Proceedings of the 20th National and 9th International ISHMT-ASME Heat and Mass Transfer Conference January 4-6, 2010, Mumbai, India

10HMTC89

Effect of Finite Electric Double Layer Conductivity on Streaming Potential and Electroviscous Effect in Nanofluidic Transport

Siddhartha Das Suman Chakraborty Department of Mechanical Engineering Department of Mechanical Engineering Indian Institute of Technology, Kharagpur-PIN- Indian Institute of Technology, Kharagpur-PIN- 721302 721302 Email:[email protected] Email:[email protected]

applications in diverse areas such as biomedical, ABSTRACT biotechnological, chemical, forensic etc [1-8]. In this paper, the role of finite Electric Because of strong dominance of surface effects Double Layer (EDL) conductivity on the and interfacial phenomena over reduced length Streaming Potential and the resulting scales, flow manipulation through exploitation of Electroviscous effect in pressure-driven electrical fields turns out to be convenient in nanofluidic ionic transport is investigated. nano-scale fluidic devices of practical relevance. Important dimensionless parameters like the Many such devices essentially function by (based on the EDL surface exploiting the formation of electrical double electrical conductivity, the bulk electrical layer (EDL) [9]), which is essentially a charged conductivity and the EDL thickness), ratio of the layer formed in vicinity of the fluid-substrate Stern Layer and the Diffuse Layer conductivities interface because of electro-chemical and the ratio of the channel height to the EDL phenomena. Under the action of driving forces, thickness are identified as important ionic charges in the mobile part of the EDL can dimensionless parameters that dictate the migrate along a preferential direction, thereby contribution of the finite EDL conductivity. For giving rise to local ionic currents. The ionic most of the combinations of these dimensionless transport in the EDL may also drag fluid parameters, the finite EDL effect lowers the elements along with the same as a consequence strength of the streaming potential and hence the of viscous shear, which may bear significant electroviscous action. However, for certain consequences with regard to the hydrodynamic choices of these parameters, particularly for transport in the narrow confinement. Critical to moderate Dukhin number and substantially large the investigation of electrokinetic transport, thus, , finite EDL conductivity effect can lies the analysis of hydrodynamics within the actually enhance these effects. EDL. Advection of mobile ions present within the EDL, in presence of a pressure-driven 1. INTRODUCTION background flow, develops a so called streaming In the present decade, investigations on potential [9] in micro and nanochannels. transport in nanofluidic confinements have Streaming potential is effectively the induced received a tremendous boost, paving the way for electric field created due to a preferential a large number of novel chemical and biological downstream accumulation of the EDL

1 counterions and is directed opposite to the differences in EDL and bulk conductivities on pressure-driven transport. Consequently, it sets nanochannel streaming field and electroviscous up a back electrokinetic transport which opposes effect calculations. Only very recently, Davidson the pressure-driven flow. The resulting reduction and Xuan [17] elaborated the effects of Stern in the volume flow rate makes the flow appear to layer conductance on electrokinetic energy have an enhanced viscosity, an artifact known as conversion in nanochannels. However, in their Electroviscous Effect. Researchers have utilized work they did not consider the effect of the the generation of streaming potential for a diffuse layer, neither they pinpointed the number of applications which include the possible variations of the streaming potential and conversion of hydrostatic pressure differences the effective viscosity (characterizing the into useful electrical energy, characterization of electroviscous effect) as a function of the interfacial charge of organic thin films, different parameters governing the finite measurement of wall charge inversion in the conductivity within the EDL. presence of multivalent ions in a nanochannel, In the present paper we pinpoint the analysis of ion transport through nanoporous possible effects of considering finite EDL membranes, designing efficient nano-fluidic electrical conductivities (different from the bulk batteries, quantifying hydrodynamic dispersion value) in altering the streaming potentials and the in nanochannels, etc. resulting electroviscous action. The EDL itself is Streaming potential is estimated by assumed to have two different conductivities; balancing the streaming current (current resulting one for the Stern Layer and another for the from flow induced downstream migration of Diffuse Layer. Based on the combination of mobile EDL ions) with the conduction current these conductivities with the bulk conductivity, (current resulting from the streaming potential along with the value of the EDL thickness, we induced reverse electro-migration of ions), under identify the relevant dimensionless parameters steady state condition. A precise determination that govern the extent of deviation of the of the conduction current, in turn, depends on an streaming potential for the cases with and accurate quantification of the conductivity of the without the considerations of varying electrical ionic solution. The standard practice is to conductivities over the channel section. characterize the conductivity by a unique value dictated by the bulk ionic concentration. This approach works fine for microchannels where 2. THEORY the EDL occupies only a negligible extent of the channel section. However, for nanochannels in which EDLs may protrude significantly into the 2.1 EDL potential and Streaming bulk, this approach may not turn out to be quite Field without Considering Finite EDL adequate. This inadequacy stems from the fact Conductivity that within the EDL the concentration of ions can We consider transport of an ionic liquid be much higher so that the corresponding value in a slit-like charged nanochannel of height 2H, of the conductivity is much larger than the bulk length L and width w. The EDL potential field conductivity. In fact, within the EDL itself, large (ψ) in the nanochannel is described in terms of differences in the ionic concentration between the net charge density ( ρe ) within the EDL as: the immobile Stern layer and the mobile diffuse layer may result in different conductivities at ∂∂22ψψ ρ different sections of the EDL. A correct e 22+=− (1) estimation of conduction current (and hence ∂∂xy ε 0ε r streaming potential and electroviscous effect estimation) in nanofluidic transport must take where ε 0 is the permittivity of free space and ε r into account these variable conductivities across is the relative permittivity of the solution. The different portions of the channel section. There charge density ρ is expressed in terms of the has been a large number of investigations e delineating the effects of finite EDL conductivity ionic number densities for symmetric electrolyte (or to be more specific, Stern Layer ( zz+−= −=z) as: conductivity) on different aspects of electrokinetic transport [10-16]. However, to the ρe =−ez( n+− n ) (2) best of our knowledge, there has been no investigation pinpointing the influences of the

2 Assuming that the ionic concentrations obey dp where is the pressure gradient, μ is the ⎛⎞ezψ dx Boltzmann distributions n= n exp ∓ ± 0 ⎜⎟ dynamic viscosity of the fluid and E is the ⎝⎠kTB S induced streaming potential field. where n0 is the bulk ionic concentration), eqs. The conduction current (Icond) results from the (1) and (2) are solved to obtain the solution of transport of the ions in response to the the potential field (under the condition that at established streaming potential, so that one may dψ write (assuming that the ions have identical ψζ= ()zeta potential and = 0 y,H=02 dy conduction velocities across the entire transverse yH= domain, i.e., the variation of the mobilities of the and the EDL thickness λ is comparable but ions depending on whether they are in the bulk always less than the channel half height) as [9]: or within the EDL is not accounted for)

⎧⎫⎡⎤⎛⎞ezζ ⎛⎞ y 2H ⎪⎪tanh-1 ⎢⎥ tanh exp−+ ⎜⎟⎜⎟ Icond =+enzu()++cond ,, + nzudy −−cond − (7) 4k T ⎪⎪⎣⎦⎢⎥⎝⎠4kB T ⎝⎠λ ∫ ψ = B ⎨⎬0 ze ⎪⎪-1 ⎡⎤⎛⎞ezζ ⎛⎞ 2H − y tanh⎢ tanh⎜⎟ exp − ⎥ ⎪⎜⎟⎪where u are the electromigration velocities ⎩⎣⎦⎢⎥⎝⎠4kB T ⎝⎠λ ⎭cond ,± (3) of the cations/anions in response to the streaming potential, and may be expressed as In presence of a pure pressure-driven transport downstream migration of the mobile EDL ions zeE u = ± S (8) give rise to what is known as the streaming cond ,± f± current. The resulting accumulation of the ions in the downstream establishes a field (called the Here f± are the friction coefficients of the ions, streaming field or streaming potential) in the which is assumed to have identical values for the upstream direction which drives a current called anions and the cations (i.e., the conduction current. The net ionic current is a f = ff==6πμ r, where r is the ionic resultant of these two effects and at steady state +− ion ion radius; here we assume, with some degree of is equal to zero. We thus have: approximation, that the cationic and anionic radii

are approximately equal to around 1.5 A0), IIIionic=+ str cond =0 (4) leading to (using eqs. 7 and 8)

The streaming current (Istr) is expressed as: 22 2H zeES Icond =+()nndy+− (9) 22HH f ∫ 0 (5) Istr==−∫∫ρ eudy ez() n+− n udy 00 Using Boltzmann distribution to express the ⎛⎞ezψ ionic concentrations (i.e., nnexp= ⎜⎟∓ ), Following the argument proposed in our ± 0 kT previous work [18], the velocity field, u, as ⎝⎠B appearing in Eq. (5), needs to be considered as a we rewrite eq. (9) as: resultant of the imposed pressure-driven transport and the opposing streaming field 22 2H 2nze0 ⎛⎞ezψ induced electroosmotic transport, such that: Icond = EdS ∫ cosh ⎜⎟y fk⎝⎠BT 0 (10) 1 dp εεζE ⎛⎞ψ 2H uu=+ u =−21 Hyy −2 −0 rS − ⎛⎞ezψ pES () ⎜⎟ = σ Ed cosh y 2μ dx μ ⎝⎠ζ bS∫ ⎜⎟ ⎝⎠kTB (6) 0 2nze22 where σ = 0 is the bulk conductivity. b f

Using eqs. (4), (5), (6) and (9) we finally obtain:

3 Using this expression, we obtain the ratio of the streaming field for the two cases (in terms of 2H ⎡⎤1 dp 2 ⎛⎞ezψ different dimensionless parameters and variables nez0 ⎢⎥−−()2sinh Hy y⎜⎟ dy ∫ ⎣⎦2μ dx ⎝⎠kB T as:) E = 0 S 22HH σεbr⎛⎞ezψψnez00εζ⎛⎞⎛⎞ezψ 22⎛⎞ψ cosh⎜⎟dy +−⎜⎟1 sinh⎜⎟dy cosh 4ψζdy+− R 1 sinh 4 ψdy ∫∫ / () 1 ⎜⎟() 2 ⎝⎠kTBBμζ⎝⎠⎝⎠kT E ∫∫ζ 00 S = 00⎝⎠ λ (11) ES ⎡⎤ ⎢⎥H ⎢⎥∫ cosh() 4ψ dy 2r Duλλ Du ⎢⎥ion 2 + ⎢⎥H 2r 2.2 Streaming Field Considering 1 HK12+−λ r⎢⎥2− ion 1+ rionH K ⎢⎥ Finite EDL Conductivity r ⎢⎥+ ∫ cosh() 4ψ dy λ Consideration of finite conductivity ⎢⎥2− ⎣⎦H within the EDL, or in other words, considering λ 2− different ionic mobilities in the bulk and the H 2 ⎛⎞ψ ++−cosh 4ψζdy R 1 sinh 4 ψdy ∫∫() 1 ⎜⎟() EDL leads to alteration in the expression of the λ 0 ⎝⎠ζ conduction current, although the streaming H current remains unchanged. (14) The modified expression for the conduction where the different dimensionless variables and current now reads: parameters are: yyH= / , ψ = ezψ /4 kB T , ζζ= ez/4 kB T ,

⎡⎤⎛⎞ezψ 8nkT00εεrB ⎢⎥cosh dy R1 = (equivalent to ionic Peclet λ ⎜⎟ kTB μσ b / σDiffE S ⎢⎥⎝⎠ IEcond=+2σ St S ⎢⎥ ∫ 22Hr− ion number, as described in our previous work [18]), λ−2rion ⎢⎥⎛⎞ezψ 2rion ⎢⎥+ cosh⎜⎟dy σσSt+ Diff ∫ kT Du = (Dukhin number denoting the ⎣⎦⎢⎥2H−λ ⎝⎠B σλb 2H−λ ⎛⎞ezψ ratio of the EDL surface conductivity and the +σbSEd∫ cosh⎜⎟y ⎝⎠kTB bulk conductivity) and Kr = σSt/σDif (ratio of the λ Stern Layer and Diffuse Layer conductivities) (12)

In eq. (12), the first two terms on RHS represent the conduction current within the Stern and 2.3 Electroviscous Considerations: Diffuse Layer and the last term accounts for the conduction current outside the EDL. Using eq. Effects of EDL Conductivity (12), along with eqs. (4), (5) and (6), we obtain Electroviscous effect is accounted for the modified expression for the streaming field by calculating the enhanced viscosity obtained by balancing the volume flow rate for pure ( E / ), accounting for the finite EDL S pressure-driven transport (with enhanced conductivity, as: viscosity) with combined pressure-driven and streaming field induced back electroosmotic 2 H ⎡⎤1 dp 2 ⎛⎞ezψ transport (with original viscosity). Thus the n0 ez −−2Hy y sinh⎜⎟ dy ∫ ⎢⎥2μ dx ()k T effective viscosity for the cases without and with / 0 ⎣⎦⎝⎠B ES = ⎡⎛⎞ezψ ⎤finite EDL conductivity effect is calculated as: ⎢⎥cosh⎜⎟ dy λ kT σ Diff ⎢⎥⎝⎠B 3 σ + ⎢⎥ dp2 H St 22()λ − r ∫ 22H− rion ion ⎢⎥2r ⎛⎞ezψ ⎢⎥ion + cosh⎜⎟ dy dx 3 ∫ kT μeff = (15) ⎣⎦⎢⎥2 H −λ ⎝⎠B 3 2H 12dp H ζε0 εrSE ⎛⎞ψ 2 H − λ +−⎜⎟1 dy σ b ⎛⎞ezψ μμdx 3 ∫ ζ ++cosh⎜⎟ dy 0 ⎝⎠ 2 ∫ kT λ ⎝⎠B (without considering EDL conductivity) 2 H nez00εεζr ⎛⎞ψψ⎛ ez ⎞ ⎜⎟1 − sinh ⎜⎟dy μζ∫ kT 0 ⎝⎠⎝⎠B (13)

4 dp2 H 3 whereas without EDL conductivity it is 2H / dx 3 ⎛⎞ezψ μeff = (16) expressed as σσ= cosh dy . 3 / 2H eff b ∫ ⎜⎟ 12dp H ζε0 εrSE ⎛⎞ψ kT +−⎜⎟1 dy 0 ⎝⎠B ∫ / μμdx 3 0 ⎝⎠ζ The difference between ES and ES originates (considering EDL conductivity) from the differences in conduction current resulting from the consideration of finite EDL Consequently the ratio of these effective conductivities, much different from the bulk viscosities is calculated as: conductivity. As zeta potential increases, the strength of the streaming field induced back εεζE 2 ⎛⎞ψ 11+−0 rS dy electroosmosis (an effect introduced in our 2 ∫⎜⎟ 2 dp⎛⎞2 H 0 ⎝⎠ζ 3 ⎛⎞ψ previous study [18]) rapidly increases / ⎜⎟ 11+−KdU ⎜⎟y μ dx 3 4 ∫ ζ outweighing the corresponding increase in the eff ==⎝⎠ 0 ⎝⎠ μ εεζE/ 22⎛⎞ψ3 ⎛⎞ψcontribution of the conduction current. These eff 1111+−+−0 rS dy K/ dy 2 ∫∫⎜⎟U ⎜⎟ two effects simultaneously balance the dp⎛⎞2 H 00⎝⎠ζ4 ⎝⎠ζ ⎜⎟ component of the streaming current caused by dx 3 ⎝⎠ the pressure-driven downstream advection of (17) mobile counterions. Consequently at larger zeta

uE potential, the influence of the conduction current ( S )max where KU = is the ratio of the is lessened, lowering the differences between ES ()u p max and E / . Larger the value of H/λ greater is the maximum electroosmotic (streaming field S (without considering finite EDL conductivity) strength of the streaming field induced back induced) velocity to the maximum pressure- electroosmosis, thereby resulting in an even / smaller difference between ES and ES . When u / ( ES ) driven velocity and K / = max is the ratio of Du number is large, the Stern Layer and Diffuse U Layer conductivities become much larger than ()u p max the bulk conductivity so that the difference the maximum electroosmotic (streaming field / between E and E , for a given H/λ and K , is (considering finite EDL conductivity) induced) S S r velocity to the maximum pressure-driven enhanced, leading to a smaller value of the / velocity. EES/ S ratio.

3. RESULTS AND DISCUSSIONS We perform the simulations with the following values of the system parameters: H=50 nm, µ=0.001 Pa-s, εr=79.8, T=300 K. E / Figure 1 depicts the variation of S with E S dimensionless zeta potential for different combinations of Du, Kr and H/λ. In inset of the figure we plot the effective conductivities (σ eff ) having units of S, for the cases with and without the EDL conductivity. With EDL effect, it is

⎡⎤⎛⎞ezψ ⎢⎥cosh⎜⎟ dy λ kT σ Diff ⎢⎝⎠B ⎥ / σσeff=+2 St ⎢⎥Figure 1: Variation of EES/ S with ∫ 22Hr− ion ()λ − 2rion ⎢⎛⎞ezψ ⎥ 2rion + cosh⎜⎟ dy dimensionless zeta potential (ezζ/(4kBT))B for ⎢∫ kT ⎥ ⎣⎢2 H −λ ⎝⎠B ⎦⎥different combinations of Du, H/λ and Kr. In the 2 H −λ inset of the figure we plot the effective ⎛⎞ ezψ conductivities (we call it σ ) (in S) for the + σ b ∫ cosh⎜⎟ dy eff ⎝⎠kTB λ cases with and without considering finite EDL conductivities

5 A very important result of this study is the effect of the relevant dimensionless parameters. / Contrary to the trends in the variations of the of Kr on the ratio EES/ S. Dependence of this / ratio on K reflects the relative contribution of μeff r EE/ / ratio, ratio does not show a the Diffuse Layer and the Stern Layer SS μeff conductivities on dictating the strength of the monotonic variation with the dimensionless zeta streaming field. The Stern Layer effect is μ / manifested simply through the value of the Stern potential. The functional form of the eff ratio Layer conductivity (σ St ). On contrary, the μeff Diffuse Layer effect is reflected through the ensures that it depends more on the individual effective Diffuse Layer conductivity variations in the streaming field values for the

⎡λ 22Hr− ion ⎤two cases (with and without considering the ( σ Diff ⎛⎞ezψ ⎛⎞ezψ), ⎢cosh⎜⎟ dy+ cosh ⎜⎟ dy⎥ / ∫∫ EDL conductivity), rather than the ratio EESS/ . ()λ − 2rkion ⎢⎝⎠BT ⎝⎠ kBT⎥ ⎣22rHion −λ ⎦Consequently the zeta potential dependence of which is thus a function of the zeta potential (i.e., this ratio follows a qualitative variation, higher the zeta potential larger is its contribution) commensurate with the variation of the also. Consideration of this effective value individual streaming field values (in the inset of ensures that the contribution of the Diffuse Layer / figure 2 we show the variation of E and E effect can be comparable or be even more than S S the Stern Layer effect, although the ratio with dimensionless zeta potential, for a given combination of values of dimensionless σ St Kr = is much greater than unity. The result parameters). For small enough zeta potentials σ Diff / (where ES is always larger than ES ), ES is that for relatively smaller values of Kr (Kr = 3, rapidly decreases with zeta potential. Similar 5, for instance), the contribution of the overall qualitative decrements (though to a much lesser EDL conductivity is enhanced, leading to a much extent) are noted for the case of E / . Thus, one smaller value of the ratio EE/ / . At larger zeta S SS may expect that over such zeta potential ranges, potential, the influence of the effective Diffuse μ / Layer conductivity, and hence the effect of the there is likely to be a decrease of the ratio eff net EDL conductivity, is enhanced leading to the μeff / maximum weakening of ES relative to ES . with increments in zeta potential. The most interesting aspect of Figure 1 is the possibility of the ratio EE/ / becoming SS more than unity for relatively small values of Du and small values of H/λ, at very large zeta potential values. To explain such behaviour we show the variation of the effective conductivities with dimensionless zeta potential, for the cases with and without the finite EDL effect (See inset of Figure 1). It is found that for extremely large values of the zeta potential, the effective conductivity for the case without considering separate EDL conductivity actually outweighs that of the case with EDL effect. Consequently / / Figure 2: Variation of viscosity ratio ( μeff/ μ eff ) the ratio EES/ S shows a value larger than one.

/ with dimensionless zeta potential (ezζ/(4kBT))B For the cases in which EES/ S approaches unity for different combinations of Du, H/λ and Kr. In for large Du (i.e., small enough H/λ or small the inset of the figure, we plot the variation of / / enough Kr), the ratio EESS/ is further enhanced ES and ES with dimensionless zeta potential for beyond 1 (for small enough Du). some of these combinations of Du, H/λ and Kr. Figure 2 gives the variation of the ratio μ / viscosity ratio ( eff ) with dimensionless zeta Both E and E / reach a maximum value at an μeff S S intermediate zeta (this zeta value may be potential (ezζ/(4kBT))B for different combinations

6 / different for ES and ES ), beyond which both of these parameters decrease with increasing zeta potential. The consequence is that the difference 4. CONCLUSIONS between E and E / decreases (thereby The present study delineates the S S significant influences of finite EDL decreasing the differences in their electroviscous conductivities (significantly different from that μ / influences), making the ratio eff increase (and dictated by bulk ionic concentrations) on the μeff streaming potential and the resulting approach a value closer to 1) with increments in electroviscous effects in nanofluidic ionic the zeta potential. In fact, for certain transport. Even within the EDL, there can be combinations of the dimensionless parameters significant differences in conductivities (depending on whether the region of interest is and at very large zeta potential values, E / may S the wall-adjacent monoionic thick Stern Layer or become greater than ES (as seen in figure 1), the outer Diffuse Layer), which in turn can bear μ / significant consequences on the streaming resulting in a eff value greater than 1. Increase potential and effective viscosity predictions. The μ eff magnitude of the combination of these in H/λ values decreases the difference between conductivity values in the two layers of the EDL / relative to the bulk value of the solution / μeff ES and ES , thereby resulting in a larger conductivity (parameterized by the Dukhin μ eff number), in addition to the relative thickness of value (i.e., a value closer to 1) corresponding to a the EDL with respect to the channel height, given value of the dimensionless zeta potential. dictate the eventual trends in the streaming With lowering of Kr values, difference between potential and electroviscous effects as attributed / ES and ES gets enhanced, enforcing a lowering to the EDL conductivity variations. When the μ / Dukhin number is significantly large (indicating of eff for a given value of the dimensionless large values of EDL conductivity) or the value of μeff H/λ is sufficiently large, the streaming potential zeta potential. Finally, a decrease in Du greatly and the electroviscous effects for the case with / varying conductivity effects within the EDL get decreases the difference between ES and ES , significantly reduced, particularly for relatively μ / / eff small values of the zeta potential. On the and can even make ES exceed ES . μeff contrary, however, for moderate Dukhin / numbers and for very large zeta potentials, variation commensurate with such E and E S S streaming potentials and the electroviscous variation is seen in figure 2. effects may get significantly augmented and can In nanoconfinements, the strength of the actually exceed those obtained without variable pressure-driven flow is substantially reduced by EDL conductivity effects. the presence of large streaming field and the resulting back electroosmotic transport. In this paper, it is illustrated that consideration of REFERENCES conductivity gradient across the different layers [1] Watson, A., 2000. Science, 289, 850. of the EDL and the bulk result in a weaker [2] Griffiths, S. K., and Nilson, R. H., 2006. strength of the streaming field. This will mean Anal. Chem., 78, 8134. that the flow strength can be significantly [3] De Leebeeck, A., and Sinton, D., 2006. augmented by choosing suitable system 27, 4999. parameters like the wall zeta potential, bulk ionic [4] Chakraborty, S., 2007. Anal. Chim. Acta 605, concentration, surface with desirable adsorptivity 175. for the counterions, size of the counterions (these [5] van der Heyden, F. H. J., Stein, D., parameters dictate the conductivities of the Stern Besteman, K., Lemay, S. G., and Dekker, Layer and the Diffuse Layer and consequently C., Phys. Rev. Lett. 96, 224502. the values of the Dukhin number and the ratio [6] Alkafeef, S. F., and Alajmi, A.F., 2007. J. K ). Hence by tailoring the Stern Layer and the r Coll. Int. Sci. 309, 253. Diffuse Layer conductivities by modifying the different physical parameters we can obtain desired nanofluidic flow behaviour.

7 [7] Daiguji, H., Oka, Y., Adachi, T., and Shirono, K., 2006. Electrochem. Commun. 8, 1796. [8] Xuan, X., and Sinton, D., 2007. Microfluid. Nanofluid. 3, 723. [9] Hunter, R. J., 1981. Zeta Potential Colloid Science. Academic Press, New York. [10] Carrique, F., Arroyo, F. J., Shilov, V. N., Cuqejo, J., Jiménez, M. L., and Delgado, A. V., 2007. J. Chem. Phys. 126, 104903. [11] Carrique, F., Arroyo, F. J., and Delgado, A. V., 2000. J. Coll. Int. Sci. 227, 212. [12] Mangelsdorf, C. S., and White, L. R., 1990. J. Chem. Soc. Farad. Trans. 86, 2859. [13] Carrique, F., Arroyo, F. J., and Delgado, A. V., 2001. J. Coll. Int. Sci. 243, 351. [14] Carrique, F., Arroyo, F. J., and Delgado, A. V., 2002. J. Coll. Int. Sci. 252, 126. [15] Rubio-Hernandez, F. J., Ruiz-Reina, E. and Gomez-Merino, A. I., 2000. J. Coll. Int. Sci. 226, 180. [16] Carrique, F., Arroyo, F. J., and Delgado, A. V., 2001. Coll. Suref. A 195, 157. [17] Davidson, C., and Xuan, X., 2008. Electrophoresis 29, 1125. [18] Chakraborty, S., and Das, S, 2008. Phys. Rev. E 77, 037303.

8