Enhanced diffusiophoresis in dead-end pores with time-dependent boundary solute concentration

The dead-end pore is a geometry common to studies of diffusiophoresis. In most works on diffusiophoresis in dead-end pores, the solute concentration at the pore inlet is quickly changed between two extreme values, yielding a steplike change in solute concentration. We explore—noting diffusiophoresis is inherently nonlinear and other boundary conditions are largely unexplored—the influence of a time-variable boundary solute concentration. We first define metrics for the efficiency of diffusiophoretic injection and withdrawal and characterize the steplike boundary condition. We then introduce a linear change in solute concentration over a variable timescale and comment on changes in efficiency. We find particle migration becomes linear in the limit of slow transitions relative to the timescale for solute diffusion and define an asymptotic efficiency. We further characterize oscillatory changes in the boundary solute concentration and the implications of the oscillation frequency, from slow (yielding linear particle migration) to fast (yielding nonlinear paricle migration) oscillation. The particle dynamics are distinct and can be tuned by adjusting the timescale for oscillation. Our findings are relevant to processes both in the laboratory, where they could inform the choice of timescale or oscillation frequency for changes in solute concentration, and in nature, where various events cause solute concentrations to change over distinct and disparate timescales.

Figure 1

Strategies for changing the solute concentration at the inlet of a dead-end pore. The step boundary, shown in (a), is typical of experiments involving diffusiophoresis in dead-end pores. A repeated step boundary condition, depicted in (b), could be used to change the concentration at the boundary between extreme values multiple times. The Y-channel configuration shown in (c) could be used to obtain an arbitrary concentration between the extreme concentrations by modulating the flow rate of each inlet. Configurations (b) and (c) could yield an oscillatory solute concentration at the pore inlet, and configuration (c) could be used to change the concentration arbitrarily with relatively slow and smooth transitions between extreme concentrations.

Figure 2

Definition of injection and withdrawal efficiency values for particle transport in a dead-end pore with a monotonic change in solute concentration at the boundary. We impose a uniform particle concentration in the pore at $t=0$. The distance the leading particle in (a) travels is the injection efficiency. The initial position of the particle that approaches $x=0$ for large $t$ in (b) is the withdrawal efficiency. We produced the images in (a) and (b) with a step boundary condition with $\beta =1/10$ and either (a) $\mathcal{M}/\mathcal{D}=1/4$ or (b) $\mathcal{M}/\mathcal{D}=-1/4$.

Figure 3

Contours of (a) injection and (b) withdrawal efficiency associated with step boundary conditions for a variety of values $\mathcal{M}/\mathcal{D}$ and $\beta$. Processes with $\beta <1$ have a higher efficiency than processes with $\beta >1$. In general, the efficiency increases as the magnitude of the diffusiophoretic mobility and the difference between final and initial solute concentrations increase. Efficiency values have converged to within $0.0001\%$.

Figure 4

Example of injection and withdrawal efficiencies for a linear change in solute concentration occurring over time $\mathrm{\Delta }t$. For this example, we consider $\varphi =1$. We calculate efficiency numerically by performing repeated simulations with smaller time steps and additional terms in truncated approximations of sums until the result has converged to within $0.001\%$ relative error. When the transition time is large relative to the timescale for solute diffusion, the process efficiency approaches the asymptote given in Eq. (9). Notably, the slow-transition efficiency does not change with $\beta \mapsto 1/\beta$ or $\mathcal{M}/\mathcal{D}\mapsto -\mathcal{M}/\mathcal{D}$. Efficiency decreases with increasing $\mathrm{\Delta }t$ if the boundary concentration decreases. Similarly, efficiency increases with increasing $\mathrm{\Delta }t$ if the boundary concentration increases. Colors are consistent with those used in Fig. 3.

Figure 5

Demonstration of linearity for a cyclical process, shown in (a), over long times. We fix the period of oscillation, $2\tau$, and assume that the effect of the initial condition on the concentration profile is negligible. We show contours of $c$ in (b), (c), and (d) and contours of $dlnc/dx$ in (e), (f), and (g). When the oscillatory timescale is comparable to the solute diffusion timescale, shown in (b) and (e) with $\tau =1$, there is considerable asymmetry about the vertical line through integer multiples of $\tau$. As $\tau$ increases—shown with $\tau =10$ and 100 in (c), (d), (f), and (g)—the contours of $dlnc/dx$ approach a configuration that is antisymmetric about the vertical line. When integrating particle trajectories with $\tau \gg 1$, the migration over either the injection or withdrawal process is negated by migration over the complementary process.

Figure 6

Demonstration of particle migration during an injection process with an oscillatory boundary. Each period of oscillation, depicted in (a), results in a net migration of particles in a fixed direction. All cases with oscillatory boundaries (except those with very slow oscillation) will eventually attain an efficiency that exceeds that associated with the step boundary condition and approaches 1 asymptotically. We demonstrate in (b) that it is likely possible to apply this to real-world systems; with $\mathcal{M}/\mathcal{D}=1/4$ and $\beta =1/10$, the efficiency associated with a sinusoidal boundary can exceed the efficiency associated with a step boundary where $t$ is $\mathcal{O}\left(10\right)$. For other boundaries with a more significant contrast between injection and withdrawal efficiencies, such as repeated step conditions, the oscillatory condition may yield a higher efficiency than a single step at even lower $t$. Notably, the best choice of frequency depends on the solute and particle species (through $\mathcal{M}/\mathcal{D}$), the magnitude of $\beta$, and the length of time over which oscillation occurs. In (b), the particle positions at $t=15$ have converged to within $0.1\%$ relative error.

Figure 7

Example of an oscillatory boundary condition for which the qualitative particle dynamics are dependent on the timescale of the boundary condition. We impose a sinusoidal boundary condition with a period of $2\tau$ and an upper bound of $\beta =100$, as shown in (a). We then consider the trajectories of particles with $\mathcal{M}/\mathcal{D}=1$ over $t\in [n\tau ,(n+2\left)\tau \right]$ for $n=0,2,4,...$. We assume the cycle has been occurring for long enough that the effect of the initial condition is negligible. We show regions of particle injection and withdrawal with a boundary to indicate the deepest particle that is withdrawn over one period (the particle that approaches $x=0$ but does not reach it), calculated to within $0.1\%$ relative error, in (b)–(e). Plot (b) shows the case where $\tau \gg 1$, for which the particle migration is linear (see Fig. 5) and the curve defining the withdrawal boundary can be found with Eq. (8). As $\tau$ decreases, so does the degree of antisymmetry in $dlnc/dx$ about integer multiples of $\tau$. This causes the particle path to distort. In this case, for $\tau \approx 1$, all of the particles initially present in the pore are injected deeper over one period, as shown in (e); no particles are withdrawn. It is possible, therefore, to achieve dramatically different particle dynamics by changing only the timescale for oscillation of the solute concentration at the pore entrance.

Figure 8

Demonstration of frequency dependence of injection and withdrawal processes; this figure complements Fig. 7. In (a), the fraction of particles remaining in the pore over time is shown for two values of $\tau$ and two values of $\beta$ and $\mathcal{M}/\mathcal{D}$. In (b), the regions of injection and withdrawal are shown. With slow oscillation, particle withdrawal is preferred; this is depicted for $\tau =1000$ in (a) and (b) with marker A. With rapid oscillation, particle injection is preferred; we show this with marker B. Notably, the withdrawal performance depends on $\beta$ and $\mathcal{M}/\mathcal{D}$. The fraction of particles remaining in the pore is accurate to within $\pm 0.005$.