J . Phys. Chem. 1987, 91, 6415-6417
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Influence of Bulk Diffusion on the Counterion Polarization in a Condensed Counterion Model Constantino Grosset Instituto de Fisica, Universidad Nacional de Tucuman, (4000) Tucuman, Argentina
and Kenneth R. Foster* Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 191 04 (Received: October 27, 1986)
Suspensions of charged particles in aqueous electrolyte exhibit a large dielectric dispersion at low frequencies. An early model by Schwarz provided a simple explanation for this phenomenon in terms of the diffusion of counterions along the surface of the particles. We reconsider this model taking into account the fact that outside the condensed counterion layer the potential must satisfy Poisson’s rather than Laplace’s equation. We show that in the case of low-conductivityelectrolytes, significant corrections are needed to the original results of Schwarz because of the diffusion of ions in the bulk electrolyte near the particles. In this case the counterion layer behaves just as a conducting shell and the large permittivity values attained at low frequencies originate from a simple capacitive charging effect. The main qualitative difference with the original results of Schwarz is that the dielectric increment has now an upper bound which cannot be surpassed no matter what values are assigned to the parameters of the model. While this bound still allows for an interpretation of some previous experimental data, it precludes any interpretation of many other results in terms of the model.
Introduction A great variety of heterogeneous materials such as rocks, wet sand, bone, biological tissues, suspensions of polyions, and DNA solutions exhibit extremely high permittivity values at low frequencies: up to 106 or more in some cases. All these systems have in common the presence of fixed charges and an electrolyte as the continuous phase. The simplest among them are suspensions of charged spherical particles, such as micron-sized latex spheres, which exhibit dielectric increments of the order of lo4 at audio frequencies. The first interpretation of this phenomenon was given by Schwarz in 1962,2 who assumed that the underlying mechanism is the redistribution of counterions. In his model the counterions are considered to be tightly bound to the surface of the charged particle so that they can only move along its surface. In 1970, Shilov and Dukhin3 reexamined this model and obtained values for the dielectric increment that are more than an order of magnitude lower than experimental results.’ Their conclusion that Schwarz’s model cannot produce a significant contribution to the low-frequency polarization was the basis for a series of later ~tudies,~-” which consider that the counterions form a diffused cloud surrounding the particle. Nevertheless, and due to the complexity of these later theories, Schwarz’s model remains the only simple explanation of the phenomenon and his results are still the only ones widely used in the interpretation of experimental data.12-15 As the model was originally developed, it had two adjustable parameters and thus considerable ability to “fit the data”. In the present work we reexamine the condensed counterion model. We show that the original results of Schwarz must be corrected in order to take into account the diffusion of ions in the bulk electrolyte surrounding the counterion layer. In contrast to the results of ref 3, our results show that this model does lead to a significant polarization at low frequencies, which is sufficient to account for the experimental results of Schwan. Nevertheless, it is unable to give an interpretation to a series of experimental results obtained with low electrolyte conductivities. Therefore, a consistent development of the model, within its own terms, shows its intrinsic limitations. *Address reprint requests to this author at the University of Pennsylvania. Consejo Nacional de Investigaciones Cientificas y Tecnicas de la Republica Argentina.
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Development The model used by Schwarz can be summarized as follows. The insulating particle is surrounded by a thin layer of counterions. Their movement is restricted to its surface and is determined by the local values of the density gradient and the tangential component of the electric field. A further assumption is that there is no exchange of ions between the counterion layer and the rest of the electrolyte. The results obtained by Schwarz for the relaxation time and the dielectric increment which characterize the low-frequency relaxation are
In these expressions R is the radius of the particle, DSuris the diffusion coefficient for the movement of counterions along its surface, v is the volume fraction occupied by the particles in the suspension, X is the surface conductivity of the counterion layer, F/m. Thus the relaxation time is solely and eo = 8.85 X determined by the diffusion of counterions along the surface of the particle, while the dielectric increment does not directly depend
(1) Schwan, H. P.; Schwarz, G.; Maczuk, J.; Pauly, H. J. Phys. Chem. 1962, 66, 2626. (2) Schwarz, G . J. Phys. Chem. 1962.66, 2636. (3) Shilov, V. N.; D u h i n , S. S. Kolloid. Zh. 1970, 32, 293. (4) Dukhin, S. S.;Shilov, V. N. Dielectric Phenomena and the Double h y e r in Dispersed Systems and Polyelectrolytes; Halsted: Jerusalem, 1974. (5) Fixman, M. J. Chem. Phys. 1980, 7 2 , 5177. (6) Chew, W. C.; Sen, P. N. J. Chem. Phys. 1982, 77, 4684. (7) Fixman, M. J . Chem. Phys. 1983, 78, 1483. (8) O’Brien, R. W. Ado. Colloid Interface Sci. 1983, 16, 281. (9) Lyklema, J.; Dukhin, S.S.; Shilov, V. N. J . Electroanal. Chem. 1983, 143, 1. (10) Chew, W. C. J. Chem. Phys. 1984, 80, 4541. (11) Mandel, M.; Odijk, T. Annu. Rev. Phys. Chem. 1984, 35, 75. (12) Sasaki, S.; Ishikawa, A.; Hanai, T. Biophys. Chem. 1981, 14, 45. (13) Cavell, E. A. S . J . Colloid Interface Sci. 1977, 62, 495. (14) Ballario, C.; Bonincontro, A.; Cametti, C. J. Colloid Interface Sci. 1976, 54, 415. (15) Ballario, C.; Bonincontro, A,; Cametti, C. J . Colloid Inferface Sci. 1979, 72, 304.
0 1987 American Chemical Society
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The Journal of Physical Chemistry, Vol. 91, No. 25, 1987
Grosse and Foster
E
n ~,l~/m]
Figure 1. (a) Schematic representation of the charge density around an insulating particle according to the solution of Laplace's equation. (b) Actual distribution of induced charges resulting from diffusion. (c) Charge density profile which is equivalent to (b).
on the conductivity of the electrolyte. The above results were obtained from the solution of Laplace's equation, which requires that the charge distributions induced by the applied field form a layer of infinitesimal thickness around the particles. This situation is represented on Figure la, in which the circle corresponds to the particle together with the counterion layer. Due to diffusion, the induced charges actually form a cloud above the surface with a thickness of the order of the Debye screening length l/xm,where Xm2
= um/('OtmD)
+ x m - l / c m + xp-'/cp
-
,/
i
e 10"
(3)
and u, is the conductivity, and D is the diffusion coefficient of the ions in the bulk electrolyte (Figure Ib). The simplest way to take this into account is suggested by the last equation in ref 16, which applies to a conducting particle surrounded by an insulating membrane and immersed in the electrolyte: d*/cs = d/cs
Figure 2. Theoretical and experimental results for the relaxation time of the suspension divided by the square of the radius of the particle, represented as a function of the conductivity of the electrolyteand calculated by using the values D = 2 X m2/s and Os,,= 0.5 X m2/s. Horizontal line, eq 1; curves, eq 5 ; full lines, eq 7. Experimental A, Ballario;I5 X, values: 0 , Schwan;' +, Sasaki;'* @, Ballari~;'~ Springer;'*0,Lim.19
(4)
In this equation, which is valid for xmR >> 1, tp and xp are the permittivity and the reciprocal of the Debye screening length in the particle, cs is the permittivity of the membrane, and d is its thickness. Equation 4 defines an equivalent thickness d*: the actual response of the system to an applied field is the same as if the insulating layer had a thickness d* and no diffusion effects existed. For homogeneous nonconducting particles, d = 0 and the last term in eq 4 is not present. Thus the volume distribution of field-induced charges can be replaced for purposes of analysis by a surface distribution located at a distance tS/(tmxm)from the particle and separated from it by a layer of permittivity ts. Choosing the value t, for this permittivity, which is otherwise arbitrary, leads to the following conclusion: the spread of the field-induced charges around a nonconducting homogeneous particle can be taken into account by considering that these charges are all located one Debye screening length from its surface (Figure IC). This result can be applied to Schwarz's model, since the particle together with the counterion layer can be reduced to a homogeneous sphere with a static conductivity equal to zero, due to the hypothesis that the counterions cannot exchange with the ions in the bulk electrolyte.2 This hypothesis is equivalent to the assumption that the counterion layer is surrounded by an infinitesimally thin insulating membrane. According to the above conclusion, the effect of bulk diffusion can be taken into account by considering that the thickness of this membrane is actually (16) Garcia, A.; Barchini, R.; Grosse, C. J . Phys. D 1985, 18, 1891.
%[s/m]
Figure 3. Theoretical and experimental results for the dielectric increment of the suspension divided by 9uR/(2 + u ) and ~ by the radius of the particle, represented as a function of the conductivity of the electrolyte and calculated by using the values D = 2 X m2/s and D,,,= 0.5 X m2/s. Horizontal lines, eq 2; curves, eq 6 ; full line, eq 8. Experimental value symbols as in Figure 2.
equal to the Debye screening length and that its permittivity is e,. A simple calculation yields R2/ (2DsIJr) T =
1+
(5)
W&"J cmXmR
'
1+ CmXmR
Thus, diffusion effects in the bulk electrolyte always decrease the relaxation time and the dielectric increment of the suspension below the values derived considering only surface diffusion of the bound counterions. For small particles in electrolytes of low conductivity eq 5 and 6 tend to the following limiting forms:
t(0) -
€(a)=
9u
(2
+ u)2emXmR
These results can be also obtained as the limit of eq 5 and 6 when the surface diffusion coefficient becomes vanishingly small. Therefore they correspond to a model in which it is assumed that the counterion layer simply behaves as a conducting surface.
J. Phys. Chem. 1987, 91, 6417-6422 The theoretical results for the relaxation time and for the dielectric increment are plotted as functions of the conductivity of the electrolyte in Figures 2 and 3, together with all the available experimental results for suspensions of polystyrene spheres in KCl electrolyte. The plots correspond to the value D,,, = 0.5 X m2/s (chosen in order to fit the experimental data of Schwan) and two different values of the surface conductivity. The scatter of the experimental points might arise in part from experimental uncertainties. It more likely reflects that the relaxation parameters do not simply depend on R and u, but rather on combinations of these variables.” Figure 2 shows that in the condensed counterion model the surface diffusion only determines the relaxation time in the limit of very high electrolyte conductivities. On the contrary, for low conductivities, the relaxation time is mainly determined by the surface conductivity of the counterion layer. The original results of Schwarz can be made to fit any experimental result for the relaxation time and the dielectric increment by a proper choice of the parameters X and D,,,. Figure 3 shows, on the contrary, that due to bulk diffusion the dielectric increment has an upper bound given by eq 8. Therefore, Schwarz’s model fundamentally cannot account for a large part of the experimental data, 5 ~ 1
Conclusion We have shown that the results originally obtained by Schwarz should be corrected to take into account the diffusion of ions in (17) Grosse, C.; Foster, K. R. J. Phys. Chem. 1987, 91, 3073. (18) Springer, M. M.; Korteweg, A.; Lyklema, J. J . Electroanal. Chem. 1983, 153, 55. (19) Lim, K.-H.; Franses, E. I. J. Colloid Interface Sci. 1986, 110, 201.
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the bulk electrolyte. When this is done, it appears that the interpretation of the polarization as a mechanism solely controlled by the surface diffusion of counterions only applies to large particles in high conductivity electrolytes. In many other cases of interest, the tightly bound counterions essentially behave as a conducting layer so that the counterion polarization reduces to a simple capacitive effect. The relaxation time is then mainly dependent on the surface conductivity while the dielectric increment is determined by the conductivity of the electrolyte. Our results formally resemble those of Shilov and Dukhin: our eq 6 is identical with eq 26 in ref 3, except for the factor ,E which is missing in this last expression. Because of this difference, the dielectric increment calculated in ref 3 was more than an order of magnitude lower than the experimental results of Schwan.l Figure 3 shows, on the contrary, reasonably good agreement between the corrected Schwarz theory and part of the experimental value^,^*^**'^ showing that a layer of tightly bound counterions which do not exchange with the bulk electrolyte can produce a significant contribution to the low-frequency relaxation. However, the dielectric increments predicted by the theory are substantially below other measured value^.^^^'^*'^ This difference is particularly significant since eq 8, which represents the upper limit of the corrected Schwarz result, has no adjustable parameters. It follows that, a t least for these systems, Schwarz’s model does not adequately represent the behavior of the counterions. A more appropriate simple model is presented in ref 17 that might be adequate to interpret experimental data. More rigorous (and far more complex) models are reviewed in ref 11.
Acknowledgment. We thank Professor H. P. Schwan for numerous discussions and suggestions about this work. This work was partially supported by Office of Naval Research Contract NO0014-86-K-0240.
Theory of Shape Transitions in Two-Dlmenslonal Phospholipid Domains D. J. Keller, J. P. Korb; and H. M. McConnell* Stauffer Laboratory for Physical Chemistry, Stanford University, Stanford, California 94305 (Received: January 26, 1987; In Final Form: June 4, 1987)
A theory is presented in which the noncircular shapes of two-dimensional solid domains of phospholipid are determined by a competition between repulsive electrostatic forces (which favor elongation of the domains) and interfacial line tension (which favors round domains). A general argument is given that, in the absence of domain fission, the crossover from circular to noncircular shapes must occur in a sharp, second-order “shape transition”. An explicit calculation is given for the special case where the noncircular shape is elliptical. When the electrostatic forces are due entirely to free charges, or perpendicular polarization, it is found that the shape transition can be described quantitatively.
I. Introduction Pure, single-component, phospholipid monolayers a t the airwater interface can exist in (at least) three two-dimensional phases: a highly expanded “gas” phase at low pressure, an intermediate fluid phase at higher pressures, and finally a dense solid phase. In passing from the fluid to the solid phase, there is a region of two-phase coexistence where domains of solid lipid emerge and grow from the fluid Perhaps the most striking feature of the solid domains is their unusual ~ h a p e . ~ -When ~ a small amount of cholesterol is added to a monolayer of dipalmitoylphosphatidylcholine (DPPC) or dimyristoylphosphatidic acid (DMPA) under certain conditions of temperature, pH, and ionic strength, it is found that the solid domains form long thin strips of uniform width. The width of the strips changes reversibly as Laboratoire de Physique de la Matiere Condensee, Ecole Polytechnique, 91 128-Palaiseau. France.
0022-365418712091-6417$01 SO10
the monolayer is compressed or the temperature is lowered and the domains increase in size. When the lipid molecules are chiral the striplike domains curl and form spirals, with the sense of the curling determined by the handedness of the lipid.5 (1) Losche, M.; Sackmann, E.; Mohwald, H. Eer. Bunsen-Ges. Phys. Chem. 1983, 87, 848-852. (2) Peters, R.; Beck, K. Proc. Natl. Acad. Sci. U.S.A. 1983, 80,
-
71R1-71R7 . - -- . - . .
(3) McConnell, H. M.; Tamm, L. K.; Weis, R. M. Proc. Narl. Acad. Sei. U.S.A. 1984 81, 3249-3253. (4) Weis, R. M.; McConnell, H. M. Nature (London) 1984,310.47-49, (5) Weis, R. M.; McConnell, H. M. J . Phys. Chem. 1985,89,4453-4459. ( 6 ) Gaub, H. E.; Moy, V. T.; McConnell, H. M. J . Phys. Chem. 1986,90, 1721-1725. (7) Moy, V. T.; Keller, D. J.; Gaub, H. E.; McConnell, H. M. J . Phys. Chem. 1986, 90, 3198-3202. (8) Moy, V. T.; Keller, D. J.; McConnell, H. M. J . Phys. Chem., to be published.
0 1987 American Chemical Society
