Abstract
The equivalence between parabolic transport equations for solute concentrations and stochastic dynamics for solute particle motion represents one of the most fertile correspondences in statistical physics originating from the work by Einstein on Brownian motion. In this article, we analyze the problems and the peculiarities of the stochastic equations of motion in microfluidic confined systems. The presence of solid boundaries leads to tensorial hydrodynamic coefficients (hydrodynamic resistance matrix) that depend also on the particle position. Singularity issues, originating from the non-integrable divergence of the entries of the resistance matrix near a solid no-slip boundary, determine some mass-transport paradoxes whenever surface phenomena, such as surface chemical reactions at the walls, are considered. These problems can be overcome by considering the occurrence of non vanishing slippage. Added-mass effects and the influence of fluid inertia in confined geometries are also briefly addressed.
Highlights
The overdamped approximation in confined geometries presents intrinsic peculiarities, just because the hydrodynamic resistance matrix depends on the position x, and in general of the orientation φ, and this raises delicate issues when the thermal fluctuations are expressed as linear superposition of increments of Wiener processes, owing to their highly singular local structure [45]
This is because particle motion at microscale is controlled by thermal stochastic fluctuations, the understanding of which would improve the performances of micofluidic devices, and could test the validity of classical physical theories down to the scale where quantum effects should start to appear
The fundamental peculiarity of microhydrodynamics in confined geometries is that fluid-particle interactions depend on the position and, owing to fluctuation-dissipation relations, these spatial nonuniformities are transferred to mass-transport properties giving rise to delicate theoretical issues and paradoxes
Summary
The main legacy of the Einsteinian theory of Brownian motion to modern physics lies in the confirmation of the atomistic nature of matter and in the equivalence between random molecular motion at the microscopic level and the macroscopic phenomenon of diffusion [1,2]. This automatically implies, due to the fluctuation-dissipation relation (4), a tensorial and position-dependent diffusivity The latter two properties have deep and non trivial implications whenever a stochastic equation of motion, in the form of Equation (3). For the sake of completeness, it should be mentioned that a position dependent effective diffusivity arises in modeling solute transport in microchannels with undulated walls, in the case the transport problem is referred exclusively to the channel axial coordinate [43,44] This is referred to as the Fick-Jacobs approximation and it is essentially a geometrical effect within an approximate transport model unrelated to any hydrodynamic interactions. While stochastic modeling of particle transport involving constant effective diffusivities is widely used in the analysis of microfluidic devices, the inclusion of hydrodynamic effects, deriving from fluid confinement, represents a completely new and unexplored field of theoretical and numerical investigation.
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