Diffusiophoresis, Batchelor scale and effective Péclet numbers

Abstract

We study the joint mixing of colloids and salt released together in a stagnation point or in a globally chaotic flow. In the presence of salt inhomogeneities, the mixing time is strongly modified depending on the sign of the diffusiophoretic coefficient $D_{dp}$. Mixing is delayed when $D_{dp}>0$ (salt-attracting configuration), or faster when $D_{dp}<0$ (salt-repelling configuration). In both configurations, as for molecular diffusion alone, large scales are barely affected in the dilating direction while the Batchelor scale for the colloids, $\ell _{c,diff}$, is strongly modified by diffusiophoresis. We propose here to measure a global effect of diffusiophoresis in the mixing process through an effective Péclet number built on this modified Batchelor scale. Whilst this small scale is obtained analytically for the stagnation point, in the case of chaotic advection, we derive it using the equation of gradients of concentration, following Raynal & Gence (Intl J. Heat Mass Transfer, vol. 40 (14), 1997, pp. 3267–3273). Comparing to numerical simulations, we show that the mixing time can be predicted by using the same function as in absence of salt, but as a function of the effective Péclet numbers computed for each configuration. The approach is shown to be valid when the ratio $D_{dp}^{2}/D_{s}D_{c}\gg 1$, where $D_{c}$ and $D_{s}$ are the diffusivities of the colloids and salt.

Macrotransport theory for diffusiophoretic colloids and chemotactic microorganisms

“…The channel wall is impenetrable to the solute and the colloid. The steady hydrodynamic flow v(r) is directed along the axial direction z and may vary in the radial direction r. The colloid C(r, z, t) and solute S(r, z, t) concentration fields are symmetrically distributed about the channel centreline, and may vary in the radial and axial directions, and in time t. For C S, which is common for colloidal or bacterial suspensions containing molecular solutes, the influence of the evolution of C on S is negligible (Lapidus & Schiller 1976;Rivero-Hudec & Lauffenburger 1986;Staffeld & Quinn 1989;Ford & Cummings 1992;Marx & Aitken 2000;Abecassis et al 2008;Tindall et al 2008;Palacci et al 2010Palacci et al , 2012Kar et al 2015;Banerjee et al 2016;Shi et al 2016;Shin et al 2016;Ault et al 2017Ault et al , 2018Peraud et al 2017;Shin et al 2017;Balu & Khair 2018;Raynal et al 2018;Raynal & Volk 2019;Shim et al 2019;Chu et al 2020a). The advection-diffusion transport of the solute is governed by…”

“…where u r (r, z, t) and u z (r, z, t) are the radial and axial components of the chemical flow, respectively. In diffusiophoresis, D c is the constant, intrinsic diffusivity of the colloid (Staffeld & Quinn 1989;Abecassis et al 2008;Palacci et al 2010Palacci et al , 2012Kar et al 2015;Banerjee et al 2016;Shi et al 2016;Shin et al 2016;Ault et al 2017Ault et al , 2018Shin et al 2017;Raynal et al 2018;Raynal & Volk 2019;Chu et al 2020a). In chemotaxis, D c is the random motility of the microorganism.…”

Mixing and unmixing induced by active camphor particles

“…For both the turbulent clustering of floaters and that of inertial particles the effective compressibility is related to some underlying coupling to the overall turbulent mixing process. In recent studies [15][16][17][18][19] our groups have shown that similar compressible effects on the transport and mixing properties of particles can be induced by the particle response (a so-called phoretic drift)-to environmental field gradients. This environment sensing strategy can be built from various phoretic phenomena: diffusiophoresis (drift induced by chemical concentration gradients), thermophoresis (drift induced by thermal gradients), electrophoresis (drift induced by electric field gradients), etc.…”

“…This environment sensing strategy can be built from various phoretic phenomena: diffusiophoresis (drift induced by chemical concentration gradients), thermophoresis (drift induced by thermal gradients), electrophoresis (drift induced by electric field gradients), etc. Experiments, simulations, and analytical models [15][16][17][18][19] show that such phoretic particles acquire an effectively compressible dynamics, even though the underlying flow is perfectly incompressible. Active (self-propelled) particles, which can be seen as an extreme case of phoretic particles (with self-generated gradients driving their drift), have also been reported to exhibit clustering when stirred by incompressible chaotic or turbulent flows [20,21].…”

“…Note that this corresponds to a solution with zero mean flux of particles transported by the flow. Actually, this term has to vanish whatever the symmetries of C(x) if glass bubbles are confined in a box as they cannot leave the domain, so that the total flux vanishes at the boundaries.Equation(19) now writes…”