7518
Langmuir 2007, 23, 7518-7525
Dielectric Behavior of Lipid Vesicles: The Case of L-r-Dipalmitoylphosphatidylcholine Vesicles as a Function of Size and Temperature G. Briganti,*,§ C. Cametti,§ F. Castelli,# and A. Raudino# Dipartimento di Fisica, UniVersita’ di Roma “La Sapienza”, Piazzale A. Moro 5, I-00185 - Rome, Italy, and INFM CRS-SOFT, Unita’ di Roma 1, Dipartimento di Scienze Chimiche, UniVersita’ di Catania, Viale A. Doria 6, 95125 - Catania, Italy ReceiVed February 14, 2007. In Final Form: April 16, 2007 We present an extensive set of radio wave dielectric relaxation spectroscopy measurements of aqueous suspensions of different size unilamellar L-R-dipalmitoylphosphatidylcholine (DPPC) vesicles, in a temperature range between 15 and 55 oC, where the lipidic bilayer experiences structural transitions from the gel to the rippled phase (at the pretransition temperature) and from the rippled to the liquid phase (at the main transition temperature). The dielectric spectra have been analyzed in the light of the Cole-Cole relaxation function, and the main dielectric parameterssthe dielectric increment ∆ and the mean relaxation frequency ω0shave been evaluated as a function of temperature. These parameters display a very complex phenomenology, depending on the structural arrangement of the lipid-water interface. The structural parameters that govern the dielectric behavior of these systems associated with the lipid bilayer have been recognized within a recent dynamic mean-field model we have proposed, aimed to predict the dipolar relaxation of an array of strongly interacting dipoles anchored to a flat or corrugated surface. They are the prefactor A(T) of the distance-dependent part of the effective dipolar interaction energy, the term Γvis, that takes into account the damping of the dipolar motion, the average dipolar distance related to the area a0 per polar head, and the bilayer thickness. The present analysis furnishes, from a phenomenological point of view, the dependence of these parameters on the temperature and on the vesicle size.
1. Introduction The radio wave dielectric properties of aqueous suspensions of mesoscopic particles (10-1000 nm in diameter) generally exhibit a very complex phenomenology, resulting in dielectric spectra composed of three (and in some cases more) different, partially overlapping contributions, arising from different polarization mechanisms, at a molecular level.1 In a sufficiently wide frequency range, dielectric spectroscopy technique can evidence the electrical polarization occurring either in the bulk of the dielectric media or at the interface between the dispersed particles and the dispersing medium. These different surface polarizations are caused by a nonuniform ion distribution induced by the external electric field close to the interfaces, or by the presence of confined charges originated from the ionization of characteristic chemical groups on the surface of the dispersed particles.2 In the case of lipid bilayers formed by zwitterionic surfactants, the surface polarization is originated by the presence of strong dipoles at the interface and, in most cases, it depends on their correlation. According to experimental evidence3-6 and theoretical predictions,7 these dipoles lay almost parallel to the lipid surface, and the motion of the head groups is essentially confined over the plane of the vesicle. At low temperatures, the lipid hydrocarbon * Corresponding author. § Universita’ di Roma “La Sapienza”. # Universita’ di Catania. (1) Kremer, K.; Scho¨nhals, A. Broadband Dielectric Spectroscopy; Springer: Berlin, 2003. (2) Clausse, M. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1983. (3) Pottel, R.; Go¨pel, K. D.; Henze, R.; Kaatze, U. Biophys. Chem. 1984, 19, 233-244. (4) Uhlendorf, V. Biophys. Chem. 1984, 20, 261-273. (5) Shepherd, J. C. W.; Bu¨ldt, G. Biochim. Biophys. Acta 1978, 514, 83-94. (6) Asami, K.; Irimajiri, A. Biophys. Biochim. Acta 1984, 769, 370-376. (7) Raudino, A.; Mauzerall, D. Biophys. J. 1986, 50, 441-449.
chains are essentially packed into a rigid and rather ordered structure, known as the gel or Lβ phase. As the temperature rises, the hydrophobic region of the bilayers assumes a fluid disordered phase, named LR, where the interior of the bilayer is composed of melted hydrocarbon chains. In some cases, a narrow range of temperature between these two phases is characterized by a welldefined different phase, known as Pβ′ or the ripple phase.8-11 This phase has mostly frozen chains and a greater in-phase order compared to that of the liquid LR phase. This order is mainly due to a permanent undulation at the bilayer-water interface with an amplitude and a wavelength on the order of magnitude of 102 nm and 103 nm, respectively.12-14 The surface undulations give rise to a further polarization mechanism that induces a new contribution to the dielectric spectra. In particular, supported by different previous investigations,15-17 the occurrence of the ripple phase produces a sharp increase in the dielectric permittivity followed by a concomitant decrease in the relaxation frequency. Moreover, the distribution of the relaxation times is sharper in this ripple phase than that in both the uncorrugated gel and liquid phases. This anomalous dielectric behavior has been partially overlooked, in spite of the attention paid to the ripple phase in the lipid bilayers because of its biological relevance. (8) Hawton, M. H.; Doane, J. W. Biophys. J. 1987, 52, 401-404. (9) Smith, G. S.; Sirota, E. B.; Safinya, C. R.; Clark, N. A. Phys. ReV. Lett. 1988, 60, 813-816. (10) Goldstein, R. G.; Leibler, S. Phys. ReV. Lett. 1988, 61, 2213-2216. (11) Carlson, J. M.; Sethna, J. P. Phys. ReV. A 1987, 36, 3359-3374. (12) Lubensky, T. C.; MacIntosh, F. C. Phys. ReV. Lett. 1993, 71, 1565-1568. (13) Heimburg, T. Biophys. J. 2000, 78, 1154-1165. (14) Katsaras, J.; Tristram-Nagle, S.; Liu, Y.; Headrick, R. L.; Fontes, E.; Mason, P. C.; Nagle, F. J. Phys. ReV. E 2000, 61, 5668-5677. (15) Kaatze, U.; Gopel, K. D.; Pottel, R. J. Phys. Chem. 1985, 89, 2565-2571. (16) Cametti, C.; De Luca, F.; Macri, M. A.; Maraviglia, B.; Sorio, S. Liquid Cryst. 1988, 3, 839-845. (17) Cametti, C.; De Luca, F.; D’Ilario, A.; Macri’, M. A.; Maraviglia, B.; Sorio, S. Liq.Cryst. 1990, 7, 571-581.
10.1021/la700314d CCC: $37.00 © 2007 American Chemical Society Published on Web 06/01/2007
Dielectric BehaVior of Lipid Vesicles
Langmuir, Vol. 23, No. 14, 2007 7519
We have recently developed a dynamic mean-field model18 aimed to calculate the electrical polarization and the dipolar relaxation of an array of strongly interacting dipoles on a bilayer surface. They mimic the polar head groups of the lipid molecules, anchored to a flat surface and rotating over either a planar surface (modeling the gel and fluid phases) or a corrugated surface (modeling the ripple phase). This model qualitatively predicts most of the observed dielectric properties for all three phases involved. In the ripple phase, the surface undulation affects the interfacial dipole correlation, giving rise to the observed dielectric response. Within the above stated model,18 we have listed four different structural parameters that can account for the main features of the dielectric behavior in the three different phases: (i) the prefactor A(T) of the distance-dependent part of the effective dipolar interaction energy, that takes into account the correlation of the dipoles lying on the interface, (ii) a term Γvis that takes into account the damping of the dipolar motion, (iii) the average dipolar distance, related to the area a0 per polar head, and (iv) the bilayer thickness d. The last two parameters define the volume of the hydrophobic chain, whereas the first two parameters define the headgroup interactions and the dynamics. These parameters characterize the permittivity in the hightemperature fluid phase, which is larger than that in the lowtemperature gel phase, caused by the reduced dipolar correlation. Moreover, this model justifies the different extents of the relaxation distribution, passing from the gel-to-ripple and from the ripple-to-liquid phases. Since this model has been developed for bilayers, it cannot contain any dependence on the vesicle size, which, on the other hand, affects the dielectric response and the thermodynamic properties of vesicles. Actually, both the dielectric increment and the relaxation frequency depend on the vesicle size18 and, in small vesicles, the ripple phase disappears. The aim of this work concerns the evaluation, on a phenomenological basis, of the model parameter dependencies on vesicle size and temperature in the three different phases: gel, ripple, and liquid phases. To this end we analyze an extensive set of radio wave dielectric measurements of aqueous suspensions containing L-R-dipalmitoylphosphatidylcholine (DPPC) vesicles of different size (from 50 to 1000 nm in diameter) in a temperature range from 15 to 60 oC, covering the gel, ripple, and liquid phases. Our results show a clear dependence of the dielectric parameters on the vesicle size, furnishing further evidence that the ripple phase in a lipid membrane depends on the balance of symmetry, geometric, and energetic effects.
2. The Dielectric Model Here, we will summarize the main characteristics of the meanfield model we have developed in detail elsewhere.18 This model describes the dynamics of a dense array of strongly interacting dipoles, which are allowed to rotate over a plane parallel to the lipid-water interface. We consider an array of almost parallel dipoles (confined over the plane of a lipid bilayer) whose instantaneous surface polarizability P(r b,t) can be written as
P(b,t) r ) µF(b)η( r b,t) r
(1)
where µ is the dipole moment modulus per lipid head group, F(r b) is their local surface density, and η(r b,t) is the polarization per unit area and unit dipole moment. The total free energy Utot contains three terms: (18) Raudino, A.; Castelli, F.; Briganti, G.; Cametti, C. J. Chem. Phys. 2001, 115, 8238-8250.
Utot ) U0 + Udip + Uint
(2)
that is, the interfacial tension term U0, the dipolar correlation energy Udip and the dipole-electric field term Uint. Under the influence of an external electric field of intensity E0, the two relevant terms Uint and Udip can be written as
Ukint ) - µFE0eiωt Udip )
µ2 2
r cos γk(b)dS r (k ) x,y,z) ∫S η(b,t)
r b′)η(,t)η( r b′,t)G( r b r - b′)dSdS′ r ∫S ∫S′ F(b)F(
(3) (4)
where γk(r b) is the angle between the kth component of the field and the tangent to the bilayer surface at the point b. r G(r b - b′) r is the dipolar interaction potential that, for dipoles at high temperatures, can be written as G(r b - b′) r = A(T)/(|r b - b′|) r 6, where A(T) measures the strength of the dipole correlation. Once the total free energy of the system is known, the relaxation of the local polarization η(r b,t) can be calculated in the framework of a Landau-Ginzburg picture:19
dη r vis dS ) ∫S F(b)Γ dt
δUtot δη
(5)
where Γvis is a phenomenological damping parameter related to the dipolar surface viscosity. Finally, the kth component of the polarization Pk is obtained from the surface integration
Pk ) µ
r b,t) r cos γk(b)dS r ∫S F(b)η(
(6)
and the complex dielectric permittivity within the classical electrodynamics formula is
1 ∂Pk k,k′ ) 1 + 2π 2 lim L Ek′f0 ∂Ek′
(7)
where Ek′ is the k′th component of the applied electric field, and L is the width of the double layer. The electric response of the system depends on the intensity of the external dielectric field parallel to the lipid bilayer surface. For a flat bilayer surface, by solving the equation for the surface polarizability and by omitting unessential numerical constants, we have previously shown18 that the characteristic dielectric parameterssthe dielectric strength per monomer, ∆flat, and the relaxation frequency, νflatsare given by
∆flat ≈ 1/a0ξ0
(8)
νflat ≈ ξ0/Γvis
(9)
where a0 is the area per polar head (or the dipole density) in the uncorrugated bilayer, and ξ0 ) µ2A(T)/a03. Experimentally, it is known that the surface density of the head groups decreases about 20% going from the gel to the fluid phase.21 Therefore, on the basis of our model, the static permittivity ∆flat turns out to be larger in the fluid phase than in the gel phase because of the reduction of the correlation among the polar head groups (governed by the parameter A(T)). Furthermore, if the viscous dissipation parameter Γvis does not change with temperature, the relaxation frequency (inversely (19) Landau, L. D.; Lifshits, E. M. Physical Kinetics; Pergamon Press: New York, 1985. (20) Havriliak, S.; Havriliak, S. J. Dielectric and Mechanical Relaxation in Materials; Hanser Publishers: Munich, Germany, 1997. (21) Needham, D.; Evans, E. Biochemistry 1988, 27, 8261-8269.
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proportional to the dielectric increment, eqs 8 and 9) must be smaller in the fluid phase than in the gel phase. In both phases, the distribution of the relaxation frequencies is expected to be monodisperse for a given bilayer orientation. If undulations, propagating along a single axis, are present on the bilayer, the permittivity is anisotropic and greater than the one associated with a planar surface. Moreover, the corrugation induces a distribution of relaxation processes, since the frequencies due to in-plane dipolar relaxations decrease with the surface roughness, while the ones due to out-of-phase relaxations increase. As a matter of fact, we have shown in a previous work18 that the mean dielectric permittivity j ) (1/3)(x + y + z), in a Cartesian coordinate system with the z axis perpendicular to the uncorrugated surface, can be written as
j(ω) = 1 + flat(0)
Qj1 1 + (ω/ωj0)2
(j ) x,y,z)
(10)
The parameter flat(0) is the static dielectric increment of a flat membrane, Qj1 describes the modulation of the static permittivity due to surface undulation, and ωj0 describes the relaxation frequencies. Explicit expressions for Qj1 and ωj0 as a function of the wavelength λ of the ripples and of the lipid surface density F0 ) a0-2 are given in ref 18. In the long wavelength limit, λH , 1, with H being the ripple amplitude, we have obtained18 2 2 Qx1 = 1 + O(λ4H4), ωx0 = ωflat 0 (1 - 1/2λ H )
Qy1
1 = 1 + λ2H2, ωy0 = ωx0 2
Qz1 =
3. Experimental 3.1. Materials. Synthetic DPPC was obtained from Fluka Chemical Co., and the purity (more than 98.9%) was assessed by two-dimensional thin-layer chromatography. 3.2. Vesicle Preparation. Unilamellar vesicles with well-defined size were prepared from the aqueous lipid mixtures by extrusion through polycarbonate membranes (nominal pore diameters of 50, 100, 200, 400, 800, and 1000 nm) by means of a Lipofast basic setup (Anvenstin). During extrusion, the temperature was kept above the main transition temperature. The absence of impurities in the lipid bilayer, the formation of the ripple phase, and the reaching of an equilibrium bilayer structure were checked by differential scanning calorimetry measurements made with a Mettler TA 3000 system. The size of the resulting vesicles was checked by means of standard dynamic light scattering measurements. In all the experiments, the fractional volume Φ of the dispersed particles was kept constant at a value of Φ ) 0.05. 3.3. Dielectric Measurements. The dielectric measurements were carried out by means of a Hewlett-Packard impedance analyzer, model HP4294A, in the frequency range of 1 kHz to 10 MHz, covering the temperature interval of 15-55 °C, where the lipid bilayer experiences different thermodynamic phases (gel, ripple, and liquid phase). The measured impedances at the input of the dielectric cell were converted into the complex dielectric constant *(ω) by means of an appropriate electrical network whose parameters were determined by calibration with standard liquids of known permittivity and electrical conductivity. The estimated uncertainty of the overall procedure is within 1-3% in the permittivity ′(ω) and within 1% in the electrical conductivity σ(ω). The complex dielectric constant *(ω) was analyzed by means of a Cole-Cole relaxation function20
(11) *(ω) ) ∞ +
σ0 ∆ + β iω 1 + (iωτ) 0
(14)
(12)
λ2H2 1 ωflat(1 + λ2F0-1 4 (1 + λ2/F ) 0 0 (1/2 - 3/4λ2F0-1)λ2H2) (13)
The presence of a z component in the surface polarization, which increases with H, induces a distribution in the relaxation frequencies. Hence, on a flat bilayer with a given orientation with respect to the external electric field, a distribution of relaxations is expected in the ripple phase, whereas a narrower relaxation frequency distribution should characterize both the gel and the liquid phases. Vesicles of large radius compared to the head group distances can be considered an ensemble of flat bilayers with different orientations. Depending on the orientation of the bilayers, different dielectric increments will be obtained, since the dipole-electric field interaction reaches its maximum for an electric field tangent to the bilayer surface and is zero when the field is perpendicular to the bilayer surface. In this case, being that the relaxation frequency is inversely proportional to the dielectric increment, a dispersion of relaxations will be present. Therefore, in the gel and liquid phases, a distribution of relaxation frequencies is expected. On the other hand, the corrugation present in the ripple phase reduces the z component of each surface polarization. Hence, a behavior opposite that of a single oriented bilayer should be expected, that is, the dielectric increment should increase and the frequency distribution of relaxations should narrow. The above predictions of the model, which agree with previous experimental results,18 compare well with the observed dependence of the dielectric increment and the relaxation frequency on the vesicle size and on the temperature.
where ∆ is the dielectric increment, ∞ is the low-frequency limit of the permittivity, τ is the relaxation time (or, conversely, the relaxation frequency ν ) 1/(2πτ)), β is a parameter that takes into account the distribution of the relaxation times, σ0 is the low-frequency limit of the electrical conductivity, ω is the angular frequency of the applied electric field and, finally, 0 is the dielectric constant of free space. Since our experiments were performed in very dilute vesicle solutions, we considered a linear relationship between the dielectric properties of the single lipid bilayer and those of the whole solution. Therefore, in our mean-field approximation, the dielectric increment of eq 14 is just that of eq 8 multiplied by the number of lipid monomers. In the following, ∆ will indicate the dielectric increment per monomer of eq 8. On the other hand, the mean relaxation frequency of the model coincides with the one present in the ColeCole equation, whereas the relaxation frequency distribution is characterized by the parameter β.
4. Results and Discussion As previously mentioned, in the case of vesicle suspensions, since the ratio between the dipolar length and the vesicle radius ranges between 1/100 and 1/2000, we can approximate the average response of each monomer by combining the responses of flat surfaces with different orientations with respect to the external field. Nevertheless, the bilayer curvature affects the dielectric relaxation, giving rise to a distribution of the relaxation frequencies. Following the previous considerations, we can still utilize the proposed analysis for our experimental results, considering an appropriate evaluation for the average dielectric increment per monomer and for the relaxation frequency . In the case of polydisperse solutions, in the limit of very high dilution, the low-frequency permittivity of a heterogeneous suspension composed by a collection of particles of radius Rk
Dielectric BehaVior of Lipid Vesicles
Langmuir, Vol. 23, No. 14, 2007 7521
and permittivity p(Rk) uniformly dispersed in a medium of permittivity m can be written, to a first approximation, as2
= m +
∑k Φkp(Rk)
(15)
where Φk ) Np(Rk)νp(Rk)/Vtot is the volume fraction, with Np(Rk) being the number of vesicles of radius Rk and volume νp(Rk), dispersed in the total volume Vtot. By defining the permittivity p(Rk) as
p(Rk) ) 1(Rk)
n(Rk)ν1(Rk)
(16)
νp(Rk)
where n(Rk) is the number of lipidic molecules in a vesicle of radius Rk (each of them having a permittivity 1(Rk) and occupying a volume ν1(Rk)), eq 15 can be written as
- m ) ∆ =
∑k Np(Rk)n(Rk) ∑k Np(Rk)n(Rk)ν1(Rk)1(Rk) Vtot
)
Figure 1. The dielectric increment ∆ of DPPC vesicles in aqueous suspension as a function of temperature, in the interval from the gel to the liquid phase. Vesicles have different average radii: (0) RH ) 50 nm; (9) RH ) 100 nm; (∆) RH ) 200 nm; (2) RH ) 400 nm; (O) RH ) 800 nm; (b) RH ) 1000 nm.
∑k
Np(Rk)n(Rk) C (17)
where C ) ∑k Np(Rk)n(Rk)/Vtot is the number concentration of lipids present in the suspension. According to the above stated picture, the overall permittivity of the vesicle suspension depends, to a first approximation, on the permittivity 1(Rk) of the single lipid in the bilayer of radius Rk, as in the case of a flat bilayer multiplied by its volume ν1(Rk) that can be approximated with the area a0 per polar head multiplied by the thickness d of the bilayer. A phenomenological analysis of the peculiar behavior of the dielectric increment ∆ as a function of temperature can be carried out on the basis of eq 8, taking into account the definition of the characteristic parameters of the model. To a first approximation, eq 17 can be rewritten as
∆ = =
a03d A(T)
(18)
Therefore, the relative change in the dielectric increment ∆ depends on three contributionssthe area a0 per polar head, the bilayer thickness d, and the dipolar interaction strength As according to the following relationship:
δa0 δd δA(T) δ∆ + ∼3 ∆ a0 d A(T)
(19)
The dielectric increment ∆ for all the different sizes of the vesicle suspensions investigated as a function of temperature, covering the three different thermodynamic phases (gel, ripple, and liquid phase), is shown in Figure 1. As can be seen, ∆ increases with the vesicle size. Additionally, close to the pretransition temperature (Tp ≈ 34 °C) and close to the main transition temperature (Tm ≈ 42 oC), a marked change in the value of ∆ is observed. In the solid phase, ∆ is, to a first approximation, independent of temperature, while, both in the ripple and in the liquid phase, it increases linearly with the temperature. The percentage variation of the dielectric increment decreases with the vesicle size, with a saturation trend going from 70% for vesicles 50 nm in radius to 50% for vesicles 800 and 1000 nm in radius.
Figure 2. The dielectric increment ∆ of DPPC vesicles in aqueous suspension measured at three different temperatures as a function of the average radius RH of the vesicle: (9) gel phase, T ) 18 °C; (b) ripple phase, T ) 35 °C; (0) liquid phase, T ) 50 °C.
It is known, from scattering experiments, that, passing from the gel to the liquid phase in large unilamellar vesicles (LUVs), the area a0 per polar head increases about 20%, whereas the thickness d of the bilayer decreases about the same quantity.22-24 As can be seen in Figure 1, for the largest vesicles investigated, the dielectric increment ∆ increases about 50% passing from the gel to the liquid phase. This means that, for LUVs in this temperature interval, a decrease of about 10% of the parameter A(T) must be expected to account for the total dipolar increment. In Figure 2, we present the dielectric increment ∆ as a function of the vesicle radius RH, at three different temperatures within the temperature interval where the three thermodynamic phases of the bilayer (gel, ripple, and liquid phase) occur. The data show a nearly linear dependence of ∆ on the radius RH of the vesicle, with a very similar slope for the gel and ripple phase and a higher slope for the liquid phase. Considering again the percentage variation of the dielectric increment given by eq 19, the (22) Janiak, M. J.; Small, D. M.; Shipley, G. G. Biochemistry 1976, 15, 45754580. (23) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351376. (24) Nagle, J. F.; Zhang, R.; Tristam-Nagle, S.; Sun, W.; Petrache, H. I. Biophys. J. 1996, 70, 1419-1431.
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dependence of ∆ on the radius RH can be ascribed to the coefficient A(T). Indeed, as we pointed out, both the area per polar head and the monomeric volume cannot appreciably change going from a 50 to 1000 nm vesicle diameter. On the other hand, the quantity A(T) is a prefactor multiplying the effective dipoledipole interaction that takes into account the dipole-dipole correlation in mean-field approximation. Therefore, being that (δA(T)/A(T)) is a negative contribution to the percentage of dielectric increment, we must conclude that it linearly decreases with the vesicle radius. The analysis of the frequency relaxation requires the definition of some average values. Equation 15 can be rewritten, by introducing the dependence on the frequency according to a Cole-Cole relaxation function (eq 14) and neglecting the highfrequency limit contribution to the dispersion, as
∑k
∆ ) - m )
Np(Rk)νp(Rk)
∆p(Rk)
Vtot
1 + (iωτk)
βk
(20)
Equation 20, again using eq 16, yields
∑k
∆ )
Np(Rk)n(Rk) ν1(Rk)1(Rk) Vtot
1 + (iωτk)βk
(21)
Under the assumption that ν1(Rk)1(Rk)/() ≈ 1, eq 21 can be rewritten as
∆ )
Vtot
∑k
Np(Rk)n(Rk) 1 + (iωτk)βk
(22)
If the Cole-Cole distribution function, 1/(1 + (iωτk)βk), for a given vesicle of size Rk, is considered as the superposition of Debye relaxation functions characterized by a set of relaxation times τj with appropriate weights Pk(τj), then
1
)
(1 + (iωτk)βk)
1
∑j Pk(τj)1 + iωτ
(23) j
For a given relaxation process τj, by defining
Pj(τj) )
∑k Np(Rk)n(Rk)Pk(τj) (24)
∑k Np(Rk)n(Rk)
and by combining eqs 16 and 17 with eq 22, the dielectric increment ∆ can be written as
∆ )
Ns Vtot
1
∑j Pj(τj)1 + (iωτ ) ) j
1 (25) Ns Vtot 1 + (iωτ0)β where Ns is the number of surfactants in solution (Ns ) ∑k Np(Rk)n(Rk)). Therefore, from eq 2, we obtain
A(T) 2π > ) = 3 τ0 a0 Γvis
