Streaming Potential near a Rotating Porous Disk - Langmuir (ACS

Aug 11, 2014 - Yaroshchuk and Luxbacher described the application of this geometry to a ... A porous sample is affixed to a nonporous cylindrical supp...
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Streaming Potential near a Rotating Porous Disk Dennis C. Prieve* and Paul J. Sides Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States ABSTRACT: Theory and experimental results for the streaming potential measured in the vicinity of a rotating porous disk-shaped sample are described. Rotation of the sample on its axis draws liquid into its face and casts it from the periphery. Advection within the sample engenders streaming current and streaming potential that are proportional to the zeta potential and the disk’s major dimensions. When Darcy’s law applies, the streaming potential is proportional to the square of the rotation at low rate but becomes invariant with rotation at high rate. The streaming potential is invariant with the sample’s permeability at low rate and is proportional to the inverse square of the permeability at high rate. These predictions were tested by determining the zeta potential and permeability of the loop side of Velcro, a sample otherwise difficult to characterize; reasonable values of −56 mV for zeta and 8.7 × 10−9 m2 for the permeability were obtained. This approach offers the ability to determine both the zeta potential and the permeability of materials having open structures. Compressing them into a porous plug is unnecessary. As part of the development of the theory, a convenient formula for a flow-weighted volume-averaged space-charge density of the porous medium, −εζ/k, was obtained, where ε is the permittivity, ζ is the zeta potential, and k is the Darcy permeability. The formula is correct when Smoluchowski’s equation and Darcy’s law are both valid. where I, ε, ζ, and μ are the measured streaming current, liquid permittivity, zeta potential, and dynamic viscosity, respectively. The parameter Ac is the effective cross-sectional area for flow and L is the length of the porous plug. The cell-constant Ac/L is well-defined for a single capillary, but requires independent measurement or a correlation for a porous body.2 The function f1(κa) corrects for the extension of the diffuse layer into the pore. The parameter κ is the reciprocal of the Debye length, and a is the effective pore radius. The value of f1(κa) is unity when κa ≫1 and takes a value between zero and unity otherwise. (See Delgado et al.2 for a careful discussion of this function that they render as [1-G(κa)]). One also can measure the streaming potential instead of the streaming current by replacing the external ammeter with a voltmeter. The streaming potential, ϕstr, is given by the following:2

1. INTRODUCTION Determination of the zeta potential of porous materials is a fundamental task in diverse technologies.1 Figure 1 shows a typical geometry used for this purpose.

ϕstr Δp

Figure 1. A common geometry for determining the zeta potential of porous material: Liquid flows through the sample and a streaming potential proportional to the pressure difference is measured by means of two Ag/AgCl electrodes.

εζ 1 f (κa) μ KL(1 + 2Du) 2

(2)

Here, the streaming potential is corrected both for the filling of the pores by the diffuse part of the double layer on the pore walls, f 2(κa), and for surface conduction of current back through the pores, where KL is the effective ionic conductivity of the liquid filling the pores, and Du, the Dukhin number, is the ratio of the surface conductivity to the product of the effective conductivity and the length L.2,3 Compression of materials like fibers and sponges, or packing of particles into a porous plug, or stacking of membranes are preparations often used for samples.1,2 In some cases, however, one might want to avoid squashing the sample into a porous

A pressure difference Δp drives liquid through the porous sample. Reversible electrodes, such as Ag/AgCl, admit streaming current or sense streaming potential. The theory by which one converts the measured current or potential to zeta potential is the same as that for a single capillary.2 The relationship between pressure difference and the streaming current is as follows:2 Istr εζ Ac = f (κ a ) Δp μ L 1 (1) © 2014 American Chemical Society

=

Received: June 5, 2014 Revised: August 8, 2014 Published: August 11, 2014 11197

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cylindrical support. Rotation of the sample draws liquid axially and expels the liquid radially. The relative motion of liquid and solid engenders a streaming current within the porous matrix and a streaming potential both within the sample and outside it. Sensors positioned near the sample and far from it detect the streaming potential outside the sample. This arrangement has several advantages. One needs only a single disk-shaped sample; no second sample is necessary. The sample size can vary from small radius (e.g., 2.5 mm) to large radius. At the small end of this range, the size of the sensor on the axis should be a fraction of the diameter of the sample. The limitations at large scale are the needs to rotate a large sample and to provide a volume of solution approximately defined by the cube of the disk’s diameter. The absence of pressurization means no seals are required. The sample can remain in its native state; compression to form a plug is unnecessary. The sample must be attached to a spindle in order to be rotated, and the sample and its attachment must withstand the shear. In exploratory experiments, we attached a 25 mm diameter sample of the loop side of Velcro to the cylindrical support, appearing in Figure 4, and measured streaming potentials when the sample was rotated in dilute salt solution (0.31 M KCl, pH = 5.6) at various speeds according to the arrangement shown in Figure 3.

plug or raveling a woven sample, and yet determine the zeta potential. Or, one might want to probe the response of the porous sample to flow parallel to its face where longitudinal flow such as through a membrane is impractical. Werner et al.4 described the use of a micro-slit geometry with variable gap, and Zimmerman et al.5 reviewed the use of tangential flow to investigate diffuse interfaces. Yaroshchuk and Luxbacher described the application of this geometry to a porous sample, as shown in Figure 2.6,7

Figure 2. A cell design for determining the zeta potential with transverse flow.

The samples, such as membranes, line the walls of a flow channel. Figure 2 is not to scale; a large ratio of length in the direction of flow to height ensures that the pressure gradient in the axial direction is independent of position normal to the direction of flow, to a good approximation. The equations, in which the measured streaming potential or streaming current is used to determine the zeta potential of the external surface and pore interior, resemble the relations appearing as eqs 1 and 2 above, as modified for the parallel geometry.6,7 The authors recommend basing the measurement on streaming current and extrapolating to zero channel width to distinguish the pore contribution from the external surface contribution. The measurement requires variation of the channel width to provide data for this extrapolation. Rotating a disk-shaped sample on its axis might be an attractive alternative method for determining the zeta potential of porous solids with an open structure. The geometry appears in Figure 3 showing a porous sample attached to a nonporous

Figure 4. A disk-shaped sample of the loop side of Industrial Strength Velcro was attached to a polyvinyl chloride cylindrical support.

The apparatus and method were otherwise the same as those used for planar samples in previous work.8,9 The results of this test appear as data in Figure 5. The fibrous sample generated much larger streaming potentials than expected for a planar sample,8 having the same radius (lower line marked “planar”) and zeta potential (−56 mV). At 360 rpm, one expects about −15 μV of streaming potential according to the theory of a planar disk, but we measured approximately −300 μV, a ratio of 30. Furthermore, the signal depended on rotation rate raised to a power larger than the 3/2 characteristic of a planar disk.8 The substantial signal obtained in this elementary test indicated that rotating a disk on its axis and measuring the streaming potential in its vicinity is a promising approach to determining the zeta potential of a porous sample; however, one must establish the relationship among streaming potential, rotation rate, zeta potential, and physical properties of the sample. Here, we present theory that relates the geometry of a porous disk and its zeta potential to the streaming potential measured during rotation of the sample on its axis. The theory has two components. First, the flow generates streaming current throughout its volume, as in the typical porous plug approach. This large volumetrically-generated streaming current returns, however, through the fixed resistance of the surrounding liquid;

Figure 3. A porous sample is affixed to a nonporous cylindrical support, immersed in the test solution, and rotated on its axis. Sensors, one nearby on the axis of the sample and one afar, detect the streaming potential. 11198

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Figure 6. Geometry of the rotating disk for this problem. The radius of the disk is R; the thickness is L, and the disk rotates at Ω radians per second. The outer plane of the disk facing the free solution is at z = L.

The porous rotating disk draws the fluid into its face and casts it radially outward. Joseph10 obtained the velocity in a “thick” rotating porous disk by modifying Darcy’s law to include centrifugal forces, as well as the apparent Coriolis force resulting from use of a rotating reference frame.

Figure 5. Streaming potential data (circles) acquired with a 25 mm disk made from the loop side of Velcro and affixed to a cylindrical sample support in 0.31 M KCl at pH 5.6 and 298 K. The 95% confidence limits are within the size of the symbol. The lowest line is expected for a planar disk having the same diameter and a zeta potential equal to −56 mV. The three upper lines are calculated from theory presented herein for the same zeta and the permeability values shown. Note the sensitivity of the theoretical lines to the permeability.

k v = − (∇ p − F ) μ

(4)

where −F = 2ρΩ × v + ρΩ × (Ω × r), ρ is the fluid density, Ω ≡ ωez is the angular velocity of rotation, r =rer + zez is the position vector and v = vrer + vθeθ + vzez is the volumetric flow rate of the fluid, per unit cross-sectional area of the porous medium, in a reference frame which is rotating with solid. Constraining the disk to be “thick” and use of Darcy’s law constitute neglect of the boundary layer at the interface between the support and sample. Later, we estimate the ratio of the boundary layer thickness to the thickness of the sample when an equation for the radial velocity is available. Combining eq 4 with incompressibility, ∇·v = 0, one obtains four scalar equations in the three unknown components of v plus p. Solving these four equations yields the following:8

increased current at constant ohmic resistance requires increased streaming potential to complete the electrokinetic circuit. Second, the porous body couples the rotation of the sample to movement of the liquid more efficiently than a planar disk, which produces a stronger dependence on rotation rate. The theory offers a testable prediction. The streaming potential depends on the square of the rotation rate at low rates (stronger than the 3/2 power characteristic of the rotating planar disk). The theory thereby allows determination of the zeta potential and permeability from a single plot involving streaming potential and rotation rate. The lines proximate to the data points in Figure 5, accounting for the thickness and permeability of the sample and calculated from the theory developed herein, demonstrate the ability of the theory to express the physics of the experiment.

vr =

NRe 1+

vθ = −

2. THEORY Fluid Velocity Profile Inside a Rotating Porous Disk. The geometry of the flow appears in Figure 6. A porous cylindrical body of small aspect ratio (radius R ≫ thickness L) is attached to a nonporous spindle, immersed in a liquid to a depth equal to its thickness, and rotated about its axis at angular rate Ω. The axis of the disk is colinear with the cylindrical coordinate z. The disk is attached to the rotating spindle at z = 0. The face of the disk in contact with free liquid is at z = L. Pressure-driven flow through porous media often obeys Darcy’s law, k v = − ∇p μ (3)

vz = −

2 NRe

rω 2

2 NRe

1+

2 NRe

NRe 2 1 + NRe

p = po +

rω 2

(5a)

(5b)

zω (5c)

ρω 2 (z 2 − L2) 2 1 + NRe

(5d)

where NRe ≡ 2kω / v and po is the pressure just outside the porous medium. The dimensionless parameter NRe corresponds to a Reynolds number. Equation 5 reveals various dependences and meets both absolute and conditional expectations. Reminiscent of the planar rotating disk, the radial and angular velocities in eqs 5a and 5b are proportional to r, and the axial velocity in eq 5c and pressure in eq 5d are independent of r. Equation 5a predicts that the radial velocity is proportional to the square of the rotation rate. The axial velocity in eq 5c vanishes at z = 0, where the porous medium contacts the impermeable solid support, as it should. No-slip at this same interface requires that vr = vθ = 0

where v is the local volume-averaged fluid velocity, k is the Darcy permeability having units of m2, μ is the dynamic fluid viscosity, and p is the local volume-averaged pressure. The details of the pore structure are volume-averaged into the coefficient k, a purely geometric factor. 11199

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solid; thus ρapp takes a sign opposite of zeta. The scaling with k can also be rationalized: for a given porosity, k (having units of length squared) scales with the square of the characteristic pore radius. Small pores (hence large k−1) have a large surface area per unit volume, which leads to large ρapp. Equation 10 is key; its product with the radial velocity in eq 5a yields an expression for the radial streaming current density flowing in the porous disk. It is not restricted to small values of zeta. Equation 10 exactly and conveniently translates flow in a porous medium to current density. Streaming Potential in the Bulk Solution. While eq 10 was derived for pressure-driven flow through porous media, we assume the same ρapp applies also to flows arising from rotation. The radial streaming current in the porous plug can be calculated by integrating over the entire porous medium having thickness L,

at z = 0. Equation 5 does not satisfy these conditions, however, because the radial velocity in a thick sample was taken as independent of z. Joseph8 obtained eq 5 under the assumption that the thickness L of the porous media substantially exceeds the thickness of a boundary layer extending from this surface into the bulk of the sample. We address this constraint later in this contribution. Finally, for a rotating porous disk, we expect the trace amount of fluid in a slightly porous solid (as k tends to zero) to undergo solid-body rotation, i.e., no motion of the fluid relative to the rotating solid. At NRe = 0, eq 5 yields vr = vθ = vz = 0 for all r and z which indeed correspond to solid-body rotation in the rotating reference frame. Effective Space Charge Density. Predicting the streaming potential outside a rotating porous disk, once the flow profile (eq 5) is available, begins by evaluating the radial advective current inside the porous medium, which in turn requires the volume-average space-charge density. Consider pressure driven flow within a single pore. A streaming potential develops along the direction of flow in order to close the electrokinetic circuit. The streaming potential is proportional to the volume-average velocity in the same way that the pressure-drop is proportional to the volume-average velocity. Eliminating the velocity between these two expressions leads to the following: ∇ϕstr =

εζ ∇p μKL

(a)

Ir (r ) =

(6)

iz|z = L = −

−KL

(7)

−εζ k

∂ϕ ∂z

ϕstr =

1+

2 NRe

εζωr L 2k

(11)

NRe εζω 1 d (rIr ) = L 2 r dr k 1 + NRe

(12)

= z=L

NRe 1+

2 NRe

εζω L k

(13)

NRe 2 1 + NRe

εζLω R k 2KL

(14)

(8)

The collection of constants, dimensions, rotation rate, and physical properties in eq 14 gives an estimate of the streaming potential that would be measured on the axis of the sample and immediately adjacent to it, in solution. Correction for Sample Conductance. Sides and Prieve12 recently derived a correction for surface conductivity in the rotating disk geometry. The correction was nil when the sample is a planar disk. As mentioned above, the rotating porous disk is analogous to the rotating planar disk for R ≫ L because the current density entering the disk at z = L is independent of radial position, as shown in eq 12. The sample thickness L, however, is macroscopic; therefore, the current leaking back through the porous sample, instead of returning through the bulk solution, cannot be ignored. The correction is obtained by analogy with the correction previously derived.10 Including this

(9)

Substituting eqs 7 and 8 into eq 9, then eliminating ∇ϕs using eq 6, and eliminating v using eq 3, ρapp =

NRe

As in the theory for the planar disk, eq 13 specifies a constant value of the electric f ield over the face of the porous disk. The solution for the streaming potential profile therefore mathematically resembles the result for the planar disk;8 hence, we multiply the current density given in eq 13 by the disk radius and divide by twice the bulk electrolyte conductivity to obtain an expression for the streaming potential on the axis at the disk’s surface relative to the potential at any point far from the sample.

The induced streaming potential given by 6 corresponds to locally zero net current or the following:

iadv + icond = 0

(b)

ρapp vr dz = −

Since the fluid just outside the porous disk is electrically neutral, the only transport mechanism available is conduction; hence the following boundary condition on Laplace’s equation holds for the fluid outside the porous disk, eq 13.

The charge density ρapp, having units of charge per unit volume, represents the amount of mobile charge (in diffuse clouds) per unit volume of porous media. We now derive a simple formula to evaluate it. The conduction of current within each pore resulting from the streaming potential is given by Ohm’s law: icond = −KL∇ϕstr

L

where we have substituted eq 10 and the radial velocity from eq 5a. The streaming current is proportional to r, which requires that charge enters the disk axially at a rate given by the surface divergence of the radial current evaluated at the disk.

often called the Helmholtz−Smoluchowski equation. This equation is usually derived for a straight, uniform, circular capillary (representing a model porous medium). Its generalization to pressure-driven flow through porous media having a more realistic geometry is attributed to Overbeek,11 who lists three requirements: (1) the flow must be laminar inside the pores; (2) the pore radius must be large compared to the Debye length; and (3) ohmic conduction in the bulk liquid inside the pores must dominate surface conductivity or the conductivity of the solid. The effective space charge density of a porous medium is that quantity ρapp, which when multiplied by the local velocity, yields the advected current density: iadv = ρapp v

∫0

(10)

Equation 10 compactly expresses the averaging of the effect of surface charge (ζ) over a complex geometry (k). The minus sign can be rationalized. The zeta potential takes the sign of charges fixed to the solid. The flow advects the oppositely charged mobile counterions in the diffuse cloud outside the 11200

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containing cylindrical fibers, concluded that the threshold Reynolds number (based on the fiber diameter), above which inertia within the pores was non-negligible, was 13. The Reynolds number for 95 mm fibers at velocity vro at 400 rpm was 12. We therefore expect the theory described herein to hold best for the Velcro sample when the rotation rate is between 100 and 400 rpm. The boundary layer is too thick below 100 rpm, and the flow within the pores is not purely viscous above 400 rpm. Equation 16 also does not contain a correction for overlapping diffuse layers because it was derived under the assumption of thin double layers. In the case of Velcro, the average distance between fibers is thousands of times greater than the approximately 18 nm Debye length for 0.31 M KCl at pH 5.6, which satisfies the thinness criterion. Correction factors such as the f1 and f 2 of eqs 1 and 2 could be adduced for this case if required. Reduction of Data of the Velcro Tests. We analyze the data shown in Figure 5 for the loop side of Velcro. Values for the dimensions and properties of Velcro (Industrial Strength Sticky-Back polyester tape) appear in Table 1. The solution

modification, the appropriate expression for the streaming potential at the outer face of the disk on its axis is as follows: ϕstr ≅

εζ ωRL NRe 1 2 k 2 1 + NRe KLg (Du p)

(15)

Dup is the porous body analogy to the Dukhin number, KpL/ KR and g(Dup) =1 + 1.52Dup + 0.135Dup2.12 Here, Kp is the effective conductivity of the porous medium, having units of bulk conductivity. Multiplying Kp by the thickness L yields a conductance, analogous to surface conductivity, that allows streaming current to leak back through the porous sample instead of flowing back outside the sample. The technical literature offers many geometry-dependent correlations for the conductivity ratio Kp/KL. For example, the ratio depends on porosity in all cases but is different for close-packed spheres, cylinders, and solids with percolating pores. Equation 15 holds when the sample is sufficiently thick that boundary layer effects are small and κa is sufficiently large that the diffuse clouds of unbalanced charge inside the pores do not overlap one another. A correction for small κa, would be required in that case, as provided by the function f 2(κa) in eq 2.

Table 1. Parameters of Velcro and Values Used to Calculate the Lines of Figure 5

3. RESULTS AND DISCUSSION Predictions of the Theory. Consider a form of eq 15 in which the definition of the Reynolds number has been reintroduced to clarify the principal dependences on system properties and operating variables. ϕstr =

εLR ζ ω2 2 2 νKL g(Du p) 1 + 4ω 2k ν

R L KL fiber diameter ζ/g(Dup) ζ k θ Kp/KL Dup g(Dup)

(16)

This expression, remarkably simple given the complexity of the flow and electrokinetic phenomena, exhibits both expected and mildly paradoxical features. The streaming potential depends directly on the zeta potential, the sample radius, and the sample thickness, as expected. The dependence of the streaming potential on the rotation rate and permeability, however, is more complicated than for flat surfaces. The streaming potential is proportional to ω2 at low rotation rate but becomes invariant with rotation rate when 4ω2k2/ν2 ≫ 1. The streaming potential is independent of the sample’s permeability at low rotation rate if Dup ≪ 1; therefore, one can determine the zeta potential of a porous material without knowing its permeability by measuring the streaming potential at low rotation rates for thin samples. This invariance with structure is analogous to pressure-driven flow through a porous plug in the sense that one need not know the permeability to determine the zeta potential, only the pressure difference. Limits of Applicability. Equation 16 should predict the streaming potential when Darcy’s law holds and when the pore size is much larger than the Debye length. We examine two main assumptions. First, eq 16 does not admit a boundary layer at the interface between the solid support and the porous body. We estimate the ratio of the maximum boundary layer thickness to the sample thickness as(vR / vroL2)1/2, where vro is the radial velocity (eq 5a) evaluated at the periphery of the disk, which takes the values 0.3 at 100 rpm and 0.1 at 400 rpm in the present case; hence, the boundary layer thickness is well inside the porous body over most of its volume for rotation rates below 400 rpm. Second, eq 16 requires that the flow within the pores be entirely viscous. Coulaud et al.,13 numerically investigating cross-flow over a slice through a domain

12.7 mm 3.5 mm 4600 μS/m 95 μm −39 mV −56 mV 8.7 × 10−9 0.93 0.89 0.24 1.38

measured measured measured measured slope, eq 20 (corrected using Dup) intercept, eq 20 from k and eq 19 from θ and eq 12 definition definition

used in the test was 0.31 M KCl at pH 5.6. The remaining parameters to be determined from streaming potential experiments were the zeta potential and permeability of the sample. The two unknown quantities, the uncorrected zeta potential ζ/g(Dup) and the permeability k, were determined by fitting eq 16 to the data for rotation rates between 100 and 400 rpm. Taking the reciprocal of eq 16, one obtains the following:

Plotting the reciprocal of the streaming potential against the reciprocal of the squared rotation rate should yield a straight line with a slope depending only on the properties of the free liquid, the dimensions and structure of the sample, and the zeta potential of the sample. Figure 7 shows the data of Figure 5 rendered as suggested by eq 17. The data were linear with the reciprocal square of the rotation rate, as predicted by the theory. The slope was 3.59 × 106 V−1s−2 and the intercept was 1090 V−1. One can calculate the uncorrected zeta potential ζ/g(Dup) from the slope according to eq 17 if the thickness and radius of the sample are known and the conductivity of the solution is measured; this calculation yielded -40 mV. With this quantity available, the only unknown in the intercept was the permeability; calculation yielded k = 8.7 × 10−9 m2. 11201

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constant in Figure 5. The higher and the lower of the three lines near the data of Figure 5 show the sensitivity of the fit to the value of the permeability. Given that permeability data often are presented on a logarithmic axis, we conclude that the streaming potential is sensitive to the value of the permeability. One can estimate simply the range of achievable Reynolds numbers governing the range of permeability that can be probed by this technique. Inspection of eq 16 indicates that a practical lower limit of NRe2 is approximately 0.1 for the permeability to affect the data, which means that 4k2ω2/v2 ≥ 0.1. A common practical range of rotation speed for commercial rotators is approximately 10k rpm. The minimum value of k in aqueous solution given these constraints is therefore greater than 10−10 m2; permeabilities much below this value would not affect the detected streaming potential. Even though the permeability cannot be determined under these circumstances, the method can be used to probe the zeta potential of the porous body as long as the assumptions underlying the theory are valid. The theory expressed in eq 16 converted the experimental data for the loop side of Velcro to reasonable values for the zeta potential and porosity, but details remain to be explored. Numerical solution of the flow equations and experiments on rigid porous media are required to quantify the effect of the momentum diffusion and inertia within the pores. For example, streaming potential measurements at rotation rates beyond 400 rpm appear in Figure 8 along with the predictions of the theory

Figure 7. A plot of the streaming potential data of Figure 5 as suggested by eq 17. The plot is linear as the theory suggests in this range of rotation rates. The slope and intercept can be combined to determine the zeta potential and permeability.

The correction for conductance of the sample, g(Dup), requires knowledge of the porosity. The technical literature provides a relationship among permeability, fiber radius b, and porosity q, eq 18.14 ⎡1 k = b2 ⎢ ⎣2

⎤2 π − 1⎥ [0.71407e−0.51854(1 − θ)] ⎦ 1−θ

(18)

The fibers of Velcro had a diameter of 95 microns. Having the permeability k, we calculated the porosity from eq 18 and found 0.931. The literature offers a relationship for the dependence of the conductivity of a random network of fibers on its porosity,15 Kp = θ 5/3 (19) K Having the porosity, we calculated Kp/K to be 0.89 from eq 19 and the correction factor g(Dup) to be 1.38 from its definition. Once the correction factor was applied, we obtained a zeta potential of −56 mV. Thus, the sample’s conductance reduced the streaming potential by approximately 38%. Meaning of the Results. Table 1 contains the parameters used in conjunction with eq 16 to obtain the line labeled “8.7” in Figure 5. The agreement between this line and the data of Figure 5 is good where the theory is applicable between 100 and 400 rpm. The theory clearly accounts for the large streaming potentials obtained for porous samples. The zeta potential (−56 mV) obtained is not far from the values for polyester found in the literature, −35 mV to −48 mV.16 The 8.7 × 10−9 m2 found for the permeability k is higher than the range 0.83−1.2 × 10−9 m2 found for “hair felt”, a fibrous material made from cattle hair and used for insulation,17 and slightly above the 7.4−7.9 found for fiberglass insulation.18 The value 0.931 for the porosity is not unreasonable. The loop side of Velcro is an open structure by design. For comparison, the hair of mammals has a porosity of 0.95−0.99 and fiberglass has a porosity of 0.88−0.93.17 Thus, the permeability, porosity, and zeta potential obtained by fitting the theory to the data are physically realistic. Using two parameters to fit an equation to data begs the question of sensitivity. The value of the permeability was lowered to 5 × 10−9 m2 and raised to 10−8 m2, all else kept

Figure 8. Data and lines representing the theory for rotation rates beyond the range of applicability. Clearly, Darcy’s law is inadequate beyond 400 rpm.

herein. The theory based solely on Darcy’s law is inadequate when the rotation rate is high. As mentioned previously, inertial flow within the pores becomes important above 400 rpm in this case; this change of flow regime enhances the shear at the surface of the fibers and hence produces a larger than expected streaming current, which accounts for the strong increase of streaming potential in Figure 8 when invariance is predicted by eq 16. The linear dependence (eq 16) of the streaming potential on thickness L and radius R distinguishes the disk geometry from the porous plug geometry; neither the thickness nor the radius of the sample appears in eq 2. Both dimensions affect the streaming potential in the present case. A larger radius increases 11202

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Author Contributions

the path length for ohmic current returning through the bulk electrolyte; hence, a larger streaming potential is necessary if the radius is increased at constant thickness. Increasing the thickness of the sample at constant radius produces more total streaming current that must travel through the electrolyte’s resistance outside the porous body through the free liquid. Passing this larger current (due to a thicker sample) through a fixed resistance (holding radius constant) requires a larger streaming potential. Adding the effect of the thickness of the sample to the quadratic dependence on rotation rate accounts for the large values of the streaming potentials obtained when rotating a porous disk. Physical Meaning of ρapp. . We evaluated ρapp from (10) in order to compute the current advected radially through the porous medium. We suggest that the formula is applicable generally and therefore deserves some direct discussion even though eq (10) is not the focus of this contribution per se. Like the local space charge density ρ appearing in Poisson’s equation, the volume-average space charge density ρapp has units of charge per unit volume. However ρapp is not the local ρ which varies strongly with position inside any single charged pore. Like the fluid velocity v or vr, the fluid pressure p, the electric potential ϕstr and the electric current densities icond and iadv appearing in (3) through (11), ρapp is a volume-averaged quantity. In particular, ρapp is a weighted average of the mobile charge density ρ found in the diffuse cloud; it does not include contributions from immobile charges such as those fixed to the solid surface or those inside an immobile Stern layer. The weighting factor is the local (not volume averaged) fluid velocity, which for thin double layers increases linearly from zero at the shear plane. In other words, ρapp is not simply the total mobile charge divided by the total volume of porous media. It is an apparent charge density that when multiplied by the volume-averaged fluid velocity gives the volume-averaged current density.

The authors contributed equally to this manuscript and have approved the final version. Notes

The authors declare the following competing financial interest(s): P.J.S. is the founder and CEO of ZetaMetrix Inc., an S Corporation that makes and sells the ZetaSpin, a device using theory referred to in this contribution.

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ACKNOWLEDGMENTS Work performed under a grant from Intel Corporation, and encouragement by Mansour Moinpur Alexander Tregub, Don Hooper, Mark Buehler, and Ashwani Rawat, provided the impetus for this contribution.

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4. CONCLUSIONS Rotating a porous disk on its axis generates a substantial streaming potential in its vicinity. The theory of the fluid mechanics and electrokinetics indicates that the streaming potential is essentially proportional to the square of the rotation rate at low rate and invariant with rotation rate at high rates. Furthermore, a change from low rotation rate to high rotation rate marks a transition from invariance of the streaming potential with permeability to dependence on the inverse square of the permeability. Fitting the theory to a preliminary test on a fibrous sample, the loop side of Velcro, showed that reasonable values of the zeta potential and permeability can be obtained from a set of measurements at different rotation rates. The range of permeabilities that can be detected is greater than approximately 10−10 m2. In principle, there is no lower limit on the range of isotropic permeability for determination of the zeta potential. This technique allows determination of the zeta potential of porous structures without the need to compress or otherwise pack the sample. Transition from viscous flow to inertial flow strongly increases the shear and leads to streaming potentials higher than predicted.

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REFERENCES

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dx.doi.org/10.1021/la5022092 | Langmuir 2014, 30, 11197−11203