Upper bounds on the separation efficiency of diffusiophoresis

TL;DR

The paper addresses the fundamental question of the maximum achievable separation of colloids via diffusiophoresis in cross-channel chemical gradients by deriving a continuum theory for the fully developed, downstream regime where advection competes with Brownian diffusion.It introduces a coupled set of advection–diffusion–reaction equations for the solute, dissociation ions, and colloids, and reduces them to a fourth-order nonlinear problem for the ionic concentration with boundary conditions that depend on whether the chemical source is liquid or gas permeation.Three key dimensionless numbers—$Da_s$, $Da_i$, and $Pe_p$—define four distinct asymptotic regimes, each predicting specific boundary-layer widths $\delta_\pm$ and corresponding particle distributions that maximize freshwater recovery; water-recovery expressions are given in terms of these boundary layers.Experimentally, CO$_2$-driven diffusiophoresis in PDMS microfluidic channels validates the theory for at least one regime, showing $n_{max}\propto Pe_p^{1/2}$ for chemo-repelled particles and highlighting practical implications for designing energy-efficient separations and estimating bounds on clean-water yield.

Abstract

The separation of colloidal particles from fluids is essential to ensure a safe global supply of drinking water yet, in the case of microscopic particles, it remains a highly energy-intensive process when using traditional filtration methods. Water cleaning through diffusiophoresis $\unicode{x2014}$spontaneous colloid migration in chemical gradients$\unicode{x2014}$ effectively circumvents the need for physical filters, representing a promising alternative. This separation process is typically realized in internal flows where a cross-channel electrolyte gradient drives particle accumulation at walls, with colloid separation slowly increasing in the streamwise direction. However, the maximum separation efficiency, achieved sufficiently downstream as diffusiophoretic migration (driving particle accumulation) is balanced by Brownian motion (inducing diffusive spreading), has not yet been characterized. In this work, we develop a theory to predict this upper bound, and derive the colloid separation efficiency by analyzing the asymptotic structure of the governing equations. We find that the mechanism by which the chemical permeates in the channel, as well as the reaction kinetics governing its dissociation into ions, play key roles in the process. Moreover, we identify four distinct regimes in which separation is controlled by different scaling laws involving a Damköhler and a Péclet number, which measure the ratio of reaction kinetics to ion diffusion and of diffusiophoresis to Brownian motion, respectively. We also confirm the scaling of one of these regimes using microfluidic experiments where separation is driven by CO$_\text{2}$ gradients. Our results shed light on pathways towards new, more efficient separations, and are also applicable to quantify colloidal accumulation in the presence of chemical gradients in more general situations.

Figures (4)

• Figure 1: Colloid separation through diffusiophoresis in field-flow fractionation. (a) Upstream, the interplay between flow advection and the permeation of the chemical results in a developing distributions of chemical species and particles. (b) Sufficiently far downstream, all fields are fully-developed and invariant in the streamwise direction.
• Figure 2: Colloid separation through diffusiophoresis strongly depends on the dissociation chemistry and type of permeating membrane. Each sub-figure represents: (a)$Da_i\ll{1}$, ions equilibrated with reservoirs [Eqs. \ref{['eq:bcs_chem_liquid_nondim']}], (b)$Da_i\ll{1}$, no ionic wall flux [Eq. \ref{['eq:bcs_chem_gas_nondim']}], (c)$Da_i\gg{1}$, ions equilibrated with reservoirs [Eq. \ref{['eq:bcs_chem_liquid_nondim']}], (d)$Da_i\gg{1}$, no ionic wall flux [Eq. \ref{['eq:bcs_chem_gas_nondim']}].
• Figure 3: Microfluidic experiments reproduce the regime of fully-developed particle concentrations using $\ce{CO2}$-driven diffusiophoresis. (a) Sketches of the top view and cross-section of the microfluidic devices. (b) Bright-field image of the acquisition window, followed by the fluorescent signal acquired at steady state for five different flow rates. Note that the white particles (in this example, c-PS particles of diameter $0.1µm$) progressively accumulate at the top wall (i.e. the sink) with a decreasing flow rate. Error bars on the bottom left corner correspond to $100µm$. (c) Corresponding fluorescent signal, normalized to yield a mean value of 1 (dashed line). Error bars indicate the standard deviation obtained from averaging the signal in time and in the streamwise direction. The data shows a thin particle-free boundary layer (i.e. an "exclusion zone") near the source of $\ce{CO2}$ that grows with a decreasing flow rate. The two curves with the lowest flow rates display a statistically negligible discrepancy, demonstrating that they correspond to the "fully-developed" regime of interest.
• Figure 4: Experimental measurements of fully-developed particle profiles reproduce the scaling predicted by theory. (a) Fluorescent signal, normalized to yield a mean value of 1, for fully-developed profiles of particles of different types and sizes, as detailed in Table \ref{['tab:particles']}. (b) Maximum particle concentration $n_\text{max}$ (at walls $y=\pm{1}$) as a function of the particle Péclet number $Pe_p$, as defined in Equation \ref{['eq:def_Pe_p']}. Note the trend $n_\text{max}\propto{Pe_p^{1/2}}$, predicted by theory, for chemo-repelled particles (c-PS and PS). For chemo-attracted particles (a-PS), the available experimental data displays significantly lower values of $n_\text{max}$, agreeing qualitatively with theory.