Theory of capillary-induced interactions beyond the superposition approximation

J. Chem. Phys. 127, 204706 (2007); https://doi.org/10.1063/1.2781420 127, 204706 © 2007 American Institute of Physics. Theory of capillary-induced interactions beyond the superposition approximation Cite as: J. Chem. Phys. 127, 204706 (2007); https://doi.org/10.1063/1.2781420 Submitted: 01 June 2007 . Accepted: 16 August 2007 . Published Online: 27 November 2007 Alvaro Domínguez, Martin Oettel, and S. Dietrich ARTICLES YOU MAY BE INTERESTED IN Force balance of particles trapped at fluid interfaces The Journal of Chemical Physics 128, 114904 (2008); https://doi.org/10.1063/1.2890035 The “Cheerios effect” American Journal of Physics 73, 817 (2005); https://doi.org/10.1119/1.1898523 The equilibrium shape of fluid-fluid interfaces: Derivation and a new numerical method for Young’s and Young-Laplace equations The Journal of Chemical Physics 141, 244702 (2014); https://doi.org/10.1063/1.4904391 Theory of capillary-induced interactions beyond the superposition approximation Alvaro Domínguez Física Teórica, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain Martin Oettel Institut für Physik, Universität Mainz, WA 331, D-55099 Mainz, Germany S. Dietrich Max-Planck-Institut für Metallforschung, Heisenbergstrasse 3, D-70569 Stuttgart, Germany and Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany 共Received 1 June 2007; accepted 16 August 2007; published online 27 November 2007兲 Within a general theoretical framework, we study the effective, deformation-induced interaction between two colloidal particles trapped at a fluid interface in the regime of small deformations. In many studies, this interaction has been computed with the ansatz that the actual interface configuration for the pair is given by the linear superposition of the interface deformations around the single particles. Here, we assess the validity of this approach and compute the leading term of the effective interaction for a large interparticle separation beyond this so-called superposition approximation. As an application, we consider the experimentally relevant case of interface deformations owing to the electrostatic field emanating from charged colloidal particles. In mechanical isolation, i.e., if the net force acting on the total system consisting of the particles plus the interface vanishes, the superposition approximation is actually invalid. The effective capillary interaction is governed by contributions beyond this approximation and turns out to be attractive. For sufficiently small surface charges on the colloids, such that linearization is strictly valid, and at asymptotically large separations, the effective interaction does not overcome the direct electrostatic repulsion between the colloidal particles. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2781420兴 I. INTRODUCTION The self-assembly of submicrometer colloidal particles at fluid 共e.g., water/air or water/oil兲interfaces has gained significant interest in view of various basic and applied issues such as the study of two-dimensional melting,1investi- gations of mesoscale structure formation,2and engineering of colloidal crystals on spherical surfaces.3The colloidal particles are trapped at the interface if the fluid phases wet the colloid only partially; this configuration is stable against thermal fluctuations, and it appears to be even the global equilibrium state, in accordance with the experimental observation that the colloids immersed in the bulk phases are attracted toward the interface.1 In order to prevent coagulation, the colloidal particles are electrically charged. The ensuing repulsive force is well understood, and at large separations dit varies like a dipoledipole interaction ⬀1/d4because the monopoles vanish due to screening by counterions in water.4,5Nevertheless, several experimental findings have led to postulating an attractive effective force between such particles with a range much larger than that of van der Waals forces6–15 共but see also Ref. 16兲. Spherical particles 共radii R=0.25–2.5 ␮ m兲at flat waterair interfaces exhibit the spontaneous formation of complicated metastable mesostructures consistent with the presence of a minimum in the effective intercolloidal potential at separations d/R⬇3–20 and with a depth of at least a few kBT. Until now, no unequivocal explanation for the appearance of these relatively long-ranged attractions is available. One possibility, which has been explored intensively in previous years, consists of an attraction mediated by the deformation of the interface 共see Fig. 1兲. This is similar to the so-called flotation force attracting particles floating at the surface of water, which is deformed by the weight of the particles.17,18 However, gravity plays no role for micrometer sized particles as described above. Instead, the electrostatic field around the charged colloids deforms the interface and gives rise to effective, capillary-induced interactions. There have been contradictory results about the properties of this effective interaction. In Refs. 12 and 19, it has been argued FIG. 1. 共Color online兲Schematic drawing of the deformation of a fluid interface by electric fields due to charged colloidal particles trapped at the interface. Counterions gather on the side of the electrolytic phase 共water兲, and a pressure field arises, which pulls on the interface and on the particles. THE JOURNAL OF CHEMICAL PHYSICS 127, 204706 共2007兲 0021-9606/2007/127共20兲/204706/17/$23.00 © 2007 American Institute of Physics127, 204706-1 in favor of an attractive force decaying like 1/d共similar to the gravity-induced flotation force兲. In Refs. 20–24, this line of thought was shown to be invalid because the decay turns out to be much faster if the physical system consisting of the colloidal particles plus the interface is mechanically isolated, i.e., if the total force acting on this system vanishes in the limit of a macroscopically extended interface with negligible borders. This is, in principle, the case for the experiments conducted in Langmuir troughs with lateral extensions several times the capillary length of the interface 共i.e., orders of magnitude larger than the colloid radius兲. In this case, the electrostatic force pulling on the interface is counterbalanced exactly by the electrostatic force pushing the particles into water. In particular, the authors of Ref. 20 argue for an attractive force decaying like 1/d7. However, this was, in turn, corrected in Refs. 21 and 22, and a repulsive force with an asymptotic decay ⬀1/d7was derived. In Ref. 22, it was noticed that this latter result is actually unreliable because the linear superposition approximation employed in these calculations is not valid if mechanical isolation holds. Furthermore, since it is the electric field E, and not the electrostatic pressure ⬀E2, which obeys a superposition principle, the superposition approximation will be unsuitable if the main contribution to the electrostatic pressure stems from cross terms in E2. This latter case was studied in Refs. 25 and 26, with the conclusion that the capillary-induced force is attractive and decays like 1/d4, which is the same asymptotic behavior as the direct dipole-dipole repulsion; whether the capillary attraction overcomes the electric repulsion must then be determined by a detailed analysis of the electrostatic problem. The total force is repulsive in the regime of small deformations of a flat interface 共equivalent to the regime of small colloidal charges兲, for which the calculations were carried out.25,26 Only for sufficiently large colloidal charges, the capillary attraction may asymptotically overcome the electrostatic repulsion, leading to a minimum in the total effective potential.25 However, further calculations—going beyond the linearization assumptions in treating the electrostatic pressure on the interface and the energy of the deformed interface—are necessary to substantiate this claim. The experiment described in Ref. 12 is peculiar in the sense that the unperturbed interface is actually that of a relatively small spherical droplet pending from a plate. The importance of this finite-size effect was studied in Refs. 22,23, and 27–29. In Refs. 23,27, and 29, it was found that the flotationlike decay 1/dis present because the plate breaks the condition of mechanical isolation, but it is quantitatively too small to explain the experimental observations.30 Here, we follow the approach of Ref. 22 in order to calculate the capillary-induced effective interaction beyond the superposition approximation in the case that there is a pressure field of the general form ⌸ ˆ共r兲acting on the interface in the limit of small deformations. The limit of small deformations corresponds to an analysis to leading order in the small dimensionless parameters ␧ ˆFand ␧ ˆ⌸, defined in Eq. 共1兲below, which are measures of the force acting on the colloidal particles and the interface, respectively. In this manner, inter alia, we provide a mathematically sound derivation of the results reported in Refs. 25 and 26. In Sec. II, we derive the exact leading-order expressions for the capillary potential and show in which respect the superposition approximation becomes inconsistent if mechanical isolation holds. In such a case, the final result depends on the stage at which the approximation is introduced and applied. In Sec. II B, we derive the leading asymptotic dependence on dof the effective interaction using only some rather general assumptions on the form of ⌸ ˆ共r兲. The result for the effective potential energy is summarized in Eq. 共27兲. In Sec. III, we analyze the particular case that the pressure field ⌸ ˆ共r兲is due to the electric field created by charged colloidal particles in a mechanically isolated system. We consider two limiting cases in order to solve the electrostatic problem: 共i兲water as a perfect conductor 共i.e., vanishing Debye length兲and 共ii兲the colloidal particles as pointlike objects 共i.e., vanishing particle radius兲. In both cases, we recover the conclusion that the effective interaction is attractive but can overcome the electric repulsion only if ␧ ˆFⲏ1, which is outside the smalldeformation regime considered here. Finally, in Sec. IV, we summarize our results and discuss their relevance in connection with the experiments described in the literature. II. FREE ENERGY OF EFFECTIVE CAPILLARY INTERACTION A. Exact results We consider Nidentical31 spherical particles trapped at a fluid interface 共see Fig. 2for N=2兲. We define the reference configuration as a flat interface in the plane z=0 and the colloids at a height such that the colloid-interface contact occurs at Young’s contact angle ␪ 苸共0, ␲ 兲. In this configuration, S ␣ denotes the circular disk delimited by the contact line on the colloid ␣ , ⳵ S ␣ is the corresponding contact line 共of radius r0=Rsin ␪ for a particle of radius R兲traced counterclockwise when viewed from the top,32 and S ˆmen is the fluid interface 共with surface tension ␥ 兲, enclosed by a boundary CLof typical size of the order of Lrepresenting, e.g., the vessel containing the system. The relative lateral positions of the colloids are kept fixed, and thus we consider only vertical displacements u ˆ共r=共x,y兲兲 of the fluid interface and of the height ⌬h ˆ ␣ of the center of the colloid ␣ relative to the plane z=0. In the reference configuration, there is a 共vertical兲force FIG. 2. Top view 共plane z=0兲of the reference configuration with two colloids. dis the fixed lateral distance between the colloid centers projected onto the plane z=0. S1and S2are disks of radius r0, the corresponding circumferences 共counterclockwise兲are ⳵ S1and ⳵ S2.The共projected兲interface is S ˆmen=R2\共S1艛S2兲. The position of any point on the plane is denoted with r; in particular, r ␣ is the position of the colloid ␣ . 204706-2 Domínguez, Oettel, and Dietrich J. Chem. Phys. 127, 204706 共2007兲 F ˆ ␣ acting on the colloid ␣ and a 共vertical兲force per unit area ⌸ ˆ共r兲on the meniscus. We define the dimensionless forces ␧ ˆF ␣ ª−F ˆ ␣ 2 ␲ ␥ r0 ,␧ ˆ⌸ª 1 2 ␲ ␥ r0 冕 S ˆmen dA⌸ ˆ,共1兲 where dA is the element of interface area. The reference configuration is the equilibrium state in the absence of forces, F ˆ ␣ ⬅0,⌸ ˆ⬅0. Within the approximation of small deviations from the reference configuration, the free energy of the system with respect to this configuration is22 F ˆ= ␥ 冕 S ˆmen dA 冋 1 2兩ⵜu ˆ兩2−1 ␥ ⌸ ˆu ˆ 册 +兺 ␣ =1 N 再 ␥ 2r0 冖 ⳵ S ␣ dᐉ关⌬h ˆ ␣ −u ˆ兴2−F ˆ ␣ ⌬h ˆ ␣ 冎 +O共␧ ˆF,␧ ˆ⌸兲3, 共2兲 where dᐉis the element of arclength. The free energy contains a contribution from the change of the contact area between the phases 共two fluid phases and the solid particles兲 and a contribution from the work done by the forces F ˆ ␣ and ⌸ ˆvia displacements from the reference configuration. The values of u ˆ共r兲and ⌬h ˆ ␣ in the equilibrium state are determined by minimizing this free energy. This leads to the following equations: ⌬h ˆ ␣ =具u ˆ典 ␣ −r0␧ ˆF ␣ ,具·典 ␣ ª 1 2 ␲ r0 冖 ⳵ S ␣ dᐉ共·兲,共3a兲 ⵜ2u ˆ=−1 ␥ ⌸ ˆ,r苸S ˆmen,共3b兲 n ␣ ·ⵜu ˆ共r兲=␧ ˆF ␣ +u ˆ共r兲−具u ˆ典 ␣ r0 ,r苸 ⳵ S ␣ ,共3c兲 u ˆ共r兲=0, r苸CL,共3d兲 where n ␣ is the unit vector in the outward normal direction of ⳵ S ␣ . Equation 共3a兲is a geometrical relationship, Eq. 共3b兲 describes a local mechanical equilibrium 共the pressure ⌸ ˆis compensated by the curvature-induced interfacial tension兲, Eq. 共3c兲describes the mechanical equilibrium of the particle 共the force F ˆ ␣ is balanced by the interfacial tension exerted at the contact line兲, and Eq. 共3d兲represents a boundary condition at the external border CL共for simplicity we take a pinned interface, but, of course, other physically reasonable boundary conditions are possible, the details of which are actually irrelevant in the limit L→⬁we shall consider22兲. The free energy functional in Eq. 共2兲evaluated at the equilibrium configuration can be simplified by using the relationships in Eqs. 共3a兲–共3d兲, and we obtain two equivalent expressions: F ˆeq =−1 2 ␥ 冕 S ˆmen dA兩ⵜu ˆ兩2+ ␲ ␥ 兺 ␣ =1 N 关具u ˆ典 ␣ 2−具u ˆ2典 ␣ −r0 2␧ ˆF ␣ 2兴 共4a兲 =− 1 2 冕 S ˆmen dA⌸ ˆu ˆ+ ␲ ␥ 兺 ␣ =1 N r0␧ ˆF ␣ 关具u ˆ典 ␣ −r0␧ ˆF ␣ 兴.共4b兲 From this point onward, we shall consider the particular case of two identical colloids. We assume that any external force acting on the system 共e.g., gravity兲is independent of the positions of the particles,33 so that symmetry arguments will allow one to simplify the expressions. We use the notations u ␣ 共r兲ªu共兩r−r ␣ 兩兲,⌸ ␣ 共r兲ª⌸共兩r−r ␣ 兩兲,␧Fª␧ ˆF ␣ , and Smen, ␣ ªR2\S ␣ for the corresponding quantities in the presence of a single colloid located at position r ␣ . This means that the function u ␣ 共r兲共 ␣ =1,2兲satisfies ⵜ2u ␣ =−1 ␥ ⌸ ␣ ,r苸Smen, ␣ ,共5a兲 n ␣ ·ⵜu ␣ 共r兲=␧F,r苸 ⳵ S ␣ ,共5b兲 u ␣ 共r兲=0, r苸CL.共5c兲 In terms of these single-colloid solutions, the configuration in the presence of two colloids can be written without loss of generality as ⌸ ˆ=⌸1+⌸2+2⌸m,共6a兲 u ˆ=u1+u2+um,共6b兲 ␧ ˆF=␧F+␧m,共6c兲 where we have introduced ␧ ˆFª␧ ˆF1=␧ ˆF2, reflecting the symmetry of the problem. The fields um共r兲and ⌸m共r兲, and the quantity ␧mintroduced this way represent the corrections to the so-called superposition approximation, which is defined by setting um=0, 共7a兲 ⌸m=0, 共7b兲 ␧m=0, 共7c兲 i.e., the effects of other particles on the single-particle configuration are neglected altogether. From Eqs. 共3a兲–共3d兲and 共5a兲–共5c兲, one can derive the following equations linking um共r兲,⌸m共r兲, and ␧m共with ␣ , ␤ =1,2兲: ⵜ2um=−2 ␥ ⌸m,r苸S ˆmen,共8a兲 n ␣ ·ⵜum−um−具um典 ␣ r0 =␧m−n ␣ ·ⵜu ␤ +u ␤ −具u ␤ 典 ␣ r0 , r苸 ⳵ S ␣ 共 ␤ ⫽ ␣ 兲,共8b兲 um共r兲=0, r苸CL.共8c兲 The superposition approximation is violated even if ⌸m=0 and ␧m=0 because of the boundary conditions at the contact lines 关Eq. 共8b兲兴, as pointed out in Ref. 22. The case of nonvanishing ⌸mwas addressed in Refs. 25 and 26. 204706-3 Theory of capillary-induced interactions J. Chem. Phys. 127, 204706 共2007兲 The capillary-induced effective interaction energy is defined as Vmen共d兲ªF ˆeq−2Feq, where 2Feq is the sum of the equilibrium free energies of the single-colloid configurations, i.e., for d→⬁.Vmen共d兲depends parametrically on the 共lateral兲separation dof the colloid centers in the reference configuration. This is not the total interaction potential, which must include, e.g., the direct electrostatic repulsion between charged colloids, not considered in expression 共2兲for the free energy. The effective interaction energy can be written as Vmen=Vsup+Vcorr, where Vsup is the result of imposing the superposition approximation 关Eqs. 共7a兲–共7c兲兴and Vcorr is the correction to this approximation. If we use expression 共4a兲, we obtain aVsup =− ␥ 冕 S ˆmen dA共ⵜu1兲·共ⵜu2兲+ ␥ 冕 S1 dA兩ⵜu2兩2 −2 ␲ ␥ 具u2 2−具u2典1 2典1,共9a兲 aVcorr =−1 2 ␥ 冕 S ˆmen dA共ⵜum兲·ⵜ共um+4u2兲 −2 ␲ ␥ 具um共um+2u2兲−具um典1具um+2u2典1典1 −2 ␲ ␥ r0 2␧m共2␧F+␧m兲.共9b兲 On the other hand, using expression 共4b兲, we obtain bVsup =− 冕 S ˆmen dA⌸1u2+ 冕 S1 dA⌸2u2+2 ␲ ␥ r0␧F具u2典1, 共10a兲 bVcorr =− 冕 S ˆmen dA关⌸2um+2⌸mu2+⌸mum兴 +2 ␲ ␥ r0具共␧F+␧m兲um+␧m共u1+u2兲典1 −2 ␲ ␥ r0 2␧m共2␧F+␧m兲.共10b兲 We emphasize that the two alternative expressions of Vsup and Vcorr are not equivalent 共but their sum Vmen is兲, and they in turn differ from Vsup as computed in Ref. 22 关see Eq. 共39兲 therein兴, which was derived by inserting the superposition ansatz directly into Eq. 共2兲here. The application of Gauss’ theorem with Eq. 共5a兲leads to aVsup −bVsup =2 ␲ ␥ 冓 r0共u1+u2兲 ⳵ u2 ⳵ n1 −共u2−具u2典1兲2 冔 1 =bVcorr −aVcorr.共11兲 The superposition approximation is inconsistent asymptotically in cases in which aVsup−bVsup does not decay more rapidly than aVsup as a function of the separation d. As remarked in Ref. 22, there are indeed relevant cases in which this consistency condition is not fulfilled 共see next section兲. B. Effective potential in the intermediate asymptotic regime r0™d™L In this section, we compute Vmen共d兲asymptotically in the intermediate range r0ⰆdⰆL. For this purpose, we have to make some restricting assumptions which, however, seem to be satisfied in the experimental setups investigated so far. First, in view of the discussion in the Introduction concerning the electrical fields, we assume the proportionality ⌸m 2⬃兩⌸1⌸2兩.共12兲 This is valid if the interface stress is quadratic in a field satisfying linear superposition in the two-particle configuration. Examples are given below by some specific electrostatic models 关see Eqs. 共31兲and 共49兲兴. Second, we assume that the single-colloid pressure ⌸decays far from the colloid as ⌸共r兲⬃r−n,n⬎4. 共13兲 In the experimentally relevant case of charged particles at water interface, it has been established both theoretically4 and experimentally34 that n=6. 共This can be understood easily: the charge of a particle induces a screening image charge in the water, so that the distant electric field is dipolar.兲The constraint n⬎4 will allow us to estimate the integrals appearing in Eqs. 共9a兲,共9b兲,共10a兲, and 共10b兲by approximating them by the contribution of the regions near the colloids. This condition excludes, however, the effect of an external electric field 共in that case, the resulting pressure does not have to decay at all兲and the case that the colloidal charge is not perfectly screened so that the distant electric field corresponds to a monopole, i.e., n=4 共this would occur if both fluid phases are dielectric, e.g., air and insulating oil兲. The quantity ␧ ˆ⌸−2␧ ˆFis the net 共vertical兲force by an external agent acting on the total system consisting of two colloids plus the interface 共see Appendix A兲. This can be, e.g., gravity 共if the colloid is large enough for it to be quantitatively relevant兲, dispersion forces by a substrate closely beneath the interface 共this effect can be modeled similarly as gravity, see Appendix B兲, or an optical tweezer pushing the colloid vertically. Since we have assumed previously that this external force is independent of the positions of the particles, it is given by the sum of the net forces in the singleparticle configuration 共i.e., if they are infinitely far apart from each other兲: ␧ ˆ⌸−2␧ ˆF=2共␧⌸−␧F兲.共14兲 From the definitions in Eqs. 共1兲and 共6a兲, we can write ␧ ˆ⌸=2共␧⌸+␧⌸m−␧12兲,共15a兲 with ␧⌸m ª 1 2 ␲ ␥ r0 冕 S ˆmen dA⌸m,␧12 ª 1 2 ␲ ␥ r0 冕 S1 dA⌸2. 共15b兲 For d→⬁, it is clear that ␧12 ⬃1 dn.共16兲 In this limit, we note that ␧⌸mreceives its main contribution from the regions around S ␣ , implying 204706-4 Domínguez, Oettel, and Dietrich J. Chem. Phys. 127, 204706 共2007兲 ␧⌸m⬃2 ␥ r0 冕 r0 ⬁ drr⌸m共r兲⬃ 1 dn/2 冕 r0 ⬁ dr r rn/2,共17兲 provided n⬎4, because 兩⌸m共r兲兩⬃冑⌸共d兲⌸共兩r−r ␣ 兩兲 in those regions which provide the dominant contribution to the integral. Therefore, from Eqs. 共6c兲and 共14兲, one obtains asymptotically ␧m=␧⌸m−␧12 ⬃␧⌸m⬃1 dn/2.共18兲 With these simplifying assumptions, we shall compute analytically the behavior of Vmen共d→⬁兲to leading order in 1/d. More precisely, on dimensional grounds35 the expansion parameter is r0/d. In principle, ⌸ ˆcan contain and thus introduce additional length scales, for example, the Debye length if the fluid phase is an electrolyte. This complicates the problem, which then has to be analyzed numerically 共see Sec. III B兲. 1. The superposition approximation Within the superposition approximation, Vmen共d兲was computed in detail in Ref. 22. Here, we sketch briefly the estimate of the asymptotic behavior of bVsup 关compare the three terms in Eq. 共10a兲兴, 冕 S ˆmen dA⌸1u2⬃2 ␲ ␥ r0␧⌸u共d兲+⌸共d兲 ⫻ 冋 2 冕 Smen,2 dAu2−1 2 ␲ r0 3␧⌸+ ␲ r0 2具u2典2 册 , 共19a兲 because the main contribution stems from the regions around S ␣ , and 冕 S1 dA⌸2u2⬃⌸共d兲u共d兲 冕 S1 dA,共19b兲 2 ␲ ␥ r0␧F具u2典1⬃2 ␲ ␥ r0␧F 冋 u共d兲−1 4 ␥ r0 2⌸共d兲 册 共19c兲 after expanding around r2=d. Thus, bVsup共d兲⬃2 ␲ ␥ r0共␧F−␧⌸兲u共d兲−⌸共d兲 ⫻ 冋 2 冕 Smen,2 dAu2+ ␲ r0 2具u2典2 册 共19d兲 because ⌸共d兲is asymptotically subdominant compared with u共d兲关see Eqs. 共13兲and 共A1兲兴. The difference aVsup −bVsup in Eq. 共11兲between the two implementations of the superposition approximation can be estimated as22 关see Eq. 共A1兲兴 冓 共u1+u2兲 ⳵ u2 ⳵ n1 冔 1 ⬇1 2r0具u1典1ⵜ2u共d兲⬀⌸共d兲⬃ 1 dn,共20a兲 具共u2−具u2典1兲2典1⬇1 2r0 2兩ⵜu共d兲兩2 ⬃ 再 1/d2共n−1兲if ␧F−␧⌸=0 1/d2if ␧F−␧⌸⫽0. 冎 共20b兲 Thus, whenever ␧F−␧⌸⫽0共i.e., the system is not mechanically isolated in the sense that there must be a force acting on the boundary CLof the interface to compensate this nonvanishing net force兲, one finds bVsup共d兲⬃2 ␲ ␥ r0共␧F−␧⌸兲u共d兲 ⬃2 ␲ ␥ r0 2共␧F−␧⌸兲2ln d L+ const, 共21兲 corresponding to an attractive force irrespective of the precise form of the function ⌸共r兲.共The additive constant, which does not affect the physical conclusions, depends on the precise form of the boundary condition at CL.兲Physically, bVsup is the work done by the net force 2 ␲ ␥ 共␧F−␧⌸兲upon a vertical shift of the subsystem consisting of one colloid plus its surrounding interface 共behaving like an “effective particle”兲 by an amount u共d兲due to the deformation induced by the second colloid. In this case 共i.e., ␧F−␧⌸⫽0兲, the difference aVsup−bVsup decays more rapidly than bVsup共d兲, and both expressions aVsup and bVsup agree asymptotically. Equation 共21兲 exhibits the same dependence on das the potential energy associated with the flotation force; this is discussed briefly in Appendix B. If ␧F−␧⌸=0 共corresponding to mechanical isolation兲, bVsup共d→⬁兲⬃1/dn关see Eq. 共13兲兴. This decay agrees with previous findings,20–22 but the reliability of this result is unclear because the difference aVsup−bVsup decays with the same power law. As a matter of fact, the amplitudes of the asymptotic decay of aVsup and bVsup differ and are, in turn, different from Eq. 共52兲in Ref. 22 because there actually neither of the two representations of F ˆeq was used. However, for the leading behavior, this is unimportant because Vmen is asymptotically dominated by the correction to the superposition approximation. 2. Beyond the superposition approximation If the interface deformation field in Eq. 共6b兲is evaluated near colloid 1, the term u2+umis dominated by u2in the absence of mechanical isolation and by umin the case of mechanical isolation 关see Eqs. 共A1兲and 共A9兲兴, u2+um⬃ 再 um共d兲⬃d−n/2if ␧F−␧⌸=0 u2共d兲⬃ln dif ␧F−␧⌸⫽0. 冎 共22兲 Thus, the superposition approximation holds if and only if the system is not mechanically isolated. Otherwise, the correction umis dominant, and for asymptotically large separations d,Vmen⬃Vcorr. Hence, in the following, we take ␧F −␧⌸=0. Since the main contributions to the integrals stem from the regions around S ␣ , one obtains the estimates 204706-5 Theory of capillary-induced interactions J. Chem. Phys. 127, 204706 共2007兲 − 冕 S ˆmen dA关⌸2um+2⌸mu2+⌸mum兴 ⬃− 冕 Smen,2 dA关⌸2um+2⌸mu2兴⬃ 1 dn/2.共23兲 Using the asymptotic decay of umgiven by Eq. 共A9兲, we find bVcorr共d兲⬃− 冕 Smen,2 dA关⌸2um+2⌸mu2兴 +2 ␲ ␥ r0具␧Fum+␧⌸mu1典1−4 ␲ ␥ r0 2␧F␧⌸m⬃1 dn/2 共24兲 because each term scales like d−n/2and there is no reason for mutual cancellations. In this expression, one can identify two distinct contributions with a simple physical meaning. After inserting Eqs. 共6b兲and 共6c兲into relationship 共3a兲, we write ⌬h ˆ2=⌬h2+⌬hmwith ⌬h2ª具u1+u2典2−r0␧F共⬃具u2典2−r0␧F+O共d−n兲兲, 共25a兲 ⌬hmª具um典2−r0␧m共⬃d−n/2兲,共25b兲 where the asymptotic decays are given by Eqs. 共18兲and 共A9兲. Accordingly, expression 共24兲shows that bVcorr is dominated asymptotically by36 共i兲the work done by the additional pressure 2⌸mif the single-colloid configuration is deformed relative to the reference configuration, that is, −兰dA2⌸mu2 +2 ␲ ␥ r0␧⌸m⌬h2, and 共ii兲the work done by the forces acting in the single-colloid configuration upon the additional displacement um, that is, −兰dA⌸2um+2 ␲ ␥ r0␧F⌬hm. The asymptotic behavior of aVcorr共d→⬁兲can be derived by estimating the behavior of the terms in Eq. 共9b兲individu- ally, as in the case of Eq. 共10b兲, with the following result: aVcorr共d兲⬃−2 ␥ 冕 Smen,2 dA共ⵜum兲·共ⵜu2兲 −4 ␲ ␥ r0 2␧F␧⌸m⬃1 dn/2.共26兲 The asymptotic decay predicted by Eqs. 共24兲and 共26兲must agree both with respect to the decay exponents and the amplitudes because the difference aVsup −bVsup decays asymptotically more rapidly 关see Eqs. 共20a兲and 共20b兲for ␧F−␧⌸ =0兴. The sign of the force described by Vcorr共d兲is not evident from the outset, but in the applications we shall consider later, it turns out to be always attractive. It is interesting to compare our result with the corresponding one in Ref. 26. After noting the equivalence um↔2u12 in the notations, one finds that Eq. 共7兲in Ref. 26 is identical with the integral term in Eq. 共26兲here. We obtain an additional term ⬀␧F␧⌸mbecause we treat the contribution by the colloid to the free energy functional in full detail, while in Ref. 26 the point-particle approximation 共r0→0兲is used from the outset 关compare Eq. 共4a兲here with Eq. 共1兲in Ref. 26兴. As a consequence, in Ref. 26 our additional term is lost as a singularity of the integral term, which is regularized there by introducing an unknown cutoff length expected to be of the order of r0共see also Refs. 21 and 22兲. The analysis of realistic models in Sec. III will show that the quantitative contribution of the term ⬀␧F␧⌸mto the effective interaction Vmen is actually larger than, but proportional to, the contribution from the other term in Eq. 共26兲. To summarize, the capillary-induced effective interaction between two colloids is given by 关compare Eq. 共13兲 defining n兴 Vmen共d兲⬃ 再 Vcorr共d兲⬃d−n/2if ␧F−␧⌸=0 Vsup共d兲⬃ln dif ␧F−␧⌸⫽0 冎 共r0Ⰶd兲. 共27兲 III. APPLICATIONS In this section, we compute Vmen共d兲for ␧F=␧⌸共mechanical isolation兲for different realistic models of ⌸ ˆderived from the solution of the electrostatic problem within various approximations. We note that the asymptotic decay of Vmen共d兲is the same as that of the direct electrostatic repulsion, so that these detailed calculations beyond the asymptotic analysis of Sec. II B are necessary in order to be able to address this fine-tuning problem and to determine whether the total force is asymptotically attractive or repulsive. A. Ideally conducting fluid phase The simplest model consists of approximating water by an ideal conductor. Formally, this corresponds to the limit of zero temperature, so that the osmotic pressure of the mobile charges accumulated at the interface vanishes and the Debye length ␬ −1 is zero 关see Eq. 共41兲兴. In this case, the electric field E ˆ=E ˆezis always normal to the interface and the pressure is given by Maxwell’s stress tensor evaluated at the insulating side of the interface 共we use Gaussian units兲, ⌸ ˆ=1 4 ␲ 冏ez· 冉 ⑀ E ˆE ˆ−1 2 ⑀ E ˆ21 冊 ·ez冏z=0+ = ⑀ 1 8 ␲ E ˆ2共z=0 +兲, 共28兲 where ⑀ 1is the dielectric constant of the insulating phase 共z⬎0兲. In the present context, the electrostatic problem of a charged sphere partially immersed in a conducting fluid has been solved numerically and semianalytically in Ref. 37. There, it has been shown that the single-colloid pressure field exhibits an integrable divergence upon approaching the three-phase contact line and that asymptotically it displays the familiar dipole behavior. The following approximate parametrization 关Eq. 共1.4兲in Ref. 37 expressed in terms of our notation兴incorporates these properties and is sufficiently accurate for our present purposes: 204706-6 Domínguez, Oettel, and Dietrich J. Chem. Phys. 127, 204706 共2007兲 ⌸共r兲= ␥ ␧F r0 b共 ␮ 兲 冋 r r0 −1 册 ␮ −1 冋 r r0 册 − ␮ −5 , 共29兲 b共 ␮ 兲ª1 6 ␮ 共 ␮ +1兲共 ␮ +2兲共 ␮ +3兲, where 0⬍ ␮ ⬍1 is a fitting parameter, the precise value of which depends on the contact angle ␪ and the dielectric constant ⑀ 1. This expression is normalized so that ␧⌸=␧F, and it corresponds to an exponent n=6 independent of the choice for ␮ . Therefore, far from a colloid, the single-colloid electric field 兩E兩=冑8 ␲ ⌸/ ⑀ 1is that of a dipole perpendicular to the interface, the strength of which is given by 兩p兩= lim r→⬁ ⑀ 1r3兩E共r兲兩 = ⑀ 1r0 3冑8 ␲ ␥ ␧Fb共 ␮ 兲 ⑀ 1r0 .共30兲 In the presence of two colloids, we take E ˆ⬇E1+E2. This approximation allows a simplification of the calculations and should not alter the physical picture significantly. The approximative character is due to possible violations of the electrostatic boundary conditions at the surfaces of the colloidal particles: The additional polarization of colloid 1 induced by E2will actually lead to an electric field E ˆ=E1 +E2+ ␦ Ein the neighborhood of colloid 1, with an induced electric field 兩 ␦ E共r,d兲兩⬃ ␹ 共r兲E2共d兲, where the “effective susceptibility” ␹ 共r兲is expected to be at most of order unity. Thus, our conclusions will be qualitatively correct with a quantitative error of a factor of order unity. Under these conditions, the field ⌸m共r兲defined by Eq. 共6a兲is given within this approximation by ⌸m共r兲=冑⌸共兩r−r1兩兲⌸共兩r−r2兩兲.共31兲 The integrals in Eq. 共26兲are computed by using the expressions in Eqs. 共A1兲and 共A9兲so that in the limit r0Ⰶd, one obtains 共for details, see Appendix C兲 ␧⌸m⬇4b共 ␮ 兲 ␮ +1␧F 冉 r0 d 冊 3 共32a兲 and 冕 Smen,2 dA共ⵜum兲·共ⵜu2兲⬇8 ␲ b共 ␮ 兲 ␮ +1 M共 ␮ 兲r0 2␧F 2 冉 r0 d 冊 3 , 共32b兲 with M共 ␮ 兲ª1 8共 ␮ +1兲b共 ␮ 兲 ⫻ 冕 0 1 dvv4 2F1共1− ␮ ,4;5;v兲2F1 冉 1− ␮ 2,1;2;v 冊 共33兲 and 2F1共 ␣ , ␤ ; ␥ ;z兲is the hypergeometric function 共see Sec. 9.1 in Ref. 38兲. We have checked numerically that M共 ␮ 兲 ⬇ ␮ /5 with a maximum error of ⬇1.3% within the range 0 艋 ␮ 艋1. The final result reads Vmen共d兲⬇aVcorr共d兲 ⬇−16 ␲ b共 ␮ 兲 ␮ +1 关1+M共 ␮ 兲兴 ␥ r0 2␧F 2 冉 r0 d 冊 3 .共34兲 This corresponds to an attractive force. We note that the dominant contribution to Vmen共d兲stems from the term proportional to ␧⌸min Eq. 共26兲; the more so the smaller ␮ is. Smaller values of ␮ correspond to an increasing importance of the electric field near the colloid 关see Eq. 共29兲兴. Vmen共d兲is to be compared with the potential energy due to the direct electrostatic repulsion Vrep共d兲of the charged colloids. The potential energy of one dipole in the field of another identical dipole is p2/共 ⑀ 1d3兲so that Vrep共d兲 ⬇p2/共2 ⑀ 1d3兲for large d.共One must divide by a factor of 2 because within our model no work is done on the image charge inside the conducting phase forming the dipole.兲Collecting the results, we find that the total interaction energy at large separations is given by Vtotal共d兲=Vrep共d兲+Vmen共d兲 ⬇4 ␲ ␥ r0 2␧Fb共 ␮ 兲 冉 r0 d 冊 3 冋 1− 4 ␮ +1关1+M共 ␮ 兲兴␧F 册 . 共35兲 Hence, we see that the attractive capillary potential is proportional to ␧F 2and the direct electrostatic repulsion is proportional to ␧F. Thus, in the limit ␧FⰆ1, on which our calculations are based, the electrostatic repulsion is always larger than the capillary attraction. The leading-order analysis of this model predicts an attraction only if the charge of the colloid is large enough so that ␧F=O共1兲. The critical value above which there is attraction is given by ␧F,crit共 ␮ 兲= ␮ +1 4关1+M共 ␮ 兲兴,共36兲 which lies in the range 1/4⬍␧F,crit ⬍5/12 for 0⬍ ␮ ⬍1. B. Finite Debye length Consider now the more general case of an upper insulating phase 共dielectric constant ⑀ 1兲and a lower electrolytic phase 共dielectric constant ⑀ 2, electrolyte concentration n0兲at a finite temperature T. The stress tensor acting on the interface is due to the difference of Maxwell’s stress tensor just above and below the interface, plus an osmotic pressure p ˆosm by the excess of ions concentrated close to the interface, ⌸ ˆ=1 4 ␲ 冏ez· 冉 ⑀ E ˆE ˆ−1 2 ⑀ E ˆ21 冊 ·ez冏z=0− z=0+ +p ˆosm.共37兲 With ⌽ ˆ共r兲denoting the electrostatic potential at the interface, E ˆz,±共r兲ªez·E ˆ共r,z=0±兲the normal component of the electric field at the interface 共with ⑀ 1E ˆz,+= ⑀ 2E ˆz,−兲, and E ˆ 储 共r,z=0兲the 共continuous兲parallel component at the interface, we have 204706-7 Theory of capillary-induced interactions J. Chem. Phys. 127, 204706 共2007兲 1 4 ␲ 冏ez· 冉 ⑀ E ˆE ˆ−1 2 ⑀ E ˆ21 冊 ·ez冏z=0− z=0+ = ⑀ 2− ⑀ 1 8 ␲ 冋 ⑀ 1 ⑀ 2 E ˆz,+ 2+E ˆ 储 2 册 , 共38兲 with ⌬n共r兲as the excess ion concentration at the interface, p ˆosm ⬇kBT⌬n=n0kBT 冋 exp 冉 e⌽ ˆ kBT 冊 + exp 冉 −e⌽ ˆ kBT 冊 −2 册 , 共39兲 assuming that the electrolyte is dilute and consists of monovalent ions. In order to solve the electrostatic problem, we introduce two simplifications: We apply the Debye-Hückel approximation for dilute electrolytes, and we approximate the extended colloid by a point charge qat its center; i.e., we retain only the monopole term of the source of the field. This “monopolar” approximation corresponds formally to the limit of vanishing contact radius, r0→0, and it can be expected to provide the correct field at distances from the colloid which are large compared to r0. This is, in fact, complementary to the limiting case ␬ r0Ⰷ1 considered in the previous section. From Eq. 共39兲,wefind p ˆosm ⬇ ⑀ 2 8 ␲ ␬ 2⌽ ˆ2,共40兲 where the screening length is given by ␬ −1 =冑 ⑀ 2kBT 8 ␲ n0e2.共41兲 For pure water, ⑀ 2=81 and n0⬇10−7 mol/l共that is, pH=7兲 lead to a screening length ⬇1 ␮ m at room temperature. We consider first the case of a single colloid corresponding to a point charge qlocated at the flat interface between an insulator and an electrolyte. Proceeding along the lines of, e.g., Refs. 4and 5,wefind ⌽共r兲=2q ⑀ 2rIa共 ␬ r兲,共42a兲 Ez,+共r兲=− 2q ⑀ 2r2Ib共 ␬ r兲,共42b兲 E 储 共r兲=−d⌽ dr er=2q ⑀ 2r2Ic共 ␬ r兲er,共42c兲 where eris the unit radial vector pointing away from the center of the colloid, and the auxiliary functions In共k兲are given by integrals over a Bessel function, Ia共k兲ª 冕 0 ⬁ dxJ0共x兲x 共 ⑀ 1/ ⑀ 2兲x+冑x2+k2,共43a兲 Ib共k兲ª ⑀ 2 ⑀ 1 冕 0 ⬁ dxJ0共x兲x冑x2+k2 共 ⑀ 1/ ⑀ 2兲x+冑x2+k2,共43b兲 Ic共k兲ªIa共k兲−kdIa dk 共k兲.共43c兲 These integrals have been computed numerically; the analytic expressions for their asymptotic behaviors are derived in Appendix D. Thus, taking ⑀ 2/ ⑀ 1Ⰷ1共the ratio ⑀ 2/ ⑀ 1is approximately 81 for water in contact with air兲in the intermediate asymptotic regime 1Ⰶ ␬ rⰆ共 ⑀ 2/ ⑀ 1兲2, one obtains the following expressions for the potential and the field components 关see Eqs. 共42a兲–共42c兲兴: ⌽共r兲⬇ 2q ⑀ 2r 冋 ⑀ 1 ⑀ 2 1 共 ␬ r兲2+e − ␬ r 册 ,共44a兲 Ez,+共r兲⬇ 2q ⑀ 2r2 冋 −1 ␬ r+冑 ␲ 2 ⑀ 1 ⑀ 2 共 ␬ r兲3/2e− ␬ r 册 ,共44b兲 E 储 共r兲⬇ 2q ⑀ 2r2 冋 3 ⑀ 1 ⑀ 2 1 共 ␬ r兲2+ ␬ re− ␬ r 册 er.共44c兲 Asymptotically, ⌽共r兲decays as 1/r3and the electrostatic interaction energy q⌽共d兲of a second charge qat a distance d from the first one decays likewise as 1/d3. This is the celebrated dipole repulsion between charged colloidal particles first conjectured in Ref. 1. However, due to the large value of the ratio ⑀ 2/ ⑀ 1, closer to the charge there is a crossover to a screened Coulomb potential ⌽共r兲⬀exp共− ␬ r兲/r. One can introduce a crossover length rcross defined from the asymptotic behavior given by Eqs. 共44a兲–共44c兲as ⑀ 1 ⑀ 2 1 共 ␬ rcross兲2=e − ␬ rcross.共45兲 This equation has solutions only if ⑀ 1/ ⑀ 2⬍共2/e兲2⬇0.54, in which case the relevant solution ␬ rcross is larger than 2 and depends only weakly, i.e., logarithmically on the precise value of the ratio ⑀ 1/ ⑀ 2. For ⑀ 2/ ⑀ 1=81, this gives ␬ rcross ⬇8.7. The parallel component E 储 exhibits the same crossover, but the normal component Ez,+ always decays algebraically and is much larger than the parallel component. Nevertheless, the contribution of Ez,+ to the pressure field ⌸共r兲is reduced by a factor ⑀ 1/ ⑀ 2关see Eq. 共38兲兴, so that ⌸共r兲will also exhibit the crossover at a distance r⬇rcross. From Eqs. 共38兲and 共39兲, one finds ⌸共r兲= ␥ r0 ␬ 2␧F 共 ␬ r兲4P共 ␬ r0兲 冋 ⑀ 1 ⑀ 2 冉 1− ⑀ 1 ⑀ 2 冊 Ib 2共 ␬ r兲 + 冉 1− ⑀ 1 ⑀ 2 冊 Ic 2共 ␬ r兲+共 ␬ r兲2Ia 2共 ␬ r兲 册 ,共46兲 where 关see Eq. 共1兲兴 ␧F=␧⌸=q2 ␬ 2 2 ␲ ⑀ 2 ␥ r0 P共 ␬ r0兲,共47兲 with the dimensionless function 204706-8 Domínguez, Oettel, and Dietrich J. Chem. Phys. 127, 204706 共2007兲 APPENDIX C: EFFECTIVE CAPILLARY POTENTIAL ENERGY IN THE CASE OF AN IDEALLY CONDUCTING FLUID In this appendix, we provide the mathematical steps leading to Eqs. 共32a兲and 共32b兲, which are valid in the limit dⰇr0and from which the effective capillary potential energy is obtained. In order to compute ␧⌸m, we insert the ansatz 关Eq. 共31兲兴 into the definition 关Eq. 共15b兲兴. In the limit d→⬁, the main contribution to the integral stems from the regions around each particle so that ␧⌸m⬇2 ␥ r0 冑⌸共d兲 冕 r0 ⬁ drr冑⌸共r兲.共C1兲 With the pressure field given by Eq. 共29兲, this reduces to ␧⌸m⬇2b共 ␮ 兲␧F 冉 r0 d 冊 3 冕 1 ⬁ dx冑共x−1兲 ␮ −1x−共 ␮ +3兲,共C2兲 and Eq. 共32a兲is obtained upon performing the integral 共Eq. 3.191.2 in Ref. 38兲. For the evaluation of the integral in Eq. 共32b兲,weuse the solutions 关Eqs. 共A1兲and 共A6a兲兴valid for ␧F−␧⌸m=0. With Eq. 共29兲, this leads to ⵜu= er ␥ r 冕 r ⬁ dss⌸共s兲 =erb共 ␮ 兲␧F r0 r 冕 r/r0 ⬁ dx共x−1兲 ␮ −1x−共 ␮ +4兲 =1 4 erb共 ␮ 兲␧F 冉 r0 r 冊 5 2F1 冉 1− ␮ ,4;5;r0 r 冊 共C3兲 and, with the ansatz 关Eq. 共31兲兴,to ⵜum⬇2er ␥ r 冕 r ⬁ dss冑⌸共d兲⌸共s兲 =2erb共 ␮ 兲␧F r0 r 冉 r0 d 冊 3 冕 r/r0 ⬁ dx共x−1兲共 ␮ −1兲/2x−共 ␮ +3兲/2 =2erb共 ␮ 兲␧F 冉 r0 d 冊 3 冉 r0 r 冊 2 2F1 冉 1− ␮ 2,1;2;r0 r 冊 .共C4兲 关Concerning the last lines in Eqs. 共C3兲and 共C4兲, see Eq. 3.194.2 in Ref. 38.兴This enables one to obtain 冕 Smen,2 dA共ⵜum兲·共ⵜu2兲 ⬇ ␲ b共 ␮ 兲2␧F 2 冉 r0 d 冊 3 冕 r0 ⬁ drr 冉 r0 r 冊 7 ⫻2F1 冉 1− ␮ ,4;5;r0 r 冊 2F1 冉 1− ␮ 2,1;2;r0 r 冊 共C5兲 so that Eqs. 共32b兲and 共33兲follow upon a change of variable in the integral. APPENDIX D: ASYMPTOTIC BEHAVIOR OF THE ELECTROSTATIC POTENTIAL AND OF THE ELECTRIC FIELD In this appendix, we discuss the asymptotic behaviors of the functions In共k兲defined in Eqs. 共43a兲–共43c兲. First, we mention that at first sight the integrals may appear to be divergent due to the weak power law decay of the integrands, which are, however, oscillatory. Actually, the integrals are regularized by an exponential, so that, for instance, Ia共k兲ªlim h→0 冕 0 ⬁ dxe−hxJ0共x兲x 共 ⑀ 1/ ⑀ 2兲x+冑x2+k2,共D1兲 reflecting the physical situation that the charge qis positioned at a 共dimensionless兲height h⬎0 above the flat interface. In the following mathematical manipulations, this regularization scheme is implied, which, unless required, we do not write explicitly to avoid a clumsy notation. Reference 4provides a method to obtain the expansion of Ia共k兲in powers of ⑀ 1/ ⑀ 2Ⰶ1. The idea is to split the integrals into a sum of two terms involving only odd or even powers of the ratio ⑀ 1/ ⑀ 2, respectively. This leads to44 Ia共k兲=e −k+ ⑀ 1 ⑀ 2 冋 ␲ 2k共I0共k兲−L0共k兲兲 −1 册 +O共 ⑀ 1/ ⑀ 2兲2,kⰆ共 ⑀ 2/ ⑀ 1兲2共D2兲 in terms of the Bessel function I0and the Struve function L0.38 We note that the validity of this expression is restricted to sufficiently small values of k. However, for the typical values of the ratio ⑀ 2/ ⑀ 1occurring in the experiments so far, this does not impose any physically relevant constraint on k. The asymptotic behavior for large kis Ia共k兲⬇e−k+ ⑀ 1 ⑀ 2 1 k2,1ⰆkⰆ共 ⑀ 2/ ⑀ 1兲2.共D3兲 As a general result,4the coefficients of the odd powers of ⑀ 1/ ⑀ 2decay algebraically for kⰇ1, while the coefficients of the even powers decay exponentially. With Eq. 共43c兲, this leads to Ic共k兲⬇ke−k+3 ⑀ 1 ⑀ 2 1 k2,1ⰆkⰆ共 ⑀ 2/ ⑀ 1兲2.共D4兲 The procedure employed in Ref. 4can be extended to analyze Ib共k兲. The radical is eliminated from the denominator of the integrand in Eq. 共43b兲, and the integral is split as follows: Ib共k兲= ⑀ 2 ⑀ 1 冕 0 ⬁ dxJ0共x兲x共x2+k2兲 关1−共 ⑀ 1/ ⑀ 2兲2兴x2+k2 − 冕 0 ⬁ dxJ0共x兲x2冑x2+k2 关1−共 ⑀ 1/ ⑀ 2兲2兴x2+k2.共D5兲 The first term involves only odd powers of ⑀ 1/ ⑀ 2, the second term only even powers. The first integral can be rewritten as 204706-15 Theory of capillary-induced interactions J. Chem. Phys. 127, 204706 共2007兲 ⑀ 2 ⑀ 1 冕 0 ⬁ dxJ0共x兲x共x2+k2兲 关1−共 ⑀ 1/ ⑀ 2兲2兴x2+k2 = ⑀ 2/ ⑀ 1 1−共 ⑀ 1/ ⑀ 2兲2 冕 0 ⬁ dxJ0共x兲x ⫻ 冋 1+k2−k2/关1−共 ⑀ 1/ ⑀ 2兲2兴 x2+k2/关1−共 ⑀ 1/ ⑀ 2兲2兴 册 .共D6兲 With the identities 共see Eqs. 6.623.2 and 6.532.4 in Ref. 38兲 冕 0 ⬁ dxJ0共x兲x=0 共D7兲 and 冕 0 ⬁ dxJ0共x兲x x2+a2=K0共a兲,共D8兲 the first integral in Eq. 共D5兲can be written as ⑀ 2 ⑀ 1 冕 0 ⬁ dxJ0共x兲x共x2+k2兲 关1−共 ⑀ 1/ ⑀ 2兲2兴x2+k2 =− ⑀ 1/ ⑀ 2 关1−共 ⑀ 1/ ⑀ 2兲2兴2k2K0共k/冑1−共 ⑀ 1/ ⑀ 2兲2兲 =−共 ⑀ 1/ ⑀ 2兲k2K0共k兲+O共 ⑀ 1/ ⑀ 2兲3.共D9兲 The second integral in Eq. 共D5兲can be written to the leading order in ⑀ 1/ ⑀ 2as 冕 0 ⬁ dxJ0共x兲x2冑x2+k2 x2+k2 = 冕 0 ⬁ dxJ0共x兲 冋 冑x2+k2−k2 冑x2+k2 册 .共D10兲 We introduce the exponential regularization as exp共 −h冑x2+k2兲and note the identity 共Eq. 6.637.1 in Ref. 38兲 冕 0 ⬁ dxJ0共x兲e−h冑x2+k2 冑x2+k2=I0 冉 k 2关冑h2+1−h兴 冊 ⫻K0 冉 k 2关冑h2+1+h兴 冊 .共D11兲 Therefore, the right-hand side of Eq. 共D10兲can be rewritten as 冕 0 ⬁ dxJ0共x兲x2冑x2+k2 x2+k2 = lim h→0 冋 ⳵ 2 ⳵ h2−k2 册 冕 0 ⬁ dxJ0共x兲e−h冑x2+k2 冑x2+k2 =1 2k2 冋 I1 冉 k 2 冊 K1 冉 k 2 冊 −I0 冉 k 2 冊 K0 冉 k 2 冊册 =kd dk 冋 kI0 冉 k 2 冊 K1 冉 k 2 冊册 .共D12兲 Therefore, from Eqs. 共D9兲and 共D12兲one finally arrives at the following expansion: Ib共k兲=1 2k2 冋 I0 冉 k 2 冊 K0 冉 k 2 冊 −I1 冉 k 2 冊 K1 冉 k 2 冊册 − ⑀ 1 ⑀ 2 k2K0共k兲 +O共 ⑀ 1/ ⑀ 2兲3.共D13兲 This expansion may also be restricted to sufficiently small values of k, although we have not found this upper bound as function of ⑀ 1/ ⑀ 2. However, as argued above, this possible constraint is expected to be physically irrelevant because, typically, ⑀ 1/ ⑀ 2is sufficiently small. The asymptotic behavior of this expression for kⰇ1is Ib共k兲⬇1 k−冑 ␲ 2 ⑀ 1 ⑀ 2 k3/2e−k,1Ⰶk, ⑀ 2/ ⑀ 1.共D14兲 1P. Pieranski, Phys. Rev. Lett. 45, 569 共1980兲. 2J. Joannopoulos, Nature 共London兲414, 257 共2001兲. 3A. Dinsmore, M. F. Hsu, M. Nikolaides, M. Márquez, A. Bausch, and D. Weitz, Science 298, 1006 共2002兲. 4A. J. Hurd, J. Phys. A 18, L1055 共1985兲. 5R. Aveyard, J. H. Clint, D. Nees, and V. N. Paunov, Langmuir 16, 1969 共2000兲. 6J. Ruiz-García, R. Gámez-Corrales, and B. I. Ivlev, Physica A 236,97 共1997兲. 7F. Ghezzi and J. Earnshaw, J. Phys.: Condens. Matter 9, L517 共1997兲. 8D. Stamou, C. Duschl, and D. Johannsmann, Phys. Rev. E 62,5263 共2000兲. 9M. Quesada-Pérez, A. Moncho-Jordá, F. Martínez-López, and R. Hidalgo-Alvarez, J. Chem. Phys. 115, 10897 共2001兲. 10 F. Ghezzi, J. C. Earnshaw, M. Finnis, and M. McCluney, J. Colloid Interface Sci. 238, 433 共2001兲. 11 S. J. Mejía-Rosales, A. Gil-Villegas, B. I. Ivlev, and J. Ruiz-García, J. Phys.: Condens. Matter 14, 4795 共2002兲. 12 M. G. Nikolaides, A. R. Bausch, M. F. Hsu, A. D. Dinsmore, M. P. Brenner, C. Gay, and D. A. Weitz, Nature 共London兲420,299共2002兲. 13 G. Tolnai, A. Agod, M. Kabai-Faix, A. L. Kovács, J. J. Ramsden, and Z. Hórvölgyi, J. Phys. Chem. B 107, 11109 共2003兲. 14 O. Gómez-Guzmán and J. Ruiz-García, J. Colloid Interface Sci. 291,1 共2005兲. 15 W. Chen, S. Tan, Z. Huang, T.-K. Ng, W. T. Ford, and P. Tong, Phys. Rev. E74, 021406 共2006兲. 16 J. C. Fernández-Toledano, A. Moncho-Jordá, F. Martínez-López, and R. Hidalgo-Alvarez, Langmuir 20, 6977 共2004兲. 17 M. M. Nicolson, Proc. Cambridge Philos. Soc. 45, 288 共1949兲. 18 D. Y. C. Chan, J. D. Henry, Jr., and L. R. White, J. Colloid Interface Sci. 79, 410 共1981兲. 19 K. D. Danov, P. A. Kralchevsky, and M. P. Boneva, Langmuir 20, 6139 共2004兲. 20 M. Megens and J. Aizenberg, Nature 共London兲424, 1014 共2003兲. 21 L. Foret and A. Würger, Phys. Rev. Lett. 92, 058302 共2004兲. 22 M. Oettel, A. Domínguez, and S. Dietrich, Phys. Rev. E 71, 051401 共2005兲. 23 A. Domínguez, M. Oettel, and S. Dietrich, J. Phys.: Condens. Matter 17, S3387 共2005兲. 24 M. Oettel, A. Domínguez, and S. Dietrich, Langmuir 22, 846 共2006兲. 25 M. Oettel, A. Domínguez, and S. Dietrich, J. Phys.: Condens. Matter 17, L337 共2005兲. 26 A. Würger and L. Foret, J. Phys. Chem. B 109, 16435 共2005兲. 27 A. Domínguez, M. Oettel, and S. Dietrich, arXiv:cond-mat/0611329. 28 A. Würger, Europhys. Lett. 75, 978 共2006兲. 29 A. Domínguez, M. Oettel, and S. Dietrich, Europhys. Lett. 77, 68002 共2007兲. 30 In this respect, Ref. 22 is incomplete, but the same conclusion about the irrelevance of this effect is reached. Reference 28 contains several important mathematical errors 共Ref. 29兲, and the corresponding conclusion erroneously disagrees with the actual irrelevance of the finite-size effect. 31 In actual experiments, there is a certain degree of polydispersity in size, shape, electric charge, etc., which we do not expect to alter the conclusions qualitatively if it remains small enough. 32 Which fluid phase is to be the top one depends on the experimental setup. Usually gravity breaks the up-down symmetry. 204706-16 Domínguez, Oettel, and Dietrich J. Chem. Phys. 127, 204706 共2007兲 33 This seems to be actually a reliable approximation for the experimental setups considered so far in the literature, but see the discussion in Sec. IV. 34 R. Aveyard, B. P. Binks, J. H. Clint, P. D. I. Fletcher, T. S. Horozov, B. Neumann, V. N. Paunov, J. Annesley, S. W. Botchway, D. Nees, A. W. Parker, A. D. Ward, and A. N. Burgess, Phys. Rev. Lett. 88, 246102 共2002兲. 35 Due to the rapid decay of ⌸ ˆfar from the particles, in the limit L→⬁, the length scale Lenters into the problem only if ␧⌸⫽␧Fand in that case just as an upper bound on dto avoid a logarithmic divergence 关see the discussion after Eq. 共A1兲兴. 36 Note that by symmetry 具u2典2=具u1典1and 具um典2=具um典1. 37 K. D. Danov and P. A. Kralchevsky, J. Colloid Interface Sci. 298, 213 共2006兲. 38 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. 共Academic, London, 1994兲. 39 Next-to-leading terms in the expansion require a more elaborate calculation than the simple Taylor expansion, which yields undefined expressions for them. 40 M. Oettel, A. Domínguez, M. Tasinkevych, and S. Dietrich, preprint, 2007. 41 K. D. Danov, P. A. Kralchevsky, and M. P. Boneva, Langmuir 22, 2653 共2006兲; this work is de facto a correction 共Ref. 24兲of previous work 共Ref. 19兲. 42 D. Frydel, S. Dietrich, and M. Oettel, Phys. Rev. Lett. 99, 118302 共2007兲. 43 S. Dietrich, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz 共Academic, London, 1988兲,Vol.12,p.1. 44 We note 共i兲a misprint in Eq. 共10兲in Ref. 4because there the term 2m should be in the numerator and 共ii兲that the series provided in that Eq. 共10兲can actually be carried out and is proportional to the error function. 204706-17 Theory of capillary-induced interactions J. Chem. Phys. 127, 204706 共2007兲