Characterization of Lecithin Cylindrical Micelles in Dilute Solution

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Langmuir 1998, 14, 7095-7103

7095

Characterization of Lecithin Cylindrical Micelles in Dilute Solution P. A. Cirkel and G. J. M. Koper* Leiden Institute of Chemistry, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands Received December 3, 1997. In Final Form: August 21, 1998

An extensive study of the polymerlike properties in dilute solutions of lecithin cylindrical micelles, with trace amounts of water, in isooctane is presented. In contrast to the current picture that this system behaves as a solution of wormlike polymers, we find that it is best described as a solution of rodlike micelles. This finding is based on the combined results of viscometry, electrooptic birefringence, dielectric spectroscopy, static and dynamic light scattering, and dynamic sedimentation. Also, the size of the micelles seems to be independent of the concentration, whereas a strong dependence is predicted for polymerlike micelles. There is a rather abrupt change in micellar size at low concentration, which can be attributed to a sphere-to-rod transition. At high water content further evidence is found for the percolation of the micelles into a connected network, as has been suggested by us recently.8

1. Introduction The characterization of polymerlike micelles is a longstanding and complex issue. In contrast to conventional polymers, of which the size is fixed, the size distribution of wormlike micelles is dictated by the solution’s thermodynamic conditions, supposedly through the end-cap energy.1 In this respect polymerlike micelles are considered to be equilibrium polymers, i.e., macromolecules that can break and recombine such that their size distribution is in an equilibrium imposed by the (monomer) concentration and temperature. To describe physical quantities of such equilibrium polymers two approaches can be used. Most parameters are measured on a time scale much smaller than that of the micellar kinetics, which means that one has to average the polymer properties of one micelle over the size distribution to obtain a so-called quenched average. On the other hand, for experimental techniques measuring very slow dynamical properties, such as long-time self-diffusion or dynamic sedimentation, one would expect the micellar kinetics to be relatively fast such that during the experiment each micelle has passed through every possible size in the distribution. In this limit an annealed average is measured, and the assumption of local thermodynamic equilibrium can be used. However, the observations in this paper indicate that for the system used here even these “slow” variables are measured on time scales shorter than that of the micellar kinetics. An additional problem for the description of polymerlike micelles is their semiflexibility. Until now, relatively few experimental studies have been devoted to polymerlike micelles in the dilute regime, mainly because of the lack of theories which incorporate both the size distribution and semiflexibility. More effort has been put into studying the semidilute regime, where in analogy with conventional polymers, the micelles overlap and most quantities exhibit a power-law dependence on the surfactant concentration. This scaling behavior can be used to study the dependence of the mean micellar length on the surfactant concentra(1) Israelachvili, J.; Mitchel, D. J.; Ninham, B. H. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525.

tion if one knows, from theory, which part of the powerlaw dependence is due to micellar growth. However, most existing theories only apply to ideal polymer solutions, whereas in the semidilute regime, deviations from the polymer behavior by micellar growth and by nonidealities, which could be expected for wormlike micelles, cannot be distinguished. In dilute solutions there is a better possibility of making this distinction, especially if the results of different techniques are combined. Recently, theories have been developed for the viscosity2 and dynamic sedimentation3 of polymerlike micelles in dilute solutions, and these can be used to determine their endcap energy. In this paper we will study cylindrical micelles in the lecithin/water/isooctane system. This is one of the few systems in which cylindrical micelles in apolar solvent are formed.4 The usage of an apolar solvent is interesting for at least two reasons: one is the absence of complicating effects by electrostatic interactions, and the other is the possibility to, with relative ease, do experiments with techniques that make use of electrical fields, such as dielectric spectroscopy and electrooptic birefringence. We will use these techniques to study the rotational correlation time of the micelles and combine these results with the results from viscometry, dynamic sedimentation, and static and dynamic light scattering. Another advantage of this system is that, recently, quite a lot of effort has been devoted to its characterization by scattering techniques, both in the dilute and semidilute regimes.5,6 Measurement of the static correlation length and osmotic compressibility by light scattering5 shows that in this system the micellar size depends not only on the surfactant concentration but also on w0, the molar ratio of added water to surfactant. There are also (2) Duyndam, A.; Odijk, T. Langmuir 1996, 12, 4718. (3) Duyndam, A.; Odijk, T. J. Chem. Phys. 1994, 100, 4569. (4) Schurtenberger, P.; Peng, Q.; Leser, M. E.; Luisi, P. L. J. Colloid Interface Sci. 1993, 156, 43. (5) Schurtenberger, P.; Cavaco, C. Langmuir 1994, 10, 100. (6) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Physica B 1997, 234, 273. (7) Jerke, G. Ph.D. Thesis, ETH Zu¨rich, 1997, nr. 12168. Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Phys. Rev. E 1997, 65, 5772.

10.1021/la971326x CCC: $15.00 © 1998 American Chemical Society Published on Web 11/19/1998

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indications that interpretation of the osmotic compressibility by renormalization group theory results in an anomalous high dependence of the micellar size on the surfactant concentration.7 This appears to be in conflict with the observations in this paper. Viscometry, dynamic sedimentation, and the rotational correlation time measured by dielectric spectroscopy indicate that there is hardly any dependence of the micellar size on the surfactant concentration. Results from the Kerr effect support these findings although like in the light scattering results, a deviant behavior is observed at very low surfactant concentrations. The data presented in this paper indicate that the key to explain this deviant behavior is related to a sphere-to-rod transition. A somewhat surprising result of the study presented in this paper is that the data are in excellent agreement with a model for rodlike particles. A model for a wormlike chain with a persistence length as was determined by neutron scattering,6 on the other hand, can hardly account for the data, in particular for the relation between the molecular weight and radius of gyration. At higher water content a qualitatively different behavior is observed, which seems to be related to the fact that the micelles are branched rather than linear. Indications for branching in this system, at higher water content, were recently found by dielectric spectroscopy for which the results could be explained by a percolation model for a connected network.8 In this paper, we will therefore compare results from the viscosity measurements at high water content with a model for dynamic percolation. 2. Experimental Section Samples were prepared with soybean lecithin from Lucas Meyer (Hamburg, Germany), type Epikuron 200. This surfactant was used without further purification so that it consists of a specific mixture of variable chain length, n, and degree of saturation, m (distribution of fatty acids, Cn:m:C16:0 ) 13.3%, C18:0 ) 3.0%, C18:1 ) 10.2%, C18:2 ) 66.9%, and C18:3 ) 6.6%).9 The samples are identified by two parameters. The first is w0, the molar ratio of added water to surfactant. For the second parameter we use C, the concentration of surfactants in g/l. Useful data to compare these with other quantities in use are the average molar weight of lecithin, 760 g/mol, and the density of lecithin, 1.056 kg/L. The water already present per mole of lecithin was determined to be approximately 0.7 mol by IR spectroscopy and Carl-Fisher titration.10 This value should be added to w0 in order to obtain the water-to-surfactant ratio in the sample. Isooctane was of analytical grade and obtained from Merck. Water was purified with a Millipore Milli-Q filtering apparatus. Special attention was given to the control of the water content in the samples: A concentration series at one w0 was performed by diluting batches from a large stock solution (0.5 or 0.25 L) to make rather large samples (25, 50, or 100 mL). Viscosity measurements were performed with an Ubbelohde capillary viscometer (Schott, Hofheim a. Ts., Germany), tube type 0a. The capillary was thermostated in a water bath, at 25 ( 0.05 °C. Calibration of the viscometer constant and Hagenbach correction factor was done by using a series of four organic solvents. The shear rate dependence was checked by performing the measurement for various samples at each w0 for three capillary diameters. The influence of the introduction of a waiting time after pumping up the sample was verified as well. None of these verifications showed a significant dependence. Efflux time measurements were repeated 10 times, which resulted in an error of typically 0.1 s. The uptake of water from the environment and evaporation of solvent from the samples limited (8) Cirkel, P. A.; van der Ploeg J. P. M.; Koper, G. J. M. Phys. Rev. E 1998, 57, 6875. (9) Shinoda, K.; Araki, M.; Sadaghiani, A.; Khar, A.; Lindman, B. J. Phys. Chem. 1991, 95, 989. (10) Scartazzini, R. Ph.D. Thesis, ETH Zu¨rich, 1990, nr. 9186.

Cirkel and Koper the accuracy. These factors were minimized by using a silicagel-filled tube at the viscometer inlet, which was contained in an Erlenmeyer filled with isooctane-soaked cotton. The data were interpreted in terms of the relative viscosity, ηr,

ηr )

η - η0 η0

(1)

By taking for η0 the viscosity of pure isooctane (i.e., 0.4736 mPas), the symbol η in this formula is the viscosity actually measured. At higher concentrations, beyond the dilute regime, measurements were performed with a Haake CV100 (strain-controlled) rotation viscometer (Haake, Karlsruhe, Germany). Shear rates between 0 and 300 s-1 were probed for the measurements at relatively low concentration. At higher concentration, the maximum in shear rate was dictated by the range in which the stress could be measured (30 Pa). At low shear rates, the rate was changed in steps of 0.01 s-1. At high concentrations, shear thinning occurred, and the viscosity was determined by a linear fit of the stress, τ, versus the strain, γ˘ , at low shear rates. At high concentration and w0 ) 2.5, a very peculiar behavior was observed: at low shear rates the data were best described by a Bingham model:

τ ) τ0 + ηγ˘

(2)

In this model τ0 denotes the yield stress, which seems to depend on the surfactant concentration by a power law with an exponent of 3. The viscosity was determined taking into account a yield stress through this expression. At high shear rates the stress exhibits a plateau, which could be related to shear banding (bands with different strains at different positions in the cell). Transient electric birefringence experiments were performed with a conventional setup11 as follows: Light from an 8 mW helium-neon laser with a wavelength of 632.8 nm travels through a high-quality Glen Thompson polarizer. This polarizer is oriented in such a way that linearly polarized light is obtained with its axis of polarization at 45° with respect to the electrical field. After passing through the sample, the beam travels first through a quarterwave plate and subsequently through a second Glen Thompson polarizer before a photomultiplier tube (PMT) converts its intensity to a voltage. The voltage of the PMT and the voltage applied on the sample are measured simultaneously by a digital oscilloscope (LeCroy 9450, Chesnut Ridge, NY). This oscilloscope can add up to 999 scans, which are subsequently transferred to a computer. The sample is contained in a quartz cuvette with two parallel platinum electrodes. The electrodes are held together at a spacing of 2.3 mm by two Teflon spacers. The optical path length is 49.3 mm. The temperature is kept constant at 25 ( 0.1 °C by circulating water from a thermostated bath through a jacket which holds the cell. The voltage is supplied by a function generator (Hameg, HM 8130, Frankfurt am Main, Germany) in combination with a high-speed power amplifier (NF Electronic instruments 4020). In this way a single pulse, as well as a train of pulses, could be generated. The response time of this setup is better than 0.1 µs. The field-free decay of the birefringence, ∆n, was fit to a stretched exponential function

∆n ∝ exp(-(t/τj)γ)

(3)

which is an empirical function for the relaxation by a distribution of relaxation times. γ is, in this case, a parameter related to the broadness of the distribution (0 < γ < 1; the lower γ the broader the distribution) and τj is the characteristic relaxation time. It is important to point out that, in general, this relaxation time is not necessarily exactly the time corresponding to a relaxation of a micelle of mean length (this is different in the procedure for dielectric spectroscopy discussed in the next paragraph). Depending on the actual shape of the length distribution these two will differ by a certain numerical factor. Dielectric spectra were recorded with an impedance analyzer, HP4194A (Hewlett-Packard, San Diego, CA), in the high(11) Stoylov, S. P. Colloid Electrooptics: Theory, Techniques, Applications; Academic Press: London, 1991.

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frequency range (100 Hz to 100 MHz). In the low-frequency range (from 1 Hz to 10 kHz) a lock-in amplifier (Stanford Research Systems, SR 510, Sunnyvale, CA), in combination with an external function generator (Hameg, HM 8130, Frankfurt am Main, Germany), was used. The analyzers were connected to cylindrical cells with various electrode separations. The measured impedance was interpreted in terms of a parallel circuit of a capacitance and a conductance, yielding both the dielectric permittivity and the conductivity versus frequency. The dielectric permittivity spectra were fit to the real part of the Cole-Cole function, which is a generalization of the Debye function12 describing the spectrum of a first-order relaxation process. The Cole-Cole function reads

 ) ∞ +

(

 s - ∞

)

1 + (iωτ)β

(4)

where ω is the circular frequency, τ the characteristic relaxation time, and s and ∞ are the permittivity at ωτ , 1 and ωτ . 1, respectively. In general, if a distribution of relaxation times is smooth and not too broad, the Cole-Cole function can be used for an almost symmetrical distribution of relaxation times. β describes the broadness of the distribution. The higher β the smaller the distribution of relaxation times, and with β ) 1 the Debye function is recovered describing one single relaxation time. We also analyzed the spectra by an inverse laplace transformation routine. This, however, did not give very satisfactory results since the variation in experimental error resulted in an inconsistent way in which the so-called smoothness parameter had to be used in order to obtain a regular variation of the relaxation time with surfactant concentration. Light scattering experiments were performed with a spectrometer consisting of an ALV goniometer and an ALV 5000 correlator. The light source was an Ar ion laser (Spectra Physics, 2000 series, Sunnyvale, CA) at the 514.5 nm line, linearly polarized perpendicular to the scattering plane. Scattering was measured at angles between 15 and 150°. The samples were centrifuged before measuring in order to remove dust particles. For dynamic light scattering, the measured reduced intensitytime correlation functions g2(t) were converted to reduced fieldtime correlation functions g1(t) via the so-called Siegert relation assuming Gaussian statistics. This field-time correlation function was submitted to a second-order cumulant analysis. The extrapolated value of the first cumulant divided by the scattering vector, q, squared for zero scattering vector was taken as the cooperative diffusion coefficient. Because the study described in this paper is concerned with dilute solutions, the cooperative diffusion coefficient equals the translational self-diffusion coefficient, D, of the micelles, which can be used to calculate the hydrodynamic radius by the Stokes-Einstein relation:

Rh )

kT 6πη0D

(5)

For static light scattering experiments the Rayleigh ratio R(q) of the solute is calculated from the ratio r(q) of the scattered intensity Isc to the intensity of the reference beam Iref. Combining with the corresponding ratios of the solvent r(sol) and the toluene standard r(st) (which was measured before and after each measurement series) and the Rayleigh ratio R(st) of toluene, the Rayleigh ratio of the solute becomes:

R(q) )

r(q) - r(sol) R(st) r(st)

(6)

From the Rayleigh ratio the form factor was determined in the usual way, by taking 9.31 × 10-5 L/g as the value for the refractive index increment. From the inverse form factor as a function of the scattering vector, a Guinier regime was determined, where it increases with the scattering vector squared. The slope of this plot was taken as 3 times the (z-averaged) radius of gyration. (12) Bo¨tcher, C. J. F.; Bordewijk, P. Theory of Electric Polarization, 2nd ed.; Elsevier Scientific Publ. Comput.: Amsterdam, 1978; Vol. II.

Dynamic sedimentation experiments were performed on an analytical ultracentrifuge (model E, Beckman Instruments, Palo Alto, CA). The setup was changed in such a way that the Schlieren pattern could be digitized by a CCD-microchip and sent to a computer, where further treatment of the image was performed. In the theory, to which we want to compare our results, sedimentation starts from a step-function density profile.9 The starting condition encountered in the experiments with ordinary cells, in which the sedimentation front starts at the upper cell boundary, is different. One of the experiments was therefore done in a two-sector cell, with one sector filled with solvent. This did not result in a significant change in the obtained propagation speed of the front. Sedimentation data were analyzed in terms of the sedimentation constant, S, obtained from the time dependence of the position of the front, r, as

S)

ln(r/r0) ω2t

(7)

where r0 denotes the position of the front at time zero. Experiments were done with rotational speeds, ω, between 2000 and 3600 rad/s. The sedimentation constant did not show any detectable dependence on rotational speed within this range. Measurements were done at 25 ( 0.1 °C.

3. Results Viscosity. The dependence of the relative viscosity on the surfactant concentration is shown in Figure 1. For all surfactant concentrations, the corrected viscosity is higher for a higher w0. This is a confirmation of the waterinduced micellar growth observed by other techniques. A qualitatively different behavior can be observed for w0 ) 0.5 and 1.5 on one hand and w0 ) 2.5 on the other hand. At lower w0, the corrected viscosity can be well described by a linear dependence on the concentration, which is not the case for the data at w0 ) 2.5. The linear dependence of the relative viscosity indicates that increasing the concentration has the effect of adding more micelles with a constant size, i.e., that the size of the micelles does not depend on the concentration. This opens the way to extract the one-particle effect in terms of the intrinsic viscosity, [η], by using the Einstein formula. The results of this analysis are shown in Tables 2 and 4. We also tried to analyze the results by the model of Duyndam and Odijk2 for the viscosity of wormlike micelles to check whether the linear dependence could be a coincidental compensation of effects. This model could indeed only be fit to the data if the exponent, by which the mean micellar size scales with the concentration, was set to a very low value.13 The data at w0 ) 2.5 could not be fit by the model in any case. For these data, the best fit to a simple function was obtained by assuming power-law dependence. In Figure 1b the viscosity measurements by the Ubbelohde viscometer are combined with results obtained by rotational viscometry at higher surfactant concentration. For w0 ) 2.5 this resulted in an S-shaped curve with scaling regimes at both high and low concentration. The exponent for the power-law behavior is lower in the low concentration regime (1.7) than in the high concentration regime (3.0). For the high concentration regime, a value of 1.9 has been reported before,14 although in that study a yield stress was not taken into account explicitly. The behavior at higher surfactant concentration is totally different for w0 ) 1.5, where the linear dependence at low surfactant concentration directly goes over into a power-law depen(13) In their model this could be done by setting the number of degrees of freedom lost upon aggregation to a very high value. (14) Schurtenberger, P.; Scartazzini, R.; Magid, L. J.; Leser, M. E.; Luisi, P. L. J. Phys. Chem. 1990, 94, 3695.

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Figure 1. (b) Relative viscosity at low surfactant concentrations. For w0 ) 0.5 and 1.5 the line represents a linear fit to the data. For w0 ) 2.5 the line represents a fit by a power law. (b) Relative viscosity at w0 ) 2.5 for both high and low surfactant concentrations. The data represented by open symbols were measured with an Ubbelohde viscometer, whereas the data represented by filled symbols were measured by rotational viscometry. φ denotes the surfactant volume fraction. Table 1. Sedimentation Constant as a Function of Surfactant Concentration at w0 ) 1.5 [lec] (g/l)

S (10-12 s)

1.14 1.90 2.66 3.42

7.645 7.159 6.074 5.597

dence with an exponent of 3.7,15 and no S-shaped regime is observed. Kerr Effect. Several examples of transient electric birefringence signals are shown in Figure 2a, in which their signal is normalized to the signal at a surfactant concentration of 7.6 g/l. For w0 ) 0.5 and 1.5, the signal relaxes to a plateau after application of the electrical field for a certain time, although at much longer times other (15) Cavaco C. Ph.D. Thesis, ETH Zu¨rich, 1994, nr. 10897 and Cavaco, C.; Schurtenberger, P. Helv. Phys. Acta 1994, 67, 227.

processes seem to occur. The birefringence at this plateau follows the Kerr law in its dependence on the electrical field, E, i.e., ∆n ) KE2. The Kerr constant, K, depends linearly on the concentration, except for the lowest surfactant concentration (i.e., 0.76 g/l); see Figure 2a. The decay after switching off the electrical field could be fit with a stretched exponential, with as best value for the broadness parameter γ ) 0.4. This value is too high when compared to the expectation based on a saddle-point approximation. Assuming an exponential length distribution,1 this kind of calculation leads to a γ of 0.25 for rigid rods16 and 0.3 for random coils.17 The disagreement with both of these values is an indication that the actual size distribution is different. It is interesting to note that a value for γ ) 0.4 is in agreement with rodlike micelles of size L for which the distribution is shaped exp(-RL2) at relatively large size. This would correspond to, for instance, a Gaussian size distribution. The characteristic relaxation time of the field-free decay, which corresponds to the rotational diffusion of micelles, has been studied in the semidilute regime before, where it was shown to depend quadratically on the surfactant concentration.17 In the dilute regime, on the other hand, it does not depend on the concentration in a systematic way. This is a confirmation of the observation from the viscosity measurement that the micellar size is concentration independent. The mean value of the relaxation time in the concentration range from 1.14 to 7.6 g/l increases with w0 and is given in Table 2. For w0 ) 2.5, the signal after switching on the electric field slows down rather than reaching a plateau (Figure 2a), indicating that there are slower processes which cannot be separated from the rotation of the micelles. The birefringence after a certain time follows the Kerr law, but the Kerr constant in this case increases more than linearly with the concentration (Figure 2a). In all cases the buildingup of the birefringence in field is slower by a factor of about 3 as compared to the field-free decay. This can be seen in Figure 2b, where on a logarithmic scale the in-field decay lags behind the outof-field decay by this factor except for very short or long times. It is a confirmation for the observation, made before in field-inversion experiments,18 that the micelles couple to the electric field by long-lived dipoles.19 Dielectric Relaxation. Two typical relaxation spectra are shown in Figure 3. In the measured dielectric spectra we distinguish two dispersions and a decrease of the permittivity at very low frequencies. This decrease strongly depends on electrode spacing; therefore, it has to be due to the diffusion of charged particles over a distance comparable to this spacing, the so-called electrode polarization. Possible slower relaxation processes will not be detected because of this electrode polarization. This effect could be separated from the rest of the spectrum by choosing a sufficiently large electrode spacing as we proposed in ref 20. At low frequencies there is a dispersion which is related to the rotational diffusion of the micelles. This dispersion could be fitted to the Cole-Cole function; see Figure 3 for an example. This is another indication that the relaxation(16) Bellini, T.; Mantagazza, F.; Piazza, R.; Degiorgio, V. Europhys. Lett. 1989, 10, 499. (17) Koper, G. J. M.; Cavaco, C.; Schurtenberger, P. In 25 Years of Nonequilibrium Statistical Mechanics; Brey, J. J., Maro, J., Rubı´, J. M., San Mighuel, M., Eds.; Springer-Verlag: Berlin, 1995. (18) Cirkel, P. A.; Stam, D. D. P. W.; Koper, G. J. M. Colloids Surf., A 1998, 140, 151. (19) Fredericq, E.; Houssier, C. Electric Dichroism and Electric Birefringence; Clarendon Press: Oxford, 1973. (20) Cirkel, P. A.; van der Ploeg, J. P. M.; Koper, G. J. M. Physica A 1997, 235, 269.

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Table 2. Experimental Data and Error Estimations at w0 ) 1.5 value error

MSeda (kg/mol)

RgSLSb (Å)

DDLSd (10-11 m2/s)

Rh,DLSc (Å)

[η]e (l/g)

τdielf (10-4 s)

τKerrg (10-4 s)

3490 50

520 25

1.63 0.12

283 21

0.0546 6.1 × 10-4

1.34 0.25

1.8 0.4

a M Sed: Average molar mass of the micelle determined by sedimentation and the diffusion coefficient from dynamic light scattering. Rg,SLS: radius of gyration from static light scattering. c Rh,DLS: hydrodynamic radius from dynamic light scattering. d DDLS: diffusion coefficient from dynamic light scattering. e [η]: intrinsic viscosity. f τdiel: rotational correlation time from dielectric spectroscopy. g τKerr: rotational correlation time from the Kerr effect.

b

Table 3. Theoretical Values of Several Parameters for Wormlike Chains and Rods roda wormlike chainb wormlike chainb

Rg (Å)

L (Å)

Dcalc (10-11 m2/s)

[η] (l/g)

τdiel (10-4 s)

τKerr (10-4 s)

520 266 520

1800 1800 5835

1.74 2.64 1.37

0.0540 0.0177 0.0488

1.356 0.177 0.911

4.1 0.50 2.7

a The values for rods were calculated according to the formulas in reference 27, with the cylinder radius 30 Å (neutron scattering) and the contour length 1800 Å (calculated from the micellar mass determined by sedimentation, which gives approximately the same result as light scattering (See text). b For wormlike chains two different contour lengths were taken for the calculations, 1800 Å from the micellar mass and 5835 Å calculated from the radius of gyration determined by light scattering (wormlike chain model25). The Kuhn length was taken from neutron scattering experiments (i.e., 300 Å). With these parameters the self-diffusion coefficient,28 intrinsic viscosity,29 and rotational correlation time30 were calculated by a model for wormlike chains.

Table 4. Comparison of the Experimental Data at Different Water/Surfactant Ratios (w0) w0 ) 0.5 w0 ) 1.5 w0 ) 2.5 (1/5 g/l)

[η] (L/g)

S (s) (at 1.5 g/l)

τdiel (s)

τKerr (s)

0.020 0.054 0.11

6.902 × 10-12 7.157 × 10-12 5.263 × 10-12

4.0 × 10-5 1.4 × 10-4 1.4 × 10-4

8.1 × 10-5 1.9 × 10-4 4 × 10-4

time distribution, reflecting the micellar size distribution, is symmetrical rather than exponential. The characteristic relaxation time of this dispersion behaves differently below and above w0 ) 2. Below w0 ) 2 this relaxation time again does not depend on the surfactant concentration for low concentrations. At higher concentrations the relaxation time increases with the concentration squared. In this respect, it has the same dependence on the surfactant concentration as the rotational correlation time measured by the Kerr effect. The full concentration dependence of this dispersion is discussed in another paper;8 here we will focus our attention on the dilute regime. The relaxation time at low concentrations increases with w0, and its mean value in the concentration range from 1.14 to 7.6 g/l is displayed in Tables 2 and 4. The concentration dependence of the increment corresponding to this dispersion is perfectly linear, as is shown in Figure 3 where the curves at different concentrations coincide if the increment is multiplied by the concentration ratio. Above w0 ) 2 the relaxation time starts to increase with a high power of concentration even at the lowest concentration studied here. The second, smaller dispersion, which appears in the spectrum at higher frequencies, is related to the rotation of the lecithin headgroup and is discussed in other papers.8,21 Light Scattering. The light scattering results are collected in Figure 4a,b. The hydrodynamic radius, Rh, and the radius of gyration, Rg, do not systematically depend on the surfactant concentration in the range from 1 to 6 g/l. This is another indication for a concentrationindependent micellar size. The averages of Rh and Rg in this range are displayed in Table 2. The value of Rh can be compared to the value of the self-diffusion coefficient determined before by FRAP (fluorescence recovery after photobleaching),22 by making use of the Stokes-Einstein equation (eq 5). In this study, the concentration depen(21) Cirkel, P. A.; van der Ploeg, J. P. M.; Koper, G. J. M. Prog. Colloid Polym. Sci. 1997, 105, 204. (22) Ott, A.; Urbach, W.; Langevin, D.; Schurtenberger, P.; Scartazzini, R.; Luisi, P. L. J. Phys: Condens. Matter 1990, 2, 5907.

dence of the self-diffusion coefficient for W0 e 2 was shown to exhibit a plateau at low concentrations. This plateau value for w0 ) 1.5 is 2.0 × 10-11 m2/s, which compares well with the value for the diffusion coefficient determined by dynamic light scattering (Table 2). The fact that the value from light scattering is somewhat lower is expected, since in light scattering the second moment of the micellar length distribution (Z or M2 average) is measured, whereas FRAP probes the first moment (M average).23 Above a surfactant concentration of 6 g/l there is a slight decrease in both the radius of gyration and the hydrodynamic radius, which is probably due to interactions. Below a surfactant concentration of 1 g/l there is a sharp decrease in both parameters. The ratio of Rg/Rh shows that the decrease is relatively significantly more important for Rg than for Rh. This ratio is 1.84 in the range from 1 to 6 g/l, whereas below 1 g/l it abruptly decreases to about 1. Sedimentation. The sedimentation of the micelles was clearly different from the sedimentation of a monodisperse sample of conventional polymers, since the Schlieren pattern became asymmetrical even after starting from a step-function density profile (corresponding to a delta peak in the middle of the Schlieren pattern). In all measurements, a clear maximum in the Schlieren pattern could be observed, of which the position, r, was followed in time. A plot of ln(r/r0) versus t always yielded straight lines, from which the sedimentation constant could be determined. The sedimentation constant turns out to depend both on w0 and on the surfactant concentration (Table 1). The theory for rodlike micelles that break up and recombine during the sedimentation process, however, predicts a sedimentation constant, which is concentration independent.3 An evaluation of the end-cap energy with this theory is therefore impossible. Several factors could be responsible for this. The most obvious explanation would be that the flexibility of the micelles in combination with an increase of the micellar size with the surfactant concentration is responsible for the concentration dependence. This, however, would always lead to an increase in the sedimentation constant with that of the surfactant concentration, whereas a decrease is observed. A decrease indicates that interactions between the micelles play a role. To evaluate this, a classical approach for the (23) Another important point in comparing data from different studies is that the amount of water present in the surfactant before adding water is different. The fact that, in this particular study, measurements at w0 ) 0, which in our case were unstable, are reported, suggests that the amount of water in their case is somewhat higher.

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Figure 3. Dielectric spectra at w0 ) 1.5. The dielectric increment of the spectrum at 1.52 g/L was multiplied by the concentration ratio with the sample at 7.6 g/L. The drawn line depicts a Cole/Cole fit, with two characteristic relaxation times, to the data.

This resulted in an apparent molecular mass, which decreases slightly with an increase in the surfactant concentration (Figure 5). This decrease is also observed for conventional polymers where it is due to interactions. As is customary for dilute polymer solutions, the interaction can be described by a virial expansion of the inverse apparent mass as follows:24

1 1 ) + 2A2c + 3A3c2 + ... Mapp M

Figure 2. (a) Decay curves of the electro-optic birefringence signal (Kerr effect) in arbitrary units. The curves were divided by both the concentration and the field strength squared. Note that the magnitude of the plateau value is perfectly linear for w0 ) 1.5 in the surfactant concentration range from 1.52 to 7.6 g/L (after the division the plateau values coincide). For a surfactant concentration of 0.76 g/L this linearity breaks down. For w0 ) 2.5 no linearity was observed in any case. (b) Decay curves of the electro-optic birefringence signal (Kerr effect) in arbitrary units.

interpretation of polymer sedimentation was used,24 in which the sedimentation constant was converted to an apparent mass, by using data about the micellar selfdiffusion coefficient, D, as follows:

Mapp )

SRT bD

(8)

Here, R denotes the gas constant (8.3 J mol-1 K-1) and T the absolute temperature (298 K). b denotes the ratio of the effective mass of the surfactant, corrected for the buoyancy of the solvent and the actual surfactant mass (taking for the density of lecithin and isooctane FL ) 1055.6 kg/m3 and FIso ) 691.9 kg/m3, respectively, so that b ) 0.3445). For the self-diffusion coefficient at w0 ) 1.5, the average value from dynamic light scattering was taken. (24) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.

(9)

Because of the low concentrations used in this study, the contribution of the third (and higher order) virial coefficient is negligible, and a second virial coefficient could be determined as A2 ) 3((1) × 10-8 L mol g-2. 4. Discussion Rodlike Structure. Since the micellar size seems to be concentration independent, comparison of the data, obtained by the various techniques used in this study, provides for a good opportunity to determine the micellar shape. To do so, we focused on the results at w0 ) 1.5. A parameter which is very sensitive to the micellar shape is the ratio between the micellar mass and the radius of gyration. For polymerlike micelles, this ratio can be compared to a model describing wormlike chains.25 From the molecular weight of a micelle the aggregation number can be calculated as N ) M/Mlec, with Mlec being the molecular weight of lecithin, i.e., 760 g/mol, so that N ) 4592. It is well established that the surface area per lecithin headgroup takes up about 70 Å21,26 in a variety of different aggregate structures. This result can be used to calculate the total surface area of a micelle: 3.21 × 105 Å2. Assuming that locally there is a cylindrical shape with a radius, determined by neutron scattering, of 30 Å7 (which is reasonable since it is about the length of a lecithin molecule), this leads to a micellar length L of about 1700 Å. The persistence length, Lp (which is two times the Kuhn length of a segment), was also determined by (25) Benoit, H.; Doty, P. J. Phys. Chem. 1953, 57, 958. (26) Small, D. The Physical Chemistry of Lipids. From Alkanes to Phospholipids; Plenum Press: New York, 1986.

Characterization of Lecithin Cylindrical Micelles

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Figure 5. Apparent molar mass calculated from the sedimentation data and the translational diffusion coefficient determined by dynamic light scattering.

Figure 4. (a) Radius of gyration (Rg) and the hydrodynamic radius (Rh) determined by light scattering for w0 ) 1.5. (b) Ratio of the radius of gyration (Rg) and the hydrodynamic radius (Rh) determined by light scattering for w0 ) 1.5.

neutron scattering.6 The value of 150 Å, combined with a total micellar length of 1700 Å, leads, for a wormlike chain,25 to a radius of gyration of

Rg ) Lp

x

[

]

2Lp Lp L -1+ 1 - (1 - e-L/Lp) ) 258 Å 3Lp L L (10)

This gives an underestimation by more than a factor of 2 compared to the radius of gyration measured by static light scattering (Table 2). On the other hand, a micelle with a radius of gyration of 520 Å and a persistence length of 150 Å would have a molar mass of 10 443 kg/mol. It is important to note that the micellar mass of 3490 kg/ mol, determined by sedimentation, is, within the experimental error, the same as the mass determined by light scattering. However, therefore, the data at very low concentration have to be disregarded6,7 (we come back to this point in the next paragraph). This suggests that the micelles are much stiffer. As an extreme case of a stiff micelle a model for rods could be taken for comparison. Interestingly enough, this leads to the correct relation between the micellar length and the radius of gyration, i.e., L ) 2x3Rg ) 1801 ( 70 Å. The ratio of the micellar

mass and radius of gyration can also be tested through the micellar volume for a certain shape. The volume of a cylindrical particle of length 1800 Å, and radius 30 Å, equals 5.1 × 10-24 m3. Combining this with the density of lecithin of 1055 kg/m3 leads to a molar mass of 3220 ( 300 kg/mol, which is well within the range of the experimental error for the mass determined by sedimentation. This encourages us to pursue further calculations assuming rodlike micelles27 and to compare these to results obtained with models for wormlike chains,28-30 with a persistence length of 150 Å. In the case of a model for rodlike micelles, the micellar length was calculated from the molecular weight and, together with the micellar radius from neutron scattering, was used to calculate the other parameters. As can be seen from the comparison between the experimental values in Table 2 and the calculated values in Table 3, the results of this analysis are very good and all parameters can be explained within their experimental error except for the rotational correlation time from the Kerr effect. The characteristic relaxation time in the latter case was fit to a stretched exponential which, as discussed in the Experimental Section, gives rise to an unknown coefficient depending on the shape of the length distribution. For wormlike chains, the data were compared to the model for two situations. First, the same analysis as in the case of rodlike micelles was performed, with the addition that the persistence length is 150 Å. The correspondence in this case is considerably worse when compared to a model for rodlike micelles. This becomes especially clear for the rotational correlation time, which is the parameter that is most sensitive to the extension of the chain. The correspondence with a model for wormlike chains can, however, be made much better if the comparison with the micellar mass is abandoned. If, for instance, the contour length is calculated from the radius of gyration (i.e., L ) 5835 Å) and a persistence length of 150 Å, the correspondence for the other parameters becomes reasonable. Nevertheless, the introduction of a considerable amount (27) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (28) Yamakawa, H.; Fuji, M. Macromolecules 1973, 6, 407. (29) Yamakawa, H.; Fuji, M. Macromolecules 1974, 7, 128. (30) Yoshizaki, T.; Yamakawa, H. J. Chem. Phys. 1984, 81, 982.

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Table 5. Calculated Micellar Mass and Contour Length for w0 ) 0.5 and 1.0 Using Model Predictions for Rodlike Particles and Values at w0 ) 1.5 W0

M (kg/mol)

L (Å)

0.5b 1.0a 1.5

2493 2896 3490

1286 1494 1801

a For w ) 1.0 the rotational correlation time from dielectric 0 spectroscopy was taken for the calculation. b For w0 ) 0.5 the average value from the intrinsic viscosity and the rotational correlation time was taken.

of flexibility worsens the general agreement if all observations are taken into account. Supportive evidence for rodlike rather than flexible particles comes from the correlation between the change in various parameters when comparing the data at w0 ) 0.5 to the data at w0 ) 1.5 (Table 4). In particular, the fact that the intrinsic viscosity increases in about the same amount as the rotational correlation time increases is incompatible with flexible polymers. The minute increase in sedimentation constant is another clue for rodlike particles. For rods, the length only affects the sedimentation constant through the logarithmic correction factor, describing end effects, as opposed to Gaussian chains for which the sedimentation constant increases with approximately the square root of length. The agreement with a model for rods allows for the calculation of the molar mass and contour length at w0 ) 0.5 and 1.0; the results of this calculation are shown in Table 5. Although the interpretation of the data in terms of rodlike micelles seems convincing, there are several factors which demand that caution be used in doing this. The most important point is that the system is rather polydisperse. The measured quantities are therefore all averages. The kind of average depends on the technique used. Unfortunately, the inclusion of polydispersity in the analysis is very complicated, and the result depends highly on the exact shape of the size distribution. However, the fact that all variables describing the dimension of the chain (Rg, D, [η], τr), measured by different techniques, can be described rather well within either a model for wormlike chains or for rodlike polymers, is a strong indication that polydispersity does not play a major role. Comparison of the dimension of the chain with the molar mass (determined by either light scattering or sedimentation) results in an error of a factor of 3 for wormlike chains with a persistence length of 150 Å, whereas the correct result is obtained for a rodlike particle. This seems, therefore, a strong indication that the particles actually are considerably stiffer than wormlike particles with a persistence length of 150 Å. In the literature, polydispersity has been taken into account explicitly for the comparison between static and dynamic light scattering data, through the ratio of the radius of gyration and the hydrodynamic radius.31 In our experiments, above a surfactant concentration of 0.75 g/l this ratio is 1.84, which compares well to the value of 2 reported before for somewhat higher concentrations.5 Application of our data with Rg/Rh ) 1.84 and Rh/d ) 4.7 (d denotes the cylindrical diameter, i.e., 60 Å) as parameters to the analysis by Mishic and Fisch,31 assuming an exponential size distribution, leads to a persistence length of more than 900 Å. In fact, under the circumstances met here (relatively low aspect ratio L/d), the result is insensitive to the inclusion of polydispersity. This analysis is again in clear disagreement with a persistence length (31) Mishic, J. R.; Fisch, M. R. J. Chem. Phys. 1990, 92, 3222.

of 150 Å, in which case an Rg/Rh of approximately 1.3 would be expected, as well as with a random coil containing a very long contour length, for which this ratio becomes 1.56.27 Another concern is that excluded-volume interactions within a micelle might play a role. In principle this could explain the apparently larger extension of the micelle relative to the calculation for (semi-) flexible chains without excluded volume effects. For a large number of classical polymers, however, it has been shown that this effect only becomes detectable if the chain length becomes larger than about 100 times the Kuhn length.32,33 The influence of interactions between micelles might also be a point of worry. However, in the concentration range studied here this appears to be unimportant for the transport properties. For instance, the viscosity does not deviate from a linear concentration dependence and the rotational correlation time does not depend on the concentration in a systematic way. For static properties, the concentration dependence can be used to quantify the interactions. This will be done in the next paragraph. Dependence on Surfactant Concentration. All parameters seem to depend on the surfactant concentration in a regular way, except at very low concentrations. Below a surfactant concentration of 0.75 g/l the most remarkable effect is the sudden drop to 1 of the ratio of Rg/Rh. This seems to be a typical indication for a sphereto-rod transition around that concentration. This interpretation is supported by the sudden decrease in the electric birefringence signal and the fact that the radius of gyration goes to 60 Å, which is exactly the value of the cylindrical diameter determined by neutron scattering. The data presented here suggest that, above the sphereto-rod transition, the micelles have a constant size independent of the surfactant concentration. This seems to be in conflict with the general idea that the micellar size increases with a certain power of concentration. This power might, however, be very low, if the number of degrees of freedom a surfactant loses upon aggregation is higher than what would be expected if only a simple mixing entropy is considered.34,35 Another idea is that one of the components of lecithin is preferentially adsorbed near the end-caps and thus dictates the size of a micelle. We are currently investigating whether this effect plays a role. This might also be a clue to explain the relative high concentration at which the sphere-to-rod transition seems to occur. Regimes with little36,37 or no38,39 dependence of the size of wormlike micelles on the surfactant concentration have indeed been observed in several other surfactant systems before. On the other hand, for the system under investigation here, it was argued from the interpretation of light-scattering results by renormalization group theory that the micellar size increases with the concentration by a power law with an exponent of 1.2.7 Such a high power cannot be explained theoretically for the growth of wormlike micelles. It seems that the growth measured in this study is very sensitive to the datapoints, which are close to the concentration region in which our data indicate a sphere-to-rod transition. This could mean that at least (32) Norisuye, T.; Fujita, H. Polym. J. 1982, 14, 143. (33) Norisuye, T.; Tsubol, A.; Teramoto, A. Polym. J. 1996, 28, 357. (34) Odijk, T. Biophys. Chem. 1991, 41, 23. (35) McMullen, W. E.; Gelbart, W. M.; Ben-Shaul, A. J. Phys. Chem. 1984, 88, 6649. (36) Wu, X.-I.; Yeung, C.; Kim, M. W.; Huang, J. S.; Ou-Yang, D. Phys. Rev. Lett. 1992, 68, 1426. (37) Berret, J.-F.; Appell, J.; Porte, G. Langmuir 1993, 9, 2851. (38) Kato, T.; Kanada, M.; Seimiya, T. Langmuir 1995, 11, 1867. (39) Steytler, D. C.; Jenta, T. R.; Robinson, B. H.; Eastoe, J.; Heenan, R. K. Langmuir 1996, 12, 1483.

Characterization of Lecithin Cylindrical Micelles

part of the abnormal growth reported there is related to this transition. At somewhat higher concentrations, the data on the osmotic compressibility reported by this group6,7 correspond very well to our data on the dynamic sedimentation. An interpretation of the apparent molecular mass as was done here would lead to approximately the same values for the mean micellar mass and second virial coefficient. From the concentration dependence of the sedimentation constant the second virial coefficient could be calculated,40 i.e., A2 ) 3 × 10-8 L mol g-2. Considering the large error due to the low absolute value, this compares reasonably well with the value for hardcore interactions between rodlike particles, i.e., 8 × 10-9 L mol g-2. The fact that the observed virial coefficient is higher indicates that there is already some effect of entanglements. The onset of this effect should be spread out in the case of wormlike micelles due to the size polydispersity. One might argue that a strong argument against the hypothesis of rodlike micelles with a concentrationindependent size is that the scaling laws in the semidilute regime are in agreement with predictions for equilibrium polymers.5,15 In fact this is not the case. For rods, most parameters are supposed to have power-law dependence over a certain concentration regime.27 For instance, the viscosity is supposed to scale with the third power and rotational correlation time scales with the concentration squared. Experiments on the system under investigation here show that for w0 ) 1.5 the viscosity scales with an exponent of 3.715 and the rotational correlation time with an exponent of 2.4 when measured by the Kerr effect17 and an exponent of 2.0 when measured by dielectric spectroscopy.8 In fact, the results from dielectric spectroscopy are in clear disagreement with the equilibriumpolymer model, which as opposed to a model for rods, predicts a dielectric increment which is concentration independent. Verification of the scaling behavior by light scattering does not exclude the interpretation by a model for rods either.41 For instance, the static correlation length is supposed to scale with a power of -0.5,42 whereas -0.7 is observed with a rather high uncertainty.5 Another argument against rodlike micelles could be that at higher concentration no liquid-crystalline phase is observed. The formation of a liquid crystal in this system will, however, be prevented by the polydispersity. Aggregation of the particles, which might well be important for this system at somewhat higher concentrations, due to the extremely high Hamaker constant for rodlike particles filled with water in an organic solvent,43 can be another factor preventing formation of a liquid crystal. This might also be an important factor in the observed deviations in scaling exponents. It is somewhat surprising that the micellar kinetics44 do not seem to affect the results from the sedimentation (40) Yamakawa, H. Modern Theory of Polymer Solutions; Harper and Row: New York, 1971. (41) DeLong, M. L.; Russo, P. S. Macromolecules 1991, 24, 6139. (42) De Gennes, P.-G.; Pincus, P.; Velasco, R. M.; Brochard, F. J. Phys. (Paris) 1976, 37, 1461. (43) Allen, M.; Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1987, 91, 2320.

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and translational diffusion experiments, since these parameters are measured on a relatively long time scale. The explanation for this could be that in dilute solutions the micellar kinetics are slower due to a difference in relaxation mechanism as compared to the semidilute regime.45 For the natural surfactant lecithin the extremely slow micellar kinetics could, as mentioned before, also be related to a certain component, which dictates both the thermodynamics and the kinetics of the system. At w0 ) 2.5, for all parameters studied in this paper a qualitatively different behavior is observed as compared to at w0 ) 0.5 and 1.5. A percolation model, under the assumption that a connected network is formed could explain the concentration dependence of the rotational correlation time and conductivity.8 In this study we tried to find supportive evidence for this hypothesis by looking at the viscosity. Qualitatively Figure 1b corresponds very well to the hypothesis of percolation for which an S shaped curve is expected with a critical surfactant concentration at the point where the viscosity changes very rapidly, corresponding to the percolation threshold. As expected, at both high and low surfactant concentration the dependence is described by scaling laws, with a lower value of the exponent at low than at high surfactant concentration.46 Unfortunately we have too little data just around the critical concentration to test a model for dynamical percolation47 in a quantitative way. 5. Conclusions This study on cylindrical micelles using classical techniques for the characterization of dilute polymer solutions revealed some rather unexpected results. Combining the experimental results from different techniques, the data can be described by a model for a solution of rods. Moreover, the size of the rodlike particles appears to be almost independent of the surfactant concentration. This is probably related to the very slow micellar kinetics, which seem to occur in this system. The micellar size distribution does not seem to be exponential, as is expected from the Israellachvili model.1 The Kerr effect measurements are in a better agreement with, for instance, a Gaussian distribution. Comparable data on other systems could be useful in order to get a better understanding of these phenomena. The need for such data is evident since, as was briefly shown in this paper, measurements on cylindrical micelles in the semidilute regime can be explained by more than one theoretical model. Acknowledgment. We thank A. Duyndam for the help with comparing the data to his models and many illuminating discussions. C Padberg is acknowledged for his indispensable help with the sedimentation experiments. LA971326X (44) For a review: Cates, M. E. In Fundamental Problems in Statistical Mechanics, 6th ed.; van Beijeren, H., Ernst, M. H., Eds.; Elsevier Science B.V.: Amsterdam, 1994. (45) Friedman, B.; O’Shaughnessy, B. Int. J. Mod. Phys. B 1994, 8, 2555. (46) Saidi, Z.; Mathew, C.; Peyrelasse, J.; Boned, C. Phys. Rev. A: At., Mol., Opt. Phys. 1990, 42, 872. (47) Grest, G. S.; Webman, J.; Safran, S. A.; Bug, A. L. R. Phys. Rev. A: At., Mol., Opt. Phys. 1986, 33, 2842. (48) Lagues, M. J. Phys., Lett. 1979, 40, 331.