Langmuir 2008, 24, 10011-10018
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Streaming Potential Generated by a Long Viscous Drop in a Capillary J. D. Sherwood* Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, U.K. ReceiVed April 15, 2008. ReVised Manuscript ReceiVed June 11, 2008 The streaming potential generated by motion of a long drop of viscosity µd ) λµ in a uniform circular capillary filled with fluid of viscosity µ is investigated by means of a model previously used to study electrophoresis of a charged mercury drop in water. The capillary wall is at potential ζc relative to the bulk fluid within it, and the surface of the drop is at potential ζd. Potentials are assumed to be sufficiently small so that the charge cloud is described by the linearized Poisson-Boltzmann equation, and the Debye length characterizing the thickness of the charge cloud is assumed to be thin compared with the gap h0 between the drop and the capillary wall. Ions in the external fluid are not allowed to discharge at the surface of the drop, and the wall of the capillary has a nonzero surface conductivity σc. The drop is assumed to be sufficiently long so that end effects can be neglected. Recirculation of fluid within the drop gives rise to an enhanced streaming current when ζd is nonzero, leading to an anomalously high streaming potential. This vanishes as the drop viscosity becomes large. If V is the velocity of the drop and γ is the coefficient of interfacial tension between the two fluids, then the capillary number is Ca ) µV/γ, and the gap varies as h0∝Ca2/3. When Ca is small, the gap h0 is small and electrical conduction along the narrow gap is dominated by the surface conductivity σc of the capillary wall, which is constant. The electrical current convected by flowing fluid is proportional to Ca, as is the change in streaming potential caused by the presence of the drop. If σc ) 0, then the electrical conductance of the gap depends on its width h0 and on the bulk fluid conductivity σ and becomes small as h0 ≈ Ca2/3 f 0. The streaming potential required to cancel the O(Ca) convection current therefore varies as Ca1/3. If σc ) 0 and the drop is rigid (λ f ∞), then the change in streaming potential over and above that expected due to the change in pressure gradient is proportional to the difference in potentials ζc - ζd.
1. Introduction Streaming potentials are caused by the motion of fluid past a charged surface. The fluid convects ions in the charge cloud adjacent to the charged surface, and if no external path is available to complete the return path for electric current, a potential builds up such that the current due to electrical conduction through the fluid is equal and opposite to the electrical convection current. The process is well understood for single-phase flow, but much less is known about the effects of two-phase flow past the charged surface.1-7 Streaming potentials are of interest as they are generated when fluid is pumped through porous rock, because the rock surface is usually charged, but we study here the simpler problem of flow through a uniform cylindrical capillary. Previous theoretical work8 has considered how streaming potentials in a capillary (with walls at potential ζc) are modified by the presence of either (i) a rigid, tightly fitting sphere, (ii) a tightly fitting spherical inviscid drop, or (iii) a long, inviscid drop.9 The inviscid drops were uncharged, but the rigid sphere was allowed to carry a surface charge. Here we consider a capillary containing a long viscous drop and allow the surface of the drop to be charged. Surface charge on such a drop may be due to the presence of ionic surfactants, in which case it is necessary to consider the flow of surfactants on the free surface, together with any flow perturbation caused by Marangoni stresses arising from * E-mail:
[email protected]. (1) Morgan, F. D.; Williams, E. R.; Madden, T. R. J. Geophys. Res. 1989, B94, 12449. (2) Antraygues, P; Aubert, M. J. Geophys. Res. 1993, B98, 22273. (3) Sprunt, E. S.; Mercer, T. B.; Djabbarah, N. F. Geophys. 1994, 59, 707. (4) Guichet, X.; Jouniaux, L.; Pozzi, J.-P. J. Geophys. Res. 2003, B108, ECV2, 2141. (5) Revil, A.; Cerepi, A. Geophys. Res. Lett. 2004, 31, L11605. (6) Revil, A.; Schwaeger, H.; Cathles III, L. M.; Manhardt, P. D. J. Geophys. Res. 1999, B104, 20033. (7) Wurmstich, B.; Morgan, F. D. Geophysics 1994, 59, 46. (8) Sherwood, J. D. Phys. Fluids 2007, 19, 053101. (9) Bretherton, F. P. J. Fluid Mech. 1961, 10, 166.
Figure 1. Bretherton drop, traveling at velocity V relative to the walls of the capillary of internal radius R. The mean velocity of the fluid is U < V. The narrow gap between the drop and capillary wall is uniform over the central section of the drop, with gap width h0 ) R - d.
nonuniform surfactant concentrations.10 This is beyond the scope of the present work. Instead, we assume that the drop is highly conducting and at constant potential ζd. Ions in the exterior fluid are unable to penetrate the drop, and no reactions take place at the drop surface. This is the model of the charged mercury drop in water, investigated by Levine and O’Brien11 and by Ohshima et al.12 Bretherton9 studied the motion of a long inviscid bubble in a capillary with internal diameter 2R (Figure 1). The bubble is surrounded by liquid with viscosity µ, and moves at velocity V. Bretherton’s capillary number is
Ca )
µV , 1 γ
(1)
where γ is the coefficient of interfacial tension between the bubble and surrounding liquid. The gap width h0 over most of the length of the long bubble is uniform and was shown by Bretherton9 to be
h0 ) RRCa2⁄3
(2)
where (10) Baygents, J. C.; Saville, D. A. J. Chem. Soc., Faraday Trans. 1991, 87, 1883. (11) Levine, S.; O’Brien, R. N. J. Colloid Interface Sci. 1973, 43, 616. (12) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1643.
10.1021/la801175n CCC: $40.75 2008 American Chemical Society Published on Web 08/20/2008
10012 Langmuir, Vol. 24, No. 18, 2008
R ) 1.337
Sherwood
(3)
More recent work13,14 indicates that Bretherton’s analysis is valid for a drop of viscosity µd ) λµ > 0 if
λ)
µd , Ca-1⁄3 µ
(4)
and because Ca , 1 the viscosity ratio λ can be large before Bretherton’s result (eq 2) fails. Higher viscosity ratios, with λ.Ca-1/3 are discussed by Hodges et al.14 Although the numerical coefficient (eq 3) holds only for λ , Ca-1/3, the gap h0 scales as Ca2/3 for all λ, with a numerical coefficient that remains O(1). Hodges et al.14 found an intermediate range of viscosity ratios Ca-1/3 , λ , Ca-2/3, for which the film thickness over the central, cylindrical section of the drop is h0 ) 3.370RCa2/3. However, this result is attained only at exceedingly small capillary numbers. In the limit λ . Ca-2/3, they found R ) 2.123, which is somewhat larger than Bretherton’s prediction (eq 3) for a low viscosity drop. From now on, we shall assume that the gap width h0 and the radius d ) R - h0 of the cylindrical drop are known and are given by eqs 2 and 3. We shall assume throughout that the potentials ζc at the capillary wall and ζd at the surface of the drop are small. As a consequence, electric stresses are small compared to viscous stresses, and hence the fluid velocity around the drop can be approximated by that around an uncharged drop. The validity of this assumption will be discussed in section 3.4. In section 2, we determine the fluid velocity in the central cylindrical section of the drop, working mainly in the frame in which the drop is at rest. In section 3, the streaming currents and potentials generated by the flowing fluid are derived. Because the potentials are small, we can appeal to linearity and obtain separately the streaming potential generated by the potential ζc on the capillary wall and that generated by the potential ζd on the surface of the drop. Finally, in section 4 we consider a hypothetical example in order to estimate typical potentials.
2. Pressure Drop along a Viscous Bretherton Drop We assume that the drop, of volume 4πa3/3, can be approximated by a long cylindrical section of length L and radius d, together with endcaps, with L . d sufficiently large that we can neglect the endcaps and hence
L≈
4a3 3d2
(5)
The densities of the drop and surrounding fluid are equal so that buoyancy can be neglected and the problem is axisymmetric. We use cylindrical polar coordinates, with the capillary cylindrical axis as the z axis. We ignore the possibility that steady motion of a high-viscosity drop may be unstable to long-wavelength capillary waves, such as are seen on infinite fluid cylinders at low Reynolds numbers.16 In the frame fixed in the walls of the capillary, the velocity of the drop is V, and the mean volumetric flow rate in the capillary is Qc ) πR2U, where U is the mean velocity of the suspending fluid. Consider now the frame in which the drop is at rest and the capillary wall moves with velocity -V. In the central, uniform portion of the drop, the velocity field must be well behaved on the axis r ) 0, and the tangential stress must be continuous at (13) Park, C. W.; Homsy, G. M. J. Fluid Mech. 1984, 139, 291. (14) Hodges, S. R.; Jensen, O. E.; Rallison, J. M. J. Fluid Mech. 2004, 501, 279. (15) Chang, H.-C. In The MEMS Handbook; Gad-el-Hak, M., Ed.; CRC Press: Boca Raton FL, 2002; Chapter 22. (16) Preziosi, L.; Chen, K.; Joseph, D. D. J. Fluid Mech. 1989, 201, 323.
Figure 2. Schematic of the fluid velocity within the central cylindrical portion of the drop in the frame moving with the drop. Additional eddies in the endcaps have been omitted.
the interface r ) d, with the same pressure gradient dp/dz inside the drop and outside. The total mass flux within the drop must be zero in this frame moving with the drop, and therefore
u)
dp (d2 - 2r2) , r 0 (59a)
∼
2a3Ca-2⁄3 , Ca f 0, σ ˆc ) 0 3πR4σR
(59b)
4. Order of Magnitude Estimates (53)
which is the usual Poiseuille solution matched to the slip condition at the capillary wall. We now consider the convection of charge by the fluid velocity (eq 52) within the charge cloud. The convected electric current is
We now make order of magnitude estimates of the streaming potentials that might be encountered in an experiment. We consider a capillary of radius R ) 1 mm with an uncharged nonconducting drop of length L ) 10 mm. The total length of the capillary is Lc ) 1 m. The fluid velocity is U ) 1 mm s-1 (19) Sherwood, J. D. J. Fluid Mech. 1980, 101, 609.
Generated Streaming Potential
≈ V, and the external fluid viscosity is µ ) 10-3 Pa s. We assume an interfacial tension of γ ) 50 mN m-1 so that the capillary number is Ca ) 2 × 10-5. Consequently, the gap width is h0 ≈ 1 µm, and 1 - dˆ2 ≈ 2hˆ0 ) 0.002. When the gap width h0 is small, the streaming potential depends strongly on the surface conductivity σc. However, the surface conductivity of a capillary wall is determined from the difference between the measured electric current and that expected from the conductivity of the fluid within the capillary and is difficult to measure accurately.20 A wide range of values have been reported for different glasses in different electrolytes and at different pH values.21 We take results for porous Sto¨ber silica: 22,20 the combined effect of ionic mobility in the diffuse charge cloud and Stern layer gave a surface conductivity on the order of 10-9 S. The surface potential was typically ζ ≈ -3kT/e ≈ -75 mV. The Debye length κ-1 ≈ 9.7 nm implies an electrolyte (KCl) concentration of 0.1 mol dm-3 and hence a conductivity of σ ≈ 1.3 S m-1 at 25 °C. The nondimensional surface conductivity is therefore σˆ c ) σc/(Rσ) ≈ 10-6 and can be neglected compared to 1 - dˆ2. Similar values for ζc and σc for various glasses are given by Jednacˇak et al.23,24 We take the permittivity of the external fluid to be ≈ 800, where 0 is the permittivity of free space. In the absence of any drop, the coupling coefficient is ζc/(µσ) ≈ 4 × 10-8. The streaming potential across the full length Lc of the capillary would be 0.3 µV, and the total pressure drop would be 8 Pa. The change in streaming potential (eq 35) due to the drop would be 1.6 µV. A train of 50 such droplets, separated from each other by a distance of 10 mm ) 10R, would change the streaming potential by 80 µV. The change in pressure ∆p due to the presence of the drop is predicted to be zero by eq 19 because end effects have been neglected. However, some idea of the magnitude of capillary effects at the ends of the drop can be obtained from Bretherton’s result (eq 20) for an inviscid drop, for which p1 - p2 ≈ 0.4 Pa. These streaming potentials would of course be larger at higher flow rates, though because surface conductivity is already negligible at U ) 1 mm s-1 we see from eq 41 that the streaming potential will grow only as Ca1/3 if the flow rate is increased within the range for which Bretherton’s gap thickness hˆ0 remains small (eq 2). If surface conduction σc is negligible, then streaming potentials are inversely proportional to the fluid conductivity σ and will increase as the electrolyte concentration is reduced. The change in streaming potential would also be increased if the drop viscosity were larger. If λ ) 10, then the change in pressure (eq 19) caused by one drop is ∆p ≈ 0.8 Pa. The change in streaming potential would be ∆φ ≈ 16 µV. If the radius R of the tube is decreased while keeping the other dimensions (e.g., the drop length L) constant, then the streaming potential and pressure will increase as R-2, though the effects of surface conductivity will ultimately become more important. Note that all of the streaming potentials estimated above are small compared to the potential ζc at the wall of the capillary, and the drop is therefore sufficiently short for eq 36 to hold.
Langmuir, Vol. 24, No. 18, 2008 10017
between streaming pressure and pressure. The results presented here for an uncharged drop of viscosity λµ show that the increase in pressure and streaming potential caused by the presence of the drop increases with λ in such a way that the coefficient of proportionality becomes a function both of λ and the flow rate (because of the variation of the gap width h0). The streaming potential generated by a pressure gradient applied to a porous material saturated with two immiscible fluid phases will therefore differ from that generated by the same pressure gradient applied to single-phase flow. Network simulations to predict the effect of long viscous drops upon streaming potentials now become interesting because the streaming potential cannot be immediately related to previously published results25,26 for the pressure within the network. The zeta potential ζd at the drop interface can have a large effect upon the streaming potential. This is in line with other studies of electrokinetic effects involving charged drops,11,12 but the results must surely depend upon the details of the model, and that used here will not always be appropriate. If insoluble surfactants cover the surface, then recirculation within the drop may be suppressed, in which case it may be more appropriate to take the limit λ f ∞ in the results presented here. Unfortunately, little is known about ζ potentials at oil-water interfaces. (See Magual et al.27 for a recent review.) The conductivity of oil is typically low,28 but electrical double layers have been detected.29 Electrophoresis can be seen in oil-based suspensions,28 and streaming currents have been observed in oil-saturated rock.30 There is much work still to be done on this topic: experiments on well-characterized fluids in cylindrical capillaries would be a worthwhile first step to understanding streaming potentials generated in the more complicated geometry of porous rock. Acknowledgment. I thank the referees for helpful comments.
Appendix: Streaming Potential with All Surfaces at the Same Potential ζ Consider incompressible fluid flowing with velocity u through a channel. All bounding surfaces of the channel are rigid (though possibly moving relative to one another) and have potential ζ. The charge density within the vanishingly thin charge cloud adjacent to the surfaces is F. Surface conduction is assumed to be negligibly small. We seek to show that the streaming potential is
φ)
ζp σµ
(60)
Previous work showed that an uncharged inviscid drop (ζd ) 0; λ ) 0) has no effect upon the coefficient of proportionality
where and σ are the permittivity and conductivity of the fluid, p is the fluid pressure, and µ is the fluid viscosity. Within each thin charge cloud we use a local coordinate y normal to surface. The fluid velocity tangential to the surface (relative to the surface) is ut ) 0 on the surface itself and (∂nut)y away from it, where ∂n denotes a derivative normal to the surface. The fluid flow convects ions within the double layer, and an electric field E ) -∇φ will be established to maintain continuity of charge. The fluid is incompressible, so
(20) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: London, 1995; Vol. 2. (21) Overbeek, J. Th. G. In Colloid Science; Kruyt, H. R., Ed.; Elsevier: Amsterdam, 1952; Vol. 1, p 205. (22) Kijlstra, J.; van Leeuwen, H. P.; Lyklema, J. Langmuir 1993, 9, 1625. (23) Jednacˇak, J; Pravdic´, V.; Haller, W. J. Colloid Interface Sci. 1974, 49, 16. (24) Jednacˇak-Bisˇc´an, J; Mikac-Dadic´, V; Pravdic´, V.; Haller, W. J. Colloid Interface Sci. 1979, 70, 18.
(25) Stark, J.; Manga, M. Transp. Porous Media 2000, 40, 201. (26) Hunt, A. G.; Manga, M. Transp. Porous Media 2003, 52, 51. (27) Magual, A.; Horva´th-Szabo´, G.; Masliyah, J. H. Langmuir 2003, 21, 8649. (28) Morrison, I. D. Colloids Surf., A 1993, 71, 1. (29) Prieve, D. C.; Hoggard, J. D.; Fu, R.; Sides, P. J.; Bethea, R. Langmuir 2008, 24, 1120. (30) Alkafeef, S. F.; Algharaib, M. K.; Alajmi, A. F. J. Colloid Interface Sci. 2006, 298, 13.
5. Concluding Remarks 8
10018 Langmuir, Vol. 24, No. 18, 2008
∇u ) ∂nun + ∇t · ut ) 0
Sherwood
(61)
where the velocity un normal to the surface is zero on y ) 0 and ∇t denotes the surface divergence operator. Hence, on the surface,
∂2nun + ∂n∇t · ut ) 0
µ∇ u ) ∇ p 2
(63)
(68)
∞ yF dy ) -ζ ∫y)0
(69)
so
(62)
The Reynolds number is assumed to be small so that flow is governed by the Stokes equation
d2φ ) -F/ dy2
Consequently, using eqs 62, 65, and 69, we find that eq 67 leads to
The fluid is incompressible, so taking the divergence of eq 63 gives
µ-1ζ∂np ) σ∂nφ
∇2p ) 0
Now consider an imposed velocity u and pressure p, and look for the streaming potential φ. Within the bulk fluid there is no charge, and hence φ must satisfy the Laplace equation ∇2φ ) 0. If the ζ potential is uniform over the entire boundary, then the electric potential (eq 60) satisfies the boundary condition (eq 70) and satisfies ∇2φ ) 0 by eq 64. Hence, eq 60 gives the streaming potential generated by the flow. Thus, if potentials are small and charge clouds are thin, then the introduction of a rigid body into the flow channel makes no change to the proportionality (eq 60) between streaming potential and pressure if the ζ potential on the body is the same as that on the rest of the channel wall. This holds whether the body is held fixed (in which case it may be considered to be part of the boundary of the channel) or moving (e.g., a force-free particle of arbitrary shape).
(64)
The surface is rigid so that in the frame of reference moving with the surface un ) 0 and ∇tun ) 0. Hence, the component of the Stokes equation normal to the surface gives, at the surface,
∂2nun ) ∇2un ) µ-1∂np
(65)
The equation of continuity for electric charge is
∇ · (σE + Fu) ) 0
(66)
We integrate eq 66 across the thickness of the thin charge cloud adjacent to one of the boundaries to obtain ∞ ∇t · (∂nut)yF dy ) -σEn ) σ∂nφ ∫y)0
(67)
because the solid surface is an insulator and fluid velocity normal to the solid surface is zero (relative to the surface). But
LA801175N
(70)
