Surface and zeta potentials of charged permeable nanocoatings
aa r X i v : . [ phy s i c s . c h e m - ph ] F e b Surface and zeta potentials of charged permeable nanocoatings
Elena F. Silkina, Naren Bag, and Olga I. Vinogradova
1, 2, ∗ Frumkin Institute of Physical Chemistry and Electrochemistry,Russian Academy of Sciences, 31 Leninsky Prospect, 119071 Moscow, Russia DWI - Leibniz Institute for Interactive Materials, Forckenbeckstr. 50, 52056 Aachen, Germany (Dated: February 18, 2021)An electrokinetic (zeta) potential of charged permeable porous films on solid supports generallyexceeds their surface potential, which often builds up to a quite high value itself. Recent workprovided a quantitative understanding of zeta potentials of thick, compared to the extension of an innerelectrostatic di ff use layer, porous films. Here, we consider porous coatings of a thickness comparableor smaller than that of the inner di ff use layer. Our theory, which is valid even when electrostaticpotentials become quite high and accounts for a finite hydrodynamic permeability of the porousmaterials, provides a framework for interpreting the di ff erence between values of surface and zetapotentials in various situations. Analytic approximations for the zeta potential in the experimentallyrelevant limits provide a simple explanation of transitions between di ff erent regimes of electro-osmoticflows, and also suggest strategies for its tuning in microfluidic applications. Keywords: Electroosmosis; porous coatings; zeta-potential; surface potential
I. INTRODUCTION
A century ago von Smoluchowski [1] proposed anequation to describe a plug electro-osmotic flow in abulk electrolyte that emerges when an electric field E isapplied at tangent to a charged solid surface. He relatedthe velocity in the bulk V ∞ to the electrokinetic (zeta)potential of the surface Z . For canonical solid surfaceswith no-slip hydrodynamic boundary condition, simplearguments lead to Z = Ψ s , where Ψ s is the surface (elec-trostatic) potential. However, the problem is not thatsimple and has been revisited in last decades. For exam-ple, even ideal solids, which are smooth, impermeable,and chemically homogeneous, can modify the hydrody-namic boundary conditions when poorly wetted [2], andthe emerging hydrophobic slippage can augment Z com-pared to the surface potential [3–6]. Furthermore, mostsolids are not ideal but rough and heterogeneous. Thiscan further change, and quite dramatically, the bound-ary conditions [7] leading to a very rich electro-osmoticbehavior and, in some situations (e.g. superhydrophobicsurfaces), providing a huge flow enhancement comparedto predicted by the Smoluchowski model [8–10].The defects or pores of the wettable solids also modifythe hydrodynamic boundary condition [11]. Besides, thelocal electro-neutrality is broken not only in the outer dif-fuse layer as it occurs for impenetrable surfaces [12, 13],but also in the inner one [14]. Moreover, even whenthe porous coating is electrostatically thick, i.e. includesa globally electro-neutral region, only mobile absorbedions can react to an applied electric field [15]. Conse-quently, the electric volume force that drives the electro-osmotic flow in the electro-neutral bulk electrolyte isnow generated inside the porous material too. This sug-gests that one can significantly impact the electro-kinetic ∗ Corresponding author: [email protected] response of the whole macroscopic system, i.e. of thebulk electrolyte, just by using various permeable nano-metric coatings at the solid support, such as polyelec-trolyte networks, multilayers, and brushes [16–20], orultrathin porous membrane films [21–23].The emerging flow is strongly coupled to the elec-trostatic potential profile that sets up self-consistently,so the latter becomes a very important consideration inelectroosmosis involving porous surfaces. Electrostaticpotentials, Ψ s and Ψ at the solid support, have beenstudied theoretically over several decades. In most ofthese studies weakly charged surfaces or thick comparedto their inner screening length porous films have beenconsidered [14, 24–26]. Very recently Silkina et al. [27] re-ported a closed-form analytic solution for Ψ , obtainedwithout a small potential assumption, which is valid forporous films of any thickness. These authors also pro-posed a general relationship between Ψ s and Ψ , butmade no attempts to derive simple asymptotic approx-imations for surface potentials that could be handledeasily.The connection between the electro-osmotic velocityand electrostatic potentials have been reported by sev-eral groups [24–26, 28–30], and these models are fre-quently invoked in the interpretation of the electroki-netic data [31]. However, despite its fundamental andpractical significance, the zeta-potential of porous sur-faces has received so far little attention, and its relationto Ψ s has remained obscure until recently. Some au-thors concluded that the zeta-potential ‘loses its signifi-cance’ [25], ‘irrelevant as a concept’ [32] or ‘is undefinedand thus nonapplicable’ [28], while others reported that Z typically exceeds Ψ s [33, 34], but did not attempt torelate their results to the inner flow and emerging liquidvelocity at the porous surface. This was taken up onlyrecently in the paper by Vinogradova et al. [15], whocarried out calculations of the zeta potential for thickcoatings of both an arbitrary volume charge density anda finite hydrodynamic permeability. These authors pre-dicted that Z is generally augmented compared to thesurface electrostatic potential, thanks to a liquid slip attheir surface emerging due to an electro-osmotic flowin the enriched by counter-ions porous films. However,this work cannot be trivially extended to the case of non-thick films, where inner electrostatic potential profilesare always, and often essentially, inhomogeneous. Theseprofiles can be calculated assuming that electrostatic po-tentials are low [14], but such an assumption becomesunrealistic in many situations. Recently, Silkina et al. [27] derived rigorous upper and lower bounds on Z ofnon-thick films, by lifting an assumption of low electro-static potential. However, we are unaware of any priorwork that investigated the connection of the zeta po-tential of non-thick films with their finite hydrodynamicpermeability.In this paper, we provide analytical solutions toelectro-osmotic flows in and outside uniformly chargednon-thick porous coatings, with the focus on their zetapotential and its relation to the surface potential. Ionicsolutions are described using the non-linear mean-fieldPoisson-Boltzmann theory. For simplicity, here we treatonly the symmetric monovalent electrolyte, but it israther straightforward to extend our results to multiva-lent ionic systems. As any approximation, the Poisson-Boltzmann formalism has its limits of validity, but italways describes very accurately the ionic distributionsfor monovalent ions in the typical concentration rangefrom 10 − to 10 − mol / L [35]. Since in this concentrationrange λ D decreases from ca. 300 down to 1 nm [36],the non-thick films we discuss are of nanometric thick-ness. We show that the nanofluidic transport inside suchfilms depends on several nanometric length scales, lead-ing to a rich macroscopic response of the whole system.In particular, we demonstrate that the zeta-potential ofnon-thick coatings becomes a property, defined by therelative values of their thickness, the Brinkman and De-bye screening lengths, and of another electrostatic length ℓ , which depends on the volume charge density, but noton the salt concentration.In Sec. II we give basic principles, brief summary ofknown relationships, and formulate the problem. So-lutions to electro-osmotic velocities and zeta-potentialsare derived in Sec. III. We illustrate the theory and vali-date it numerically in Sec. IV. Implications for the use ofnon-thick porous films to enhance electro-osmotic flowsat di ff erent salt concentration are discussed in Sec. V,followed by concluding remarks in Sec. VI II. MODEL, GOVERNING EQUATIONS, ANDSUMMARY OF KNOWN RELATIONSHIPS
The system geometry is shown in Fig. 1. The proper-ties of the sketched heterogeneous supported film are, ofcourse, related to its internal structure and can be evalu-ated in specific situations, but here we do not try to solve
FIG. 1. Permeable non-thick coating of thickness H and posi-tive volume charge density ̺ in contact with a bulk electrolytesolution of concentration c ∞ . The wall, Ψ , and surface, Ψ s ,electrostatic potentials are build up self-consistently and de-fined, besides H , by two lengths, λ D ∝ c − / ∞ and ℓ ∝ ̺ − / .The application of a tangential electric field, E , leads to anelectro-osmotic flow of solvent that depends on the Brinkmanscreening length Λ . The velocity at the surface of a porous filmis V s , and that in the bulk is V ∞ . the problem at the scale of the individual pores. Instead,motivated by the theory of heterogeneous media [37, 38],we replace such a real coating by an imaginary homoge-neous one, which ‘macroscopically’ behaves in the sameway and possess e ff ective properties, such as a volumecharge density or a hydrodynamic permeability. Thus,we consider a homogeneous permeable film of a thick-ness H , which sets a length scale for our problem, of avolume charge density ̺ , taken positive without loss ofgenerality.The film is in contact with a semi-infinite 1:1 elec-trolyte of bulk ionic concentration c ∞ , permittivity ε ,and dynamic viscosity η . Ions obey Boltzmann distri-bution, c ± ( z ) = c ∞ exp( ∓ ψ ( z )), where ψ ( z ) = e Ψ ( z ) / ( k B T )is the dimensionless electrostatic potential, e is the ele-mentary positive charge, k B is the Boltzmann constant, T is a temperature, and the upper (lower) sign correspondsto the cations (anions). In the bulk, i.e. far away fromthe coating ( z → ∞ ), an electrolyte solution is electro-neutral, c ± ( z ) = c ∞ , and ψ ( ∞ ) →
0. The inverse Debyescreening length of an electrolyte solution, κ ≡ λ − D , isdefined as usually, κ = πℓ B c ∞ , with the Bjerrum length ℓ B = e ε k B T . The Debye length defines a new (electro-static) length scale and is the measure of the thickness ofthe outer di ff use layer, where the local electro-neutralityis broken. We emphasize that it is independent on thefilm charge.The system subjects to a weak tangential electric field E , so that in steady state ψ ( z ) is independent of thefluid flow. For our geometry the concentration gradi-ents at every location are perpendicular to the directionof the flow, it is therefore legitimate to neglect advection.Consequently, the dimensionless velocity of an electro-osmotic flow, v ( z ) = πℓ B η e E V ( z ), satisfies the generalizedStokes equation [39] v ′′ i , o − K v i , o Θ ( H − z ) = ψ ′′ i , o + κ ρ Θ ( H − z ) , (1)where ′ denotes d / dz , with the index { i , o } standing for“in” ( z ≤ H ) and “out” ( z ≥ H ), Θ ( z ) is the Heavisidestep function, K = Λ − is the inverse Brinkman length,and ρ = ̺ c ∞ . For small volume charge and / or highelectrolyte concentration ρ is small and below we refersuch coatings to as weakly charged. For large volumecharge and / or dilute electrolyte solutions ρ is large andwe term these films strongly charged. The Brinkmanlength can theoretically vary from 0 to ∞ . In the latter(idealized) case an additional dissipation in the porousfilm is neglected. If so, the hydrodynamic permeabilityof the porous film reaches its highest possible limit and ∝ H [15]. In the former case of vanishing Λ the additionaldissipation inside the coating is so high that the porousfilm permeability ( ∝ Λ ) tends to zero, i.e. the innerflow is fully suppressed. In reality, however Λ is finiteand defined by the parameters of the porous film, such,for example, as volume fraction of the solid, size of thepores, and their geometry.At the wall we apply a classical no-slip condition, v = v i (0) =
0, and at the surface the condition of continuityof velocity, v i ( H ) = v o ( H ), and shear rate, v ′ i ( H ) = v ′ o ( H ),is imposed. Far from the surface, the solution of Eq.(1)should satisfy v ′ o → z → ∞ to provide a plug flow.The velocity v ∞ at z → ∞ is constant and equal to − ζ .The dimensionless zeta-potential, ζ = e Z / ( k B T ) isgiven by [15, 27] ζ = ψ s − v s , (2)where ψ s = e Ψ s / ( k B T ) and v s = v ( H ). Note that − v s rep-resents the velocity jump inside the porous film. Anysituation where the value of the tangential componentof velocity appears to be di ff erent from that of the solidsurface is normally termed slip [7]. Therefore, − v s repre-sents the (positive definite) slip velocity of liquid at thefilm surface, z = H . As a side note, since the film withan outer di ff use layer is much thinner than any of themacroscopic dimensions, the bulk liquid also appears toslip, but with the velocity − v ∞ . By this reason in colloidscience − v ∞ is often termed an apparent electro-osmoticslip velocity. [40]. To distinguish between real and ap-parent slip, and recalling that − v ∞ = ζ , below we willrefer this (dimensionless) apparent slip to as ζ .Silkina et al. [27] carried out calculations in the limit ofzero and infinite K H and concluded that for films of anarbitrary thickness at K H → − v s ≃ ∆ ψ + ρ ( κ H ) , ζ ≃ ψ + ρ ( κ H ) , (3)and − v s ≃ , ζ ≃ ψ s (4)when K H → ∞ . Here ψ = e Ψ / ( k B T ) = ψ (0) is thewall potential, ψ s = ψ ( H ) is the surface potential, and ∆ ψ = ψ − ψ s is the drop of the electrostatic potential inthe coating. TABLE I. Various limits of electrostatic “thickness” for a porousfilm of a (geometric) thickness H . Electrostatic Weakly charged Highly chargedthickness films ( ρ ≪
1) films ( ρ ≫ κ H ≫ κ H √ ρ ≫ κ H κ H √ ρ κ H ≪ κ H √ ρ ≪ K H .It follows from Eq.(1) that to calculate electro-osmoticvelocity we have to find the distribution of electrostaticpotentials that satisfy the nonlinear Poisson-Boltzmannequation ψ ′′ i , o = κ (cid:0) sinh ψ i , o − ρ Θ ( H − z ) (cid:1) , (5)and to obtain simple expressions for ψ , ψ s , and ∆ ψ . Weassume that the wall is uncharged, ψ ′ i (0) =
0, and set ψ i ( H ) = ψ o ( H ) and ψ ′ i ( H ) = ψ ′ o ( H ) at the surface of thecoating.The solution of Eq.(5) satisfying ψ o → ψ ′ o → z → ∞ is the same as for an impenetrable wall of thesame ψ s [35] ψ o ( z ) = h γ e − κ ( z − H ) i , (6)where γ = tanh ψ s κ H (1 + ρ ) / ≫ ff use layer, withan extended ‘bulk’ electro-neutral region (where intrin-sic coating charge is completely screened by absorbedelectrolyte ions, is formed). The potential in this regionis usually referred to as the Donnan potential, ψ D . Notethat Eq.(5) immediately suggests that ψ D = arsinh( ρ )since in the electro-neutral area ψ ′′ i vanishes. A system-atic treatment of the influence of the Brinkman length onthe zeta-potential of thick films was contained in a paperpublished by Vinogradova et al. [15]. Here we will focuson the case of films of κ H (1 + ρ ) / ρ ≪ κ H = O (1) or smaller. Themore interesting strongly charged coatings of ρ ≫ κ H √ ρ κ H √ ρ ≪
1. For convenience in Table I wegive a summary of criteria defining di ff erent limits foran electrostatic thickness.Non-thick films do not contain an electro-neutral por-tion, where the intrinsic volume charge is fully screenedby absorbed ions. Consequently, their ψ given by [27] ψ ≃ ln + ( ρκ H ) + ρκ H p + ( κ H ) (1 + ρ )2 + ρ ( κ H ) . (8)is smaller than the Donnan potential.The surface potential, ψ s , and the potential drop in thefilm, ∆ ψ = ψ − ψ s , are related to ψ as [6, 27] ψ s ≡ ψ − cosh ψ − ρ , ∆ ψ = cosh ψ − ρ . (9)The inner ψ -profile of a non-thick film is given by [27] ψ i ( z ) ≃ ψ − ρ κ z ) [1 − F ] , (10)where F = sinh ψ ρ (11) represents the fraction of the screened film intrinsiccharge at z = ψ s ≃ ψ − ρ κ H ) [1 − F ] . (12) III. ELECTROSTATIC POTENTIALS VS.ZETA-POTENTIAL
The expression for an outer velocity can be writtenas [15, 27] v o ( z ) = v s + ψ o ( z ) − ψ s , (13)where ψ o is given by Eq.(6) and ψ s obeys Eq.(12). There-fore, in order to obtain a detailed information concerningzeta-potential a calculation of v s arising due to the innerflow is required.We have calculated the inner velocity profile by solv-ing Eq.(1) with ψ i satisfying Eq.(10) and prescribedboundary conditions, and obtained that v i is given by v i = ρ (cid:18) κ K (cid:19) − ∆ ψ ( K H ) ! (cid:16) e −K z − (cid:17) + sinh K z cosh K H " ρ (cid:18) κ K (cid:19) − ∆ ψ ( K H ) ! e −K H − ∆ ψ K H . (14)so that at the surface v s = h ρ ( κ H ) − ∆ ψ i (1 + tanh K H ) e −K H − K H ) − ∆ ψ tanh K H K H , (15)Eq.(15) can be used for any values of ρ and K H , and inthe limits of K H → ∞ reduces to Eqs.(3) and (4).When K H is small, Eq.(15) can be expanded about K H =
0, and to second order we obtain − v s ≃ ∆ ψ − ( K H ) ! + ρ ( κ H ) − K H ) ! (16)The first term in Eq.(16) is associated with the reductionof the potential, ∆ ψ , in the porous film, but also dependson K H . The second term is associated with a body force ρκ that drives the inner flow. Both terms reduce with K H leading to deviations from the upper value of − v s defined by Eq.(3). Using then (2) we conclude that the ζ -potential can be approximated by ζ ≃ ψ − ∆ ψ ( K H ) + ρ ( κ H ) − K H ) ! (17) Expanding v s in Eq.(15) at large K H we find − v s ≃ ∆ ψ K H + ρ (cid:18) κ K (cid:19) (18)Eq.(18) indicates that v s is a superposition of a flow thatis linear in ∆ ψ and of a plug flow, ρ (cid:18) κ K (cid:19) . Then it followsfrom Eq.(2) that ζ ≃ ψ s + ∆ ψ K H + ρ (cid:18) κ K (cid:19) (19)Thus, our treatment clarifies that at a given K H , aslip velocity − v s (and a consequent ζ ) can be enhancedby generating larger ∆ ψ and / or when ρ ( κ H ) is large.When both are small, − v s ≃ ζ ≃ ψ s .The value of ψ can be generally calculated from Eq.(8),which then allows to find ψ s and ∆ ψ from Eq.(9). Usingstandard manipulations we derive ∆ ψ ≃ ρ ( κ H ) + ρ ( κ H ) + ( κ H ) (1 − ρ ) − κ H p + ( κ H ) (1 + ρ )2 + ( ρκ H ) + ρκ H p + ( κ H ) (1 + ρ ) (20)and F ≃ + ( ρκ H ) ρ + ( ρκ H ) − ρ (cid:16) + ( ρκ H ) + ρκ H p + ( κ H ) (1 + ρ ) (cid:17) (21)These two last equations are expected to be very ac-curate, but are quite cumbersome. Fortunately, in somelimits they can be dramatically simplified leading to verysimple analytic solutions for ζ . We discuss now sepa-rately two limits, depending on how strong the dimen-sionless volume charge density is. A. Weakly charged coatings ( ρ ≪ ) At small ρ one can expand ψ given by Eq.(8) intoa series about ρ =
0, and we conclude that a sensibleapproximation for ψ should be ψ ≃ ρκ H p + ( κ H ) − κ H . (22)Note that ψ is linear in ρ , but is a non-linear functionof κ H since to derive Eq.(22) we do not make an addi-tional assumption that κ H ≪
1. Consequently, this andfollowing equations of this subsection should be valideven when κ H = O (1).Expanding Eq.(9) at small ψ and substituting Eq.(22)we obtain ∆ ψ ≃ ψ ρ ≃ ρ ( κ H ) ( p + ( κ H ) − κ H ) , (23)which together with (22) leads to ψ s ≃ ρκ H (cid:16) p + ( κ H ) (2 + ( κ H ) ) − κ H (4 + ( κ H ) ) (cid:17) (24)Note that imposing the condition of small κ H one caneasily recover the known result of the linearized Poisson-Boltzmann theory (see Appendix A) ψ ≃ ρκ H (cid:18) − κ H (cid:19) , ψ s ≃ ρκ H (1 − κ H ) , (25)which suggests that the ψ -profile is almost constantthroughout a weakly charged thin film.Expanding Eq.(11) at small ψ and using Eq.(22) weget F ≃ ψ ρ = κ H ( p + ( κ H ) − κ H )2 + O ( ρ ) (26) We remark that in this low ρ regime to leading order F does not depend on ρ , and is finite even if ρ →
0, where ψ ≃
0. At first sight this is somewhat surprising, butwe recall that our dimensionless charge density is intro-duced by dividing the real one by the salt concentration,so that a nearly vanishing ρ simply implies that the (non-thick) film is enriched by counter-ions that partly screenits intrinsic charge.It is clear that ψ , ψ s , and ∆ ψ are small, so is v s givenby Eq.(16). Consequently, ζ is also generally small andwe do not discuss it here in detail. However, it wouldbe worthwhile to mention that an upper bound on ζ inthis case is ζ ≃ ρκ H p + ( κ H ) , (27)which together with (24) gives ζ/ψ s ≃ + κ H − ( κ H ) ! .Thus, the electro-osmotic flow in the bulk can potentiallybe enhanced in more than two times compared to theSmoluchowski case. B. Strongly charged coatings ( ρ ≫ ) For strongly charged coatings Eq.(8) reduces to [27] ψ ≃ ρκ H ! − ln + ρ ( κ H ) ! . (28)Straightforward calculations show that Eqs.(20) and(21) can be transformed to ∆ ψ ≃ ρ ( κ H ) + ρ ( κ H ) , (29) F ≃ + ( ρκ H ) ρ + ( ρκ H ) ≃ ∆ ψ + ρ (2 + ρ ( κ H ) ) , (30)indicating that ∆ ψ ≃ F when ρ is large.Two limits can now be distinguished depending onthe value of ρ ( κ H ) .
1. The limit of ρ ( κ H ) ≪ We recall that since ρ is large, the film becomes thinwhen κ H √ ρ ≪ quasi-thin ). We fur-ther remark that in this limit the first term in Eq.(28)dominates, so that it can be further simplified to give ψ ≃ ρκ H ! − ρ ( κ H ) ∆ ψ ≃ ρ ( κ H ) , (32)leading to ψ s ≃ ρκ H ! − ρ ( κ H ) (33)When ρκ H is small, Eqs.(31) and (33) reduce toEqs.(25), which implies that they can also be employedwhen ρ is small, provided κ H is not large.From Eq.(17) we then find that for small K H the zeta-potential can be approximated as ζ ≃ ρκ H ! − ρ ( κ H ) ( K H ) , (34)which leads to ζ ≃ ρκ H ! (35)when K H →
0. However, using (33) we obtain ζ − ψ s ≤ ρ ( κ H ) , which is small in this limit. Therefore,our asymptotic arguments suggest that even in the caseof extremely large hydrodynamic permeability of theporous layer, the di ff erence between ζ and ψ s cannot besignificant. Thus a knowledge of ψ s should be su ffi cientto provide a realistic evaluation of ζ (and vice versa).Nevertheless, for completeness we mention that at large K H from (19) one can obtain ζ ≃ ρκ H ! + ρ (cid:18) κ K (cid:19) (1 + K H − ( K H ) ) , (36)which tends to ψ s given by Eq.(33) when K H → ∞ .
2. The limit of ρ ( κ H ) ≫ This limit is close to, but weaker of, the condition for athick film κ H √ ρ ≫ quasi-thick . For large ρ ( κ H ) , Eqs.(28) and (29) can be further sim-plified to ψ ≃ ln(2 ρ ) − ρ ( κ H ) , ∆ ψ ≃ − ρ ( κ H ) (37)which gives the same ψ s as for thick films [15, 41] ψ s ≃ ln(2 ρ ) − K H ≪ ζ ≃ ln(2 ρ ) + ρ ( κ H ) − K H ) ! , (39)which suggests that for quasi-thick films ζ can becomevery large and significantly exceeds ψ s .Substitution of Eqs.(37) and (38) into (19) for K H ≫ ζ ≃ ln (cid:0) ρ (cid:1) − + ρ (cid:18) κ K (cid:19) + K H (40) IV. NUMERICAL RESULTS AND DISCUSSION
It is of considerable interest to compare exact numer-ical data with our analytical theory and to determinethe regimes of validity of asymptotic results. Here wefirst present results of numerical solutions of Eq.(5) withprescribed boundary conditions, using the collocationmethod [42]. We then solve numerically the system ofEqs.(1) and (5). The exact numerical solutions will bepresented together with calculations from the asymp-totic approximations derived in Sec.III. ρ ψ , ψ s FIG. 2. Potentials at wall (solid lines) and surface (dashed)as a function of ρ computed for fixed κ H = . κ H = . ψ from Eq.(A3). .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 . ρκH . . . ψ , ψ s FIG. 3. The data sets for ψ and ψ s obtained at smaller values of ρ reproduced from Fig. 2 and plotted as a function of ρκ H . Theupper set of curves and symbols shows κ H = .
1, the lowerone corresponds to κ H = . In Fig. 2 we plot ψ and ψ s , computed using κ H = . .
1, as a function of ρ . It is well seen that for a thin-ner film ψ ≃ ψ s up to ρκ H ≃
3. On increasing ρ further ∆ ψ increases slowly. For a thicker film of κ H = . ∆ ψ growsmuch faster as ρ is increased. The theoretical curves cal-culated from Eqs.(22) and (24) are also included in Fig. 2.The fits are quite good for ρ ≤
2, but at larger ρ there issome discrepancy, especially for κ H = .
8, and the the-oretical potentials predicted by low ρ approximationsbecome higher than computed. Note, however, that for κ H = . ρ ≤
2. Toexamine its significance more closely, the initial portionsof the ψ -profiles from Fig. 2 are reproduced in Fig. 3, butnow plotted as a function of ρκ H . An overall conclu-sion from this plot is that the approximations derivedin Sec.III A are very accurate when ρκ H ≤
1. We nowreturn to Fig. 2 and focus on the large ρ portions of thecurves. As reported by Silkina et al. [27], Eq.(8) fits veryaccurately the numerical data for ψ at any ρ , so doesmore elegant (28), except for ρ ≤
1, where some verysmall discrepancy is observed. Calculations with ourparameters fully confirm this conclusion, so that we donot show these data. Instead, we include ψ calculatedfrom Eqs.(31) and (37) that correspond to small and large ρ ( κ H ) . It is well seen that for ρ ≥
10 the agreement withnumerical data is excellent in both cases. Also includedis ψ s from (38) and (33), and we see that these asymptoticapproximations coincide with the numerical data.Fig. 4 shows the electrostatic potential drop, ∆ ψ , insidethe film computed for κ H = . ρ . The degree of screened intrinsic charge at the wall, F , calculated numerically for the same values of κ H isalso plotted. It is seen that ∆ ψ first increases linearlywith ρ and, when ρ is getting su ffi ciently large, slowlyapproaches to unity for a film of κ H = .
8. However, ina chosen interval of ρ the potential drop of a thinner filmof κ H = . ρ . . . ∆ ψ , F FIG. 4. ∆ ψ (solid curves) and F (dashed curves) vs ρ com-puted using κ H = . . curves are well fitted by Eq. (23), and the nonlinear onesare reasonably well described by Eq.(29). Also includedin Fig. 4 are the curves for F computed using the samevalues of κ H . For strongly charged coatings F ≃ ∆ ψ ,confirming predictions of Eq.(30). When ρ = F isfinite and its value is given by (26). This equation alsopredicts a parabolic growth of F at small ρ , which is wellseen in Fig. 4. z/H − − − v , − ψ FIG. 5. The profiles v computed using κ H = . ρ =
50 with K H = . , , and 5 (solid curves from top to bottom). Dashedline shows the electrostatic potential profile taken with thenegative sign, − ψ . Filled circles show predictions of Eqs.(13)and (14). Open circles correspond to v s calculated from Eq. (15). We now turn to the electro-osmotic velocity. The veloc-ity profiles computed using three K H in the range from0.1 (small) to 5 (relatively large) are shown in Fig. 5. Theyhave been obtained using κ H = . ρ =
50. Notethat with these parameters κ H √ ρ ≃ . ρκ H =
15, and ρ ( κ H ) = .
5, so in our terms we deal with a non-thickhighly charged film of moderate value of ρ ( κ H ) . Alsoincluded is the computed ψ -profile for this film. As de-scribed in Sec. II, the electrostatic potential of a non-thickfilm is generally nonuniform throughout the system. Itsmaximum value (at the wall) reaches about 4.5, indicat-ing that nonlinear electrostatic e ff ects become significant.The theoretical curves calculated from Eq.(13) for v o us-ing v s defined by Eq.(15) and from Eq.(14) for v i coincidewith the numerical data. It can be seen that on reducing K H the value of − v increases. All outer velocity profilesare of the same shape that is set by ψ o , indicating that thedramatic increase in − v o upon decreasing K H is inducedby changes in v s only. At very large z / H the curves for v o saturate to v ∞ = − ζ (not shown). − K H ζ FIG. 6. Zeta potential as a function of K H computed for κ H = . ρ = ψ s . Open and filled circles show predictionsof Eqs.(35) and (36). Open and filled squares are obtained usingEqs.(39) and (40). Fig. 6 intends to indicate the range of ζ that is encoun-tered at di ff erent K H . For this numerical example weuse films of κ H = . ρ =
20. With these parame-ters ρ ( κ H ) = . ff er signifi-cantly and correspond to di ff erent limits (or quasi-thinand quasi-thick films) described in Sec. III B, but the sur-face potentials, which are also shown in Fig. 6, are quiteclose (and not small). In the chosen range of values of K H , which are neither too small nor quite large, zeta po-tentials of both films reduce strictly monotonically. Wesee that the value of ζ is much larger for the quasi-thickfilm of κ H = .
8, where ζ can exceed ψ s in several times.For a quasi-thin film of κ H = . ψ s , but not much. The parts of the ζ -curvescorresponding to K H ≤ K H ≥
2, the decay of ζ is well consistent with predictions of Eqs.(36) and (40),indicating that the latter are also valid well outside therange of their formal applicability. As is usual, ζ → ψ s as K H → ∞ .We now fix κ H = . ζ as a functionof ρ using K H = . ζ given by Eqs.(3) and (4). The computed at finite K H zeta potentials are naturally confined between these twovalues. For small ρ ( κ H ) we observe a rapid increaseof ζ with ρ that is well described by Eqs. (34) and (36).As ρ is increased, ρ ( κ H ) is shifted to a large value and ρ ζ FIG. 7. Zeta-potential ζ computed for a film of κ H = . ρ for fixed K H = . ζ . formulas (39) and (40) become very accurate. V. TOWARDS SWITCHING SURFACE AND ZETAPOTENTIALS BY SALT
So far we have considered ψ s , ψ , and ζ using di-mensionless variables, such as ρ , κ H , K H , and theircombinations. Additional insight into the problem canbe gleaned by expressing ζ as a function of characteris-tic length scales. These are the geometric length H , thehydrodynamic one Λ , and, of course, the electrostaticlength λ D . We recall that a useful formula for 1:1 elec-trolyte is [36] λ D [nm] = . √ c ∞ [mol / L] , (41)and the dependence of ψ s and ζ on λ D in the equationsbelow reflects their dependence on c ∞ . The later is oftenprobed in electrokinetic experiments, where a decreaseof both potentials with salt is observed [33, 43, 44]. Westress, however, that the measurements have been oftenconducted by using only a very narrow range of rela-tively large c ∞ since existing linear theories could notprovide a reasonable interpretation of data at low con-centrations, where potentials are high.It is also convenient to introduce a new electrostaticlength of the problem ℓ = r e4 πℓ B ̺ ∝ ̺ − / , (42)which is inversely proportional to the square root of thevolume charge density, but does not depend on the bulksalt concentration.The definition of dimensionless ρ can then be refor-mulated as ρ = (cid:18) λ D ℓ (cid:19) (43)This suggests that it is the ratio of two electrostatic lengthscales of the problem that determines whether coatingsare weakly or strongly charged. It is clear that an inter-esting “cross-over” behavior must occur for some inter-mediate values c ∞ that corresponds to λ D ≃ ℓ . We returnto this important point below.Condition (7) of a non-thick film then becomes H λ D + (cid:18) λ D ℓ (cid:19) ! / , (44)i.e. H /λ D H /ℓ H /ℓ and does not depends on salt. Therefore, such filmsare thin when H /ℓ ≪
1, but weakly charged films arethin when H /λ D ≪ ψ s , ψ , and ζ can then be related to ℓ as ρκ H = λ D H ℓ , ρ ( κ H ) = (cid:18) H ℓ (cid:19) , ρ (cid:18) κ K (cid:19) = (cid:18) Λ ℓ (cid:19) (45)Eqs.(45) illustrate that there exist several length scales,lying always in the nanometric range, which determinedi ff erent regimes of the electro-osmotic flow. Anotherimportant conclusion from Eqs.(45) is that ρ ( κ H ) and ρ ( κ/ K ) do not depend on the salt concentration in thebulk. Accordingly, the dependence on salt is hidden onlyin ρκ H , which is the function of λ D .We now present some results illustrating the role oflength scales and showing that for films of a given H /ℓ the electrostatic regimes (of thin and thick films,or highly and weakly charged coatings) can be tuned bythe concentration of salt. Let us now keep fixed H = ℓ = H /ℓ = ℓ correspondsto ̺ =
360 and 10 kC / m , and we note that our largervalue of ̺ is close to the maximal one reported in exper-iments [45, 46]. In our concentration range λ D /ℓ reducesfrom 60 down to 0.6 for the film of ℓ = ℓ =
30 nm. It is easy to checkthat with the chosen parameters both model films fall toa category of non-thick.We begin with the treatment of ψ s obtained from thenumerical solution of Eq.(5). Fig. 8 summarize di ff er-ent regimes in the ( H /ℓ, H /λ D ) plain, where the magni-tude of computed ψ s is reflected by color. The small-est and largest values of H /ℓ in this diagram coincidewith those of the two model films specified above, andthe range of H /λ D corresponds to c ∞ from 10 − to 10 − mol / L. It is now useful to divide the ( H /ℓ, H /λ D ) planeinto two regions, of weakly and strongly charged films,where the above scaling expressions for ψ s approxi-mately hold. We first remark that the conditions ofweakly and highly charged coatings summarized in Ta-ble I coincide when ρ =
1. Consequently, we include inFig. 8 (dotted) straight line that corresponds to λ D /ℓ = ρ =
1) separating weakly and .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . H/λ D H / ℓ quasi-thinquasi-thick strongly chargedweakly charged ψ s FIG. 8. Schematic representation of various electrostaticregimes for a non-thick porous film. The colorbar values of ψ s ascend from top to bottom. The diagram is plotted in the ( H /ℓ , H /λ D ) plane. Dotted line separates the regions, where coatingsobey a linear theory, and where they can only be describedusing a non-linear theory. Dashed curve separates regions ofquasi-thick and quasi-thin films as discussed in the text. Opentriangle marks the point of c ∞ = c △ ∞ . highly charged surfaces. When H /ℓ is below this linea simple linear theory can be employed. However, forlarger H /ℓ the Poisson-Boltzmann equation (5) cannotbe linearized. Apart from this line, another crossoverlocus, H λ D + (cid:18) λ D ℓ (cid:19) ! / = ψ s should crossover smoothly fromone electrostatic regime to another. We can now con-clude that in very dilute solutions both films are highlycharged. However, at low salt the film of ℓ = ℓ =
30 nm is quasi-thin. Ifwe increase H /λ D (increase c ∞ ) for a film of ℓ =
30, wemove to a situation of weakly charged quasi-thin films.The intersection of the horizontal line H /ℓ = . λ D /ℓ = c △ ∞ . On increasing H /λ D further this film be-comes weakly charged quasi-thick. The film of ℓ = H /λ D and becomes weakly chargedat c N ∞ that is defined by the the intersection of the line H /ℓ = λ D /ℓ = ψ s ≃ (cid:18) λ D ℓ (cid:19) + ln 2 − λ D /ℓ . However, when the highly-charged film is quasi-thin, ψ s obeys Eq.(33) that can berewritten as ψ s ≃ (cid:18) λ D H ℓ (cid:19) − (cid:18) H ℓ (cid:19) (47)Thus, in this situation ψ s is defined by both λ D /ℓ and H /ℓ .0At high salt, where both films become weakly chargedand quasi-thick, to calculate ψ s one can use (A7), whichgives ψ s ≃ (cid:18) λ D ℓ (cid:19) ∝ ̺ c − ∞ , (48)i.e. the surface potential is again controlled solely by λ D /ℓ . − − − − − c ∞ , mol/L0 . . . . ψ s FIG. 9. ψ s vs c ∞ computed using H =
15 nm and ℓ = λ D = ℓ . In Fig. 9 we plot ψ s vs. c ∞ for these two specimen ex-amples of the films. The surface potential is quite highat c ∞ ≃ − mol / L (ca. 198 and 78 mV) and reduces withsalt. At larger concentrations ψ s becomes smaller thanunity and practically vanishes when c ∞ ≥ − mol / L.To specify better the branches of low and high concen-trations, in Fig. 9 we have marked c N ∞ and c △ ∞ by blackand open triangles. For an upper curve computed us-ing ℓ = c N ∞ ≃ . × − mol / L,and for a lower, of ℓ =
30 nm, at c △ ∞ ≃ − mol / L. Thecorresponding surface potentials are ψ s ≃ . . λ D /ℓ =
1, both films are of low surface po-tentials. The first film is quasi-thick as discussed above,and the branch of the curve with c ∞ ≥ c N ∞ is well fittedby Eq.(48). We recall that at c △ ∞ the second film still re-mains quasi-thin and becomes quasi-thick, where (48)should be strictly valid, only when c ∞ ≃ . × − mol / L(see Fig. 8). We see, however, that the fit is quite goodfor c ∞ ≥ c △ ∞ , although at concentrations smaller than4 . × − mol / L there is some discrepancy, and Eq.(48)slightly overestimates ψ s . Also included in Fig. 9 aretheoretical calculations for low salt concentrations. Wesee that at c ∞ smaller than c N ∞ Eq.(46) is very accurate fora curve of ℓ = ℓ =
30 nm, Eq.(47) providesan excellent fit to numerical data. Finally, we would liketo stress that it is impossible to generate a very high ψ s just by increasing ̺ . This is well seen in Fig. 9, where theupper curve corresponds to the film with 36 times larger ̺ than that for a film corresponding to a lower curve.The ratio of the values surface potentials for these twocoatings is always smaller. Its largest value is equal to 18, as follows from Eq.(48) for the high salt regime, where ψ s is small. However, when ψ s is large, its amplificationwith ̺ is very weak (only about 2 when c ∞ ≃ − mol / L). . . H / Λ H / λ D ζ FIG. 10. ζ plotted as a function of two variables, H /λ D and H / Λ , for a coating of thickness H =
15 nm using ℓ = ℓ =
30 nm (bottom surface). In the latter case ζ ≃ ψ s as discussed in the text. We are now on a position to calculate ζ , which gener-ally depends on the Brinkman length Λ , and to contrast ζ to ψ s . In Fig. 10 we plot ζ as a function of two vari-ables, H /λ D and H / Λ , for two porous coatings discussedabove. We recall that they are of the same thickness,but their values of ℓ are di ff erent. An overall conclusionfrom this three dimensional plot is that for a film of ℓ = H / Λ .However, for a coating of ℓ =
30 nm the e ff ect of H / Λ on ζ , if any, is not discernible at the scale of Fig. 10. Indeed,as discussed in Sec. III B 1, in this case even at the infi-nite Brinkman length the zeta potential exceeds ψ s , butvery slightly (see also the lower curve in Fig. 6). Simplecalculations show that ζ − ψ s ≤ ( H /ℓ ) , which is equal to0.25, i.e. very small, when ℓ = ζ ≃ ψ s and can be evaluated, depending on c ∞ that tunes an electrostatic regime, either using Eq.(47)or Eq.(48). Consequently, below we focus only on thequasi-thick film of ℓ = ζ for a filmof ℓ = Λ =
30 and 3.75 nm. Thesegive H / Λ = . ζ . We see that thedi ff erence between the upper and lower bounds is quitelarge. The numerical ζ -curves at finite Λ are confinedbetween these bounds, and are of the same shape as ψ s ,but shifted towards higher values that grow with Λ until ζ reaches its upper attainable limit. When c ∞ ≤ c N ∞ thesurface potential ψ s is given by (46). At small H / Λ theexpression for the zeta potential can be obtained from1 − − − − − c ∞ , mol/L051015 ζ FIG. 11. ζ vs c ∞ computed using Λ =
30 nm (upper solidcurve) and 3.75 nm (lower solid curve) for a film of H = ℓ = ζ . Eq. (39) ζ ≃ ψ s + (cid:18) H ℓ (cid:19) " − (cid:18) H Λ (cid:19) + , (49)and when H / Λ is large, it follows from (40) that ζ ≃ ψ s + (cid:18) Λ ℓ (cid:19) + Λ H (50)In Sec. II we have clarified that the hydrodynamic per-meability of the porous films ∝ H at low H / Λ and ∝ Λ when H / Λ . Thus, Eqs.(49) and (50) point strongly thatthe ratio of the hydrodynamic permeability to ℓ is animportant parameter controlling ζ . At larger concentra-tions, c ∞ ≥ c N ∞ , small ψ s is given by Eq.(48). Using thenEqs.(17) and (19) for small and large H / Λ we derive ζ ≃ ψ s " − (cid:18) H Λ (cid:19) + (cid:18) H ℓ (cid:19) " − (cid:18) H Λ (cid:19) , (51) ζ ≃ ψ s (cid:18) + Λ H (cid:19) + (cid:18) Λ ℓ (cid:19) (52)We remark that again the ratio of the hydrodynamic per-meability to ℓ becomes an important factor that deter-mines the amplification of ζ compared to ψ s . The calcu-lations from Eqs.(49)-(52) are also included in Fig. 9 andwe see that provide an excellent fit to numerical data.Thus, for quasi-thick films of a finite hydrodynamicpermeability ζ , ψ s . As follows from Eqs.(49) - (52), be-sides ψ s (that can be tuned by varying the concentrationof salt) the value of ζ also reflects H / Λ and depends onthe ratio of the hydrodynamic permeability to ℓ . VI. CONCLUDING REMARKS
We have presented a theory of surface and zeta poten-tials of non-thick porous coatings, i.e. those of a thick- ness H comparable or smaller than that of the inner dif-fuse layer, of a finite hydrodynamic permeability. Ourmean-field theory led to a number of asymptotic approx-imations, which are both simple and very accurate, andcan easily be used to predict or to interpret ψ s and ζ indi ff erent regimes, including situations when non-linearelectrostatic e ff ects become significant.The main results of our work can be summarized asfollows. We have introduced an electrostatic length scale ℓ ∝ ̺ − / and demonstrated that depending on its valuetwo di ff erent scenarios occur. In the high salt concentra-tion regimes, ℓ > λ D ∝ c − / ∞ , the non-thick porous filmsare weakly charged and their electrostatic properties canbe described by linearized equations. These films e ff ec-tively behave either as thin or thick depending on the val-ues of H /λ D and H /ℓ . We have also stressed the connec-tion between the zeta potential and the Brinkman length,which is a characteristics of the hydrodynamic perme-ability of the porous film. Interestingly, the Brinkmanlength contribution to ζ permits to augment it comparedto ψ s only if ( H /ℓ ) is large, i.e. when films are quasi-thick.Overall we conclude that tuning fluid transport insidea nanometric non-thick coating can dramatically a ff ectthe whole response of the large system to an appliedelectric field. Such a tuning can be achieved modifyingits internal structure and charge density, or by varyingfilm thickness, or concentration of an external salt solu-tion. ACKNOWLEDGMENTS
This work was supported by the Ministry of Scienceand Higher Education of the Russian Federation and bythe German Research Foundation (grant 243 / AUTHOR’S CONTRIBUTION
E.F.S. developed numerical codes, performed com-putations, and prepared the figures. N.B. participatedin theoretical calculations. O.I.V. designed and super-vised the project, developed the theory, and wrote themanuscript.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study areavailable within the article.2
Appendix A: The limit of low potentials
For completeness, in this Appendix we briefly dis-cuss the case of low electrostatic potentials( ψ ≤ ψ ′′ i , o = κ (cid:2) ψ i , o − ρ Θ ( H − z ) (cid:3) (A1)Note that this case has been considered before byOhshima and Ohki [14]. Here we present a compactderivation of expressions for ψ and ψ s in our (di ff er-ent) variables and complete the consideration by givingapproximate expressions for ∆ ψ , F . We also obtain anupper limit of ζ .Integrating Eq. (A1) with prescribed boundary condi-tions (see Sec. II) one can easily obtain ψ i = cosh( κ z ) (cid:0) ψ − ρ (cid:1) + ρ, ψ o = ψ s e − κ ( z − H ) , (A2)which leads to ψ = ρ (cid:16) − e − κ H (cid:17) (A3) ψ s = ρ sinh( κ H ) e − κ H (A4) It is then straightforward to obtain ∆ ψ = ρ (1 − e − κ H ) , (A5)and F = − e − κ H (A6)We recall that these equations are valid at any κ H . Atsmall κ H they transform to Eq.(25), but at large κ H theyreduce to ψ ≃ ρ, ψ s ≃ ρ/ κ H ≥
5, and the second when κ H ≥
2. In other words, they describes not only thethick films, but also valid for some non-thick ones thatcan be termed quasi-thick.Interestingly, low potential films satisfying (A7) canpotentially generate a high zeta-potential. Its upperachievable limit can be obtained using Eq.(3) and is givenby ζ ≃ ρ + ρ ( κ H ) , (A8)The last equation coincides with that for weakly chargedthick films [15]. Dividing (A8) by (A4) we concludethat for low potential thick and quasi-thick films ζ/ψ s ≃ + ( κ H ) . [1] M. von Smoluchowski, Handbuch der Electrizit¨at und desMagnetism. Vol. 2 , edited by L. Graetz (Barth, J. A., Leipzig,1921) pp. 366–428.[2] O. I. Vinogradova, Int. J. Miner. Proc. , 31 (1999).[3] V. M. Muller, I. P. Sergeeva, V. D. Sobolev, and N. V.Churaev, Colloid J. USSR , 606 (1986).[4] L. Joly, C. Ybert, E. Trizac, and L. Bocquet, Phys. Rev. Lett. , 257805 (2004).[5] S. R. Maduar, A. V. Belyaev, V. Lobaskin, and O. I. Vino-gradova, Phys. Rev. Lett. , 118301 (2015).[6] E. F. Silkina, E. S. Asmolov, and O. I. Vinogradova, Phys.Chem. Chem. Phys. , 23036 (2019).[7] O. I. Vinogradova and A. V. Belyaev, J. Phys.: Condens.Matter , 184104 (2011).[8] S. S. Bahga, O. I. Vinogradova, and M. Z. Bazant, J. FluidMech. , 245 (2010).[9] T. M. Squires, Phys. Fluids , 092105 (2008).[10] A. V. Belyaev and O. I. Vinogradova, Phys. Rev. Lett. ,098301 (2011).[11] G. S. Beavers and D. D. Joseph, J. Fluid Mech. , 197(1967).[12] J. L. Anderson, Annu. Rev. Fluid Mech. , 61 (1989).[13] L. Bocquet and E. Charlaix, Chem. Soc. Rev. , 1073 (2010).[14] H. Ohshima and S. Ohki, Biophys. J. , 673 (1985).[15] O. I. Vinogradova, E. F. Silkina, N. Bag, and E. S. Asmolov,Phys. Fluids , 102105 (2020). [16] M. Cohen Stuart, W. T. S. Huck, J. Genzer, M. M ¨uller,C. Ober, M. Stamm, G. B. Sukhorukov, I. Szleifer, V. V.Tsukruk, M. Urban, F. Winnik, S. Zauscher, I. Lizunov,and S. Minko, Nature Mater , 101 (2010).[17] B. Chollet, M. Li, E. Martwong, B. Bresson, C. Fretigny,P. Tabeling, and Y. Tran, ACS Appl. Mater. Interfaces ,11729 (2016).[18] M. Ballau ff and O. Borisov, Current Opinion Colloid In-terface Science , 316 (2006).[19] O. I. Vinogradova, O. V. Lebedeva, K. Vasilev, H. Gong,J. Garcia-Turiel, and B. S. Kim, Biomacromolecules , 1495(2005).[20] S. Das, M. Banik, G. Chen, S. Sinha, and R. Mukherjee,Soft Matter , 8550 (2015).[21] A. van den Berg and M. Wessling, Nature , 726 (2007).[22] M. P. Pina, R. Mallada, M. Arruebo, M. Urbiztondo,N. Navascues, O. de la Iglesia, and J. Santamaria, Mi-croporous and Mesoporous Materials , 19 (2011).[23] M. Lukatskaya, B. Dunn, and Y. Gogotsi, Nat Commun ,12647 (2016).[24] E. Donath and V. Pastushenko, Bioelectrochem. Bioener-getics , 543 (1979).[25] H. Ohshima and T. Kondo, J. Colloid Interface Sci. ,443 (1990).[26] H. Ohshima, Adv. Colloid Interface Sci. , 189 (1995).[27] E. F. Silkina, N. Bag, and O. I. Vinogradova, Phys. Rev. Fluids , 123701 (2020).[28] J. F. L. Duval and H. P. van Leeuwen, Langmuir , 10324(2004).[29] J. F. L. Duval, Langmuir , 3247 (2005).[30] H. Ohshima, Theory of colloid and interfacial electric phenom-ena (Elsevier, 2006).[31] A. C. Barbati and B. J. Kirby, Soft Matter , 10598 (2012).[32] L. P. Yezek and H. P. van Leeuwen, J. Colloid Interface Sci. , 243 (2004).[33] V. D. Sobolev, A. N. Filippov, T. A. Vorob’eva, and I. P.Sergeeva, Colloid J. , 677 (2017).[34] G. Chen and S. Das, J. Colloid Interface Sci , 357 (2015).[35] D. Andelman, “Soft Condensed Matter Physics in Molec-ular and Cell Biology,” (Taylor & Francis, New York, 2006)Chap. 6.[36] J. N. Israelachvili, Intermolecular and Surface Forces , 3rd ed.(Academic Press, 2011).[37] K. Z. Markov, in
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