Effective forces between colloids at interfaces induced by capillary wavelike fluctuations

Abstract

Abstract. -We calculate the effective force mediated by thermally excited capillary waves between spherical or disklike colloids trapped at a fluid interface. This Casimir type interaction is shown to depend sensitively on the boundary conditions imposed at the three-phase contact line. For large distances between the colloids an unexpected cancellation of attractive and repulsive contributions is observed leading to a fluctuation force which decays algebraically very rapidly. For small separations the resulting force is rather strong and it may play an important role in two-dimensional colloid aggregation if direct van der Waals forces are weak.Introduction and synopsis. -The effective forces between rigid objects immersed in a fluctuating medium have attracted a steadily growing interest because their understanding allows one to design and tune them by choosing suitable media and boundary conditions and by varying the thermodynamic state of the medium. Possible applications range from micromechanical systems to colloidal suspensions and embedded biological macromolecules. Accordingly, these fluctuations may be the zero-temperature, long-ranged quantum fluctuations of the electromagnetic fields giving rise to the original Casimir effect [1] between flat or corrugated immersed metallic bodies [2,3]. Other examples for fluctuation induced longranged effective forces between immersed objects involve media such as bulk fluids near their critical point [4], membranes [5] or interfaces [6].In this work we investigate the latter manifestation of thermal Casimir forces for nanocolloids floating at surface-tension dominated liquid-vapor or liquid-liquid interfaces where they are effectively trapped. In the presence of charges their mutual interactions often exhibit unexpected strong and long-ranged attractions [7] inducing mesoscale pattern formation. However, these unusual patterns, which, e.g., are of interest for optical applications once they are fixated on a solid substrate, appear also for neutral nanocolloids [8]. In spite of some progress [9], the nature of these effective forces between the colloids is not yet fully understood. This pertains in particular to the role of fluctuations for these types of observations. Therefore the present effort aims at determining the fluctuation induced contribution to these forces generated by the inevitable thermally excited capillary waves of the fluid interface. As a first step the colloids are taken to be spherical or disklike and electroneutral. We shall

Effective Forces Induced by a Fluctuating Interface: Exact Results

“…Note that these forces are of much longer range than the electrostatic ones induced by photon fluctuations, termed van der Waals [27] or the fluctuation-induced Casimirtype repulsion between extended objects in similar systems. This 1/(r log r) result has also been obtained by Lehle, Oettel and Dietrich [16] in a continuum model for a particular, but very physically reasonable case of contact of the interface with two extended objects; these we may think of as spheres each with two hemispherical surfaces having different wetting properties which keep the sphere at the interface at an orientation and immersion which makes the equatorial line separating the two wetting region sit exactly in the interface, as mentioned at the beginning of this Letter. The case we have considered is an extreme one in that the colloidal sphere has been treated as though it were a point (or a unit cube on the dual lattice).…”

“…The interest of such questions stems not only from the various practical applications [8,11,12] but also from the basic requirement to understand the nature of the effective forces between the colloidal particles [13,14]. In particular, the important role of the aforementioned capillary fluctuations is clear; substantial progress has recently been made [15,16], but there remain a number of issues.…”

Ellipsoidal particles at fluid interfaces

“…In that respect, colloids at fluids interfaces appear to be a possible realization of a two-dimensional system exhibiting the Casimir effect. For spheres and disks, these forces have been calculated in references [19,20]. The large-distance behavior of these forces depends sensitively on the boundary conditions at the three-phase contact line whereas for close distances a strong attraction similar to van-der-Waals forces has been found, independent of the type of boundary condition.…”

Effective field theory approach to fluctuation-induced forces between colloids at an interface

“…(7) that every pair of field occurrences in these many-point correlators carries a factor β −1 = k B T , each quadratic term in β H can be counted as a factor β 0 , each quartic term as β −1 , and so on, in the term of expansion (6) they appear in. From this we see that the traditionally encountered form F = k B Tf ({r a }) of fluctuation-induced interactions [7][8][9][10][11]24,28,29] stems from quadratic terms in H. This allows the interpretation of the interactions as those between induced "capillary charges," arising from thermal fluctuations around each particle and subsequent polarizations due to the deformation that propagates from those. Similarly, one can see that higher-than-quadratic terms in H give rise to an excess of factors of k B T in the cumulants in Eq.…”

“…(3) sifts out only permissible field shapes h(r) [7][8][9][10][11]14,24,28]. In this article we follow another path, which encodes the constraints in additional terms to the free-surface Hamiltonian.…”