Browsing by Author "Guzowski, Jan Jerzy"

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Capillary interactions between colloidal particles at curved fluid interfaces
(2010) Guzowski, Jan Jerzy; Dietrich, Siegfried (Prof. Dr.)
The subject of the thesis is the behavior of colloidal particles at liquid-gas interfaces in the situation when the particles are partially wetted by the liquid. First, we have investigated the case of a flat interface. We have derived exact expressions for the free energy depending on the immersion of the particle in the liquid phase. We have found that the exact results are almost perfectly reproduced by a linearized theory for small deformations of the flat interface with the amplitude renormalized in order to match the solution of the full non-linear problem far away from the particle. Furthermore, we have compared the above macroscopic approach with a microscopic calculations based on the mean-field density functional theory for the fluid surrounding the particle being characterized by long-range intermolecular forces and assuming so-called sharp-kink of the density profile at the interface. From these results we have drawn a general conclusion that the macroscopic linear theory is sufficient for the purpose of calculating the capillary forces even for sub-micrometer particles and that the effect of the long-range intermolecular forces enters then only through the surface tensions. Next, we have investigated the interaction free energy of two heavy spheres at a flat fluid-fluid interface in presence of gravity. We have assumed that the contact lines at the particles are pinned which leads to the asymptotic result which coincides with the result obtained by Oettel (2005) for the particles with free contact lines (fixed contact angles). This leads to a conclusion that, for the leading asymptotic behavior, the mechanism of the attachment of the particles to the interface is irrelevant. In order to set up a more general theoretical framework, we have introduced the analogy between capillarity and electrostatics, in which the small deformations of an initially flat interface play the role of the electrostatic potential and the external pressure can be interpreted as the capillary charge'' distribution. In the case of a deformation induced by a particle the analogy can be used to identify the capillary monopole with the total external force acting on the particle and the capillary dipole with the total external torque. As a consequence, a free particle of arbitrary shape corresponds to a quadrupole. In this picture the asymptotic results for the interaction energy between particles subjected to external forces or between free ellipsoidal particles, reported in the literature, can be easily explained in terms of electrostatics and multipole expansion. Subsequently, we have studied the spherical interfaces. We have considered small deformations of a spherical droplet subjected to an external pressure field . In the limit of large droplets we have obtained a relation between the spherical multipoles associated with a particle and the capillary multipoles for the identical particle at a flat interface. We have derived general expressions for the interaction potentials between arbitrary-order multipoles at an arbitrary angular separation. We have shown that the result for monopoles reproduces the Green's function derived by Morse and Witten in 1993. Additionally, we have obtained a closed expression for point-quadrupoles. Finally, we have approached the problem of a single spherical particle at the surface of a sessile droplet. In the case of the particle being at the drop apex we have used the axial symmetry in order to obtain exact analytic solutions for the droplet shape and expressions for the surface free energy as a function of the elevation of the particle above the substrate. In the cases without axial symmetry we have taken into account the fact that the condition of balance of forces acting on the droplet in the lateral direction requires either a fixed lateral position of the center of mass of the droplet (model A) or a pinned contact line at the substrate (model B). Using a perturbation theory for small deformations of the reference cap-like spherical shape of the droplet we have derived the free energy functional incorporating the liquid-substrate surface free energy. The effects of the particle pulled (or pushed) by an external force and of the fixed center of mass have been incorporated by introducing effective pressure fields. In terms of those fields, the linear Young-Laplace equation governing the small deformations, has been derived . We have shown that in the limit of a small particle the free energy of the sessile droplet can be expressed in terms of the Green's functions satisfying the boundary conditions at the substrate corresponding to either a free or a pinned contact line and it does not depend on the size of the particle but only on the pulling force, the contact angle at the substrate, and on the angular position of the particle. For contact angle at the substrate 90 degrees we have exploited an analogue of the method of images known from electrostatics in order to calculate the surface free energy (in excess over the surface free energy of the reference configuration of a drop shape given by a spherical cap) analytically. Because in this case the reference droplet forms a half of a sphere the boundary conditions at the substrate can be fulfilled by introducing an image particle at the virtual hemisphere below the substrate surface (such that the union of the actual and the virtual droplet forms a full sphere). Further analysis shows that due to the conditions of force balance and volume constraint the Green's function requires additional terms, but they do not change the results qualitatively. Using the analytical results for the Green's functions in the case we have also calculated pair-potentials for two particles at arbitrary angular positions at the droplet and analyzed possible minimum free energy configurations. The analytical results have been compared with the results of the numerical minimization of the free energy functionals for a spherical and for an ellipsoidal particle at a sessile droplet. In the case of a spherical particle pulled (or pushed) by an external force we have found an almost perfect agreement with the predictions of the perturbation theory. For this particular geometry a pinned contact line corresponds to Dirichlet boundary conditions and a free contact line with fixed contact angle to Neumann boundary conditions. The type of boundary conditions determines the sign of the capillary monopole associated with the image particle at the virtual hemisphere and therefore the free energy, which is proportional to the product of the capillary charges of the original particle and its image, can change sign, too. Besides the known phenomena of attraction of a particle to a free contact line and repulsion from a pinned one, we have observed a local free energy minimum for the particle being located at the drop apex or at a characteristic intermediate angle, respectively. This peculiarity can be traced back to a non-monotonic behavior of the Green's functions for a free droplet, which is a consequence of interplay between the deformations of the droplet and the volume constraint. In the case of force-free ellipsoidal particles we have obtained monotonic free energy landscapes, in qualitative and partially quantitative agreement with the point-quadrupole approximation. Particularly, the theoretically predicted monotonic dependence of the free energy on distance of the particle from the contact line and scaling with the droplet radius has been confirmed. We have argued that in the case of a pinned contact line at the substrate the ellipsoidal particle gets trapped at the drop apex in an energy well typically exceeding by far the thermal energy and therefore this effect could be observed in an experiment. As an outlook, the pair potential for point-quadrupoles at a free droplet could be used in order to derive the corresponding pair-potential in the case of a sessile droplet, which could be of significant practical importance, because any force- and torque-free particle of non-spherical shape trapped at the surface of a drop corresponds to a capillary quadrupole. In much more general terms, it is also still a matter of a future research to extend the theory of capillary interactions beyond flat and spherical interfaces towards general curved interfaces.