Langmuir 1985,1, 83-90
83
Sedimentation Potential in Aqueous Electrolytes Bruce J. Marlow? and Robert L. Rowell* Department of Chemistry, University of Massachusetts, Amherst, Massachusetts 01003 Received March 16, 1984. In Final Form: August 9, 1984 The validity of the Smoluchowski treatment of the sedimentation potential (Dorn effect) is explored in relation to the recent cell-model treatment of Levine et al. which is applicable at higher volume fraction where both electrical and hydrodynamic interactions must be taken into account. An apparatus was constructed for the rapid successive measurement of the sedimentation potential gradient, the particle volume concentration, and the specific volume conductivity of the medium. Measurements on a model system of 97-~mglass microspheres dispersed in aqueous electrolytes showed that the Smoluchowskitheory was followed below a volume fraction of 0.018 and that the potentials obtained compared to those obtained from streaming potential measurements at the glass-water interface characterized by an iep at pH 3.69. At higher volume fraction where the surface conduction correction was -1.1% the Levine cell-model theory was required to explain a -28% deviation from the linear approximation theory.
Introduction When charged particles move relative to a liquid medium by the action of mechanical forces an electric field is induced along the direction of motion. In a settling dilute dispersion the electric fields from the individual particles Ei add linearly to produce a macroscopic electric field E or E = CEi (1) i
The direction of the field is such that it retards sedimentation. Increasing particle concentrations leads tQ electrical and hydrodynamic particle-particle interactions and E is no longer a linear summation of the individual fields. Dorn' was the first to observe that as a dispersion settled in a column an electric potential developed between two points along the length of the column. Since that time there has been only modest effort aimed at experimental verification2-15of the f i t theoretical treatment of the Dorn effect by Smoluchowski.16 Our objective was to perform an in-depth experimental study of the Dorn effect to produce a comprehensiveset of datal7 which could be used to evaluate the Smoluchowski treatment as well as to bridge the gap with later and more rigorous approaches.1"26 To accomplish this we chose a model system composed of glass microspheres dispersed in aqueous electrolytes. The system allowed us to concentrate solely on the flow of the continuous phase past the particles and its effect on the electric double layer.
Theory As charged particles move relative to a liquid due to sedimentation, ions from the bulk flow into the lower part of the double layer, tangentially around the particle, and then return to the bulk solution via the upper half of the double layer. The net flow of electric current across a given plane by particle convection must balance that due to electrical conduction in the steady state. Thus, to relate the macroscopic induced field E to pertinent electrokinetic parameters one must equate the convection current to the conduction current. If one assumes the (i) particles are spherical, nonconducting, and monodisperse, (ii) laminar flow around the particles occurs (Reynolds number < l), (iii) interparticle interactions are negligible, (iv) surface conduction is negligible, and (v) the double-layer thickness 1 / is~small compared to the particle radius a (KU >> l), then one obtains the classical result of Smoluchowski16 Present address: Standard Oil Co. (Indiana), Amoco Research Center, Naperville, IL 60566.
0743-7463/85/2401-0083$01.50/0
where eo is the permitivity of free space, D the dimensionless dielectric constant, { the { potential, g the acceleration due to gravity, r#J the particle volume fraction, and p2 the particle density and pl, A, and q are the continuous phase density, specific volume conductivity, and viscosity, respectively. Thus, for a given system the sedimentation potential gradient E, is directly proportional to the b potential and particle volume fraction but is inversely proportional to the bulk solution specific volume conductivity. It can be shown17 that the condition of monodisperse particles can be relaxed if one has knowledge of the volume-average radius and nonspherical particles can be utilized provided one has knowledge of the Stokes' frictional factor. The Smoluchowskiresult is valid for all potentials provided the above conditions prevail. (1)Dorn, E. Ann. Phys. (Leipzig) 1880,lO(3), 46. (2)Billiter, J. Ann. Phys. (Leipzig) 1903,11 (4),902. Sitzungsber Akad. Wiss Wien, Math.-Naturwiss.Kl. 1904,I13 (2), 861. (3)Makelt, E. Dissertation, Dresden, 1909. Freundlich, H.; Makelt, E. Z . Elektrochem. 1909,15, 161. (4)Stock, J. Bull. Int. Akad. Pol. Sci. Lett., Cl.Sci. Math. Nat., Ser. A 1913,131;1914,95. (5)Burton, E. J.; Currie, J. E. Phios. Mag. 1925,49 (6),194. (6)Bull, H.B. J . Phys. Chem. 1929,33,656. (7)Quist, J. D.; Washburn, E. R. J. Am. Chem. SOC.1940,62,3169. (8)Elton, G. A. H.; Peace, J. B. J . Chem. SOC.1956,22. (9)Elton, G. A. H.; Peace, J. B. J . Chem. SOC.1960,2186. (10)Elton, G. A. H.; Peace, J. B. Proc. Int. Conf. Surf. Act. 2nd 1957, 4, 177. (11)Benton, D. P.; Elton, G. A. H. J . Chem. Soc. 1953,2096. 1953,preprint No. (12)Elton, G. A. H.; Mitchell, J. W. J . Chem. SOC. 741. (13)Bordi, S.;Rucci, G.; Papechi, G. Ann. Chim. 1963,53 (7), 934. (14)Matukura, Y. Jpn. J . Appl. Phys. 1967,3 (17),409. (15)Oyabu, Y.; Yasumori, Y. Talanta 1972,19,423. (16)Von Smoluchowski, M. In "Graetz Handbuch der Electrizitiit und des Magnetismus"; VEB Georg Thieme: Barth, Leipzig, 1921;Vol 11, p 385. (17)Marlow, B. J. Ph.D. Thesis, University of Massachusetts, Amherst, MA, 1982. (18)Levine, S.;Neale, G.; Epstein, J. Colloid Interface Sci. 1976,57, 424-437. (19)Kuwabara, S.;J . Phys. SOC.Jpn. 1959,14,527. Shilov, V. N. Colloid Journal USSR 43 (Engl. (20)Zharkikh, N. 0.; Transl.) 1981, (6),865. (21)Levine, S.;Neale, G. J . Colloid Interface Sci. 1974,47,520. (22)Levine, S.;Neale, G. J. Colloid Interface Sci. 1979,49,330. (23)Booth, F.J . Phys. Chem. 1954,22, 1956. (24)Hermans, J. J. Philos. Mag. 1938,26,650. (25)Dukhin, S.S.In "Research in Surface Forces"; Deryaguin, B. Y., Ed.; Consultants Bureasu: New York, 1966;Vol 11, p 54;1966;Vol 111, p 306. (26)Saville, D. A. Adu. Colloid Interface Sci. 1982,16, 267.
0 1985 American Chemical Society
84 Langmuir, Vol. 1, No. 1, 1985
Marlow and Rowel1
Booth23has developed a theory for the sedimentation potential for arbitrary values of KU. Deryaguin and Dukhin,25using the same system of equations and boundary conditions as Booth, have shown that Booth’s method is restricted to small potentials and PBclet number. The PBclet number reflects the relative importance of connective vs. molecular diffusion and is defined as the product of the particle radius and velocity divided by the ion diffusion coefficient. Booth’s result reduces to Smoluchowski’s when the above assumptions hold and the mobilities of all the ions are equal. Deryaguin and Dukhin25have shown that Smoluchowski’s formula retains its value even when the ions differ in mobility provided the PBclet number is large, which is true for large particles or slow counterions, and the above assumptions hold. Recent theoretical work by SavilleZ6elucidates the shortcomings of previous theoretical work. The treatment considers dilute colloids and can be reduced to the Smoluchowski theory for the appropriate conditions. When particle-particle interactions occur the theory of Smoluchowski is no longer valid since it fails to account for both electrical and hydrodynamic interactions. Levine et have developed a general theory where both 4 and KU can be varied arbitrarily. The theory is based on the Kuwabara cell m0de1.l~The following general expression was obtained
EL = ( 9 € ~ ~ ’ 4 / ( 2 X ~ ‘ ~ ) y ( ~ a , 4 ) (3)
I
b F a r a d a y cage
Figure 1. Sedimentation potential apparatus: R particle reservoir, N 20-turn needle-valve stopcock, E Ag/Agbl electrodes, SIand S,6-mm2slits, P CdS photocell, incident light, SP sample port, and B bucking circuit.
tials ( e { / ( k T ) < 1). Smoluchowski’s theory is valid for all potentials but is limited to the above restrictions. Thus, to account for deviations from the Smoluchowski theory at higher particle concentrations the cell-model theory of Levine can be used for low potentials and arbitrary KU. For high potentials and arbitrary K U , a general solution is not u’ = V o ( 1 + (€ocT/a)2(1/(X77))H(Ka,4)l-’ (4) yet available and Levine’s theory is only a first approximation. where H(Ku,@) is a complicated function behaving similar The correction factor to Stokes’ law developed from the ) is shown in Figure 4 of ref 18. The settling to Y ( K u , ~and cell model is strictly valid for situations where the Reynolds velocity of the uncharged particles in a dispersion is related number is less than unity (laminar flow). For Reynolds to the settling velocity of an isolated uncharged particle numbers 3 or less, Stokes law is still a good first apu by” proximation. Beyond these limits turbulent motion of the V o = v / n = 2a2g(p2- P1)/(977Q) (5) liquid around the particles occurs and Stokes’ law is no longer applicable. Deryaguin and DukhinZ5discuss the where 0 is a hydrodynamic correction term to Stokes’ law effect of Reynolds number on the sedimentation potential and is given by in detail. Experimental observations of electrophoretic mobility Q = 5/(5 - 94113 + 54 - 4’) (6) in concentrated systems showed excellent agreement with Substitution of eq 5-7 into eq 4 gives the cell-model theory21B22 which encouraged the present investigation of the applicability of the theory to the Dorn EL = EJ(1 - (941’3/5) + 4 - (4*/5))/(1 + effect. The need for a controlled set of sedimentation ( ~ o ~ r / a ) 2 ( l / ( t l X ) ) H ( ~ ~ , ~(7) ))Jy(~ ~,4) experiments was established by the theoretical treatment of Levine et al. who found that the predictions of their As 4 0 and Ka 03, T ( K U , ~and ) H ( K u , ~ ) 1, giving theory were qualitatively consistent “with the rather sparse EL = + ( ~ O ~ C / ~ ) ~ ( ~ / ( V M ) I (8) experimental data appearing in the literature”.l* For large particles (a > 50 Fm)i n aqueous electrolytes (A Experimental Section > lo-* mho/cm) and { < 0.2 V where u’ is the actual sedimentation velocity and ~ ( K u , $ ) a complicated function of KU and 4 shown in Figure 3 of ref 18. The actual settling velocity u’ was shown to be related to the settling velocity for the same particles in the absence of charge uo through
-
-
-
-
(€o€C/a)2(l/(77X))1O'O R. An OmniScribe recorder was used when the resistance of the medium was lowered significantly (high electrolyte concentration)because of its low input impedance of 1 MR. A short unimpeded path to the electrode on the high impedance side of the multimeter was provided and the bucking circuit for hacking off the asymmetry potential was on the low terminal side and grounded. The sedimentation column was 80 cm long with a 3-em diameter and all joints were made of Teflon to avoid contamination by greases. Two small holes 2 mm in diameter were placed 25 cm apart 25 em down from the particle reservoir. The electrode seat could not alter the contour of the column because it was found that eddy current would develop. The entire cell was Faraday caged to avoid stray potentials. The model system was glass microspheres (see Figure 3) dispersed in aqueous electrolytes which conforms to the above requirements. The glass microspheres were obtained from Jaygo Inc. of Hawthorne, NJ. The chemical and physical properties of the microspheres are shown in Table I. The microspheres contained a high lead content, which made the particles essentially nonconducting and raised the relative refractive index to 8-10, which increased the sensitivity of the turbidity measurement. The size and density of the microspheres resulted in an appreciable sedimentation velocity (Reynolds number 2-3) and enhance-
-
(27) Brown, A. S. J. Am. Chem. Soe. 1934.56.696.
Figure 3. Photomicrograph of high-lead glass microspheres.
Diameter luml
Figure 4. Particle size distribution of glass microspheres;mean diameter is 97 3 pm. Table I. Chemical and Physical Properties of Jaygo Leaded-Glass Microspheres compd eompn, 90 Chemical Prooerties SiO. 58
Physical Properties density 2.840 + 0.004 g/cm3 -IO-" mholcm conductivity transformation temperature 424 modulus of elasticity 6200 kPa/mm2 modulus of rigidity 2560 kPa/mm2 rockwell hardness DIN 50103 47 refractive index 8-10 mean particle diameter 97*3pm ment of the sedimentation potential gradient. The particles were sieved to 9C-106-fim diameters and sized on a microscope with a reticule graduated in l0-fim divisions. The particle-size histogram is shown in Figure 4, and an average diameter of 97 3 pm was observed. Thus, the microspheres in aqueous electrolytes conform well with the theoretical requirements in that xa >> 1, Peclet number > 1, Reynolds number < 3, and the particles are nonconducting,
*
86 Langmuir, Vol. 1, No. 1, 1985
Marlow and Rowel1
5-
-
4-
E >. E 3-
5
-E
w I
>F
21-
v,=o
0 0
20
I
1
40
Time (mini
I
2 @%
a
I
3
Figure 5. Typical strip-chart record of data collection showing steady-state sedimentation potential V, at flow volume fractions 6,with background asymmetry potential V, reduced to near zero.
Figure 6. Comparison of the experimental and theoretical dependence of the sedimentation potential gradient on volume fraction for glass microspheres in lo4 M KCl at pH 7.00 and 25 "C.
spherical, and relatively monodisperse and when KCl is used the mobilities of all the ions are approximately equal. The sedimentation potential gradient E was obtained from
in Figure 5. This procedure allowed potential gradients to be reproduced to within 6% and volume fractions to within 10%.
E = (V, - V,)/h
Results and Discussion To assess at what point the prediction of the Smoluchowski theory fails, Le., (dE/dh) # constant, the sedi-
(11)
where V , is the observed potential for a given particle concentration, V , the asymmetry potential usually backed off to zero electronically or graphically, and h the distance between the electrodes. The specific volume conductivityof the aqueous solutions was determined with a Horizon 1484A conductivity meter. The pure water had a conductivity of 2.5 pmho/cm. I t was obtained by passing distilled water through an ion exchange and then doubly distilling in a Corning AG1-B still. The glass microspheres were washed with dilute nitric acid and then water. They were stored over water for a minimum of 14 days. They were then washed, dried, added to the desired electrolyte, sonicated at 60 HZ until no visual evidence of aggregation existed, and then allowed to equilibrate until a steady conductivity of the supernatant was observed. It was found that the conductivity increased when the particles were added to the water by 2.2 pmho L/(g cm) presumably due to dissociation of surface groups (ApH -0.2). The dispersions were prepared in a concentrated regime to minimize the effect of impurities (large area available). In dilute electrolyte solutions the change in conductivity over the bulk value was large where as in concentrated electrolyte the change was small. Thus, in dilute electrolyte, the concentration of ions is known only to a fiit approximation. Also, dilute electrolytes are complicated by adsorbed COzas well as specific cation adsorption with the column walls. Although only the conductivity of the equilibrated solution is required to obtain the potential, the comparison of our [ potentials as a function of electrolyte concentration with literature values is only a first approximation in dilute electrolyte. The pH was adjusted to 7.00 i 0.03 with small quantities of the electrolytes acidic or basic counterpart; e.g., for KC1 solutions the pH was adjusted with either HCl or KOH. When pH studies were done, the pH was adjusted by the same procedure. All measurementswere performed at room temperature, which was 25 k 3 O C . The temperature drift over the time of an experiment (
