Hydrodynamic Interactions Between Solid Particles at a Fluid–Gas Interface - Drops and Bubbles in Contact with Solid Surfaces - page 95

allowed to study in great detail phenomena like the hydrodynamic enhancement of
self- and collective diffusion [22-26], two-dimensional melting- and freezing tran-
sitions via an intermediate hexatic phase of quasi-long range orientational order
[27], elastic properties and the overdamped phonon band structure of Q2D col-
loidal crystals [28, 29], and the partial clustering [30] and glass formation [31, 32]
in binary magnetic mixtures. To describe quantitatively the lateral particle diffusion
properties, as well as the overdamped lattice dynamics (see, e.g., [33-36]), requires
to know precisely the lateral hydrodynamic mobility tensors of the spheres in pres-
ence of the liquid-gas interface. The mobility tensors are thus an essential input in
theoretical and dynamic computer simulation studies of in-plane diffusion and non-
equilibrium microstructures. Most likely, also the vitrification dynamics of dense
magnetic monolayers is affected by the form of the hydrodynamic mobilities, even
though a comparison of experimental data with recent two-dimensional mode cou-
pling theory calculations suggests the hydrodynamic influence to be rather small
[37].
In typical monolayers of super-paramagnetic spheres the long-range magnetic
dipole repulsions render particle distances shorter than r< 6 a very unlikely. There-
fore, the long-distance form of the Q2D mobility matrix of two spheres should
suffice to describe the hydrodynamics of the translational (and rotationally con-
strained) motion of spheres along the interface. In this article, we explain how such
a far-field expression is constructed and we estimate its precision by comparing
it with the precise values of the mobility coefficients, evaluated by the multipole
expansion.
The present review is organized as follows. In Section B, we describe how to use
the spherical multipole method to determine the low-Reynolds-number HI between
colloidal spheres of radius a , confined to lateral motion along a planar fluid-gas
interface, and with the sphere surfaces permanently touching the interface. In par-
ticular, we show how to construct the many-sphere Q2D mobility tensors using
the method of images [38], in combination with an irreducible multipole expan-
sion method [39], adapted to the present system symmetry. The mobility matrix is
constructed in terms of a multiple scattering series [40-43]. In Section C, explicit
numerical results are presented for the translational and rotational single-sphere
mobilities. In Section D, we provide explicit results for the long-distance contribu-
tions to the self- and distinct Q2D hydrodynamic mobility tensors of two spheres
up to order 1 /r 3 in the interparticle distance r . An alternative far-field expression,
corresponding to a particle approximation, is presented in Section E. To keep the
point-particles at a fixed distance from the interface, additional constraint forces are
introduced. In Section F, the two-sphere far-field Q2D mobility is shown to differ
significantly from the corresponding 3D Rotne-Prager expression for an unbounded
fluid. Furthermore, both versions of the long-distance asymptotic mobilities are
compared with the accurate numerical values, evaluated by the multipole expan-
sion for test configurations of two spherical particles.