3 Results

In a cylindrical domain, the profiles $\unicode[STIX]{x1D711}(r)$ and $\unicode[STIX]{x1D6F9}(r)$ of (2.8) and (2.9) are the linear combinations

where $\text{I}_{n}$ is the modified Bessel function of the first kind of order $n$ , such that the wavenumbers $q_{i}$ ( $i=1,2$ ) obey the biquadratic equation

The amplitudes $a_{i}$ and $b_{i}$ follow from symmetry at $r=0$ and the BCs given above at the channel surface $r=R$ . Also making use of the identity $((q_{1}\unicode[STIX]{x1D706}_{D})^{2}-1)((q_{2}\unicode[STIX]{x1D706}_{D})^{2}-1)=g_{D}^{2}/(4\unicode[STIX]{x03C0}l_{B}C)$ , which follows from (3.3), the amplitudes read

We plot in figure 2(a) the resulting composition and potential profiles for a water–acetonitrile mixture with $n_{0}=50$  mM of NaCl ( $\unicode[STIX]{x1D706}_{D}=1.16$  nm, $g^{+}\approx 0$ and $g^{-}\approx 4$ (Kalidas et al. Reference Kalidas, Hefter and Marcus2000; Marcus Reference Marcus2007)) at room temperature ( $\unicode[STIX]{x1D712}=1.86$ , $\unicode[STIX]{x1D709}=1.31$  nm) inside a channel with radius $R=20\unicode[STIX]{x1D706}_{D}$ . The mixture physical properties are given in the caption of figure 2. We assume that these properties are independent of temperature, except for the mixture interdiffusion constant $D^{m}$ , which follows from $D^{m}=(6\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709})^{-1}$ (Kawasaki Reference Kawasaki1970). The ionic diffusion constants are taken to be $D^{\pm }=1\times 10^{-9}~\text{m}^{2}~\text{s}^{-1}$ .

Figure 2.

(a) Composition $\unicode[STIX]{x1D711}(r)$ and scaled potential $\unicode[STIX]{x1D6F9}(r)$ at the channel centre ( $z=0$ ) and (b) the corresponding scaled cation and anion profiles, for a water–acetonitrile mixture with a Debye length $\unicode[STIX]{x1D706}_{D}=1.16$  nm and a correlation length $\unicode[STIX]{x1D709}=1.31$  nm. The solid lines represent the numerical solution and the dashed curves represent the linear theory (see text). We plot the profiles near the channel surface, where here and in all other figures the channel radius is $R=20\unicode[STIX]{x1D706}_{D}$ . The channel surface is uncharged ( $\unicode[STIX]{x1D70E}=0$ ) and hydrophilic, $\unicode[STIX]{x1D6FE}=0.2C$ , corresponding to a tension of ${\approx}7~\text{mN}~\text{m}^{-1}$ . The mixture with $\bar{\unicode[STIX]{x1D700}}=58$ and at a temperature of 293 K ( $\unicode[STIX]{x1D712}=1.86$ ) contains 50 mM of NaCl ( $g^{+}=0,g^{-}=4$ ). For the mixture properties, we use the critical temperature $T_{c}=272$  K and a molecular length $a=3.6$  Å, estimated from a critical density of $900~\text{kg}~\text{m}^{-3}$ and a viscosity of 0.7 mPa s (Sazonov et al. Reference Sazonov, Shaw, Skrzecz, Lisov and Sazonov2007; Wohlfarth Reference Wohlfarth2009). The liquid permeability is given by $\unicode[STIX]{x1D700}(\unicode[STIX]{x1D711})/\unicode[STIX]{x1D700}_{0}=\unicode[STIX]{x1D700}_{acetonitrile}/2+\unicode[STIX]{x1D700}_{water}/2+(\unicode[STIX]{x1D700}_{water}-\unicode[STIX]{x1D700}_{acetonitrile})\unicode[STIX]{x1D711}$ , where $\unicode[STIX]{x1D700}_{0}$ is the vacuum permittivity, and we used $\unicode[STIX]{x1D700}_{acetonitrile}=36.6$ and $\unicode[STIX]{x1D700}_{water}=79.5$ .

The channel surface is uncharged ( $\unicode[STIX]{x1D70E}=0$ ), but is hydrophilic ( $\unicode[STIX]{x1D6FE}=0.2C>0$ ), resulting in the adsorption of water near the wall. Since the anions are hydrophilic, the water ‘drags’ them along, and hence the anions also effectively adsorb at the surface; see the ionic profiles in figure 2(b). Although the cations are indifferent to the local composition, they also adsorb at the surface due to electrostatics, but are distributed more broadly. The result is an electric double layer at an uncharged surface (Samin & Tsori Reference Samin and Tsori2013; Samin et al. Reference Samin, Hod, Melamed, Gottlieb and Tsori2014), even though this layer is overall charge neutral. Figure 2 shows good agreement between the full numerical solutions and the linear theory. The numerical profiles of $\unicode[STIX]{x1D711}$ and $n^{\pm }$ decay to values that slightly differ from their bulk values due to a weak nonlinear effect induced by solvation (Samin & Tsori Reference Samin and Tsori2016). The thickness of the charged layer at the surface, $\unicode[STIX]{x1D70C}(r)=2en_{0}(g_{D}\unicode[STIX]{x1D711}-\unicode[STIX]{x1D6F9})$ , could be estimated in experiments from the characteristic lengths $q_{i}^{-1}$ . These length scales are to be estimated from (3.3) given data for $\unicode[STIX]{x1D706}_{D}$ , $\unicode[STIX]{x1D709}$ and $g_{D}$ .

Although the fluid is overall neutral, the negative ion layer is more strongly adsorbed at the wall. When an external axial electric field is applied, this layer then serves as an effective surface charge, and the adjacent positively charged layer as an effective electric double layer. The result is an induced fluid flow, similar to EOF. The steady-state axial flow profile $v_{z}(r)$ is obtained by putting (3.1) and (3.2) into (2.10), and imposing the no-slip BC at $r=R$ and symmetry at $r=0$ ,

which for the parameters of figure 2 is plotted in figure 3 (labelled with $\unicode[STIX]{x1D70E}=0$ ), showing reasonable agreement between the numerical solution and (3.6). Indeed, a flow is generated by the overall neutral but locally charged layer of fluid near the wall. Figure 3 shows that $v_{z}(r)$ quickly saturates to a plug-like flow of a few $\text{mm}~\text{s}^{-1}$ . We term this solvation-induced phenomenon solvo-osmotic flow (SOF). This flow is similar and comparable in magnitude to simple EOF, shown in figure 3 by the curve for a neat solvent with the same properties as the mixture, but near a weakly charged surface with $\unicode[STIX]{x1D70E}=-0.012~\text{nm}^{-2}$ .

Solvo-osmotic flow can either combine with or work against EOF, as shown by two curves in figure 3, where we used the parameter of figure 2, but with non-zero surface charges, $\unicode[STIX]{x1D70E}=\pm 0.012~\text{nm}^{-2}$ . For $\unicode[STIX]{x1D70E}<0$ , the result is an enhanced flow with the velocity at the channel centre slightly smaller than the sum of SOF and EOF on their own. In a neat solvent, when $\unicode[STIX]{x1D70E}$ changes sign, one expects a simple change of sign also for the velocity. However, here SOF opposes EOF for $\unicode[STIX]{x1D70E}>0$ and, for the parameters of figure 3, the result is that inside most of the channel $v_{z}$ remains positive, but with a much smaller magnitude. There is, however, a weak backflow close to the channel wall resulting from the radial component of the body force in the fluid becoming significant in this case.

Figure 3.

Axial velocity profiles $v_{z}(r)$ at an applied field of $E_{0}=2.5~\text{V}~\text{mm}^{-1}$ for several scenarios; see the labels on the figure. All other parameters are the same as in figure 2. The numerical results (solid curves) are in good agreement with the linear theory (dashed curves), where in (3.6) we used $E_{z}$ from the numerical calculation. Crosses were calculated using (3.8).

In the large- $R$ limit where $R\gg \unicode[STIX]{x1D706}_{D},\unicode[STIX]{x1D709}$ , the fluid moves as a plug, $v_{z}=\text{const}.$ , and one defines the electro-osmotic mobility of the fluid as $\unicode[STIX]{x1D707}_{e}=v_{z}/E_{z}$ . In this limit, we find

Noting that the biquadratic equation (3.3) implies $q_{1}q_{2}=(\unicode[STIX]{x1D709}\unicode[STIX]{x1D706}_{D})^{-1}$ and $q_{1}^{2}+q_{2}^{2}=\unicode[STIX]{x1D709}^{-2}+\unicode[STIX]{x1D706}_{D}^{-2}[1-g_{D}^{2}/(4\unicode[STIX]{x03C0}l_{B}C)]$ , and assuming that also $\unicode[STIX]{x1D706}_{D}/\unicode[STIX]{x1D709}\gg |g_{D}|/(4\unicode[STIX]{x03C0}l_{B}C)$ , equation (3.7) reduces to the modified Helmholtz–Smoluchowski equation

Equation (3.8) is the main result of this paper, where the term in brackets is the zeta potential $\unicode[STIX]{x1D701}$ , with the shear plane located at $r=R$ . Only the first term $\propto \unicode[STIX]{x1D70E}$ in (3.8) exists for a neat solvent (Keh & Tseng Reference Keh and Tseng2001). In a mixture, $\unicode[STIX]{x1D701}$ is determined also by the properties of the salt and surface through the second term $\propto g_{D}\unicode[STIX]{x1D6FE}$ in (3.8), which plays the role of an effective surface charge. The attractive feature of (3.8) is the ability to easily tune the fluid mobility through $g_{D}$ without making any surface modification. For example, by replacing the anion $\text{Cl}^{-}$ with the hydrophobic anion $\text{BPh}_{4}^{-}$ , we have $g_{D}\approx 6$ (Marcus Reference Marcus2007) (instead of $g_{D}=-2$ ), which would then change $\unicode[STIX]{x1D707}_{e}$ drastically, in some cases even its sign. As we show below, the dependence of $\unicode[STIX]{x1D707}_{e}$ on $\unicode[STIX]{x1D709}$ in mixtures opens the possibility to tune the mobility also with temperature, since $\unicode[STIX]{x1D709}$ diverges near the mixture critical temperature. Nevertheless, we stress that $\unicode[STIX]{x1D709}\sim a$ for completely miscible mixtures (the $\unicode[STIX]{x1D712}\rightarrow 0$ limit in our theory). Hence, equation (3.8) should be applicable to mixtures in general, and, as the crosses in figure 3 confirm, it is in good agreement with the numerical results for $v_{z}(0)$ .

Figure 4.

Dependence of the fluid mobility $\unicode[STIX]{x1D707}_{e}$ on (a) the surface–solvent coupling $\unicode[STIX]{x1D6FE}$ , (b) the surface charge $e\unicode[STIX]{x1D70E}$ , (c) the salt concentration $n_{0}$ and (d) the temperature $T$ , for three types of electrolytic mixture (see text). Symbols correspond to numerical results and lines to (3.8). In (b–d), we plot also $\unicode[STIX]{x1D707}_{e}$ for a neat solvent with the same physical properties as the mixtures (circles). In (a–c), $T=293$ K is fixed, while in (a,b,d) we keep $n_{0}=50$ mM fixed. For the mixture data sets in (b–d), we took $\unicode[STIX]{x1D6FE}=0.2C$ , and in (a,c,d), we took $\unicode[STIX]{x1D70E}=0$ . For the neat solvent in (c,d), we used $\unicode[STIX]{x1D70E}=0.012~\text{nm}^{-2}$ . The inset of (b) shows the shift in mobility between the mixtures and the neat solvent.

In figure 4, we show the effects of varying some of the free parameters of our model on the fluid mobility, and compare numerical results (symbols) with (3.8) (lines). We compare three types of electrolytic mixture, one containing NaCl (diamonds) as before and two containing so-called antagonistic salts in which one of the ions is hydrophilic and the other hydrophobic. In order to single out the role of $g_{D}$ , we choose $g_{S}=0$ by setting $g^{-}=-g^{+}=2$ (squares) and 4 (triangles). Figure 4(a) shows the pure SOF mobility as a function of the surface field $\unicode[STIX]{x1D6FE}$ , revealing that the linear theory becomes an excellent approximation for small enough $|\unicode[STIX]{x1D6FE}|$ , for which $\unicode[STIX]{x1D711}\ll 1$ holds, and that $\unicode[STIX]{x1D707}_{e}$ increases with $\unicode[STIX]{x1D6FE}$ and $g_{D}$ , as expected from (3.8). It should be noticed that the sign of $\unicode[STIX]{x1D707}_{e}$ depends on the wetting properties of the surface. For the salts having the same solubility contrast $g_{D}$ but different overall solubility $g_{S}$ (squares and diamonds), we find significant differences in the mobility only for large enough $|\unicode[STIX]{x1D6FE}|$ . In this regime, the entropy decrease of the ions near the wall hinders their adsorption, leading to a reduced mobility compared with the linear theory.

In figure 4(b–d), we also contrast the electrolytic mixtures with a neat solvent (circles) having the same physical properties as the mixtures. In figure 4(b), we examine the effect of adding a fixed surface charge $\unicode[STIX]{x1D70E}$ . Clearly, here SOF increases the mobility compared with regular EOF in a neat solvent. The inset of figure 4(b) shows that the shift in mobility is almost constant and increases with $g_{D}$ for the antagonistic salts, as expected from (3.8). The mobility shift is not constant for NaCl, rather it increases with $\unicode[STIX]{x1D70E}$ , and the result is an asymmetry in the mobility when $\unicode[STIX]{x1D70E}$ changes sign, an effect that is absent in neat solvents. This effect originates from the nonlinear coupling between electrostatics and solvation (Samin & Tsori Reference Samin and Tsori2012), and is expressed by the next-order term $\propto g_{S}\unicode[STIX]{x1D711}\unicode[STIX]{x1D6F9}$ in the expansion of $\unicode[STIX]{x1D6FF}n^{\pm }$ . This contribution vanishes for our ideal antagonistic salts, but should be relevant for the majority of realistic salts. The relative shift in mobility due to SOF is significant for the weakly to moderately charged surfaces we considered here, which is a common situation for hydrophobic surfaces (Jing & Bhushan Reference Jing and Bhushan2013), or in hydrophilic surfaces near their iso-electric point. For typically highly charged hydrophilic surfaces, the contribution of SOF may be small, although this may not always be the case since both $g_{D}$ and $\unicode[STIX]{x1D6FE}$ can be much larger than the values used in this work.

In figure 4(c,d), we show the influence of the salt concentration and temperature respectively on mobility. In both panels, we compare electrolytic mixtures in an uncharged channel with a neat solvent in a channel with $\unicode[STIX]{x1D70E}=0.012~\text{nm}^{-2}$ . Strikingly, figure 4(c) shows that the mobility increases with $n_{0}$ for the mixtures, while it decreases for the neat solvent. This behaviour follows from the dependence of $\unicode[STIX]{x1D707}_{e}$ on $\unicode[STIX]{x1D706}_{D}\propto n_{0}^{-1/2}$ in (3.8). For constant $\unicode[STIX]{x1D70E}$ , the contribution to the surface potential from the first term in (3.8) becomes smaller when $\unicode[STIX]{x1D706}_{D}$ decreases, while the second term becomes larger. Figure 4(c) suggests that for $n_{0}\sim 100$  mM, the contribution to the mobility from solvation could be dominant when the surface is weakly charged. The SOF is enhanced by increasing the ionic strength since this leads also to increased ion adsorption at the wall, which results in a larger effective surface charge. The opposite dependence on $n_{0}$ between the two terms in (3.8) can therefore lead to a minimum in the mobility of mixtures at charged surfaces, which is confirmed by numerical calculations (not shown).

Figure 4(d) demonstrates that, in mixtures, temperature can be used to effectively tune the mobility, whereas its effect is much smaller in neat solvents. The reason is that the correlation length $\unicode[STIX]{x1D709}$ diverges as $T_{c}$ is approached from the one-phase region at the critical composition. In (3.8), the mobility increases with $\unicode[STIX]{x1D709}$ , or equivalently when the temperature is lowered towards $T_{c}$ , which is confirmed by the numerical results in figure 4(d). Here, the agreement with the linear theory is not as good since $\unicode[STIX]{x1D711}$ is no longer small when $T_{c}$ is approached. The numerical results show that $\unicode[STIX]{x1D707}_{e}$ changes by as much at 60 % in a temperature window of 20 K, whereas it essentially remains constant for the neat solvent. We did not account, however, for the change of $\unicode[STIX]{x1D700}$ with temperature. This effect will be at play for both mixtures and neat solvents, and should be smaller since, for example, the permittivity of water changes only by approximately 10 % in the same temperature window.