Electrostatic interaction of particles trapped at fluid interfaces: Effects of geometry and wetting properties

The electrostatic interaction between pairs of spherical or macroscopically long, parallel cylindrical colloids trapped at fluid interfaces is studied theoretically for the case of small inter-particle separations. Starting from the effective interaction between two planar walls and by using the Derjaguin approximation, we address the issue of how the electrostatic interaction between such particles is influenced by their curvatures and by the wetting contact angle at their surfaces. Regarding the influence of curvature, our findings suggest that the discrepancies between linear and nonlinear Poisson-Boltzmann theory, which have been noticed before for planar walls, also occur for spheres and macroscopically long, parallel cylinders, though their magnitude depends on the wetting contact angle. Concerning the influence of the wetting contact angle $\theta$ simple relations are obtained for equally sized particles which indicate that the inter-particle force varies significantly with $\theta$ only within an interval around $90^\circ$. This interval depends on the Debye length of the fluids and on the size of the particles but not on their shape. For unequally sized particles, a more complicated relation is obtained for the variation of the inter-particle force with the wetting contact angle.

Cross-sectional view in the y = 0 plane of a system with two spherical or parallel cylindrical colloids (yellow discs) floating at a fluid interface indicated by the horizontal line. The projection of the center of the left particle onto the interface is chosen as the center (0, 0, 0) of the Cartesian coordinate system used to describe the system. The fluid medium below the interface is called medium "1" and the one above the interface medium "2". The particles are chemically identical, such that the contact surfaces with one of the media carry the same charge densities and the same wetting contact angles θ. The particles may differ in size with radius R 1 on the left and R 2 on the right particle. The horizontal distance of the particles is characterized by the width L of the gap between both particles as depicted by the vertical dashed lines. The equilibrium heights of the centers of the left and right particles from the interface are given by |D 1 | and |D 2 |, respectively, with θ ∈ [0, π] determining the sign of D 1 and D 2 according to the relations D 1 = −R 1 cos θ and D 2 = −R 2 cos θ.
exceeds several orders of k B T [2,10]. Wetting properties are crucial not only for the adsorption of a single particle at an interface, but also for the interaction between several of them. For example, whereas very hydrophobic silica particles (θ ≥ 129 • ) form well-ordered monolayer structures (with inter-particle separations of several particle diameter) at octane-water interfaces, less hydrophobic (θ ≤ 115 • ) particles fail to do so [11,12]. This can be attributed to dif-ferent strengths of the repulsive electrostatic force, which acts mainly through the oil-phase because the electrostatic field is well screened inside the aqueous phase at the high salt concentrations used. Moreover, the capillary interaction due to the overlap of the interface deformation field around each particle also depends on the contact angle θ. However, here we disregard deformations of the interface, which can be significant if the particle surfaces are rough [13] or the particles are large (radius 10 µm) [14][15][16][17][18][19], and we focus only on the electrostatic interaction between the colloids.
Whereas the electrostatic interaction between particles trapped at a fluid interface has been studied extensively since the pioneering studies by Pieranski [20] and Hurd [21], most of the investigations deal with the case of particles situated far away from each other. At long distances the electrostatic pair-interaction takes the particularly simple form of an interaction between two electric dipoles, which are generated by the asymmetric counterion distribution at the particle surfaces in contact with the two fluid phases. It has been shown that in this case the linearization of the Poisson-Boltzmann (PB) theory is applicable [21]. Recent studies, directed towards the opposite limit of small inter-particle separations, have been performed within the linearized PB theory [22,23] or by considering a flat plate geometry [22,24] in order to simplify the problem. Whereas the former approximation is often violated at short separations, the latter represents the ideal situation of a contact angle of exactly 90 • and the absence of particle curvature. However, in reality, the contact angle θ can vary significantly to either side of 90 • [2,10,19,25] and whether the particle curvature plays any important role at short inter-particle separations still remains to be addressed within the nonlinear PB theory.
Accordingly, in this contribution, we investigate the electrostatic interaction between spherical and parallel cylindrical colloids with an arbitrary contact angle appearing at a fluid interface. As long as the size of the particles is sufficiently larger than both the length scale of the interaction and the inter-particle separation, which is usually the case for close by particles we are interested in, one can apply the Derjaguin approximation (DA) using results of the corresponding case of planar walls [26]. Having recently solved this two-plate problem exactly (i.e., without using the superposition approximation) within the nonlinear PB theory [24], we proceed one step further and compute the force between a pair of spheres or parallel cylinders. A similar approach has already been employed and proved to be valid in this context [27,28]. However, the present study differs from those treatments in several aspects. Whereas Ref. [27] deals with macroscopically long cylinders trapped at an oil-water interface and having a constant surface potential, which is most suitable for metallic particles, we consider dielectric particles described by constant charge densities at their surfaces. Moreover, to keep our analysis general and to be consistent with experimental observations [29][30][31][32][33], we consider the particle surfaces to be charged in both fluid phases, which is not the case in Ref. [27]. On the other hand, Ref. [28] describes the interaction between spherical particles by using the superposition approximation, which has been shown to be qualitatively wrong for small inter-particle separations [22] and which discards any interaction between the particle-water surface of one particle and the particle-oil surface of the other particle. As explained in the next section, this latter contribution to the interaction energy is included in our calculation via the line contribution.

II. MODEL AND FORMALISM
As depicted in Fig. 1, we consider two particles with radii R 1 and R 2 placed at a fluid-fluid interface described by a threedimensional Cartesian coordinate system. The projection of the center of the left particle onto the interface is chosen as the origin (0, 0, 0) of the coordinate system. The particles are either two spheres or two macroscopically long, parallel cylinders with axes in y-direction; their cross-sections in the plane y = 0 are shown in Fig. 1. The fluid-fluid interface is indicated by the horizontal line at x = 0. Although the particles can differ in size, they are taken to be chemically identical such that the surfaces of both particles in contact with the same fluid phase carry the same surface charge density, and that the contact angle θ is the same for both particles. This is a simplifying assumption because chemically identical particles in general need not to be equally charged [34]. In equilibrium the centers of the left and the right particle are located at x = D 1 and x = D 2 , respectively. Depending upon the contact angle θ ∈ [0, π], both D 1 = −R 1 cos θ and D 2 = −R 2 cos θ can be negative (for θ < π 2 ; the case considered in Fig. 1) as well as positive (for θ > π 2 ). In between the particles a gap of width L occurs (see the vertical dashed lines in Fig. 1) so that the horizontal center-to-center distance is L + R 1 + R 2 . The fluid phase below (above) the interface occupying the half-space x < 0 (x > 0) is denoted by medium "1" ("2"). Both fluids are modeled as structureless, continuous media with dielectric constant ε i = ε r,i ε 0 , i ∈ {1, 2}, where ε r,i is the relative premittivity of medium i and ε 0 is the vacuum permittivity. The ionic strength of added salt in medium i ∈ {1, 2} is denoted by I i . The corresponding Debye screening length in each medium is given by is the vacuum Bjerrum length with e > 0, k B , and T being the elementary charge, the Boltzmann constant, and the absolute temperature, respectively. Within our description at the meanfield level, the length scale of interest is the Debye length as both the local charge density and the electrostatic interaction vary on this scale. Phenomena which occur on smaller length scales, e.g., the structuring of liquids on the molecular length scale, are not considered. Here we consider medium "2" to be the less polar phase in the sense that κ −1 2 > κ −1 1 . With this the criteria for the applicability of the DA are that both radii R 1 and R 2 have to be much larger than κ −1 2 and L. Within the DA, one basically decomposes the two interacting particles into infinitesimal surface elements. Assuming that the elementary surface pieces, which face each other, interact like flat parallel surfaces, the total interaction between the two curved objects is obtained via integration over the whole surface. Here, however, the particle surfaces are homogeneously charged only separately inside each medium, the properties of which in general differ. As a result, a three-phase contact line is formed where a particle surface intersects the fluid interface; two such contact lines on opposing particles interact as well. But this does not introduce any additional constraint for applying the DA. In the spirit of the DA, each of these two contact lines can be divided in infinitesimal pieces and the total contribution due to the line interaction can be obtained as long as the interaction between two parallel lines is known. Therefore, in order to apply the DA, one needs to know the interaction of parallel flat surfaces dipped into medium "1" or medium "2" and the interaction between two parallel three-phase contact lines. These are exactly the quantities we calculated in Ref. [24] numerically by solving the nonlinear PB equation. To be more precise, the relevant quantities, as defined in Ref. [24], are ω γ,i (r), which is the interaction energy per total surface area between two parallel, planar surfaces dipped at a distance r into medium i ∈ {1, 2}, and ω τ (r), which is the interaction energy per total line length between two parallel three-phase contact lines at a distance r. Please note that the interaction of the surface of one particle in contact with medium "1" and that of the other particle in contact with medium "2" is included in the line contribution ω τ (r). In order to tackle the problem efficiently we fit superpositions of exponential contributions to the the numerically obtained data for ω γ,1 (r), ω γ,2 (r), and ω τ (r) from Ref. [24]: and Due to the asymptotic decays ω γ,i (r → ∞) ∼ exp(−κ i r) and ω τ (r → ∞) ∼ exp(−κ 2 r) (note the convention κ −1 2 > κ −1 1 ), the slowest of the decay rates b i in Eq. (1) equals κ 1 and the slowest of the decay rates d i in Eq. (2) as well as the slowest of the decay rates h i in Eq. (3) equal κ 2 . All three interactions, i.e., ω γ,1 , ω γ,2 , and ω τ result in forces onto the particles, which can be obtained by taking the negative derivative with respect to the appropriate distance between the facing surface or line elements, followed by integrating over the particles. Due to the geometry of the problem, the electrostatic force between the particles acts only in the horizontal z-direction. Please note that the movement of the particles in the vertical x-direction is suppressed by the steep and strong trapping potential [2]. In the following, we denote the z-component of the electrostatic force, which the left particle in Fig. 1 exerts on the right one, by F(L), and we decompose it, according to F(L) = F 1 (L) + F 2 (L) + F 3 (L), into the surface contribution F 1 (L) due to the surface interaction ω γ,1 in medium "1", the surface contribution F 2 (L) due to the surface interaction ω γ,2 in medium "2", and the line contribution F 3 (L) due to the line interaction ω τ .

III. RESULTS AND DISCUSSION
In this section we discuss the variation of the force F(L) between the particles as function of the particle sizes (radii R 1 and R 2 ), the particle separation L, and the contact angle θ. First, the results for a pair of spheres are presented and then those for a pair of parallel cylinders. For our discussion we consider a typical experimental set-up consisting of polystyrene particles placed at a water-lutidine (2,6-dimethylpyridine) interface at temperature T = 313 K. The added salt is NaI with bulk ionic strengths I 1 = 1 mM and I 2 = 0.85 mM. The relative permittivities are ε r,1 = 72 for the water-rich phase (medium "1") and ε r,2 = 62 for the lutidine-rich phase (medium "2"). The chemically identical particles are assumed to be similarly charged; the magnitude of the surface charge density in contact with the aqueous phase is σ 1 = 0.1 e/nm 2 and that in contact with the oil-phase is σ 2 = 0.01 e/nm 2 . Differences in the solubilities of the ions in the two fluids result in a potential difference between the bulk of the two media, which is called the Donnan potential or Galvani potential difference [35], and which, for our system, is assumed to be 1 k B T/e. These numbers correspond to a standard set of parameters as used in Ref. [24]. They are either taken or estimated from various experimental studies [30,[36][37][38][39][40][41]. It is worth mentioning that all numerical examples presented here have been chosen such that the conditions for applying the DA are satisfied.

A. Spheres
In the case of two interacting spheres at a fluid interface, after performing the surface and line integrations, for the three distinct, lateral force contributions the following expressions are obtained: where erf(x) denotes the error function [42], and  unequally-sized (2κ 1 R 1 = κ 1 R 2 ≈ 100) particles with varying θ, (e) unequally-sized (10κ 1 R 1 = κ 1 R 2 ≈ 300) particles with varying θ, and (f) equally-sized (κ 1 R ≈ 100) particles with θ = 90 • within linear and nonlinear PB theory. As shown by panel (a) and its inset, the force increases linearly with increasing R and decays exponentially with increasing separation κ 1 L. Panels (b) and (c) suggest that, for equally-sized spheres, the force increases significantly with decreasing contact angle θ only within an interval around 90 • . Outside this interval the force remains de facto constant and the interval of θ, across which the force actually varies, widens upon decreasing κ 1 R. For unequally-sized spheres, if the size asymmetry is moderate, the force may increase as well as decrease if the contact angle deviates from 90 • (panel (d)). However, if the size contrast is high, the force becomes weaker once θ is slightly shifted away from 90 • in either direction (panel (e)). From panel (f) and the inset therein one can infer that the discrepancy between the linear and the nonlinear results diminishes with increasing θ.
with the scaled separation κ 1 L for different system parameters are shown in Fig. 2. Here β = 1/ (k B T ) is the inverse thermal energy. As one can infer from Fig. 2(a), for equally-sized spheres (R 1 = R 2 = R) with a contact angle of θ = 90 • , the effective force scales linearly with the size of the particles and decays exponentially with increasing separation between the particles. Both the linear scaling with R and the exponential decay with L can be directly inferred from the inset of Fig. 2(a) where the ratio of the dimensionless force βF/κ 1 to the scaled radius κ 1 R is plotted as a function of the scaled   Fig. 1) of the force F(L), per length L y in the y-direction, due to the electrostatic interaction between a pair of parallel cylindrical colloids, expressed in units of 10 2 κ 2 1 /β, as function of their scaled separation κ 1 L for (a) equally-sized (R 1 = R 2 = R) cylinders of varying radius with contact angle θ = 90 • , (b) equally-sized (κ 1 R ≈ 100) particles with varying θ, (c) equally-sized (κ 1 R ≈ 30) particles with varying θ, (d) unequally-sized (2κ 1 R 1 = κ 1 R 2 ≈ 100) particles with varying θ, (e) unequally-sized (10κ 1 R 1 = κ 1 R 2 ≈ 300) particles with varying θ, and (f) equally-sized (κ 1 R ≈ 100) particles with θ = 90 • within linear and nonlinear PB theory. As shown by panel (a) and its inset, the force increases ∝ √ R with increasing R and decays exponentially with increasing separation κ 1 L. Panels (b) and (c) suggest that, for equally-sized cylinders, the force increases significantly with decreasing contact angle θ only within an interval around 90 • . Outside this interval the force remains de facto constant and the interval of θ, across which the force actually varies, widens upon decreasing κ 1 R. For unequally-sized cylinders, if the size asymmetry is moderate, the force may increase as well as decrease if the contact angle deviates from 90 • (panel (d)). However, if the size contrast is high, the force becomes weaker once θ is slightly shifted away from 90 • in either direction (panel (e)).
From panel (f) and the inset therein one can infer that the discrepancy between the linear and the nonlinear results diminishes with increasing θ. separation κ 1 L using a semi-logarithmic scale, revealing data collapse. Figures 2(b) and 2(c) display the variation of the scaled force with contact angle θ for equally-sized particles. As one can see, for κ 1 R ≈ 100 (b), the force increases with decreasing contact angle but de facto it varies only within the interval 80 • < θ < 100 • . A similar phenomenon is observed for κ 1 R ≈ 30 (c) albeit with variation in a slightly broader interval 75 • < θ < 105 • . Figures 2(d) and 2(e) show the variation of the force with contact angle θ as function of separation distance κ 1 L for unequally-sized spheres. For the relatively small size-asymmetry in Fig. 2(d), the force between the particles can increase in a certain range of separation between the particles as the contact angle θ is decreased slightly from 90 • downwards. Otherwise the force is weaker than the one obtained for a neutral wetting situation θ = 90 • . For the larger size-asymmetry in Fig. 2(e), the force becomes weaker as soon as the contact angle θ differs from 90 • . However, in both cases, as considered in Figs. 2(d) and 2(e), the force between the particles decreases upon increasing θ in the interval θ > 90 • . Finally, in Fig. 2(f), we compare the force obtained within the linearized and the nonlinear PB theory for equally sized spheres. As one can see from the plot, the forces differ by almost an order of magnitude even at large separations such as κ 1 L ≈ 6 for θ = 90 • . It turns out that with increasing contact angle θ this difference diminishes; see the inset of Fig. 2(f), where both curves are of almost the same magnitude. This is expected since the portion of the particles immersed in the more polar phase, for which the electrostatic interaction is stronger because κ 1 ≈ κ 2 κ 1 ≈ 0.1059 nm −1 , κ 2 ≈ 0.1053 nm −1 and σ 1 σ 2 , decreases with increasing contact angle. Note that this situation differs from the one in Refs. [11,12] in which κ 1 κ 2 . A similar discrepancy appeared while comparing the interactions within the linear and nonlinear theory for parallel flat surfaces [24]. Thus taking into account particle curvature does not significantly change this result.
All these observations are in accordance with the force expressions given in Eqs. (4)- (6). For micron-sized particles considered here, in most of the cases the line contribution to the total interaction is negligible. Therefore, the total force F(L) is dominated by the surface contributions F 1 (L) (Eq. (4)) and F 2 (L) (Eq. (5)). For θ = 90 • , which is the case considered in Fig. 2(a)  c i exp (−d i L), respectively, which transparently explain the linear variation of the force with the particle size R (the coefficients a i and c i do not depend on R) and its exponential decay as function of the separation L. The decay rate in the limit of large distances is determined by the smaller of the two Debye lengths κ −1 1 and κ −1 2 (please note that for our system κ −1 1 ≈ κ −1 2 ). For R 1 = R 2 = R with an arbitrary contact angle θ, one has D 1 − D 2 = 0 so that the dependence on θ appears in Eqs. (4) and (5) only through the terms involving the error functions, which reduce to 1 − erf − cos θ √ b i R and 1 + erf − cos θ √ d i R , respectively. Since the error function levels off to | erf(x)| ≈ 1 for |x| 2, the variation of the force with respect to θ in Eq. (4) saturates once the slowest decay rate b i , which in the present case is κ 1 , satisfies the inequality Similarly, in Eq. (5) the saturation is obtained once the slowest decay rate d i , i.e., κ 2 satisfies the inequality Note that once the slowest decay rates b i and d i satisfy these conditions, all the other decay rates will do so, too. For κ 1 R ≈ 100 (and therefore, κ 2 R ≈ 100 as κ 1 ≈ κ 2 ), both Eqs. (7) and (8) predict that the force varies appreciably only within the interval 78 • ≤ θ ≤ 102 • , which one precisely observes in Fig. 2(b). Decreasing the contact angle θ from π 2 implies that the particles become more hydrophilic. Consequently, the contribution F 1 (L) to the total force increases while F 2 (L) decreases upon decreasing θ. Finally, at θ ≈ 78 • the former attains a non-zero finite value and the latter vanishes (please note the different signs in front of the error functions in Eqs. (4) and (5)). On the other hand, increasing the contact angle θ from π 2 implies that the particles become more hydrophobic. As a result, the contribution F 2 (L) increases and F 1 (L) decreases with the former attaining a non-zero finite value while the latter is vanishing at θ ≈ 102 • . Upon decreasing κ 1 R, the interval of θ over which the force varies broadens as can be inferred from Fig. 2(c). For unequal particles sizes, i.e., for R 1 R 2 , the dependence on θ in Eqs. (4) and (5) originate from both the exponential and the error function. For moderate sizeasymmetry, like the one considered in Fig. 2(d), a competition between these two functions determines the variation of the force with θ. However, for the extremely asymmetric case considered in Fig. 2(e), the difference D 1 − D 2 is large and the exponential terms dominate as soon as θ differs slightly from 90 • .

B. Cylinders
For macroscopically long, parallel cylinders, the expressions for the lateral force contributions, expressed per length L y in the y-direction, are given by: and We note that for geometrical reasons these expressions are slightly different from those obtained for spheres in Eqs. (4)- (6). In particular, the contact lines for cylinders are just straight lines and in order to obtain F 3 (L) there is no need to use the DA. Figure 3 shows the variation of the z-component of the total force F(L) = F 1 (L) + F 2 (L) + F 3 (L), per length L y in the y-direction and in units of 10 2 κ 2 1 /β, which the left cylinder exerts on the right one as function of the scaled separation κ 1 L for the sizes R 1 and R 2 and for the contact angle θ. Except for a few features, the findings are qualitatively the same as those obtained for spheres in Fig. 2. For example, for R 1 = R 2 = R and θ = 90 • , the force between two cylinders also decays exponentially with varying separation L between them and increases with increasing size R, but for the cylinders the increase is proportional to √ R; see Fig. 3(a) and the inset therein. This is evident from the prefactors of the exponential functions in Eqs. (9) and (10). The variation of the force with respect to θ as well as Eqs. (7) and (8) remain the same as for the spheres, because the θ-dependent terms in Eqs. (9) and (10) have exactly the same form as in Eqs. (4) and (5). This behavior is confirmed by Figs. 3(b)-3(e). Finally, the comparison of the effective force for θ = 90 • within linearized and nonlinear PB theory reveals a significant discrepancy between the predictions of the two approaches, which becomes smaller for larger contact angles θ, i.e., as the portion of the particles, dipped into the more polar phase, decreases (see Fig. 3(f)).

IV. CONCLUSIONS
To conclude, by using the Derjaguin approximation and a fitting procedure for numerical results for the effective inter-action between parallel, planar surfaces in contact with two demixed fluids in between, we have calculated the force due to the electrostatic interaction between pairs of spheres or cylinders at close distance from each other at a fluid-fluid interface. The comparison between the results obtained within linear and nonlinear PB theory shows that the former overestimates the force both for spheres and for cylinders, even at distances of several Debye lengths. Concerning the results within the nonlinear theory, we have investigated the effects of varying the sizes and the contact angle of the particles. Our general study is applicable also to pairs of particles which differ in size. For equally-sized spheres and cylinders the force always decays exponentially with increasing separation, and it scales ∝ R for spheres and ∝ √ R for cylinders, where R is the common radius of the particles. Importantly, for equally-sized particles (both spherical and cylindrical) we have found an interval around the contact angle of 90 • , beyond which the force de facto does not vary. We have also obtained simple relations (Eqs. (7) and (8)) involving the Debye lengths of the two media and the radii of the particles for calculating the width of this interval. These robust results can be expected to be useful for describing more general or complex particle interactions at fluid interfaces, which is important for various application perspectives of such systems.