Advective-diffusive spreading of diffusiophoretic colloids under transient solute gradients

Abstract

We analytically calculate the one-dimensional advective-diffusive spreading of a point source of diffusiophoretic (DP) colloids, driven by the simultaneous diffusion of a Gaussian solute patch. The spreading of the DP colloids depends critically on the ratio of the DP mobility, M (which can be positive or negative), to the solute diffusivity, D_s. For instance, we demonstrate, for the first time, that solute-repelling colloids (M < 0) undergo long-time super-diffusive transport for M/D_s < −1. In contrast, the spreading of strongly solute-attracting colloids (M/D_s ≫ 1) can be spatially arrested over long periods on the solute diffusion timescale, due to a balance between colloid diffusion and DP under the evolving solute gradient. Further, a patch of the translating solute acts as a “shuttle” that rapidly transports the colloids relative to their diffusive timescale. Finally, we use numerical computations to show that the above behaviors persist for three-dimensional, radially symmetric DP spreading. The results presented here could guide the use of DP colloids for microscale particle sorting, deposition, and delivery.

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1 Introduction

Diffusiophoresis (DP) is the deterministic drift of one species induced by the concentration gradient of another species.^1–3 A common instance is DP of a colloidal particle in an electrolyte gradient, where the DP colloid velocity u_DP is proportional to the gradient of the logarithm of the solute (ion) concentration S; that is, u_DP = M∇ log S.⁴ Prior studies focused on calculating the DP mobility M with respect to the interactions between the colloid and the surrounding ions, showing that M can be positive or negative, corresponding to DP driving colloids up (solute-attracting) or down (solute-repelling) the solute gradient, respectively.^1,2,4–8 Recently, there has been intense interest in utilizing DP to affect microscale colloid transport, including modification of the Batchelor scale for mixing^9,10 and chaotic advection;^11–13 log-sensing osmotic traps and particle manipulation;^14–18 and transport in dead-end pores.^19–22 Underlying these applications is the nonlinear relation between the DP colloid velocity and the solute gradient.

The advective-diffusive spreading of DP colloids is particularly rich precisely due to the nonlinear dependence of u_DP on S. For instance, Ault et al.²¹ calculated similarity solutions for the entrainment (extraction) of DP colloids into (from) a dead-end pore, under the assumption that the transport is one-dimensional along the axis of the pore. The pore initially is filled uniformly with a solute and is devoid of colloids. Colloids continuously enter the pore via a boundary with a prescribed colloid concentration. The same boundary also allows solute exchange by which a solute gradient within the pore is developed to cause colloid DP. In the limit of negligible colloid diffusivity, the method of characteristics was applied to predict the DP colloid motion. This group later extended their analysis to quantify particle focusing under pressure-driven and diffusioosmotic flows.²² Raynal et al.⁹ analyzed the spreading of a coincident deposition of Gaussian patches of solute and colloids in a two-dimensional linear flow, demonstrating that a patch of solute-attracting colloids first undergoes spatial compression before eventually spreading in the flow. Subsequently, an effective Péclet number for quantifying the mixing of colloids and salt in the same setting as in ref. 9 as well as in a sinusoidal patch under a two-dimensional periodic flow was defined.¹⁰

In this article, we analyze the advective-diffusive spreading of a point source of DP colloids, driven by the simultaneous diffusion of a Gaussian solute patch. In Section 2, we calculate the one-dimensional DP spreading of a point source of colloids under the influence of a diffusing solute patch whose centroid is coincident with that of the colloid distribution. Exact solutions of the solute and colloid evolution are derived to demonstrate, for the first time, two novel phenomena: super-diffusion of solute-repelling colloids and arrested spreading (which might be interpreted as “extreme sub-diffusion”) of strongly solute-attracting colloids. An asymptotic analysis is conducted to unify the long-time response for a variety of initial solute profiles. In Section 3, we illustrate how super-diffusion and arrested spreading could enhance the extraction and delivery of DP colloids when coupled with a spatially non-coincident deposition of solute and colloids. In Section 4, we show that a non-coincident deposition of the solute that undergoes advection could shuttle colloids over large distances rapidly compared to their diffusive transport. In Section 5, we use numerical computations to show that the above behaviors persist for three-dimensional DP spreading. We summarize this study and offer ideas for future work in Section 6.

2 One-dimensional coincident DP spreading

We first focus on the simplest physical picture of one-dimensional DP transport, mimicking that in a long, uniform microchannel. This will allow us to derive exact solutions, whose behavior we show persists in higher spatial dimensions. Thus, consider DP of a point source of colloids in an unbounded domain −∞ < x < ∞ in the absence of solvent flow. For a colloid concentration C much less than that of the solute S, the influence of the evolution of C on S is negligible. This approximation originates from Anderson, Prieve, and co-workers,^2,4,5,23 who formally derived the relation u_DP = M∇ log S first proposed by Derjaguin and co-workers.¹ The chief physical assumption of the relation is based on an ideal solution such that the ionic fluxes are given by the Nernst–Planck equation, which gives very good predictions for electrolyte solutions up to millimolar concentration near moderately charged surfaces of zeta potential ζ ∼ O(kT/e) (k the Boltzmann constant, T the absolute temperature, and e the unit charge). The effects of non-ideal solution such as steric repulsion due to finite ion size, which are important in concentrated electrolytes or nano-confined systems, will generate additional chemical potentials and hence fluxes to the classical Nernst–Planck equation. Relevant discussions and detailed analyses can be found in ref. 6 and 7. Recent experiments^13–16,20 have shown excellent agreements with theoretical predictions^{9,10,12,21,22} of DP colloid transport based on the present approximations. Hence, DP of the colloids is due to a simultaneous, spatially coincident source of the solute whose diffusive transport is governed bywhere D_s is the solute diffusivity and t is the time. For a Gaussian distribution of the solute centered at x = 0 and deposited at t = 0,where is the lengthscale characterizing the initial solute width [σ_s²(0) is the initial solute variance], the solution to eqn (1) iswith centroid and variance²⁴andwhere Q_s is the mass of the solute per unit area and α(t) = 1 + 4D_st/l². Eqn (1) yields the DP velocity u_DP = M∂(log S)/∂x = −2Mx/(α(t)l²) in the advection-diffusion equation for the colloids:where D_c is the diffusivity of the colloid. Note that u_DP is spatially non-uniform and hence represents a “compressible” velocity field. Thus, eqn (6) becomeswhose solution for a point source of colloids at x = 0 and t = 0, C₀(x,0) = Q_cδ(x)δ(t), is‡withwhere Q_c is the mass of colloids per unit area, δ is the Dirac delta, and D_eff = D_cD_s/(M + D_s) is an effective diffusivity which also appears in the effective Péclet number of previous studies.^9,11,12 Consistent with prior work, DP colloid evolution [eqn (8)] is not obtained simply via replacing D_s by D_eff in the solute evolution [eqn (3)], due to the compressibility of the DP velocity. In other words, DP spreading cannot be simply treated as pure diffusion with an effective diffusivity. The centroid and the variance of the colloid distribution are given by eqn (4) and (5) with S replaced by C.

We plot the time evolution of S, C, and σ_c² in Fig. 1. The variance is

where long-time asymptotics at t ≫ l²/D_s are shown on the right of the half bracket. Henceforth we choose an experimentally relevant ratio D_c/D_s = 10⁻², which corresponds to colloids of size O(10⁻²–10⁻¹) μm for small ion solutes of D_s = O(10³–10²) μm² s⁻¹. All phenomena discussed below persist for D_c/D_s ≪ 1. The evolution of the colloids depends critically on M/D_s [Fig. 1(a)], which is typically O(1) and can be positive or negative.^16,20 For M/D_s < −1,§ the solute-repelling DP leads to super-diffusion of colloids at long times: σ_c² ∼ t^−M/D_s, indicating that rapid spreading of colloids can be achieved by DP. The prediction of super-diffusion is the major, original conclusion of our work. For instance, at t/(l²/D_s) = 10², the variance for M/D_s = −5 is O(10¹⁰) times larger than that for M/D_s = 0. For illustrative purpose in a situation where D_s = O(10³) μm² s⁻¹ and l = O(10) μm, DP colloids will spread over a characteristic distance σ_c = O(10²) μm in 1 s for M/D_s = −5 compared to σ_c = O(1) μm for M/D_s = 0. For −1 ≤ M/D_s ≤ 1, DP has a relatively minor impact on the diffusive spreading of colloids. For M/D_s = 10³,¶ the colloids first undergo regular diffusion at short times [t/(l²/D_s) ≲ 10⁻³]; σ_c² grows linearly with time. Subsequently, the DP forces that drive the colloids up the solute gradient dominate over diffusion of colloids. Solute-attracting DP inhibits the diffusive spread of colloids such that over a sizable period of time, 5 × 10⁻⁴ ≲ t/(l²/D_s) ≲ 5 × 10⁻², the colloid spreading is spatially arrested, which might be interpreted as an extreme sub-diffusion, in contrast to the super-diffusive behavior mentioned above. At t/(l²/D_s) ≳ 10⁻¹ the solute gradient is sufficiently weak—the arrest is relaxed—and the colloid distribution recovers a diffusive spreading with an effective diffusion coefficient D_eff, highlighting that DP affects the long-time transport. As a general remark, the time interval of arrested spreading increases with M/D_s. Fig. 1(b) illustrates the difference in the broadening of the colloid distribution between different values of M/D_s at t/(l²/D_s) = 10². For arrested spreading (M/D_s = 10³) the colloid distribution broadens negligibly; the colloids spread more than 10² times the width of the initial solute distribution in the presence of super-diffusion (M/D_s = −2). The spreading occurs on the solute diffusion time, which is much faster than the colloid diffusion time since D_c/D_s is small. The centroids of the colloid distributions remain stationary, μ_c = 0.

Fig. 1 (a) Time evolution of σ_c²/l² for D_c/D_s = 0.01 and various M/D_s. From top to bottom: M/D_s = −5, −2,0 (red dotted), 1, 10, 10², and 10³. Super-diffusion of colloids occurs at long times for M/D_s < −1 whereas arrested spreading or “extreme sub-diffusion” occurs at intermediate times for M/D_s ≫ 1. (b) C/(Q_c/l) (red solid: M/D_s = −2; green solid: M/D_s = 0; and blue solid: M/D_s = 10³) and S/(Q_s/l) (black dotted) for D_c/D_s = 0.01 and various M/D_s at t/(l²/D_s) = 10². Colloids undergoing super-diffusion (M/D_s = −2) exhibit significant spreading in contrast to the case with arrested spreading (M/D_s = 10³) or without DP (M/D_s = 0).

The transition of colloid evolution from regular diffusion for M/D_s ≥ −1 to super-diffusion for M/D_s < −1 at long times is expected for a broad class of initial colloid C₀(x,0) and solute S₀(x,0) distributions. By the method of steepest descent,²⁶S₀(x,0) asymptotes to a Gaussian

at long times, provided thatis a convergent integral. All information of the initial solute distribution is encompassed in A(t). However, u_DP ∼ Mx/2D_st at long times because the time-dependent term of α(t) dominates; that is u_DP is independent of A(t) at long times. Hence, the influence of the initial solute distribution on the DP spreading is lost. Hence, the long-time colloid transport equation is

Since eqn (13) is the long-time asymptote of eqn (7), we expect the transition from regular to super-diffusion to be generic for a variety of initial solute profiles. We illustrate this by numerically solving the DP spreading of an initially smoothed point source of colloids subject to a Laplace distribution of the solute [Fig. 2(b)]:

andwhere is the lengthscale of the initial colloid distribution. Numerical solutions are obtained by a standard forward-in-time-central-in-space finite-difference scheme.²⁷ The transition is clearly demonstrated in Fig. 2(a), which leads to significantly different long-time colloid spreading between the case with and without super-diffusion [Fig. 2(c)]. Notice too that the solute profile eventually becomes Gaussian.

Fig. 2 (a) Evolution of σ_c²/l² for D_c/D_s = 0.01 and various M/D_s for an initial Laplace distribution of the solute. Red dotted: M/D_s = −2; blue solid: M/D_s = 1. (b) Initial colloid and solute distributions, C/(Q_c/l) (red solid) and S/(Q_s/l) (black dotted), for D_c/D_s = 0.01. (c) C/(Q_c/l) (red solid: M/D_s = −2; blue solid: M/D_s = 1) and S/(Q_s/l) (black dotted) for D_c/D_s = 0.01 and various M/D_s at t/(l²/D_s) = 10². Transition from regular diffusion for M/D_s ≥ −1 to super-diffusion for M/D_s < −1 at long times is not limited to an initial point source of colloids and a Gaussian solute profile.

3 One-dimensional non-coincident DP spreading

Next, we demonstrate how a spatially non-coincident source of solute and colloids can enhance colloid extraction and delivery when coupled with arrested spreading and super-diffusion responses. The diffusion of a Gaussian distribution of the solute centered at x = x₀ and deposited at t = 0,is described by eqn (3) with x being replaced by x − x₀. The corresponding colloid transport equation is identical to eqn (7) except that x in the prefactor of the ∂C/∂x term is replaced by x − x₀. The solution for a point source of colloids at x = 0 and t = 0 is then||where , and the centroid is

μ_c(t) = x₀γ(t). (18)

The variance is the same as eqn (10).

The time evolution of S, C, μ_c, and σ_c² are plotted in Fig. 3. First consider the solute-attracting case of M/D_s = 10³ [Fig. 3(a) and (b); see supplementary videos, ESI†]. As the solute is deposited at x₀/l = 5, colloids are attracted towards the solute centroid—a “soluto-inertial beaconing” effect that has been observed experimentally.^14,17 Here, we show that this beaconing can be coupled to arrested spreading to achieve a high recovery of colloids from a remote location: while the colloid distribution broadens slightly at short times [t/(l²/D_s) ≲ 10⁻³], the spreading of colloids is subsequently arrested and major spreading only occurs upon colloids arrival at the solute centroid. Note that the evolutions of σ_c² with t are the same regardless of x₀. The evolutions of μ_c with t for different x₀ can be collapsed onto a single curve upon dividing by x₀, as apparent from eqn (18).

Fig. 3 Time evolution of (a) C/(Q_c/l) (red solid) and S/(Q_s/l) (black dotted) and (b) σ_c²/l² (red solid) and μ_c/l (blue dotted) for M/D_s = 10³, D_c/D_s = 0.01 and x₀/l = 5. By arrested spreading, the majority spreading of colloids occurs upon it being attracted towards the solute centroid, leading to high recovery of colloids. Time evolution of (c) C/(Q_c/l) (red solid) and S/(Q_s/l) (black dotted) and (d) σ_c²/l² (red solid) and μ_c/l (blue dotted) for M/D_s = −2, D_c/D_s = 0.01 and x₀/l = −1. Colloids are transported and spread over a large distance due to repelling solute and super-diffusion, suggesting enhanced colloid delivery. In (a) and (c) the time labels in the dotted and solid boxes refer to the solute and colloid distribution, respectively, for different t/(l²/D_s). In (b) and (d), the arrows point at the y-axes to which the lines refer. These evolutions can be seen clearly in the supplementary videos, (ESI†).

Fig. 3(c) and (d) (see supplementary videos, ESI†) show the solute-repelling case of M/D_s = −2. Here, the colloids are driven away from the solute deposited at x₀/l = −1. The colloids spread diffusively initially after which super-diffusion takes over at long times. Notice the large distance traveled by the colloids, whose centroid reaches x/l = 400 at t/(l²/D_s) = 10². If DP was absent, the colloids would spread diffusively about x = 0 and a negligible amount would reach x/l = 400 at t/(l²/D_s) = 10². Thus, the present results suggest a means for enhanced colloid delivery via DP. Previous remarks concerning Fig. 3(b) on the evolution of σ_c² and μ_c with t and the collapse of the latter onto a single curve also holds for Fig. 3(d).

4 One-dimensional non-coincident DP spreading under a translating solute

Here, we analyze colloid transport by a spatially non-coincident release of the solute that is undergoing advection upon its deposition. The colloids are free from solvent advection. This case mimics and generalizes a recent laboratory setup: a “soluto-inertial beacon” that emits a long-lasting solute flux.¹⁷ Although in ref. 17 the beacon was not translating, this could be realized by a beacon that sediments under gravity while the colloids are neutrally buoyant. Consider a Gaussian distribution of the solute centered at x = x₀ and deposited at t = 0 that is now advected with a constant velocity w upon its release. Since w is a spatial and temporal constant, the effect of the solvent flow is simply to advect the centroid of the solute Gaussian, without distortion of its shape into a fore-aft asymmetric form. This statement is valid for one-dimensional advective-diffusive transport. Note, in our suggestion of a sedimenting beacon the (three-dimensional) solvent velocity field around the beacon could lead to a loss of the fore-aft symmetry of solute release. Here, the solute evolution is described by eqn (3) with x being replaced by x − x₀ − wt such that the solute centroid μ_s(t) = x₀ + wt. The corresponding colloid DP transport equation is identical to eqn (7) except that x in the prefactor of the ∂C/∂x term is replaced by x − x₀ − wt. The solution for a point source of colloids at x = 0 and t = 0 is**wherewith Pe = wl/D_s, a Péclet number describing the relative importance of solute advection to diffusion. The centroid of the colloids is

μ_c(t) = x₀γ(t) − κ(t)l. (21)

The variance is the same as eqn (10).

We plot the time evolution of μ_c, σ_c², and μ_s for Pe = 1 in Fig. 4. With M/D_s = 10³ [Fig. 4; see supplementary videos, ESI†], colloids are attracted towards the initial solute centroid regardless of x₀. Strongly attractive DP arrests spreading of colloids at early times, and the majority of spreading occurs at long times over which colloids are advected with the solute [μ_c ∼ μ_s at t/(l²/D_s) ≫ 1]. Here, the solute acts to guide and shuttle the movement, spreading, and deposition of colloids. Controlled deposition can also be achieved by solute-repelling DP [Fig. 5] where the deposition distance may be much larger than that for solute-attracting DP. However, a slight difference in M/D_s may lead to a qualitatively different colloid spreading with repelling DP. For instance, for M/D_s = −1.5 the solute and colloids travel in the opposite direction, which is in contrast to that for M/D_s = −2.5 despite the same x₀/l = −1 [Fig. 5(a)]. This occurs because, with M/D_s = −1.5, solute advection is strong relative to the DP velocity at early times such that the solute centroid overtakes that of the colloids. Regardless of the sign and magnitude of w, such a bi-directional transport of solute and colloids is observed for a spatially coincident deposition with solute advection [Fig. 5(b); M/D_s = −1.5, x₀ = 0].

Fig. 4 Time evolution of σ_c²/l² (red crosses), μ_s/l (blue circles), and μ_c/l (solid green) for M/D_s = 10³, D_c/D_s = 0.01, and Pe = 1. (a) x₀/l = 20; (b) x₀/l = −20. In both cases, colloids are first attracted towards the solute centroid and then shuttled by the solute. The evolutions in (a) can be seen clearly in the supplementary videos, (ESI†).

Fig. 5 Time evolution of σ_c²/l² (red crosses and solid triangles), μ_s/l (blue circles and triangles), and μ_c/l (green solid and dotted lines) for D_c/D_s = 0.01, and Pe = 1. The two panels share the same set of data for M/D_s = −1.5 and x₀/l = −1 for comparison. In (a), changing M/D_s from −1.5 to −2.5 while keeping x₀/l fixed causes the solute and colloids to travel in the same direction. In (b), a bi-directional transport of the solute and colloids is always observed for a spatially coincident deposition (x₀/l = 0).

5 Three-dimensional DP spreading

Lastly, we show that the novel super-diffusion and arrested spreading responses identified in this work persist for DP in higher dimensions. This is done by generalizing the analysis in Section 2 to three dimensions. The equation governing the diffusive transport of the solute in three dimensions iswhere ∇² here is the Laplace operator in the spherical coordinate (r,θ,Φ) specifying the radial direction r, the polar angle θ, and the azimuthal angle Φ. For a Gaussian distribution of the solute centered at r = 0 and deposited at t = 0,the solution iswhere Q_s here is the mass of the solute. Eqn (24) yields the DP velocity u_DP = M∇ log S = −2Mr/(α(t)l²)i_r (i_r is the radial unit vector) in the advection-diffusion equation for the colloids:where ∇· is the divergence in spherical coordinates. Thus, eqn (25) becomes, under the assumption of a radially symmetric spreading,

An exact solution of eqn (26) for a point source of colloids at r = 0 and t = 0 is not obtainable by the method employed in the one-dimensional case. Thus, we solve eqn (26) numerically using the built-in partial differential equation solver pdepe in MATLAB and plot the time evolution of σ_c² in Fig. 6. As in Fig. 1(a), super-diffusion is observed at long times t ≫ l²/D_s for M/D_s < −1; and the spreading of colloids is spatially arrested—a constant variance over time—at intermediate times for M/D_s ≫ 1. Observations made in Sections 3 and 4, in particular the shuttling response, are also expected to hold in three dimensions.

Fig. 6 Time evolution of σ_c²/l² in three dimensions for D_c/D_s = 0.01 and various M/D_s. From top to bottom: M/D_s = −2, 0 (red dotted), 1, 10, 10², and 10³. Super-diffusion of colloids occurs at long times for M/D_s < −1 whereas arrested spreading or “extreme sub-diffusion” occurs at intermediate times for M/D_s ≫ 1.

6 Conclusion

We have analyzed the advective-diffusive spreading of DP colloids under transient solute gradients, revealing three novel phenomena: namely, super-diffusion, arrested spreading (or extreme sub-diffusion), and shuttling. Exact solutions are derived for a one-dimensional point source of colloids in a Gaussian patch of solute for three cases: a spatially coincident and non-coincident deposition, and a non-coincident deposition with a translating solute patch. We have shown that DP radically alters the spreading of colloids, which is critically dependent on the ratio of the DP mobility to solute diffusivity. We have demonstrated that solute-attracting DP can arrest the colloid distribution spreading over a sizable period of time, suggesting a novel mechanism for particle trapping. In contrast, solute-repelling DP leads to super-diffusion of colloids at long times: a means for rapid spreading. An asymptotic analysis indicates that the transition from regular to super-diffusion at long times is generic for a variety of initial solute profiles. For a spatially non-coincident solute and colloid deposition, we have presented two schemes for particle transport: the first to recover a high yield of solute-attracting colloids from a distant location and the second to achieve a long-range, rapid transport of solute-repelling colloids. Further, we have illustrated that a translating solute acts to shuttle the movement of colloids. We closed by demonstrating the generality of the three novel transport phenomena to DP in three spatial dimensions.

As noted in Section 2, we assume that the solute evolution is unaffected by the spreading of the colloid distribution, as is typical in modeling DP colloid transport. Intuitively, or formally from the Onsager reciprocal relation, one may expect that the gradient in the colloid concentration would affect the solute distribution via generation of a cross-flux. However, as demonstrated in experiments on DP transport^13–16,20 and recent analyses of multi-component ion transport,^28,29 the effect of such a cross-flux, and hence the influence of the colloid motion on the solute distribution, is negligible when the colloid concentration is small compared to that of the solute. This is the relevant limit of our study, in which the colloids are micron-scale and the solute (i.e. ions) is sub-nanometer, and the colloid concentration is orders of magnitude less than the solute concentration. Nevertheless, the effect of cross-fluxes on DP spreading could be relevant to systems where the two species (i.e. the colloid and solute) are of similar size^3,30 and concentration.^28,29 We will, however, leave this interesting effect for future studies.

The present findings will enable new techniques for colloid transport at microscopic scales, with applications in particle separation, sensing, and deposition. Further modeling work could focus on multiple solute and colloid sources and the stability of the spreading fronts.

Conflicts of interest

There are no conflicts to declare.

Appendix: derivations and solutions for one-dimensional DP spreading

Here, we derive eqn (19) for the colloid concentration in the case of a non-coincident solute deposition, which subsequently undergoes advection with velocity w.³¹Eqn (19) can readily be simplified to obtain the solution for the other two specific cases, namely coincident [eqn (8)] and non-coincident deposition [eqn (17)], both in the absence of solvent flow.

To obtain eqn (19), we first recall the corresponding advection-diffusion equation

and equate it with the following formThe functions [scr A, script letter A](t), [scr B, script letter B], and [capital script C](x,t) are identified as

[scr A, script letter A](t) = 1, (29)

On further equating [scr B, script letter B][capital script C](x,t) = [scr D, script letter D]x[scr E, script letter E](t) + [scr B, script letter B][scr F, script letter F](t), the functions [scr D, script letter D], [scr E, script letter E](t), and [scr F, script letter F](t) are identified as

[scr E, script letter E](t) = α⁻¹(t), (33)

Eqn (27) can then be written asvia the transformationwhere

To proceed, an initial condition is assumed

C(X,0) = C[script letter H](X), (38)

and the following three transformations are invokedAs a result, eqn (35) can be written as the diffusion equationwhere the initial condition becomes

K(η,T = 0) = C[script letter H](η). (43)

Thus, eqn (19) can be obtained from eqn (42) using the standard method of Green's function.³²

Appendix: a sample physical system showing super-diffusion

Here, we suggest a physical system in which super-diffusion (M/D_s < −1) and arrested spreading (M/D_s ≫ −1) could occur. In the case of charged colloids in a binary electrolyte, the DP mobility M takes the form:²³where the first and the second terms on the right-hand side of eqn (44) are the electrophoretic and chemiphoretic contributions to diffusiophoresis, respectively.^3,14,23 Here, Ψ = zeζ/kT is the normalized zeta potential of colloids ζ, l_B = (ze)²/4πεkT is the Bjerrum length (e the unit charge, z the valence of the ions, k the Boltzmann constant, T the absolute temperature, and ε the dielectric constant of water), and G = tanh(Ψ/4). The factor R ≡ (D₊ − D₋)/(D₊ + D₋) is defined using the diffusivity of the cations (D₊) and anions (D₋). Solute diffusivity is computed by the Nernst–Hartley formula D_s = 2D₊D₋/(D₊ + D₋).^14,33 For dissociation of phosphoric acid into H⁺ (D₊ = 9.31 × 10³ μm² s⁻¹) and H₂PO₄⁻ (D₋ = 0.85 × 10³ μm² s⁻¹),³ the evolution of M/D_s with ζ at T = 363 K (η = 0.32 × 10⁻³ N s m⁻²) is shown in Fig. 7. Super-diffusion occurs for ζ < −30 mV whereas weakly arrested spreading [M/D_s ≈ O(10)] is seen for strongly positive ζ. Although strongly arrested spreading is not common in typical settings, it is potentially attainable in some physico-chemical systems^21,25 and the present work prompts experiments to seek other systems for exploiting this phenomenon.

Fig. 7 Evolution of M/D_s with ζ for D₊ = 9.31 × 10³ μm² s⁻¹ and D₋ = 0.85 × 10³ μm² s⁻¹ at T = 363 K. Blue dotted: normalized chemiporetic mobility; red dotted: normalized electrophoretic mobility; black solid: normalized diffusiophoretic mobility due to the sum of its chemiphoretic and electrophoretic contributions. Super-diffusion is attainable for ζ < −30 mV.

Acknowledgements

This work was funded by Procter & Gamble.

References

1. S. S. Dukhin and B. V. Derjaguin, Electrokinetic phenomena, Wiley, New York, 1974.
2. J. L. Anderson, Annu. Rev. Fluid Mech., 1989, 21, 61–99.
3. D. Velegol, A. Garg, R. Guha, A. Kar and M. Kumar, Soft Matter, 2016, 12, 4686–4703.
4. D. C. Prieve, J. L. Anderson, J. P. Ebel and M. E. Lowell, J. Fluid Mech., 1984, 148, 247–269.
5. J. L. Anderson, M. E. Lowell and D. C. Prieve, J. Fluid Mech., 1982, 117, 107–121.
6. R. F. Stout and A. S. Khair, Phys. Rev. Fluids, 2017, 2, 014201.
7. D. C. Prieve, S. M. Malone, A. S. Khair, R. F. Stout and M. Y. Kanj, Proc. Natl. Acad. Sci. U. S. A., 2019, 116, 18257–18262.
8. A. Gupta, B. Rallabandi and H. A. Stone, Phy. Rev. Fluids, 2019, 4, 043702.
9. F. Raynal, M. Bourgoin, C. Cottin-Bizonne, C. Ybert and R. Volk, J. Fluid Mech., 2018, 847, 228–243.
10. F. Raynal and R. Volk, J. Fluid Mech., 2019, 876, 818–829.
11. J. Deseigne, C. Cottin-Bizonne, A. D. Stroock, L. Bocquet and C. Ybert, Soft Matter, 2014, 10, 4795–4799.
12. R. Volk, C. Mauger, M. Bourgoin, C. Cottin-Bizonne, C. Ybert and F. Raynal, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2014, 90, 013027.
13. C. Mauger, R. Volk, N. Machicoane, M. Bourgoin, C. Cottin-Bizonne, C. Ybert and F. Raynal, Phy. Rev. Fluids, 2016, 1, 034001.
14. B. Abecassis, C. Cottin-Bizonne, C. Ybert, A. Ajdari and L. Bocquet, New J. Phys., 2009, 11, 075022.
15. J. Palacci, B. Abecassis, C. Cottin-Bizonne, C. Ybert and L. Bocquet, Phys. Rev. Lett., 2010, 104, 138302.
16. J. Palacci, C. Cottin-Bizonne, C. Ybert and L. Bocquet, Soft Matter, 2012, 8, 980–994.
17. A. Banerjee, I. Williams, R. N. Azevedo, M. E. Helgeson and T. M. Squires, Proc. Natl. Acad. Sci. U. S. A., 2016, 113, 8612–8617.
18. N. Shi, R. Nery-Azevedo, A. I. Abdel-Fattah and T. M. Squires, Phys. Rev. Lett., 2016, 117, 258001.
19. A. Kar, T. Chiang, I. O. Rivera, A. Sen and D. Velegol, ACS Nano, 2015, 9, 746–753.
20. S. Shin, E. Um, B. Sabass, J. T. Ault, M. Rahimi, P. B. Warren and H. A. Stone, Proc. Natl. Acad. Sci. U. S. A., 2016, 113, 257–261.
21. J. T. Ault, P. B. Warren, S. Shin and H. A. Stone, Soft Matter, 2017, 13, 9015–9023.
22. J. T. Ault, S. Shin and H. A. Stone, J. Fluid Mech., 2018, 854, 420–448.
23. D. C. Prieve and R. Roman, J. Chem. Soc., Faraday Trans. 2, 1987, 83, 1287–1306.
24. R. Aris, Proc. R. Soc. London, Ser. A, 1956, 235, 67–77.
25. R. P. Sear and P. B. Warren, Phys. Rev. E, 2017, 96, 062602.
26. C. M. Bender and S. A. Orszag, Advanced mathematics methods for scientists and engineers, Springer-Verlag, New York, 1999.
27. J. H. Ferziger and M. Peric, Computational methods for fluid dynamics, Springer-Verlag, New York, 2002.
28. J. Peraud, A. J. Nonaka, J. B. Bell, A. Donev and A. L. Garcia, Proc. Natl. Acad. Sci. U. S. A., 2017, 114, 10829–10833.
29. B. Balu and A. S. Khair, Soft Matter, 2018, 14, 8267–8275.
30. J. F. Brady, J. Fluid Mech., 2011, 667, 216–259.
31. A. Sanskrityayn and N. Kumar, J. Earth Syst. Sci., 2016, 125, 1713–1723.
32. M. D. Greenberg, Applications of Green's functions in science and engineering, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.
33. J. Josserand, G. Lagger, H. Jensen, R. Ferrigno and H. H. Girault, J. Electroanal. Chem., 2003, 546, 1–13.

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Footnotes

† Electronic supplementary information (ESI) available: The colloid and solute evolutions in Fig. 3(a), (c), and 4(a). See DOI: 10.1039/c9sm01938c

‡ See Appendix for the derivation.

§ See Appendix for a setting where super-diffusion can be realized.

¶ M/D_s = 10³ is admittedly large for colloids in an ionic solution, but potentially attainable in some physico-chemical systems, such as near a liquid–liquid de-mixing critical point.^21,25

|| See Appendix for the derivation.

** See Appendix for the derivation.

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