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Electrodynamics of Liposome Dispersions R. Barchini, H. P. van Leeuwen,* and J. Lyklema Laboratory for Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands Received December 14, 1999. In Final Form: July 11, 2000 This work presents a theoretical and experimental study of the electrodynamic properties of liposome dispersions. The experiments focus on measuring the electrophoretic mobility and the low-frequency dielectric spectra of unilamellar dioleoylphosphatidylglycerol vesicles suspended in 1-1 and 2-1 electrolytes. The theoretical part presents a relatively simple model for the dynamics of the electric double layer surrounding the liposome. It includes an analytical expression for the frequency-dependent permittivity, which successfully accounts for the experimental spectra. The resulting parameters are consistent with the electrophoretic mobility of the liposome. In the range of electrolyte concentrations investigated, we do observe some effect of the counterion valency, but there is no other ion specificity in the double-layer dynamics.
1. Introduction Understanding fundamental biological cell processes constitutes one of the major goals of current biophysicochemical research. The functioning of cells is to a large extent related to electrochemical processes that occur at and across their membranes. In particular, the negative electric charge of a biomembrane plays a very important role because it generates differences in ion concentrations between its two sides. The corresponding transmembrane potential influences the conformation and functioning of many molecules that in turn may affect the translocation of charged species. Moreover, the binding of active ionic species also affects the aggregation behavior of cells. Because the backbone structure of cell membranes is the lipid bilayer, liposomes are widely used as model systems. At neutral pH, liposomes consisting of acidic phospholipids possess a negative charge due to their dissociated groups (in our case, -PO4--) and hence they provide good models for investigating membrane/ion interactions. In this work we present a study involving dioleoylphosphatidylglycerol (DOPG) bilayer vesicles, immersed in an aqueous electrolyte solution, as model membranes. To facilitate the theoretical interpretation of the experimental data, special care was taken in preparing simple welldefined model dispersions. The experimental technique and the pertaining physicochemical model analyze the polarizing effect of an applied alternating current (ac) electric field in the 600 Hz-500 kHz frequency range. The ensuing polarization is a measure of the extent of redistribution of charges in the double layer. This may strongly depend on the presence of ions in the stagnant part of the double layer and their ability to move along the surface of the liposomes. One of the objectives of the present paper is therefore to determine the mobilities of counterions adsorbed in the stagnant layer. The correctness of the interpretation of the dielectric results is verified by checking the consistency with electrophoretic mobility data and by comparison with independent literature data. Given the complex nature of the liposome/solution interphase, we have used a rather phenomenological model of the double-layer dynamics. It is based on the thin double* To whom correspondence should be addressed. Fax: (NL) 317483777.
layer approximation and fast exchange of counterions between stagnant and diffuse layers, governed by electroneutrality. 2. Experimental Methods and Results 2.1. Liposome Preparation. DOPG was purchased from Avanti Polar Lipids, Alabaster, AL. According to the procedure explained in ref 1, the phospholipid was dissolved in a chloroform/ methanol mixture (3/1 v/v) and dried in a rotary evaporator under reduced pressure. The final evaporation to dryness was carried out overnight under high vacuum. The film was then hydrated by vortex-mixing in a NaCl solution and the obtained suspension frozen and thawed 10 times using liquid N2 and a water bath at 25 °C, respectively. The resulting multilamellar vesicle dispersion was then extruded through a stainless steel extruder with two stacked polycarbonate filters (Nucleopore, 100 nm), and this was repeated 10 times. This procedure ensures the formation of a highly monodisperse dispersion of unilamellar vesicles.1,2 Under the given conditions of preparation, the obtained liposome size essentially depends on the pressure applied during extrusion. With these vesicle dispersions, the external electrolyte was adjusted to the desired final concentration. The electrolytes considered in this study were NaCl, CaCl2, CdCl2, and CuCl2, purchased from Sigma. The aqueous medium was always prepared with distilled deionized water. The concentrations used are indicated in Table 1. The pH of the samples was around 6. To check the effect of the vesicle interior, experiments were run with 1 mM and 0.1 mM NaCl as the internal electrolyte. Vesicle dispersions were kept under nitrogen, refrigerated, and all determinations were carried out within a week after preparation. 2.2. Sizing. Liposome diameters were determined by dynamic light scattering. The instrumental setup consists of an ALV5000 correlator and a scattering device with an ALV-125 goniometer and a multiline Lexel Ar-laser source. The data were collected at a scattering angle of 90° for a wavelength of 514.5 nm. The measuring temperature was 25 °C. Before the sizing, each sample was diluted by adding outer electrolyte. The measured mean radii a are indicated in Table 1. The polydispersity was less than 1%. From the liposome radius and the total lipid content of the samples the vesicle volume fraction v could be computed. The results are given in Table 1. The final lipid concentration in the samples was determined by a modified Barlett3 procedure for (1) New, R. R. C., Ed. Liposomes, A Practical Approach; Oxford University Press: Oxford, 1994. (2) Mayer, L. D.; Hope, M. J.; Cullis, P. R. Biochim Biophys. Acta 1986, 858, 161. (3) Barlett, G. R. J. Biol. Chem. 1959, 234, 466.
10.1021/la991640m CCC: $19.00 © 2000 American Chemical Society Published on Web 09/29/2000
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Table 1. Characteristics of Vesicle Dispersions Used sample
intravesicular electrolyte
extravesicular electrolyte
I II III
1 mM NaCl 0.1 mM NaCl 0.1 mM NaCl
IV
0.1 mM NaCl
V
0.1 mM NaCl
1 mM NaCl 1 mM NaCl 0.1 mM NaCl + 0.45 mM CdCl2 0.1 mM NaCl + 0.45 mM CaCl2 0.1 mM NaCl + 0.45 mM CuCl2
radius a (nm)
volume fraction v (%)
mobility u (10-6m2/Vs)
61 ( 2 60 ( 2 71 ( 2
4.80 5.20 4.51
-4.15 ( 0.08 -4.19 ( 0.08 -1.90 ( 0.05
74 ( 3
4.38
-2.14 ( 0.06
68 ( 3
5.39
-1.98 ( 0.05
phospholipid phosphorus analysis. After analyzing published experimental and numerical simulation results for the headhead spacing,4,5 the headgroup area per lipid molecule was taken as 0.72 nm2. The thickness d of the lipid bilayer was determined by a technique different from those normally used: we measured the wavelength dependence of the light transmitted by a dilute liposome sample (UV/vis spectrophotomer, Hitachi). For a dilute sample, the transmittance is additively determined by the light scattered by all individual liposomes (no interference effects). A plane electromagnetic wave of wavelength λ and intensity Io, after traveling over a distance x through the sample, has an intensity It given by:
It ) Io exp
[
]
3Q(λ,a,d,ns,nm)vxX(θ,a) 4a
(1)
where X(θ,a) denotes a correction factor due to the detector’s finite field of view.6 The quantity of interest is the scattering efficiency Q, defined as the total scattering cross-section divided by the geometric cross-section of the vesicle. By taking the ratio of the logarithms of the transmitted intensities at two different wavelengths, it is possible to compare the calculated scattering efficiency ratios with those measured (from the intensities). The theoretical values were calculated assuming the vesicle to be a coated particle, where the coating is taken as a hydrocarbon chain shell. The scattering efficiency depends on the wavelength, the vesicle radius, the thickness of the shell, and the refractive indices of the three different zones. We assumed a refractive index nm ) 1.33 for the inside and outside medium of the liposome (intra- and extravesicular electrolytes), and ns ) 1.55 for the hydrocarbon chain shell. The calculation was repeated as a function of the thickness d until a best fit was achieved. Figure 1 shows such a comparison for the case of the dispersion in CaCl2. For all other samples studied the best agreement also corresponds to a shell thickness of 6 nm. The results given in this paper are based on the Mie solution of the scattering problem according to a numerical code for coated particles as given by Bohren and Huffman.7 2.3. Electrophoresis. Mobility measurements were performed using a Malvern Zetasizer 3 (Malvern Instuments, U.K.). The measuring mode, described in ref 8. makes use of an ac field that effectively suppresses electroosmosis. This approach allows more rapid determinations because it is no longer necessary to measure the position of the stationary layers. Table 1 reports the measured mobilities. In all cases, the net charge of the liposomes is negative, but the presence of divalent counterions considerably reduces the mobility. This could be indicative of a stronger attachment of such counterions and concomitantly stronger friction in their motion. 2.4. Dielectric Response. The low-frequency dielectric properties of the liposome samples were measured in the frequency range 600 Hz-500 kHz using a four-electrode dielectric (4) Nagle, J. F.; Zhang, R.; Tristam-Nagle, S.; Sun, W.; Petrache, H. I.; Suter, R. M. Biophys. J. 1996, 70, 1419. (5) Lau, A.; McLaughlin, A.; McLaughlin, S. Biochim. Biophys. Acta 1981, 645, 279. (6) Deepak, A.; Box, M. A. Appl. Opt. 1978, 17, 2900. (7) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; John Wiley & Sons: New York, 1983. (8) Minor, M.; van der Linde, A. J.; van Leeuwen, H. P.; Lyklema, J. J. Colloid Interface Sci. 1997, 189, 370.
Figure 1. Comparison of predicted (continuous line) and measured scattering efficiencies Q(λ,a,d,ns,nm) relative to its value at 350 nm, taken arbitrarily as the reference (points). The measured values correspond to a dilute liposome dispersion in CaCl2. The calculation corresponds to a bilayer thickness of 6 nm. spectrometer connected to a Solartron impedance analyzer.9 The cell was thermostated at 25 °C. The static conductivities were independently measured with a Knick bridge Konduktometer 702. The polarization induced by the externally applied ac field is described with a complex dielectric permittivity:
ˆ (ω) ) (ω) -
iK(ω) ωo
(2)
where ω represents the angular frequency, ο the absolute permittivity of free space, the relative permittivity (real part of ˆ ), and Κ the conductivity (imaginary part of ˆ ) of the sample. To obtain a quantity directly related to the contribution of the liposomes and their double layers, for each sample the extravesicular electrolyte was measured as the reference, and the corresponding dielectric signal subtracted from the signal of the vesicle dispersion. We also verified the linearity of the dependence of the spectra with respect to the volume fraction by repeating the measurements at different dilutions. Figure 2 shows the measured spectra for a dispersion in NaCl. To emphasize the measured relaxation, the results are presented in terms of the increments ∆ and ∆K as compared with arbitrary, but fixed, reference frequencies. The ordinates of the curves represent the differences between the increments for the dispersion and those for the reference solution:
∆(ω) ) [(ω) - (500 kHz)]disp - [(ω) - (500 kHz)]ref (3) ∆K(ω) ) [K(ω) - K(600 Hz)]disp - [K(ω) - K(600 Hz)]ref (4) Conductivity and permittivity are interdependent: they can be converted into one another with the Kramers-Kronig rela(9) Kijlstra, J.; Wegh, R. A.; van Leeuwen, H. P. J. Electroanal. Chem. 1994, 366, 37.
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Figure 4. Measured spectra for the dispersions corresponding to samples II, III, IV, V. The roman numbers refer to Table 1. Table 2. Empirical Relaxation Parameters Resulting from Fitting Experimental Data to Eqs 5 and 6
Figure 2. The frequency-dependent permittivity (top) and conductivity (bottom) increments as defined in eqs 3 and 4. The experimental data correspond to system II (liposomes in NaCl solution).
sample
∆/V (103)
τV(µs)
/ ∆C-C (103)
τC-C (µs)
RC-C
II III IV V
4.13 ( 0.06 3.81 ( 0.06 3.61 ( 0.06 3.44 ( 0.06
1.5 ( 0.1 2.3 ( 0.1 2.0 ( 0.1 2.0 ( 0.1
3.7 ( 0.1 3.6 ( 0.1 3.4 ( 0.1 3.3 ( 0.1
5.5 ( 0.6 7.3 ( 0.9 6.7 ( 0.9 6.6 ( 0.9
0.52 ( 0.04 0.55 ( 0.03 0.56 ( 0.04 0.56 ( 0.04
the normalized permittivity ∆/v is the appropriate quantity for a comparative study. Within experimental error the two sets of results are identical. The implication is that for the studied frequencies the external field is essentially unable to penetrate the membrane, and therefore the intravesicular ions do not significantly contribute to the polarization. The spectra for the 2-1 electrolytes are shown in Figure 4, which for sake of comparison also includes the 1-1 electrolyte curve. A first quantitative estimation of the main parameters of the curves (relaxation amplitude ∆* and relaxation time τ) can be obtained by fitting the data to a relaxation pattern as proposed by Vogel and Pauly,11 also reported in ref 12
∆/v ∆(ω) ) v (1 + ωτv)(1 + xωτv)
(5)
Another fit of the dielectric data with a known empirical relaxation function can be done with a Cole-Cole-type of spectral function:10
[
]
/ ∆C-C ∆(ω) ) Re v (1 + i ωτC-C)1-RC-C
Figure 3. Effect of the intravesicular electrolyte in a liposome dispersion. b, system I (1 mM NaCl inside and outside the liposome); 9, system II (0.1 mM NaCl inside, 1 mM NaCl outside the liposome). tion.10 Because, in the frequency range covered, experimental accuracy is better in permittivity, results will be presented and analyzed essentially on the basis of that component of eq 2. The overall accuracy of the permittivity data is on the order of 5%. Figure 3 compares the normalized dielectric permittivity of two samples with different intravesicular electrolytes. As different samples had slightly different liposome concentrations, (10) Bo¨ttcher, C. J. F.; Bordewijk, P. Theory of Electric Polarization; Elsevier: Amsterdam, 1978; Vol. II.
(6)
The errors with this latter fitting are larger than those obtained with eq 5. Results are summarized in Table 2. The fitted ColeCole parameter RC-C reflects the width of the relaxation. The value fitted for it is characteristic of processes generally associated with double-layer relaxations.13 The experimental amplitudes are also better described by eq 5. The principal relaxation time is smaller for system II (defined in Table 1) than for the systems containing 2-1 electrolytes. We shall come back to the relaxation times in section 4.2. Table 2 demonstrates that there is almost no difference between the spectra of the three different 2-1 electrolytes. Otherwise stated, the liposome dispersion system exhibits no significant divalent ion specificity. There is, however, a small but significant difference between 1-1 and 2-1 electrolytes. (11) Vogel, E.; Pauly, H. J. Colloid Interface Sci. 1988, 89, 3830. (12) Lyklema, J. Fundamentals of Interface and Colloid Science. Solid-Liquid Interfaces; Academic Press: London, 1995; Vol. II, Ch. 4. (13) Barchini, R.; Saville, D. A. J. Colloid Interface Sci. 1995, 173, 86.
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In accordance with the results presented in section 2.4, Figure 3, we will disregard the structure of the liposome and consider it as an isolating sphere with a homogeneous surface charge, made up by the phosphorus groups, and the surrounding double layer. Hence, the liposome is characterized by its radius a and a frequency-independent permittivity. For the electrolyte solution outside the liposome, that is, the external medium, we need to specify the permittivity m and the bulk conductivity Km. Taking into account the asymmetry of the electrolyte,
is in fact the real liposome charge density. Studies on PG monolayers16 also confirm the relatively low values of the proton binding constants and the negligible amounts of protons at the liposome/medium interface at a pH of 6. A direct consequence of the very high surface charge is the negligible contribution of co-ions to the surface conduction: Their negative adsorption is practically complete in the diffuse part of the double layer and the Stern layer does not contain co-ions. The present model does not require specification of the finite width and the molecular nature of the Stern region. The effects of this layer enter the equations in integrated form, through the concentration of counterions Γ+ and their surface mobility us. The outer boundary of the Stern layer, where the boundary conditions will be applied, will be indicated as r ) a′. The solution side of the liposome system is thus modeled as consisting of essentially two zones that correspond to r e a′ and r > a′. 3.1. Dielectric Theory. Application of an external electric field Eo exp(iωt) to a dispersion of charged particles generally leads to redistribution of ions in the double layer. The fixed structural liposome surface charge density σo does not change at the frequencies considered in this work (δσo ) 0). Diffusion and rotation of lipid molecules take place at higher frequencies,17,18 but for our DOPG lipids the influence on the charge distribution may be neglected. We will therefore consider the surface charge density as uniform over the liposome surface. As a result of the applied field, the concentrations outside the surface layer change by δn+/-exp(iωt). The induced counterion concentration changes in the Stern layer result in a corresponding charge density variation:
Km ) e(z+n+u+ + z-n-u-)
δσs exp(iωt) ) z+ e δΓ+ exp(iωt)
Notice that we are comparing systems with approximately the same extravesicular conductivity, and therefore with approximately the same extension of the ionic cloud around the liposome (Debye length κ-1). The effect of 3-1 electrolytes could not be investigated because they destabilize the liposomes. To achieve a comprehensive interpretation of the dielectric results, a fit to an empirical relaxation equation is only a first semiquantitative step. More meaningful is the development and application of a theory that can account for the observed relaxation in terms of the physical properties of the system. To that end, we first attempted to use a model for double polarization in which the dispersion is analyzed in terms of the polarization of the diffuse layer only. Such a limiting case can be derived from the general expressions presented in ref 12. However, using this approach, the predicted dielectric increments are systematically far too small. This led us to a more detailed assessment of the applicability of various dynamic double-layer approaches to the present case of liposome dispersions. As outlined in sections 4.1 and 4.2, the most satisfactory explanation is obtained by a more global model in which the counterion concentration in the stagnant layer is not a priori related to the concentration in solution by some specified isotherm.
3. Electrodynamic Model
(7)
where e is the elementary charge, z+/- the absolute values of the ion valencies, u+/- the mobilities, and n+/- the equilibrium bulk concentrations of the ions, respectively. The electric double layer is usually divided into two parts, that is, the Stern layer, encompassing all complications due to finite ion size, specific adsorption, etc., and the Gouy or diffuse layer, assumed to start at the outer Helmholtz plane. The Stern layer is more or less identical to the electrokinetically stagnant layer as far as macroscopic tangential flow is concerned.14 The implication is that there is no electroosmosis in this layer. However, in our model both the diffuse layer ions and the ions in the Stern layer can participate in tangential movements along the liposome/outer-electrolyte interface. Therefore, the relative surface conduction can be divided into two contributions described in terms of two dimensionless quantities, viz. the Dukhin numbers for the corresponding parts of the counterion surface conductivities:
Dus )
Ksσ Kdσ ; Dud ) aKm aKm
(8)
The subscripts σ and m stand for surface and bulk contributions; the superscripts s and d refer to the two kinds of contributions, that is, from Stern and diffuse layers. The charges of the Stern and diffuse layers compensate the liposome surface charge density σο. For the given head spacing on DOPG (see section 2.2), σο ≈ -22 µC/cm2. At the pH values considered in our study, no protons are bound to the phosphate groups (pKa ) 2.9),15 so that this (14) Lyklema, J. Colloids Surf., A 1994, 92, 41. (15) Cevc, G. Biochim. Biophys. Acta 1990, 1031, 311.
(9)
The changes due to the imposed field are assumed to be relatively small (δn+/- , n+/-, δΓ+, Γ+) so that the problem can be solved retaining only terms that are linear in the applied field. According to the theory of polarization of a relatively thin double layer, the oscillating diffuse charge can be regarded as being in continuous equilibrium with the surface charge. This pseudoequilibrium double-layer approximation can be understood in terms of fast ion adsorption/desorption kinetics over the entire double layer.19 The local electroneutrality condition for each surface segment together with its diffuse layer implies that
∫a′∞(z+δn+ - z-δn-) dr
z+ δΓ+ ) -
(10)
The condition is very general and relaxes the need to specify the adsorption equilibrium. Most studies published in the literature assume a Langmuir-type adsorption isotherm.20-23 The present case of counterion adsorption (16) Graham, I. S.; Cohen, J. A.; Zuckermann, M. J. J. Colloid Interface Sci. 1990, 135, 335. (17) Smith, G.; Shekunov, B. Yu.; Shen, J.; Duffy, A. P.; Anwar, J.; Wakerly, M. G.; Chakrabarti, R. Pharm. Res. 1996, 13, 1181. (18) Haible, A.; Nimtz, G.; Pelster, R.; Jaggi, R. Phys. Rev. E 1998, 57, 4838. (19) Dukhin, S. S.; Shilov, V. N. Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes; Wiley: New York, 1974. (20) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1986, 114, 32. (21) Rosen, L. A.; Baygents, J. C.; Saville, D. A. J. Chem. Phys. 1993, 98, 4183. (22) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1990, 86, 2859. (23) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1998, 94, 2583.
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at the lipid/solution interphase is certainly more complicated and therefore we have chosen to look only at the changes in Γ+, as derived from eq 10, irrespective of the adsorption model by which they were established. All of the field-induced quantities depend on the radial distance r, the polar angle θ, and the time. In what follows we will use shorthand notations such as, for example, for the electric potential change δψ(r,θ,t) ) δψ exp(iωt). The quantities of interest can be calculated from the Poisson and the continuity equations. The first one establishes:
(z+δn+ - z-δn-) om
∇2δψ ) -e
which the following solution can be written: 2
δn+/- )
[
(11)
δψ ) - Eor +
J+/- ) -n+/-z+/-u+/- e grad δψ z+/- eD+/- grad δn+/- (13)
C
u+/-kBT z+/-e
(14)
with kB the Boltzmann constant and T the absolute temperature. The effect of fluid motion close to the liposomes will enter the equations through an electroosmotic contribution that is treated in a similar way as in ref 25. Within the frame of our simplified treatment, the solvent velocity profile and the ensuing electroosmotic term will not be explicitly given, although their contributions are contained in the boundary conditions. This rather phenomenological way of dealing with convective polarization is not expected to lead to considerable errors in the evaluation of the dispersion dielectric properties. Using a slightly different approach to the one in this paper, Fixman26 showed that the inclusion of convection is especially important for quantitative accuracy only in the case of large particles with low to relatively moderate zeta potentials. And, as will be shown later, these conditions are not met in the case of the liposomes studied in this work. Combining eqs 11-14 together with the bulk electroneutrality condition:
z+ n+ ) z- n-
(15)
yields a differential equation for the ion densities, from (24) Kijlstra, J.; van Leeuwen, H. P.; Lyklema, J. J. Chem. Soc., Faraday Trans. 1992, 88, 3441. (25) O’Brien, R. W. J. Colloid Interface Sci. 1983, 92, 204. (26) Fixman, M. J. Chem. Phys. 1983, 78, 1483. (27) Teubner, M. J. Colloid Interface Sci. 1983, 92, 284. (28) Grosse, C.; Tirado, M.; Pieper, W.; Pottel, R. J. Colloid Interface Sci. 1998, 205, 26.
1 λjr
λj r
∑ j)1
e z- z+
omλj2 z-
r2
)
cosθ (16)
1
+
λj2r2
- βj A j ×
1
cosθ for r > a′ (17)
λjr
and:
δψ ) -Fr cosθ for r e a′
(18)
The electric polarization of the liposome is described through the frequency-dependent dipolar term coefficient C. Equations 17 and 18 contain four unknown coefficients (C, A1, A2, F) that are determined according to the following set of boundary conditions: (a) The potential is continuous at the boundary a′:
where the diffusion coefficients D+/- can be expressed in terms of the ion mobilities through the Einstein relation:
D+/- )
+
2 2
( ) ( )] 2
-
exp(-λjr)
(12)
with J+/- the current densities. As has been explained before,12,19 the set of equations simplifies outside the relatively thin double layer. In this so-called “far field”, the ionic flux divergences lose the convective contribution because of the supposed incompressibility of the electrolyte solution. The diffusion and conduction terms remain:
(
1
The details of the derivation as well as the definition of the different parameters in eq 16 are included in Appendix A. With the solutions for δn at hand via eqs A2-A7, the potential change δψ outside the model liposome can be obtained from the Poisson eq 11:
and the ion conservation equations are formulated as:
div J+/- ) iωz+/- eδn+/-
Aj+/- exp(-λjr) ∑ j)1
δψ|+a′ ) δψ|-a′
(19)
(b) The presence of the surface charge in the Stern layer gives rise to a discontinuity in the displacement field:
|
|
∂δψ ∂δψ -om + op ) z+ eδΓ+ ∂r +a′ ∂r -a′
(20)
(c) Co-ions are not allowed to enter the Stern layer zone
[
]|
∂ -z- e nδψ + δn∂r kBT
)0
(21)
+a′
(d) The ion conservation condition applies to the flux of counterions toward and along the surface of the liposome
[
∂ z+ e n+ δψ + δn+ ∂r kBT
(
]|
-
+a′
)
[
2Dud z+ e n+ δψ + δn+ a kBT
(
)
]|
-
+a′
2 e n+ z+ u+ 2Ds 1 1+ Dus δψ|a - i ω + 2 δΓ+ ) 0 kBT ua D+
(22) where Ds represents the diffusion coefficient of the counterions in the Stern layer. The Dukhin number Dud accounts for the tangential ion transport including a contribution from electroosmosis.12 Our experimental samples III and IV with different types of divalent counterions contain a relatively small amount of monovalent cations; for these systems the boundary conditions 21 and 22 apply for the counterion with the highest absolute valency. This is a valid approach for the case of high surface potential systems, as has been discussed by O’Brien and Hunter.29 (29) O’Brien, R. W.; Hunter, R. J. Can. J. Chem. 1981, 59, 1878.
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Langmuir, Vol. 16, No. 22, 2000 8243
After straightforward but tedious calculation, dipolar coefficient C can be expressed in terms of already introduced parameters of the system (see pendix B). Eventually the dielectric properties of dispersion are obtained from:
(
)
C ˆ ˆ (ω) ) ˆ m(ω) 1 + 3v Eoa3
the the Apthe
With the above equations, we calculated the electrokinetic potential for all the prepared samples. These potentials were used to evaluate the vesicle mobility using a reinterpretation of the theoretical expression by O’Brien and Hunter:29
(23)
u Due ) yd + (γ - yd) uo 1 + Due
The range of validity of eq 23, with C defined as in eq B1, is restricted by the fact that we have assumed that the Stern layer adjusts itself instantaneously to changes in the diffuse double layer. Consequently, when the frequency of the field approaches the Maxwell-Wagner regime, eq B1 cannot be directly used. We have also assumed that the permittivity and conductivity of the bulk electrolyte are independent of the frequency. At higher frequencies this assumption is no longer valid. However, these limitations do not apply in the frequency window of our experiments. Finally, our treatment of surface conduction parameters holds for sufficiently large values of κa, which for the systems in this study is O(10). 3.2. Mobility Theory. The surface concentration of ions in the unperturbed Stern layer (Γ+) together with the surface charge density of the liposome (σo) determine the electrokinetic charge density σd (σd ) σo - z+ eΓ+). From this charge, the electrokinetic potential at the Stern layer boundary (outer Helmholtz plane) can be estimated. The equations governing their relation have been clearly described by, among others, Ohshima.30,31 For a charged colloidal particle with Stern potential ψd in a 1-1 electrolyte solution, the diffuse charge density is given by:
σd )
( ){ [ [
]
2omκkBT 1 yd 2 sinh 1+ + e 2 κa cosh2(yd/4) d 1/2 1 8 ln[cosh(y /4)] (κa)2
sinh2(yd/2)
]}
{
omκkBT 4 (3 - p)q - 3 pq 1 + + e κa (pq)2 q+1 4 6 ln + ln(1 - p) 2 2 2 (κa) (pq)
[ (
)
]}
(24)
1/2
(25)
In these equations, yd represents the dimensionless Stern potential:
yd ) eψd/kBT
(26)
and:
( )
(27)
[32 exp(y ) + 31]
(28)
yd 2
p ) 1 - exp q)
d
1/2
where the mobility is normalized by the factor
uo )
omkBT ηe
(30)
Here, η is the water viscosity and γ takes the values -1.3862 and -0.4748 for the 1-1 and 2-1 electrolytes, respectively.29 In eq 29 the effect of the double layer is taken into account through the Dukhin number. According to the model presented in section 3, this should include the contributions from the Stern and diffuse layers. We have therefore introduced an effective Dukhin number, defined as:
z+n+u+ + z-n-u2z+n+u+
Due ) (Dud + Dus)
(31)
where the asymmetry of the electrolyte is taken into account. The mobility factor also accounts for the fact that, whereas only counterions contribute to Kσ, both counterand co-ions contribute to Km. 4. Discussion
whereas for a 2-1 electrolyte solution
σd )
(29)
In eqs 24 and 25 the corresponding counterion valency appears through the κ variable (eq A3). (30) Electrical Phenomena at Interfaces. Fundamentals, Measurements and Applications; Ohshima, H., Furusawa, K., Eds.; Surfactant Science Series, 2nd ed.; Marcel Dekker: New York, 1998; Vol. 76, Ch.1. (31) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 17.
4.1. Assessment of Model. The model presented in section 3.1 describes a low-frequency relaxation process with characteristics depending on the dynamic doublelayer properties of the liposome. The salient feature of the model is the inclusion of a thin Stern layer where counterions are allowed to migrate tangentially (eq 22). The relaxation in the Stern layer is coupled to the relaxation in the diffuse layer through condition 10. To understand the implications of the model, let us first consider some different limiting situations. First we note that in the absence of Stern layer conductivity, that is, for Dus ) 0, the model essentially reduces to the diffuse layer limit.12 A relaxation process is predicted in the frequency range of interest but, as stated before, the ensuing theoretical dielectric increments underestimate the experimentally measured values for the liposome dispersions. Figure 5, curves iii and iv, illustrates this behavior. These curves were calculated using the characteristic values of the dispersion system II (in NaCl). The comparison should be made with curve i, for which the parameters reproduce the measured dielectric spectrum. The other extreme limiting case corresponds to a surface conduction mechanism due solely to the Stern counterions (Dud ) 0). The first model of this type was presented by Schwartz32 with the additional condition of absence of exchange between the Stern and diffuse layers. However, it has been well established in the literature that it is generally necessary to include the coupling between bound and diffuse counterions. In our model, this is accounted (32) Schwartz, G. J. Chem. Phys. 1962, 66, 2636.
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Figure 5. Calculated dielectric spectra for dispersion sample II using different values of the Dukhin numbers. Curve i, Dud ) 2.8, Dus ) 4.7, which reproduces the experimentally measured spectra; curve ii, Dud ) 5.0, Dus ) 4.7; curve iii, Dud ) 2.8, Dus ) 0; curve iv, Dud ) 50, Dus ) 0; curve v, Dud ) 0.
for by the local equilibrium condition 10. Then, in cases where the diffuse layer does not generate a surface conduction effect (Dud ) 0), the Stern ions also do not contribute to the polarization and hence the low-frequency relaxation disappears (curve v in Figure 5). The diffuse layer response prevails over that of the Stern layer. The tangential oscillations of the diffuse atmosphere are responsible for the relaxation process, but quantitatively this relaxation is influenced by the polarization of the bound Stern ions. The effect of Dud can be inferred from curves i and ii in Figure 5. For a given Dus, increasing Dud decreases the dielectric increment but increases the static conductivity (not shown in Figure 5). On the other hand, for a fixed value of Dud, increasing Dus increases the dielectric increment. A theoretical treatment similar to the one presented here has been given by Grosse et al.28 The main difference lies in the condition for the Stern layer concentration changes. Grosse et al.’s choice of boundary condition 25 in ref 28 is not substantiated. Nevertheless it can be interpreted in terms of a site adsorption model such as discussed by Saville and colleagues21 and Mangelsdorf and White.22 Grosse et al.’s eq 25, in our notation
δn+/n+ ) δΓ+/Γ+
(32)
implies that the model is only applicable to cases of very weak counterion adsorption. The condition can also be interpreted as a thermodynamic equilibrium between the Stern layer and the adjacent diffuse layer: it states the constancy of the chemical parts of the electrochemical potentials. Obviously, our eq 10 is less restrictive. It is also important to point out that Grosse et al.’s model does not allow for tangential flow of the diffuse counterions. If, anyway, this model is used to fit our experimental results, a Stern layer charge exceeding the total surface charge of the liposome is required. This would imply a reversal of the sign of the liposome mobility, which is in contradiction to the experimental findings. The model proposed here considers the presence of a finite surface charge density of counterions, so it should be also compared with the dynamic Stern layer model.20-24 The different variants of that approach consider an adsorption isotherm to relate counterion concentrations
in the Stern and the diffuse layers. The models are elaborated with all rigor and the solutions cannot be expressed in a simple analytical way. Moreover, the dynamic Stern layer model does not predict experimentally observed relaxation frequencies, which has been reported in the literature as a shortcoming of the model. 21 Another approach12 that includes the Stern layer contribution in the dielectric response considers the effect of the total Dukhin number on the far-field potential. Still, the resulting permittivity underestimates our dielectric increments, so an alternative treatment of the double layer is necessary. Finally we should mention the work of Razilov and Dukhin.33 These authors consider two extreme conditions: free exchange and no exchange of ions between Stern and diffuse layers. In the first case, the predicted dielectric increments underestimate our measured ones. For the second situation, not considered in our model, the oscillating charge in the double layer is simply assumed to follow the Schwarz model. It is difficult to make any further comparisons because the authors do not present the complete frequency dependence of the permittivity. 4.2. Interpretation of Liposome Spectra. In applying our model to the analysis of the experimental spectra, we have taken p ) 4, typical for the permittivity of hydrocarbon chains, and m ) 78, for the permittivity of water. The parameters of the model were adjusted until the result was consistent with the measured static conductivities as well as with the electrophoretic mobilities of the liposomes. The choice of the Dukhin number Dud is crucial for reproducing the static conductivity values. The Dukhin number Dus proved to be a central parameter for explaining the measured dielectric increments. This number depends on the stagnant layer surface conductivity Κσs, defined as:
Kσs ) ez+Γ+us
(33)
The fitting of Dus requires the values of the two parameters Γ+ and us. This problem was solved by complementing the dielectric data with electrophoretic mobility results. In Figure 6 we present two examples of the fitting of the theoretical eqs 23/B1 to our experimental data. The model properly accounts for the dielectric spectra of all the studied dispersions. The agreement between experiments and theory suggests that the main polarization mechanism is adequately described by the proposed model. The fitted parameters are shown in Table 3. There are essentially three of them: (a) the Dukhin number Dud, (b) the concentration of counterions in the Stern layer Γ+, expressed as the effective electrokinetic surface charge density σd, and (c) the surface counterion mobility us, normalized with respect to the bulk counterion mobility. The last two parameters determine the Dukhin number Dus, also included in Table 3. The last columns of that table compare the measured and calculated scaled liposome mobilities. The consistency of the results strongly supports the values fitted for the parameters of the dielectric model. Taking into account the experimental errors, the uncertainty of the fitted parameters is about 5%. Systems III-V can be described with double-layer parameters that are very similar to each other. Practically no distinction can be made between the different divalent counterions. For the case of the divalent counterions, the best fit to the experimental data was achieved by specifying (33) Razilov, I. A.; Dukhin, S. S. Colloid J. 1995, 57, 364.
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The dielectric relaxation of the system is simultaneously related to the concentration polarization of the diffuse layer and the bound counterion relaxation of the Stern layer. The influence of the latter can be seen directly through the amplitude of the dielectric relaxation. For the influence of the diffuse ions, we recall that the relaxation disappears when Dud ) 0, which could mean that the radial exchange of ions between Stern and diffuse layers dominates the relaxation. The interplay of the two parts of the double layer can be analyzed through the relaxation times. For the Stern layer the characteristic time for the redistribution of ions is given by τs ) a2/2Ds as defined in eq B9. For the concentration polarization of the diffuse layer as a whole, the characteristic time
τd ≈ a2/D
(34)
depends on the effective ionic diffusion coefficient in the bulk:
D)
Figure 6. Permittivity data for systems II (top) and III (bottom) and curves fitted according to eqs 23/B1. The values of the fitted parameters are indicated in Table 3. Table 3. Dukhin Number Dud, Electrokinetic Charge Density σd, and Surface Mobility of Counterions us, Expressed in Terms of Mobility Values in Bulk,40 Resulting from Fitting Experimental Data to Eqs 23/B1a sample
Dud
σd (µC/cm2)
us
Dus
u/uo (calc)
u/uo (exp)
II III IV V
2.8 4.2 3.8 4.0
-15.0 -16.5 -17.3 -16.9
uNa 0.6 uCd 0.6 uCa 0.6 uCu
4.7 2.3 2.0 2.3
-2.10 -0.97 -1.06 -1.00
-2.09 -0.95 -1.07 -0.99
a From these values the calculated Dukhin number Dus and the scaled mobility u/uo (calc), from eq 29, are also shown. To facilitate comparison the scaled experimental values u/uo (exp) are included.
a surface counterion mobility smaller than in the bulk. When compared with system II, and considering the adjusted Dukhin numbers, we observe an inversion of the relative contributions of Stern and diffuse layers. The data also show that for electrolyte concentrations at the millimolar level, some 25% of the countercharge is located in the stagnant layer. For the case of system II (Na+ ions) this finding is in good agreement with the computations of Graham et al.,16 predicting that around 20% of Na+ counterions are bound to the PG headgroups. The results can also be set against the continuous (diffuse layer) potential profile of a charged particle in an indifferent electrolyte solution, as resulting from approximate theoretical expressions.31 Consistency with the calculated surface charge is obtained if the stagnant layer is situated at some 0.2 nm from the liposome surface. Without going into detailed consideration of the structure of the Stern layer, this can be taken as a reasonable outcome that further supports our interpretation of the dynamic double layer at the liposome/medium interface.
(z+ + z-)D+D(z+D+ + z-D-)
(35)
Comparison of the values for τs (calculated from Table 3) and τd shows that diffuse and Stern layers have similar relaxation times. Changes in the ion distributions in both Stern and diffuse layers are accomplished on the same time scale, allowing the local equilibrium condition 10 to be fulfilled. With regard to the relaxation time of the measured dispersions, it is not easy to obtain an explicit theoretical expression for τ. We have therefore considered the estimations recorded in Table 2. Depending on the empirical expression used, we can say that for system II 1.5 < τ < 5 µs, whereas for the other systems 2 < τ < 7 µs. These ranges of values agree with the calculated τs and τd. It must be stressed that the relaxation times are in principle independent of the Dukhin number. The general conclusion is also that the relaxation time is a less sensitive parameter than the relaxation amplitude to assess a certain double-layer model. As a last remark we should mention that our model explicitly takes into account the differences in ionic diffusion coefficients of anion and cation. Although for 1-1 electrolytes the difference hardly affects the dielectric behavior of the system, we found that this is not the case for 2-1 electrolytes. 5. Conclusion We have presented an electrodynamic study of liposome dispersions in aqueous electrolytes with emphasis on the measurement and interpretation of their dielectric properties. The proposed theoretical model disregards the presence of the intravesicular electrolyte since it was experimentally verified that it does not play a role in the polarization of the liposomes. This observation was supported by a theoretical analysis in which we derived the dipolar moment of an onion-like sphere, with a shell representing the lipid bilayer. The latter was characterized by the measured lipid bilayer thickness (see section 2.2) and a permittivity of 4. The calculation showed no influence of the ions in the intravesicular region. Furthermore, the model assumes the presence of a Stern layer with mobile ions. Our experimental results call for this assumption. Comparison between the theoretically predicted spectra, eqs 23/B1, and the experimental data shows that the present model is able to provide a consistent explanation for the dielectric spectra. The static conductivity and the electrophoretic mobility of the liposome
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dispersion studied were also successfully explained within the context of the presented model. The results of this study indicate that the Stern layer ions undergo tangential transport. The existence of a surface current around liposomes originating from the mobility of counterions in the Stern layer was also found in ref 34. It is interesting to compare our data for the ratio us/u+ with those obtained by Matsumura and colleagues35 for egg yolk phosphatidylcholine vesicles using a completely different procedure based on the difference between the isoelectric point and the isoconductive point. For monovalent counterions we find about 1.0 for Na+, and about 0.6 for Cd2+, Ca2+, and Cu2+, whereas Matsumura and colleagues34,35 report ≈1 for Cs+, ≈0.6 for Be2+, and ≈0.8 for Ca2+. Given the very different approaches, we consider this agreement as gratifying; at the same time it supports the feasibility of our model. Similar data were found for latex suspensions36 and corroborate the findings of molecular dynamics simulations.37 Such behavior might be explained on the basis of movement of the surface counterions impeded by their interaction with the particle surface. On a molecular scale the surface of the liposome consists of regions of lower and higher charge densities where the counterions could remain trapped. There is evidence for this from dielectrophoretic measurements on biological cells.38 The effect could be more important for the divalent counterions because there is also some evidence of a 2-1 stoichiometry of their binding to the liposome headgroups.16 The influence of the chemical nature of the ion (divalent case) appears to be of secondary relevance. The trend seems to be more general because it is also observed for lattices and for SiO2.34 Contrary to other models,39 our results also show that the low-frequency relaxation is very sensitive to the detailed ion distribution in the proximity of the charged particle. The implications of our approach need further investigation. One of the deficiencies of the present model stems from the mathematical idealization of the Stern layer. Work with other lipid molecules with different structural charges, electrolytes (including acids), and electrolyte concentrations is in progress. Acknowledgment. This work has profited from helpful discussions with Marcel Minor. We are thankful for the financial support from the Graduate School M&T, Wageningen University. We are also indebted to the staff of the Institute of Biomembranes, Utrecht University for their advice regarding the liposome preparation.
where
χ2 )
2z2-e2nomkBT
(A2)
Note that χ2 represents the inverse-squared Debye length κ2 for a 1-1 electrolyte. In the general case, the two quantities are related through:
χ2 )
2z-
κ2
(z+ + z-)
(A3)
For differential equations of type A1, the general solution can be written as: 2
δn+/- )
Aj+/- exp(-λjr) ∑ j)1
(
1
1
+
λj2r2
λjr
)
cosθ (A4)
This kind of solution has already been used27,28 for simpler cases of eq A1. The angular dependence stems from the axisymmetry of the problem. The coefficients Aj, for the co- and counterion distributions are related through
Aj- ) βjAj+
(A5)
where the factors β are the roots of the equation
[
(
z-χ2βj2 - (z+ - z-)χ2 + 2z-
)]
iω iω β - z+χ2 ) 0 D+ D- j (A6)
The coefficients λj can also be expressed in terms of βj
λj2 )
(
)
z+ χ2 iω + - βj z2 D+
(A7)
Appendix A Equations 11 to 15 determine the following differential equations for the field dependent ion density variations:
∇2 δn+/- )
(
)
z-/+ 2 z+/- 2 iω χ + δn+/- χ δn-/+ 2zD+/2z-
(A1)
(34) Matsumura, H.; Verbich, S. V.; Dukhin, S. S. Colloids Surf., A 1999, 159, 271. (35) Verbich, S. V.; Dukhin, S. S.; Matsumura, H. J. Dispersion Sci. Technol. 1999, 20, 83. (36) Lo¨bbus, M. Untersuchungen zum elektrochemischen Verhalten von oxidischen und polymeren Kolloiddisperionen in Elektrolytloesungen. Ph.D. Thesis, University of Jena, 1999. (37) Lyklema, J.; Rovillard, S.; DeConinck, J. Langmuir 1998, 14, 5659. (38) Paul, R.; Kaler, K. V. I. S.; Jones, T. B. J. Phys. Chem. 1993, 97, 4745. (39) Arroyo, F. J.; Carrique, F.; Bellini, T.; Delgado, A. V. J. Colloid Interface Sci. 1999, 210, 194.
Appendix B The analytical expression for the dipolar field coefficient C derived from eqs 17 to 22 is
{
Km C ) p - m + mh ( - mS2 + pR2) + 3 i ωom 2 m Eoa Km V [h ( - mS1 + pR1) - h2(m - mS2 + i ωom N 1 m 2Km pR2)] {p + 2m mh ( - mS2 + pR2) i ωom 2 m 2Km V [h ( - mS1 + pR1) - h2(m - mS2 + i ωom D 1 m pR2)]} (B1)
}/
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{
where the different symbols are defined as:
uu+ + u-
(B2)
z+/z- - βj j ) 1,2 βj
(B3)
m) hj ) Rj ) Sj )
{
VD ) (z-/z+)[(1 - m)(1 - Dud) + Dus] + m g2 -
λja + 1 (λja)2 + 2(λja + 1) (λja)2 2
(λja) + 2(λja + 1)
j ) 1,2
(B4)
j ) 1,2
(B5)
}/{
2Km(z-/z+) s Du mh2R2 iωom
- g1 + g2 +
2Km(z-/z+) × iωom
Dus[h1R1 + h2R2]
}
(B7)
In the last two expressions, the functions gj are given by
1 + iωτs 1 + 2 Dud Rj - (z-/z+)hjβjSj iωτs gj ) j ) 1,2 (B8) βj
VN ) (z-/z+)[(1 - m)(1 - 2Dud) - 2Dus] + m g2 -
}/{
2Km(z-/z+) s Du mh2R2 iωom
2Km(z-/z+) - g1 + g2 + × iωom Dus[h1R1 + h2R2]
}
with
τs ) a2/2Ds
(B6) LA991640M
(B9)
