be independent of the value of the constant c^{∞}_{1}, see, cf., eqn (46) and (48).) By combining eqn (37) with the flux boundary condition (eqn (6)) at the surface of the particle and by projecting the lhs and the rhs of eqn (6) onto the Legendre polynomial P_{n}(ω), one arrives at the following set of linear equations determining the coefficients B_{n}:

with the convention B_{−1} = 0. With ω_{0} := cosη_{0} defining, as discussed above, the edge between the active and the passive parts of the surface, the coefficients f_{n} are given in terms of the activity function f(θ) (eqn (6) in the main text):

This infinitely large system of linear equations is solved by truncating it at a sufficiently large index n = N, followed by a numerical treatment. In practice, the truncation, as well as the series representation, are converging very fast as long as L/R ≳ 1.1. We have found that in most cases N = 50 is sufficient for providing accurate results. This procedure is analogous to the ones used in ref. 46 and 63.

The solution of the diffusion problem (eqn (1)–(6)) can be expressed in terms of Legendre polynomials P_{n} as^{65}

(36) |

Inserting these two series representations into the boundary conditions at infinity (eqn (2) and (4)) leads to

(37a) |

Â_{n} = _{n} = λB_{n}, A_{n} = λαB_{n}, | (37b) |

(38) |

(39) |

A.3 Solution of the auxiliary hydrodynamics problem.
The auxiliary problem consists of a passive spherical particle, i.e., there is no chemical reaction, with a no slip boundary condition at its surface. The center of the particle is located at z = L and moves with velocity Û = e_{z}Û through fluid 1 along the direction normal to the flat fluid–fluid interface.

This stream function can be represented in bi-polar coordinates:^{64,65}

a similar expression, but with different coefficients _{n}, _{n}, _{n}, and _{n}, holds for ξ < 0. In these equations

denotes the Gegenbauer polynomial of order n and degree −1/2.^{64} The dimensionless coefficients K_{n}, L_{n}, M_{n}, and N_{n}, as well as the hatted ones, depend on ξ_{0} (but not on η_{0}) and are determined by the boundary conditions for the velocity field. The requirement of a finite flow everywhere and the boundary conditions at infinity and at the interface lead to

and

where μ_{1,2} denote the respective viscosities of the two fluid phases. By combining these relations with the no-slip and no-impenetrability conditions at the surface of the particle, the coefficients L_{n} and N_{n} are obtained as

and

where

and

With noting that μ_{2}/μ_{1} = 1/α, the derivation of the stream function for the auxiliary problem is complete.

The corresponding solution for the velocity field û(x) of the incompressible Stokes equations can be expressed in terms of a stream function Ψ_{aux}(x) = ÛR^{2}ψ_{aux}(x) as^{55}

(40) |

(41) |

(42) |

_{n} = _{n} = −_{n} = −_{n}, n ≥ 1, | (43a) |

(43b) |

(43c) |

(44a) |

(44b) |

(45a) |

χ^{(2)}_{n} = −e^{−(n−1/2)ξ0} + e^{−(n+3/2)ξ0}, | (45b) |

(45c) |

(45d) |

(45e) |

(45f) |

A.4 Integral over the phoretic slip in bi-polar coordinates.
We start the calculation of the integral over the phoretic slip in eqn (14) by noting that: (i) in terms of the bi-polar coordinates the normal to the surface Σ_{p} of the particle is given by n = −e_{ξ}; (ii) the tangent plane to the surface of the particle is spanned by the unit vectors e_{η} and e_{ϕ}; and (iii) since the present problems exhibit axial symmetry, the solute number density as well as the stress tensor of the auxiliary problem are independent of ϕ (see the previous subsections). Thus, the phoretic slip is given by v_{p} := −b∇_{‖}c(ξ_{0},η) = −b(h_{η}^{−1}|_{ξ0})∂_{η}c(ξ_{0},η)e_{η}, where denote the so-called scale (metric) factors corresponding to the η and ξ coordinates, respectively (see eqn (34) and recall the abbreviation ω = cosη). With this eqn (14) in the main text takes the form

where denotes the scale factor corresponding to the ϕ coordinate.

with L_{n} and N_{n} given by eqn (44).

where (see eqn (36)); the coefficients A_{n} and B_{n} are determined from eqn (38) and (39), as well as from the relation A_{n} = (λα)B_{n}.

The flow components u_{ξ} and u_{η} are obtained from the stream function Ψ_{aux} (derived in the previous section) as (see Ch. 4-4 in ref. 55 and eqn (34))

and

which concludes the calculation.

(46) |

The terms on the right hand side of eqn (46) are calculated as follows. First, is determined from the stream function Ψ_{aux} = ÛR^{2}ψ_{aux}(x) as^{59,64–66}

(47) |

Second, the derivative of the number density at the surface of the particle is given by

(48) |

The contraction e_{η}··e_{ξ} of the stress tensor at the surface of the particle (which is immersed in fluid “1”) is calculated as follows (see also ref. 38). By writing the hydrodynamic flow in the auxiliary problem as û = û_{ξ}e_{ξ} + û_{η}e_{η} and by using the representation of the dyadic product ∇v_{aux} (which one needs in order to be able to calculate the stress tensor in the auxiliary problem) in terms of the general orthogonal curvilinear coordinates provided in ref. 55 (see Appendix A-7, eqn (A-7.7) therein), one arrives at (recalling h_{ξ} = h_{η} and e_{η}·e_{ξ} = 0)

(49) |

(50a) |

(50b) |

_{im}(x, y, z) = _{d}(x, y, z) − p(x, y, z), | (51) |

(52) |

(53) |

r = (x, y, L + z), R = (x, y, 2L + z). | (54) |

_{d}(x, y, z) = _{d}(−x, −y, z), | (55) |

_{d,x} = _{d,y} = 0, |

∂_{x}_{d} = ∂_{y}_{d} = 0. | (56) |

(57) |

(58) |

(59) |

With n given by

n = (sinθcosϕ, sinθsinϕ, cosθ) | (60) |

(61) |

Fig. 4 Number density profiles at the surface of a homogeneously covered active particle near a fluid–fluid interface for λα = 5 and L/R = 2 (panel (a)), L/R = 5 (panel (b)), and L/R = 10 (panel (c)), normalized by the number density c_{0} = QR/D_{1} at the surface of the same particle suspended in a homogeneous unbounded fluid medium (λα = 1). The red lines correspond to the exact solution (eqn (36)) whereas the blue lines provide the far-field approximation (eqn (23)). The concentration is highest (lowest) at the north (south) pole which is distant (close) to the interface. The far-field approximation underestimates (overestimates) c(θ) on the northern (southern) hemisphere even at large values L/R (panel (c)). Upon increasing L/R → ∞ (i.e., moving the particle farther from the interface) both curves flatten and approach the value c(θ)/c_{0} ≡ 1, as expected for a homogeneously active sphere. |

(62) |

(63) |

(64) |

(65) |

(66) |

(67) |

• At zeroth order, , in the expansion only the “source” terms, which are independent of L, contribute to the slip velocity:

(68) |

• At first order, , in the expansion there is no contribution to the slip velocity:

v_{p,1}(θ)e_{θ}·e_{z} = 0. | (69) |

• At second order, , in the expansion the contribution to the slip velocity reads

(70) |

In order to obtain eqn (70) we have used the relation

(71) |

(72) |

(73) |

Alternatively, for a purely dipolar contribution eqn (70) reduces to eqn (30) in the main text:

(74) |

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## Footnotes |

† Here we assume that the advection of the products and reactants by the hydrodynamic flow is negligible compared with their diffusion so that motion of the particle due to a spontaneous symmetry breaking^{49,50} does not occur. |

‡ We remind that eqn (23) has been obtained by accounting only for those images needed to enforce the boundary conditions at the fluid interface. Additional images are needed to enforce the boundary condition on the surface of the colloidal particle.^{33} In order to keep the model simple we disregard the latter ones which, in principle, provide additional contributions to eqn (23). Such approximate results are sufficient to describe the qualitative behavior of active colloids close to fluid interfaces (as discussed in the main text). |

§ The velocity U_{z} is the instantaneous velocity an active particle will attain when its center is at distance L apart from the interface. Under the assumptions of the model (fast diffusion of solute, quasi-steady state instantaneously attained) the dynamics of the system is in the overdamped regime. Hence, the velocity of the particle does not depend on initial conditions and equals the one that would be observed by fixing an active colloid at position L (for example by an optical trap) and suddenly releasing it. |

¶ The amplitude of the image of a dipole can be obtained by recalling that a dipole is obtained from a pair of point sources of opposite magnitude located at a distance d apart (see ref. 46 for the derivation of the magnitude of the image of a point source). Accordingly, the image of a dipole with dipole moment P located at z = h is a dipole, located at z = −h, with dipole moment (see Appendix D). |

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