Langmuir 1996, 12, 3425-3430
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A Study of Polymer Depletion Layers by Electrophoresis: The Influence of Viscosity Profiles and the Nonlinearity of the Poisson-Boltzmann Equation E. Donath,† A. Krabi,‡ G. Allan,§ and B. Vincent*,§ Department of Biology, Humboldt University Berlin, Berlin, Germany, Max Planck Institute for Colloid and Interface Research, Berlin, Germany, and School of Chemistry, University of Bristol, Cantock’s Close, Bristol, U.K. Received November 10, 1995. In Final Form: April 10, 1996X
Electrophoretic measurements of lipid liposomes in aqueous solutions of high molecular weight dextrans yielded a hydrodynamic depletion layer thickness of approximately 1 nm, assuming an exponential viscosity profile. This is significantly less than the corresponding radius of gyration of the dextran molecules in solution. The theory of electrokinetics in the presence of depletion layers has been extended to the case of nonlinear effects and arbitrary viscosity profiles. It is shown that the underlying functional character of the viscosity profile has only a small effect on the mobility. On the other hand, the depletion effect is very sensitive to the thickness of the depletion layer and to a shift of the plane of shear. The nonlocal character of the viscosity is also discussed.
Introduction Detailed understanding of the behavior of polymers at interfaces is of increasing interest in technological applications, as well as in basic research.1-3 Depletion of polymer molecules at a solid/solution interface takes place if the loss of configurational entropy of the polymer chains near the interface is not balanced by a favorable enthalpy change. In that case a polymer depletion layer is formed where the polymer segment density decreases toward the surface. This is accompanied by a concomitant position-dependent decrease of the osmotic pressure. For two approaching colloidal particles depletion attraction between the particles may result in reversible aggregation.4,5 Some aspects of depletion aggregation can be treated by reversible thermodynamics.6,7 A number of experimental studies have been reported in which the structure and extent of depletion layers have been characterized.8-11 It has been known for some time that unexpectedly high electrophoretic mobilities of particles can be found in solutions of some nonadsorbing soluble polymers.12-14 This †
Humboldt University Berlin. Max Planck Institute for Colloid and Interface Research. § University of Bristol. X Abstract published in Advance ACS Abstracts, June 1, 1996. ‡
(1) Fleer, G. J.; Scheutjens, J. M. H. M.; Cohen Stuart, M. A.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman and Hall: London, 1993. (2) Feigin, R. I.; Napper, D. H. J. Colloid Interface Sci. 1980, 75, 525. (3) Napper, D. H. Polymeric Stabilisation of Colloidal Dispersions; Academic Press: New York/London, 1983. (4) Fleer, G. J.; Scheutjens, J. M. H. M.; Cohen Stuart, M. A. Colloids Surf. 1988, 31, 1. (5) Vincent, B.; Edwards, J.; Emmett, S.; Croot, R. Colloids Surf. 1988, 31, 267. (6) Fleer, G. J.; Scheutjens, J. M. H. M.; Vincent, B. In Polymer Adsorption and Dispersion Stability; ACS Symp. Ser. 240; Goddard, E. D., Vincent, B., Eds.; American Chemical Society: Washington, DC, 1984; p 245. (7) Vincent, B. Colloids Surf. 1990, 50, 241. (8) Li-in-on, F. K.; Vincent, B.; Waite, F. A. J. Colloid Interface Sci. 1987, 116, 305. (9) Cowell, C.; Li-in-on, F. K.; Vincent, B. J. Chem. Soc., Faraday Trans. 1978, 74, 337. (10) Emmett, S.; Vincent, B. Phase Trans. 1990, 21, 197. (11) Vincent, B.; Edwards, J.; Emmett, S.; Jones, D. A. R. Colloids Surf. 1986, 18, 261. (12) Brooks, D. E.; Seaman, G. V. F. J. Colloid Interface Sci. 1973, 43, 670.
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effect has been explained by the existence of a depletion layer with a reduced viscosity as compared to the bulk solution viscosity value.15 Within the framework of the linear approximation of the Poisson-Boltzmann equation, and assuming an exponential viscosity profile in the depletion layer, an analytical expression for the electrophoretic mobility as a function of the thickness of the depletion layer has been derived for both “smooth” and “hairy” particles.16-19 Reasonable agreement between the theory and experiment has been demonstrated for mobility of human erythrocytes and liposomes in aqueous solutions of poly(ethylene glycol) and dextran.20 In general, the thickness of the depletion layer was found to be of the order of, or slightly less than, the radius of gyration of the polymer in solution. However, a crucial limitation of this initial theory is the artificial character of the assumed viscosity profile, which did not take into account any consideration of the statistical mechanics of polymers at interfaces.1 It was chosen purely for ease of analytical integration of the corresponding mobility expression. On the other hand, it is not expected that this approximation will deviate grossly from the real situation. In an earlier paper20 we reported electrophoresis measurements of liposomes in dextran solutions of high molecular weight, which were carried out over a large range of ionic strengths. These data are summarized in Figure 1. It can be clearly seen that the previously-used, linear approximation of the depletion effect (eq 16 in ref 20) fails completely to fit the data. Even when linearization was not assumed (eq 1 in ref 16), the data could only be fitted using a very small value for the depletion layer thickness, which is much less than the corresponding value of the radius of gyration of the polymer in solution. The question arises as to whether these poor fits are a result (13) Brooks, D. E. J. Colloid Interface Sci. 1973, 43, 687. (14) Snabre, P.; Mills, P. Colloid Polym. Sci. 1985, 263, 494. (15) Ba¨umler, H.; Donath, E. Stud. Biophys. 1987, 120, 113. (16) Pratsch, L.; Donath, E. Stud. Biophys. 1988, 123, 101. (17) Donath, E.; Pratsch, L.; Ba¨umler, H.; Voigt, A.; Taeger, M. Stud. Biophys. 1989, 130, 117. (18) Ba¨umler, H.; Donath, E.; Pratsch, L.; Lerche, D. In Hemorheologie et Agregation Erythrocytaire; Stoltz, J. F., Donner, M., Copley, A. L., Eds.; Editions Medicales Internationales: Cachan Cedex, 1991; p 24. (19) Donath, E.; Kuzmin, P.; Krabi, A.; Voigt, A. Colloid Polym. Sci. 1993, 271, 930. (20) Krabi, A.; Donath, E. Colloids Surf., A: Physicochem. Eng. Aspects 1994, 92, 175.
© 1996 American Chemical Society
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of the shear plane. Previously we performed the integration explicitly.16-19 This restricted us to the use of the linearized Poisson-Boltzmann equation, together with an assumed exponential viscosity profile21
η(x) )
ηP ηP 1+ - 1 e-x/λ η0
(
)
(5)
Here ηP is the viscosity of the bulk polymer solution and η0 the viscosity at x ) 0. λ is the parameter describing the effective thickness of the depletion layer. The halfwidth of the depletion layer, x1/2, depends, additionally, on the bulk viscosity. Figure 1. Electrophoretic mobility ratio of liposomes of 200 nm diameter measured in 500 000 dextran (4 wt %) as a function of ionic strength. Liposomes from PC, containing 20% PS. pH 7.4. The solid line represents the best fit according to the previous (linearized) theory.20 The dashed and dash-dotted curves correspond to an analytical linear (Poisson-Boltzmann) solution with an exponential viscosity profile16 with the thickness parameter of 15 and 1 nm, respectively. The corresponding values for the viscosity at the interface were 1.75 and 1 times the solvent viscosity, respectively. The dotted line is the expected mobility ratio if depletion is not present.
of the assumed artificial viscosity profile and also to what extent the inverse problem of establishing the viscosity profile from the mobility data, measured as a function of ionic strength, can be solved. In this work a numerical integration of the position-dependent viscosity form of the Navier-Stokes equation has been performed. Various profiles have been investigated and compared with some new experimental data. Theory In the Smoluchowski limit, of large smooth particles, the Navier-Stokes equation describing the electroosmotic profile, v(x), adjacent to the particle surface, taking into account the position dependent viscosity, η(x), may be expressed as follows:
-EF(x) )
d d η(x) v(x) dx dx
[
]
(1)
Here, x denotes the coordinate directed perpendicular to the surface with its origin at the surface, E is the electric field strength, and F(x) is the spatial density of the mobile charges in the double layer. The boundary conditions for eq 1 are
v(∞) bounded, v(x0) ) 0
(2)
2
F(x) d ψ(x) ) 2 r 0 dx
(3)
into eq 1 and subsequent integration results in the electrophoretic mobility, b (eq 4). ψ(x) is the electric potential and r and 0 are the relative and absolute permittivities, of the medium and free space, respectively.
v(∞) ) -r0 E
ηP + η0 η0
(6)
Previous applications of the electrokinetic theory of depletion have been to biological systems, mainly erythrocytes. Therefore, emphasis was put on depletion at hairy surfaces.18,19 A shift of the shear plane in the polymercontaining solution has not been considered. Cases occur frequently where the shear plane is displaced some distance from the surface of colloidal particles, e.g., particles carrying adsorbed or grafted polymer layers. For this reason an explicit expression for the ratio of the mobilities, as a function of the Debye-Hu¨ckel length, κ, is provided here.
bS η0 ) bP ηS
-x0/λ
e
e
-x0/λ
-
1 - η0/ηP [1 + κλ(1 - e-x0/λ)] 1 + κλ
(7)
Here, x0 denotes the position of the shear plane, bS is the mobility of the particles in the absence of the polymer and bP denotes the mobility in the presence of the polymer, and ηS denotes the viscosity of the solvent. It should be stressed that the assumed exponential viscosity profile starts at x ) 0. If other forms for the viscosity are chosen and if the nonlinear Poisson-Boltzmann equation is used, then explicit solutions of eq 4 cannot be found. It becomes necessary to resort to numerical integration. In the current work the well-known solution to the PoissonBoltzmann equation for flat plates and symmetric electrolytes was used:
[ ]
ze0ψ 4kT 1 x(ψ) ) - ln κ ze0ψ0 tanh 4kT tanh
Substituting the Poisson equation
b)-
x1/2 ) λ ln
dψ ∫ψ(x0 )η[(x(ψ)] 0
(4)
ψ(x0) is the zeta potential, where x0 denotes the position
(8)
where ψ0 is the surface potential (at x ) 0), z is the valency of the (symmetrical) electrolyte ions, e0 is the elementary charge, k is the Boltzmann constant, and T is the absolute temperature. We present analyses for the following viscosity profiles: a step profile, eq 9, a linear profile, eq 10, and a hyperbolic profile, eq 11 (this is the form derived by de Gennes for the polymer segment density distribution in a depletion layer.22 (21) Starov, V. M.; Churaev, N. V. Colloid J. 1982, 41, 297 (in Russian). (22) de Gennes, P. G. Macromolecules 1981, 14, 1637.
Polymer Depletion Layers by Electrophoresis
{
η η(x) ) ηS P
η(x) )
{
xeλ x>λ
ηP - ηS x+ηS λ ηP
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(9)
xeλ
(10)
x>λ
η(x) ) η(c)cbulk tanh2 (x/λ)
(11)
η(c) is an experimentally determined function describing the relationship between polymer concentration and bulk viscosity. In contrast to classical electrophoresis theory, where the viscosity is taken to be independent of position from the surface, eq 1 demonstrates the breakdown of the concept of the similarity of the hydrodynamic and the electric field.23 Consequently, it is to be expected that the effect of a polymer depletion layer on the electrophoretic mobility depends on the surface charge density if the nonlinearity of the Poisson-Boltzmann equation is taken into account. Furthermore, as eq 4 demonstrates, the electrophoretic mobility is given by an integral over the viscosity profile. Thus, it has to be determined to what extent, if any, variations in the viscosity profile might result in specific dependences of the mobility on the Debye length. It may be the case that different viscosity profiles result in similar mobility-ionic strength dependences. The solution of this problem is of crucial importance to the analysis of the experimental data.
Figure 2. Comparison of the electric potential decay as a function of distance from the interface in the linear and nonlinear approximations of the Poisson-Boltzmann equation. For the sake of comparison the same surface potential was used in the linear and nonlinear descriptions. The respective surface charge densities are as follows: curve 1, 0.05 C/m2; curve 2, 0.01 C/m2; curve 3, 0.0001 C/m2.
Experimental Section Materials. L-R-Phosphatidylcholine (PC), extracted from frozen egg yolk, and L-R-phosphatidylserine (PS), from bovine brain, were obtained from Sigma Chemical Co. Dextran with the molecular weights 110 000 and 500 000 and potassium chloride were from Fluka and used as supplied. The molecular weight distribution for each of the dextran samples was determined using gel permeation chromatography. The polydispersity ratios of the 110 000 and 500 000 dextran were MW/MN ) 1.53 and 3.51, respectively. Multilamellar liposomes were prepared by evaporation of a chloroform solution and subsequent gentle shaking of the remaining lipids in the electrolyte solution. They were converted to a more uniform size by means of an extruder (Lipex Biomembranes Inc.). The pore diameter of the Nucleopore filters used was 200 nm. The extrusion was carried out at 52 °C, above the gel-to-liquid crystalline phase transition temperature of the lipids. Aqueous solutions of KCl of different ionic strength were prepared in doubly-distilled water, adjusted to pH 7.4, and buffered with TRIS (tris(dimethylamino)methane) (1 mM). Electrophoresis. The electrophoretic mobility of the liposomes was measured using a Zetasizer 4 (Malvern Instruments, United Kingdom). All the measurements were performed at 24 °C. Liposomes with the required surface charge density were prepared by adjusting the ratio of the charged lipid (PS) to the neutral lipid (PC) in the initial chloroform solution. Electrophoretic mobility values (bS) were determined for liposomes containing 20% PS, over the range of electrolyte concentrations from 0.1 mM to 120 mM KCl. Dextran solutions of the appropriate concentration were prepared and their viscosities were measured and evaluated using a “Viscoboy” capillary viscometer (ex Lauda). Samples for the electrophoresis measurements were prepared by the addition of liposome stock solution to the polymer solution, to give a final lipid concentration of 0.2 mg/mL. The mobility values (bP) of the liposomes in the polymer solutions were determined and the electrophoresis results are presented as the ratio bS/bP. (23) Dukhin, S. S.; Deryaguin, B. V. Electrophoresis; Nauka: Moscow, 1976; p 40 (in Russian).
Figure 3. Effect of surface charge density on the mobility ratio in the presence of depletion. Charge densities correspond to Figure 2. In all curves an exponential viscosity profile with λ ) 5 nm and an initial viscosity of 1.5 times the solvent viscosity was assumed.
Results and Discussion Nonlinearity of the Poisson-Boltzmann Equation. The nonlinear solution for the potential profile in the case of a symmetric electrolyte is given by eq 8. The main consequences of the nonlinearity are that, firstly, with increasing surface charge density, the surface potential increases less and, secondly, that, at small distances from the surface, the decay against κ is steeper, as compared to predictions of linear double layer theory as illustrated in Figure 2. This second effect has the ramification of a greater contribution of the double layer region in the immediate vicinity of the interface. In the case of depletion, increasing surface charge density will cause the viscosity closer to the interface to have an increasingly dominant effect. Therefore, the depletion effect on electrophoresis should increase with the charge density. This point is demonstrated in Figure 3, where the ratio of the electrophoretic mobility of the control to that in the presence of polymer is plotted as a function of the Debye length, for three different surface charge densities. In these cases the exponential profile, given in eq 5, was used. A strong effect of increasing surface charge density is seen on the electrophoretic mobility in the polymer solution. As expected, the additional nonlinear
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Figure 4. Viscosity profiles according to eqs 9-11: curve 1, exponential; curve 2, hyperbolic; curve 3, linear profile. λ ) 5 nm in all cases.
Figure 5. Effect of the viscosity profiles given in Figure 4 on the theoretical depletion effect: curve 1, hyperbolic; curve 2, exponential; curve 3, linear profile. Surface charge density ) 0.01 C/m2. λ ) 5 nm.
contribution to the surface charge density increases with decreasing ionic concentration, i.e., increasing Debye screening distance. Viscosity Profile. Let us now explore the effect of the different viscosity profiles referred to earlier on the depletion effect. In Figure 4 three different viscosity profiles are plotted: the exponential profile given by eq 5 (curve 1), the hyperbolic viscosity profile provided in eq 11 (curve 2), and the linear profile given by eq 10 (curve 3). The characteristic length of the depletion, λ, is taken as 5 nm in each case. Due to the different functional behavior the apparent rate of increase of the viscosity as a function of distance from the surface is significantly different, although λ is the same. The slowest increase of viscosity is observed for the exponential profile, but it starts at an initially higher level, 1.5 times the solvent viscosity. The corresponding depletion effect on the electrophoretic mobility is shown in Figure 5. While, qualitatively, the depletion effect as a function of ionic strength, is similar for the three profiles, quantitatively there are strong differences. This depletion effect is chiefly determined by the form of the viscosity profile at small distances from the interface (see Figures 4 and 5). This is clearly observed when comparing curves 1 and 2 (Figure 5). The hyperbolic profile yields a higher viscosity at larger distances, while closer to the interface the viscosity is smaller, compared to the exponential profile. On the other hand, the depletion effect on mobility for the tanh profile is, over the whole range of ionic strengths, significantly
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Figure 6. Influence of the thickness parameter λ on the theoretical depletion effect. All curves correspond to an exponential profile (initial viscosity is equivalent to the solvent viscosity) with λ ) 15, 5, and 1 nm (curves 1-3). Surface charge density ) 0.01 C/m2.
larger than for the exponential profile. This underlines the importance of the viscosity behavior close to the surface. The result is readily understood considering the exponential distribution of the spatial charge density in the diffuse part of the electric double layer. On comparison of the qualitative behavior of the three depletion curves, it becomes clear that it would be almost impossible to deduce the actual form of the viscosity profile by inspection of the experimental data. On the other hand, the characteristic length parameter of the particular chosen viscosity profile should be readily obtained from experimental data. A theoretical plot demonstrating the effect of the length parameter, λ, on depletion is shown in Figure 6, where we have plotted the depletion effect for an exponential viscosity distribution for λ ) 1, 5, and 15 nm. It can be seen that the depletion effect is very sensitive to the length parameter, λ. With increasing λ, the depletion effect on the mobility naturally disappears at smaller values of the ionic strength. Comparison with Experimental Data. To compare the theoretically predicted depletion effect with experiments, the model described in the Experimental Section with liposomes was used. Electrophoretic mobility measurements where performed using both the 110 000 and 500 000 dextran samples. Special attention was given to measurements over a very large range of ionic strength. This allowed a wide range of values for the ratio of the polymer radius of gyration to the Debye length to be investigated. The experimental data for the two polymers are given in Figure 7. Somewhat surprisingly, the electrophoretic depletion effect does not appear to depend significantly on the molecular weight of the dextrans. However, different polymer concentrations were used, which yielded approximately the same bulk viscosity (4% for 500 000 and 6% for 110 000). One may also perhaps attribute this coincidence of the depletion effect on mobility for the two molecular weights used to some extent on the relatively broad distribution of the 500 000 dextran sample. We now explore different viscosity profiles and adjust the respective characteristic length parameters to fit the experimental data. The nonlinearity was incorporated in the following manner. From the electrophoretic mobility of the control the surface potential was calculated and used for the integration of eq 4 in the presence of the polymer. The insert in Figure 7 presents the three viscosity profiles: a step, a linear, and an exponential
Polymer Depletion Layers by Electrophoresis
Figure 7. Comparison of experimental data with theoretical fits as a function of the various viscosity profiles provided in the insert: solid squares, 4 wt % 500 000 dextran; solid triangles, 6 wt % 110 000 dextran. An indication of the experimental errors is given in Figure 1. Solid line, exponential profile with thickness λ ) 1 nm; dotted line, linear profile with thickness λ ) 3.5 nm; dash-dotted line, step profile with λ ) 1 nm. ηS and ηP denote the solvent and bulk polymer viscosity, respectively.
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plane, of 0.5 or 1.0 nm thickness, results in a decrease of the depletion effect on mobility. This decrease is greater at high ionic strengths where the shift of the shear plane becomes comparable to the Debye length. On the contrary, at low salt concentration the mobility tends to become independent of the position of the shear plane, as only at high ionic strengths is a significant portion of the counterions located behind the shear plane. Generally, if the shift of the shear plane is not too large, it results in the mobility ratio varying less over a given range of ionic strength. Another aspect of the hydrodynamic screening of a portion of the counterions is that, if the shear plane is further away from the interface, a non-monotonous behavior of the mobility ratio is observed. This effect shows up at very high salt concentrations where a significant number of counterions become excluded from the range of electroosmotic accessibility. One obvious reason for such a shift in the shear plane when polymer is present, compared to the control system, is if adsorption of polymer occurs. The depletion layer would exist outside the adsorbed layer of molecules. Although, adsorption cannot be ruled out a priori in the current system, one would expect a much thicker adsorption layer (and hence shift of the shear plane) than that investigated here. Conclusions
Figure 8. Theoretical effect of a shift of the shear plane, in addition to depletion, on the electrophoretic mobility ratio. All three curves correspond to a step profile with thickness λ ) 1.2 nm. The respective distance the shear plane is shifted for each curve is shown in the figure.
profile. When the length parameter (λ) was appropriately adjusted, the theoretical calculated depletion effect for the three curves coincided and matched the experimental data almost perfectly, within the experimental uncertainty, as demonstrated by the set of theoretical curves plotted in Figure 7. The following characteristic length parameters were used: 1 nm for the exponential profile, 1 nm for the step profile, and 3.5 nm for the linear profile. The linear profile with λ ) 3.5 nm yielded almost the same viscosity profile as the exponential profile with λ ) 1 nm. According to eq 6 both half-widths are almost equal. It cannot be excluded a priori that the shear plane may not correspond to the particle (in this case liposome) surface. We have therefore studied the influence of a possible shift (x0) of the shear plane on the effect of depletion on the mobility. Equation 4 was reintegrated, assuming the step viscosity profile (eq 9). This was chosen since the fit of the various viscosity profiles to the experimental data in Figure 7 is of approximately equal quality. Thus, the least complex profile is the most suitable one to investigate other effects, such as a shift in the shear plane. The result is given in Figure 8. It can be seen that even a relatively small shift of the shear
Electrophoretic mobility measurements have been made with lipid liposomes in dextran solutions of two molecular weights, over a wide range of ionic strengths, in order to compare the new theoretical developments of the depletion effect on the electrophoretic mobility presented here with experimental behavior. It has been shown that the new experimental data could be well described by the new approach, for a depletion layer of the polymer near the interface of approximately 1 nm apparent thickness. On comparison of the data obtained with both the 110 000 and 500 000 dextran samples, and normalization of the results to the respective bulk viscosity values, it was found that both plots were virtually coincident, as shown in Figure 7. This naturally led, within the experimental error, to identical estimates of the depletion layer thicknesses. It has to be emphasized that the value obtained for the depletion layer thickness (∼1 nm) is much less than the radius of gyration of the two dextrans used, which are 8.9 and 19 nm, respectively.20 This needs to be accounted for. At present it is not known which function would best describe the viscosity profile within the depletion layer. Analytical solutions for the polymer segment density profiles are available from statistical mechanics.24 However, since the polymer chains occupy a finite space comparable in dimensions with the thickness of the electric double layer, it is ultimately not correct to conclude anything from the time-averaged segment density profile about the viscosity profile. The viscosity, which is a kinetic coefficient, is determined by the interaction of the whole molecule with the solvent and other polymer molecules and, consequently, is a nonlocal parameter, whereas the segment density profile is local. Strictly speaking, the hydrodynamics should be treated nonlocally, which would mean replacing the NavierStokes equation (eq 1) by a nonlocal equation. At present, such a new concept, regarding electrophoresis in the presence of polymers, is not available. But it is clear that the main consequence of the nonlocal character of the hydrodynamics near the interface must be a significantly (24) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979; p 255.
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smaller hydrodynamic thickness of the depletion layer, as compared to the characteristic thickness of the segment density profile. This is simply a result of the fact that a polymer molecule as a whole is present both in the bulk and in the depletion layer simultaneously. It therefore experiences interactions of the bulk hydrodynamics as well. One might conjecture that this nonlocal character of the viscosity increases with increasing molecular weight. We believe that, to a certain extent, these considerations explain the surprisingly small hydrodynamic thickness of the depletion layer found. Another factor to be considered regarding the coincidence of the two thickness values estimated for both molecular weight samples is the width of the respective molecular weight distributions. There is, in fact, significant overlap of the two molecular weight distributions. The electrophoretic mobility is given by the electric field strength weighted integration over the inverse of the viscosity profile (eq 4). This causes a limited sensitivity of the electrophoretic mobility to the underlying assumed viscosity profile. Different profiles may yield similar mobilities (see Figure 7). In particular, the viscosity at larger distances from the surface, where the potential has decreased to small values, cannot be accurately assessed from the mobility measurements. This may be seen from the effect of the viscosity profiles in Figure 4 on the mobility ratios in Figure 5. In the limit of the assumed viscosity profile the mobility-ionic strength curves appeared to be quite sensitive to the hydrodynamic thickness of the depletion layer. Hence, this parameter can be obtained with greater accuracy than the details of the viscosity profile in the outer regions of the double layer. Naturally, different assumed profiles resulted in different estimates of the thickness parameter. Given the arbitrary nature of selection of the assumed viscosity profile, we recommend
Donath et al.
the use of either the step or exponential profiles, since these are characterized by only one free parameter, i.e., the thickness parameter, λ. The analysis of the experimental data further showed that it is best to assume that the profile starts with the value for the solvent directly at the interface. When we previously applied20 a linearized equation, two parameters were necessary. It is also important to include the nonlinearity of the electric potential profile. Otherwise two systems with identical depletion layers but with small and high surface charge would give apparently different depletion layer thicknesses and an erroneous value for the depletion layer thickness. It has also been shown that, even a small shift of the shear plane in the polymer system, as compared to the control, can have a significant influence of the depletion effect on the mobility. Such a situation can be easily distinguished by either a nonmonotonous behavior of the bS/bP-(κ-1) plot at high ionic strength or by a mobility value in the presence of the polymer that is smaller than that predicted by the bulk viscosity at low ionic strength. Experimental evidence for such situations has been found and is currently being prepared for publication. In summary, we hope that the achievements in the interpretation of electrophoretic depletion measurements made in this study give rise to further applications of electrokinetics to depletion layer measurements. Acknowledgment. The authors thank the DAADARC exchange program for financial support. E. Donath acknowledges the support of a grant of Deutsche Forschungsgemeinschaft (Do 410 1-1). G. C. Allan and B. Vincent acknowledge the support of the DTI “Colloid Technology” Programme. LA9510238
