Analysis of improved Nernst–Planck–Poisson models of compressible isothermal electrolytes

We consider an improved Nernst–Planck–Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non-equilibrium. The elastic deformation of the medium, that induces an inherent coupling of mass and momentum transport, is taken into account. The model consists of convection–diffusion–reaction equations for the constituents of the mixture, of the Navier–Stokes equation for the barycentric velocity and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross-diffusion phenomena must occur, and the mobility matrix (Onsager matrix) has a non-trivial kernel. In this paper, we establish the existence of a global-in-time weak solution, allowing for a general structure of the mobility tensor and for chemical reactions with fast nonlinear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time.

The grain boundary mobility tensor

The grain-boundary (GB) mobility relates the GB velocity to the driving force. While the GB velocity is normally associated with motion of the GB normal to the GB plane, there is often a tangential motion of one grain with respect to the other across a GB; i.e., the GB velocity is a vector. GB motion can be driven by a jump in chemical potential across a GB or by shear applied parallel to the GB plane; the driving force has three components. Hence, the GB mobility must be a tensor (the off-diagonal components indicate shear coupling). Performing molecular dynamics (MD) simulations on a symmetric-tilt GB in copper, we demonstrate that all six components of the GB mobility tensor are nonzero (the mobility tensor is symmetric, as required by Onsager). We demonstrate that some of these mobility components increase with temperature, while, surprisingly, others decrease. We develop a disconnection dynamics-based statistical model that suggests that GB mobilities follow an Arrhenius relation with respect to temperature T below a critical temperatureTcand decrease as1/Tabove it.Tcis related to the operative disconnection mode(s) and its (their) energetics. For any GB, which disconnection modes dominate depends on the nature of the driving force and the mobility component of interest. Finally, we examine the impact of the generalization of the mobility for applications in classical capillarity-driven grain growth. We demonstrate that stress generation during GB migration (shear coupling) necessarily slows grain growth and reduces GB mobility in polycrystals.

On Diffusion Through Soft Filter

Diffusion through soft polymer filters is a nonlinear process: the increase of the pressure on the filtrating liquid does not trigger the proportional increase of the flux through the filter. There are two sources of nonlinearity: the diffusivity properties of the filter and its high deformability. In the present work we use a theoretical formulation coupling large deformations and diffusion to describe a liquid flux through a polymeric filter. Two key factors making the present formulation simple are the molecular incompressibility condition and the nonlinear mobility tensor. The developed model is calibrated based on the experiments on toluene-rubber filtration.

Effective slip boundary conditions for arbitrary periodic surfaces: the surface mobility tensor

In a variety of applications, most notably microfluidics design, slip-based boundary conditions have been sought to characterize fluid flow over patterned surfaces. We focus on laminar shear flows over surfaces with periodic height fluctuations and/or fluctuating Navier scalar slip properties. We derive a general formula for the ‘effective slip’, which describes equivalent fluid motion at the mean surface as depicted by the linear velocity profile that arises far from it. We show that the slip and the applied stress are related linearly through a tensorial mobility matrix, and the method of domain perturbation is then used to derive an approximate formula for the mobility law directly in terms of surface properties. The specific accuracy of the approximation is detailed, and the mobility relation is then utilized to address several questions, such as the determination of optimal surface shapes and the effect of random surface fluctuations on fluid slip.

Anisotropic electro-osmotic flow over super-hydrophobic surfaces

Patterned surfaces with large effective slip lengths, such as super-hydrophobic surfaces containing trapped gas bubbles, have the potential to greatly enhance electrokinetic phenomena. Existing theories assume either homogeneous flat surfaces or patterned surfaces with thin double layers (compared with the texture correlation length) and thus predict simple surface-averaged, isotropic flows (independent of orientation). By analysing electro-osmotic flows over striped slip-stick surfaces with arbitrary double-layer thickness, we show that surface anisotropy generally leads to a tensorial electro-osmotic mobility and subtle, nonlinear averaging of surface properties. Interestingly, the electro-osmotic mobility tensor is not simply related to the hydrodynamic slip tensor, except in special cases. Our results imply that significantly enhanced electro-osmotic flows over super-hydrophobic surfaces are possible, but only with charged liquid–gas interfaces.

Drag on a sphere in slow shear flow

A general closed form for the mobility tensor of a sphere moving in a fluid in stationary homogeneous flow is derived using the induced force method up to the first order in the square root of the Reynolds number based on the velocity gradient of the unperturbed flow. The closed form for the mobility tensor is valid for the time-dependent case as well as for the stationary case. As a special case, we calculate it explicitly for a simple shear flow. The result for the x, z-component for the stationary case, which gives the lift force, agrees with the value calculated by Saffman.