# Diffusiophoresis of charged colloidal particles in the limit of very high salinity

^{a}Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213;^{b}Reservoir Engineering Technology Division, Exploration and Petroleum Engineering Center Advanced Research Center, Saudi Aramco, Dhahran 31311, Saudi Arabia;^{c}Center for Integrative Petroleum Research, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

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Edited by Snezhana I. Abarzhi, The University of Western Australia, Perth, Australia, and accepted by Editorial Board Member David A. Weitz May 19, 2018 (received for review October 1, 2017)

## Significance

Contact between low- and high-salinity water occurs in many applications including enhanced oil recovery (EOR), desalination by reverse osmosis, or electric-power production (“blue energy”) by pressure-retarded osmosis. The resulting salinity gradients can propel particles which cause fouling of osmosis membranes. In EOR the gradient can transport functionalized nanoparticles into the porous oil-bearing rock. Once transported, the nanoparticles undertake in situ sensing and intervention activities in the rock. At very high salinity, the long-range interaction needed for diffusiophoresis or electrophoresis of charged particles is expected to vanish. Nonetheless, we observed measurable speeds at salinities approaching saturation.

## Abstract

Diffusiophoresis is the migration of a colloidal particle through a viscous fluid, caused by a gradient in concentration of some molecular solute; a long-range physical interaction between the particle and solute molecules is required. In the case of a charged particle and an ionic solute (e.g., table salt, NaCl), previous studies have predicted and experimentally verified the speed for very low salt concentrations at which the salt solution behaves ideally. The current study presents a study of diffusiophoresis at much higher salt concentrations (approaching the solubility limit). At such large salt concentrations, electrostatic interactions are almost completely screened, thus eliminating the long-range interaction required for diffusiophoresis; moreover, the high volume fraction occupied by ions makes the solution highly nonideal. Diffusiophoretic speeds were found to be measurable, albeit much smaller than for the same gradient at low salt concentrations.

Diffusiophoresis is the steady motion of a rigid particle through a viscous solution, propelled by a gradient in concentration of a molecular solute which experiences a long-range physical interaction with the particle. In this paper and virtually all past studies of diffusiophoresis (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–21), we are concerned with charged colloidal particles and ionic solutes, for which the interaction is electrostatic. Some descriptions of these past studies can be found in *SI Appendix*.

Recently diffusiophoresis has again become a popular research topic (22⇓⇓–25) owing to the realization that this transport mechanism occurs in a wide variety of natural and man-made settings. Two commercial processes of current interest are desalination by shock electrodialysis (26) and electric power generation by pressure-retarded osmosis (27). In both of these, diffusiophoresis of naturally occurring colloidal particles can contribute to fouling of the membrane.

Our motivation for this study is the use of nanoparticles in enhanced oil recovery (EOR). During EOR, water is injected into an oil reservoir to displace the oil, which is trapped in porous rock together with a highly concentrated brine. The contact of this brine with the injected water creates a salinity gradient which can transport functionalized nanoparticles into the rock. Any nanoparticles thus transported can undertake in situ sensing and intervention activities in the porous rock (28, 29).

An important distinction of the EOR application is the extreme salinity. At very high concentration of ions, electrostatic interactions are expected to be completely screened, thus eliminating the long-range interaction needed for diffusiophoresis. Moreover, a significant volume fraction of ions guarantees that steric interactions will make the solution nonideal. Does diffusiophoresis still occur? If so, what is the effect of steric interactions?

Treating molecules as hard spheres of finite volume, Carnahan and Starling (30) used statistical mechanics to obtain an accurate equation of state for pure fluids. This approach was later generalized to multicomponent mixtures by Boublik (31) and Mansoori et al. (32). Biesheuvel and van Soestbergen (33) and Bazant et al. (34) explored the implications of this thermodynamic model for steric interactions between ions on the equilibrium structure of the counterion cloud next to a charged surface. Khair and Squires (35) explored the implications on electrophoresis of colloidal particles, while Stout and Khair (36) did the same for diffusiophoresis.

Here we present measurements of diffusiophoresis in solutions having a high concentration of salt. As with much lower salt concentrations, we find the dominant contribution to be electrophoresis of particles in the electric field generated by the salt gradient. Thus, we also measured electrophoretic mobilities of the same particles (as a function of salt concentration) as well as the diffusion potential drop across a membrane separating two well-stirred solutions having different salt concentrations. These results are compared with predictions from the recent theory of Stout and Khair (36).

The relevance of this work to the key topic of this special-feature issue is twofold. First, diffusiophoresis is nonequilibrium transport of colloidal particles at the interface created by contacting two solutions having different salt concentrations. Second, a wide range of length scales govern the behavior: The electrostatic interactions between the particle and any ion have a range of 1 nm, the particle size used in our experiments is 210 nm, and the gradient is formed by separating two different well-stirred solutions by a porous membrane having a thickness of 10 μm. In an oil reservoir, the gradient is generated by pumping water into an injection well which can be located kilometers from the production well.

## Experiment

The deposition experiments used an apparatus similar to that of Lin and Prieve (13) whose schematic is shown in Fig. 1. The salt cell holds 175 cm^{3} of the brine having the higher concentration and is open to the atmosphere above. The salt cell (with membrane attached) is immersed in a 2-L glass beaker, which is also open to the atmosphere. The larger vessel is then filled to the same level as the salt cell with a 0.5 wt % latex added to the salt solution with the lower concentration. Additional details can be found in *SI Appendix*.

## Theory

A rigorous treatment of the coupling between ion fluxes in concentrated solutions requires consideration of the Stefan–Maxwell-like equations which involve *n*^{2} transport coefficients, where *n* is the number of components (37). In this preliminary study, we use a simple extension of the dilute solution theory in which steric interactions are considered and coupling is primarily through the induced electric field. For a 1–1 electrolyte like NaCl, the electric field **E** induced by a salt gradient is given by (see derivation leading to *SI Appendix*, Eq. **S4**)*C* is the local salt concentration, ψ is the electrostatic potential, *e* is the charge on a proton, *D*_{+} and *D*_{−} are the cation and anion diffusion coefficients, and *kT*) for nonideal solutions (*SI Appendix*, Eqs. **S5** and **S6**).

### Ideal Solutions.

A quantitative theory is available (11, 13, 17) for predicting the diffusiophoretic velocity of a charged colloidal sphere through a viscous, 1–1 electrolyte solution at infinite dilution. In the limit in which the Debye length is infinitesimal compared with the size of the colloidal particle (a reasonable assumption for the current experiments), the diffusiophoretic velocity of a charged colloidal particle caused by a gradient in concentration *C* of a 1–1 electrolyte is (11)*C*. The first term in [**2**] represents Smoluchowski’s equation for the electrophoretic velocity in an externally applied electric field **E**. The second term arises because the gradient in salt concentration also generates (inside the counterion cloud) a gradient in osmotic pressure along the surface of the particle.

If we take the dot-product of both sides of [**1**] by some arbitrary but differential (in length) displacement vector, then integrate along an arbitrary contour connecting some point in the well-stirred bulk of the low-concentration (*C*_{lo}) reservoir to a second point in the well-stirred bulk of the high-concentration (*C*_{hi}) reservoir, we obtain the diffusion potential difference which arises between the two reservoirs. For an ideal solution with

### Nonideal Solutions.

While all solutions behave ideally at infinite dilution, at higher concentrations complications arise from physical interactions between components, such as steric and electrostatic interactions. While Debye–Huckel theory does a good job of predicting activity coefficients of strong electrolytes for concentrations up to 0.01 M (38), at still higher concentrations more advanced theories of electrostatic correlations are needed (39). (M is an abbreviation for mol/dm^{3}.) Using the approach of Storey and Bazant (40), Stout and Khair (41) showed that electrostatic correlations could even lead to a reversal in the direction of electrophoresis (i.e., the coefficient of the first term in [**2**]). For nonideal solutions, the simpler theory in [**1**] leads to*SI Appendix*, Eqs. **S5** and **S6**).

### Electrophoretic Mobility.

At high salt concentrations, the Debye length becomes much less than the 210-nm diameter of our latex particles; then, Smoluchowski’s equation for the electrophoretic mobility is expected to apply:*m*_{e} appears as the coefficient of **E** in the first term of [**2**]. To predict how *m*_{e} depends on salt concentration *C*, we will assume the charge density σ on the surface of the particle remains fixed as the salt concentration is increased.

When the Debye length is thin compared with the particle radius, we can relate σ and ζ using the Gouy–Chapman equation for a 1–1 electrolyte:**5**] and [**6**]:*C*) of a 4 M NaCl solution is almost double that of dilute solutions (43), while the permittivity ε(*C*) is about half (44). So, we have also corrected for these effects, although steric interactions, electrostatic correlations, and the viscoelectric effect have not been considered.

## Results and Discussion

### Diffusiophoretic Rates and Diffusion Potentials.

Fig. 2 summarizes some typical results for the measured deposit mass at various exposure times using two sets of NaCl concentrations. The control experiment (4 M NaCl on both sides of the membrane) typically yielded about 1–2 mg of deposit, independent of exposure time. This represents the dried contents of thin films of liquid which cling to either side of the membrane simply as a result of wetting.

The linear increase in deposit weight with time in response to the salt gradient suggests a constant rate of deposition. It further suggests that the growing latex film imposes no significant resistance to diffusion of NaCl through the membrane and the latex film. The regressed rate of deposition (*dm/dt* = 0.61 mg/min) can be converted into a particle speed *V* using *dm*/*dt* = *cVA*, where *A* = 13 cm^{2} is the area of the membrane exposed to the latex and *c* = 5 g/L is the mass concentration of latex. The resulting migration speed is *V* = 1.6 μm/s.

Fig. 3 shows the effect of the concentration ratio. The higher concentration remains fixed at 4 M while the lower concentration varies between 0.4 M and 4 M. The open squares represent measurements of the diffusion potential across the membrane. According to [**3**], this is expected to be proportional to the logarithm of the concentration ratio. For this reason the horizontal axis of Fig. 3 was made logarithmic. To within experimental error, the measured diffusion potentials appear to be directly proportional to **3**]. This will be discussed further in the next section.

The filled circles in Fig. 3 represent measurements of the deposit mass. While the deposit increases monotonically with the concentration ratio, the increase is not directly proportional to **2**] and [**3**] (assuming the only thing changing with salt concentration is the local electric field **E** induced by the gradient). Also changing with salt concentration is the coefficient of **E** in [**2**] which represents the electrophoretic mobility of the colloidal particles.

Fig. 4 summarizes changes in the measured electrophoretic mobility with salt concentration. The electrophoretic mobility is essentially independent of salt concentration for concentrations below about 0.01 M. Above 0.01 M, the absolute value of mobility decreases monotonically to zero.

If the deposition rate is proportional to the electrophoretic velocity of the latex particles in the electric field induced by the salt gradient (corresponding to the first term in [**2**]), then the deposit amount should correlate with the product of the electrophoretic mobility and the diffusion potential induced by the salt gradient. One question is, for the purpose of predicting the deposition amounts in Fig. 3, at what salt concentration should the mobility be evaluated (4 M, 0.4 M, or some value in between)?

The correct answer is the salt concentration at the outer edge of the growing deposit. To estimate this value, we would need to evaluate each resistance to diffusion of the salt: the porous membrane, the porous latex deposit, and the two diffusion boundary-layers arising from stirring the bulk solutions on either side of the membrane. The linear growth of the deposit with time implies that the resistance of the growing deposit is negligible; if the boundary-layer resistance on the low-concentration side of the membrane is also negligible, then the salt concentration at the outer edge of the growing deposit is approximately equal to that on the low-concentration side of the membrane.

Fig. 5 shows a strong linear relationship between the deposit mass and the product of mobility (taken from Fig. 4 at the lower salt concentration) and diffusion potential. This strong correlation supports the hypothesis that the rate of deposition is determined by the first term in [**2**]: electrophoresis in the induced electric field. This was also the conclusion reached by Lin and Prieve (13) at much lower concentrations.

The deposition rates are also quite similar to those of Lin and Prieve (13) despite the much higher salt concentrations used here. For a concentration ratio of 10:1, Fig. 3 reports about 21 mg of deposit in 30 min (0.7 mg/min), compared with 4 mg deposited in 4 min (1.0 mg/min) in figure 7 in ref. 13 for a 10:1 ratio of NaCl.

The main difference between Fig. 5 and the corresponding figure 6 in ref. 13 is that they obtained a linear relationship without including mobility. The electrophoretic mobility of their latex did not depend significantly on salt concentration (at least for the lower salt concentrations used by them). This is also true for our latex at lower salt concentrations (Fig. 4). However, at the much higher salt concentrations used by us the mobility decreases sharply with salt concentration. This decrease needs to be considered—along with changes in the diffusion potential—to get the linear correlation shown in Fig. 5.

### Electrophoretic Mobility.

The solid curve in Fig. 4 represents [**7**], which is based on the Smoluchowski and Gouy–Chapman equations. The surface charge density σ was treated as an adjustable parameter. The best-fitting value of 0.07 C/m^{2} corresponds to an average spacing of 1.5 nm between elemental charges on the surface of the particle. Since many hydrated adsorbing ions are smaller than 1.5 nm, this average spacing seems reasonable if the charges are adsorbed on a planar interface.

The fit of the solid curve to the experimental data are quite good for *C* > 0.1 M, which is remarkable because the Gouy–Chapman equation used to derive [**7**] assumes a thermodynamically ideal solution; this is unlikely to be valid for such large concentrations of NaCl. A similar conclusion was reached by Garg et al. (45).

The failure of the solid curve to predict the plateau in electrophoretic mobilities for very low concentrations is expected. At these low concentrations, the Debye length becomes comparable to or larger than the particle radius and neither Smoluchowski’s equation [**5**] nor the Gouy–Chapman equation [**6**] would apply. While corresponding theories at low concentrations are available for small zeta potentials, an electrophoretic mobility of −5.5 (μm/s) per (V/cm) corresponds to a zeta potential more negative than −100 mV, which is not small. The available theories valid for large zetas are only numerical.

### Diffusiophoretic Velocity.

Although Stout and Khair (36) have developed a new theory for predicting the diffusiophoretic velocity under conditions of high salinity, we have made no effort to compare the preliminary measurements of deposition rates with these predictions. The main reason is that the local concentration gradient—which controls the induced electric field [**1**] and the diffusiophoretic velocity [**2**]— is not well-defined in these experiments (see the discussion above regarding at what salt concentration the mobility should be evaluated).

Instead of a direct comparison between measured and predicted velocities, we have established a strong correlation between the measured deposition rate and the measured diffusion potential (Fig. 5) as the salinity gradient is varied. This cross-correlation eliminates the need to establish the concentration gradient, which directly affects both of the correlated quantities (although not necessarily equally). This was done as part of a preliminary investigation of the effect of high salinity.

### Diffusion Potential.

We have tested the ability of the hard-sphere theory to predict the diffusion potential. According to Stout and Khair (36), the most dramatic effect of steric interactions is to increase the electric field induced by the salt gradient. The diffusion potential is simply the integral of this electric field between the two bulk solutions and has the advantage of being directly measurable and relatively easy to predict. To predict the diffusion potential from [**4**], we need to assign values to the diameters of sodium and chloride ions (i.e., the *a*_{+} and *a*_{−} appearing in *SI Appendix*, Eqs. **S5** and **S6**). In the paragraphs below, we discuss three different choices.

### Hydrated Ion Size.

Sizes of many hydrated ions, reported in over 400 different studies, were reviewed by Ohtaki and Radnai (46). The sizes for sodium and chloride ions are summarized in Fig. 6. Different symbols denote different tables in the review article, which in turn usually denote different techniques (e.g., neutron scattering, X-ray scattering, molecular dynamics simulations, etc.). While a range of sizes is reported for either ion, all of the chloride ion radii lie above the dashed line while all of the sodium ion radii lie below the dashed line. We will return to this inequality after the following discussion.

For the Curve #1 in Fig. 7, we chose the ion sizes at the lowest ratio of water/salt: a radius of 0.320 nm (*a*_{−} = 0.640 nm) for chloride ions and of 0.241 nm (*a*_{+} = 0.482 nm) for sodium ions. This water/salt ratio is near that corresponding to *C*_{hi} = 4 M; also, these radii reasonably represent the means of the distributions of radii reported for each ion. According to the table titled “Ionic Conductivity and Diffusion at Infinite Dilution” in ref. 43, the diffusion coefficients of Na^{+} and Cl^{−} at 25 °C are 1.334 × 10^{−9} m^{2}/s and 2.031 × 10^{−9} m^{2}/s, respectively. These values were used for *D*_{+} and *D*_{−} in both curves of Fig. 7.

These *a*_{+} and *a*_{−} values lead to a volume fraction of 0.472 for the most concentrated solution (4 M). Such a large volume fraction suggests that the effects of volume exclusion should be very significant. Indeed, the diffusion potentials shown as Curve #1 in Fig. 7 are almost an order of magnitude larger than those shown as the dashed line. However, our measurements of diffusion potential are actually smaller than the predictions for an ideal solution, rather than larger.

Are the hydrated ion sizes the most appropriate for this calculation? In section 3.5.4 in ref. 38 Bockris and Reddy discuss which ion size is most appropriate for extending the validity of the Debye–Huckel theory. In the next paragraph, we paraphrase their discussion.

### Crystal Lattice Spacing.

Another measure of ion size is the lattice spacing in a crystal. Of course, in a crystal the ions are completely unhydrated and so this measure of size represents a lower bound for the size of ions dissolved in water, which are hydrated. Bockris and Reddy (38) regard the hydrated ion size to be an upper bound, arguing that (upon close approach) two oppositely charged ions will experience strong attraction which can compress the hydration shell around either ion. In support of this argument is the value of their best-fitting ion size: In table 3.9 in ref. 38 they report *a* = 0.40 nm for NaCl, which is considerably smaller than the sum of the radii of the hydrated ions (0.56 nm). They observe that taking *a* = 0.40 nm “gives an almost exact agreement up to 0.02 M.” Of course, this concentration is still a factor of 200 smaller than *C*_{hi} in our experiments.

### Hydrodynamic Size.

A third measure of ion size is the hydrodynamic size inferred by applying Stokes’ law to measurements of ion mobility or by applying the Stokes–Einstein equation to measurements of diffusion coefficients:*a*_{i} is assumed to be very large compared with the molecules of the fluid. When the sphere represents small ions like Na^{+} and Cl^{−} in liquid water, this assumption is not met. So we really should not expect [**8**] to be quantitatively correct. Nonetheless, [**8**] works surprisingly well for nonionic solutes and within a factor of two or so for ionic solutes in dilute solutions (47).

At the large salt concentrations of our experiments, the significant volume fraction occupied by the electrolyte increases the viscosity μ of the solution and decreases the diffusion coefficients. However, only ratios of diffusion coefficient appear in [**4**]; if increased viscosity reduced both *D*_{+} and *D*_{−} by the same factor—as expected from [**8**]—the diffusion potential calculated from [**4**] will be unchanged. However, other complications such as the coupling in the Stefan–Maxwell equations could profoundly change this picture (48). We have made no effort to correct *D*_{+} and *D*_{−} for concentration.

Using the values of *D*_{+} and *D*_{−} cited above and μ = 0.893 mPa⋅s, [**8**] yields hydrodynamic diameters of *a*_{+} = 0.367 nm and *a*_{−} = 0.241 nm. Then, [**4**] yields Curve #2 in Fig. 7. Since both hydrodynamic diameters are smaller than the corresponding hydrated diameters, the maximum volume fraction corresponding to *C*_{hi} is only 0.080 (instead of 0.472). Then, it is not surprising that the effect of steric repulsion on the diffusion potentials is also much less for Curve #2 compared with Curve #1.

Unexpected is the direction of change caused by steric repulsion, which decreases the diffusion potential for Curve #2 in Fig. 7 but increases the diffusion potential for Curve #1. This reversal in direction can be attributed to a reversal in the relative sizes of the two ions: For Curve #1 the chloride ion is larger than the sodium; for Curve #2, the chloride ion is smaller than the sodium. If we keep the two ion sizes the same as in Curve #2 but interchange which ion is assigned the larger size, the 15% decrease in diffusion potential becomes a 60% increase. Clearly, the relative size of the two ions determines whether steric repulsion increases or decreases the diffusion potential.

As we just observed, our values of ion diffusion coefficients satisfy *D*_{+} < *D*_{−} and [**8**] implies that the hydrodynamic diameters satisfy *a*_{+} > *a*_{−}. This inequality is not consistent with the relative sizes of the two hydrated ions reported in Fig. 6. Our values for *D*_{+} and *D*_{−} are inferred from measurements of limiting ion conductance at infinite dilution. However, even at infinite dilution the ions are hydrated, so hydration alone does not account for the difference in the relative sizes of the two ions from these two experiments.

One speculation for the difference in relative ion sizes is that an electric field must be applied to measure the ionic conductivity, but no electric field is applied during neutron or X-ray scattering experiments. Perhaps electromigration of ions through the viscous fluid strips away part of the hydration shell. Since the orientation of water molecules in the solvation layer is reversed by the opposite charge of Na^{+} (compared with Cl^{−}), the energy holding the water molecules in the layer might be different for the two ions—which could lead to a reversal in the inequality between ion sizes. Moreover, the unhydrated ion sizes are different, which could also contribute to a difference in hydration energy. However, if the hydration shell is being partially removed by electromigration, then one would expect ion mobility to depend on electric field strength, but at low electric field strengths no such dependence is observed.

## Conclusions

A gradient in NaCl concentration induces migration of anionic latex particles toward higher NaCl concentration, which can be attributed to diffusiophoresis. For a given gradient in concentration, the rate of diffusiophoresis was found to be about the same as reported in the earlier study by Lin and Prieve (13) despite the salt concentrations’ being larger by a factor of 100.

The observed dependence of the rate of latex deposition on the salt concentration ratio suggests that diffusiophoresis is dominated by electromigration of the charged latex in an electric field induced by the salt diffusion. The main difference from the earlier study is that the electrophoretic mobility of the latex was found to decrease sharply with concentration (instead of remaining constant) when the salt concentration is higher than 0.03 M, becoming vanishingly small at 4 M. To our surprise, this decrease in mobility is predicted by the ideal-solution theory if changes in viscosity and permittivity are considered.

Also measured was the electrical potential arising across the membrane. This diffusion potential was found to be proportional to the logarithm of the ratio of salt concentrations on either side of the membrane. The proportionality constant was of the same sign but about half of that predicted using ideal-solution theory. We also compared the experimental diffusion potentials with those predicted by Stout and Khair (36), who considered steric repulsion between ions of finite size. Depending on which of the two hydrated ions is larger, the predictions of diffusion potential can either be increased or decreased relative to predictions for ideal solutions (zero size). Our measurements of the diffusion potential are consistent with Na^{+} being larger than Cl^{−}, which is also consistent with sizes inferred from limiting ionic conductance measurements but is inconsistent with the relative ion sizes inferred by many studies employing scattering techniques. We offer some speculation on why scattering might lead to different relative ion sizes compared with limiting ionic conductance.

## Acknowledgments

This work was supported by the Exploration and Petroleum Engineering Center Advanced Research Center, Saudi Aramco.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: dcprieve{at}cmu.edu. ↵

^{2}Present address: Sequencing Instrument Research and Development, Roche Sequencing Solutions, Santa Clara, CA 95050.

Author contributions: D.C.P., A.S.K., and M.Y.K. designed research; S.M.M., A.S.K., and R.F.S. performed research; D.C.P., A.S.K., and M.Y.K. analyzed data; and D.C.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. S.I.A. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1701391115/-/DCSupplemental.

Published under the PNAS license.

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