Diffusiophoresis in porous media saturated with a mixture of electrolytes

Abstract

Current theories of diffusiophoresis in porous media are limited to a porous medium saturated with a valence symmetric electrolyte. A predictive model for diffusiophoresis in porous media saturated with a valence asymmetric electrolyte, or a general mixture of valence symmetric and asymmetric electrolytes, is lacking. To close this knowledge gap, in this work we develop a mathematical model, based upon the regular perturbation method and numerical integration, to compute the diffusiophoretic mobility of a colloid in porous media saturated with a general mixture of electrolytes. We model the electrokinetics using the Poisson–Nernst–Planck equations and the fluid transport in porous media using the Brinkman equation with an electric body force. We report three novel key findings. First, we demonstrate that, in the same electrolyte concentration gradient, lowering the permeability of the porous medium can significantly weaken the colloid diffusiophoretic motion. Second, we show that, surprisingly, by using a valence asymmetric electrolyte the colloid diffusiophoretic motion in a denser porous medium can be stronger than that in a less dense porous medium saturated with a symmetric electrolyte. Third, we demonstrate that varying the composition of an electrolyte mixture does not only change the strength of the colloid diffusiophoretic motion drastically, but also qualitatively its direction. The model developed from this work can be used to understand and predict natural phenomena such as intracellular transport, as well as design technological applications such as enhanced oil recovery, nanoparticle drug delivery, and colloidal species separation.

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1 Introduction

Diffusiophoresis is the deterministic motion of a colloid induced by a surrounding solute concentration gradient.^1–7 Diffusiophoresis in an ionic solute concentration gradient comprises the chemiphoretic and electrophoretic transport: the chemiphoretic transport is driven by an osmotic pressure gradient and the electrophoretic transport is driven by an electric field; both the osmotic pressure gradient and the electric field are induced by the solute concentration gradient. Diffusiophoresis is pivotal in natural settings such as intracellular transport^8–11 and is increasingly utilized in applications such as enhanced oil recovery,^12–15 nanoparticle drug delivery,^16,17 and colloidal species separation.^18–41 The diffusiophoretic velocity follows u = M∇ log n, where n is the ion number density and M is the diffusiophoretic mobility. A positive (negative) M corresponds to a colloid driven up or down the solute concentration gradient. The mobility is the key to quantify diffusiophoresis and encompasses physical properties of the colloid and the surrounding media. The objective of this article is to develop a predictive model of diffusiophoresis that accounts for the surrounding porous media and a mixture of valence symmetric and asymmetric electrolytes, thereby uncovering and predicting their impacts on diffusiophoresis.

Current theories of diffusiophoresis in a valence asymmetric electrolyte and a mixture of valence symmetric and asymmetric electrolytes have ignored the presence of porous media. Pawar et al.⁴² developed the first theory of diffusiophoresis in a valence asymmetric electrolyte, where the colloid electric double layer thickness κ⁻¹ is infinitesimally thin relative to the colloid radius a, i.e., κa ≫ 1. Bhattacharyya et al.^43,44 derived a general model that accounts for arbitrary values of the colloid surface potential ζ and κa. Wilson et al.⁴⁵ measured the colloid diffusiophoretic mobility in a valence asymmetric electrolyte in experiments and obtained good agreement with prior theories. Chiang and Velegol⁴⁶ developed the first theory of diffusiophoresis in a mixture of electrolytes. Their theory assumes that κa ≫ 1 and ζ is low compared to the thermal voltage kT/e, where k is the Boltzmann constant, T is the absolute temperature, and e is the proton charge. Shi et al.¹⁹ derived a more general expression for the diffusiophoretic mobility, by relaxing the assumption of ζ ≪ kT/e. Gupta et al.⁴⁷ and Ohshima⁴⁸ presented approximate expressions for the mobility in the limit of ζ ≪ 1, ζ ≫ 1, and κa ≥ 50. Samanta et al.⁴⁹ derived a general model that accounts for κa ≥ 50 and arbitrary values of ζ.

Current theories of diffusiophoresis in porous media are not applicable to a valence asymmetric electrolyte nor a mixture of electrolytes, and are limited to a valence symmetric electrolyte, i.e., two ionic species with valence z₁ = −z₂ = z. Recently, our group pioneered a theory for diffusiophoresis in porous media.⁵⁰ The theory models the hydrodynamic interactions between the colloid and the porous media, which act to dampen the colloid motion as the porous medium permeability decreases. The model accurately captures diffusiophoresis experiments by Doan et al.¹⁷ with no adjustable parameters, where collagen gels are used as the porous media. Bhaskar and Bhattacharyya⁵¹ incorporated the effect of ion size into the theory, showing quantitatively different predictions. Somasundar et al.⁵² demonstrated the feasibility of diffusiophoresis in another porous medium, by driving colloids through porous bacterial films using a concentration gradient of a valence symmetric electrolyte. Jotkar et al.^53,54 and Alipour et al.⁵⁵ conducted pore-scale simulations and microfluidic experiments to examine the effect of diffusiophoresis on the hydrodynamic dispersion in porous media with different degrees of water saturation and geometric disorder. However, currently it is lacking a model that can simulate diffusiophoresis in porous media saturated with a valence asymmetric electrolyte or a general mixture of valence symmetric and asymmetric electrolytes.

In this work, we develop a mathematical model to predict the colloid diffusiophoretic mobility in a porous medium saturated with a general mixture of valence symmetric and asymmetric electrolytes. We employ the Poisson–Nernst–Planck equations to model the ion transport and electric potential distribution, and the Brinkman equation with an electric body force to model the fluid transport in a porous medium. We solve the equations using a combination of the regular perturbation method and numerical integration. We report three novel key findings that highlight the coupled effects of porous media and electrolyte mixtures on diffusiophoresis. First, we show that in the same electrolyte concentration gradient the colloid diffusiophoretic motion is significantly weaker in a less permeable porous medium. This is consistent with the fundamental nature of porous media in dampening hydrodynamics,^56,57 and generalizes the same conclusion that was made in prior work for a valence symmetric electrolyte^17,50–55 to a valence asymmetric electrolyte and a general mixture of electrolytes. Second, we show that by using a valence asymmetric electrolyte, surprisingly, diffusiophoresis in a denser porous medium can be stronger than that in a less dense porous medium filled with a valence symmetric electrolyte. This demonstrates the competition between electrokinetics and hydrodynamics, and offers new insights to generate strong diffusiophoresis in porous media using a valence asymmetric electrolyte. Third, we show that, in a mixture of electrolytes, not just the magnitude of the colloid diffusiophoretic motion but qualitatively its direction can change by varying the mixture composition. This highlights the novelty of the present work to leverage valence asymmetric electrolyte and electrolyte mixtures to generate a richer set of diffusiophoresis responses.

The rest of this article is structured as follows. In Section 2, we present the problem formulation for the electrokinetic equations and the diffusiophoretic mobility. In Section 3, we present the results and discussion that elucidate the above-mentioned three key findings. In Section 4, we conclude this study.

2 Problem formulation

Consider a non-conducting colloid of radius a and a constant surface charge density q (or a constant surface potential ζ) in a static porous medium with a constant permeability l² and screening length l, subject to a constant concentration gradient of an electrolyte mixture ∇n_i^∞ with i = 1, 2, …, N (Fig. 1). By symmetry, the colloid translates with a constant diffusiophoretic velocity U parallel to ∇n_i^∞ along the z-direction, where U is to be determined. A reference frame moving with U is adopted.

Fig. 1 A concentration gradient of a mixture of N ionic species, ∇n_i^∞ with i = 1, 2, …, N, induces the diffusiophoretic motion of a colloid of radius a in a static porous medium with a constant screening length l. The colloid diffusiophoretic velocity U is parallel to ∇n_i^∞. (a) A wholistic view of the colloid and the porous medium. (b) The adopted spherical coordinate and moving reference frame.

At steady state, ionic species conservation requires that⁵⁸

where j_i, D_i, n_i, and z_i are the flux density, diffusivity, number density, and valence of the i-species, respectively; ϕ is the electric potential, and u is velocity of the electrolyte mixture. The electric potential follows the Poisson equation⁵⁸where ε and ρ are the permittivity and space charge density of the electrolyte mixture, respectively. The fluid dynamics in the porous medium is governed by the continuity equation and the Brinkman equation with an electric body force^{50,51,59–62}

∇·u = 0 and 0 = −∇p + η∇²u − ρ∇ϕ − ηl⁻²(u + U), (3)

where p is the pressure and η is the dynamic viscosity of the electrolyte mixture. Modeling the presence of porous media in diffusiophoresis using the Brinkman equation differentiates the present model from the majority of prior work which modeled diffusiophoresis in a free electrolyte solution (in the absence of porous media) using the Stokes equation. The Brinkman equation recovers the Stokes equation in a continuous manner as the screening length l approaches infinity. For example, the term −ηl⁻²(u + U) accounts for the frictional force exerted by the porous medium on the fluid. As l → ∞, this frictional force vanishes, and the Brinkman equation recovers the Stokes equation.

Eqn (1)–(3) are specified by the following boundary conditions. At the colloid surface at r = a, no slip and no penetration of the solvent prescribe u = 0. No penetration of the ionic species prescribes n·j_i = 0, where n is the unit normal vector pointing away from the colloid surface. The colloid surface charge density q or surface potential ζ could be specified as −n·ε∇ϕ = q or ϕ = ζ. Far from the colloid at r → ∞, u → −U and p → p^∞, where p^∞ is a reference constant pressure. The ion number densities and their imposed gradients are represented by n_i → n_i^∞ + ∇n_i^∞·r, where n_i^∞ is the constant bulk electrolyte concentration and the position vector r is anchored at the centroid of the colloid. To main bulk electroneutrality, the electrolyte concentration gradient induces an electric field E = −∇ϕ = (kT/e)βG,^1–3 where G = (∇n₁^∞)/n₁^∞ = …=(∇n_N^∞)/n_N^∞ and . The ion diffusivity ratio β controls the magnitude of E and thus the electrophoretic contribution in diffusiophoresis.

In typical diffusiophoresis,^1–3,5,6 the electrolyte concentration gradient at the size of the colloid is much smaller than the background concentration, α = |G|a ≪ 1. Using the regular perturbation method, we expand the dependent variables in α as

where is twice the ionic strength of the mixture, and quantities with a superscript 0 (and 1) are associated with the equilibrium (and nonequilibrium) state, where an electrolyte concentration gradient and diffusiophoresis are absent (and present). Quantities with carets are dimensionless. We non-dimensionalize lengths by the colloid radius a and the surface charge density by εkT/(ea). Substituting eqn (4) into the governing equations and boundary conditions yield a set of differential equations at different orders of α. Below we present the O(1) perturbation equations and the O(α) perturbation equations for computing the diffusiophoretic velocity and mobility.

2.1 O(1) perturbation equations

The O(1) equations pertains to the equilibrium state where there is no electrolyte concentration gradient and fluid flow, u⁰ = 0. The ionic species conservation and Poisson equation arewhere [small kappa, Greek, circumflex] = κa with κ⁻¹ = (εkT/e²n^∞)^1/2 being the Debye length. The boundary condition at the colloid surface at r̂ = 1 is or [small phi, Greek, circumflex]⁰ = [small zeta, Greek, circumflex]. The far-field boundary condition at r̂ → ∞ are n̂⁰_i → n̂_i^∞ and [small phi, Greek, circumflex]⁰ → 0. Integrating eqn (5) with the boundary condition at r̂ → ∞ gives the Boltzmann distribution of the ionic species n̂⁰_i = n̂_i^∞ exp(−z_i[small phi, Greek, circumflex]⁰). Substituting this result into eqn (6) gives the Poisson–Boltzmann equation⁵⁸

To solve eqn (7) in a finite computational domain, we consider a sphere of radius R̂ that is concentric to and encloses the colloid.^50,63–65 At a sufficiently large R̂ = 1 + 20/[small kappa, Greek, circumflex] as in this work, the electric potential decays to zero asymptotically as (1/r̂)exp(−[small kappa, Greek, circumflex]r̂). The far-field boundary conditions that were at r̂ → ∞ are transformed to d[small phi, Greek, circumflex]⁰/dr̂ + ([small kappa, Greek, circumflex] + 1/R̂)[small phi, Greek, circumflex]⁰ = 0 at r̂ = R̂. Eqn (7) is solved using the standard shooting method with the boundary condition at r̂ = 1 and the transformed far-field boundary condition.

2.2 O(α) perturbation equations

The O(α) equations pertains to the nonequilibrium state. We convert the O(α) equations to a set of ordinary differential equations by introducing the [small psi, Greek, circumflex]_i¹ potential, n̂_i¹ = −z_in̂⁰_i([small psi, Greek, circumflex]_i¹ + [small phi, Greek, circumflex]¹),^50,63,64 and writing the dependent variables as û_r̂¹(r̂, θ) = −(2/r̂)ĥ cos θ, û_θ¹(r̂, θ) = (1/r̂)[d(r̂ĥ)/dr̂]sin θ, and [small psi, Greek, circumflex]_i¹(r̂, θ) = [small psi, Greek, circumflex]_i¹ cos θ, where û_r̂¹ and û_θ¹ are the radial and angular components of û¹, respectively. The O(α) equations now reduce to solving for ĥ = ĥ(r̂) and [small psi, Greek, circumflex]_i¹ = [small psi, Greek, circumflex]_i¹(r̂), which are governed bywhere γ = a/l and the Peclet number Pe_i ≡ εk²T²/(ηe²D_i) describes the ratio of the advective to diffusive transport of the i-species. The boundary conditions at the colloid surface at r̂ = 1 are d[small psi, Greek, circumflex]_i¹/dr̂ = 0, ĥ = 0, and dĥ/dr̂ = 0. The far-field boundary conditions at r̂ → ∞ are [small psi, Greek, circumflex]_i¹ → (β − 1/z_i)r̂ and ĥ → M̂r̂/2. The non-dimensionalized diffusiophoretic mobility M̂ relates to the dimensional diffusiophoretic mobility M via

To solve eqn (8) and (9) in a finite computational domain, it again requires transforming the far-field boundary conditions at r̂ → ∞ to ones at r̂ = R̂. First, in the asymptotic limit of a vanishing electric potential at a sufficiently large R̂, the far-field boundary condition for [small psi, Greek, circumflex]_i¹ that was at r̂ → ∞ is transformed to be

Second, the net hydrodynamic and electric forces are zero on the large sphere of radius R̂ that encloses a freely suspended colloid. A sufficiently large R̂ guarantees that the sphere is electric force-free.^50,63–65 Thus, the constraint reduces to that the sphere is hydrodynamic force-free: , where is the Newtonian stress tensor and I is the identity tensor. Rearranging eqn (9) at r̂ = R̂ with the hydrodynamic force-free condition yields

where λ = −2γ² + 3γ²R̂³ + 9γR̂² + 9R̂. In sum, we solve eqn (8) and (9) for ĥ and [small psi, Greek, circumflex]_i¹ subject to the boundary conditions d[small psi, Greek, circumflex]_i¹/dr̂ = 0, ĥ = 0, and dĥ/dr̂ = 0 at r̂ = 1, in addition to the boundary conditions (11) to (13) at r̂ = R̂, using the solver NDSolve in Wolfram Mathematica and the solver bvp4c in MATLAB. We obtain the diffusiophoretic mobility from eqn (14). We have validated the present model by recovering results from prior work. Details are presented in Appendix A.

3 Results and discussion

Current knowledge of diffusiophoresis in porous media is limited to porous media saturated with a valence symmetric electrolyte.^17,50–55 In this section, we compute and discuss novel results diffusiophoresis in porous media saturated with a valence asymmetric electrolyte, in addition to a mixture of valence symmetric and asymmetric electrolytes. Model inputs taken from typical experiments are listed in Table 1. The zeta potential ζ ∈ [−100, 100] mV corresponds to [small zeta, Greek, circumflex] ∈ [−4, 4]. The porous medium pore diameter is at least about 1.5 times the colloid diameter, which allows the passage of the colloid. In Section 3.1, we demonstrate the first and the second key findings of this work. Namely, in the same electrolyte, lowering the permeability of the porous medium significantly weakens the colloid diffusiophoretic motion. Also, surprisingly, in an valence asymmetric electrolyte, diffusiophoresis in a denser porous medium can be stronger than that in a less dense porous medium saturated with a valence symmetric electrolyte. In Section 3.2, we demonstrate the third key finding of this work. Namely, in a mixture of electrolytes, not just the magnitude but also the direction of the colloid motion can change by varying the mixture composition.

Table 1 Input parameters for computation

Parameters	Values	Ion diffusivities^66	Values
Colloid radius (a)^17,20,52	100 nm	Potassium (DK^+)	1.96 × 10^−9 m^2 s^−1
Ionic strength (I)^17,20	[0.25, 1] mM	Sodium (DNa^+)	1.33 × 10^−9 m^2 s^−1
Screening length (l)^17,20,67,68	≥50 nm	Barium (DBa^2+)	0.847 × 10^−9 m^2 s^−1
Temperature (T)	298 K	Magnesium (DMg^2+)	0.706 × 10^−9 m^2 s^−1
Permittivity (ε)	6.95 × 10^−10 F m^−1	Aluminum (DAl^3+)	0.541 × 10^−9 m^2 s^−1
Dynamic viscosity (η)	0.891 × 10^−3 kg m^−1 s^−1	Lanthanum (DLa^3+)	0.619 × 10^−9 m^2 s^−1
Surface potential (ζ)^17,20,52,58,69–73	[−100, 100] mV	Chloride (DCl^−)	2.03 × 10^−9 m^2 s^−1

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3.1 Diffusiophoresis in porous media saturated with a valence asymmetric electrolyte

Let us first examine diffusiophoresis in porous media saturated with a AlCl₃ solution. Fig. 2(a) shows the non-dimensionalized diffusiophoretic mobility M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex]. The ionic strength of the solution is I = 0.25 mM. Let us first focus on the black line with l/a ≫ 1, which corresponds to an infinitely permeable porous media, i.e., an electrolyte solution in the absence of porous media. Here, the first observation is that for a negatively charged colloid M̂ is positive, whereas for a positively charged colloid M̂ is negative. This is consistent with the classical understanding of diffusiophoresis as follows.^1–4 Let us first consider a negatively charged colloid as shown in Fig. 2(b). A negative β implies that the induced electric field E is pointing in the negative z-direction. The field E drives positive counterions to the left and generates an electroosmotic flow in the same direction, causing the colloid to undergo electrophoresis to the right. Meanwhile, the osmotic pressure gradient induced by the electrolyte concentration gradient generates a chemiosmotic flow to the left, causing the colloid to undergo chemiphoresis to the right. Since diffusiophoresis is the sum of electrophoresis and chemiphoresis, diffusiophoresis acts in the positive z-direction and M̂ is positive. In contrast, for a positively charged colloid as shown in Fig. 2(c), electrophoresis which acts to the left outweighs chemiphoresis which acts to the right. This results in diffusiophoresis acting in the negative z-direction and M̂ is negative.

Fig. 2 (a) The non-dimensionalized diffusiophoretic mobility M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex] in different porous media saturated with a AlCl₃ solution of ionic strength I = 0.25 mM. Black line: an electrolyte solution in the absence of porous media (the ratio of the screening length to the colloid radius l/a ≫ 1). Blue line: a less dense porous medium (l/a = 1). Green line: a denser porous medium (l/a = 0.5). (b) A negatively charged colloid undergoing diffusiophoresis moving in the positive z-direction (to the right). The induced electric field E points to the negative z-direction. Both the electroosmotic flow (EOF) and chemiosmotic flow (COF) transport ions in the negative z-direction. Both electrophoresis (EP) and chemiphoresis (CP) in diffusiophoresis move the colloid in the positive z-direction. (c) A positively charged colloid undergoing diffusiophoresis moving in the negative z-direction (to the left). The EOF transports ions in the positive z-direction and the COF transports ions in the negative z-direction. CP is outweighed by EP which acts in the negative z-direction. (d) M̂ versus [small zeta, Greek, circumflex] in different porous media saturated with a BaCl₂ solution. (e) M̂ versus [small zeta, Greek, circumflex] in different porous media saturated with a NaCl solution. The color scheme and the ionic strength of (d) and (e) are the same as (a).

The second observation of the black line in Fig. 2(a) is that for a negatively charged colloid the magnitude of M̂ experiences a decay at [small zeta, Greek, circumflex] ≤ 1, whereas for a positively charged colloid the magnitude of M̂ is monotonically increasing. This can be understood by examining the ionic transport between the two cases. Let us first consider a negatively charged colloid by referring to Fig. 2(b). Both the electroosmotic and chemiosmotic flow are transporting positive counterions downstream to the colloid to the left. Due to coulombic attraction, these counterions will slow down the colloid motion that is to the right.^4,63 The stronger the colloid's charge, the stronger this slowing-down effect to the colloid. Therefore, M̂ decays as [small zeta, Greek, circumflex] becomes more negative. Next, let us consider a positively charged colloid by referring to Fig. 2(c). In contrast to Fig. 2(b), in Fig. 2(c) only the electroosmotic flow is transporting negative counterions downstream to the colloid to the right, whereas the chemiosmotic flow is transporting counterions upstream to the colloid to the left. As a result, the slowing-down effect to the colloid due to coulombic attraction between the counterions and the colloid is weak. Therefore, the magnitude of M̂ does not experience a decay but grows monotonically.

The first key finding of this work arises from an overview of Fig. 2(a), where decreasing l/a decreases the magnitude of M̂ significantly. For example, at [small zeta, Greek, circumflex] = 2, decreasing l/a (from black to green line) decreases the magnitude of M̂ from 0.459 to 0.298. Physically, decreasing l/a implies decreasing the permeability of the porous medium to the transport of the electrolyte and the colloid. That is, the presence of porous media is to weaken the magnitude of diffusioosmosis (and its constituting electroosmosis and chemiosmosis), and therefore the magnitude of the resulting diffusiophoresis (and its constituting electrophoresis and chemiphoresis). As the permeability decreases, the hydrodynamic drag to the electrolyte and the colloid increases, and therefore M̂ decreases. This key finding is consistent with the fundamental nature of porous media in weakening hydrodynamics.^56,57 Also, this key finding and the two observations in the previous paragraphs hold in a lower valence, valence asymmetric electrolyte BaCl₂ as shown in Fig. 2(d) and in a valence symmetric electrolyte NaCl as shown in Fig. 2(e).

Next, let us turn to Fig. 3. We consider a porous medium with l = 75 nm. In three difference cases, the porous medium is saturated with three different electrolyte solutions, AlCl₃, BaCl₂, and NaCl. The ionic strength of the three solutions are the same as I = 0.25 mM. The third observation is that, for a negatively charged colloid, the decay in the diffusiophoretic mobility M̂ occurs at a lower colloid surface potential [small zeta, Greek, circumflex] in a higher valence electrolyte than in a lower valence electrolyte. For example, the decay in M̂ for AlCl₃ (black line) occurs at [small zeta, Greek, circumflex] ≤ −1, whereas the decay for NaCl (green line) occurs at [small zeta, Greek, circumflex] ≤ −3.5. This can be understood by recalling the explanation for the second observation. Specifically, the decay is due to the coulombic attraction between the colloid and the downstream counterions. For AlCl₃, the counterions are Al³⁺, which have a higher charge density than the counterions Na⁺ of NaCl. Hence, at the same [small zeta, Greek, circumflex], the coulombic attraction between the colloid and Al³⁺ is stronger than that between the colloid and Na⁺. As a result, the decay in M̂ occurs at a weaker [small zeta, Greek, circumflex] for AlCl₃.

Fig. 3 The non-dimensionalized diffusiophoretic mobility M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex] in the same porous media (the ratio of the screening length to the colloid radius l/a = 0.75) saturated with different electrolyte solutions of ionic strength I = 0.25 mM. Black line: AlCl₃. Blue line: BaCl₂. Green line: NaCl.

The fourth observation is from the range of a positive [small zeta, Greek, circumflex] in Fig. 3, where M̂ can be increasingly enhanced by a more valence asymmetric electrolyte. The enhancement in M̂ is significant. For example, at [small zeta, Greek, circumflex] = 2, M̂ for NaCl is −0.137 (green line). In a BaCl₂ solution, M̂ has a 86% increase and equals −0.254 (blue line). In a AlCl₃ solution, M̂ has a 158% increase and equals −0.354 (black line). As noted in the first observation, the electrophoretic component of diffusiophoresis dominates for a positively charged colloid. The driving force for electrophoresis is the induced electric field E = (kT/e)βG, which is proportional to the ion diffusivity ratio β. Thus, as the magnitude of β increases from |β| = 0.207 (NaCl) to |β| = 0.318 (BaCl₂) and |β| = 0.408 (AlCl₃), the magnitude of M̂ increases.

Next, let us turn to Fig. 4. The black line denotes a denser porous medium (l/a = 0.5) saturated with a AlCl₃ solution and the blue line denotes a less dense porous medium (l/a = 0.75) saturated with a NaCl solution. The solution ionic strength I = 0.25 mM is the same in the two cases. Here, the second key finding of this work is that the magnitude of the diffusiophoretic mobility M̂ in a denser porous medium can be stronger than that in a less dense porous medium. For example, for a positive colloid surface potential [small zeta, Greek, circumflex], the magnitude of the black line is larger than that of the blue line. This is a rather surprising and non-intuitive result, as one may expect that the colloid movement in a denser porous medium is always weaker than that in a less dense porous medium due to the stronger hydrodynamic drag to the fluid and the colloid. However, results here show that, by employing a valence asymmetric electrolyte, diffusiophoresis in a denser porous medium can be much stronger than that in a less dense porous medium saturated with a valence symmetric electrolyte. Physically, this is because the enhancement to the electrophoretic component of diffusiophoresis by the valence asymmetric electrolyte outweighs the hydrodynamic drag due to the porous media. This finding offers new insights into using valence asymmetric electrolyte to generate strong diffusiophoresis in porous media.

Fig. 4 The non-dimensionalized diffusiophoretic mobility M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex] in different porous media saturated with different electrolyte solutions of ionic strength I = 0.25 mM. Black line: a denser porous medium (the ratio of the screening length to the colloid radius l/a = 0.5) saturated with a AlCl₃ solution. Blue line: a less dense porous medium (l/a = 0.75) saturated with a NaCl solution.

3.2 Diffusiophoresis in porous media saturated with a mixture of valence symmetric and asymmetric electrolytes

Let us examine diffusiophoresis in porous media saturated with solution mixtures of p molar of KCl and q molar of LaCl₃ with different compositions f = q/(p + q), where f ∈ [0, 1] with f = 0 and f = 1 representing a pure KCl solution and a pure LaCl₃ solution, respectively. Fig. 5(a) shows the diffusiophoretic mobility M̂ versus the colloid surface potential [small zeta, Greek, circumflex]. The screening length is l = 75 nm and the ionic strength of the solution is I = 1 mM. Let us first focus on the black and the red line with f = 1 and f = 0, respectively. The first observation is that the variation of M̂ versus [small zeta, Greek, circumflex] in a pure LaCl₃ solution (f = 1; black line) is qualitatively different from that in a pure KCl solution (f = 0; red line). Specifically, in a pure LaCl₃ solution, M̂ is positive for a negative [small zeta, Greek, circumflex] and is negative for a positive [small zeta, Greek, circumflex]. This follows the same trend and explanation as for AlCl₃ in the first observation in Section 3.1 and is not repeated here. In contrast, in a pure KCl solution, M̂ is positive regardless of the sign of [small zeta, Greek, circumflex]. This is understood by noting the very small value β = −0.0188, which reflects the negligibly small electrophoretic contribution in diffusiophoresis. Hence, the colloid undergoes diffusiophoresis chiefly due to chemiphoresis that drives the colloid in the positive z-direction and thus M̂ is positive.

Fig. 5 (a) The non-dimensionalized diffusiophoretic mobility M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex] in the same porous media (the ratio of the screening length to the colloid radius l/a = 0.75) saturated with solution mixtures of p molar of KCl and q molar of LaCl₃ with different compositions f = q/(p + q). The solution ionic strength I = 1 mM. Red line: f = 0. Green line: f = 0.33. Blue line: f = 0.5. Grey line: f = 0.67. Black line: f = 1. (b) M̂ versus [small zeta, Greek, circumflex] in the same porous media (l/a = 0.75) saturated with solution mixtures of p molar of KCl and q molar of MgCl₂ with different compositions f = q/(p + q). The color scheme and the ionic strength are the same as (a). (c) The ratio of the contribution to the mobility by the asymmetric electrolyte to that by the mixture M̂_LaCl3/M̂ and M̂_MgCl2/M̂ versus f. Blue line: M̂_LaCl3/M̂. Black line: M̂_MgCl2/M̂.

The third key finding of this work is that varying the mixture composition f can change not just the magnitude of the colloid diffusiophoretic motion drastically, but also qualitatively its direction; the latter is reflected from the sign change of M̂ for a fixed positive [small zeta, Greek, circumflex] as f increases. In a mixture of electrolytes, M̂ is not simply the superposition nor the weighted average of the contributions from its individual electrolyte. For example, in Fig. 5(a) at [small zeta, Greek, circumflex] = 3, M̂_f=0 = 0.240 (red line) and M̂_f=1 = −0.328 (black line), but M̂_f=0.5 = −0.179 (blue line) ≠ (M̂_{f = 0} + M̂_f=1)/2. In fact, M̂ depends nonlinearly on f. This is because M̂ is a result of the nonlinearly coupled Poisson–Nernst–Planck and Brinkman eqn (1)–(3). Because of this intricate coupling, M̂ can vary non-monotonically with f. For example, at [small zeta, Greek, circumflex] = −2.8, M̂_f=0 = 0.283 (red line); as f increases, M̂ first decreases and then increases. In the special case at [small zeta, Greek, circumflex] = −2.4, M̂_f=0 (red line) can even be identical to M̂_f=1 (black line), meaning that a colloid undergoes the same diffusiophoretic motion regardless of the electrolyte identity. These observations are not restricted to a KCl/LaCl₃ mixture but is general to other electrolyte mixtures. For example, Fig. 5(b) shows a similar set of diffusiophoretic response in a KCl/MgCl₂ mixture as in Fig. 5(a). These observations underline the importance of this work to solve the coupled Poisson–Nernst–Planck–Brinkman equations to predict diffusiophoresis.

A scaling analysis illustrates that the nonlinearity between M̂ and f depends on the ion valence z. First, referring to the far-field boundary condition below eqn (9), M̂ ∼ ĥ. Further, from eqn (9), , where Î is the non-dimensionalized mixture ionic strength. Combining these relations, M̂ ∼ Î. Second, recall that f = q/(p + q). Stoichiometry requires that the ionic strength of KCl is IKCl = (1/2)(zK+2p + zCl−2p) M = p M, with zK+ = −zCl− = 1. The ionic strength of LaCl3 is ILaCl3 = (1/2)(zLa3+2q + zCl−2q) M = 6q M, with zLa3+ = 3 and zCl− = −1. Note that ILaCl3 increases nonlinearly with an increase in the ion valence zLa3+2. Thus, I = IKCl + ILaCl3 = p + 6q M. Upon non-dimensionalization, ÎLaCl3 = 6q̂ and Î = q̂(5f + 1)/f. Recall that M̂ ∼ Î, thus M̂LaCl3 ∼ ÎLaCl3, where M̂LaCl3 is the contribution of LaCl3 to the mobility of the mixture M̂. Thus, it is established that

where eqn (16) is obtained for a KCl/MgCl2 mixture following a similar procedure as eqn (13). Fig. 5(c) shows the nonlinear relations of M̂LaCl3/M̂ and M̂MgCl2/M̂ to f. Note that M̂LaCl3/M̂ (blue line) shows a stronger increase than M̂MgCl2/M̂ (black line) at the same f. This is because the nonlinearity in the ion valence for zLa3+2 = 9 is stronger than that for zMg2+2 = 4. This explains why in Fig. 5(a) and (b) the sensitivity of M̂ to a change in f in a KCl/LaCl3 mixture is stronger than that in a KCl/MgCl2 mixture.

Before closing, we demonstrate the generality of the first two key findings of this work, which were demonstrated in Section 3.1 with a valence asymmetric electrolyte, in an electrolyte mixture. First, recall the first key finding that the magnitude of diffusiophoresis weakens significantly as the porous medium permeability decreases. Fig. 6(a) shows the diffusiophoretic mobility M̂ versus the colloid surface potential [small zeta, Greek, circumflex] in different porous media saturated with a KCl/LaCl₃ mixture of composition f = 0.5. The first key finding is confirmed that the magnitude of M̂ decreases substantially as l/a decreases (from black to green line). In fact, the first key finding holds not just in a mixture of two electrolytes. Fig. 6(b) shows M̂ versus [small zeta, Greek, circumflex] in different porous media saturated with a phosphate-buffered saline (PBS) solution, which is a mixture of NaCl, KCl, Na₂HPO₄, and KH₂PO₄. The detailed ionic composition of the PBS solution is given in Appendix B. Again, the first key finding is confirmed by the decreasing magnitude of M̂ with decreasing l/a (from black to green line).

Fig. 6 (a) The non-dimensionalized diffusiophoretic mobility M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex] in different porous media saturated with a solution mixture of p molar of KCl and q molar of LaCl₃ with a composition f = q/(p + q) = 0.5. The solution ionic strength I = 1 mM. Black line: an electrolyte solution in the absence of porous media (the ratio of the screening length to the colloid radius l/a ≫ 1). Blue line: a less dense porous medium l/a = 1. Green line: a denser porous medium l/a = 0.5. (b) M̂ versus [small zeta, Greek, circumflex] in different porous media saturated with a phosphate-buffered saline (PBS) solution. The color scheme and the ionic strength are the same as (a). (c) M̂ versus [small zeta, Greek, circumflex] in different porous media saturated with different KCl/LaCl₃ mixtures of I = 1 mM. Black line: a denser porous medium saturated with a higher fraction of valence asymmetric electrolyte l/a = 0.5 and f = 1. Blue line: a less dense porous medium saturated with a lower fraction of valence asymmetric electrolyte l/a = 0.75 and f = 0.33.

Lastly, recall the second key finding that, with a valence asymmetric electrolyte, diffusiophoresis in a denser porous medium can be stronger than diffusiophoresis in a less dense porous medium saturated with a valence symmetric electrolyte. Fig. 6(c) shows M̂ versus [small zeta, Greek, circumflex] in different porous media saturated with different KCl/LaCl₃ mixtures. The black line corresponds to a denser porous medium (l/a = 0.5) saturated with a mixture with a higher fraction of LaCl₃ (f = 1), whereas the blue line corresponds to a less dense porous medium (l/a = 0.75) saturated with a mixture with a lower fraction of LaCl₃ (f = 0.33). The second key finding is confirmed by observing that for positive [small zeta, Greek, circumflex] the magnitude of M̂ of the black line can indeed be larger than that of the blue line. In sum, these results underline the novelty of this work and the importance of leveraging valence asymmetric electrolytes and electrolyte mixtures to achieve richer responses in diffusiophoresis in porous media.

4 Conclusions

In this work, we have developed a predictive model for diffusiophoresis in porous media saturated with a general mixture of electrolytes. Prior models have focused either on diffusiophoresis in a mixture of electrolytes in the absence of porous media,^{19,42–44,46–49} or diffusiophoresis in porous media saturated with a valence symmetric electrolyte.^17,50–55 The present model is novel in that it accounts for the coupled effects of porous media and electrolyte mixtures. We have computed the diffusiophoretic mobilities in porous media saturated with a valence asymmetric electrolyte as well as in porous media saturated with a mixture of valence symmetric and asymmetric electrolytes. We summarize the three key findings below.

The first key finding of this work is that, in the same electrolyte solution, decreasing the porous medium permeability weakens the colloid diffusiophoretic motion significantly. This is consistent with the fundamental nature of porous media which provides a larger hydrodynamic hindrance to the colloid and the suspending fluid, and thus, weakens diffusiophoresis. This key finding is consistent with the same conclusion drawn in prior work for diffusiophoresis in porous media saturated with a valence symmetric electrolyte.^17,50–52 Here, we have generalized this key finding to diffusiophoresis in porous media saturated with a valence asymmetric electrolyte and a general mixture of electrolytes.

The second key finding is that, by utilizing a valence asymmetric electrolyte, diffusiophoresis in a denser porous medium can be stronger than that in a less dense porous medium saturated with a valence symmetric electrolyte. This is contrary to what one might expect that diffusiophoresis in a denser porous medium is always weaker. In fact, this surprising result is due to the fact that valence asymmetric electrolyte can generate a stronger electrophoretic motion of the colloid, which outweighs the hydrodynamic drag due to the porous medium. We have further generalized this key finding by showing that diffusiophoresis in a denser porous medium saturated with a mixture of higher fraction of asymmetric electrolyte can be stronger than that in a less dense porous medium saturated with a mixture of lower fraction of asymmetric electrolyte.

The third key finding is that varying the mixture composition can change not only the magnitude of the diffusiophoretic motion significantly, but also qualitatively its direction. The diffusiophoretic mobility in an electrolyte mixture is not simply the superposition nor the weighted average of the contributions from its individual electrolyte. This is because diffusiophoresis is governed by the nonlinearly coupled Poisson–Nernst–Planck–Brinkman equations. This underlines the value of the present work to determine the colloid diffusiophoretic motion by solving the coupled governing equations. Together, the present model and these key findings will enable fundamental understanding of diffusiophoresis in porous media and predict a richer set of diffusiophoresis responses in the presence of an electrolyte mixture.

Data availability

All data that supports this study is included in this article.

Conflicts of interest

There are no conflicts of interest to declare.

Appendices

Appendix A: Model validations

We validate the present model by recovering results from prior work. First, Fig. 7 shows that the present model recovers Fig. 4 in Sambamoorthy and Chu,⁵⁰ for diffusiophoresis in different porous media saturated with a NaCl solution. Second, Fig. 8 shows that the present model recovers Fig. 3 in Gupta et al.,⁴⁷ for diffusiophoresis in different valence asymmetric electrolyte solutions in the limit of a thin electric double layer and in the absence of porous media. Here, the diffusiophoresis results have the same magnitude and an opposite sign as the diffusioosmosis results in Gupta et al.,⁴⁷ since diffusiophoresis is the exact opposite of diffusioosmosis in the limit of a thin electric double layer. Third, Fig. 9 shows that the present model recovers Fig. 9 in Prieve and Roman,⁶⁵ for diffusiophoresis in a NaCl solution in the absence of porous media. Lastly, Fig. 10 shows that the present model recovers Fig. 4 in Prieve et al.,¹ for diffusiophoresis in different valence symmetric electrolyte solutions in the limit of a thin electric double layer and in the absence of porous media.

Fig. 7 The non-dimensionalized diffusiophoretic mobility M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex] in different porous media saturated with a NaCl solution with the Debye–Huckel approximation. Lines: predictions by the present model. Symbols: predictions by Sambamoorthy and Chu.⁵⁰ Black: the ratio of the screening length to the colloid radius l/a ≫ 1. Grey: l/a = 4. Blue: l/a = 1. Green: l/a = 0.1. Orange: l/a = 0.02.

Fig. 8 The non-dimensionalized diffusiophoretic mobility M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex] in different valence asymmetric electrolyte solutions in the limit of a thin electric double layer (the ratio of the colloid radius to the Debye length [small kappa, Greek, circumflex] ≫ 1) and in the absence of porous media (the ratio of the screening length to the colloid radius l/a ≫ 1). Lines: predictions by the present model. Symbols: predictions by Gupta et al.⁴⁷ Black: H₂SO₄. Blue: Na₂SO₄. Green: CaCl₂. Orange: Mg(HCO₃)₂.

Fig. 9 The non-dimensionalized diffusiophoretic mobility (3/2)M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex] in a NaCl solution in the absence of porous media (the ratio of the screening length to the colloid radius l/a ≫ 1). Lines: predictions by the present model. Symbols: predictions by Prieve and Roman.⁶⁵ Black: the ratio of the colloid radius to the Debye length [small kappa, Greek, circumflex] ≫ 1. Grey: [small kappa, Greek, circumflex] = 1000. Purple: [small kappa, Greek, circumflex] = 100. Blue: [small kappa, Greek, circumflex] = 10. Cyan: [small kappa, Greek, circumflex] = 1. Green: [small kappa, Greek, circumflex] = 0.1. Orange: [small kappa, Greek, circumflex] = 0.

Fig. 10 The non-dimensionalized diffusiophoretic mobility 2M̂ versus the non-dimensionalized colloid surface potential [small zeta, Greek, circumflex] in different valence symmetric electrolyte solutions in the limit of a thin electric double layer (the ratio of the colloid radius to the Debye length [small kappa, Greek, circumflex] ≫ 1) and in the absence of porous media (the ratio of the screening length to the colloid radius l/a ≫ 1). Lines: predictions by the present model. Symbols: predictions by Prieve et al.¹ Black: NaCl. Blue: KCl. Green: NH₄F.

Appendix B: Ionic composition of phosphate-buffered saline solution

A 1× phosphate-buffered saline (PBS) solution comprises 137 mM NaCl, 2.7 mM KCl, 10 mM Na₂HPO₄, and 1.8 mM KH₂PO₄. From dissociation, there are 157 mM Na⁺, 139.7 mM Cl⁻, 4.5 mM K⁺, 10 mM HPO₄²⁻, and 1.8 mM H₂PO₄⁻. Dihydrogen phosphate ion H₂PO₄⁻ and monohydrogen phosphate ions HPO₄²⁻ have a chemical equilibrium, H₂PO₄⁻ ⇌ H⁺ + HPO₄²⁻, with an equilibrium constant of 10^−4.21 mM and diffusivities D_HPO4²⁻ = 0.439 × 10⁻⁹ m² s⁻¹ and D_H2PO4⁻ = 0.879 × 10⁻⁹ m² s⁻¹.^66,74 Thus, there is 1.11 × 10⁻⁵ mM H⁺, which is negligible compared to the five dominant species, namely, Na⁺, K⁺, Cl⁻, HPO₄²⁻, and H₂PO₄⁻. To obtain the concentrations of the five dominant species at I = 1 mM as in Fig. 6(b), we first obtain the ratios of the concentrations of the five species to the ionic strength of the mixture as

The above ratios remain unchanged for PBS solutions at other ionic strengths. Thus, the concentrations of the five dominant species at I = 1 mM can be obtained by multiplying the above ratios to I = 1 mM, giving 0.915 mM Na⁺, 0.0262 mM K⁺, 0.815 mM Cl⁻, 0.0583 mM HPO₄²⁻, and 0.0105 mM H₂PO₄⁻.

Acknowledgements

H. C. W. Chu acknowledges the funding support from University of Florida. This work was supported by the donors of ACS Petroleum Research Fund under Doctoral New Investigator Grant 66915-DNI9. H. C. W. C served as Principal Investigator on ACS PRF 66915-DNI9 that provided support for S. S. The authors acknowledge the fruitful discussion with Professor Anthony J. C. Ladd.

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