We discuss an electro-osmotic flow near charged porous coatings of a finite hydrodynamic permeability, impregnated with an outer electrolyte solution. It is shown that their electrokinetic (zeta) potential is generally augmented compared to the surface electrostatic potential, thanks to a large liquid slip at their surface emerging due to an electro-osmotic flow in the enriched by counter-ion porous films. The inner flow shows a very rich behavior controlled by the volume charge density of the coating, its Brinkman length, and the concentration of added salt. Interestingly, even for a relatively small Brinkman length, the zeta-potential can, in some cases, become huge, providing a very fast outer flow in the bulk electrolyte. When the Brinkman length is large enough, the zeta-potential could be extremely high, even at practically vanishing surface potential. To describe the slip velocity in a simple manner, we introduce a concept of an electro-osmotic slip length and demonstrate that the latter is always defined by the hydrodynamic permeability of the porous film and also, depending on the regime, either by its volume charge density or by the salt concentration. These results provide a framework for the rational design of porous coatings to enhance electrokinetic phenomena, and for tuning their properties by adjusting bulk electrolyte concentrations, with direct applications in microfluidics.
I. INTRODUCTION
When an electric field E is applied tangent to a charged surface, an electro-osmotic flow of an electrolyte solution is induced.1 The successful understanding of electro-osmosis, due to Smoluchowski,2 was a triumph of 20th century colloid physics. Smoluchowski clarified that the electro-osmotic flow takes its origin in the adjacent diffuse layer of counter-ions and argued that the finite V∞ outside of the diffuse layer is given by2
with permittivity of the solution ε, its dynamic viscosity η, and the so-called (electro-hydrodynamic) zeta-potential Z of the surface, where the no-slip boundary condition is postulated. This postulate implies, via the Stokes equation, that Z must be equal to the surface (electrostatic) potential Ψs.
This classical subject of colloid and interface science is currently widely used in a microfluidics that requires manipulating fluids in thin channels.3 However, the no-slip surfaces and a consequent concept of Z as Ψs that form the basis of the classical theory are of limited applicability. Since typical values of Ψs are of the order of a few tens of mV, to achieve velocities of a few millimeters per second, a high-voltage supply is required, which is an obvious impediment for the use.4 Consequently, a search for mechanisms for generating a large zeta-potential in the low-voltage situation is one of the important challenges in the modern microfluidics. A natural and promising strategy would be a generation of a finite liquid velocity at the surface that may help to augment Z to a very large value. An increase in V∞ can be then quantified by , which we refer below to as an amplification factor.
One avenue to increase Z and is to employ a hydrodynamic (hydrophobic) slippage.5 This is usually quantified by the hydrodynamic slip length b, which can be of the order of tens of nanometers,6–9 but not much more. Simple arguments show that V∞ can be amplified by a factor of10–12
where dimensionless ψs = eΨs/(kBT) and is the inverse Debye screening length. This can provide a discernible flow enhancement in relatively concentrated solutions of small λD,13 but not in dilute solutions where λD is of the order of hundreds of nanometers. Equation (2) implies that adsorbed at the slippery surface, charges are immobile, but if they move in response to the field, its second term could be significantly reduced.10,14 Equation (2) with κb ≫ 1 suggests that a massive amplification of electro-osmotic flow can be achieved over super-hydrophobic (Cassie) surfaces with trapped gas bubbles that significantly enhance the hydrodynamic slip.15–17 During the last decade, several authors concluded that this is, indeed, possible, but only with charged liquid–gas interfaces.18–20
Another avenue could be to employ charged porous coatings that are permeable to water and ions, such as polyelectrolyte networks, ion-exchange resins, silica gels, porous membranes, polyelectrolyte multilayers and brushes, and, in fact, any hydrophilic microtextured surfaces in the Wenzel state. Brinkman21 proposed a modification of the Darcy law to accommodate situations involving shear rates of an outer fluid at the surface of the porous medium. This led to the generalized Stokes equation for a (tangent) pressure-driven flow and to the concept of the Brinkman screening length, Λ, defined as the square root of the hydrodynamic (Darcy) permeability of the medium. Such an approach is widely employed in hydrodynamics of porous media leading, in particular, to the prediction of the exponential decay of a fluid velocity at the permeable surface to the (finite) Darcy velocity inside the porous medium. As proposed by Beavers and Joseph,22 this velocity drop is proportional to the shear rate of an outer fluid and can be very large.23 Despite the obvious importance of an emerging liquid slip at the porous surface for a potential amplification of electro-osmotic flows, fundamental understanding of this did not begin to emerge until quite recently. In general, while considerable progress has been made over the last few decades in understanding the equilibrium properties of porous surfaces in electrolyte solutions, their electro-hydrodynamic properties are relatively less well understood. There is some literature describing attempts to provide a satisfactory theory of electro-osmosis near porous surfaces. We mention below what we believe are the more relevant contributions.
Donath and Pastushenko24 appear to have been the first to study theoretically the electro-osmotic velocity near a porous permeable film. These authors addressed themselves to the case of low potentials and calculated the velocity by including into the Stokes equation inside the film the so-called “friction coefficient,” which is obviously equivalent to Λ2. One of the main results of this pioneering work is that Ψs of porous surfaces does not define unambiguously the flow properties. As a consequence, V∞ does not vanish at high salt concentrations, where Ψs ≃ 0, as it would be for impermeable walls. The authors, however, failed to propose a physical interpretation of these results.
A more systematic treatment of the influence of the “friction coefficient” on electro-osmosis was contained in a paper published by Ohshima and Kondo.25 These authors relaxed the low potential assumption and proposed an expression relating V∞ to the integral of an electro-static potential. However, they concluded that a substitution of the potential profile to this expression gives “results too complex for practical use” so that the case of only Λ/H ≪ 1, where H is the thickness of the porous film, was resolved explicitly. For this situation, Ohshima and Kondo25 predicted that V∞ is controlled, besides Ψs, by the Brinkman and inner Debye screening lengths and also by the Donnan potential of the porous medium, but did not present any calculations illustrating or verifying their theoretical results. A similar remark applies to a paper by Ohshima26 that generalized his prior work25 to several configurations, such as a neutral coating on a charged wall and more. The author generally confirmed an earlier conclusion about finite V∞ at high salt concentrations24 but did not attempt to relate his results to the inner flow and emerging liquid velocity at the interface.
Subsequent attempts at improving the description of electro-osmotic flow near porous surfaces have been concerned mostly the lifting of the standard assumption of the uniform volume charge density and “friction coefficient” for some specific polymer systems. Duval and van Leeuwen27 and Duval28 introduced a concept of the “diffuse soft layer,” where both parameters decrease linearly from constant “bulk” values to zero, and presented a solution for velocities, which is expressed by an infinite series. These authors argued that such a model improves the fit of the low salt data, compared to prior theoretical work,25,26 although they have ignored that experimentalists used a linear version of the theory that is unsuitable for dilute solutions since it overestimates the electrostatic potential. Consequently, the questions whether and how should one take into account the possible non-uniformity of the porous material near an interface remain open, but we consider papers27,28 to be important contributions.
While the existing theories of electro-osmosis near porous systems24–29 are frequently invoked in the interpretation of the electro-osmotic velocity, its relation to the zeta potential has remained somewhat obscure. Following von Smoluchowski,2 some authors25–27 termed the static potential at the location of the no-slip boundary condition “a zeta-potential” and concluded that it “loses its significance”25 or “is undefined and thus nonapplicable.”27 However, such a definition of Z is a consequence of the no-slip postulate that is unsuitable for porous surfaces. Although the authors do not recognize this, their results simply imply that for permeable interfaces, Z ≥ Ψs (which is equivalent to ). More recent calculations also indicate that Z typically exceeds Ψs.30,31 Neither paper addresses itself to the issues of the surface slip. This was taken up only recently in the paper by Silkina, Bag, and Vinogradova32 who carried out calculations in the limit of infinite Brinkman length, Λ → ∞, in an attempt to obtain a proper understanding of an upper bound on achievable zeta-potential. These authors concluded that Z of a porous surface can potentially exhibit an enhancement by an order of magnitude or more due to the emergence of a large surface slip, thanks to the fluid flow in a porous film enriched by counter-ions. Nevertheless, general principles to control Z of porous surfaces of a finite Λ are not established yet, and we also gained the impression that several crucial aspects of the electro-hydrodynamics of the porous interface have been given so far insufficient attention.
In this paper, we give a general theoretical description of electro-osmosis near porous surfaces with the focus on the zeta-potential and flow amplification. We consider planar, uniformly charged porous films placed in contact with a reservoir of electrolyte solution. Our analytic theory provides an explanation of the variation in electro-osmotic velocity with the volume charge density of the coating, ionic concentration, and Brinkman length. It is very well suited to the exact calculations replacing numerical work, as well as has the merit of yielding useful (approximate) analytical results in some limits. These features are especially advantageous when one is attempting to calculate velocities at arbitrary values of parameters. Our results provide new insight into the physics of electro-osmosis, the zeta-potential, and the nature of electro-osmotic slip at the porous interface.
Our paper is arranged as follows: In Sec. II, some general considerations concerning the velocities of a liquid, zeta-potential, and flow amplification are presented. The summary of theoretical relationships for an electrostatic potential is given, and the concept of an electro-osmotic slip length is introduced. Section III describes the theoretical results. In Sec. III A, the case of , where κi is the inner screening length and , is described, and the exact and approximate equations for the slip velocity, zeta-potential, amplification factor, and slip length are derived. The analogous results for the case of are presented in Sec. III B. Section IV contains the results of numerical calculations of the velocity profiles in both large and small Brinkman length situations, validating the theoretical predictions. Numerical and approximate theoretical results for zeta-potentials are presented and compared with surface potentials and velocities at the surface. Finally, the amplification factor and slip length are discussed and contrasted. The issues of implications of these results and the applicability range of our asymptotic approximations are also addressed. In Sec. V, the results for the zeta-potential and slip length, plotted vs salt concentration, are presented for specific porous films. We conclude in Sec. VI with a further discussion of our results and their possible relevance for electro-kinetic experiments.
II. GENERAL CONSIDERATIONS
The system geometry is shown in Fig. 1. Rather than trying to solve the problem at the scale of the individual pores, it is appropriate to consider the “macroscale” situation of the imaginary smooth and homogeneous coating that mimics the actual (porous heterogeneous) one and has the same effective properties. We, thus, consider the homogeneous permeable film of a thickness H (that will be the reference length scale of our problem) and a fixed volume charge density ϱ, taken positive without loss of generality, on a solid support. The film is in contact with a semi-infinite 1:1 electrolyte of concentration c∞, permittivity ε, and dynamic viscosity η. Ions obey Boltzmann distribution, c±(z) = c∞ exp(∓ψ(z)), where ψ(z) = eΨ(z)/(kBT) is the dimensionless electrostatic potential, e is the elementary positive charge, kB is the Boltzmann constant, T is a temperature, and the upper (lower) sign corresponds to the cations (anions). The inverse Debye screening length of an electrolyte solution, , is defined as usually, κ2 = 8πℓBc∞, with the Bjerrum length . The Debye length defines a new (electrostatic) length scale and is the measure of the thickness of the outer diffuse layer.
Charged porous film of thickness H in contact with an electrolyte solution. Anions and cations are denoted with bright and dark circles. The film is permeable for ions and solvents so that the Donnan, ψD, and surface, ψs, static potentials are established self-consistently. An outer electrostatic diffuse layer of a thickness, which is of the order of Debye screening length, λD, is formed in the vicinity of the porous film. The application of a tangential electric field, E, leads to an electro-osmotic flow of a solvent (shown by arrows) that depends on the Brinkman screening length Λ. Due to the emerging velocity at the surface, its dynamic zeta-potential ζ ≥ ψs.
Charged porous film of thickness H in contact with an electrolyte solution. Anions and cations are denoted with bright and dark circles. The film is permeable for ions and solvents so that the Donnan, ψD, and surface, ψs, static potentials are established self-consistently. An outer electrostatic diffuse layer of a thickness, which is of the order of Debye screening length, λD, is formed in the vicinity of the porous film. The application of a tangential electric field, E, leads to an electro-osmotic flow of a solvent (shown by arrows) that depends on the Brinkman screening length Λ. Due to the emerging velocity at the surface, its dynamic zeta-potential ζ ≥ ψs.
The ions of an outer electrolyte can permeate inside the porous film, giving rise to their homogeneous equilibrium distribution in the system, with the enrichment of anions in the film. We consider here only thick, compared to the inner diffuse layer, films with an extended “bulk” electro-neutral region (where the intrinsic coating charge is completely screened by the absorbed electrolyte ions). The potential in this region is referred below to as the Donnan potential, ψD. The surface potential ψs is defined at z = H.
The system subjects to a weak tangential electric field E so that in the steady state, ψ(z) is independent of the fluid flow and satisfies the nonlinear Poisson–Boltzmann equation
where ′ denotes d/dz, with the index {i, o} standing for “in” (z ≤ H) and “out” (z ≥ H), Θ(z) is the Heaviside step function, and . We will term films of ρ ≪ 1 weakly charged and those of ρ ≫ 1 highly charged.
Note that Eq. (3) immediately suggests that
where ψ0 = ψi(0), since in the electro-neutral area vanishes. The potential drops in the inner diffuse layer as
Equation (5) was first derived for the weakly charged films by Ohshima and Ohki33 and recently strictly justified for coatings of any ρ.32 Here, Δψ = ψD − ψs depends on ρ only,34
and the inner screening length
is the function of both κ and ρ. The criterion of a thick film we consider here is then
In the outer diffuse layer, the potential decays from
down to zero in the bulk electrolyte as35
with .
In the limits of small and large ρ, the above equations can be simplified. When ρ ≪ 1, Eqs. (6), (7), and (9) reduce to33
If ρ ≫ 1, they transform to34
For our geometry, the concentration gradients at every location are perpendicular to the direction of the flow, and it is therefore legitimate to neglect advection. Consequently, the dimensionless velocity of an electro-osmotic flow, , satisfies the generalized Stokes equation
where is the inverse Brinkman length. The equation contains multiple length scales due to the different physical effects involved.
At the wall, we apply a classical no-slip condition, 0 = i(0) = 0, and at the surface, the condition of continuity of velocity, i(H) = o(H), and shear rate, (H) = (H), is imposed. Far from the surface, z → ∞, the solution of Eq. (13) should satisfy (∞) = 0 to provide a plug flow. Its constant, i.e., independent of z, velocity is denoted below as ∞.
The outer -profile is given by
where ψo is defined by Eq. (10) and
where s = (H) is the liquid velocity at the surface and ζ = eZ/(kBT) is the dimensionless zeta-potential. Note that it follows from Eq. (15) that the enhanced electro-osmotic mobility is expected due to large ψs, which is an equilibrium property of the system, as well as large s that reflects the hydrodynamic permeability of the porous coating. The amplification factor can be then expressed as
The outer problem, thus, reduces to the calculation of (negative) s.
The velocity jump inside the porous film Δ = 0 − s = −s. In common applications, the film is much thinner than any of the macroscopic dimensions. Therefore, the liquid appears to slip at the velocity − s along the surface of a porous coating. It is then possible to define an effective (positive definite) electro-osmotic slip length using the boundary condition −Δ = o′(H), where the outer liquid velocity is proportional to the shear strain rate via the slip length. In such a definition, b is the distance from the surface at which the outer flow profile extrapolates to zero. From Eq. (14), it follows that (H) = (H). Therefore, the shear rate is
and we obtain
Substituting s from the latter relation into Eq. (16), we recover Eq. (2). This indicates that an electro-osmotic flow near porous films is identical to that near slippery impermeable surfaces with immobile surface charges, but κb and ψs are established self-consistently. Using ψs given by Eqs. (11) and (12), it is possible to obtain sensible approximations in the limits of weakly and highly charged films. When ρ ≪ 1,
and for ρ ≫ 1, standard manipulations yield
where e is the base of the natural logarithm.
III. SURFACE SLIP, ZETA-POTENTIAL, AND FLOW AMPLIFICATION
In order to obtain a detailed information concerning the zeta-potential, outer flow amplification, and slip length, a calculation of s arising due to the inner flow is required. We have obtained the inner velocity profiles by solving Eq. (13) with ψi satisfying Eq. (5) and the prescribed boundary conditions. Below, we consider two distinct cases, of and , that lead to different forms of the solution for i and s.
A. The case of
In these circumstances,
where
Note that Eq. (21) is invalid when since both C2 and C3 diverge. This special case should be treated separately, and we will return to this point in Sec. III B. Equations (21)–(23) allow us to obtain the following expression for an emerging slip velocity:
where the potential drop in the film Δψ is given by Eq. (6) and κi is expressed by Eq. (7). Equation (24) can be used for thick films of any ρ and . As follows from Eq. (15), once s is determined, the zeta potential can be immediately obtained using ζ = ψs − s, with the surface potential expressed by Eq. (9).
Equation (24) can be simplified in the limits of large and small Brinkman lengths. Below, we discuss these two situations.
1. Large Brinkman length ()
We first consider a somewhat idealized case of the very large Brinkman length Λ, where an additional dissipation in the porous film can be neglected. Note that it is by no means obvious that a homogenization, i.e., replacement of a heterogeneous coating by a homogeneous one that “behaves” in the same manner, could indeed lead to a very large Λ. This question remains open and requires further studies. This case is, however, very instructive since it can be seen as an upper bound on the electro-osmotic velocity that constrains its attainable (largest) value.
For small , the inner velocity given by Eq. (21) can be expanded in series for small and and to leading order
The first term is associated with the reduction in potential in the inner diffuse layer and is equal to −(ψD − ψi) [see Eq. (5)]. This contribution to −i decreases exponentially from 0 to Δψ with increasing z/H from 0 to 1. The second term is associated with a body force ρκ2 that drives the inner flow [see Eq. (13)] by acting on the mobile ions accumulated in the Donnan portion of the film. This contribution resembles the usual no-slip parabolic Poiseuille flow. One can define a hydrodynamic permeability of such a film as a ratio of the flow rate (expressed per unit film width) and the driving force. Performing integration from 0 to H of the expression in brackets in Eq. (25) and dividing by H, we find that the hydrodynamic permeability of the film is equal to H2/3, i.e., is varying as the square of its thickness, but does not depend on the Brinkman length.
It follows from Eq. (25) that
Since Δψ ≤ 1, the second term should dominate even at moderate ρ.
For the weakly charged films, ρ ≪ 1, the last equation is reduced to
and
We remark that in this low ρ regime, and κb do not depend on ρ and are finite even if ρ → 0, where ψs ≃ 0. At first sight, this is somewhat surprising, but we recall that our dimensionless charge density is introduced by dividing the real one by the salt concentration so that a nearly vanishing ρ simply implies that the film is enriched by counter-ions that practically fully screen its intrinsic charge. It then becomes almost self-evident that these absorbed mobile ions should induce some inner flow and slip velocity. Indeed, Eqs. (27) and (28) predict that large slip velocity and zeta-potential can be generated even by relatively weakly charged thick films, i.e., when the impact of diffuse layers on the electro-osmosis is marginal. This suggests that an outer flow takes it origin mostly in the “bulk” portion of the coating, where the total electroneutrality condition is satisfied. It is tempting to speculate that one can significantly amplify ∞, making the porous film thicker. However, when the film becomes thick enough, the condition violates and Eq. (25) is no longer valid.
For the highly charged films, ρ ≫ 1, using (12), we find
The corresponding zeta-potential and flow amplification can then be found from Eqs. (12), (15), and (16),
Note that in this limit of high intrinsic volume charge density, the functions ζ, , and κb depend on ρ and κH, and these dependences are nonlinear. Another important remark would be that the contributions of ρ and κH are decoupled.
2. Small Brinkman length ()
Let us now consider the limit of small Brinkman length or . It is generally accepted that, in such a situation, the slip velocity nearly vanishes, and ψs ≃ ζ with . This limit is, therefore, often seen as a lower bound on the electro-osmotic velocity that constrains its attainable minimal value.32 Below we demonstrate that, despite a common belief, in some situations, one can generate a strong flow even with small Λ.
We first note that in this situation, C3 given by Eq. (23) reduces to
The i-profile
can then be easily ascertained to find
Equation (37) indicates that s is a superposition of a flow in the inner diffuse layer that is linear in Δψ, but now also depends on , and of a plug flow emerging in the Donnan region. The later, naturally, satisfies Eq. (13) when and ,
where we have introduced
From Eq. (38), one can easily calculate the hydrodynamic permeability defined in Sec. III A 1 and obtain that it is equal to Λ2, i.e., coincides with the Darcy permeability of the porous medium in a pressure-driven flow.
Two limits can now be distinguished depending on the value of ρ as we did in Sec. III A 1.
The limit of small volume charge density, ρ ≪ 1, yields
and Eq. (19) gives
When ϰ ≪ 1, s ≃ −ρϰ/2, i.e., nearly vanishes, κb ≃ ϰ, , and ψs ≃ ζ ≃ ρ/2 ≪ 1. Thus, the electro-osmotic flow is equivalent to that expected for impermeable surfaces of the same ψs. However, when ϰ ≫ 1, s ≃ −ρϰ2, , and ζ ≃ ρϰ2 ≃ −s. The overall conclusion from these is that ζ can be quite large despite small ρ and reflects mostly s, but not the surface potential. The main contribution to ζ comes from the Donnan region, where the potential and the ionic concentrations are almost constant, but not from the inner diffuse layer.
Repeating the above calculations for the limit of ρ ≫ 1 [using Eqs. (12) and (20) instead of Eqs. (11) and (19)] yields different approximate expressions
We see that in this regime, electro-osmosis is controlled by . For small , we get , ζ ≃ ψs, , and . The sole role of a porous film is thus to set ψs.
When , the large surface slip s ≃ −ρϰ2 and zeta-potential ζ ≃ ρϰ2 + ln(2ρ) are generated. We remark that both s and ψs contribute to the value of ζ. The amplification factor , indicating that the flow is significantly enhanced. Finally, . Overall, we conclude that one can generate a very strong electro-osmotic flow even when the Brinkman length is small, provided both ρ and are large, i.e., when the Brinkman length of the highly charged film is larger than the Debye screening length.
B. The case of
When , the inner velocity is given by
and the slip velocity at a thick film is then equal to
i.e., the slip velocity depends on ρ only and −s ≤ 3/2.
Employing the same arguments as in Sec. III A, we find approximations for weakly and highly charged films. In the limit of ρ ≪ 1, both the slip velocity and zeta-potential are linear in ρ,
but the amplification factor and κb are constant,
If ρ ≫ 1, a maximum possible slip velocity for the case is reached and we derive that ζ, showing a weak logarithmic growth with ρ, reflects mostly ψs,
The amplification factor and κb decrease with ρ,
IV. NUMERICAL RESULTS AND DISCUSSION
It is of considerable interest to compare the exact numerical data with our analytical theory and to determine the regimes of validity of these asymptotic results. Here, we present the results of numerical solutions of the system of Eqs. (3) and (13) with the prescribed boundary conditions, following the approach based on the collocation method,36 together with specific calculations using asymptotic approximations.
A. Velocity profiles
We begin by studying the velocity profiles at fixed ρ = 5, κH = 3, and several in the range of 0.1–10. The numerical results are shown in Fig. 2. Also included are the theoretical curves calculated from Eq. (21) for i and (14) for o [with ∞ defined by Eq. (15)]. In the later case, we used Eq. (24) to calculate s. These exact theoretical results fully coincide with the numerical data. The calculations of s in the limits of small and large are also included in Fig. 2 showing their coincidence with numerical solutions using and 10. It can be seen that on reducing , the value of − increases. All outer velocity profiles are of precisely the same shape, set by ψo [see Eq. (14)], so that the dramatic increase in −∞ upon decreasing is induced by changes in s only. The latter are associated with the inner flows discussed in Sec. III.
The profiles computed using κH = 3 and ρ = 5 with , and 10 (solid curves from top to bottom). Filled circles show the predictions of Eqs. (14) and (21). Open circles correspond to s calculated from Eqs. (26) and (37).
To examine a significance of inner flows more closely, in Fig. 3, we plot the velocity profiles computed using . These results refer to fixed κH, taken the same as in Fig. 2, but we now vary ρ from 0.5 to 8. As z/H is increased, − increases strictly monotonically for all ρ, mostly in the inner (film) region, until it saturates in the bulk electrolyte. The parabolic shape of the inner profile is well seen in numerical examples with ρ ≥ 2. The magnitude of velocity also increases on increasing ρ, confirming the predictions of Sec. III A 1. Finally, we mention that calculations made from Eq. (25) for i and (14) for o fit perfectly the numerical data, and so do s obtained from Eqs. (27) and (31).
The -profiles computed using κH = 3 and ρ = 8, 2, 0.5 (solid curves from top to bottom), and . The filled circles show the predictions of Eqs. (14) and (25). The open circles indicate s calculated from Eqs. (27) and (31).
The results of the identical computations, but made for large , together with theoretical , from Eqs. (14) and (36) are included in Fig. 4. The absolute value of velocity shows a weakly monotonic increase with z/H and saturates at infinity, i.e., outside of the outer diffuse layer. The inner velocity shows a distinct plateau, where the constant i is given by Eq. (38). On increasing z/H further, −i increases up to −s, and the velocity jump in the interface layer is equal to as follows from Eq. (37). Then, in the outer region, −o increases from −s until −∞ in the electroneutral bulk. It can be seen that even at large , the slip velocity makes a discernible contribution to the flow enhancement, provided ρ is large enough. This is exactly what we have proposed in Sec. III A 2. Note, however, that the absolute values of velocities are an order of magnitude smaller than in Fig. 3.
B. Zeta-potential vs slip velocity
We now turn to the zeta-potential of surfaces. The main issue we address is how to enhance ζ by generating a large slip velocity at the surface.
Figure 5, plotted in a log–log scale, is intended to indicate the ranges of −s and ζ = −∞ that are encountered at different . For this numerical example, we use a film of κH = 3, as before, and explore only the case of a large ρ = 10, where the flow enhancement is more pronounced. The surface potential, ψs, of such a film is also shown in Fig. 5, and it is seen that it is quite small. ζ takes its maximal (constant, i.e., independent of the Brinkman length) values at . This part of the curve is well described by Eq. (32), pointing out that this asymptotic approximation has validity well beyond the range of the original assumptions (see Sec. III A). We remark and stress that for the parameters chosen for this specimen example, the upper value of ζ is several tens of times larger than ψs. Zeta-potential then reduces and meets ψs tangentially at very large . When , this decay is well consistent with the predictions of Eq. (45); thus, the latter is also valid well outside the range of its formal applicability. One important conclusion from Fig. 5 is that ζ ≃ 10 at and can be twice larger than ψs even when . Another conclusion we would like to highlight is that when , ζ ≃ −s, but when is in the range of ∼4–20, the zeta-potential is dictated both by ψs and s. Finally, we recall that these asymptotic approximations have been derived by assuming . The (smooth) numerical curves shown in Fig. 5 include ζ and −s defined at so that we have also calculated their values from Eq. (53) and see that they well agree with the numerical data.
Zeta-potential ζ (solid curve) and slip velocity −s (dashed curve) computed for a film of κH = 3 and ρ = 10 as a function of . The surface potential ψs is shown by the dashed-dotted line. The open circles indicate calculations from Eq. (32). The filled circles are obtained using Eqs. (44) and (45). The squares show the predictions of Eq. (53).
Zeta-potential ζ (solid curve) and slip velocity −s (dashed curve) computed for a film of κH = 3 and ρ = 10 as a function of . The surface potential ψs is shown by the dashed-dotted line. The open circles indicate calculations from Eq. (32). The filled circles are obtained using Eqs. (44) and (45). The squares show the predictions of Eq. (53).
In the special case of , the zeta potential and slip velocity depend on ρ only (see Sec. III B). If we keep κH = 3 fixed and increase ρ from 0 to 10 imposing the condition , the value of will increase with ρ from 3 to ∼9.5. We thus obtain the situation illustrated in Fig. 6. As ρ is increased, both ζ and ψs increase, but note that ζ always exceeds ψs. One can see that at very small ρ, the contribution of slip velocity to ζ is significant so that it becomes several times larger than ψs. This part of the curves is well described by Eq. (51). Already when ρ ≃ 3, −s saturates, but ζ increases further, solely since ψs continues to increase with ρ. Asymptotic approximations [Eqs. (53)], although obtained for ρ ≫ 1, fit our numerical results well, when ρ ≥ 3.
Zeta-potential ζ (solid curve) and slip velocity −s (dashed curve) vs ρ, computed for the case of using κH = 3. The surface potential ψs is shown by the dashed-dotted line. The open squares show calculations from Eq. (51), and filled squares are obtained from Eq. (53).
C. Amplification factor
Next, we examine the amplification factor that characterizes an enhancement of an outer plug flow relative to what is generated near the surfaces of ζ ≃ ψs. Since is defined by Eq. (16), one can immediately calculate it using the results of Sec. IV B. Our theory suggests several distinct routes to increase , which implies that there is a choice of appropriate variables for illustrating them.
The above results (see Fig. 5) indicate that our asymptotic equations could be applicable for calculating at certain values of , intermediate between those appropriate to Figs. 3 and 4. We now set and 3 that may be considered as intermediate, and in Fig. 7, we show computed using ρ as a variable. It is interesting to note that the amplification factor is finite at ρ = 0, when ψs vanishes. It increases with ρ for both , and when ρ ≥ 5, the growth appears as linear. The flow is amplified in both cases, but much stronger when leading to . Overall, the agreement with asymptotic expressions obtained for small and large is very good.
The amplification factor computed as a function of ρ for a film of κH = 3 using (solid curve) and (dashed curve). The open and filled squares show the predictions of Eqs. (29) and (33). The open and filled circles show the results of calculations from Eqs. (42) and (46).
Another parameter that controls is ϰ given by Eq. (39), which is the ratio of the Brinkman length to the Debye length. When it is small, only a marginal flow amplification could be expected, and this immediately interprets the lower curve in Fig. 7, where ϰ = 1. However, when this ratio is large, the flow should be significantly amplified. This is exactly what we observe for the upper curve in Fig. 7 corresponding to ϰ = 10 and relatively small . As follows from Eq. (42), in the case of ϰ ≫ 1, a very large can be expected even when the Brinkman screening length and ρ are relatively small. This is illustrated in Fig. 8, where we plot as a function of ϰ for several ρ. In a lin–log scale of Fig. 8, the curves appear (shifted and scaled) sigmoid, i.e., having a characteristic inclined “S”-shape with a bell shaped first derivative (not shown), and are constrained between and its maximal, insensitive to a further increase in ϰ, value. The latter increases with ρ, and we see that the data for ρ = 0.5 and 8 are well fitted by Eqs. (29) and (33), respectively. As predicted in Sec. III A 2, at a relatively small ρ, we see an order of magnitude flow amplification when ϰ ≥ 8. The lowermost branches of these two curves are in agreement with Eqs. (42) and (46), and Eqs. (52) and (54) are equally valid for the case of . Finally, we note that is pretty large already when ϰ = O(1).
The amplification factor computed as a function of for a film of κH = 3 using ρ = 8, 2, and 0.5 (solid curves from top to bottom). The filled circles show calculations from Eqs. (42) and (46), and the open circles show calculations from Eqs. (29) and (33). The squares indicate from Eqs. (52) and (54).
The amplification factor computed as a function of for a film of κH = 3 using ρ = 8, 2, and 0.5 (solid curves from top to bottom). The filled circles show calculations from Eqs. (42) and (46), and the open circles show calculations from Eqs. (29) and (33). The squares indicate from Eqs. (52) and (54).
The volume charge density ρ and ϰ are parameters that control . Equation (46) indicates that their combination, , could be an equally useful variable and provides an additional insight into the problem. In Fig. 9, we plot vs . The numerical calculations are made using quite small and moderate ρ, and we conclude that both curves are again sigmoidal. Note that the inflection point is located at smaller if ρ = 0.5. Another unexpected result is that the amplification factor for a weakly charged film is larger than that for a coating of ρ = 6 until the two curves meet at certain value of (≃6 with these parameters). Only on increasing further, a stronger charged film provides a better amplification of the flow. The agreement with asymptotic approximations is again very good, and we omit a detailed discussion of that.
The amplification factor as a function of computed for κH = 3 and ρ = 0.5 (dashed line) and ρ = 6 (solid line). The open circles show calculations from Eqs. (29) and (33). The filled circles are obtained from Eqs. (42) and (46). The squares indicate calculations from Eqs. (52) and (54).
D. Electro-osmotic slip length
In Sec. II, we suggested an equivalent interpretation of flow enhancements in terms of an electro-osmotic slip length, which is established self-consistently. In contrast to , which characterizes the flow enhancement in the bulk, the slip length is the property of the interface itself and is related to by Eq. (2). Since the main reason for a flow enhancement is the ratio of b to Debye length, below we investigate the variation in κb in response to changes in ρ and .
The numerical results for κb as a function of ρ, obtained with different , are shown in Fig. 10. The asymptotic approximations are very well verified for large and small values of , both for a limit of vanishing ρ [Eqs. (30) and (34)] and for highly charged films [Eqs. (43) and (47)]. It is well seen in Fig. 10 that κb of porous surfaces increases with a volume charge density, and we remark that at low , it increases more rapidly with increasing ρ, especially for the weakly charged films.
Scaled slip length κb computed for a film of κH = 3 as a function of ρ using , 1, and 10 (solid curves from top to bottom). The open and filled squares show calculations from Eqs. (30) and (34). The open and filled circles indicate the results of Eqs. (43) and (47).
By varying at fixed ρ, it is possible to obtain the curves for κb shown in Fig. 11. The curves plotted in a lin–log scale resemble (shifted and scaled) inverse sigmoids smoothly decaying from small down to large regimes on increasing from ∼0.3 to 2. Outside of this transient range of , the asymptotic approximations appear to adequately describe numerical data.
Scaled slip length κb computed for a film of κH = 3 as a function of using ρ = 10, 5, and 0.5 (solid curves from top to bottom). The open and filled squares show calculations from Eqs. (30) and (34). The open and filled circles indicate the results obtained from Eqs. (43) and (47).
V. TOWARD TUNING ZETA-POTENTIAL BY SALT
So far, we have considered the electro-osmotic properties in terms of dimensionless generic parameters, such as ρ, κH, , and their combinations. Additional insight into the problem can be gleaned by calculating ζ as a function of c∞ ∝ κ2 at given H, Λ, and ϱ.
Let us now keep fixed H = 100 nm and set Λ = 200 nm and 20 nm, which correspond to and 5. We also keep fixed ϱ = 150 kC/m3, which is close to that reported in the experiment by Duval et al.,37 and vary c∞ from 10−5 mol/L to 10−2 mol/L. Upon increasing c∞ in this range, ρ is reduced from about 78 down to 0.08 so that the regimes of highly and weakly charged coatings can be tuned simply by adjusting the concentration of salt, and κiH is increased from about 3 to 33. Thus, the criterion of a thick film [Eq. (8)] is fulfilled strictly in our examples, except for concentrations that are very close to 10−5 mol/L.
The computed ζ is shown in Fig. 12 together with ψs. We recall that ψs can be considered as ζ of a reference surface of Λ = 0. The surface potential reduces from 4 (Ψs ≃ 100 mV) practically to zero as c∞ increases, leading to a suppression of a flow. In dilute solutions, where the film is highly charged, the decay is logarithmic, , and appears as linear in this lin–log plot.32 Since the coating becomes weakly charged in more concentrated solutions, , as follows from Eq. (11). When Λ is finite, the zeta-potential becomes larger, and the curves computed using Λ = 20 nm and 200 nm look like the shifted ψs-curve.
Zeta-potential ζ vs c∞, computed using H = 100 nm and ϱ = 150 kC/m3 with Λ = 200 nm (solid curve) and Λ = 20 nm (dashed curve). The surface potential ψs is shown by the dashed-dotted line. The open circles are obtained from Eqs. (55) and (57). The filled circles show calculations from Eqs. (56) and (58). The large squares correspond to calculations using the second term of Eqs. (56) and (58) only. The filled and open triangles indicate experimental data.30,38
Zeta-potential ζ vs c∞, computed using H = 100 nm and ϱ = 150 kC/m3 with Λ = 200 nm (solid curve) and Λ = 20 nm (dashed curve). The surface potential ψs is shown by the dashed-dotted line. The open circles are obtained from Eqs. (55) and (57). The filled circles show calculations from Eqs. (56) and (58). The large squares correspond to calculations using the second term of Eqs. (56) and (58) only. The filled and open triangles indicate experimental data.30,38
In the situation of Λ = 200 nm and low salt concentrations, we can use Eq. (32) to obtain
This equation is equivalent to that derived by Silkina, Bag, and Vinogradova32 for the case of Λ → ∞. From Eq. (28),
which describes a salt dependence of ζ in concentrated solutions and large Brinkman lengths. For sufficiently large c∞, the first term is negligibly small, and ζ becomes constant. The nature of this constant, i.e., of the second term in Eqs. (55) and (56), is apparent. Set Δψ = 0, Eq. (26) then yields a value for − s, and we conclude that the latter depends only on the volume charge density and hydrodynamic permeability. Of course, Eqs. (55) and (56) are very approximate, when Λ is not too large compared to H, but it provides us with some guidance. Indeed, it is seen in Fig. 12 that they overestimate numerical data for Λ = 200 nm. A more precise description requires accounting of a higher-order term into expansion (25), which is beyond the scope of the present work.
When Λ = 20 nm, for dilute solutions, we can use Eq. (45). An order of magnitude estimate shows that with our parameters, so that the contribution of the first term is practically compensated by that of the last one so that
For concentrated solutions, Eq. (41) should be employed, and using that at high salt concentrations, ϰ becomes large, and we find
Figure 12 shows that calculations from Eqs. (57) and (58) are in excellent agreement with the numerical results for Λ = 20 nm. The second term in Eqs. (57) and (58) represents −s given by Eq. (40) if and only if Δψ = 0, i.e., at high salt concentrations. We again conclude that it is defined by the volume charge density and hydrodynamic (now Darcy) permeability, and note that in such a situation, the large zeta-potential reflects solely an electro-osmotic plug flow inside the film (the latter is well shown in Fig. 4).
There have been experimental reports of a phenomenon of a finite electrokinetic mobility in concentrated solutions for various systems, including the cell surface coated with a layer of charged glycoproteins and glycolipids,24 “hairy” polystyrene latexes,39 and more,40,41 but it has never been interpreted in this fashion. Sobolev et al.30 recently reported zeta-potential measurements for a set of polyacrylonitrile hollow fiber membranes by varying c∞ from 10−4 mol/L to 10−3 mol/L and concluded that ζ reduces with salt. Their experimental results, included in Fig. 12, are similar to the portion of our numerical curve corresponding to their concentration range, but the values of ζ are below our calculations due to the smaller experimental ψs. Note that their experimental values of increase from about 1.5 to 2 and well reproduce the trend we predicted, but are of smaller values than those calculated with our parameters theoretically (not shown). Lorenzetti et al.38 obtained ζ vs c∞ curves for nanotubular surfaces, proposed as the coating material for Ti body implants, by varying c∞ from 10−3 mol/L to 10−2 mol/L. As a side note, the authors38 did not attempt to make connection with the volume charge density and Λ, and so the status of their theory remains obscure. Their experimental results are, however, consistent with Eq. (58) and practically coincide with our numerical curve.
Such a behavior can be interpreted in terms of slip length, which is related to by Eq. (2). Figure 13 shows that κb increases with the salt concentration, and there are again two distinct regimes. At low concentrations,
Here, the first equation corresponds to large and the second to small Λ. We see that κb increases linearly with the hydrodynamic permeability of the coating and that (or ). At high concentrations,
i.e., in a high salt regime κb ∝ c∞ (or ) and does not depend on the volume charge density of the film. The linear increase with the hydrodynamic permeability remains. Note that in this situation, ζ ≃ −s ≃ κbϱ/(4ec∞) and . Thus, at high salt concentrations, both κb and are actually very large, up to 102–103 as shown in Fig. 13.
Scaled slip length κb computed as a function of c∞ for a film of H = 100 nm and ϱ = 150 kC/m3 using Λ = 200 nm (solid curve) and 20 nm (dashed curve). The open circles show calculations from Eq. (59). The filled circles are obtained from Eq. (60).
VI. CONCLUDING REMARKS
Our model, which is probably the simplest realistic model for the electro-osmosis near supported porous films that one might contemplate, provides considerable insight into different regimes of the flow and suggests several routes for its enhancement. Many of our results will have validity beyond some specific assumptions underlying the analysis, as demonstrated by numerical calculations.
The main results of our work can be summarized as follows: The Smoluchowski theory of electro-osmosis cannot be employed for the description of the zeta-potential of the porous surface, except for some specific cases in the situation when Λ ≪ H. Specifically, to provide ζ ≃ ψs for the weakly charged films, the Brinkman length Λ should be much smaller than λD, but for the highly charged coatings, the condition should be fulfilled. In all other cases, ζ may be significantly augmented, compared to the surface potential due to the liquid slip at the porous surface. Even when Λ is small, ζ can become very large, provided both and . The slip velocity emerges due to the electro-osmotic flow inside the porous film, and depending on the value of Λ, two different scenarios occur. For large Λ, an inner flow resembles the usual parabolic Poiseuille flow leading to a very large slip velocity, but for small Λ, it is similar to the Darcy plug velocity in a pressure-driven flow. Thus, for porous media, the origin of electro-osmotic flows should be attributed both to diffuse layers and the absorbed ions. The former mechanism is, of course, traditional, while the latter is specific for porous surfaces only. It is responsible for a flow amplification and dominates at high salt concentrations.
We have proposed at the beginning that it may be convenient to quantify the slip velocity at the surface in terms of slip length. Such an approach has been successfully employed in describing flows in various polymer systems,42–44 and those over hydrophobic surfaces, where the slip length b reflects the wettability but does not depend on the salt concentration.45,46 Joly et al.46 also suggested that b of impermeable surfaces could slightly decrease with the surface charge density. By contrast, the electro-osmotic slip length of permeable surfaces is proportional to the square root of ϱ at low salt concentrations and increases with the square root of c∞ in rather concentrated electrolyte solutions.
As mentioned in Sec. I, for microfluidic applications, it would be important to achieve velocities of a few millimeters per second at a low-voltage. Returning to dimensional variables and using the results of Sec. V, we conclude that when the Brinkman length is large, . Then, using the dynamic viscosity of water and typical (low) E = −3 kV/m, we obtain Vs ≃ 2.3 mm/s for a film of Λ = 200 nm described in Sec. V. For the second film analyzed in Sec. V, of Λ = 20 nm, and Vs ≃ 0.2 mm/s. This velocity, of course, could be enhanced for films of larger ϱ, but probably not much. The largest value of ϱ we found in the literature was about 400 kC/m3 (for polyacrylamide gels),47 which has potential to provide only twice faster slip velocity compared to our examples. To increase Vs, a more efficient strategy would be increasing the hydrodynamic permeability, and we will return to this point below.
Several of our theoretical predictions could be tested by experiment. We have already mentioned the work of Donath and Pastushenko24 who found a finite V∞ at high salt concentrations, as well as the papers by Sobolev et al.30 and Lorenzetti et al.38 who reported the same trends as we predict here, but they used a very narrow range of c∞. The concentration must vary in the range described in Sec. V and Fig. 12 to find a transition between highly and weakly charged film regimes. It would be of some interest to find ϱ from the slope of the ζ-curve at low salt concentrations, from Eq. (57). Once it is known, the value of Λ can be obtained using Eq. (57), from the high salt plateau, as well as the salt dependence of ψs can be easily determined using the expressions given in Sec. V [and then may be verified by using Eq. (6)]. It should be possible to obtain ψs from electrostatic force measurements using the atomic force microscope,48 but this would invoke, besides tedious experiments, a very complicated data analysis with computational efforts29 so that electrokinetic measurements of ψs appear easier and more promising.
The hydrodynamic permeability depends on the porous texture (volume fraction of the solid, size of the pores, and their geometry). The experimental examples discussed in the present paper correspond to quite dense textures, where Λ tends to a small value. However, the implications of porosity for the inner electro-osmosis, which is especially important for inducing a very large zeta-potential, so far remain unexplored. Thanks to techniques coming from microelectronics and 3D printing, one can fabricate porous films, structured in a very well controlled way (often at the micro- and nanometer scale). It would be of some interest to measure the zeta-potential of more dilute textures such as arrays of pillars or periodic honeycomb structures, where larger Λ could be expected. To guide the optimal design of porous coatings to increase Λ and for a consequent electro-osmotic flow amplification, it would be timely to develop the theoretical approaches, similar to those known in superhydrophobic microfluidics.49
Our strategy can be extended to describe the amplification of a variety of electrokinetic phenomena,1 including the classical streaming current and less widely known diffusio-osmosis, where the flow is driven by the gradient of salt concentration that produces the electric field. The former has already been studied for some porous systems, such as hydrogel films37 and membranes.30 It would appear that the trends that are observed are consistent with our theoretical description of the zeta-potential. We suggest that further analysis of these measurements, or similar measurements with other porous films in the above-specified concentration range should employ our interpretation of zeta-potential, rather than the Smoluchowski approach since the latter makes a no-slip assumption, which is not generally valid for porous surfaces. The information about diffusio-osmotic flow near porous surfaces is still rather scarce. Although we are not aware of any direct measurements, some indirect experiments have recently revealed extremely strong and long-range diffusio-osmotic flows in the presence of porous surfaces that have consequences for the remote control of particles assemblies50 and even for laundry cleaning.51 Interestingly, the logarithmic decay fits well the diffusio-osmotic velocities obtained for real porous materials in dilute solutions by Feldmann et al.50 The measurements of diffusio-phoresis of latex particles by Prieve et al.52 also lend some support to the picture of the slip velocity that is presented here. A systematic study of the diffusio-osmotic flow amplification by porous surfaces appears to be very timely and would constitute a significant extension of the present work.
DATA AVAILABILITY
The data that support the findings of this study are available within the article.
ACKNOWLEDGMENTS
This work was supported by the Ministry of Science and Higher Education of the Russian Federation and by the German Research Foundation (Grant No. VI 243/4-2) within the Priority Programme “Microswimmers – From Single Particle Motion to Collective Behaviour” (SPP 1726).