Boundary Layer near the Surface of a Solid Body and Low- Frequency Dielectric Dispersion BY S. S . DUKHIN Institute of Colloid and Water Chemistry Academy of Sciences of the Ukrainian S.S.R. Kiev U.S.S.R. Received 19th March 1970 The paper deals with the possibility of studying the stagnant part of the boundary layer on the basis of effects due to polarization of the electrochemical double layer (DL). A theory is developed for polarization of the diffuse part of a thin DL of a spherical particle in a direct and alternating field allowing for the effect of ion flow through the stagnant part of the DL. Formulae are obtained on the basis of this theory for electrophoresis complicated by polarization and for the large low- frequency dielectric dispersion E(o). Calculation of the stagnant layer thickness and the ionic mobilities of this layer is shown to be possible on the basis of electrophoretic measurement on three fractions of spherical particles with identical values of $d and Z,.Using only one fraction we can obtain similar information from the low-frequency section of E(o). The advisability of investigating electrokinetic and related electro-surface pheno- mena in disperse systems in connection with the boundary layer problem has been acknowledged and substantiated by systematic studies 2-4 which have already yielded valuable results. We shall consider certain new means for studying the boundary layer by electric methods. Since terminology in this new field of research has not yet been established we shall adhere to Deryaguin’s model-system ideas and terminology.We shall call a boundary layer one in which anomaly of structure and the structural-sensitive properties of the liquid are observed The fact that within a layer of some thickness the liquid loses fluidity and does not participate in hydrodynamic processes is only one of many manifestations This layer is called stagnant. Such effects as thermo-osmosis and capillary osmosis indicate that the thickness of the boundary layer may exceed that of the stagnant layer. It is natural to compare the thickness of the stagnant layer with the distance from the surface to the slip plane a conception based on electrokinetic phenomena. The stagnant layer thickness may then be estimated by the difference in the values of the Stern $d and electrokinetic potentials. Simultaneous determination of the ( and $* potentials have been made for an oil-water i n t e r f a ~ e ~ with an absorption monolayer of ionogenic surfactant and for spherical micelles.lo In both cases the slip plane coincides with the surface enveloping the hydrated charges of the long- chain ions.This does not however affect the possibility of their being a stagnant polymolecular layer of polar liquid on a solid surface. Some difference between the [ and $d potentials may be due to the viscoelectric effect l1 and the roughness of the surface.12 Later research,l** l 3 however indicates that the viscoelectric constant was overestimated by more than one order in ref. (1 1). Hence with moderate values of t,bd and of the ionic force the viscoelectric effect of this anomaly of liquid structure. 158 S . S . DUKHIN 159 cannot cause a perceptible difference between the t,bd and c potentials and hinder the estimation of the stagnant layer thickness.Since the time of Freundlich who introduced the concept of the stagnant layer it was believed that within this layer the ions as well as the liquid were immobile so that only the mobile part of the diffuse layer contributes to the surface conduct- ance. This assumption is part of Bikerman’s l4 surface conductance theory and the Overbeek-Booth double-layer polarization theory and has persisted up to recently.16 There are no grounds for this assumption and it should be discarded. It has long been noted that normal values of ionic mobilities are retained in hydro- dynamically immobile ge1s.l’ The physical meaning of viscosity differs * depending on whether we consider the macroscopic flow of the system or the movement of small molecular particles through the same medium.The structural frame of the polymer excludes the possibility of macroscopic movement of liquid but does not hinder the thermal motion of the liquid molecules and ions,l* and consequently cannot prevent ion migration in an electric field. Similarly structure formation of a polar liquid under the effect of a surface gives rise to a hydrodynamically stagnant layer. But the intensive thermal motion of the liquid and the ion is maintained as evidenced from nuclear resonance data. Accordingly all diffuse layer ions are to be considered mobile and we shall call the diffuse-layer charge the mobile charge in distinction to the electrokinetic charge which is only part of the mobile charge.Simultaneous determination of the mobile and electrokinetic charges yields information about the ion content in the stagnant layer from which the difference between the I)d and 5 potentials and the stagnant layer thickness may be calculated with the aid of the Gouy-Chapman theory. Information about the mobile charge can be obtained from surface conductance measurements but the question then arises as to the possible deviation from the normal of ionic mobilities in the stagnant layer. The difference between the mobile and electrokinetic charges may also be due to surface roughness. This difference should not however be sensitive to temperature variations as is the case in structure formation of the liquid which determines the stagnant layer thickness. Fridrikhsberg measured three magnitudes characterizing the electric properties of BaSO microcrystal surfaces the surface conductance streaming potential and ionic adsorption.Comparing the results he concluded that the tangential electro- migration ionic currents permeate the stagnant layer as well as the mobile part of the double layer that the magnitude of ionic mobility in it is close to the bulk magnitudes of the mobilities and its thickness is approximately 15 A. The precision of Fridrikhsberg’s method of composite electro-surface measure- ments is decreased because capillary-porous systems were used. It is difficult to substantiate rigorously the possibility of calculating the specific surface conductance 7ca and the [ potential from macroscopic measurements on capillary-porous systems.Moreover when the contribution of the surface conductance to total conductance of the system becomes perceptible a necessary condition for measuring IP there is considerable double-layer polarization a substantial effect of which cannot be allowed for practically. However the effect of double-layer polarization which complicates investigation of the electrokinetic properties of capillary-porous system affords new opportunities for studying the stagnant layer in dilute dispersions. Development along these lines was apparently retarded because the cumbersome mathematical apparatus in the Overbeek-Booth theory of double-layer polarization of spherical particles made it impossible to take into account the difference between the I)d and c potentials mani- fested in the effect of the ionic stream through the stagnant layer on polarization and 160 BOUNDARY LAYERS AND DIELECTRIC DISPERSION electrophoretic movement.The mathematics of double-layer polarization of particles of regular shape has recently been considerably simplified 20* 21 in the important special case of a thin double layer. The new mathematical technique permits extension of the theory and in particular to base theory on various models of the colloid micelle taking into account the difference between the $d and 5 potentials. Under the effect of an external electric field tangential flows of diffuse layer ions arise which redistribute them over the surface; the double layer is deformed and polarized and thus departs from the original spherical symmetrical structure. Steady tangential flows of double-layer ions are maintained through ion exchange with the contiguous volume of electrolyte.The steady exchange of cations and anions is possible only when concentration differences arising beyond the double layer as well. A change in electrolyte concentration along the outer double-layer boundary causes a change in its thickness the latter becoming less when the electrolyte concentration is higher and increases with a decrease in electrolyte concentration. An essential simplification in thin double-layer theory2 is that although the double layer as a whole is in a non-equilibrium state equilibrium is maintained locally between the given double-layer area and the contiguous volume of electrolyte. For each double layer area i.e. at fixed 8 Boltzmann's formula retains its validity connecting the ionic concentration inside the double layer C* (X$) with the concentra- tion at its outer boundary c'(8) and with the change in potential across the double layer 4( X,8) where 8 is the angle with the external field E X is the distance to the surface Z* is the electrovalence.The deviation of 4(X,8) from the initial spherically symmetrical 4*(X) caused by the concentration change c*(O) can be expressed in general form through c*(O) on the basis of the local equilibrium between the given double-layer area and the contiguous volume of electrolyte. Functions are derived in ref. (20)-(21) describing the spatial potential and charge distribution in the polarized double layer and the contiguous electrolyte volume involved in the process of exchange with the double layer. This yielded a formula 22 for electrophoresis velocity complicated by double- layer polarization U,.We present the formula for dimensionless electrophoretic mobility in a symmetrical electrolyte z+ = z- = z = 1 C*(X,8) = c*(@ exp (Tzfe4(X,8)/kT) (1) 3&2 + 6m) sinh2 (f/4) + 41 + 6 In cosh (5"/4)[(2 + 6m) sinh (i"/2) - 3mf+ 2g] - - > (2) rca + (8 + 24m) sinh2 ([/4) - 24m In cosh (&I) + 4q where q = p(cosh@,/2) -cosh(i/2)) g = p(sinh(&,/2) - sinh(i/2)) &d = (e$d/kT) = (ec/kT>* ql and are the viscosity and dielectric constant of the liquid p is the ratio of the diffusion coefficients in the stagnant layer and in the bulk m is the dimensionless parameter given in ref. (15) (20) a is the radius of the sphere zc-I is the double layer thickness. The second term always negative characterizes the decrease in velocity due to polarization.Factors q and g characterize the effect of ionic currents through the S . S . DUKHIN 161 stagnant layer on polarization and electrophoresis. If the electrophoretic movement is measured on three fractions of spherical particles with surfaces of identical nature (i.e. 5 and $d is the same for all three fractions and Ica is known) we obtain three equations with three unknown from which we determine ( $d andp. Since with double-layer polarization the particle acquires an induced dipole moment the dielectric constant of the suspension should differ from that of the medium not only because of the trivial effect due to the difference between the dielectric constants of the medium and the particle. If the frequency of the alternating field is not great so that local equilibrium can be established between the given double layer area and the contiguous electrolyte volume the polarization mechanism in the alternating field is the same as in the static which enabled the author together with Shilov 23 to extend the thin double-layer polarization theory to the case of an alter- nating field of moderate frequency.Polarization of the particle proved to be associated with the ion concentration decrease along the outer double-layer boundary which varies periodically i n time with a certain lag behind the imposed field. This lag in phase is mathematically expressed in the fact that polarizability i.e. the ratio of the dipole moment to the external field cP is a complex variable if the time dependence of the alternating field is described by the exponent of the complex argument .Owing to the presence of conductance the continuous phase (electrolyte) is also polarized with a perceptible lag in phase so that its dielectric constant is also complex E = c1 -i4nK,/o being proportional to the conductance of the continuous phase K1. The dielectric increment caused by introduction into the continuous phase of colloidal particles with volume-ratio n may be represented by means of the Maxwell- Wagner theory 24 On multiplying Im EY; by Im a* a component of Re AE* of the suspension arises which increases considerably with decrease in m since the latter is accompanied by an unlimited increase in Im 8:. The complicated formula presented in ref. (25) for the static dielectric increment Re A$ i.e.Re AE* with m+O is simplified for highly charged particles when exp ( $ d / 2 ) 9 1 AE* = 3n~Ta". (3) A huge rise in Re AC-* was observed in the low-frequency band by Schwann et aZ.,26 working with monodisperse suspensions of spherical polystyrene particles in an aqueous solution of KCl. For the special case of a = 0.094pm n = 0.3 the experimentally found relationships Re E*(w) (curve 1) and Im E*(m) (curve 2) are presented in fig. 1. Assuming $d = c p = 1 Ica = 60 (in accordance with the electrolyte conductance given in ref. (26) and using the mcasured value of Re E*(m) at co-+O equal to 2370 we arrived at the conclusion that $d = 3.5 for the particles. Accepting this value of t j d we then calculated Im ?(a) (curve 3). A certain dis- crepancy between the experimental and theoretical curves of Im E*(m) may be due to the fact that at the volume-ratio of the particles used in the experiment their diffusion atmospheres greatly overlapped a complication which cannot be readily taken into consideration in the theory and hence was neglected.For the interpretation of their experimental data Schwann and collaborators used Schwarz's hypothesis 24 about the peculiar behaviour of counterions which they called " bound ". These ions are said to be capable of redistribution under the effect SPI-F 162 BOUNDARY LAYERS AND DIELECTRlC DlSPERSlON of the field along the particle surface without moving away from it i.e. the possibility of ion exchange with the disperse medium is excluded. Such behaviour of the double- layer ions has not yet been explained on double-layer theory; moreover in later papers the investigators note the inadmissibility of such an idealization.28 The double-layer model accepted by Schwarz assumes that the outer part of the double 2 0 0 0 1000 layer is formed by bound Within the frame work of mental results.- - - 3 0 0 0 1200 I ions only and there is accordingly no diffusion layer. this model Schwarz satisfactorily interprets the experi- 8 0 0 6 00 2 0 0 I 1 I I I I 0 I 10 100 I000 frequency (kHz) FIG. 1 .-Dispersion of dielectric constant of dispersion polystyrene spherical particles ; experimental data 26 and theory of polarization of diffused part of double layer 2 5 ; 1 and 2 Re Ai* and Im AE* from data of ref. (26) ; 3 Im AE* theoretical curve.25 Even if there is no diffuse atmosphere in an equilibrium double layer as assumed by Schwarz on polarization of the bound counterion layer a diffuse atmosphere appears locally compensating the polarization charge of the bound ions.Since Schwarz did not have at his disposal a theory of the polarization of the diffuse part of the double layer his theory did not take into account the potential jump in this diffuse atmosphere and its effect on the tangential transfer of the bound ions. The correction of this error in a paper by Shilov and the author 2 5 dealing with Schwarz’s model indicates that in Schwarz’s theory Re AE*(co) is over-estimated by more than one order i.e. there can be no agreement between Schwarz’s theory and Schwann’s experiment. Ions bound according to Schwarz’s theory affect A P but cannot contribute to 5. Hence experimental proof of the equality of [ and t,bd potentials by the electrophoresis data and dielectric measurements is at the same time proof of the non-existence of Schwarz’s bound ions.Chelidze 29 measured U and i*(co) for almost nionodispersed diluted latex suspensions of nairite (a = 0.13 pm 7ca - 15) and chloroprene (a = 0.45 pm ica = 60) in aqueous solutions obtaining values of 3400 and 1400 for Re A$/n. According to these experimental data cSM calculated according to Smoluchowski’s formula equals 2.3 ; 4 and T calculated from the solution of the system of equations (2) and (4) equal 3.5 and 3.5. For chloroprene SSM = 2.5 qd = 3.2 f = 3. S . S . DUKHIN 163 The difference between $d and c, and the agreement within limits of experimental error of $d and 5 calculated from eqn (2) and (4) corroborate the correctness of the theory of double-layer polarization in a direct and alternating field its effect on electrophoresis and Ai?.Along with the difference of more than one order between the experimental values 26 of Re A$ and those calculated from the refined formula for A 2 derived for the Schwarz Chelidze’s experiments 29 indicate that Schwarz’s model does not agree with the facts at least with respect to the investigated systems and that AE*(u) measurements constitute a promising niethod for measuring the $d potential. In an experimental verification of the DLVO theory 13* 30 or the electroviscous effect on monodisperse suspensions it is advisable to measure the large low-frequency dielectric dispersion rather than electrophoresis. In addition agreement of 5 and $d indicates that in the investigated systems boundary layers are either absent or so thin as compared to K - ~ that they cannot be detected with the present experimental precision.To secure detection and measurement of the boundary layer thickness by the recommended method latices should be used for which the presence of boundary layers is more probable (those studied in ref. (5) for instance) and I C - ~ should be decreased by raising the electrolyte concentration. If not only Re A$ but the entire low-frequency section of the dispersion curve is used it is possible to determine the change in ionic mobilities in the boundary layer along with 5 and $d. J. Th. G. Overbeek Pure Appl. Chem. 1965 10 359. D. A. Fridrikhsberg and V. Y . Barkovsky Kolloid Zhur. 1964 26 722. N. Bondarenko S.Nerpin Bulletin RILEM 1964 29 13 ; N. Bondarenko S. Nerpin Ztzt. Congr. Pure Appl. Chem. (Moscow 1965) thesis A and B. S. V. Nerpin and A. F. Chudnovsky Soil Phys. 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