Within nonlinear Poisson-Boltzmann theory we calculate the pair and triplet interactions between charged colloidal spheres, specifically in the nonlinear regime of low salt concentrations and high charges. We find repulsive pair interactions and attractive triplet interactions. Within a van der Waals-like mean-field theory we estimate in which parameter regime a gas-liquid coexistence is to be expected.

We use video microscopy to follow the phase-space trajectory of a two-dimensional colloidal model liquid and calculate three-point correlation functions from the measured particle configurations. Approaching the fluid-solid transition by increasing the strength of the pair-interaction potential, one observes the gradual formation of a crystal-like local order due to triplet correlations, while being still deep inside the fluid phase. Furthermore, we show that in a strongly interacting system the BornGreen equation can be satisfied only with the full triplet correlation function but not with threebody distribution functions obtained from superposing pair-correlations (Kirkwood superposition approximation). PACS numbers:Our current understanding of the structure of simple fluids is based on the n-body distribution functions g (n) , measuring the probability density of finding two, three, and more particles at specified positions in space. When the total potential energy of a liquid is given by a sum of pair-potentials, all of its thermodynamic properties can be calculated by means of the pair-correlation function g(r) ≡ g (2) (r) and its density (ρ) and temperature (T ) derivatives. However, the latter two quantities, ∂g(r)/∂ρ and ∂g(r)/∂T , explicitly depend on the triplet correlation function, even if the particle interactions are only pair-wise additive [1]. Explicit knowledge of triplet correlations is also required in perturbation theories for static fluid properties [2], in theories of transport properties [3], of solvent reorganization processes around solutes [4], of systems under shear-flow [5], but also to understand the structural properties of a 2D amorphous system [6]. Most of our knowledge on triplet correlations come from computer simulation studies of hard-spherefluids [7], Lennard-Jones fluids [8,9] and electrolyte systems [10]. In the overwhelming majority, these papers are concerned with testing Kirkwood's superposition approximation (KSA) [11] for the triplet distribution function. By contrast, semi-analytical theories for g (3) beyond the KSA are rather rare [9,12]. However, despite the long-standing theoretical interest in its properties, it has never been possible to measure three-particle correlations directly. Indirect ways to identify higher-order correlations in scattering data have been suggested for instance in [13]. An alternative, but also indirect way to obtain experimental information on g (3) is based on the relationship between the isothermal pressure derivative of the fluid structure factor ∂S(q)/∂P and the triplet distribution function [1], a relationship which has been systematically exploited by Egelstaff and co-workers in rare-gas systems [14]. The present Letter reports on the first direct measurement of g (3) in a two-dimensional colloidal model liquid with well-defined pair-interaction potentials.The preparation of the samples and the experiments FIG. 1:A typical image (500 × 380 µm) of our twodimensional colloidal model system with paramagnetic colloids of d = 4.7µm diam...

Abstract. -Two-and three-particle correlation functions are computed from video-microscopy data of two-dimensional suspensions of charged colloids and inverted to derive the pair and three-body interaction potentials between the colloidal particles. Our method allows to resolve the full spatial dependence of the three-body potentials. Examining colloidal systems at different colloid densities, we find density-independent, attractive three-body potentials, with a minimum of a few kT that is most pronounced in the equilateral triangle configuration.A simple liquid is a liquid consisting of particles which interact with pair potentials only dependent on the particle separations. A charge-stabilized colloidal suspension, by contrast, is a complex fluid: here the inter-particle interaction is governed by the inhomogeneous distribution of electrolyte ions between the highly charged colloids. Only in certain limiting cases, is it possible to integrate out the ionic degrees of freedom, leading to an effective colloid-colloid potential of the Yukawa form. This pair potential is an essential element of the classic DerjaguinLandau-Verwey-Overbeek (DLVO) theory of interactions in charge-stabilized colloids [1,2]. In general, however, the interaction between colloids in a charge-stabilized suspension may not be expected to be pairwise additive: many-body interactions among the colloids can be important, primarily under low salt conditions and for highly charged colloids. A good example is the three-body interaction potential which has received some attention recently: it has been theoretically predicted [3][4][5], and also experimentally observed [6,7]. In [6, 7] a video-microscopy experiment is described, examining in what way the presence of a third charged colloid in the neighborhood of a pair of colloids affects their mutual interaction. By decomposing the measured total interaction potential in an appropriate way, the three-particle interaction potentials could be extracted and successfully compared to theoretical Poisson-Boltzmann calculations.Examining only three isolated particles remains a somewhat artificial situation, considering that, in reality, colloids live together in a suspension of finite density. Therefore, in order to prove that three-body potentials are present also in concentrated colloidal suspensions, one has to observe three-body forces "at work", that is, one has to infer the shape and magnitude of the three-body potential from analyzing the particle coordinates of a large number of colloids which are together in a concentrated suspension. In this letter, we compute correlation functions and extract from these functions the whole three-body interaction potentials among the colloids. Since from two-particle correlations one obtains microscopic information only on the level of pair interactions, we have to analyze pair and, in addition, triplet correlation Letters (EPL) ; 69 (2005), 3. -S. 468-474 https://dx

We study the effect of lipid demixing on the electrostatic interaction of two oppositely-charged membranes in solution, modeled here as an incompressible two-dimensional fluid mixture of neutral and charged mobile lipids. We calculate, within linear and nonlinear Poisson-Boltzmann theory, the membrane separation at which the net electrostatic force between the membranes vanishes, for a variety of different system parameters. According to Parsegian and Gingell, contact between oppositely-charged surfaces in an electrolyte is possible only if the two surfaces have exactly the same charge density (sigma(1) = -sigma(2)). If this condition is not fulfilled, the surfaces can repel each other, even though they are oppositely charged. In our model of a membrane, the lipidic charge distribution on the membrane surface is not homogeneous and frozen, but the lipids are allowed to freely move within the plane of the membrane. We show that lipid demixing allows contact between membranes even if there is a certain charge mismatch, /sigma(1)/ not equal /sigma(2)/, and that in certain limiting cases, contact is always possible, regardless of the value of sigma(1)/sigma(2) (if sigma(1)/sigma(2) < 0). We furthermore find that of the two interacting membranes, only one membrane shows a major rearrangement of lipids, whereas the other remains in exactly the same state it has in isolation and that, at zero-disjoining pressure, the electrostatic mean-field potential between the membranes follows a Gouy-Chapman potential from the more strongly charged membrane up to the point of the other, more weakly charged membrane.

Three-body distribution functions in classical fluids have been theoretically investigated many times, but have never been measured directly. We present experimental three-point correlation functions that are computed from particle configurations measured by means of video-microscopy in two types of quasi-two-dimensional colloidal model fluids: a system of charged colloidal particles and a system of paramagnetic colloids. In the first system the particles interact via a Yukawa potential, in the second via a potential Γ/r 3 . Varying the particles density in the charged system, or, the interaction strength Γ in the magnetic system, one can systematically explore how triplet correlations behave if the coupling between the particles changes. We find for both systems very similar results: on increasing the coupling between the particles one observes the gradual formation of a crystal-like local order due to triplet correlations, even though the system is still deep inside the fluid phase. These are mainly packing effects as is evident from the close resemblance between the results for the two systems having completely different pair-interaction potentials. To demonstrate that triplet correlations are significant not only locally, but also when integrated over the whole volume we consider the Born-Green equation and show that in a strongly interacting system this equation can be satisfied only with the full triplet correlation function but not with three-body distribution functions obtained from superposing pair-correlations (Kirkwood superposition approximation).