Confinement effects on diffusiophoretic self-propellers

J. Chem. Phys. 130, 194702 (2009); https://doi.org/10.1063/1.3133239 130, 194702 © 2009 American Institute of Physics. Confinement effects on diffusiophoretic selfpropellers Cite as: J. Chem. Phys. 130, 194702 (2009); https://doi.org/10.1063/1.3133239 Submitted: 02 February 2009 • Accepted: 22 April 2009 • Published Online: 15 May 2009 M. N. Popescu, S. Dietrich and G. Oshanin ARTICLES YOU MAY BE INTERESTED IN Self-diffusiophoretic colloidal propulsion near a solid boundary Physics of Fluids 28, 053107 (2016); https://doi.org/10.1063/1.4948398 Diffusiophoretic self-propulsion of colloids driven by a surface reaction: The sub-micron particle regime for exponential and van der Waals interactions Physics of Fluids 25, 012001 (2013); https://doi.org/10.1063/1.4772978 Life at low Reynolds number American Journal of Physics 45, 3 (1977); https://doi.org/10.1119/1.10903 Confinement effects on diffusiophoretic self-propellers M. N. Popescu,1,a兲S. Dietrich,2,b兲and G. Oshanin3,c兲 1Ian Wark Research Institute, University of South Australia, 5095 Adelaide, South Australia, Australia 2Max-Planck-Institut für Metallforschung, Heisenbergstr. 3, 70569 Stuttgart, Germany and Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany 3Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie (Paris 6)-4, Place Jussieu, 75252 Paris, France 共Received 2 February 2009; accepted 22 April 2009; published online 15 May 2009兲 We study theoretically the effects of spatial confinement on the phoretic motion of a dissolved particle driven by composition gradients generated by chemical reactions of their solvent, which are active only on certain parts of the particle surface. We show that the presence of confining walls increases in a similar way both the composition gradients and the viscous friction, and the overall result of these competing effects is an increase in the phoretic velocity of the particle. For the case of steric repulsion only between the particle and the product molecules of the chemical reactions, the absolute value of the velocity remains nonetheless rather small. © 2009 American Institute of Physics.关DOI: 10.1063/1.3133239兴 I. INTRODUCTION Recent years have witnessed a growing technological, experimental, and theoretical interest in scaling standard machinery down to micro- and nanoscales as needed for the development of “lab on a chip” devices. For applications in, e.g., drug-delivery systems or micromechanics one of the most challenging problems at this stage is to develop ways to enable small-scale objects to perform autonomous, controlled motion.1,2Although the research in this area is still in its early stages, several such proposals have already been tested experimentally 共see, e.g., Refs. 3–6兲. A review of the recent progress in this field can be found in Ref. 2. Ismagilov et al.3proposed a design of self-propelling devices based on an asymmetric decoration of the surface of small objects by catalytic, active sites promoting a chemical reaction in the surrounding liquid medium. This asymmetric distribution can provide motility through a variety of mechanisms, such as surface tension gradients and/or cyclic adsorption and desorption.2,7The use of an asymmetric surface distribution of a catalyst has been further proposed for an autonomous diffusiophoretic motion emerging as a result of self-created density gradients.5An experimental realization of such systems, using platinum coated polystyrene spheres, has been recently reported.6The issue of designing optimal distributions of a catalyst for spherical and rodlike particles has also been approached.8 For most of the applications in biological systems or in lab on a chip-type devices, one has to deal with a complicated internal structure of the systems, such as networks of narrow channels or pores and various impenetrable impurities. In some situations, one may even encounter a quasi two- or one-dimensional behavior, like in the cases of bacteria motion on planar nutrient substrates9and of the motion of small particles within a biological membrane10 or in a monolayer adsorbed on a three-dimensional 共3D兲liquid subphase.11 Thus spatial confinement is a relevant feature so that the assumption of the presence of an unconfined 3D bulk reactive solvent1,2,5may break down. Intuitively, one expects that spatial confinement does influence the resulting motion of self-propelling objects like the ones discussed above. However a priori it is not clear if such effects are significant. Starting from the model used in Ref. 5, here we study the effects of spatial confinement on the phoretic motion of a particle that generates number density gradients of the product molecules emerging from the chemical reactions in the depleting Newtonian liquid solvent. We shall focus on the simple case in which the particle and the 3D solution of solvent and product molecules are bounded by a spherical shell because 共i兲this simple geometry allows for an exact solution and 共ii兲it has been experimentally shown for a variety of phoretic systems that confinement effects are dominated by the smallest confining length scale.12 Thus this geometry has a paradigmatic character. Similar to the earlier studies,1,2,4–6our work is based on adopting the standard theory of phoresis for the present case, in which the gradients are self-generated rather than being produced and maintained by external sources. In doing so, one is bound to make a number of assumptions that are either already present in the classical theory of phoresis, or arise as a result of mapping the description of such “active” surface particles onto the framework of a theory developed to describe the case of inert particles immersed in a predefined, externally controlled concentration gradient. Since in literature these assumptions are often overlooked or not spelled out explicitly, we consider it as necessary to provide also a critical discussion of the significant assumptions involved by a兲Electronic mail: [email protected]. Also at Max-Planck- Institut für Metallforschung, Heisenbergstr. 3, 70569 Stuttgart, Germany. b兲Electronic mail: [email protected]. c兲Electronic mail: [email protected]. Also at Max-Planck-Institut für Metallforschung, Heisenbergstr. 3, 70569 Stuttgart, Germany. THE JOURNAL OF CHEMICAL PHYSICS 130, 194702 共2009兲 0021-9606/2009/130共19兲/194702/16/$25.00 © 2009 American Institute of Physics130, 194702-1 this mapping, as well as of some of those assumptions implicitly contained in the standard theory of phoresis. Accordingly, the outline of this paper is as follows. In Sec. II we define the model and discuss some general aspects of systems with self-generated motion, with particular emphasis on the assumptions involved in adopting the standard theory of phoresis. Section III is devoted to the computation of the diffusiophoretic velocity. This includes the calculation of the diffusiophoretic slip velocity and the determination of the phoretic hydrodynamic flows and density profiles of the product molecules around a self-propelling particle. The results are discussed in Sec. IV, and we conclude in Sec. V with a brief summary of our results and general conclusions. Important details of our calculations are presented in Appendixes A–C. II. THE MODEL A. General aspects The system we consider is shown in Fig. 1. It consists of an impermeable, spherical particle of radius Rwith a pointlike catalytic site 共black dot in Fig. 1兲on its surface, which promotes the chemical conversion of a surrounding solvent into product molecules of diameter a共small hatched circles in Fig. 1兲. The particle and the surrounding solution 共solvent plus the solute, i.e., the product molecules兲of viscosity ␮ are enclosed in a concentric, impermeable, spherical shell of radius R1= ␩ R共 ␩ ⬎1兲. In general, such a chemical conversion of the solvent gives rise to several types of product molecules. Here we shall focus on the particular case in which the chemical conversion of a solvent molecule 共A兲leads to two molecules only 共A⬘and B兲, one very similar in size and properties with the solvent itself 共A⬘⬇A兲, the other one 共B兲different 关A→ cat A⬘+B兴; in the following this latter is denoted as “product molecule” and plays the role of a solute in the solvent. In other words, we consider a situation in which the net result of the chemical conversion can be approximated as the generation of a solute, solely, and the reaction does not lead to a solvent depletion near the catalytic site which otherwise would reduce the current of solvent toward the catalytic site acting as a solvent sink. For example, this is approximately the case for the Pt catalyzed decomposition of hydrogen peroxide 共H2O2兲in aqueous solution into water 共H2O兲and oxygen 共O2兲molecules, as discussed in Refs. 1,2, and 6.In these experimental studies the oxygen plays the role of the product molecule the properties of which differ significantly from those of the solvent. Here the solvent is actually a binary liquid mixture of H2O and H2O2for which H2Ois passive and does not participate in the chemical conversion. We note that actually it is rather difficult to assess whether or not H2O and H2O2can be treated as being the same in such systems, and probably the answer would be on a case by case basis. This is not only because the H2O-particle and H2O2-particle interactions may be quite different for different materials 共the simplest example is exactly the catalytic decomposition of H2O2on Pt兲, but also because of the sensitivity of phoresis to the details of the solvent-mediated interactions—and the weakly acidic nature of H2O2may play an important role here. In providing this example as a possible realization of a catalyzed reaction A→ cat A+Bwe had to rely on: 共i兲the statement in Ref. 2共second entry, bottom of left column on page 13427 therein兲, which is based on some previously published results by Phibbs and Giguère13 that hydrogen peroxide and water in contact with Au have almost identical interfacial tensions 共this translates into similar particle-solvent interactions兲, and thus that there is no additional solvent-density gradient to be considered, and 共ii兲on the implicit statement in the same Ref. 2关second entry, Eq. 共8兲therein兴that the effective interaction of the O2molecule with the Au surface of the rod does not depend on the composition of the solvent H2O–H2O2mixture. We thus assume that the reaction at the catalytic site 共located at rs=−Rez, where ezis the unit vector of the z-axis兲 acts effectively only as a pointlike source of product molecules5of diameter awhich are diffusing in the solvent with diffusion coefficient D. In passing we note that this is in contrast to the situation considered in Ref. 7, where the catalytic site acts both as a source for the product and as a sink for the solvent, i.e., the generation of a product molecule is accompanied by the annihilation of a solvent one so that A→ cat Bwhich implies that Bcan only be a different configuration of Abecause it must have the same chemical constituents. In the situation considered here, the amplitude of the production rate of Bis denoted as B共t兲. We shall neglect any interaction between the product molecules. Thus the number density of product molecules is considered to be so low that among themselves they behave like an ideal gas. There is an interaction potential between the product molecules and the moving particle 共see Appendix A兲, which includes the impermeability condition at the surface of particle. The interactions between the product molecules and the solvent are accounted for in an effective way via the Stokes–Einstein FIG. 1. An impermeable, spherical particle of radius Rwith a pointlike catalytic site at rs=−Rez共depicted as a black dot兲on its surface, enclosed by a concentric, impermeable, spherical wall of radius R1= ␩ R共 ␩ ⬎1兲.The product molecules of diameter aare shown as small horizontally hatched circles. 194702-2 Popescu, Dietrich, and Oshanin J. Chem. Phys. 130, 194702 共2009兲 expression D=kBT/共3 ␲ ␮ a兲for the diffusion coefficient Dof the product molecules,14 where kBis the Boltzmann constant and Tis the temperature. B. Discussion of the model in the context of the standard theory of phoresis The presence of a source of solute 共product molecules兲 on the surface of the particle creates a nonuniform and timedependent distribution of solute in the solution 共see Fig. 1兲, i.e., a nonuniform composition of the solution. Because the understanding of the way in which such a nonuniform composition gives rise to phoretic motion is not straightforward, we discuss here in some detail how the general model introduced in Sec. II A can be set into the context of the standard theory of phoresis.12 A recent clear exposition of the basic concepts and a discussion of different phoretic transport scenarios can be also found in Ref. 15. If the production rate of the source is not very large and the diffusion coefficient of the product molecules through the solvent is not too small, the solute density varies smoothly in space and slowly in time. This justifies the assumption of local equilibrium and the definition of a position- and timedependent chemical potentials of the solution 共the spatial gradients of which describe the diffusion of the solute away from the source兲. In a quasi steady-state of the solution, corresponding to slow time variations of the composition, the net solute current is approximately zero and the solute density gradients are balanced by pressure gradients 关see also Eq. 共B3兲in Ref. 15兴. Because in the absence of external body forces the mechanical equilibrium for a liquid solution is generally established much faster than the chemical one 共see, e.g., Ref. 16兲, it is physically plausible to assume that far from the boundaries 共i.e., far from the wall and far from the particle surface兲the pressure adjusts instantaneously to accommodate the spatially varying solute density profile15 and keeps varying slowly with time 共on time scales much larger than the diffusion time ⬃R2/D兲because the overall density of the solution increases in time. In other words, the pressure is determined by the solute density profile and is obtained from the equation of state of the solvent-solute binary mixture considered as a thermodynamic system in local equilibrium described by the corresponding free energy density. Clearly, the situation is different in the region of contact between the solution and the particle surface 共and also of contact with the wall兲where the interactions of the particle with the product molecules 共solute兲and with the solvent molecules are relevant. This interaction disturbs the distribution of solute molecules, net currents of solute and solvent result, and the pressure field becomes a quantity determined by the incompressibility requirement for the hydrodynamic flow, and not by an equation of state. We shall discuss this important point below. At a given time the spatial variations of the number density ␳ of the solute and of the number density ␳ solv of the solvent are characterized as follows: ␳ solv is constant throughout the solution apart from the close vicinity of the particle surface and of the surface of the confining sphere. There ␳ solv vanishes and reaches its constant bulk value via density oscillations which are induced by local packing effects caused by the finite diameter asolv of the solvent particles. Away from any kind of surface phase transition the approach of the bulk value occurs exponentially on the scale of the bulk correlation length ␰ solv of the solvent. 共The presence of long-ranged dispersion forces causes power-law decays; however, their amplitude is small and is neglected in the present context.兲Away from bulk phase transitions ␰ solv is comparable with the range of the interaction potential between the solvent molecules, which in turn is proportional to asolv with a prefactor of order unity. Within this picture ␳ solv does not vary along the surface of the particle. The solute number density ␳ is characterized by two important features. On the scale of the system size RⰇa, ␳ varies due to the diffusion process. On the much smaller length scale of the solute diameter a, ␳ also varies near the particle surface and near the surface of the confinement as discussed above for the solvent. Since, however, according to our earlier assumption the solute particles can be considered to form an ideal gas, near the walls ␳ varies proportional to exp共− ␤ ⌿兲, where ⌿is the effective substrate potential 共see Appendix A兲in the sense that it describes the interaction of the solute molecule with the substrate in the presence of the solvent, and ␤ =1/kBT. Typically the range ␦ of ⌿is proportional to the solute diameter a. Accordingly the solute molecules interact directly with the particle only if they are within a thin surface film of thickness ␦ , which is assumed to not be deformed by the motion of the particle.12 The standard theory of phoresis assumes a, ␦ Ⰷasolv so that within this surface film of thickness ␦ the solvent can be considered as a continuum with constant ␳ solv. We note that this assumption is rather convenient from a computational point of view. However, this assumption is at odds with the basic underlying model according to which the solute particle is created from a solvent particle via a catalytic reaction. Under these circumstances one expects a⬇asolv 共see the above example of the reaction 2H2O2→ cat 2H2O+O2兲.Inthe absence of a more detailed theory of phoresis, which treats the sizes of the solvent and solute particles on the same footing, we proceed along the lines of the standard theory. Nonetheless we point out that there is an urgent need for improvement. It might be that due to the fact that continuum hydrodynamics remains quantitatively reliable down to surprisingly small length scales 共here asolv兲, it is conceivable that the aforementioned continuum description yields reliable results, too. However this remains unproven. According to the standard theory a hydrodynamic description applies within the aforementioned surface film. Within this picture the solute molecules and their effective interaction with the particle are replaced by a corresponding distribution of “point forces” acting on the solvent in the film. Within the limitations of such an approach the hydrodynamic description of the solution naturally splits into that of an “inner” region formed by the surface film and that of the “outer” region formed by the exterior space beyond the surface film. Following Refs. 8,12, and 17, the ensuing asymmetric, nonuniform solute number density ␳ 共r,t兲 around the particle will give rise, within the surface film 194702-3 Confined diffusiophoretic self-propellers J. Chem. Phys. 130, 194702 共2009兲 only, to a pressure gradient along the surface of the particle. This is because the surface film is very thin on the scale of the system size Rso that the equilibration of the composition profile of the solution within the surface film can be assumed to be fast along the direction normal to the surface of the self-propelling particle compared with the diffusional relaxation time of the composition gradient along its surface, which typically involves a length scale of the order of the particle size R. Therefore, within the surface film the solute density in the direction normal to the surface of the particle is given by a Boltzmann distribution corresponding to the local equilibrium configuration in the presence of the effective interaction potential between the particle and the solute molecules, with a prefactor which depends on the position along the surface of the particle 共for details, see Appendix A, which builds on Ref. 12兲. Mechanical equilibrium of the solvent within the surface film along the direction normal to the surface of the particle 共no flow along this direction兲requires the pressure gradient along the normal to be equal to the body force densities due to the effective particle-solute interactions. Therefore the pressure within the surface film differs from the outer pressure field by an “osmotic pressure” term, i.e., a term proportional to the extra solute density ␳ in excess to the constant solvent density ␳ solv. Since this osmotic pressure varies along the surface of the particle, there is a gradient of pressure along the surface of the particle. This lateral pressure gradient is not balanced by any body force and thus generates shear stress within the surface film. Therefore it induces hydrodynamic flow of the solution along the surface of the particle and entails motion of the particle with a velocity V共t兲. Because the system has azimuthal symmetry, the motion is along the z-axis, i.e., V =Vez. The hydrodynamic flow of the solution, the diffusive transport of the solute, and the coupling between the two giving rise to phoretic motion of the particle are analyzed in detail in Sec. III 共see also the Appendixes A–C兲. III. COMPUTATION OF THE DIFFUSIOPHORETIC VELOCITY Our calculation of the diffusiophoretic velocity proceeds along the lines of Ref. 12 and is based on dividing the problem into an inner one—within the film around the particle surface discussed in Sec. II B, and an outer one—beyond the range of the effective interaction between the solutes and the particle. In view of the arguments presented in Sec. II, we base our analysis on the following assumptions. 共i兲The chemical reaction leads to a change in the solute density only. The thickness of the surface film, defined by the range of the effective interaction between the product molecules 共solute兲and the particle, is much smaller than the particle radius R. The spatial variations of the solute number density along the surface of the particle occur over length scales of the order of R, which allows one to use the approximation of a locally planar interface. 共ii兲The number density of product molecules is sufficiently low so that the solute can be viewed as an ideal gas and the solvent-solute mixture behaves like an ideal dilute solution. In this case, the pressure gradient is simply proportional to the gradient of the number density of the product particles 共see Appendix A兲. 共iii兲Temporal variations of the number density of the product molecules due to their creation occur on time scales much longer than those needed for the fluid flow 共as seen from the moving particle兲to relax to a steady state corresponding to the number density profile at that moment. 共iv兲The flow field of the solution within the surface film can be described by the laws of hydrodynamics. Additionally, we assume that both the Reynolds number Re⯝ ␳ ˜ solvVR/ ␮ , where ␳ ˜ solv is the mass density of the solvent, and the Peclet numbers Pe⯝VR/Dare small such that one can approximate the hydrodynamic description with the creeping flow 共Stokes兲equations and disregard the convection of the solute compared to its diffusive transport. Here we have assumed that the magnitude of the hydrodynamic flow uis similar to that of the phoretic velocity V; this assumption is supported a posteriori by the fact that for our system the phoretic velocity is basically the average of the slip velocity over the surface of the particle 关see, c.f., Eqs. 共1兲and 共10兲兴. For a micron size particle moving through water 共density ␳ ˜ solv=103kg/m3, viscosity ␮ =10−3 Pa s兲 with a velocity of the order of ␮ m/s, which is typical for phoresis, one has Re⯝10−6. For the diffusion at room temperature 共kBTroom⬃4⫻10−21 J兲of O2共a⬃10−10 m兲in H2O2共 ␮ ⯝10−3 Pa s兲, the Stokes–Einstein relation leads to an estimate D⬃4⫻10−9 m2/s for the diffusion coefficient 共in agreement with Ref. 2兲, and thus Pe⯝10−3. Therefore the latter assumptions are justified. Note that if one uses R1 rather than Ras a characteristic length scale, the above results imply that the Re and Pe numbers remain both very small as long as ␩ ⱗ10. A. Diffusiophoretic slip velocity As discussed in Sec. II, the pressure gradient along the surface of the particle, induced by its interaction with the nonuniform distribution ␳ 共r,t兲of product molecules, leads to flow of the solution relative to the particle. As shown in Appendix A, the hydrodynamic flow within the surface film translates into a 共phoretic兲slip velocity, vs共r,t兲=−bⵜs ␳ 共r,t兲for 兩r兩=R+,共1兲 at the outer edge R+ⲏRof the surface film as a boundary condition 共BC兲for the hydrodynamic flow in the outer region. In this equation ⵜsdenotes the projection of the gradient operator onto the tangential planes of the surface of the particle, ⵜs=e ␪ 1 r ⳵ ⳵ ␪ +e ␾ 1 rsin ␪ ⳵ ⳵ ␾ ,共2兲 where e ␪ and e ␾ are the polar and azimuthal unit vectors, respectively, while 194702-4 Popescu, Dietrich, and Oshanin J. Chem. Phys. 130, 194702 共2009兲 b=kBT ␮ ⌳共3兲 is an effective “mobility,” and ␭=冑兩⌳兩is a characteristic length scale. The latter is determined by the effective interaction potential ⌿between the particle and the product molecules12 关which determines their distribution within the surface film along the direction y ˆnormal to the particle surface within the local coordinate system 共Fig. 3兲or the radial direction here兴, ⌳= 冕 0 ⬁ dy ˆy ˆ共e− ␤ ⌿共y ˆ兲−1兲.共4兲 In the case of a purely steric repulsive interaction, one has12 ⌳=−a2/8 so that ␭=a/共2冑2兲. For a single catalytically active site as shown in Fig. 1 ␳ and thus vsdo not depend on the azimuthal angle ␾ . B. Phoretic hydrodynamic flow We focus here on the quasistatic approximation 共iii兲 described above and the limit of low Re numbers. In the laboratory frame, in which the particle moves with a yet unknown velocity V共t; ␩ 兲, the hydrodynamic flow u ⬅u共r;t, ␩ 兲at time tin the domain beyond the surface film around the particle obeys the steady-state force-free Stokes equations ⵜ·⌸ ˆ=0, ⵜ·u=0, R+⬍兩r兩⬍R1,共5兲 where ⌸ ˆ=−pI ˆ+ ␮ ⌺ ˆis the pressure tensor, pis the hydrostatic pressure, and ⌺ ˆis the shear stress tensor, i.e., ⌺ ␣␤ = ⳵ u ␣ / ⳵ x ␤ + ⳵ u ␤ / ⳵ x ␣ , subject to BCs of no slip at the wall at R1and prescribed slip velocity on the surface R+, i.e., 兩u兩兩r兩=R+=V共t; ␩ 兲+vs,兩u兩兩r兩=R1=0. 共6兲 Note that the parametric dependences of the flow field uon t and ␩ stem from the BCs. Note that because for the outer problem the variations of the flow field and those of the number density of the product molecules are over length scales that are much larger than ␦ , one can replace everywhere in the calculations the 共unknown兲value of R+by R; we shall use this approximation in the following. The solution of the Stokes equations with BCs on spherical surfaces is based on expressing both the flow field uand the pressure field pas series of solid harmonics18,19 Kl共r, ␪ 兲 共l苸Z兲, which are the eigenfunctions of the 3D Laplace operator. In Appendix B we provide a brief outline of the general method for solving Eq. 共5兲in spherical coordinates and derive the solution obeying the BCs given by Eq. 共6兲. Equation 共6兲at R1corresponds to a no-slip condition imposed on a wall fixed with respect to the laboratory frame. This deserves further discussion. In Eq. 共6兲the BC is that u at the surface of the moving particle, i.e., 兩u兩兩r−Rp共t兲兩=R+, where Rp共t兲=兰0 tV共t⬘兲dt⬘is the position of the particle center, takes the value V共t; ␩ 兲+vs. As long as 兩Rp共t兲兩ⰆR+, concentricity holds and one obtains the first of the two BCs given by Eq. 共6兲. If, however, the velocity Vis not small, we readjust the center of R1in order to impose concentricity using the following “protocol:” the particle is allowed to move for a short time ␦ twith the instantaneous velocity V共t兲while the spherical shell at R1is fixed, after which the spherical shell is displaced by V共t兲 ␦ tto its new fixed position in such a way that it does not perturb significantly the density and flow fields; this procedure is then repeated. It is not clear to which extent this latter assumption, which allows us to obtain an analytical solution, can be realized experimentally. However it is expected that the results we derive for the present geometrical setup are relevant also for more general geometries,20 such as a particle moving along a channel in 3D, for which the constraint that the confinement moves with the particle is not needed. We note that in the general case of a nonspherical particle, or of a particle with nonuniform surface properties 共e.g., a spatially varying effective mobility b兲the particle can also rotate and the BC at the particle surface, Eq. 共6兲, should include a term accounting for a rigid-body rotation with angular velocity ⍀. However, this angular velocity turns out to be identically zero in the case of a spherical particle with an effective mobility bwhich is constant on its surface;12,21 therefore we completely disregard it here. C. Phoretic velocity The velocity V共t; ␩ 兲of the particle is obtained by requiring that the hydrodynamic force F= 冕冕 兩r兩=R ⌸ ˆerdS,共7兲 where eris the radial unit vector and dS the surface area element on the spherical surface 兩r兩=R共note that we replaced R+by R, as discussed in the previous subsection兲, exerted by the fluid on the composite domain particle plus surface film vanishes 关see the vector identities in Eq. 共B7兲,共i兲–共iii兲below this equation, and 共r⫻共ⵜ⫻u兲兲 ␣ =x ␤ ⳵ ␣ u ␤ −x ␤ ⳵ ␤ u ␣ , with summation over ␤ 兴: 冕冕 兩r兩=R dS 冋 −per+ ␮ 冉 ⳵ u ⳵ r− u r 冊 + ␮ rⵜ共ru兲 册 =0. 共8兲 If, as discussed above, a rotational motion with angular velocity ⍀共t; ␩ 兲would have to be considered, too, this will be determined by the additional requirement that the motion is not only force-free but also torque free.12 This is again due to the fact that there are no net forces acting on the object composed of the particle and its surface film. The above argument for determining the velocity Vwas discussed in detail by Anderson 共see Ref. 12 and references therein兲, but it is often overlooked and replaced by the incorrect argument of a balance between a drag force—i.e., the integral of the nonuniform osmotic pressure proportional to the density of solute 共see Appendix A兲—exerted on the particle and a Stokes-like viscous friction from the solvent 共see, e.g., Refs. 2,22, and 23兲. It is important to realize that the occurrence of composition gradients in the solution does not give rise, by itself, to an osmotic pressure 共see also Ref. 15兲, in contrast to such an assumption made in Ref. 22. Such gradients will simply lead to diffusion of the product molecules, while the pressure in the solvent will adjust to accom- 194702-5 Confined diffusiophoretic self-propellers J. Chem. Phys. 130, 194702 共2009兲 modate the spatially varying chemical potential corresponding to the quasistationary density profile 共mildly time dependent due to the overall increase in the solute number density, in the case of the confined system兲, reflecting mechanical equilibrium.15 As a matter of fact, the origin of this osmotic pressure resides in the interaction between the particle and the product molecules, i.e., it requires the explicit consideration of the effective interaction potential between the particle and the product molecule 共for a detailed illuminating discussion of this point see Refs. 12 and 17兲. An intuitive argument that the use of such a “Stokes formula,” which stems from a standard “drag balanced by viscous friction” type of reasoning, is incorrect can be formulated as follows. At distances far from the particle the flow field should look like that produced by the superposition of a point force fat the origin, i.e., the center of the particle, which is the integrated 共over the particle surface兲effective “product molecules on particle” interaction 共the forces f ˆDin Fig. 3in Appendix A兲, and a distribution of effective point forces oriented radially 共of the particle acting on the product molecules兲in a small shell region around the surface of the particle 共which is the aforementioned surface film兲, which upon integration gives exactly −f. Note that for repulsive effective interactions and for an axisymmetric distribution of product molecules with an increasing density toward the source at z=−R, as in Fig. 1,f=兰Sf ˆDdS is oriented into the positive z-direction; for attractive interactions, fwould be oriented into the negative z-direction. This can be seen as follows. For repulsive interactions, each of the forces f ˆDis oriented radially inwards; since the density of the product molecules is increasing toward the source located on the lower hemisphere, the magnitude and the projection onto the z-direction of the force due to the domain Dlocated at any 0⬍ ␪ ⬍ ␲ /2 is smaller than the one in the corresponding domain Dlocated at ␲ - ␪ . Thus the contribution into the positive z-direction from the lower hemisphere will dominate. For attractive interactions, the argument is simply reversed. This is in accordance with the following two statements regarding the characteristics of the far-field 共i.e., on length scales over which the particle plus surface film are seen as pointlike兲hydrodynamic flow: 共i兲In the absence of external body forces such as gravity or centrifugal forces the motion of the particle plus its surface film is net force-free. 共ii兲The generated flow corresponds, within a first approximation, to that produced by a “force dipole” 关which is the superposition of a distribution of force dipoles 共f ˆDacting on the center and −f ˆDon the product molecules in D兲as in Fig. 3in Appendix A兴, or a higher order “force multipole,” e.g., quadrupole 共if it happens that the net force dipole vanishes too兲15 at the origin, i.e., the position of the center of the particle rather than to the one due to a point force, which would be the case for an object uniformly dragged against the viscous Stokes friction 共see, e.g., Ref. 24兲. For an unbounded system, these forces translate into a radial decay of the phoretic flow field proportional to r−2 共force dipole兲or r−3 共force quadrupole兲, in contrast to the decay proportional to r−1 corresponding to a point force; for a bounded system, the differences between the flow fields cannot be any longer summarized by such a simple criterion as different power laws for the radial decay, but they remain significant nevertheless. These features of the hydrodynamic flow are discussed in more detail in Appendix B 共see also, c.f., Fig. 4兲. Using the expansion of the velocity and pressure fields in terms of the solid harmonics Kᐉ共see Appendix B兲, the hydrodynamic force on the particle, defined by Eq. 共7兲and expressed as on the left hand side of Eq. 共8兲, reduces to F=4 ␲ p ˜ −2 ⵜ关rP1共cos ␪ 兲兴,共9兲 where p ˜ −2 is the coefficient of K−2 in the expansion of the pressure18 关Eqs. 共B2兲and 共B3兲兴, and P1is the Legendre polynomial of order one. All other terms vanish since the corresponding integrals are exactly equal to zero. Thus the requirement of a vanishing Fimplies p ˜ −2=0, which leads to 共see Appendix B兲 V共t兲= ␹ 1共 ␩ 兲b R 冕 0 ␲ d ␪ sin ␪ cos ␪ ␳ 共R, ␪ ,t; ␩ 兲,共10兲 where ␹ 1共 ␩ 兲=1−5 2 ␩ 2−1 ␩ 5−1.共11兲 The structure of the expression on the right hand side 共rhs兲of Eq. 共10兲clarifies the meaning of the factor ␹ 1共 ␩ 兲, which varies between zero, at ␩ →1, and one, at ␩ →⬁共see Fig. 2兲. Without this factor, one has on the rhs the phoretic velocity in the unbounded space due to a source which generates a composition profile ␳ ⬁such that ␳ ⬁共R, ␪ ,t兲⬅ ␳ 共R, ␪ ,t; ␩ 兲at all times t, as derived in Ref. 5关with ⌳→−␭2entering into b; see the text around Eq. 共4兲兴. Thus ␹ 1共 ␩ 兲is a “hydrodynamic wall-correction” factor which quantifies and summarizes the effects solely due to the confinement induced changes in the characteristics of the solvent flow. Note that ␹ 1共 ␩ 兲⬍1 means that the hydrodynamic effects due to confinement will tend to decrease the velocity from the value corresponding to an unbounded system. D. Density profile of the product molecules and diffusiophoretic velocity According to Eq. 共10兲, knowledge of the density ␳ 共兩r兩 =R+,t; ␩ 兲of product molecules completely determines the 1 3 5 0 1 2 η χ,χ,χ 12 χ χ χ 2 1 FIG. 2. The long-time asymptotic velocity V共⬁兲共 ␩ 兲/V0⬅ ␹ 共 ␩ 兲⬎1共solid line兲as a function of the confinement ␩ 共see Fig. 1兲and the hydrodynamic and diffusion wall-correction factors ␹ 1共 ␩ 兲⬍1共dotted line兲and ␹ 2共 ␩ 兲⬎1 共dashed line兲. 194702-6 Popescu, Dietrich, and Oshanin J. Chem. Phys. 130, 194702 共2009兲 velocity Vof the particle as a function of ␩ ⬍⬁ for confined systems and thus allows one to quantify the effects of confinement on the resulting phoretic motion. Within the assumptions that the diffusion of product molecules is fast compared to the convection by the solvent flow, i.e., in the limit of small Peclet numbers, and that the product distribution ␳ 共r,t兲is undisturbed by the flow, i.e., neglecting any so-called polarization effects of the surface film,12 the time evolution of the number density ␳ 共r,t兲of product molecules around the moving particle is governed, in the comoving frame, by the diffusion equation ⳵ t ␳ =Dⵜ2 ␳ +B共t兲 ␦ 共r−rs兲,R+⬍兩r兩⬍R1.共12兲 This equation is to be solved subject to the initial condition 共IC兲of zero density of product molecules and to the BCs of zero normal current at the confining wall 共assuming that the latter is comoved, without perturbing the solution, such that at all times it remains concentric with the particle兲and at the outer edge r=R+of the surface film. Actually, the latter condition should be imposed at the surface of the particle, but we shall consider only the case of a thin surface film so that due to its small volume the transport of solute to the surface film is negligible compared to the transport in the outer region and the zero normal current condition at R+is a good approximation. Hence, the IC and BCs are as follows: ␳ 共r,0兲=0, 冏 冉 ⳵␳ 共r,t兲 ⳵ r 冊 冏兩r兩=R+,R1 =0. 共13兲 We remark that the linearity of Eqs. 共12兲and 共10兲with respect to ␳ implies that by superposition the solution of the present problem can be extended to the case of an arbitrary spatial arrangement of several catalytic sites on the particle surface. This will allow one to find an optimal decoration for providing stability against particle rotations 共caused by thermal noise兲, which can otherwise spoil the unidirectional motion 共see Ref. 8for such examples of designed optimal surface distributions兲. Equation 共12兲subject to the IC and BC conditions given by Eq. 共13兲is solved by using the Laplace transform. Similar to the approximation employed in Sec. III B, from this point further in the calculations, we replace R+by Reverywhere because the difference ␦ 共the thickness of the surface film兲 between the two is negligible on the macroscopic length scales characterizing the transport in the outer region. The solution is obtained as a series in products of Legendre polynomials and modified spherical Bessel functions of the first and third kind, respectively; the details are provided in Appendix C. We focus on the particular case in which after its start the activity of the catalytic site is time independent, i.e., B共t兲is constant: B共t兲=H共t兲/ ␶ f, where H共t兲is the Heaviside step function and ␶ fis the average production time for the creation of a product molecule. The Laplace transform of the integral in Eq. 共10兲is computed by using the Laplace transformed density ␳ 共r, ␨ ; ␩ 兲and the fact that cos ␪ =P1共cos ␪ 兲. Formally inverting the Laplace transform, one obtains the diffusiophoretic velocity as a function of time and confinement, V 冉 s=Dt R2; ␩ 冊 =b ␹ 1共 ␩ 兲 ␲ 2R2D ␶ f 兩共L−1关⌽ ¯ 共 ␰ ; ␩ 兲兴兲兩s,共14兲 where ⌽ ¯ 共 ␰ ; ␩ 兲⬅i3/2共冑 ␰ 兲 冑 ␰ 关 ␣ ˆ1i3/2共冑 ␰ 兲+共 ␤ ˆ1−1兲k3/2共冑 ␰ 兲兴,共15兲 iᐉ+1/2共z兲=冑 ␲ /共2z兲Iᐉ+1/2共z兲and kᐉ+1/2共z兲=冑 ␲ /共2z兲Kᐉ+1/2共z兲 are modified spherical Bessel functions of the first and third kind, respectively,25 and the dimensionless coefficients ␣ ˆ1共冑 ␰ , ␩ 兲and ␤ ˆ1共冑 ␰ , ␩ 兲are fixed by imposing the BCs 关see Eq. 共C8兲and Appendix C with ␰ = ␨ R2/Dso that ␰ ,⌽ ¯ , and L−1关⌽ ¯ 兴are dimensionless兴. IV. DISCUSSION The complicated structure of the function ⌽ ¯ 共 ␨ ; ␩ 兲makes it rather laborious to carry out the inverse Laplace transform so that the full time dependence of the velocity V共t兲cannot be derived easily. However, the asymptotic 共s=tD/R2Ⰷ1兲 value of the velocity V共⬁兲共 ␩ 兲ªlims→⬁V共s; ␩ 兲can be straightforwardly derived by using the inversion formula for the Laplace transform26 by noticing that ⌽ ¯ 共 ␰ ; ␩ 兲has a simple pole at ␰ =0, which determines the asymptotic value of the velocity as the residue of ⌽ ¯ 共 ␰ ; ␩ 兲at ␰ =0.26 The existence of such a constant asymptotic value, which at first glance seems to be in conflict with the fact that the density ␳ 共r;t兲does not reach a steady state 共since we consider a closed system with a source continuously producing noninteracting pointlike particles兲is due to the fact that the phoretic motion is determined solely by the gradient of the number density of the product molecules along the surface of the particle. At long times, the total density of product molecules is large and any redistribution 共which would lead to a change of the density gradient兲proceeds on very slow time scales. 共Ultimately the density of product molecules becomes so high that they no longer behave as an ideal gas and the latter assumption breaks down.兲This yields V共⬁兲共 ␩ 兲=b ␹ 1共 ␩ 兲 ␲ 2R2D ␶ f Res ␰ =0关⌽ ¯ 共 ␰ ; ␩ 兲兴 =V0 ␹ 共 ␩ 兲,共16兲 where V0=−b/4 ␲ R2D ␶ f=V共⬁兲共 ␩ →⬁兲is the asymptotic velocity in the case of an unbounded system and 关see Eq. 共11兲兴 ␹ 共 ␩ 兲= ␹ 1共 ␩ 兲 ␩ 3+2 ␩ 3−1ª ␹ 1共 ␩ 兲 ␹ 2共 ␩ 兲,∀ ␩ ⬎1. 共17兲 Thus ␹ 2共 ␩ 兲=共 ␩ 3+2兲/共 ␩ 3−1兲is a “diffusion wall-correction” factor, and ␹ 共 ␩ 兲quantifies the combined wall effects, which in the present case factorize into a hydrodynamic and a diffusion contribution. These two contributions oppose each other such that the confinement in hydrodynamics decreases the particle velocity 共 ␹ 1⬍1兲, whereas the confinement of the diffusion enhances it 共 ␹ 2⬎1兲共see Fig. 2兲; the latter dominates so that ␹ ⬎1 and there is an overall enhancement of the velocity 共see Fig. 2兲. Assuming the Stokes–Einstein relation for the diffusion coefficient of the product molecules in the solvent 共see 194702-7 Confined diffusiophoretic self-propellers J. Chem. Phys. 130, 194702 共2009兲 Sec. I兲and using the expression Eq. 共3兲for the mobility b, the velocity V0关Eq. 共16兲兴can be rewritten as V0=−3 4 ⌳ 兩⌳兩 冉 ␭ R 冊 2a ␶ f .共18兲 Surprisingly, this expression has no explicit dependence on the viscosity of the solvent or on the temperature. In fact, the dependence on viscosity drops out upon using the Stokes– Einstein relation. In turn, the dependence on Tis hidden in the temperature dependence of the 共effective兲interaction between the particle and the product molecules which determines the thickness of the surface film, i.e., the length scale ␭关see Eq. 共4兲兴. In the case of only steric repulsion between the particle and the product molecules, for a particle of radius R=1 ␮ m关100 nm兴, a product molecule with diameter a=1.0 nm, ␭=1.0 nm, and a reaction rate 共following Ref. 5兲 of 1/ ␶ f=25 kHz, one finds the velocity V0⯝10 pm/s 关1 nm/s兴; consequently V共⬁兲共 ␩ 兲is of the same order of magnitude. Note that these values are much smaller than the estimate V0=O共 ␮ m/s兲of Ref. 5, which is based on a significantly larger value ␭2⯝10−15 m2than the one ␭2 ⯝10−18 m2corresponding to the reasonable estimate a =1 nm for the diameter of the product molecule. However, this estimate corresponds to the case of a single catalytic site. One can argue that several such single reaction sites distributed closely around rsmay lead to a significant increase in the velocity 共intuitively, by a factor equal to the surface density of such reaction sites兲. Thus the resulting velocities may eventually be closer to the experimentally reported values2,6 V0=O共 ␮ m/s兲for objects with a ␮ m2area covered by catalyst if the surface density of reaction sites is of the order of nm−2. Moreover, if the effective interaction between the particle and the product molecules is attractive, the ratio 共␭/a兲2 can be very large.12 Another possibility leading to a significant increase is that of a classical hydrodynamic slip BC with a large slip length at the surface of the particle for the flow within the surface film, assuming that 共see Appendix A兲 classical hydrodynamics provides an accurate description for the solvent flow even at such small scales.17 关According to the text following Eq. 共A5兲, for the derivation of the effective mobility bin the phoretic slip velocity 关Eqs. 共1兲,共3兲, and 共A6兲兴, we have used a no-slip BC for the flow in the surface film at the particle surface. This condition can be generalized to a slip condition characterized by a slip length.兴Such hydrodynamic slip may be, e.g., due to roughness of the particle surface, and its effect on the phoretic motion can be intuitively understood as follows. The slip over the surface of the particle facilitates the solvent flow within the surface film as compared to the case of zero slip 共a “lubrication” effect兲, and thus for the same pressure gradient along the surface as in the case of zero slip the flow velocity in the surface film will be increased for nonzero slip lengths. The phoretic slip vs, which is the flow velocity at the outer boundary of the surface film, is thus also increased compared to its value in the case studied here so far that there is a no-slip condition at the surface of the particle; this leads to an increase 共by up to orders of magnitude17兲of the phoretic velocity of the particle. The velocity V共⬁兲共 ␩ 兲of the particle can be positive or negative, depending on the details of the interaction potential. This means that the particle can travel either following the gradient of solute density, or against it. According to Eqs. 共16兲and 共18兲, as well as the definition of the effective mobility b, the sign of the phoretic velocity is opposite to the sign of the parameter b共or, equivalently, that of the parameter ⌳兲, which in turn depends, inter alia, on the attractive or repulsive character of the effective interactions between the particle and the solute molecules 关see Eq. 共4兲兴. Equations 共1兲 and 共2兲show that the sign of the phoretic slip velocity, i.e., of the polar component vs·e ␪ is also opposite to the one of b, because for our system ⳵ ␪ ␳ is positive 共see Fig. 1兲.Inthe case of hard-core interactions only one has ⌳⬍0 and thus b⬍0; this implies V共⬁兲共 ␩ 兲⬎0, i.e., the particle moves in the direction of ezand thus away from the catalytic site, while the slip velocity points in the same direction as e ␪ , which has a negative z-component so that the flow around the particle is also oriented toward the catalytic site.12 The confinement does not change the direction of V, which for an unbounded system is, as expected, in agreement with Ref. 5. A simple intuitive explanation for the direction of the motion of the particle follows from the observation that there is no net force acting on the composite consisting of the particle plus the surface film so that there is conservation of momentum. Because the solvent flow around the particle is in the negative z-direction 关see Fig. 4共a兲兴, the particle should move in the positive z-direction. Note that while this simple argument is clear in the unbounded case, for the confined system, it breaks down because the formation of vortices 关see Fig. 4共c兲兴 implies that there are spatial regions where the flow is in the positive z-direction. Thus a quantitative analysis is required. If b⬎0, on the contrary, V共⬁兲共 ␩ 兲⬍0, which means that the particle will move toward the source of the product molecules. This is in contrast to the results presented in Ref. 22, which predict motion always away from the source. As discussed before the reason for this discrepancy is the ignorance in Ref. 22 of the mediating role of the solvent and its assumption of a mapping of the nonuniform density of the solute onto a nonuniform osmotic pressure acting on the particle 共i.e., assuming that more product molecules impinge on the particle from the side with higher density兲, rather than mapping onto gradients in the solution pressure along the surface of the particle within its surface film. This opposing directionality in the case b⬎0 thus provides a simple criterion for an experimental discrimination between the predictions of the two proposals. Spatial confinement leads to an overall enhancement of the phoretic motion, because 兩V0兩⬍兩V共⬁兲共 ␩ 兲兩 for all finite ␩ . As shown in Fig. 2, even at moderate values of ␩ the effect of the confinement is important, e.g., at ␩ ⯝2 there is a ca 8%increase in the velocity compared to the unbounded case, and at ␩ ⯝1.5 the increase reaches ca 25%. For ␩ →1 the velocity stays finite. This is, of course, an artifact which stems from the assumption that the product molecules are pointlike. Actually, there is a lower cutoff R1 c, where R1 c−Ris of the order of the hard-core diameter aof the product molecules, below which this assumption breaks down and Eq. 共16兲is no longer valid. However, for aⰆR, which is a rea- 194702-8 Popescu, Dietrich, and Oshanin J. Chem. Phys. 130, 194702 共2009兲 motion are closing back and then “slide” along the surface of the particle. Since closed-form analytical expressions are available in these unbounded cases,12,18 the qualitative differences noticed above can be correlated with the different characteristics of the flows. In the case of phoretic motion the flow field decays as r−3 at large distances rfrom the particle, which is a “multipole” far-field corresponding to the action of a force-quadrupole disturbance, in contrast to the r−1 decay in the osmotic case, which is a monopole far-field corresponding to the action of a point force 共“Stokeslet”兲disturbance. The faster decay of the velocity field in the first case means that a boundary at R1will perturb the phoretic flow much less. This explains why for the phoretic motion the viscous friction increases with the confinement weaker than what would be expected from naively using a Stokes-friction concept. The flow fields remain qualitatively different even in the presence of the confinement 共in particular, with respect to the opposite direction of flow along the z-axis兲, although the differences are not as pronounced as in the unbounded case. A further qualitative difference between the two confined flows is that in 共d兲a vortex structure is formed fully detached from the particle whereas in 共c兲the vortex involves flow along the surface of the particle. Due to the presence of the boundary at R1, all the terms ⬃rᐉin the expansions of Eq. 共B15兲are present in the solution and an analysis of the flow in terms of fundamental solutions, as for the unbounded case, is no longer possible. Note that since ⵜ⫻uhom=ⵜ⌼and ⌼ =0 关see the text following Eqs. 共B7兲and 共B19f兲兴one has ⵜ⫻u=ⵜ⫻upso that with Eqs. 共B5兲and 共B6兲one obtains ⵜ⫻u=1 ␮ e ␾ 兺 ᐉ苸Z p ˜ ᐉ ᐉ+1 ⳵ Kᐉ共r, ␪ 兲 ⳵ ␪ ,共B22兲 which is in general nonzero 关e.g., for osmotic flow p ˜ −2⫽0 共Eq. 共9兲兲兴. Thus in agreement with Fig. 4the flow field can contain vortices. APPENDIX C: CALCULATION OF THE NUMBER DENSITY OF PRODUCT MOLECULES Equation 共12兲, subject of the IC and BC conditions given in Eq. 共13兲, is solved by using the Laplace transform and the inversion theorem26 f ¯ 共 ␨ 兲⬅L关f兴= 冕 0 ⬁ dte− ␨ tf共t兲共C1兲 and f共t兲⬅L−1关f ¯ 共 ␨ 兲兴 ª1 2 ␲ i 冕 ␥ −i⬁ ␥ +i⬁ d ␨ e ␨ tf ¯ 共 ␨ 兲,共C2兲 respectively, where f ¯ 共 ␨ 兲共the Laplace transformed quantities are indicated by an overbar兲is assumed to be well defined for ␨ 苸R+and ␥ 苸Ris sufficiently large such that all singularities of f ¯ 共 ␨ 兲lie to the left of the integration path. By taking the Laplace transform of Eqs. 共12兲and 共13兲and by using the IC condition of zero number density of the product molecules, one finds that the Laplace transformed density is given by ␳ ¯ 共r, ␨ 兲=−a ¯ 共 ␨ 兲G ¯ 共r,q兲,共C3兲 where a ¯ 共 ␨ 兲=B ¯ 共 ␨ 兲/Dand G ¯ 共r,q兲is Green’s function for the Helmholtz operator ⵜ2−q2,q=冑 ␨ /D⬎0, satisfying the BCs of vanishing normal derivative at 兩r兩=R,R1. Decomposing G ¯ 共r,q兲as G ¯ 共r,q兲=Gs共r,q;rs兲+g共r,q兲, where the singular part Gsis the free space Green’s function for the Helmholtz operator 共which is known in any spatial dimension D, see, e.g., Ref. 28兲, the initial problem is reduced to that of finding the solution gof the homogeneous Helmholtz equation subject to the BCs 冏 冉 ⳵ g共r,q兲 ⳵ r 冊 冏兩r兩=R,R1 =−冏 冉 ⳵ Gs共r,q;rs兲 ⳵ r 冊 冏兩r兩=R,R1 .共C4兲 In 3D the singular part of Green’s function is given by28 Gs共r,q;rs兲=− q 4 ␲ e−q兩r−rs兩 q兩r−rs兩,共C5兲 while the regular part 共i.e., the general solution of the homogeneous Helmholtz equation兲can be written as g共r, ␪ ,q兲=兺 ᐉⱖ0 关 ␣ ᐉiᐉ+1/2共qr兲+ ␤ ᐉkᐉ+1/2共qr兲兴Pᐉ共cos ␪ 兲, 共C6兲 where iᐉ+1/2共z兲=冑 ␲ /共2z兲Iᐉ+1/2共z兲and kᐉ+1/2共z兲 =冑 ␲ /共2z兲Kᐉ+1/2共z兲are the modified spherical Bessel functions of the first and third kind, respectively,25 Pᐉis the Legendre polynomial of degree ᐉ, while the coefficients ␣ ᐉ and ␤ ᐉwill be fixed to fulfill the BCs. Using one of the addition theorems for the Bessel functions26 and noting that the angle between rand rsis ␲ − ␪ 共see Fig. 1兲,Gscan be rewritten as Gs共r,q;rs兲=− 兺 ᐉⱖ0 cᐉ共q兲iᐉ+1/2共qr⬍兲kᐉ+1/2共qr⬎兲Pᐉ共cos ␪ 兲, 共C7兲 where cᐉ共q兲=共−1兲ᐉ共2ᐉ+1兲q/共2 ␲ 2兲,r⬍=min共r,rs兲, and r⬎ =max共r,rs兲. By combining Eqs. 共C6兲and 共C7兲and the BCs 关Eq. 共C4兲兴, by noting that rs=R+ ⑀ , ⑀ 0共i.e., the source of particles is on the surface of the particle兲, and by rewriting the coefficients as ␣ ᐉ⬅cᐉiᐉ+1/2共qR兲 ␣ ˜ ᐉand ␤ ᐉ ⬅cᐉiᐉ+1/2共qR兲 ␤ ˜ ᐉ, the dimensionless coefficients ␣ ˜ ᐉ共qR兲and ␤ ˜ ᐉ共qR兲are determined by the solution of the following closed system of two linear equations with two unknowns 共 ␣ ˜ ᐉand ␤ ˜ ᐉ兲: ␣ ˜ ᐉ+ ␤ ˜ ᐉwᐉ+1/2共qR兲=vᐉ+1/2共qR兲,共C8a兲 ␣ ˜ ᐉ+ ␤ ˜ ᐉwᐉ+1/2共 ␩ qR兲=wᐉ+1/2共 ␩ qR兲,共C8b兲 where vᐉ+1/2共z兲ªkᐉ+1/2共z兲/iᐉ+1/2共z兲and wᐉ+1/2共z兲 ª关dkᐉ+1/2共z兲/dz兴/关diᐉ+1/2共z兲/dz兴. The solution ␣ ˆᐉand ␤ ˆᐉacquires a dependence on ␩ via Eq. 共C8b兲: ␣ ˆᐉ= ␣ ˆᐉ共qR, ␩ 兲and ␤ ˆᐉ= ␤ ˆᐉ共qR, ␩ 兲, with qR=R冑 ␨ /D. This determines Green’s function G ¯ 共r,q兲and thus the Laplace transformed density ␳ ¯ 共r, ␨ 兲关Eq. 共C3兲兴from which, in principle, the density ␳ ¯ 共r,t兲 is obtained via the inverse Laplace transformation. Note that 194702-15 Confined diffusiophoretic self-propellers J. Chem. Phys. 130, 194702 共2009兲 although the final result is in the form of a series, the quantities in which we are interested, in particular, the phoretic velocity V共t兲关Eq. 共10兲兴, involve integrals over the polar angle ␪ of the product between this series and a specific ᐉ Legendre polynomial Pᐉ共cos ␪ 兲关ᐉ=1 in the case of the velocity, ᐉ=0 in the case of the total number of product molecules 共see below兲兴 and thus only one of the terms from the series will contribute. We make the following three remarks: 共i兲Since for qR⬎0 one has lim ␩ →⬁wᐉ+1/2共 ␩ qR兲=0, in the limit ␩ →⬁one finds ␣ ¯ ᐉ=0 and Eq. 共C8a兲reduces to the corresponding BC in Ref. 5, i.e., as expected the solution for the unbounded case is recovered. 共ii兲Eqs. 共C8a兲and 共C8b兲will not coincide in the limit ␩ →1 because even in this limit of extreme confinement a source singularity is present at rsand is picked up by Gs共q兲关Eq. 共C7兲兴. 共iii兲The Laplace transform N ¯ 共p兲of the total number of product particles in the system at time tis N ¯ 共p=Dq2兲 =2 ␲ 冕 R R1 drr2 冕 0 ␲ d ␪ sin ␪ P0共cos ␪ 兲关−a ¯ 共p兲兴G ¯ 共q兲 =−4 ␲ a ¯ 共p兲c0共q兲i1/2共qR兲 ⫻ 冕 R R1 drr2关 ␣ ¯ 0i1/2共qr兲+共 ␤ ¯ 0−1兲k1/2共qr兲兴 =Da ¯ 共p兲/p⇒N共t兲= 冕 0 t dt⬘A共t⬘兲,共C9兲 i.e., as expected N共t兲is given by the time integral of the production rate, providing a welcome consistency check. 1W. F. Paxton, A. Sen, and T. E. Mallouk, Chem.-Eur. J. 11, 6462 共2005兲. 2W. F. Paxton, S. Sundararajan, T. E. Mallouk, and A. Sen, Angew. Chem., Int. 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