Diffusiophoresis: from dilute to concentrated electrolytes

Abstract

Electrolytic diffusiophoresis is the movement of colloidal particles in response to a concentration gradient of an electrolyte. The diffusiophoretic velocity v_DP is typically predicted through the relation v_DP = D_DP ∇log c_s, where D_DP is the diffusiophoretic mobility and c_s is the concentration of the electrolyte. The logarithmic dependence of v_DP on c_s may suggest that the strength of diffusiophoretic motion is insensitive to the magnitude of the electrolyte concentration. In this article, we emphasize that D_DP is intimately coupled with c_s for all electrolyte concentrations. For dilute electrolytes, the finite double layer thickness effects are significant such that D_DP decreases with a decrease in c_s. In contrast, for concentrated electrolytes, charge screening could result in a decrease in D_DP with an increase in c_s. Therefore, we predict a maximum in D_DP with c_s for moderate electrolyte concentrations. We also show that for typical colloids and electrolytes , where D_s is the solute ambipolar diffusivity. To validate our model, we conduct microfluidic experiments with a wide range of electrolyte concentrations. The experimental data also reveals a maximum in D_DP with c_s, in agreement with our predictions. Our results have important implications in the broad areas of electrokinetics, lab-on-a-chip, active colloidal transport and biophysics.

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1 Introduction

The concentration gradient of an electrolyte induces a motion of charged colloidal particles through the phenomenon of diffusiophoresis.^1–6 Since diffusiophoresis enables control of colloidal transport, it has been exploited for applications in active transport,^7–10 membraneless water filtration,¹¹ zeta potential measurement,¹² delivery or extraction of particles to a dead-end pore,^13,14 colloidal focusing or trapping,^15–17 among others. Fundamental investigations have focused on understanding the effect of surfactant concentration gradients,¹⁸ high salinity,¹⁹ ion valence^20–22 and multiple electrolytes^20,23,24 on the diffusiophoresis of colloidal particles.

In electrolytic diffusiophoresis, the diffusiophoretic velocity v_DP is given by v_DP = D_DP ∇log c_s (ref. 3), where D_DP is the diffusiophoretic mobility and c_s is the electrolyte concentration. This expression has been utilized for a wide variety of experimental and theoretical studies.^{10–16,20,22,25–27} Since D_DP is typically assumed to be constant, the logarithmic dependence suggests that v_DP is insensitive to the magnitude of electrolyte concentration. For instance, if there are two concentration fields where one varies from 0.01 mM to 1 mM and the other varies from 10 mM to 1 M, and the conditions are such that both the fields have identical ∇log c_s, the above relation implies that the diffusiophoretic response will remain the same. In fact, in some scenarios, the logarithmic dependence can even predict a ballistic motion of colloidal particles where the particle transport is orders of magnitude faster than the diffusive transport of solute.^25,27 Therefore, in this article, we focus on the assumption that D_DP is constant and investigate the impact of a concentration dependent diffusiophoretic mobility, which is consistent with theory for predicting the influence of electrolyte concentration, on the aforementioned predictions.

The principal conclusion of our analysis is that assuming D_DP to be constant may lead to inaccurate conclusions since D_DP is a strong function of electrolyte concentration c_s. In the dilute limit, the finite double layer thickness effects become significant such that D_DP decreases with a decrease in c_s. In contrast, for the concentrated limit, charge screening could become significant¹⁹ and an increase in c_s could result in a lower D_DP. Upon inclusion of all these effects, we demonstrate that D_DPversus c_s displays a maximum for moderate electrolyte concentrations. We calculate achievable D_DP values for typical colloids and electrolytes and observe that , where D_s is the solute ambipolar diffusivity. We validate our predictions through experiments in a dead-end pore configuration where we vary electrolyte concentration by four orders of magnitude, while keeping ∇log c_s constant.

2 Mathematical details

We consider a binary 1 : 1 electrolyte (e.g. NaCl and KCl) where the ion concentration is denoted by c_s(x,t). We assume that the colloidal particles of radius a and concentration n_p are present in a concentration gradient of electrolyte ∇c_s(x,t) (Fig. 1). The transport of c_s is governed bywhere v_fluid is the fluid phase velocity and D_s is the electrolyte ambipolar diffusivity.^28,29 For a 1 : 1 electrolyte, , where D₊ and D₋ are diffusivities of the cations and the anions respectively. To describe the conservation of particles, we writewhere v_p is the particle velocity and D_p is the diffusivity of the particle. The particle velocity is given by^12,16,26,27

v_p = v_DP + v_fluid, (3)

where v_DP is the induced diffusiophoretic velocity and is estimated as³

v_DP = D_DP ∇log c_s, (4)

where D_DP is the diffusiophoretic mobility. Prieve et al.³ showed that for a spherical particle D_DP is of the form (for a 1 : 1 electrolyte)where ε is the electrical permittivity, k_B is the Boltzmann constant, T is temperature, e is the charge on an electron, μ is the fluid phase viscosity, is the Debye length and ζ is the dimensionless zeta potential scaled by the thermal potential . The numerator, i.e., u₀(ζ) is the leading-order term and is evaluated as^2,3,20where . The first term in eqn (6) is the electrophoretic contribution and the second term is the chemiphoretic contribution. We note that for |ζ| ≫ 1, u₀(ζ) is linear in |ζ|. We also note that u₀(ζ), and by extension D_DP, could be positive or negative,^3,20i.e., the particle can move up or down the external gradient.

Fig. 1 Colloidal particles of radius a in a solute concentration gradient, i.e., ∇c_s(x,t). The diffusiophoretic velocity of the particles is given as v_DP = D_DP∇ log c_s. We investigate the effect of finite values and the effect of different surface boundary conditions on D_DP.

The term in the denominator, i.e., is the correction,³ where , is the Péclet number. Since the expression of u₁(ζ,Pe) involves several integral terms and series expansions, we only summarize the main features here and refer the readers to the details provided in the Appendix and ref. 3 (see pp. 266–267, eqn (B1)–(B12)). The value of u₁(ζ,Pe) is always negative such that the correction typically decreases D_DP. More importantly, the correction can become significant even for .^3,14 Finally, the value of u₁(ζ,Pe) is exponential in |ζ|; see Fig. 5 in ref. 3. We also note that since u₁ is always negative, when u₀ is also negative, eqn (5) may breakdown as may approach zero. However, the negative value of u₀ is only observed in a very small potential window,^3,20i.e., when the electrophoretic and chemiphoretic contributions compete with each other. Therefore, eqn (5) will be likely valid in most circumstances. Nonetheless, in this article, we only utilized eqn (5) for u₀ > 0, in which limit eqn (5) is always valid.

Depending on the surface chemistry, the dimensionless zeta potential ζ may further depend on c_s. We assume that ζ_ref is a reference zeta potential at a specified concentration of the salt c_ref such that . The commonly described boundary conditions are constant potential (CP) and constant charge (CC). Mathematically, the CP boundary condition reads

ζ = ζ_ref, (7)

where the zeta potential is independent of salt concentration. For the CC boundary condition, q = −εn·∇ψ|_surf, where q is the surface charge density, ε is the electrical permittivity, n is the unit normal vector to the surface and ψ is the electrical potential. The standard Gouy–Chapman solution for isolated surfaces (i.e., dilute suspensions) yields . Therefore, the CC boundary condition becomeswhere the zeta potential increases with a decrease in salt concentration to maintain a constant surface charge. We note that . Also, we recognize that the Gouy–Chapman solution is the leading order solution for a spherical geometry. The correction³ can be included in the expression of q. However, the correction is negligible for typical parameter values and thus has not been included here. Eqn (8) suggests³⁰ that for ζ ≪ 1, . For ζ ≫ 1, .

We acknowledge that both the CP and CC are idealized boundary conditions and may not be able to capture the details of the colloidal surface chemistry. Nonetheless, CP and CC boundary conditions help identify the range of diffusiophoretic mobilities to be expected in common experiments. In addition to CC and CP, a charge regulation boundary condition is also employed where the surface charge can include both mobile and immobile charges.^31–34 However, since the charge regulation boundary conditions needs additional parameters, we did not include it in our analysis. Finally, electrical permittivity and viscosity might also be influenced for very concentrated electrolytes^19,21 but these effects haven't been incorporated here since we consider c_s ≲ 1 M.

3 Diffusiophoretic mobility

Using eqn (5)–(8), we summarize the effect of finite double layer thickness and different boundary conditions on D_DP in Fig. 2. We utilize the parameter values of an aqueous NaCl solution at room temperature, i.e., D₊ = 1.33 × 10⁻⁹ m² s⁻¹, D₋ = 2.03 × 10⁻⁹ m² s⁻¹, ε = 6.9 × 10⁻¹⁰ F m⁻¹, k_B = 1.38 × 10⁻²³ J K⁻¹, T = 298 K, e = 1.6 × 10⁻¹⁹ C and μ = 10⁻³ Pa s. In addition, we assume a = 0.5 μm, c_ref = 5 mM and ζ_ref = − 3 (i.e., a zeta potential of about −75 mV).¹² The results for the CP boundary condition without including the finite double layer thickness effect predicts a constant D_DP, which is the most widely used assumption in the diffusiophoresis literature.^{10,12,15–17,22,25–27} The results for the CC boundary condition show a monotonically decaying value of D_DP with c_s since the dimensionless ζ potential monotonically decreases with an increase in c_s; see eqn (8). However, when we include the effect of finite double layer thickness, D_DP decreases for dilute concentrations for both the CP and CC boundary conditions. In fact, since the value of is exponential in |ζ|, the decrease is larger for the CC boundary condition. Therefore, the CC boundary condition with finite double layer effects predicts a maximum in D_DP with c_s. We note that the influence of D_DP on c_s for the CP boundary condition is through only. In contrast, for the CC boundary condition, the dependence of D_DP on c_s is through both and ζ. For the assumed physical parameters, c_s = 0.1 mM implies . For both the CP and CC boundary conditions, D_DP decreases significantly for c_s = 0.1 mM even though the value of is significantly smaller than O(1); see Fig. 2. Therefore, finite double layer thickness effects can be significant even for . We reiterate that the surface chemistry of real surfaces might be a combination of the CC and CP boundary conditions which implies that the change in D_DP value around the maximum value may be smaller than the change predicted by the CC boundary condition alone.

Fig. 2 Dependence of electrolyte concentration c_s on diffusiophoretic mobility D_DP as given by eqn (5)–(8). The physical parameters correspond to that of an aqueous NaCl solution, i.e., D₊ = 1.33 × 10⁻⁹ m² s⁻¹, D₋ = 2.03 × 10⁻⁹ m² s⁻¹, ε = 6.9 × 10⁻¹⁰ F m⁻¹, k_B = 1.38 × 10⁻²³ J K⁻¹, T = 298 K, e = 1.6 × 10⁻¹⁹ C and μ = 10⁻³ Pa s. In addition, we assume a = 0.5 μm, c_ref = 5 mM and ζ_ref = −3. We note that increases as c_s decreases. For the aforementioned physical parameters, for c_s = 0.1 mM and for c_s = 100 mM.

For , we remark that the values of for the entire range of c_s; see Fig. 2. This trend is intuitive since the particle motion is induced by solute gradients and the electrolyte establishes D_s. However, some recent reports have utilized .^25,27,35 To make clear typical ranges of diffusiophoretic mobilities in different electrolyte solutions, we seek to verify if is applicable to all electrolytes. We summarize the maximum D_DP values assuming the CC boundary condition and for 16 different binary salts; see Fig. 3. To determine the maximum value of D_DP for each electrolyte, we adjusted the sign of the zeta potential such that βζ > 0 to ensure that the electrophoretic term and the chemiphoretic term are additive; see eqn (6). We assumed ζ_ref = ±3 (corresponding to ±75 mV) at a = 0.5 μm and c_ref = 5 mM. We note that these are relatively favorable conditions since typical colloidal zeta potentials measured experimentally are lower than ±75 to ±100 mV.³⁶ We find that the majority of the electrolytes still satisfy . The only electrolyte that displays is H⁺H₂PO₄⁻; see Fig. 3. However, H⁺H₂PO₄⁻ is likely to be found in aqueous solutions with HPO₄²⁻ and PO₄³⁻ ions when phosphoric acid disassociates. As indicated in Fig. 2, we typically find that the maximum value of D_DP is obtained for c_s = O(1) mM, which helps identify the range of concentration values where diffusiophoresis is most effective. We also repeated the analysis assuming different zeta potential values, i.e., ζ_ref = ±4 (±100 mV) at c_ref = 5 mM and obtained similar results where 14 out of the 16 electrolytes showed , except K⁺H₂PO₄⁻ and H⁺H₂PO₄⁻. We also obtain the same trends with the CP boundary condition; see Fig. 6 in the Appendix. Therefore, is likely to be valid for the majority of the electrolytes.

Fig. 3 Summary of maximum values for 16 different electrolytes based on the constant charge boundary condition and while including finite double layer effects. We adjusted the sign of the zeta potential such that βζ > 0 to ensure that the electrophoretic term and the chemiphoretic term are additive; see eqn (6). We assume ζ_ref = ±3 (corresponding to ±75 mV) at a = 0.5 μm and c_ref = 5 mM. The diffusivity values for cations and anions are taken from ref. 2.

4 Diffusiophoretic response to the spread of a Gaussian solute

We investigate the scenario where colloidal particles respond diffusiophoretically to a constant mass of solute diffusing in space. We assume that the initial distribution of solute is Gaussian with a width of [small script l]₀. We consider a one-dimensional problem such that c_s(x,t), v_DP = v_DPe_x and v_fluid = 0. We define and . The solution of eqn (1) yieldswhere c_s(X = 0,τ = 0) = c₀. Further, we define the dimensionless velocity V_DP = v_DP[small script l]₀/D_s. By utilizing eqn (5), we write

By using eqn (9) in eqn (10), we obtain

Assuming [small script l]₀ = 1 mm, t = 60 s, c₀ = 11.8 mM, ζ_ref = −3, c_ref = 5 mM and using the parameter values that correspond to aqueous solution of NaCl (provided earlier), we report c_s(X,τ = 0.1) (Fig. 4(a)), (Fig. 4(b)) and |V_DP|(X,τ = 0.1) (Fig. 4(c)) for both constant potential and constant charge boundary conditions. The distribution of c_s shows that the concentration gradient is significant only for |X| ≲ 2 (Fig. 4(a)). The values of can be quite large; see Fig. 4(b). However, the values of V_DP are monotonically increasing for models; see eqn (11) and Fig. 4(b). In fact, even for X = 6, i.e., the region where solute has not yet diffused, the predictions with suggest that the velocity can be significant. Furthermore, the predictions suggest a ballistic motion for X = O(10²). Clearly, should not be ignored in this physical system since and even a value of could significantly influence D_DP; see Fig. 2. Therefore, upon inclusion of finite double layer thickness effects, V_DP sharply drops beyond |X| > 2 (where ), and the ballistic motion vanishes for both CP and CC boundary conditions.

Fig. 4 Diffusiophoretic response to the spread of a Gaussian solute. (a) for τ = 0.1 and c₀ = 11.8 mM. (b) estimated based on c_s(X,0.1) and c₀ = 11.8 mM. (c) Prediction of the dimensionless diffusiophoretic velocity v_DP by using eqn (11) for different models. The models without the effect of finite double layer thickness predict a monotonically increasing velocity profile even in the region where the electrolyte concentration gradients are negligible. The models with the effect of finite double layer thickness predict that the velocity drops to zero for large X. (d) We modify the problem by adding a background solute concentration such that for τ = 0.1, c₀ = 11.8 mM and c_b = 0.1 μM. (e) estimated based on c_s(X,0.1), c₀ = 11.8 mM and c_b = 0.1 μM. (f) Prediction of the dimensionless diffusiophoretic velocity v_DP by using eqn (13) for different models. Physical parameters correspond to that an aqueous solution of NaCl where . Curves are plotted assuming a = 0.5 μm, ζ_ref = −3 and c_ref = 5 mM.

Recently, this particular configuration and its variants have been investigated in detail^25–27 while using the CP boundary condition with and . Specifically, in ref. 25, the authors solved for n_p(x,t) through eqn (2) numerically and demonstrated that for , the variance in n_p(x,t) scales super-linearly with time, a feature the authors described as super-diffusive. We believe that the super-diffusive regime will be challenging to obtain experimentally from diffusiophoresis alone because . In addition, the finite-double layer effects will significantly reduce the velocity magnitude; see Fig. 4(c). In summary, although the aforementioned studies provide useful insights into the diffusiophoretic phenomena, we believe the inclusion of finite double layer effects and imposing is likely be more reflective of experimental trends.

A variant of the above problem is to add a background chemical concentration since aqueous solutions usually possess ionic concentration of 0.1 μM, i.e., the concentration of ions at pH = 7. We modify the concentration field by adding a constant background concentration as

where c_b is the background concentration. By using eqn (12) to evaluate and substituting in eqn (10), we obtain

We plot the results for the same parameters used previously with c_b = 0.1 μM; see Fig. 4(d)–(f). Since for X = O(10), , V_DP decreases for large values of X in all scenarios. However, even if the predictions agree qualitatively for all scenarios, they disagree quantitatively, which is what we focus on in the next section.

5 The dead-end pore geometry

We now focus on the dead-end pore geometry^{12–14,22,23,35} to quantitatively investigate the differences between different models and to compare the model predictions with experiments. In this setup, a dead-end pore of length [small script l] is initially filled with a solution of electrolyte and colloidal particles; see Fig. 5(a).

Fig. 5 The dead-end pore geometry. (a) Schematic of the problem setup. The dead-end pore of length [small script l] is filled with a solution with electrolyte concentration c_s = c_pore and the scaled particle concentration n_p = 1. Next, at t = 0, we bring the solution in the pore in contact with a reservoir where the electrolyte concentration c_s = c_bulk and n_p = 0. Due to diffusiophoresis, the particles are compacted inside the pore. The value of is kept constant across all experiments and models. (b) n_p(X,τ) is evaluated from different models obtained by numerically solving eqn (15). X_peak(τ) is obtained by finding the locations where n_p is maximum. X_peakversus for different models and for c_pore = 1 mM. (c) Experimental snapshots at t = 300 s for a range of c_pore values and [small script l] = 1 mm. Scale bar is 100 μm. The top of each image represents the mouth of the pore. At larger concentrations, the accumulation of colloids near the mouth of the pore is attributed to charge screening. (d) Comparison of the X_peak values between experiments and different models for a range of c_pore values at τ = 0.5. Physical parameters in the model correspond to that of an aqueous solution of NaCl where . Curves for modeling trends are plotted assuming c_pore = 10⁻¹–10³ mM, [small script l] = 1 mm, D_p = 2 × 10⁻¹³ m² s⁻¹, a = 0.5 μm, c_ref = 5 mM and ζ_ref = −3, i.e., the parameter values consistent with the experiments. The experimental error bars are evaluated based on 3–4 independent experiments.

We assume that the configuration can be described through a one-dimensional model such that v_fluid = 0. The initial concentration of electrolyte c_s(0 ≤ x ≤ [small script l],t = 0) = c_pore and particles n_p(0 ≤ x ≤ [small script l], t = 0) = 1, where the particle concentration has been appropriately scaled. For t > 0, the solution inside the pore is brought in contact with a reservoir where c_s(0,t) = c_bulk and n_p(0,t) = 0. Due to the diffusiophoretic motion of the particles, the particles get compacted inside the pore; see Fig. 5(a).

The electrolyte concentration can be described as²⁸

where , and . Next, we non-dimensionalize eqn (2) to getwhere . We evaluate using eqn (14) and we numerically integrate eqn (15) using the method of lines and an implicit scheme with n_p(X,0) = 1, n_p(0,τ) = 0 and . We utilized a grid with spacing δX = 2.5 × 10⁻³ and a time step δτ = 10⁻³. The values of physical parameters used are c_pore = 10⁻¹–10³ mM, , [small script l] = 1 mm, D_p = 2 × 10⁻¹³ m² s⁻¹, a = 0.5 μm, c_ref = 5 mM and ζ_ref = −3. The remaining physical parameters are the same as that of an aqueous NaCl solution (provided earlier).

Next, we focus on the predictions of n_p(X,τ) obtained by integrating eqn (15). For each τ, we define the location X_peak(τ) as the location where n_p is maximum.²² The effect of different boundary conditions and models is provided in Fig. 5(b). Since the motion of particles is diffusive, X_peakversus is linear for τ ≲ 1 and for all models; see Fig. 5(b).²² However, for longer times, finite pore-size effects become significant and the X_peak profiles start to deviate from the linear behavior.²² We note there are quantitative differences between the models.

We use a dead-end pore geometry (Fig. 5(a)) to perform compaction experiments²² with polystyrene (PS; Invitrogen) particles of diameter 1 μm with volume fraction 2.6 × 10⁻⁴ in NaCl solution. Microfluidic channels are prepared by standard soft lithography, and the width, height, and the length of the main channel and the pores, respectively, are W = 750 μm, H = 150 μm and L = 5 cm, and w = 100 μm, h = 50 μm and [small script l] = 1 mm.²³ As described in Fig. 5(a), we initially fill the pores with PS particles suspended in NaCl solution of concentration c_pore. Next, we introduce an air bubble into the main channel at a volumetric flow rate of 350 μL h⁻¹, which is followed by the second NaCl solution of concentration c_bulk (without particles). Once the two solutions come in contact with each other, the mean flow rate is reduced to 20 μL h⁻¹, corresponding to a mean flow speed 〈u〉 = 50 μm s⁻¹ (syringe pump; Harvard Apparatus). Every experiment is repeated 3–4 times to gain confidence in the quantitative measurements. We vary c_pore = 10⁻¹, 1, 10¹, 10², 10³ mM and fix . By fixing the concentration ratio for different experiments, we examine the role of ion concentrations on D_DP of PS particles while keeping the form of ∇log c_s identical for all experiments; see eqn (14). We note that since , the lowest electrolyte concentration utilized in the experiment is 10⁻² mM. Furthermore, since c_bulk ≤ c_s(x,t) ≤ c_pore, a background ion concentration of 0.1 μM, such as in eqn (12) and (13), is unlikely to significantly influence our dead-end pore analysis.

We obtain fluorescent images with an inverted microscope (Leica DMI4000B) and analyze the peak positions X_peak at t = 300 s (see Appendix). Fig. 5(c) shows fluorescent images of the dead-end pores from experiments for different c_pore. First, we note that the diffusiophoretic motion does result in compaction of colloidal particles; see Fig. 5(c). However, X_peak is dependent on the value of c_pore. If D_DP was independent of c_s, the microscopic images would have been identical across the entire range of c_pore. Clearly, this is not the case.

We now compare the predictions from different models with the experimental data; see Fig. 5(d). We find that the predicted trends for X_peak from different models are similar to that of D_DP; see Fig. 2. However, the quantitative differences between the models are smaller since the dependence of X_peak with D_DP is sub-linear.^22,35 The experimental analysis of X_peak with c_pore shows a maximum, similar to the model with the CC boundary condition with finite . Since the polystyrene particles employed in experiments are latex colloids,³⁷ the charge regulation boundary is the most appropriate, i.e., the boundary condition which is a combination of the CP and the CC boundary conditions. Therefore, the decrease in the experimental X_peak values is less drastic as compared to the CC model. Finally, we also note that the maximum in X_peak is for the concentrations of O(1) mM (note that c(X,τ) ≤ c_pore), consistent with our model. We acknowledge that there are quantitative differences between the experimental values and the model predictions, especially in predicting the distribution of particle concentration; see Fig. 7(c) in the Appendix. The disagreements arise due to the diffusioosmosis from the channel walls.^12,23 Another source of error is the charge screening effect at high salinity conditions due to which some particles stick to the wall (see Appendix), an effect that is not captured in the model. Finally, there are convection effects near the mouth of the pore,³⁸ which are currently ignored in the analysis. Nonetheless, our experimental results show that D_DP is not constant and possesses a maximum with c_s.

6 Conclusions

We conclude that diffusiophoretic mobility varies significantly with the electrolyte concentration for typical experimental conditions. For dilute electrolytes, the diffusiophoretic mobility decreases due to finite double layer effects. For concentrated electrolytes, the mobility decreases due to charge screening. Therefore, we observe a maximum in diffusiophoretic mobility for electrolyte concentrations around a few mM. Furthermore, we show that diffusiophoretic mobility is typically smaller than the solute ambipolar diffusivity. We also show that incorporating the finite double layer thickness effects, the diffusiophoretic response to the spread of a Gaussian solute does not yield a ballistic motion. Moreover, for the dead-end pore geometry, we find that experiments also predict a maximum in the diffusiophoretic mobility with ion concentration, in agreement with our modeling predictions.

Looking forward, our results suggest that to achieve maximum diffusiophoretic transport rates in experiments, it is advisable to have c_s = O(1) − O(10) mM, at least for a = O(1) μm. Furthermore, the condition will help identify the physical scenarios where diffusiophoresis is likely to be significant. Moreover, a precise measurement of diffusiophoretic mobilities might assist in classifying the surface chemistry of the particles, i.e., constant potential, constant charge or charge regulation.

We recently estimated the leading-order diffusiophoretic mobility, i.e., u₀(ζ) in eqn (5), for a mixture of multivalent electrolytes.²⁰ Our results here motivate the need to evaluate D_DP for a mixture of electrolytes because the values of u₁(ζ,Pe) and will need to be appropriately modified. Since electrolytic diffusiophoresis has potential applications in delivery or extraction of particles to dead-end pore,^13,14 colloidal focusing or trapping^15–17 and lab-on-a-chip devices,^11,12 our results emphasize the need to consider the finite double layer effects in regions with low ion concentrations.

Conflicts of interest

There are no conflicts to declare.

Appendix: description of u₁(ζ,Pe)

To complete the description of D_DP in eqn (5), u₁(ζ,Pe) is evaluated aswhere for ζ > 0, F_n(ζ) are evaluated numerically as where , and . To evaluate F_n(ζ) for ζ < 0, one can exploit the relation F_n(−ζ) = (−1)ⁿF_n(ζ). We refer the readers to ref. 3 for the details of the derivation.

Appendix: D_DP for constant potential boundary condition

We repeat the analysis presented in Fig. 3 but with the constant potential boundary condition. The results are presented in Fig. 6. The analysis further underscores that .

Fig. 6 Summary of maximum values for 16 different electrolytes based on the constant potential boundary condition and while including finite double layer effects. We adjusted the sign of the zeta potential such that βζ > 0 to ensure that the electrophoretic term and the chemiphoretic term are additive; see eqn (6). We assume ζ_ref = ±3 (corresponding to ±75 mV) at a = 0.5 μm and c_ref = 5 mM. The diffusivity values for cations and anions are taken from ref. 2.

Appendix: analysis of experiments

To obtain the X_peak values (reported in Fig. 5(d)), we first evaluate the width-averaged intensity along the length of the pore; see Fig. 7(a). We conduct every experiment 3–4 times and report the average values. Next, X_peak is determined as the local maximum that appears after the boundary of exclusion zones; see Fig. 7(b). We note that when c_pore = 1 M, the axial variation in intensity is smaller because particles get attached to the wall due to charge screening; see Fig. 5(c) and 7(b). We also provide a direct comparison of the experimentally obtained particle concentration distribution with the numerical results obtained by solving eqn (15) for the CC boundary condition and ; see Fig. 7(c).

Fig. 7 Procedure to evaluate X_peak at t = 300 s from the intensity plots along X. (a) We measure the gray values along the pore using the region of interest (ROI, 80 μm X 990 μm), i.e., the region indicated with the dashed box. The top side of ROI is aligned with the inlet of the pore, and the other three sides are 10 μm away from the pore walls. The images are shown for c_bulk = 1 mM and c_pore = 10 mM. The horizontal and vertical scale bars are, respectively, 50 μm and 100 μm. (b) Typical intensity plots for a range of c_pore values at t = 300 s. The peak location X_peak is defined as the local maximum that appears after the boundary of exclusion zones. The presented plots are averaged values from 3–4 independent experiments (corresponding to 11–15 pores). A moving average of period 10 is applied to reduce the noise in the intensity data. (c) Comparison of . The experimental data does not reach unity because of the moving average. The numerical data is obtained from the constant charge model while including the finite double layer thickness effects.

Acknowledgements

We thank the Andlinger Center for Energy and the Environment at Princeton University and the NSF grant CBET-1702693 for financial support for our research. We thank Prof. Robert K. Prud'homme for helpful discussions. We also thank the two anonymous referees for their insightful comments.

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