Langmuir 2000, 16, 1593-1601
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Measurements of Streaming Potential for Mica Covered by Colloid Particles Maria Zembala and Zbigniew Adamczyk* Institute of Catalysis and Surface Chemistry Polish Academy of Science, Niezapominajek, 30-239 Cracow, Poland Received May 18, 1999. In Final Form: October 25, 1999 The zeta potentials of natural mica, bare and covered by positively charged latex particles of micrometer size range, were determined by the streaming potential method. Measurements were carried out using the parallel-plate channel formed by clamping together two mica sheets separated by a Teflon spacer. The dependence of streaming potential on surface coverage (Θ) of adsorbed particles was determined for various ionic strengths and particle sizes. The coverage was determined directly by optical and electron microscope counting procedures. It was found that the negative streaming potential for bare mica Es was significantly increased by the presence of adsorbed particles. The dependence of the reduced streaming potential Esp/Es (where Esp is the zeta potential in the presence of particles) on Θ exhibited an universal behavior being independent of particle size and ionic strength. These experimental data were interpreted in terms of a theoretical model postulating that the streaming potential change was due to flow damping over the interface and by charge transport from the double-layer surrounding adsorbed particles. In contrast with previous approaches no assumption of the slip (shear) plane shift upon particle adsorption was made. By exploiting the experimental results it was suggested that colloid and bioparticle adsorption kinetics can be studied in situ using the streaming potential method.
Introduction Adsorption of colloid and bioparticles at solid/liquid interfaces is of large significance in many practical processes, e.g., various filtration procedures, papermaking, electrophoresis, chromatography, etc. Learning about kinetics and mechanisms of particle adsorption phenomena is also relevant for polymer and colloid science, biophysics, and medicine, enabling one to control protein and cell separation, enzyme immobilization, thrombosis, biofouling of membranes and artificial organs, etc. An obstacle in studying quantitatively adsorption processes of colloids and bioparticles at solid surfaces is the lack of direct experimental methods of surface coverage determination working in situ. Usually indirect methods are applied like ellipsometry,1,2 reflectometry,3-5 total internal reflection fluorescence,6,7 Fourier transform infrared spectroscopy,8,9 concentration depletion method,10,11 or the radiotracer techniques using labeled particles.12-14 The disadvantage of the above methods is (1) Golander, C. G.; Kiss, E. J. Colloid Interface Sci. 1988, 121, 240. (2) Wahlgren, M.; Arnebrant, Th.; Lunstrom, I. J. Colloid Interface Sci. 1995, 175, 506. (3) Schaaf, P.; Dejardin, Ph.; Schmitt, A. Langmuir 1987, 3, 1131. (4) Elgersma, A. V.; Zsom, R. L. J.; Lyklema, J.; Norde, W. Colloids Surf. 1992, 65, 17. (5) Bohmer, M. R.; van der Zeew, E. A.; Cooper, G. J. M. J. Colloid Interface Sci. 1998, 197, 242. (6) Hlady, V.; Andrade, J. D. Colloids Surf. 1988, 32, 359. (7) Fraaije, J. G. E. M.; Kleijn, M.; van der Graaf, M.; Dijt, J. C. Biophys. J. 1990, 57, 965. (8) Castillo, E. J.; Koenig, J. L.; Anderson, J. M. Biomaterials 1986, 7, 89. (9) Chittur, K. K.; Flink, D. J.; Leininger R. I.; Hudson, T. B. J. Colloid Interface Sci. 1986, 111, 419. (10) Vincent, B.; Young, C. A.; Tadros, T. F. Faraday Discuss. Chem. Soc. 1978, 65, 296. (11) Mura-Galelli, M. J.; Voegel, J. C.; Behr, S.; Bres, E. F.; Schaaf, P. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 5557. (12) Chan, B. M. C.; Brash, J. L. J. Colloid Interface Sci. 1981, 8, 2217. (13) van Dulm, P.; Norde, W. J. Colloid Interface Sci. 1983, 91, 248. (14) Boumaza, F.; Dejardin, Ph.; Yan, F.; Baudin, F.; Holl, Y. Biophys. Chem. 1992, 42, 87.
that they become sensitive enough for higher coverage only, preferably close to the saturation coverage. Recently, atomic force microscopy (AFM) tapping mode was applied for examining adsorbed particle density.15,16 However, this method gives satisfactory results when working ex situ only, after drying the sample. This may produce a considerable distortion of a particle monolayer. Another class of indirect methods, more sensitive for low coverage region, is based on measurements of streaming potential changes induced by particle adsorption. Many experiments of this type involving protein solution were performed in the parallel-plate channel arrangement 4,17,18 or using a single capillary,19 which is considerably more convenient from an experimental viewpoint. Although the streaming potential method performs well for low coverage where other methods fail, it can only be used as a relative method due to the lack of well-established theory connecting streaming potential changes with particle coverage. As a consequence, adsorption kinetics can only be studied quantitatively by the streaming potential method if independent measurements of surface coverage are carried out. This may decrease accuracy of the data obtained because the surface coverage and streaming potential measurements are done under different conditions. Recently Hayes20 used this procedure (based on scanning electron microscope measurements of particle coverage) for determining the zeta potential of an aminopropylsilane-modified glass surface covered with 90 nm diameter silica particles. The results were interpreted in terms of empirical models postulating a gradual shift of the slip (shear) plane with particle coverage. (15) Johnson, C. A.; Lenhoff, A. M. J. Colloid Interface Sci. 1996, 179, 587. (16) Semmler, M.; Mann, E. K.; Ricka, J.; Boerkovec, M. Langmuir 1998, 15, 5127. (17) Norde, W.; Rouwendal, E. J. Colloid Interface Sci. 1990, 139, 169. (18) Scales, P. J.; Grieser, F.; Healy, Th. W.; White, L. R.; Chan, D. Y. C. Langmuir 1992, 8, 965. (19) Zembala, M.; Dejardin, Ph. Colloids Surf., B 1994, 3, 119. (20) Hayes, R. A. Colloids Surf., A 1999, 146, 89.
10.1021/la9905970 CCC: $19.00 © 2000 American Chemical Society Published on Web 12/14/1999
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Preliminary results for mica covered in situ by a precisely controlled amount of colloid particles (polystyrene latex) were reported in ref 21. In this case the absolute surface concentration of adsorbed particles was known a priori. The experimental data were interpreted in terms of a new theoretical model postulating no shift of the slipping plane upon particle adsorption. Instead, the changes in streaming current (potential) due to particle adsorption were attributed to (i) damping of fluid velocity in their vicinity resulting in decreased charge transport from the double layer at the interface and (ii) transport of charge from the double layer surrounding adsorbed particles. It was also demonstrated theoretically that the zeta potential of particle-covered surfaces cannot be unequivocally determined from the streaming potential measurements. This can only be achieved if independent determination of particle zeta potential is carried out, e.g., by the microelectrophoretic method. The aim of this work was to test the validity of the theoretical model by presenting more extensive experimental data obtained for the model system of natural mica covered by a controlled amount of polystyrene latex particles of various sizes. The results obtained in the welldefined model systems can be used for more complex systems involving, e.g., proteins. Then, protein adsorption kinetics could quantitatively be studied by measuring the streaming potential vs time dependencies without relying on tedious coverage scaling. Material and Methods Natural ruby mica sheets supplied by Mica and Micanite Supplies, Ltd. (England), were used without any pretreatment. Thin sheets were freshly cleaved for each experiment. Three samples of positively charged polystyrene latexes (L+39, L+57, and L+53) were synthesized according to the method described in refs 22 and 23 without addition of a surfactant. In the case of L+39 latex, the 1,2-azo-bis(2-methylpropamidinum) dichloride (ABA) initiator was used,22 whereas the L+53 and L+59 latexes were produced using the N,N′-dimethyl-4,4′-azobis4-cyano-1-methylpiperidine (DACMP) initiator.23 The latex suspensions were charge stabilized by positive amino groups stemming from the initiators. After synthesis, the latexes were purified by steam distillation under decreased pressure and by prolong membrane filtration until the conductivity of filtrate reached the value of water used for washing. The filtrates were also checked for the presence of traces of initiator and nonreacted styrene using UV spectroscopy. The size of larger particles determined by ZM Coulter Counter was found to be 1.13 µm for the L+39 and 0.55 µm for the L+57 samples. The size of the smallest latex, L+53 equal to 0.15 µm, was determined by Coulter N4 Plus Sizer. The zeta potential of latex particles was obtained by microelectrophoresis using the Brookhaven zeta-sizer (Zeta Pals). For suspensions in 10-4 M KCl solutions this potential was found to be +20 mV for L+39, +24 mV for L+57, and +25 mV for L+53 latex samples. For sake of convenience the relevant physicochemical characteristics of these latexes are collected in Table 1. Table 1 latex
d (µm)
ζ (mV)* in 10-4 M KCl
L+39 L+57 L+53
1.13 0.55 0.15
+20 +24 +25
The crystallized KCl was used for preparing the electrolyte solutions with triply distilled water. (21) Adamczyk, Z.; Warszyn´ski, P.; Zembala, M. Bull. Pol. Acad. Chem., in press 1999. (22) Goodwin, J. W.; Ottewill, R. H.; Pelton, R. Colloid Polym. Sci. 1979, 257, 61. (23) Blaakmeer, J.; Fleer, G. J. Colloids Surf. 1989, 36, 439.
Figure 1. Schematic view of the setup used for the streaming potential measurement: 1, the cell; 2, electrodes for streaming potential measurements; 3, electrometer; 4, electrodes for cell resistance measurements; 5, conductivity cell; 6, conductometer. Streaming potential of mica sheets, bare and covered by latex particles, was determined in the plane-parallel cell shown schematically in Figure 1. The cell, being in principle the improved version of previous ones used in refs 4, 17 and 18, consisted of two polished Teflon blocks, one of them containing two rectangular inlet and outlet compartments. Two thin mica sheets were placed between the Teflon blocks separated by a Teflon gasket of the thickness of 0.025 cm serving as a spacer (see Figure 1). The parallel plate channel of dimensions 2b × 2c × L ) 0.025 × 0.33 × 6.0 cm was formed by clamping together the Teflon blocks with two mica sheets and the spacer between them, using a press under constant torque conditions. The whole setup was placed inside the earthened Faraday cage to avoid any disturbances stemming from external electric fields. The streaming potential Es, occurring when an electrolyte was flowing through the cell under regulated and constant hydrostatic pressure difference, was measured using the two Ag/AgCl electrodes. These electrodes were located in two glass tubes having a direct contact with inlet and outlet compartments of the cell. A series of streaming potential measurements was done for at least five various pressures forcing the flow through the cell. A Keithley 6512 electrometer having the input resistance of more than 200 TΩ was used for potential measurements. The high resistance allowed potential measurements to be carried out at practically zero current conditions. Another pair of electrodes situated outside the Teflon cell (along the flow path) was used for determination of the cell resistance Rcell. This parameter was determined after the streaming potential measurements for each electrolyte solution studied and for each experiment. The surfaces covered with particles were produced in situ by filling the channel with an appropriate latex suspension of varied particle concentration (typically in the range of 108-1010 cm-3). The electrolyte concentration in the particle deposition process was 10-4 M (adjusted by KCl addition) and pH ) 5.8. The fluid flow rate during the deposition experiments was kept as low as possible in order to create the diffusion-controlled transport conditions. The particle deposition time was typically 30 min. Then, the cell was washed with pure 10-4 M KCl solution without allowing air to enter the channel. When the outlet solution was free of latex particles, the streaming potential measurements were performed using the KCl solutions of appropriate concentration. After the streaming potential measurements were finished, the cell was washed with water and opened and mica sheets were dried in the air. The dry mica sheets with deposited larger latex particles were examined under an optical microscope, whereas for the smallest latex (L+53), the particle counting was done using a scanning microscope. Particle surface concentration was obtained by determining the number of particles Np found over equal sized surface areas (of geometrical area ∆S equal typically a few thousand µm2) chosen at random over the interface.
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It should be mentioned that the series solution given by eq 2 is more convenient than the double cosine series solution derived by Yang and Li.24 The fluid velocity gradients at the channel walls (referred also to as wall shear rates), i.e., for y ) -b, and z ) -c are given by the expression
G0y
Figure 2. Schematic view of the parallel-plate channel showing the flow distribution. The total number of particles used in surface concentration determination was typically 1000-3000. The averaged surface concentration N h p was calculated in the usual way as the mean value of 〈Np/∆S〉. The averaged surface coverage was calculated h p, where a is the particle radius. as Θ ) πa2N Special precautions were undertaken to ensure the high purity of the studied latex suspensions. The purity level was checked by controlling conductivity and UV spectrum of the filtrate obtained from the membrane filtration. Additionally, it was also proven that none of the filtrates, at concentrations exceeding those used in deposition experiments, was influencing zeta potential of bare mica.
)
b
G0z ) uch 16 c π2R
V ) uchu j (yj,zj)ix
(1)
where ix is the unit vector of the x-axis (see Figure 2) and u j (yj,zj) is the dimensionless fluid velocity component parallel to this axis given by the series
32 π3
∑
n)0
(2n + 1)π
(-1)
n
(2n + 1)3
(2n + 1)π cos
2
cosh yj
2R
zj
(2n + 1)π cosh
u ; uch
uch )
(2)
(2n + 1)2
(2n + 1)2
∆P b2 L 2η
zj ) z/c (3)
where u is the dimensional fluid velocity, ∆P/L is the hydrostatic pressure gradient along the channel, and η is the fluid dynamic viscosity.
2
2R
2R
yj tanh
]
zj
(2n + 1) cosh
(2n + 1)π cos
(2n + 1)π 2R
The values of G0y and G0z averaged over the corresponding channel wall are
〈G0y 〉 )
1
1 G0y (zj) dzj ) ∫ -1 2
uch
〈G0z 〉 )
[
2-
32R π3
∞
∑ n)0
]
(2n + 1)π
1 (2n + 1)3
tanh
2R
1
1 G0z (yj) dyj ) ∫ -1 2
uch 32 c Rπ
∞
∑ 3 n)0
1 (2n + 1)3
(2n + 1)π tanh
2R
(5)
The above equations are valid for an arbitrary value of the parameter R ) b/c describing the ratio of the depth to the width of the channel including the case R ) 1 (a channel of square cross section). However, under usual experimental conditions this parameter assumes values much smaller than unity. Then, the expressions for 〈G0y 〉 and 〈G0z 〉 simplify to
uch 32 uch - 3 b π c
〈G0z 〉 =
2R
yj ) y/b
a ) b/c
n)0
〈G0y 〉 = 2
The dimensionless variables occurring in eqs 1 and 2 are defined as
u j)
∑ n)0
∑
π2
1
(-1)n
∞
b
1. Fluid Flow Field in Rectangular Channel. Let us consider a fully developed, two-dimensional laminar flow in a channel of rectangular cross section 2b × 2c (see Figure 2). The governing Navier-Stokes equations can be solved in this case analytically by using the variable separation, technique which results in the following expression for the fluid velocity18,21
∞
2-
∞
16
(4)
The Theoretical Model
u j ) 1 - yj2 -
[
uch
(2n + 1) cosh
32 uch π3 b
(6)
It should be noted that the above equations are only valid for Newtonian, uncharged fluid. It can be shown, however, that in the case when the double-layer thickness remains much smaller than the channel dimensions and the charge distribution on channel walls is quasi-uniform, the secondary electroviscous flow in the channel can be neglected, so the above expressions retain their validity. 2. Determining the Streaming Potential for a Rectangular Channel. Knowing the fluid velocity field in the channel, one can calculate the amount of charge leaving the channel due to convection per unit time (convective current) from the constitutive expression21 (24) Yang, C.; Li, D. J. Colloid Interface Sci. 1997, 194, 95.
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I)
Zembala and Adamczyk
∫∫Fe V‚dS ) uchbc ∫∫Fe(yj,zj) uj (yj,zj) dyj dzj S
(7)
S
where S is the channel cross section perpendicular to the N zinbi is x axis, dS is the surface element vector, Fe ) e∑i)1 the electric charge density, e is the elementary charge, zi is the valency of ith ion, and nbi is the bulk concentration of this ion. Substituting the expression for Fe from the PoissonBoltzmann equation and assuming a linear velocity increase at the boundary surfaces, one can evaluate the above double integral explicitly (in the case of thin double layers) obtaining the general expression for I in the form21
I) ζ1
c
〈G0y 〉ζ1 +
b
[ ( )
Ro
π
1- 1-
π
〈G0z 〉ζ2 )
ζ2 16R
ζ1
∞
∑ 3 n)0
(2n + 1)3
π
]
(2n + 1)π
1
tanh
2R
(8)
where is the dielectric constant of water, ζ1 and ζ2 are the zeta potentials of the channel walls (of the x,z and x,y planes, respectively), and Ro ) Iπb/2uchc ) πη/bc(∆P/L) can be treated as the dynamic resistance of the channel. It should be mentioned that in our approach the zeta potentials correspond to the electrostatic potential in the slip plane displaced from the solid wall by adsorbed ion diameter, i.e., generally much less than 1 nm. No presence of a stagnant gel-like layer is postulated in contrast with the Lyklema et al.25 approach. Equation 8 represents the general solution valid for arbitrary channel shape (including the case of a square cross-section channel). In practice, instead of I, one is measuring the potential difference Es (called the streaming potential) between the inlet and the outlet from the channel and the channel ohmic resistance Rc. Then, equating I with Es/Rc (Ohmic current) one can express eq 8 as
ζc )
[ ( )
ζ1 1 - 1 -
ζ2 16R
ζ1 π3
∞
∑ n)0
1
]
(2n + 1)π
) 2R EsRo (9) ζ1f(R,ζ2/ζ1) ) Rc
(2n + 1)3
tanh
[
ζ1 1 -
]
Es πη 16R 16R ζ2 + 3 = ζ + 0(R) ) (10) 3 Rc ∆P π π ζ1 bc L
By introducing the specific conductivity of the solution λ ) L/4Rcbc, one can convert eq 10 to the Smoluchowski expression,26 i.e.
ζ)
4πηλ Es ∆P
(11)
3. Streaming Potential for Particle Covered Surfaces. The above-discussed expressions can only be used for interpretation of experimental results obtained for smooth interfaces in the absence of adsorbed particles. When adsorbed particles are present at the interface, the streaming current can still be determined from eq 7, which can be formulated as
∆Ip ) Ip - I )
∫∫(Fe*V* - FeV)‚dS
(12)
S
where Ip is the streaming current for particle covered surface, I is the current for bare surface given by eq 8, and Fe* and V* are the perturbed (due to adsorbed particles) electric charge density and fluid velocity in the S plane, respectively. These quantities obviously depend on particle coverage and distribution over the surface. Therefore, it is impossible to evaluate the surface integral in the general case because this would require a solution of the PoissonBoltzmann and Navier-Stokes equations for multibody problems. Due to lack of appropriate coordinate systems and nonlinearity of the Poisson-Boltzmann equation, no analytical solutions of general validity were found yet. One can only derive numerical solutions for the case of one particle attached to the surface using tedious finitedifference relaxation techniques as discussed in ref 27. Similarly, no solution of the hydrodynamic problem exist except for a single sphere immersed in the simple shear flow. In this case, exact solution was derived by O’Neill28 in terms of Bessel functions using the cylindrical coordinate system. An analogous solution in bispherical coordinates was derived in ref 29 in the form of an infinite series of Legendre polynomials. Knowing the solution for one particle, one can construct solutions for multiparticle systems assuming the additivity principle. As far as the electrostatic problem is concerned, the additivity rule (also called the linear superposition method LSA) remains appropriate if the separation distances between particles are larger than the double-layer thickness. On the other hand, it is difficult to assess the distance range when fluid velocity fields remain additive, although they seem comparable with particle diameter. By accepting additivity, the following general expression for the excess streaming current ∆Ip was derived in ref 21
where ζc can be treated as the effective zeta potential of the channel and f(R,ζ2/ζ1) is the universal correction function. All the variables on the right-hand side of eq 9, which can be treated as the generalized Smoluchowski equation,26 are experimentally accessible so the effective zeta potential ζc ) ζ1f(R,ζ2/ζ1) can unequivocally be determined. However, the zeta potentials ζ1 and ζ2 of separate channel walls cannot be determined a priori by a single streaming potential measurement. This would become feasible if two independent measurements are carried out with different channel geometry (when the ratio of height to width is changed). Another possibility would be to construct channels as thin as possible, characterized by very small R. Then, eq 9 reduces to
where δy is the variation of the transverse coordinate, u* is the velocity component parallel to the x axis, Θ ) πa2N is the dimensionless surface concentration (coverage), and N is the number of particles adsorbed per unit area. The volume integral in eq 12a should be carried out over the
(25) Lyklema, J.; Rovillard, S.; De Coninck, J. Langmuir 1998, 14, 5659. (26) von Smoluchowski, M. Bull. Acad. Sci. Cracovie, Classe Sci. Math. Natur. 1903, 1, 182.
(27) Warszyn´ski, P.; Adamczyk, Z. J. Colloid Interface Sci. 1997, 187, 283. (28) O’Neill, M. E. Chem. Eng. Sci. 1968, 23, 1293. (29) Goren, S. L.; O’Neill, M. E. Chem. Eng. Sci. 1971, 26, 325.
δ ∆Ip ) Ip - I ) Θ
∫∫∫(Fe*u* - Feu) dV
δy πa2
(12a)
V
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region where the electric charge density or fluid velocity deviates from the corresponding unperturbed values, u and F. It is also assumed that δy is much larger than the particle dimension but small enough so the unperturbed fluid velocity u does not vary within δy. Implementation of eq 12a for thick double layer seems difficult, because the volume integral can only be evaluated numerically. However, it was shown in ref 21 that the electrostatic problem can be simplified in the case κa > 1 (double-layer thickness smaller than particle radius). Then, accumulation of charge occurs within thin layers adjacent to the interface and particles so the volume integral occurring in eq 12a can be expressed via two surface integrals calculated separately over the interface and the particle. Consequently, the expression for the excess streaming current for a surface element becomes21
δ∆Ip ) Θ G0(ζiCi + ζpCp) δy 4π
(13)
where G0 is the unperturbed wall shear rate (which is dependent on the transverse coordinates (z, y)) ζi and ζp are the zeta potentials of the particle and interface, respectively, and the constants Ci and Cp are given by
Ci )
1 πa2
Cp )
( ) G0
∫∫ G0i - 1 Si
1 πa2
dSi
G0p
∫∫ G0 dSp
(14)
Sp
G0i ) (∂u*/∂z)i and G0p ) (∂u*/∂z)p are the local (perturbed) shear rates at the interface and particle, respectively. It should be mentioned that eq 13 is valid for the case when the coverage of all channel walls remains the same. The dimensionless function Ci describes the purely hydrodynamic effect of flow shear rate decrease over the interface as a result of the presence of adsorbed particles, whereas the function Cp is a measure of the flow shear rate averaged over adsorbed particles. It should be mentioned that the Cp definition is valid for arbitrary flow distribution, whereas the Ci expression is only valid for additive velocity fields. It is interesting to observe that by formulating eqs 13 and 14, no shift in the position of the slip plane due to adsorbed particles was postulated. In this respect our model differs completely from that discussed by Hayes et al.20 It was also shown in ref 21 that the Ci constant is connected with the hydrodynamic force acting on the adsorbed particle through the dependence
F ) πηa2G0Ci ) G0aRh
(15)
where Rh ) πηa|Ci| is the hydrodynamic resistance coefficient. By use of the hydrodynamic solutions of O’Neill28 and Goren and O’Neill,29 the double integrals in eq 14 can be evaluated in an exact way with the result21
Ci ) C0i ) -10.21 Cp ) C0p ) 6.51
(16)
As mentioned, eq 16 remains exact as long as the flow perturbations stemming from individual particles remain additive. This assumption is fulfilled for low coverages
only. For higher coverages the flow additivity rule certainly breaks down and the values of Ci and Cp are expected to decrease. It can also be predicted that the change in Ci should be more pronounced than the change in the Cp constant. However, it is not possible to solve this manybody problem in any exact way. One may only introduce correction functions in analogy to the flow model parameter Af used for describing 3D flows in packed bed columns or filtration mats.30,31 Then, the Ci and Cp functions can be formally expressed as
Ci ) C0i Ai(Θ) Cp ) C0p Ap (Θ)
(17)
where the correction functions (flow model parameters) Ai and Ap are dependent on the surface coverage Θ. By definition, Ai ) Ap f 1 when Θ f 0. Substituting eq 17 into eq 13 and averaging over all channel walls (-c < z < c and -b < y < b), one obtains the following expression for the excess current
∆Ip ) [c 〈G0y 〉 ζ1 + b 〈G0z 〉 ζ2] C0i Ai(Θ)Θ + π ζ [c〈G0y 〉 + b〈G0z 〉]C0p Ap(Θ)Θ ) π p ζp ζ1 f(R,ζ2/ζ1)C0i Ai(Θ)Θ + C0p Ap(Θ)Θ (18) R0 R0 where 〈G0y 〉 and 〈G0z 〉 are the averaged wall shear rates defined by eq 5. Equation 18 represents the general expression for the current change due to particle adsorption in a rectangular channel. Note that ∆Ip does not depend explicitly on the particle radius a. Equation 18 can also be expressed in the equivalent form as
ζa ) C0p Ap(Θ)Θζp + [1 - |C0i |Ai(Θ)Θ]ζc
(18a)
where ζa ) IpR0 ) 4Espπηλ/∆P is the “apparent” zeta potential of the channel in the presence of deposited particles. No shift in the shear plane was postulated by deriving eq 18a. Equation 18a resembles formally the expression postulated by Hayes,20 who assumed that
ζa ) Θζp + (1 - Θ) exp(-κdeff)ζc where deff is the empirical parameter characterizing the effective “displacement” of the shear plane. Equation 18 can also be expressed in the form more suitable for interpretation of experimental results
Ip c〈G0y 〉 + b〈G0z 〉 ) 1 - |C0i |Ai(Θ)Θ + ζp C0p Ap(Θ)Θ 0 0 I c〈Gy 〉ζ1 + b〈Gz 〉ζ2 (19) Assuming that the Ohmic resistance of the channel is not affected by adsorbed particles, one can identify the Ip/I ratio with the streaming potential ratio Esp/Es where Esp is the streaming potential in the presence of particles. This assumption can be validated by the fact that the relative Ohmic resistance increase due to adsorbed (30) Happel, J. AIChE J. 1958, 4, 197. (31) Kubawara, S. J. Phys. Soc. Jpn. 1959, 14, 527.
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particles is of the order Θa/b. This is a very small quantity for any situation of experimental interest. It can easily be deduced from eq 19 that for neutral (uncharged) particles, one has always
Ip Esp ) ) 1 - |C0i |Ai(Θ)Θ I Es
(20)
Note that in this case the Ip/I or Esp/Es ratio does not depend on any parameters characterizing the channel shape, its surface properties, or flow rate (governed by pressure difference). Therefore, this equation is valid for a channel of arbitrary shape (circular, elliptic, or rectangular) provided that the cross section does not vary with the longitudinal coordinate x. This conclusion has a considerable practical impact indicating that one can determine the Ai(Θ) function empirically for arbitrary surface coverage just by measuring the ratio of the streaming currents or streaming potentials. Once this universal function is known, one can determine particle surface coverage in other experimental systems, when direct observation of particles is not feasible. The only limitation is the requirement that the double layer thickness should be much smaller than particle radius, which is fulfilled for higher electrolyte concentrations. When ζ1 ) ζ2 ) ζ, eq 19 reduces to the simple form
ζp Ip Esp ) ) 1 - |C0i |Ai(Θ)Θ + C0p Ap(Θ)Θ (21) I Es ζ In this case the Ap(Θ) function can be determined empirically from the Ip/I (or Esp/Es) dependence if ζ and the function Ai(Θ) is known (from experiments carried out for weakly charged particles) as well as ζp (from e.g., microelectrophoretic measurements). Another possibility of determining the Ap(Θ) function would arise if the streaming current (or potential) is measured for weakly charged surfaces (when ζ f 0). Then eq 18 becomes
∆Ip ) Ip )
1 ζ C0 A (Θ)Θ R0 p p p
(22)
On the other hand, for low coverages, when Ap(Θ) f 1, eq 22 can be used for determining the zeta potential of particles from the dependence
ζp )
4πηλ ∆Esp 6.51∆P ∆Θ
(23)
where ∆Esp/∆Θ is the slope of the Esp vs Θ dependence in the low coverage limit. Note, that eq 23, valid for channels of arbitrary shape, does not involve explicitly particle size. Results and Discussion To check the performance of the cell, a series of experiments was carried out for bare mica (uncovered by particles). In these experiments the streaming potential was measured as a function of solution ionic strength changed within 10-5-10-3 M by addition of KCl. The pH of these experiments was kept constant, equal to 5.6. The cell resistance Rcell measured in each experiment was used for determination a correction function f(1/Rcellλ)) describing a deviation of the cell conductivity from that resulting from bulk conductivity λ. This function incorporates inter alia the surface conductivity effects expecting to play a
Figure 3. Zeta potential of pure mica in KCl solution of various concentrations: b (full circles with error bars), our results; 2, data of Scales et al.18
role for lower ionic strength. The zeta potential of mica ζ1 was calculated from eq 9 (with the appropriate correction for temperature dependence of viscosity and of dielectric constant) by assuming that the zeta potential of Teflon ζ2 ) ζ1. It should be mentioned that the uncertainty in ζ2 exerted very little effect on calculated values of mica zeta potential because the term 16R/π3 was equal 0.04 for our channel geometry (R ) 0.076). This was additionally checked by performing a separate series of experiments with a two times thicker spacer. The results were found to be the same within the experimental error bounds. In this way we found ζ1 ) -101 mV for 10-5 M, -95 mV for 10-4 M, and -81 mV for 10-3 M. For sake of convenience the dependence of ζ1 on ionic strength for bare mica is plotted in Figure 3. As can be seen our data agree within experimental error with data previously reported by Scales et al.18 who worked with the rectangular cell having the dimensions 0.01 × 2.2 × 7.5 cm. Slightly larger deviation occurs at lowest ionic strengths (I < 10-4 M) when the surface conductivity effect may decrease the accuracy of experimental measurements. In the case of particle-covered surfaces, streaming potential measurements were carried out according to the procedure described above. In the first step, particles were adsorbed to a desired surface concentration under a diffusion-controlled transport mechanism. In this way the gradient of particle surface concentration along the channel, appearing as a result of convective effects,32 was minimized. This was confirmed in a series of experiments were adsorbed particle distributions over mica surface were checked with respect to both the longitudinal (x) and transverse (y, z) coordinates. These distributions were found to be statistically uniform as can be seen in the series of micrograph presented in Figure 4 (L+57 sample, I ) 10-4 M, particle coverage 0.013, 0.042, and 0.090, respectively). The coverage range studied in the series of experiments discussed hereafter was limited to 0-0.28 since attaining higher coverages required an excessive suspension concentration, especially for low ionic strength. After the particle deposition procedure was completed, the streaming potential measurement was carried out in (32) Adamczyk, Z.; Siwek, B.; Zembala, M.; Belouschek, P. Adv. Colloid Interface Sci. 1994, 48, 151.
Streaming Potential for Mica with Colloid Particles
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Figure 5. Dependence of the slope Esp/∆P (left side axis, empty points) and the apparent zeta potential (right side axis, full points) on particle coverage for ionic strength 10-4 M.
Figure 6. Same as Figure 5 but for ionic strength equal to 10-3 M.
Figure 4. Micrograph of latex particles (L+57) adsorbed on mica from 10-4 M KCl: (a) Θ ) 0.013; (b) Θ ) 0.042; (c) Θ ) 0.090.
situ. As mentioned previously, for each experiment the dependence of Esp on the hydrostatic pressure gradient was determined in order to increase the experimental precision. These experimental results, showing the dependence of the slope Esp/∆P as a function of particle coverage Θ, are presented in Figure 5, for I ) 10-4 M, and in Figure 6 for I ) 10-3 M. The apparent zeta potential of the channel ζa ) (Esp/∆P)4πηλ/ is also plotted in these figures. It should be mentioned that in all cases the κa parameter was larger than unity, ranging from 2.5 for the L+53 sample and I ) 10-4 M to 56 for the L+39 sample and I ) 10-3 M. As one can observe in these figures the presence of adsorbed particles exerted a significant effect on the slope Esp/∆P (and apparent zeta potential) which changed sign for Θ as low as 0.15. It should also be noted that the increase in the ionic strength exerted little effect on the Esp/∆P vs Θ dependence. These observations are in agreement with our theoretical model exposed above predicting a significant increase in Esp/∆P with particle coverage, which should be independent of ionic strength for κa . 1.
A quantitative comparison with experiments can be carried out when plotting the universal dependence Esp/Es vs Θ, where Es is known from previous measurements for bare mica. The data obtained for the three latex samples and ionic strengths 10-4 and 10-3 M are collected in Figure 7. As can be seen, the data seem to exhibit a universal character, being independent of particle size and ionic strength in accordance with the theoretical predictions. It should also be noted that for the low coverage range the experimental data are well reflected by the linear model (broken line in Figure 7) derived from eq 21 by assuming the flow parameters Ai and Ap equal 1, i.e.
Esp Es
) 1 - 10.21Θ - 6.51
||
ζp Θ ζ
where ζp is the average value of the zeta potential determined by electrophoresis (see Table 1). It can be seen in Figure 7 that the linear model performs well for coverages smaller than 0.05. It should be noted, however, that even for these small coverages the relative change in the streaming potential can reach 50%. This result has a considerable theoretical significance confirming the thesis that the streaming potential changes in the presence of neutral or slightly charged particles (in our case the |ζp/ζ| ratio was about 0.25) are induced by flow
1600
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Zembala and Adamczyk
Figure 7. Dependence of Esp/Es on surface coverage of particles Θ. The points denote experimental results obtained for positive latex particles: L+39 (0), L+57 (O), L+53 (4), and various KCl concentrations, i.e., 10-5 (gray), 10-4 (full), and 10-3 M (empty). The broken line represents the theoretical results stemming from the linear model, when Ap ) Ai ) 1 for Θ < 0.1 and Ai ) 0 otherwise, the continuous line shows the theoretical results calculated from eq 21 with Ap ) 1. Ai ) (1 - eCi0Θ)|C0i |Θ.
damping in the thin layer adjacent to the interface (whose thickness is comparable with the double-layer extension). Moreover, since according to eq 15 the hydrodynamic resistance coefficient of particles adsorbed at surfaces is equal to |C0i | πηa, our measurements seem to confirm the calculations of O’Neill28 concerning the flow around a spherical particle at the interface immersed in a simple shear flow. For higher coverages, however, the experimental results shown in Figure 7 deviate from theoretical predictions stemming from the linear model. This can be interpreted, in accordance with previous discussion, as due to the fact that the flow fields stemming from individual particles are no longer additive, so the function Ai decreases below unity. At higher surface coverages, Ai is expected to vanish since the flow intensity at the interface becomes negligible. Considering this, an improved linear model can be derived by postulating that Ai ) 1 for Θ < 1|C0i | (coverage less than 0.1) and zero otherwise. The results stemming from this approximation are also shown in Figure 7. Another, nonlinear interpolating function for Ai(Θ) can be derived by noting that the probability of finding an empty surface area ∆S over the interface covered by particles can be approximated by e-(∆S/πa2)Θ.33 As a consequence, the streaming current Ip/I (or potential Esp/Es) ratio should be proportional to this quantity. Furthermore, from the requirement that the initial slope of this dependence equals C0i ) -10.21, one can deduce that the contribution due to fluid damping at the interface is
Esp/Es ) Ip/I ) e
Ci0Θ
Considering also the contribution stemming from adsorbed particles which should not be modified too significantly (33) Adamczyk, Z.; Weron´ski, P. J. Chem. Phys. 1998, 108, 9851.
for the coverage range shown in Figure 7, one obtains the interpolating function 0
Esp/Es ) Ip/I ) eCi Θ +
ζp 0 CΘ ζ p
As one can see in Figure 7 this function reflects well our experimental data for the entire range of particle coverages studied. This seems to have a practical impact because once knowing Ai(Θ) one can determine quantitatively surface coverage of colloid particles just by measuring the streaming current (or potential) ratio, i.e., Ip/I or Esp/Es. Then, for uncharged or slightly charged particles one has
Θ)
| |
||
Esp Ip 1 1 ) ln ln I Es C0i C0i
To generalize this expression to larger zeta potentials of particles, one needs, however, the Ai(Θ) function which can be determined from zeta potential measurements at noncharged channels. Experiments of this kind are planned in the future. Concluding Remarks It was demonstrated that for a rectangular channel of arbitrary height-to-width ratio, R ) b/c, one can determine the effective zeta potential only, given by the expression
ζc ) ζ 1 +
[ ( ) 1- 1-
ζ2 16R
ζ1 π3
∞
∑ n)0
1 (2n + 1)3
]
(2n + 1)π tanh
2R
Contributions stemming from separate walls can be determined if at least two independent measurements for different channel geometry are carried out.
Streaming Potential for Mica with Colloid Particles
It was also predicted theoretically that adsorption of colloid particles will influence significantly the apparent zeta potential of the channel which can be calculated from the equation
ζa ) C0p Ap(Θ)Θζp + [1 - |C0i |Ai(Θ)]ζc In contrast to previous approaches, no assumption of the slip plane shift was necessary to formulate this equation. In the case of thin double layers (κa > 1) the C0p and C0i constants were found to be 6.51 and -10.21, respectively. However, the correction functions Ap(Θ) and Ai(Θ) describing the multibody effects due to adsorbed particles could not be evaluated theoretically. For neutral or slightly charged interfaces, the ζa/ζc or Esp/Es dependence was predicted to be a universal function of particle coverage Θ, independent of particle size, ionic strength, shape of the channel, etc., according to the expression
Langmuir, Vol. 16, No. 4, 2000 1601
ζp )
4π λη ∆Esp 6.51 ∆P ∆Θ
The above theoretical predictions were verified experimentally by using monodisperse suspensions of positively charged latex particles adsorbed at natural mica surface. It was found, in accordance with theoretical predictions, that the streaming potential of the channel was significantly affected by the presence of adsorbed particles. The experimental dependence of Esp/Es on Θ was found to exhibit a universal behavior, being independent of particle size κa and ionic strength. The initial slope of this dependence was close to the theoretical value of -10.21. Additionally, it was found that the linear model characterized by Ap ) 1 and Ai ) 1 for Θ < 0.1 and Ai ) 0 otherwise, can be used as a reasonable estimate of experimental data for the entire range of coverages. A much better agreement with experimental data was found when using for Ai the simple interpolating function 0
Ai(Θ) )
Esp
ζa ) ) 1 - |C0i |Ai(Θ)Θ ζc Es This relationship can be used for determining surface coverage of arbitrary-sized particles (if κa > 1) just by measuring experimentally the streaming potential ratio. The precision of this measurement is expected rather high for small coverages due to the large value of |C0i |. On the other hand, it was theoretically suggested that for neutral or slightly charged surfaces one can determine the zeta potential of adsorbed particles from the equation
1 - eCi Θ |C0i |Θ
By exploiting these findings, the streaming potential method can be used for determining in situ surface coverage of adsorbed particles (of arbitrary size) both for charged and uncharged surfaces. Acknowledgment. This work was partially supported by Copernicus Grant No. ERBIC 15 CT-98-0121. The authors thank Dr. P. Warszyn´ski for helpful discussions. LA9905970
