Effect of Geometry on Electrokinetic ... - ACS Publications

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Effect of Geometry on Electrokinetic Characterization of Solid Surfaces Abhijeet Kumar,*,†,‡ Jochen Kleinen,† Joachim Venzmer,† and Tatiana Gambaryan-Roisman*,† †

Institute for Technical Thermodynamics and Center of Smart Interfaces, TU Darmstadt, 64287 Darmstadt, Germany Research Interfacial Technology, Evonik Nutrition & Care GmbH, 45127 Essen, Germany

‡

S Supporting Information *

ABSTRACT: An analytical approach is presented to describe pressure-driven streaming current (Istr) and streaming potential (Ustr) generation in geometrically complex samples, for which the classical Helmholtz−Smoluchowski (H−S) equation is known to be inaccurate. The new approach is valid under the same prerequisite conditions that are used for the development of the H−S equation, that is, the electrical double layers (EDLs) are sufficiently thin and surface conductivity and electroviscous effects are negligible. The analytical methodology is developed using linear velocity profiles to describe liquid flow inside of EDLs and using simplifying approximations to describe macroscopic flow. At first, a general expression is obtained to describe the Istr generated in different cross sections of an arbitrarily shaped sample. Thereafter, assuming that the generated Ustr varies only along the pressure-gradient direction, an expression describing the variation of generated Ustr along the sample length is obtained. These expressions describing Istr and Ustr generation constitute the theoretical foundation of this work, which is first applied to a set of three nonuniform cross-sectional capillaries and thereafter to a square array of cylindrical fibers (model porous media) for both parallel and transverse fiber orientation cases. Although analytical solutions cannot be obtained for real porous substrates because of their random structure, the new theory provides useful insights into the effect of important factors such as fiber orientation, sample porosity, and sample dimensions. The solutions obtained for the model porous media are used to device strategies for more accurate zeta potential determination of porous fiber plugs. The new approach could be thus useful in resolving the long-standing problem of sample geometry dependence of zeta potential measurements. where ρe is the electric charge density (nonzero only inside of EDLs),20 V is the macroscopic velocity field, and dAc is a directed surface element of the cross section Ac along the pressure-gradient direction. Because of the unidirectional transport of counterions with liquid flow, an electric potential difference, known as the streaming potential Ustr, develops across the sample. The developed Ustr induces a conduction current flow through the electrolyte opposite to the direction of the liquid flow. The Istr is balanced by the conduction current Ic, and the value of the developed Ustr is obtained by applying Ohm’s law for conduction current flow. The classical relationship that is commonly used to convert measured Istr and Ustr values into substrate zeta potentials is known as the Helmholtz−Smoluchowski (H−S) equation. It has been derived analytically for uniform cross-sectional capillaries by applying eq 1 and Ohm’s law.21 The equation is, however, valid only under the conditions that are as follows:

1. INTRODUCTION Electrokinetic characterization via streaming current and streaming potential measurements is one of the most widely used techniques for determining the zeta potential of macroscopic solid surfaces. It is thus an indispensable research tool in several application domains such as paper,1−3 textiles,4−8 membranes,9−11 biomaterials,12−14 soil and minerals, and so forth.15−17 The measurement technique involves the flow of an electrolyte solution of known conductivity through capillaries, channels, or porous structures formed out of the solid substrate18 under an applied pressure difference (ΔP), as shown in Figure 1A. When electrolytes flow at charged solid surfaces, the flow inside of electrical double layers (EDLs) results in convective transport of counterions in the flow direction, which constitutes the streaming current (Istr). The total Istr along the pressure-gradient direction through a sample cross section Ac is given by the basic electrokinetic equation,19 as given below Istr =

∬A ρe V·dA c

Received: February 9, 2017 Revised: June 25, 2017 Published: June 28, 2017

(1)

c

© XXXX American Chemical Society

A

DOI: 10.1021/acs.langmuir.7b00344 Langmuir XXXX, XXX, XXX−XXX

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Figure 1. (A) Experimental scheme used for streaming current/potential characterization of samples and (B) flow diagram of processes leading to streaming current/potential generation. The steps that could be affected by the sample geometry are marked with red and labeled as 1−3. The existing theories/equations that can be used to describe these steps are also mentioned alongside.

where ϕ is the sample porosity and AT is the total crosssectional area of the sample. In experimental works investigating the applicability of the parallel capillary model,18,28−30 it was identified that although the linear variation of Ustr and Istr with ΔP is observed with most samples, the experimentally measured Ustr and Istr values could deviate significantly from the values given by eqs 4 and 5. Furthermore, this approach fails to explain the capillary shape dependence of Istr and Ustr generation in the case of nonuniform cross-sectional capillaries.18,31 In the general case of porous fiber plugs, it does not capture the variations of Istr and Ustr with fiber orientation and packing density.32,33 A common approach used for incorporating sample geometry effects on Istr generation, referred to as “H−S approximation”, involves experimentally estimating the socalled “cell constant” (the ratio L/Ac in eq 2) via sample electrical resistance measurements.18,34 Although the parameters L and Ac are well-defined for a uniform capillary, they are complex parameters for nonuniform geometries, and hence, apparent values are used, which are evaluated indirectly from sample electrical resistance values, as given below

(i) the EDL thickness is sufficiently small compared with the capillary radius,21−23 (ii) the electric charge density inside of EDLs is unperturbed by the liquid flow,24 (iii) the electroviscous effect, that is, the effect of EDL on liquid viscosity, is negligible,25,26 and (iv) surface conductivity effects are negligible. The same above conditions also need to be satisfied for the analytical approach presented in this work to be applicable. Given that the above conditions are satisfied, the Istr and Ustr generated in straight capillary geometries could be described by the H−S equation Istr =

ϵoϵrs Ac ΔPζ μ L

(2)

Ustr =

ϵoϵrs ΔPζ μκ

(3)

where ϵo and ϵrs are the vacuum permittivity and dielectric number of the electrolyte, respectively, Ac and L are the capillary cross-sectional area and length, respectively, κ and μ are the bulk electrical conductivity and dynamic viscosity of the electrolyte solution, respectively, ΔP is the pressure difference applied across the capillary, and ζ is the solid surface zeta potential. Although the H−S equation has been obtained analytically only for uniform cross-sectional capillaries, the characterization of most real-life substrates involves geometrically complex samples such as membranes, textiles, minerals, and so forth, where electrolytes are flown through nonuniform and irregular channels. An important question therefore has been whether, in such cases, the H−S equation is still applicable. Overbeek27 suggested that if pressure-driven liquid flow inside of porous samples is laminar and if pore sizes remain larger than the EDL thickness, a simplistic extension of the H−S equation could still be used. In this approach, nonidealities induced by irregular sample geometries are ignored, and the porous sample is treated as a bundle of parallel straight capillaries. The equations that are hence obtained are given below Istr =

ϵoϵrs A Tϕ ΔPζ μ L

(4)

Ustr =

ϵoϵrs ΔPζ μκ

(5)

R=

1 Lapp κ A app

(6)

where R is the experimentally measured sample electrical resistance and Lapp/Aapp is the apparent value of the cell constant for the nonuniformly shaped sample, which is subsequently substituted in eq 2 to obtain the generated Istr. Although widely used, the idea underlying the H−S approximation that the cell constant as determined via sample electrical resistance measurements could be used to obtain the generated Istr is not theoretically precise. Furthermore, the H−S approximation does not account for the sample geometry dependence of Ustr generation and is thus not an all-inclusive approach. In earlier attempts to resolve the above discrepancy, the use of empirical functions of plug packing densities, referred to as shape correction factors, has been proposed.35−37 In other works, the so-called “formation factors”38,39 have been used to account for the sample geometry effects. Another recent practice has been to lump the nonidealities arising from the geometry effects into effective quantities such as effective porosity (ϕeff) and effective zeta potential (ζeff).40 In spite of a lack of clear insights into the factors on which such effective B


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Figure 2. (A) Velocity profile (shown in green) and electric charge density profile (shown in pink) inside of EDLs near charged solid surfaces. (B) Schematic representation of an arbitrarily shaped porous sample. (C) Schematic representation of the forces acting on an infinitesimally thin liquid layer of thickness dz located at height z along the pressure-gradient direction. (D) Schematic representation of the currents flowing in and out of the thin liquid layer because of streaming current and bulk conduction current flow.

case of zeta potential determination of porous fiber plugs. Strategies are devised for optimizing the measurement conditions for improving the accuracy of zeta potential measurements. Finally, the findings are summarized in the Conclusions section.

quantities depend, this approach has been widely used in characterization of membranes.41,42 An analytical scheme is therefore required, which incorporates all possible geometry effects on Istr and Ustr generation, and developing the same forms the main objective of this work. In principle, once the velocity field has been obtained, it could be coupled with eq 1 to obtain the generated Istr followed by an application of Ohm’s law to obtain the generated Ustr. However, resolving these theories to obtain usable relationships could be very challenging even for moderately nonideal sample geometries. This paper is organized into a total of four sections. After Introduction, a general theory is presented in the second section wherein a fundamental microscale relationship to describe Istr generation in different cross sections of a sample is established. Thereafter, this fundamental relationship is applied to an arbitrarily shaped porous substrate, where, using a force balance on thin liquid layers in the direction of pressure difference, the generated Istr is related to the pressure gradient inside of the sample. Furthermore, assuming that the generated Ustr varies only along the pressure-gradient direction, a general expression is obtained to relate the generated Ustr to the total pressure difference applied across the sample. This analytical approach provides a framework that accounts for the sample geometry dependence of Istr and Ustr generation. In the third section, the general expressions for Istr and Ustr generation are applied to obtain analytical solutions for a set of three nonuniform cross-sectional capillaries and for a square array of cylindrical fibers (model porous media). The pressure gradients inside of sample geometries are obtained by using simplifying approximations to describe the macroscopic liquid flow. On the basis of the results obtained for model porous media, some insights are obtained into the practically important

2. THEORY 2.1. Origin of Geometry Effects on Istr and Ustr Generation. To understand the reasons underlying the dependence of generated Istr and Ustr on the sample geometry, it is helpful to consider a flow diagram of all steps leading to Istr and Ustr generation, as shown in Figure 1B. Among the steps involved, those steps that might be affected by the sample geometry are labeled as steps 1−3 on the flowchart and are discussed below. First, the geometry of porous samples has a direct influence on the characteristics of a pressure-driven liquid flow (step 1 in Figure 1B). Although the velocity field (V) could be in principle obtained by solving the Navier−Stokes equation using a no-slip condition at solid−liquid interfaces, obtaining analytical solutions is very difficult for nontrivial geometries. Simplifications such as the creeping flow and lubrication approximations43,44 could be utilized under certain special conditions. Apart from the direct effect of sample geometry on the velocity field, the orientation of solid surfaces with respect to the pressure-gradient direction also influences Istr generation as determined by the dot product term V·dAc in eq 1 (step 2 in Figure 1B). The geometry of porous substrates affects bulk conduction current flow through confined liquids and hence could influence Ustr generation (step 3 in Figure 1B). We assume that the surface conductivity effects are negligible. In such cases, C


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Langmuir the nonideal porous substrate geometry affects only the macroscopic current flow characteristics, and an effective electrical conductivity term (κeff) is usually defined to account for it.45 However, whether such an effective electrical conductivity (κeff) should be used while describing Ustr generation is discussed in section 2.3.2. We demonstrate that because our theory employs a pore−scale relationship to describe Ustr generation, it is the bulk electrical conductivity that should be used. 2.2. Fundamental Microscale Relationship for Istr Generation. Given that a pressure difference (ΔP) is applied across a porous sample in the z-direction, let Lc(z) be the intersection line of the solid−liquid interface with a sample cross section Ac located at a level of z along the pressuregradient direction. A length element dLc of the intersection line Lc(z) with the local surface tangent along the local liquid flow direction (τ̂) inclined at an angle θ with respect to the zdirection is shown in Figure 2A. The total streaming current (Istr(z)) through the cross section Ac could be estimated via summation of infinitesimal streaming currents (dIstr) generated because of the flow inside of EDLs adjoining the length elements dLc constituting Lc(z). This approach is valid if the EDL thickness is much smaller than the pore radius. The streaming current Istr(z) is obtained via a path integral of the expression for dIstr along the intersection line Lc(z). dIstr can be expressed as a function of the solid surface zeta potential (ζ), local hydrodynamic wall shear stress (τwall), and local orientation of the solid surface with respect to pressuregradient direction (θ), following the procedure outlined below. In the case of thin EDLs, the liquid flow velocity inside of the EDL adjoining dLc can be expressed as a linear function of the local wall shear stress (τwall)

⎛τ ⎞ Vτ(y)τ ̂ = ⎜ wall ⎟y ⎝ μ ⎠

dIstr =

=

∫0

δ EDL

Istr(z) =

∫0

yρe (y) dy

(8)

where δEDL is the thickness of the EDL and ρe(y) is the electric charge density inside of the EDL. δ EDL

yρe (y) dy in eq 8 depends only on EDL The integral ∫ 0 properties and could be determined independently using the Poisson−Boltzmann equation to describe ρe(y), as shown by Adamczyk et al.24 and also shown in the Supporting Information section

∫0

δ EDL

[PAc]z − [PAc]z + dz +  net upward pressure force

yρe (y) dy = ϵoϵrsζ

ϵoϵrsζ μ

∮L (z) τwall·ẑ dLc c

(11)

It should be kept in mind that the above relationship for Istr generation is valid under the same prerequisite conditions that are required for the H−S equation. First, eq 7 is valid only if the EDLs are thin, that is, δEDL is small in comparison to the pore radius. Second, the electric charge density inside of EDLs should be unperturbed by the liquid flow.24 Furthermore, the electroviscous effect, that is, the effect of EDLs on liquid viscosity, should be negligible. In an earlier work, it has been demonstrated that the electroviscous effect is insignificant when the factor a/δEDL, where a is the pore radius, does not lie between 0.3 and 30.46,47 Therefore, if the pore radius is much greater than δEDL, the electroviscous effect could be safely neglected. When using this theory in applications, it should be therefore ensured that this condition is met. This condition would be satisfied in measurements involving macroscopic substrates where the pore radius is typically on the order of micrometers, whereas the used electrolyte solution has an EDL thickness on the order of nanometers. 2.3. General Expressions for an Arbitrary Porous Sample. In this section, we consider a porous sample of length L with arbitrary internal geometry and with the external form of an object with nonuniform cross-sectional area, as shown schematically in Figure 2B. Thereafter, general expressions to describe Istr and Ustr generation in such geometries are obtained. 2.3.1. Streaming Current. It is evident from eq 11 that the streaming current generated in a certain sample cross section is dependent on the shear stresses and the surface orientations at the solid−liquid interface line. Thus, for an irregularly shaped sample, Istr could vary along the sample length from one cross section to another. To relate the streaming current Istr given by eq 11 to the applied pressure difference ΔP, we consider a balance of forces along the pressure-gradient direction on thin liquid layers of thickness dz located between two cross-sectional planes at the levels of z and z + dz (Figure 2B). The shear and normal reaction forces exerted by the surrounding walls and the pressure forces exerted by the liquid above and below the thin layer are accounted for. The total shear force exerted on the thin liquid layer opposite to the pressure-gradient direction is obtained via an integral of the z-component of shear forces acting on the solid−liquid interface. For understanding the force balance, the thin liquid layer is visualized, as shown in Figure 2C. The balance of forces acting on the thin layer in the z-direction is given below

(7)

δ EDL

∮L (z) dIstr = c

ρe (y)Vτ(y)τ ·̂ z ̂ dLc dy

τwall·ẑ dLc μ

(10)

The total streaming current Istr(z) in the z-direction through the cross section located at the level of z could be obtained via an integral of eq 10 along the solid−liquid interface line Lc(z)

where τwall is the local wall shear stress vector, Vτ is the liquid flow velocity tangential to the solid surface, τ̂ is the unit vector along the local liquid flow direction, and y is the distance from the solid surface along the local surface normal (n̂), as shown in Figure 2A. By making use of eq 7 to describe the liquid flow velocities inside of EDLs, the streaming current dIstr resulting from the convective transport of counterions in the EDL adjoining the solid−liquid interface length element dLc is determined dIstr =

ϵoϵrsζ τwall·z ̂ dLc μ

(9)

=

P dA c ⏟ force upward contact

∮L (z) τwall·ẑ cosdzθ dLc

c  

Upon substituting the value of the integral from eq 9 in eq 8, a simplified expression is obtained for dIstr

downward shear force

D

(12) DOI: 10.1021/acs.langmuir.7b00344 Langmuir XXXX, XXX, XXX−XXX

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Figure 3. (A−C) Geometry of nonuniform cross-sectional capillaries for which analytical solutions are obtained, (D) variation of the generated streaming current along the length of the capillaries, and (E) variation of Ustr shape correction factor (Uscf) with the size ratio γ1(Ri/Ro) at a fixed size ratio γ2(Ro/L) = 0.1 for the capillaries. For an ideal capillary with uniform cross-sectional area, Uscf is equal to 1.

where P is the pressure, Ac is the area of the liquid in the cross section, and the line integral on the right gives the total shear force acting on the liquid layer opposite to the pressuregradient direction. It is, however, to be noted that such a force balance is valid only under such flow conditions, wherein the creeping flow approximation is applicable. The above force balance could be consequently simplified to the following form − dP A c (z ) = dz

∮L (z) c

1 τwall·z ̂ dLc cos θ

Figure 2D. Both the streaming current and the bulk conduction current described using Ohm’s law are accounted for ⎤ ⎡ ⎛ ∂ψ ⎞ κ ⎜ ⎟ dA c ⎥ [Istr]z + ⎢ ⎦ z + dz ⎣ Ac ⎝ ∂z ⎠  

∬

total inward current flow

⎤ ⎡ ⎛ ∂ψ ⎞ κ ⎜ ⎟ dA c ⎥ = [Istr]z + dz + ⎢ ⎝ ⎠ ⎦ ⎣ Ac ∂z z

∬

(13)

total outward current flow

where ψ is the bulk electric potential developed as a response to streaming current flow. It is important to note that the electric potential ψ is not necessarily uniform in a certain cross section. Therefore, the total bulk conduction current flowing through a certain cross section is obtained via a surface integral. Although the bulk electric potential ψ might also vary perpendicular to the z-direction inside of the cross sections, the main objective of our analysis is obtaining the measurable electric potential difference developed across the sample length along the pressure-gradient (z) direction. For this purpose, we assume that the generated electric potential varies only along the pressure-gradient direction and that it is uniform in each cross section, that is, ψ|z = Ustr(z). Upon simplifying eq 16, assuming uniform electric potential in each cross section, the following relationship is obtained

Thereafter, eq 11 is divided by eq 13 Istr(z) =

τ ·z ̂ dLc ϵoϵrsζ −dP ∮Lc(z) wall Ac 1 μ dz ∮ τ ·z ̂ dLc L (z) cos θ wall c

(14)

Equation 14 is the general expression that can be used to describe Istr generation in different cross sections of a sample. An important property of eq 14 is that if all solid surfaces in a particular sample cross section are identically oriented and make the same angle with the pressure-gradient direction, the 1/cos θ term could be taken outside of the path integral in the denominator. Equation 14 could then be further simplified Istr(z) =

ϵoϵrsζ − dP Ac cos θ dz μ

(16)

(15)

dIstr(z) d2Ustr(z) = κA c dz dz 2

2.3.2. Streaming Potential. To estimate the streaming potential generated across the sample in the pressure-gradient direction, we consider a balance of currents flowing into and out of a thin liquid layer between z and z + dz, as shown in

(17)

Upon integrating eq 17 with respect to z, the following relation is obtained E


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Langmuir Istr(z) = κAc

dUstr(z) dz

Additionally, the shapes of these capillaries could be defined in terms of only two nondimensional size ratios, γ1(Ri/Ro) and γ2(Ro/L). It is assumed that for the considered capillaries the ratio Ro − R i is much less than 1, in which case the lubrication L approximation can be used to describe the pressure-driven macroscopic liquid flow inside of the capillaries. This condition is frequently fulfilled in substrates such as membranes and textiles.52 The results obtained for the different capillary shapes are contrasted to demonstrate the capillary shape dependence of Istr and Ustr generation. 3.1.1. Capillary A. We consider the capillary having a conical shape as shown in Figure 3A with a zeta potential ζ and a pressure difference ΔP applied across it. Eqs 15 and 20 are used to obtain the expressions for the variation of streaming currents/potentials along the capillary length. Under lubrication approximation, the velocity profile along the pressure-gradient direction in a capillary cross section with radius Rz located at a level of z along the capillary length is given by

(18)

The integration constant is equal to zero because at the extreme ends of the sample (z = 0 & z = L), the values of the streaming current (Istr(z)) and the conduction current dUstr dz

(κA (z) ) are exactly equal because no net current enters c

or leaves the system. The electrical conductivity term used in eq 18 is the bulk conductivity (κ) and not the effective electrical conductivity of the porous structure (κeff). Equation 18 is a microscale relationship, whereas κeff is a macroscopically defined effective parameter; therefore, κeff is not required in eq 18. This means that step 3 as depicted in Figure 1B does not have an effect on Ustr generation in our case. The generated Ustr can be obtained by integrating eq 18 along the sample length after substituting the value of Istr(z) from eq 14 z

ϵϵ ζ Ustr(z) = o rs μκ

∫ 0

τ ·z ̂ dLc −dP ∮Lc(z) wall dz 1 dz ∮ τwall·z ̂ dLc θ cos L (z) c

1 ∂ ⎛ ∂Vz ⎞ 1 dP ⎜r ⎟= , r ∂r ⎝ ∂r ⎠ μ dz

(19)

The above general expression could be further simplified in the case of porous samples in which all solid surfaces in a particular cross section are oriented at the same angle (θ) with respect to the z-direction ϵϵ ζ Ustr(z) = o rs μκ

∫0

z

− dP cos θ dz dz

Vz(r ) = −

1 dP (R z 2 − r 2 ) 4μ dz (21)

The volumetric flow rate (Q) through a capillary cross section could be determined, as shown below Q=

(20)

∫0

Rz

Vz(r )2πr dr = −

π R z 4 dP 8μ dz

(22)

Upon expressing the cross-sectional radius Rz as a function of

It is important to note that in the development of the above theory surface conductivity effects have been neglected, and it is assumed that the back current flow takes place entirely through the bulk electrolyte. The relative significance of surface conductivity effects is usually quantified in terms of the dimensionless Dukhin number (Du) that is related to the surface and bulk conductivities.20,21 The contribution of surface conductivity to current flow is negligible when Du ≪ 1.48 When surface conductivity effects are not negligible, their role in back current flow should be duly accounted for.49−51 The surface conductivity includes contributions from both the diffuse double layer and the stagnant layer. 20,21 The contribution of the diffuse double layer to surface conductivity is negligible when the EDL thickness is sufficiently low and the magnitude of ζ is low.20 The contribution of the stagnant layer to surface conductivity could, however, not be easily correlated with other physical parameters in the framework of the present theory.21 Therefore, when applying the proposed theory in practice, it should be determined beforehand whether the surface conductivity effects are negligible at the conditions employed during the measurements.

the distance along the capillary length z as R o −

( R −L R )z, and o

i

integrating eq 22 along the capillary length (0 ≤ z ≤ L), the volumetric flow rate Q is obtained in terms of the applied pressure difference ΔP. The obtained value of Q is thereafter substituted in eq 22 to obtain the pressure gradient as a function of z −

Ro4 dP ΔP 1 = dz L fA (R i /R o) R z 4

where fA (R i /R o) =

(1 + γ1 + 3γ13

γ12)

(23)

with γ1 = Ri/Ro.

Then, the pressure gradient from eq 23 is substituted in eq 15 to obtain Istr through different cross sections of the capillary Istr(z) =

Ro4 ϵoϵrsΔPζ π cos θA fA (R i /R o) R z 2 μL

where cos θA is a constant and is given by

(24) L (R o − R i)2 + L2

.

Ustr developed in the capillary is obtained by substituting the expressions for the pressure gradient (eq 23) and cos θA into eq 20

3. APPLICATION TO MODEL SAMPLE GEOMETRIES 3.1. Capillaries with Nonuniform Cross Section. In this section, the simplified general expressions for Istr and Ustr generation in porous samples, as given by eqs 15 and 20, respectively, are applied to a set of three axi-symmetric capillaries with nonuniform cross-sectional area, as shown in Figure 3A−C. These capillary geometries are chosen as their sizes could be described using the same three physical dimensions, namely, Ro (diameter of the wider end), Ri (diameter of the narrower end), and L (length of the capillary).

Ustr(z) =

ϵoϵrsΔPζ Ro4 1 μκ fA (R i /R o) L

∫0

z

cos θA dz Rz4

(25)

The total Ustr developed across the capillary is obtained by using z = L as the upper integration limit in eq 25, and upon simplification, it could be expressed in terms of nondimensional size ratios γ1(Ri/Ro) and γ2(Ro/L) F


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Langmuir Ustr =

ϵoϵrsΔPζ ϵ ϵ ΔPζ cos θA = o rs μκ μκ

1 ⎛ 3 + (γ + γ 2 + γ 3) 1 1 1 1 + 4γ2 2(1 − γ1)2 ϵoϵrsΔPζ ⎜ Ustr = ⎜ μκ ⎜ 3 + γ1 + γ12 + γ13 ⎝

1 1 + γ2 2(1 − γ1)2 (26)

(32)

The analytical methodology used for capillary A could also be extended to the capillaries B and C to first obtain expressions for −dP/dz and thereafter for Istr(z) and Ustr(z). 3.1.2. Capillary B. The pressure gradient along the length of capillary B is given by −

Ro4 dP ΔP 1 = dz L fB (R i /R o) R z 4

where fB (R i = R o) =

3.1.4. Capillary Shape Dependence of I str and U str Generation. The variation of the generated Istr in capillaries A, B, and C, as given by eqs 24, 28, and 31, respectively, is plotted in Figure 3D for a particular set of values of Ro, Ri, and L (30, 10, and 300 μm, respectively). The values of Ro, Ri, and R −R L are thus chosen to ensure that o L i ≪ 1, such that the lubrication approximation is applicable. It is easily identifiable from Figure 3D that the contrasting shapes of capillaries lead to significant differences in the generated Istr. Eqs 26, 30, and 32 give the total Ustr generated across the capillaries A, B, and C, respectively. For each capillary, it is possible to define a shape correction factor (Uscf) as the ratio of the analytically obtained Ustr and that given by the H−S equation Ustr‑HS

(27)

1 + γ1 + γ12 + 3γ13 6γ13

with γ1 = Ri/Ro and Rz/Ro

( R −L R )(z − L2 ) for L/2 ≤ z

for 0 ≤ z ≤ L/2 and R z = R o − 2

o

i

≤ L. Using eq 15, Istr through different cross sections along the capillary length is expressed as below

Uscf =

ϵ ϵ ΔPζ Ro4 π Istr(z) = o rs cos θB μL fB (R i /R o) R z 2

(28) 4(R o − R i)2 + L2

for L/2 ≤ z ≤ L. The total Ustr developed across capillary B is evaluated and expressed in terms of nondimensional size ratios γ1(Ri/Ro) and γ2(Ro/L), as shown below 1 ⎛ 3γ 3 + (1 + γ + γ 2) 1 1 1 + 4γ2 2(1 − γ1)2 ϵoϵrsΔPζ ⎜ 1 Ustr = ⎜ μκ ⎜ 1 + γ1 + γ12 + 3γ13 ⎝

⎞ ⎟ ⎟ ⎟ ⎠ (29)

3.1.3. Capillary C. The pressure gradient along the length of capillary C is given by −

Ro4 dP ΔP 1 = dz L fC (R i /R o) R z 4

where fC (R i /R o) = o

)

= Uscf (γ1 , γ2) (33)

with γ 1 = R i /R o and

( R −L R )z for 0 ≤ z ≤ L/2 and R

R z = Ro − 2

(

Ustr ϵoϵrsΔPζ μκ

(30)

3 + γ1 + γ12 + γ13 6γ14

Ustr = Ustr‐HS

The extent to which the capillary shapes affect the generated Ustr could be identified by plotting the function Uscf(γ1, γ2) for the capillaries A−C, as shown in Figure 3E. For this plot, γ2 is kept fixed at 0.1, whereas γ1 is varied from 0.2 to 1. It can be easily identified from Figure 3E that Ustr generated in capillaries not only changes significantly with the size ratio γ1 but is also dependent on the capillary shape (A, B, or C). The results demonstrate that both Istr and Ustr generation could be significantly sensitive to geometry, and these effects should thus be duly accounted for to achieve better accuracy in measurements. This effect is expected to increase significantly R −R for larger values of o i . L 3.2. Square Fiber Array (Model Porous Media). 3.2.1. Flow Parallel to Fiber Axes. We consider a porous structure in the form of a square array of cylindrical fibers with dimensions of L × L × W (as shown in Figure 4). Considering that a pressure difference of ΔP is applied across the fiber array

L

where cos θB = 1 for 0 ≤ z ≤ L/2 and cos θB =

⎞ ⎟ ⎟ ⎟ ⎠

i

z

= Ri for L/2

≤ z ≤ L. As in the case of capillaries A and B, the Istr through different cross sections of capillary C is obtained using eq 15, as given below Istr(z) =

Ro4 ϵoϵrsΔPζ π cos θC fC (R i /R o) R z 2 μL

where cos θC =

L 4(R o − R i)2 + L2

(31)

for 0 ≤ z ≤ L/2 and cos θC = 1

for L/2 ≤ z ≤ L. The total Ustr developed across capillary C could also be expressed in terms of nondimensional size ratios γ1(Ri/Ro) and γ2(Ro/L), as shown below

Figure 4. Square array of cylindrical fibers with pressure difference applied parallel to the fiber axes. G


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Figure 5. (A) Square array of cylindrical fibers with pressure difference applied normal to the fiber axes, (B) geometry of unit cells constituting the fiber array, and variation of the generated streaming current (C) and streaming potential (D) along the length of the unit cell.

in the direction parallel to the fiber axes, the generated Istr and Ustr could be evaluated using the general expressions given by eqs 15 and 20. As the sample geometry is unchanged along the pressuregradient direction, the pressure gradient is uniform along the sample length and is equal to ΔP/W. Furthermore, as the liquid flow at all solid surfaces is aligned with the pressure-gradient direction, the cos θ term in eqs 15 and 20 is equal to 1. Upon simplification, it is found that Istr and Ustr generated in the fiber array are the same as given by the parallel capillary model (eqs 4 and 5). This leads us to the general conclusion that the parallel capillary model holds irrespective of the type of arrangement of fibers (i.e., square/hexagonal/random), as long as all fiber axes are oriented parallel to the pressure-gradient direction. Adamczyk et al.24 established the same result for straight channels with different cross-sectional geometries. 3.2.2. Flow Normal to Fiber Axes. In this section, we consider a square array of fibers with a pressure difference of ΔP being applied perpendicular to the fiber axes, as shown in Figure 5A. To simplify our analysis, we consider the square fiber array to be made up of identical unit cells, as shown in Figure 5B. The side length of the unit cell is S, and the fiber radius is equal to R. If the number of unit cells along the sample length in the pressure-gradient direction is n, the pressure difference across a single unit cell is ΔPs = ΔP/n. The macroscopic liquid flow velocity inside such a unit cell can be approximated by the following relationship53 Vz(y) = −

1 dP 2 (δz − y 2 ) 2μ dz

δz = S /2 − R2 − z 2 for 0 ≤ z ≤ R and δz = S/2 for R ≤ z ≤ S/2, as shown in Figure 5B. In the derivation of eq 34, it is assumed that the flow velocity at the borderline of two laterally adjoining unit cells is zero.53 As the unit cell is symmetrical along its horizontal centerline, the liquid flow characteristics in the lower and upper halves of the unit cell are identical. It is therefore sufficient to carry out the analysis only for the lower half of the unit cell (0 ≤ z ≤ S/ 2) and use equivalent values for the upper half (S/2 ≤ z ≤ S). The volumetric liquid flow rate (Q) through a certain unit cell cross section could be obtained, as shown below Q=

+δz

∫−δ

Vz(y)W dy = −

z

2δz 3 dP W 3μ dz

(35)

Given that ΔPs is the total pressure difference across the unit cell, ΔPs could be expressed in terms of the volumetric liquid flow rate (Q) by integrating eq 35 along the unit cell length 0 ≤ z≤S ΔPs =

3μQ ⎧ ⎛ ⎨2⎜ 2W ⎩ ⎝ ⎪

⎪

∫0

R

1 dz + δz 3

∫R

S /2

⎞⎫ 1 ⎟⎬ d z δz 3 ⎠⎭ ⎪

⎪

(36)

The pressure gradients in different unit cell cross sections could be obtained by substituting the value of Q from eq 36 in eq 35 −

(34)

where δz is the half-width of the liquid part of a unit cell cross section located at a height of z along its length with

dP 1 R2 = ΔPs dz fs (S /R ) δz 3

(37)

where fs(S/R) is a function of the size ratio S/R and has been provided in its complete form in the Appendix. H


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It could be remarked from eq 41 that the total Ustr generated across the unit cell is independent of the fiber radius and dependent only on the size ratio S/R, which is in turn related to the porosity (ϕ) of the fiber array

A similar approach, as described above, has been used by Tamayol and Bahrami53 to square fiber arrays for the estimation of viscous permeability, and a good agreement with experimental data was obtained. This shows that the applicability of the approximate solution for describing the pressure-driven liquid flow inside of the unit cells is good. With the pressure gradients inside of the unit cell known, the generated Istr inside of the unit cell could be obtained using eq 15. The other two variables in eq 15 are cos θs and Ac. The cos θs term that represents the solid surface orientation with respect to the z-direction could be expressed as a function of the nondimensional height z/R cos θs =

R2 − z 2 = R

1 − (z /R )2

S = R

(42)

Thus, Ustr developed across the unit cell could be thus expressed only as a function of porosity (ϕ) Ustr =

ϵoϵrsΔPsζ hs(ϕ) μκ

where hs(ϕ) = 2

(38)

(43)

gs*(S / R ) fs (S / R )

. S/R=

π 1−ϕ

3.2.3. Effect of Fiber Orientation and Sample Porosity. In this section, the variation of Istr and Ustr generated in the square array of fiber with sample porosity (ϕ) is plotted for both parallel and transverse orientation of fibers, as shown in Figure 6A,B. It is identifiable from the plots that the generation of both Istr and Ustr is highly sensitive to both fiber orientation and porosity. 3.2.3.1. Effect on Streaming Current. Because in the case of transverse fiber orientation the generated Istr does not have a uniform value for all cross sections along the unit cell length, we plot both (i) the maximum Istr in the unit cell (Imax str ) and (ii)

The Ac term in eq 15, which is the area of liquid part in a unit cell cross section, is equal to δzW. Upon substituting the expressions for −dP/dz, cos θs, and Ac in eq 15, Istr generated in the region 0 ≤ z ≤ R of the unit cell could be obtained. In the unit cell region R ≤ z ≤ S/2, Istr is equal to 0 because of the complete absence of solid surfaces. Istr generated in different parts of the unit cell is given by the following expression Istr(z /R ) = ⎧ 2 ⎪ ϵoϵrsΔPsζ W if 0 ≤ z ≤ R ⎪ fs (S /R ) μ ⎪ ⎨ 1 − (z /R )2 , ⎪ 2 2 ⎪ (S /2R − 1 − (z /R ) ) ⎪ if R ≤ z ≤ S /2 ⎩ 0,

π 1−ϕ

(39)

The generated Ustr in the unit cell (0 ≤ z ≤ S/2) could similarly be obtained using eq 20 Ustr(z /R ) = ⎧ ϵ ϵ ΔP ζ g (z /R , S /R ) ⎪ o rs s s , if 0 ≤ z ≤ R fs (S /R ) ⎪ ⎪ μκ ⎨ ⎪ ϵoϵrsΔPsζ gs(z /R , S /R ) , if R ≤ z ≤ S /2 ⎪ fs (S /R ) ⎪ μκ ⎩ z/R=1

(40)

where gs(z/R, S/R) is a function of the nondimensional height z/R and the size ratio S/R. It has been provided in its complete form in the Appendix. Eqs 39 and 40 allow us to directly estimate the variation of generated Istr and Ustr inside of the unit cells. This variation is plotted for a particular value of the size ratio S/R = 3, as shown in Figure 5C,D. It could be identified from Figure 5C that the generated Istr is highest in the cross section where the separation between the fibers is minimum. The gradient of Ustr with respect to z is maximum for the same cross section. The total Ustr generated across the unit cell (0 ≤ z ≤ S) is twice the Ustr generated across its lower half (0 ≤ z ≤ S/2) and could be expressed as Ustr = 2

ϵoϵrsΔPsζ gs*(S /R ) fs (S /R ) μκ

Figure 6. Variation of the generated streaming current (maximum and average value) (A) and the generated streaming potential (B) with porosity in square fiber arrays for parallel and transverse flow directions.

(41)

where gs*(S/R) = gs(z/R, S/R)|(z/R)=1. I


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Langmuir the length-averaged Istr value (Iavg str ). It is observed from Figure 6A that Istr generated in the case of parallel fiber orientation is avg lower than Imax str but greater than Istr for the case of transverse avg fiber orientation. Although both Imax str and Istr vary almost linearly with the sample porosity just like in the case of parallel max fiber orientation, Iavg str is much lower in magnitude than Istr . 3.2.3.2. Effect on Streaming Potential. As described earlier, in the case of parallel fiber orientation, the generated Ustr is equal to the value given by the H−S equation (Ustr‑HS) and hence does not change with porosity. In the case of transverse fiber orientation, the generated Ustr varies significantly with the sample porosity as can be observed from Figure 6B. It can be further noted that the difference between Ustr generated in the two cases of parallel and transverse fiber orientation increases with the sample porosity. This indicates that in measurements involving porous substrates, the deviation of measured Ustr from Ustr‑HS because of the geometrical nonidealities would be low for samples with low porosity. 3.2.4. Effect of Sample Dimensions. In the case of transverse fiber orientation, if the number of sample unit cells along the direction of the applied pressure gradient is n and ΔP is the total pressure difference applied across the sample, the pressure drop across a single unit cell (ΔPs) equals ΔP/n. Similarly, if Utstr is the total streaming potential generated across the sample, the streaming potential generated across a single unit cell is Utstr/n. Upon substituting for Ustr and ΔPs in eqs 39 and 43 in terms of Utstr and ΔP, the dependence of the generated Itstr and Utstr on ΔP, n, and W could be identified. 3.2.4.1. Effect on Streaming Current. It is easily identifiable from eq 39 that Istr generated inside of the unit cells decreases linearly with the number of unit cells along the pressuregradient direction (n). However, increasing the width of the sample (W) leads to a proportional increase in Istr generated inside of the unit cells. Thus, at fixed sample porosity, the magnitude of the streaming current is sensitive to the ratio n/ W. This inference is analogous to the prediction of H−S equation that the generated Istr is proportional to the crosssectional area and inversely proportional to the sample length. 3.2.4.2. Effect on Streaming Potential. The relationship given by eq 43 shows that Ustr generated across the unit cells is only dependent on the pressure difference across the unit cells (ΔPs) and porosity (ϕ). Furthermore, when ΔPs and Ustr are substituted in terms of ΔP and Ustrt in eq 43, the number of unit cell factor (n) gets canceled from both sides. As a result, Utstr and ΔP are relatable via the same equation as Ustr and ΔP (eq 43). It could be thus established that at constant porosity the generated Utstr is independent of the physical dimensions of the porous sample. 3.3. Implications of Obtained Results in Practical Use. 3.3.1. Validity of the Presented Theory in Practical Applications. As stated in the Introduction, the theory presented in this work has the same prerequisite conditions as the H−S equation. Therefore, when considering the implications of this theory in practical cases, it is necessary to ensure that these prerequisite conditions are satisfied. The most important prerequisite condition is that the EDL thickness (δEDL) should be small in comparison to pore sizes in the substrate. When the pore sizes (a) are known, this condition could be satisfied by adjusting the bulk electrical conductivity

The other important prerequisite condition is that the surface conductivity effects should be negligible. If the absence of surface conductivity effects in a substrate has been shown independently, it is possible to directly apply our theoretical treatment. However, if it is not already known, it is important to ascertain whether the surface conductivity effects are present and whether they could be neglected at the conditions chosen for the experiments. For a substrate of known geometry, one can identify whether the surface conductivity effects are present via the measurements of effective conductivities of the electrolyte inside of the substrate. The measured effective conductivity value is compared with the value that is obtained theoretically for the substrate under the assumption that the surface conductivity effects are absent. When the measured effective conductivity is higher than the theoretically predicted value, the presence of surface conductivity effects is implied. In the case of real substrates with complex geometries such as fiber plugs, a precise theoretical estimation of the effective conductivity requires information about the geometric features such as plug porosity and fiber orientation.54 When dealing with fiber plugs, another commonly used approach is to compare the measured effective conductivities obtained at different packing densities with the bulk electrical conductivity.33 If the measured effective conductivity increases with packing density and surpasses the bulk electrical conductivity, the presence of surface conductivity is implied. In such cases, the bulk electrical conductivity could be increased until the effect of surface conductivity is minimized. Measuring the streaming potential at such a high bulk electrical conductivity level could be thus a way of eliminating the effect of surface conductivity on the final zeta potential measurement. 3.3.2. Differences between Istr and Ustr Approaches. The theoretical analysis carried out in this work demonstrates that the generation of both Istr and Ustr is intricately linked to sample geometry. Under a certain applied pressure difference, Istr generation is shown to be dependent both on the intrinsic geometrical features of the sample such as porosity and fiber orientation and its physical dimensions. The theoretical results, however, indicate that Ustr generation is dependent only on the intrinsic geometric features of the sample and is unaffected by the sample size. This inference implies that Ustr measurements would be reproducible even on setups with different measurement cells given that the samples used have the same intrinsic geometry. For instance, in the case of single fiber plugs, this could be done by maintaining the same packing density in different samples. This conclusion will, however, need to be experimentally verified. It is in contrast to Istr measurements where the measurement cell size plays a direct role and needs to be accounted for as well. Another practically important aspect that is brought to light is that the generated Istr does not have a single uniform value for the sample and could vary along the sample length from one cross section to another. However, in the experiments where Istr values are externally detected, a single value is obtained for the generated Istr.18 In such cases, the dependence of externally detected Istr on the real Istr values in different sample cross sections is not very well-understood. Therefore, in order to ensure the reliability of ζ determination via Istr measurements, there is a need for further theoretical work in order to resolve the problem. 3.3.3. Optimization of ζ Determination via Ustr Measurements. When using Ustr measurements to estimate ζ, an important question is whether the accuracy of ζ estimation

δ

≪ 1. Furthermore, if this condition is satisfied, such that EDL a the electroviscous effects could also be neglected.46,47 J


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Langmuir could be improved by optimizing the sample geometry. Such an optimization could be aimed at minimizing the deviation of generated Ustr from that given by the H−S equation (Ustr‑HS). This question is especially relevant when dealing with samples such as porous fiber plugs where the packing density can be varied to change the sample geometrical features such as fiber orientation and sample porosity. Using the obtained results, three different approaches are suggested for optimizing the fiber plug geometry for more precise ζ determination. The first approach is based on the results obtained for Ustr generation in square fiber arrays, as shown in Figure 6A. It could be remarked from Figure 6A that the difference in Ustr generation in the two contrasting cases of parallel and transverse fiber orientation is reduced at low sample porosity. Therefore, using highly dense fiber plugs with low porosity could be one possible way of minimizing the deviation of generated Ustr from Ustr‑HS. Although theoretically viable, this approach would not be easily implementable in practice because (i) compressing single fibers into such highly dense plugs could be very difficult and (ii) obtaining appreciable liquid flow rates through such dense plugs would require much higher pressure differences than attainable in presently used commercial devices.3 As most commercial devices use vacuum chambers, the maximum applied pressure difference is typically limited to 1 atm. In the second approach for the estimation of ζ, we utilize the general observation from eq 19 that a nonideal sample geometry leads to only negative deviations of generated Ustr from Ustr‑HS (detailed explanation in the Supporting Information section). Thus, if measurements are carried out for a range of fiber packing density values, the maximum obtained value would be closest to Ustr‑HS. The highest obtained Ustr values should hence be used for zeta potential determination when using the H−S equation. This approach could be more useful in experiments, as it provides us a way of working within the experimental limitations pointed out in the last paragraph. An experimental demonstration of this methodology using polyester fibers is included in the Supporting Information section. As the third approach, which might be useful in cases where an extremely precise determination of ζ is desired, we suggest the direct usage of eq 19 as the relationship between ζ and Ustr. This would of course be restricted to those cases for which the earlier described conditions required for the applicability of our theory are satisfied. Furthermore, this would require a full characterization of the substrate geometry. If the surfaces in sample cross sections are not identically oriented, it would be necessary to obtain accurate liquid flow velocity profiles and shear stresses inside of the substrate. Although obtaining analytical solutions might not be feasible for real substrates, numerical simulations could be implemented using representational 3D models55 to obtain a very precise relationship between ζ and Ustr. This could be a subject of future investigations.

a general expression is obtained to describe the Istr generated in different cross sections of the sample. Thereafter, assuming that the generated Ustr varies only along the pressure-gradient direction, an expression is obtained to describe the variation of the generated Ustr along the sample length. The general expressions describing Istr and Ustr generation form the basic theoretical framework of this work, which is applied to several substrate geometries. The considered geometries include a set of three nonuniform cross-sectional capillaries and a square array of fibers (model porous media) for both parallel and transverse fiber orientation. The obtained results are used to illustrate that the nonideal sample geometries induce deviations in the generated Istr and Ustr from the values given by the H−S equation. The implications of the obtained results for experimental zeta potential determination are discussed. For the practically important case of fiber plugs, strategies are suggested, which can be utilized to make more precise ζ measurements. The established analytical approach thus furthers our understanding of the physical phenomena underlying the electrokinetic zeta potential characterization of solid surfaces. In future works, the theoretical framework could be made more robust by considering the surface conductivity effects and possible variations of Ustr normal to the pressuregradient direction. Furthermore, the proposed theory could be applied to real porous substrates such as membranes, textiles, minerals, and so forth via simulations using representational 3D models.55

4. CONCLUSIONS This work presents an analytical treatment of pressure-driven generation of streaming current (Istr) and streaming potential (Ustr) in samples that are geometrically complex and for which the classical H−S equation is known to be inaccurate. The prerequisite conditions required for the validity of the new analytical treatment are the same as those required for the H−S equation. First, using linear velocity profiles inside of the EDLs,

■

■

APPENDIX The expressions for the functions fs(S/R) and gs(z/R, S/R) are as given below 1

∫

fs (S /R ) = 2

0

1

(

S 2R

1 − x2

−

3

)

⎛ R ⎞2 dx + 8⎜ ⎟ ⎝S⎠

⎛ R ⎞3 − 16⎜ ⎟ ⎝S⎠

(44) z/R

g s (z / R , S / R ) =

■

∫ 0

1 − x2

(

S 2R

−

1−x

2

3

)

dx (45)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b00344. Analysis and demonstration of the methodology for optimization of apparent zeta potential determination via Ustr measurements (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.K.). *E-mail: [email protected] (G.-R.). ORCID

Abhijeet Kumar: 0000-0002-7050-6328 K


Article

Langmuir Notes

(17) Luong, D. T.; Sprik, R. Streaming Potential and Electroosmosis Measurements to Characterize Porous Materials. Geophysics 2013, 2013, 1−8. (18) Luxbacher, T. The Zeta Guide: Principles of the Streaming Potential Technique; Anton Paar GmbH, 2014; pp 24−25. (19) Ciriacks, J. A.; Williams, D. G. A happel flow modelstreaming current relationship for determining the zeta potential of cylindrical fibers. J. Colloid Interface Sci. 1968, 26, 446−456. (20) Delgado, A. V.; González-Caballero, F.; Hunter, R. J.; Koopal, L. K.; Lyklema, J. Measurement and interpretation of electrokinetic phenomena. J. Colloid Interface Sci. 2007, 309, 194−224. (21) Lyklema, J. Electrokinetics after Smoluchowski. Colloids Surf., A 2003, 222, 5−14. (22) Christoforou, C. C.; Westermann-Clark, G. B.; Anderson, J. L. The streaming potential and inadequacies of the Helmholtz equation. J. Colloid Interface Sci. 1985, 106, 1−11. (23) Rice, C. L.; Whitehead, R. Electrokinetic Flow in a Narrow Cylindrical Capillary. J. Phys. Chem. 1965, 69, 4017−4024. (24) Adamczyk, Z.; Sadlej, K.; Wajnryb, E.; Nattich, M.; EkielJeżewska, M. L.; Bławzdziewicz, J. Streaming potential studies of colloid, polyelectrolyte and protein deposition. Adv. Colloid Interface Sci. 2010, 153, 1−29. (25) Oldham, I. B.; Young, F. J.; Osterle, J. F. Streaming potential in small capillaries. J. Colloid Sci. 1963, 18, 328−336. (26) Elton, G. A. H. Electroviscosity. I. The Flow of Liquids between Surfaces in Close Proximity. Proc. R. Soc. London, Ser. A 1948, 194, 259−274. (27) Overbeek, J. T. G. Electrokinetic Phenomena. In Colloid Science; Elsevier: Amsterdam, 1952; Vol. 1, pp 194−244. (28) Rodier, E.; Dodds, J. Streaming Current Measuring for Determining the Zeta Potential of Granular Particles. Part. Part. Syst. Charact. 1995, 12, 198−203. (29) Johnson, P. R. A Comparison of Streaming and Microelectrophoresis Methods for Obtaining the ζ Potential of Granular Porous Media Surfaces. J. Colloid Interface Sci. 1999, 209, 264−267. (30) Bukšek, H.; Luxbacher, T.; Petrinić, I. Zeta potential determination of polymeric materials using two differently designed measuring cells of an electrokinetic analyzer. Acta Chim. Slov. 2010, 57, 700−706. (31) Chun, M.-S.; Lee, S.-Y.; Yang, S.-M. Estimation of zeta potential by electrokinetic analysis of ionic fluid flows through a divergent microchannel. J. Colloid Interface Sci. 2003, 266, 120−126. (32) Ciriacks, J. A. An Investigation of the Streaming Current Method for Determining the Zeta Potential of Fibers. Doctor’s Dissertation, The Institute of Paper Chemistry, 1967; 24−25. (33) Jacobasch, H.-J.; Bauböck, G.; Schurz, J. Problems and results of zeta-potential measurements on fibers. Colloid Polym. Sci. 1985, 263, 3−24. (34) Szymczyk, A.; Fievet, P.; Foissy, A. Electrokinetic Characterization of Porous Plugs from Streaming Potential Coupled with Electrical Resistance Measurements. J. Colloid Interface Sci. 2002, 255, 323−331. (35) Mason, S. G.; Goring, D. A. I. Electrokinetic Properties of Cellulose Fibers: I. Stream Potential and Electro-osmosis. Can. J. Res., Sect. B 1950, 28, 307−322. (36) Biefer, G. J.; Mason, S. Electrokinetic streaming, viscous flow and electrical conduction in inter-fibre networks. The pore orientation factor. Trans. Faraday Soc. 1959, 55, 1239−1245. (37) Chang, M. Y.; Robertson, A. A. Zeta potential measurements of fibres. D-C. Streaming current method. Can. J. Chem. Eng. 1967, 45, 66−71. (38) Hasselbrink, E., Jr.; Hunter, M.; Even, W., Jr.; Irvin, J. Sandia Report-Microscale Zeta Potential Evaluations Using Streaming Current Measurements; Sandia National Laboratories, 2001; pp 15−23. (39) Glover, P. W. J.; Walker, E.; Jackson, M. D. Streaming-potential coefficient of reservoir rock: A theoretical model. Geophysics 2012, 77, D17−D43. (40) Yaroshchuk, A.; Luxbacher, T. Interpretation of Electrokinetic Measurements with Porous Films: Role of Electric Conductance and

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We are thankful to Prof. Dr. Ing. Peter Stephan (Technical Thermodynamics, TU Darmstadt), Vignesh Thammana Gurumurthy (Institute of Fluid Mechanics and Aerodynamics, TU Darmstadt), and Dr. Bastian Arlt (Anton Paar GmbH) for their invaluable suggestions and stimulating discussions. The study has been prepared in the framework of the Marie Curie Initial Training Network Complex Wetting Phenomena (CoWet), grant agreement no. 607861.

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