Shengji
Zhang
a and
Henry C. W.
Chu
*b
aDepartment of Chemical Engineering, University of Florida, Gainesville, FL 32611, USA
bDepartment of Chemical Engineering and Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA. E-mail: h.chu@ufl.edu
First published on 1st October 2024
Existing theories can predict separately the effects of hydrodynamic slip and ion–ion electrostatic correlations on diffusioosmosis. However, a predictive model for the coupled dynamics of hydrodynamic slip and electrostatic correlations in diffusioosmosis is lacking. In this work, we develop a mathematical model to compute the diffusioosmotic mobilities of valence-symmetric electrolytes in a charged parallel-plate channel. We employ the Navier slip condition to model the hydrodynamic slip at the channel walls and the modified Poisson equation to model ion–ion electrostatic correlations. We report two key findings arising from the competition between electrostatic correlations and hydrodynamic slip, which radically change diffusioosmosis. First, in a divalent electrolyte, a minute hydrodynamic slip defers the reversal in the direction of diffusioosmosis caused by electrostatic correlations to a higher concentration. Hydrodynamic slip can even eliminate the diffusioosmosis reversal in a monovalent electrolyte. Second, electrostatic correlations limit the change in the mobility due to hydrodynamic slip, by hindering the slip-enhanced ionic transport via surface charge overscreening. Electrostatic correlations can reduce the change in the mobility by ≈60% in a monovalent electrolyte, whereas the stronger electrostatic correlations in a divalent electrolyte can even reduce the change by an order of magnitude. The model developed from this work can be used to understand and predict diffusioosmosis in natural settings such as metamorphic transformation, in addition to that in applications such as colloidal species separation, nanoparticle drug delivery, and enhanced oil recovery.
Current diffusioosmosis theories investigate the effect of hydrodynamic slip in the absence of ion–ion electrostatic correlations.31–35 Hydrodynamic slip occurs over a low surface energy material such as organosilane or fluorine. Physically, the lower the surface energy, the stronger the repulsion between the fluid and the surface and the weaker the fluid can wet a surface. Thus, a fluid dissipates less energy due to the lower friction over a low surface energy, hydrodynamically slipping surface. Hydrodynamic slip is characterized by the slip length, defined as the distance into a surface where the flow velocity profile would extrapolate to zero (Fig. 1).36–41 The slip length can be extracted from various experimental measurements, including the fluid flow rate using molecular dye tracers and confocal microscopy, the fluid velocity profile by micro-particle image velocimetry, and the shear stress by atomic force microscopy.40,41 Ajdari and Bocquet32 pioneered the study of diffusioosmosis in the presence of hydrodynamic slip. Ignoring the convective ionic flux, they predicted that diffusioosmosis over a hydrophobic surface with a slip length b can be enhanced by a factor of (1 + 2bκ) relative to that over a non-slipping surface, where κ−1 is the Debye length. For a smooth hydrophobic surface where b ∼ O(1)–O(102) nm42–47 and a 1 M monovalent electrolyte solution where κ−1 ∼ O(0.1) nm,48 their results indicated that the enhancement can be as large as a thousand times. Hoshyargar et al.34 extended the prior model32 by incorporating the convective ionic flux. They showed that the convective ionic flux in fact controls the electric field induced by the solute concentration gradient and thus sets an upper bound for the enhancement of diffusioosmosis by the hydrodynamic slip. However, hydrodynamic slip still plays a significant role in diffusioosmosis in that it can enhance diffusioosmosis by up to an order of magnitude relative to that over a non-slipping surface.
Our recent study examined the effect of ion–ion electrostatic correlations on diffusioosmosis in the absence of hydrodynamic slip.49 Ion–ion electrostatic correlations refer to the Coulombic interactions between an ion pair, which are important in concentrated monovalent electrolytes or dilute and concentrated multivalent electrolytes.50,51 Electrostatic correlations cause overscreening of the surface charges. Overscreening means that the charge of the surface is overcompensated by a first layer of counterions, and the first layer of counterions is overcompensated by a second layer of coions, and so on. Overscreening manifests in sign oscillations in the space charge density normal to a charged surface. Overscreening can be modeled by molecular dynamics simulations, integral approaches, and statistical theories.52–61 For overscreening over a metal surface, the recently developed density–potential–polarization functional theory can further account for the coupling between the metal electronic effects and ions-layering effects in electrolytes.62 Although these approaches can model overscreening very accurately by capturing individual oscillation peaks in the space charge density,63 their high computational cost has largely restricted them to modeling static systems. Without capturing individual oscillation peaks, Storey and Bazant51 developed a continuum-level, modified Poisson equation that can model overscreening effectively. The modified Poisson equation can be coupled with other continuum-level transport equations, such as the Nernst–Planck and Stokes equation, to efficiently model dynamical transport processes.49,51,64–66 Employing the modified Poisson equation, our recent study showed that overscreening causes a unique reversal in the direction of diffusioosmosis,49 which cannot be captured by prior models that ignore electrostatic correlations. The diffusioosmosis reversal occurs at O(0.1) M in a monovalent electrolyte and at an even lower concentration O(10−3)–O(10−2) M in a divalent electrolyte, thus highlighting its practical significance. However, there is currently a lack of a predictive model for diffusioosmosis that considers the coupling of electrostatic correlations and hydrodynamic slip.
In this work, we develop a mathematical model to predict the diffusioosmotic mobility in a charged parallel-plate channel under the influence of ion–ion electrostatic correlations and hydrodynamic slip. We employ the modified Poisson equation51 to model electrostatic correlations and the Navier slip condition to model the hydrodynamic slip.36–41 We report two key findings which highlight the radical impacts of the coupling of hydrodynamic slip and electrostatic correlations on diffusioosmosis. The effects of hydrodynamic slip and electrostatic correlations are competitive in nature. First, we show that, relative to diffusioosmosis over a no-slip surface, a minute hydrodynamic slip can postpone the occurrence of the diffusioosmosis reversal caused by electrostatic correlations in a divalent electrolyte from O(10−2) M to a much higher concentration at O(10−1) M. Such a small hydrodynamic slip can even eliminate the diffusioosmosis reversal in a monovalent electrolyte, where the effect of hydrodynamic slip outweighs electrostatic correlations. Second, we show that electrostatic correlations can reduce the change in the mobility due to hydrodynamic slip by ≈60% in a monovalent electrolyte, and can even reduce the change by an order of magnitude in a divalent electrolyte. We identified that electrostatic correlations strengthen the attraction between the oppositely charged ions near the charged surface. The hindered ionic transport by electrostatic correlations competes with the enhanced ionic transport by hydrodynamic slip, resulting in a weakened modification of diffusioosmosis by slip. These two key findings demonstrate that the coupling of hydrodynamic slip and electrostatic correlations not only impacts diffusioosmosis by changing its magnitude significantly, but also qualitatively reverses its direction.
The rest of this article is structured as follows. In Section 2, we present the problem formulation for computing the electric potential, induced electric field, diffusioosmotic velocity, and diffusioosmotic mobility. In Section 3, we present our results and elaborate on the two above-mentioned key findings. In Section 4, we summarize this study and offer ideas for future work.
(1) |
(2) |
We have non-dimensionalized eqn (2) and equations hereafter using the following schemes, where quantities with carets are non-dimensional: ŷ = y/H, = ϕ/(kT/e), = κH, and = ρ/(n∞ez). The Debye length is the characteristic lengthscale over which the space charge density varies. The quantity δc = κlc contains the ion–ion electrostatic correlation length lc = z2e2/(4πεkT), which characterizes the significance of electrostatic correlations: correlations are absent at δc = 0 whereas they are prominent at δc ≳ 1.50,51 The boundary conditions for eqn (2) are as follows: the surface charge density is specified d/dŷ = − = −qeH/(εkT) and the mean-field charge density is flat d3/dŷ3 = 0 at ŷ = 0; and the electric potential and its derivative decay to zero smoothly, = 0 and d/dŷ = 0, at ŷ = 1.51 The condition = 0 is guaranteed for ≥ 10 in this work.
(3) |
(4) |
(5) |
We have validated the present model by recovering the results from prior work. First, in the absence of ion–ion electrostatic correlations (δc = 0) and hydrodynamic slip ( = 0), the present model recovers the results from Ma and Keh.68 Details of this validation were presented in our prior work.49 Second, in the absence of ion–ion electrostatic correlations (δc = 0) but in the presence of hydrodynamic slip ( ≠ 0), the present model recovers the results from Hoshyargar et al.34 Third, in the presence of ion–ion electrostatic correlations (δc ≠ 0) but in the absence of hydrodynamic slip ( = 0), the present model can recover the results from Zhang and Chu.49 Some of these validations are presented in Section 3.1 to 3.3 to contrast with the new results of this work that concern diffusioosmosis in the presence of ion–ion electrostatic correlations (δc ≠ 0) and hydrodynamic slip ( ≠ 0).
Fig. 3(b) shows the variation of in the presence of electrostatic correlations (δc ≠ 0). The second observation is that, in the absence of hydrodynamic slip ( = 0; the black line), electrostatic correlations cause a reversal in the direction of diffusioosmosis, manifested in a sign change in the mobility at C ≈ 0.4 M.49 Physically, electrostatic correlations are prominent in a concentrated electrolyte at C = 1 M. Electrostatic correlations overscreen the channel surface charge and cause a sign change in the space charge density away from the channel wall, as shown at ŷ ≈ 8 × 10−3 in Fig. 3(d). This in turn causes a reversal in the direction of the diffusioosmotic flow û away from the channel wall, as shown at ŷ ≈ 8 × 10−3 in Fig. 3(f). Thus, averaging û in Fig. 3(f) across the channel yields the positive mobility at C = 1 M in Fig. 3(b), which is in contrast to the negative mobility at C = 10−3 M. This sign change in the mobility across low to high electrolyte concentrations illustrates the diffusioosmosis reversal caused by electrostatic correlations. Note that overscreening and diffusioosmosis reversal are absent when electrostatic correlations are ignored [Fig. 3(a), (c) and (e)].
Next, let us examine a key result arising from the coupled effects of hydrodynamic slip and electrostatic correlations on diffusioosmosis. Fig. 3(b) shows that a minute hydrodynamic slip can eliminate the diffusioosmotic flow reversal caused by electrostatic correlations. For instance, the mobility is negative at all C for ≥ 10−2. This can be understood by analyzing Fig. 3(f), which shows the diffusioosmosis velocity û at C = 1 M. Specifically, the negative û at the channel wall is significantly enhanced by hydrodynamic slip. Fluid momentum is transferred from the wall to the bulk via viscous drag. In the presence of slip, in the bulk the enhanced negative momentum by slip outweighs the positive momentum by electrostatic correlations, leading to a negative û across the channel at all ŷ. Averaging û across the channel yields the negative mobility in Fig. 3(b) for ≥ 10−2 at C = 1 M. Thus, the same sign of the mobility across low to high electrolyte concentrations signifies that a minute hydrodynamic slip can eliminate the diffusioosmosis reversal caused by electrostatic correlations. This demonstrates the competitive nature between the effects of hydrodynamic slip and electrostatic correlations on diffusioosmosis. We remark that, given that = b/H with H = 100 nm, = 0.01 corresponds to a slip length b = 1 nm, which is typical for common natural and synthetic materials.42–47 This implies that the correlation-induced diffusioosmosis reversal in a monovalent electrolyte occurs only over rough surfaces with b < 1 nm.
Next, let us examine a key result arising from the coupled effects of hydrodynamic slip and electrostatic correlations on diffusioosmosis. Comparing with Fig. 4(a) and (b) shows that electrostatic correlations can significantly reduce the change in the mobility due to hydrodynamic slip by ≈60%. For example, at C = 1 M, Fig. 4(a) shows that the mobilities at = 10−3 and = 1 are −0.15 and −6.27 respectively, yielding a difference of −6.12 in the absence of electrostatic correlations. In contrast, in the presence of electrostatic correlations, Fig. 4(b) shows that the mobilities at = 10−3 and = 1 are 0.09 and −2.30 respectively, yielding a difference of −2.39. To understand the reduced hydrodynamic slip-induced change in the mobility by electrostatic correlations, recall that electrostatic correlations and overscreening lead to the first layer of positive counterions adjacent to the charged surface being overcompensated by a second layer of negative coions, as shown in Fig. 4(d). Due to Coulombic attractions, the second layer of coions attracts the first layer of counterions and therefore weakens the slip-enhanced diffusioosmosis [the effect (i) in the previous paragraph]. The counteracting electroosmosis induced by the slip-enhanced diffusioosmosis [the effect (ii)] is also weakened. Thus, electrostatic correlations weaken the overall diffusioosmosis and significantly reduce the change in the mobility by hydrodynamic slip. This again demonstrates the competitive nature between the effects of hydrodynamic slip and electrostatic correlations on diffusioosmosis.
In Appendix A, we present the same analyses as in Section 3.1 for a monovalent electrolyte with β = 0.5. All the observations in Section 3.1 for β = −0.5 persist when β = 0.5, demonstrating the generality of the observations and the underlying physics.
Fig. 5 Diffusioosmosis of a divalent electrolyte with ion diffusivity ratio β = −0.1. The channel surface charge density q = 0.014 C m−2 corresponds to a channel surface potential ζ = 50 mV at a bulk electrolyte molar concentration C = 10−3 M. Panels (a), (c), and (e) ignore ion–ion electrostatic correlations (δc = 0). Panels (b), (d), and (f) account for electrostatic correlations (δc = κlc). Other figure captions are the same as those in Fig. 3. |
However, the key result arising from the coupling of hydrodynamic slip and electrostatic correlations in Section 3.1.1 does not persist for a divalent electrolyte. Specifically, Fig. 3(b) shows that a minute hydrodynamic slip ( ≥ 10−2) can eliminate the diffusioosmosis reversal of a monovalent electrolyte caused by electrostatic correlations. In contrast, Fig. 5(b) shows that even a large hydrodynamic slip ( = 1) cannot cancel the diffusioosmosis reversal of a divalent electrolyte but can only defer its occurrence to a higher concentration. This can be understood by analyzing Fig. 5(f), which shows the diffusioosmosis velocity û at C = 1 M. The positive û at the channel wall is enhanced by hydrodynamic slip. However, electrostatic correlations cause a sign reversal in the space charge density [Fig. 5(d)], which is sufficiently strong to cause a sign reversal in û and thus a negative û at ŷ ≳ 2 × 10−3 for all , as shown in Fig. 5(f). Averaging û across the channel yields the negative at C = 1 M for all in Fig. 5(b). Thus, switching from a positive to a negative across low to high electrolyte concentrations signifies that even a large hydrodynamic slip cannot cancel the diffusioosmosis reversal of a divalent electrolyte, but can only postpone its occurrence to a higher concentration. This implies that the correlation-induced diffusioosmosis reversal in a divalent electrolyte is likely present in most common natural and synthetic materials where ≤ O(1). Note that, despite the different responses shown here for a divalent electrolyte compared to a monovalent electrolyte in Section 3.1.1, their origin is common and is the competition between hydrodynamic slip and electrostatic correlations in diffusioosmosis.
Fig. 6 Diffusioosmosis of a divalent electrolyte with ion diffusivity ratio β = −0.1. The channel surface charge density q = 0.014 C m−2 corresponds to a channel surface potential ζ = 50 mV at a bulk electrolyte molar concentration C = 10−3 M. Panels (a) ignores ion–ion electrostatic correlations (δc = 0). Panels (b) accounts for electrostatic correlations (δc = κlc). Other figure captions are the same as those in Fig. 4(a) and (b). |
Next, we show that the coupled effect of hydrodynamic slip and electrostatic correlations presented in Section 3.1.2 is radically intensified here in a divalent electrolyte. For example, at C = 1 M, Fig. 6(a) shows that the mobilities at = 10−3 and = 1 are 0.10 and 4.13 respectively, yielding a difference of 4.03 in the absence of electrostatic correlations. In the presence of electrostatic correlations, Fig. 6(b) shows that the mobilities at = 10−3 and = 1 are −0.85 and −0.42 respectively, yielding a difference of 0.43. That is, in a divalent electrolyte, electrostatic correlations can reduce the change in the mobility due to slip by an order of magnitude. This is in contrast to the weaker ≈60% reduction in a monovalent electrolyte in Section 3.1.2. The stronger reduction in a divalent electrolyte is due to the stronger electrostatic correlations, represented by a larger δc = κlc.50,51 For instance, at C = 1 M, δc = 18.8 in a divalent electrolyte whereas δc = 2.4 in a monovalent electrolyte. Stronger electrostatic correlations give rise to stronger attractions between the second layer of coions and the first layer of counterions adjacent to the overscreened charged surface. This causes a stronger hindrance to the first layer of counterions and a stronger reduction in the slip-enhanced diffusioosmosis. Thus, electrostatic correlations can reduce the change in the mobility due to hydrodynamic slip more significantly in a divalent electrolyte compared to a monovalent electrolyte. This again confirms the competition between the effects of hydrodynamic slip and electrostatic correlations on diffusioosmosis.
Fig. 7 Diffusioosmotic flow direction of monovalent electrolytes versus the channel surface charge density q and ion diffusivity ratio β. Ion–ion electrostatic correlations are accounted for (δc = κlc). (a): Bulk electrolyte molar concentration C = 10−3 M and slip length = 0. (b): C = 10−3 M and = 1. (c): C = 1 M and = 0. (d): C = 1 M and = 1. Domains with a positive (negative) mobility M denote that diffusioosmosis is from a region of low (high) to high (low) electrolyte concentration. Circle symbols in (a) and (c) correspond to the mobility at C = 10−3 M and C = 1 M along the line = 0 in Fig. 3(b). Star symbols in (c) and (d) correspond to the mobility with = 0 and = 1 at C = 1 M in Fig. 3(b). |
Fig. 8 Diffusioosmotic flow direction of divalent electrolytes versus the channel surface charge density q and ion diffusivity ratio β. Ion–ion electrostatic correlations are accounted for (δc = κlc). Other figure captions are the same as those in Fig. 7. The shaded regions in (d) are absent in Fig. 7(d), owing to the stronger electrostatic correlations in a divalent electrolyte than a monovalent electrolyte. |
Let us first examine Fig. 7(a) and (b), which show diffusioosmosis of a dilute electrolyte at C = 10−3 M in a non-slipping channel ( = 0) and a slipping channel ( = 1), respectively. Electrostatic correlations are negligible in both panels, where δc = 0.07 ≪ 1.50,51 Thus, comparing the two panels demonstrates the effect of hydrodynamic slip. On comparison, panel (b) shows that, in the absence of electrostatic correlations, hydrodynamic slip has a minor quantitative effect on shifting the values of q and β at which a sign change in the mobility occurs.
Next, let us examine Fig. 7(c), which shows diffusioosmosis of a concentrated electrolyte at C = 1 M in a non-slipping channel ( = 0). Electrostatic correlations are significant in Fig. 7(c), where δc = 2.4 > 1.50,51 Comparing Fig. 7(c) and (a), in the absence of hydrodynamic slip, electrostatic correlations can cause a reversal in the direction of diffusioosmosis, manifested in a sign change in the mobility as the electrolyte concentration increases [from panel (a) to (c)] while q and β are fixed. This echoes the second observation in Section 3.1.1. For example, the two circle symbols in Fig. 7(a) and (c) correspond to the mobility at C = 10−3 M and C = 1 M along the line = 0 in Fig. 3(b).
Next, let us examine Fig. 7(d), which shows diffusioosmosis of a concentrated electrolyte at C = 1 M in a slipping channel ( = 1). In a concentrated electrolyte, one may expect that the distribution of the +M and −M regions in Fig. 7(d) would be the same as that in Fig. 7(c). However, the distribution in Fig. 7(d) is qualitatively different from that in Fig. 7(c) and is in fact reversed. This difference is due to the strong hydrodynamic slip in Fig. 7(d), which cancels the diffusioosmosis reversal caused by electrostatic correlations. This echoes the coupled and competitive effect of hydrodynamic slip and electrostatic correlations on diffusioosmosis, as discussed in Section 3.1.1. For example, the two star symbols in Fig. 7(c) and (d) correspond to the mobility with = 0 and = 1 at C = 1 M in Fig. 3(b).
The above discussions regarding Fig. 7(a)–(d) for monovalent electrolytes also apply to Fig. 8(a)–(d) for divalent electrolytes. For brevity, we do not repeat them. Instead, we elaborate on a qualitative difference between Fig. 8(d) and 7(d). Fig. 8(d) shows the mobilities for divalent electrolytes in the presence of electrostatic correlations and hydrodynamic slip. Fig. 8(d) displays two shaded domains which are absent in Fig. 7(d). In these domains, the mobility is negative when |q| ≳ 3 × 10−3 C m−2. Physically, these domains arise, since electrostatic correlations are strong over highly charged surfaces and are more prominent in a divalent electrolyte than in a monovalent electrolyte. As a result, when |q| ≳ 3 × 10−3 C m−2, the sign of the mobility follows that in a system with strong electrostatic correlations [Fig. 8(c)], that is, a negative mobility. In contrast, when |q| ≲ 3 × 10−3 C m−2, electrostatic correlations are weak and thus the sign of the mobility follows that in a system with negligible electrostatic correlations and strong hydrodynamic slip [Fig. 8(b)]. For instance, the mobility is positive for a negative β and positive q. This again demonstrates the competition between electrostatic correlations and hydrodynamic slip in diffusioosmosis.
The experimental setup comprises a H-shape, micro- and nanochannel network, with the nanochannel connected to two side microchannels. The width W and length L of the nanochannel are at least two orders of magnitude larger than its height 2H, so that the species and fluid transport inside the nanochannel can be accurately approximated as one-dimensional. A high concentration electrolyte solution with fluorescent molecular dye tracers of concentration C0Dye is flowed through the left microchannel. A low concentration electrolyte solution without dye tracers is flowed through the right microchannel. Thus, an electrolyte concentration gradient is set up across the nanochannel. The electrolyte concentration gradient drives a diffusioosmotic flow with the tracers through the nanochannel. The pressure across the nanochannel is balanced using high precision pressure pumps, so that the fluid flow inside the nanochannel is chiefly due to diffusioosmosis and the flow due to a pressure gradient is negligible. The fluorescence intensity signal is recorded using confocal microscopy. The fluorescence intensity is proportional to the dye tracer concentration CDye(x).
At steady state, the transport of the dye is governed by the steady convection diffusion equation, Dd2CDye/dx2 − umdCDye/dx = 0, where D is the dye diffusivity and um is the mean diffusioosmotic flow velocity. Solving the equation with the boundary conditions CDye = C0Dye at x = 0 (left end of the nanochannel) and CDye = 0 at x = L (right end of the nanochannel) gives CDye = C1(D/um)exp(umx/D) + C2, with C1 = −C0Dyeum/[Dexp(umL/D) − D] and C2 = (um/D)exp(umL/D)/[Dexp(umL/D) − D]. The tracer diffusivity D is known from the Nernst–Einstein relation.97 The unknown um can be determined by using it as a fitting parameter to best-fit the profile of CDye from experiments. Once um is determined, we recall that the volumetric flow rate Q = umW(2H) with um = M∇x|logn∞| and n∞ = 103AC, where A is the Avogadro constant.‡ The flow rate can be integrated from x = 0 to x = L to obtain Q = 2M(WH/L)log[C(L)/C(0)]. Since Q, W, H, L, C(L), and C(0) are known, the only unknown mobility M can be determined and compared with model predictions.
Our first key finding arising from the coupling of hydrodynamic slip and electrostatic correlations is that hydrodynamic slip can eliminate the reversal in the direction of diffusioosmosis caused by electrostatic correlations in a monovalent electrolyte. Specifically, in the absence of slip, a prior study49 found that electrostatic correlations cause overscreening of the channel surface charges, reverse the direction of fluid momentum away from the channel surface, and lead to a reversal in the net diffusioosmosis averaged across the channel. In this work, we have shown that, in a monovalent electrolyte, hydrodynamic slip enhances the fluid momentum and outweighs that in the opposite direction caused by electrostatic correlations. As a result, in a slipping channel, the direction of diffusioosmosis, represented by the sign of the mobility, remains unchanged across low to high electrolyte concentrations. This signifies that a minute slip length as small as O(1) nm can eliminate the diffusioosmosis reversal caused by electrostatic correlations. However, such an elimination is absent in a divalent electrolyte. Specifically, electrostatic correlations in a divalent electrolyte are stronger than those in a monovalent electrolyte of the same concentration. Thus, the slip-enhanced fluid momentum is insufficient to outweigh that in the opposite direction caused by electrostatic correlations. As a result, the direction of diffusioosmosis and the sign of the mobility change across low to high electrolyte concentrations. Even a large slip length of O(102) nm cannot cancel the diffusioosmosis reversal but can only postpone its occurrence to a higher concentration.
Our second key finding arising from the coupling of hydrodynamic slip and electrostatic correlations is that electrostatic correlations can radically diminish the slip-induced change in the mobility by an order of magnitude in a divalent electrolyte, compared to the same system that ignores electrostatic correlations. Specifically, the mobility is set by a balance of two effects, namely, (i) a slip-enhanced ionic transport and diffusioosmosis and (ii) a counteracting electroosmosis induced by the slip-enhanced diffusioosmosis. We have shown that electrostatic correlations cause strong Coulombic attractions between the oppositely charged ions near the channel surface. In a divalent electrolyte, these attractions significantly weaken the effects (i) and (ii), resulting in an order-of-magnitude reduction in the change in the mobility by hydrodynamic slip. However, the reduction is more gentle in a monovalent electrolyte. This is because electrostatic correlations in a monovalent electrolyte are weaker than those in a divalent electrolyte. Thus, electrostatic correlations are not strong enough to significantly weaken the effects (i) and (ii). As a result, compared to the same system that ignores electrostatic correlations, electrostatic correlations only reduce the slip-induced change in the mobility by ≈60% in a monovalent electrolyte.
Many interesting questions remain for future work. For example, the present model can incorporate a concentration-dependent solution permittivity. Recall that the modified Poisson equation in the present work assumes a constant permittivity. In general, the solution permittivity decreases with an increasing electrolyte concentration, a phenomenon known as dielectric decrement.82,95,98–100 Incorporating a concentration-dependent permittivity will expand the applicability of the present model to electrolyte solutions of concentrations beyond a few molars. However, it is expected that dielectric decrement will have insignificant effects on the results and conclusions of the present work. This is reasoned as follows. The solution permittivity varies linearly with the electrolyte concentration as ε = εw − (γ+C+ + γ−C−), where εw is the solvent (water) permittivity, γ+ and γ− are the coefficient of dielectric decrement of the cationic and anionic species, respectively, and C+ and C− are the molar concentrations of the cationic and anionic species, respectively.95 For water at 293 K, εw = 80.1. For NaCl, γ+ = 8 M−1 and γ− = 3 M−1. Let us consider C+ = C− = 1 M, since the key observations of this work arise at 1 M or lower. Thus, the dielectric decrement is (γ+C+ + γ−C−) = 11, which is about an order of magnitude smaller than εw. Hence, dielectric decrement will have insignificant effects on the results and conclusions of the present work. This is consistent with prior work,51,60,65,76–79 where the predictions by the modified Poisson equation are in good agreement with Monte Carlo and density functional theory simulations in electrolyte solutions of concentrations up to 1 M, and likely up to a few molars. In addition, a prior model which assumes a constant solution permittivity accurately captures the experimentally measured electrophoretic mobility in electrolyte solutions up to 1 M.64 This provides strong support for the reasonable assumption of a constant solution permittivity to predict the new diffusioosmotic flow responses in the present work.
Analogous to Fig. 3(a), Fig. 9(a) shows the variation of with the bulk electrolyte concentration C in the absence of ion–ion electrostatic correlations (δc = 0). The first observation is that the amplification of the mobility by hydrodynamic slip (non-zero slip length ) is more prominent in a concentrated electrolyte (C = 1 M) than in a dilute electrolyte (C = 10−3 M). Analogous to Fig. 3(b), Fig. 9(b) shows the variation of in the presence of electrostatic correlations (δc ≠ 0). The second observation is that, in the absence of hydrodynamic slip ( = 0), electrostatic correlations cause a reversal in the direction of diffusioosmosis, manifested in a sign change in the mobility at C ≈ 0.4 M. As a key result arising from the coupled effects of hydrodynamic slip and electrostatic correlations, Fig. 9(b) shows that a minute hydrodynamic slip can eliminate the diffusioosmotic flow reversal caused by electrostatic correlations.
Fig. 9 Diffusioosmosis of a monovalent electrolyte with ion diffusivity ratio β = 0.5. The channel surface charge density q = −0.014 C m−2 corresponds to a channel surface potential ζ = −100 mV at a bulk electrolyte molar concentration C = 10−3 M. Panels (a) and (c) ignore ion–ion electrostatic correlations (δc = 0). Panels (b) and (d) account for electrostatic correlations (δc = κlc). (a) and (b): The diffusioosmotic mobility versus C. Color schemes follow Fig. 3. (c) and (d): versus the slip length . Color schemes follow Fig. 4. |
Analogous to Fig. 4(a), Fig. 9(c) shows the variation of with in the absence of electrostatic correlations. The first observation is that the amplification of by hydrodynamic slip is bounded, as shown by the plateau in the limit of large of each line. A key result arising from the coupled effects of hydrodynamic slip and electrostatic correlations is that, when comparing Fig. 9(c) and (d) at C = 1 M, the latter shows that electrostatic correlations can significantly reduce the change in the mobility due to hydrodynamic slip. This is analogous to comparing Fig. 4(a) and (b).
Footnotes |
† The zero current condition will require modifications if the system is connected to a closed circuit, e.g., the two ends of the electrolyte concentration gradient are connected to an external resistor.85,86 |
‡ Lee et al.96 defined um = −M∇x|logn∞| whereas our definition does not have the minus sign. Both definitions are correct as long as the theory and experiments consistently use the same definition. |
This journal is © The Royal Society of Chemistry 2024 |