Molecular Diffusion in a Living Network - Langmuir (ACS Publications)

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Molecular Diffusion in a Living Network L. Ambrosone,† R. Angelico,*,† A. Ceglie,† U. Olsson,‡ and G. Palazzo†,§ Consorzio per lo sviluppo dei Sistemi a Grande Interfase (CSGI) c/o, Universita` del Molise (DISTAAM), v. De Sanctis, I-861 00 Campobasso, Italy, Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, S-221 00 Lund, Sweden, and Dipartimento di Chimica, Universita` di Bari, v. Orabona 4, I-70126 Bari, Italy Received April 9, 2001. In Final Form: July 23, 2001 We report on the diffusion of a surfactant confined in a branched cylindrical “micellar” network, formed by lecithin and small amounts of water in the solvent isooctane. By means of the pulsed field gradient 1H NMR technique, the measured surfactant mean square displacement, 〈z2〉, allows for a detailed investigation on the microstructure of the micellar network. Our results show that the structure depends weakly on the micellar volume fraction, Φ, and strongly on the water-to-lecithin molar ratio, W0. We have studied the lecithin diffusion along two different oil dilution lines, corresponding to different water-to-lecithin molar ratios, 2 and 3. The time window in the diffusion experiments was varied in the range from 50 ms to 1 s. At W0 ) 3, a Gaussian diffusion, characterized by a mean square displacement varying linearly with time, was observed for all concentrations and all observation times investigated. Furthermore, the selfdiffusion coefficient was found to be independent of the concentration in the micellar volume fraction range studied from Φ ) 0.1 to Φ ) 0.38. The value of the diffusion coefficient is approximately 1/3 of the value of the lateral diffusion coefficient, Dc. At the second dilution line, W0 ) 2, the situation is markedly different. At lower concentrations (Φ < 0.11), we found at shorter times a mean square displacement 〈z2〉 scaling as t1/2 consistent with curvilinear diffusion. For longer times, there was a crossover to a Gaussian diffusion with 〈z2〉 ∝ t. The observation time where there is a crossover from curvilinear to a Gaussian diffusion shifts to shorter times with increasing Φ. At higher concentrations, only a Gaussian diffusion was observed within the experimental time window. The diffusion coefficient evaluated from the Gaussian regime increases linearly with Φ, the value varying from Dc/100 to Dc/20. The high diffusion coefficients evaluated at W0 ) 3 clearly indicate that the structure is a branched micellar network where the curvilinear distance along the cylindrical micelles between two branch points is smaller than the persistence length. At W0 ) 2, the data can also be interpreted in terms of a branched network, however with a much smaller density of branch points. The branching density increases with increasing Φ. Finally, the measured water diffusion along the two oil dilution lines was found to be Gaussian with a time-independent, single diffusion coefficient. The dominating mechanism for the water diffusion was found to be the motion inside the giant wormlike reverse micelles mediated by an interaggregate exchange with a characteristic time of the order of microseconds.

1. Introduction Surfactant solutions that form extremely long wormlike aggregates in the micrometer range are known to be equilibrium polymers or living polymers.1 They are linear aggregates whose length is fixed not chemically but thermodynamically. They reach a dynamic equilibrium where the self-assembly aggregates have a finite lifetime and can break and reconnect2 on a time scale which is dependent on the system and on the physicochemical conditions. In the semidilute regime, wormlike micelles entangle to build up continuous three-dimensional structures. These supermolecular structures exhibit strong gel properties, very reminiscent of the viscoelastic behavior of transient polymer networks. In recent years, the static and dynamic properties of flexible and entangled micelles have been extensively investigated experimentally both in water-rich3,4 and in organic solvent rich systems.5-7 * Corresponding author: R. Angelico, Universita` del Molise, DISTAAM, Via De Sanctis, I-861 00, Campobasso, Italy. Phone: +39-0874404647. Fax: +39-0874-404652. E-mail: [email protected]. † Universita ` del Molise (DISTAAM). ‡ Lund University. § Universita ` di Bari. (1) See, for instance: Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869 and references therein. (2) Cates, M. E. Macromolecules 1987, 20, 2282. Cates, M. E. J. Phys. (Paris) 1988, 49, 1593. (3) Lequeux, F. Curr. Opin. Colloid Interface Sci. 1996, 1, 341 and references therein.

The rheological and transport properties of the micellar network strongly depend on the possible cross-linking of the wormlike micelles leading to a truly cross-linked network (living networks) rather than an entangled network (living polymers). A statistical description of the occurrence of cross-links versus entanglements in semidilute solutions of polymer-like micelles has recently been proposed.8 It should be emphasized that the effect of the intermicellar connections, for example, in a flow experiment, is very different from that of classical interpolymer cross-links. Due to the liquid nature of the micelles, micellar branching leads to a decrease in the low-shear viscosity.9 On the other hand, the experimental observation of a decrease in viscosity of such surfactant solutions not always is due to the presence of branch points. For example, temporary cross-links, formed when two micelles cross, would also give rise to a decreasing viscosity10 as (4) Shikata, T.; Imai, S.; Morishima, Y. Langmuir 1998, 14, 2020. (5) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Phys. Rev. E 1997, 56, 5775. (6) Angelico, R.; Balinov, B.; Ceglie, A.; Olsson, U.; Palazzo, G.; Soderman, O. Langmuir 1999, 15, 1679. (7) Shchipunov, Y. A.; Hoffmann, H. Rheol. Acta 2000, 39, 542. (8) For binary systems, see: (a) Drye, T. J.; Cates, M. E. J. Chem. Phys. 1992, 96, 1367. (b) Elleuch, K.; Lequeux, F.; Pfeuty, P. J. Phys. I France 1995, 5, 465. (c) Cristobal, G.; Rouch, J.; Cure´ly, J.; Panizza, P. Physica A 1999, 268, 50. (9) Lequeux, F. Europhys. Lett. 1992, 19, 675. (10) Appel, J.; Porte, G.; Khatory, A.; Kern, F.; Candau, S. J. J. Phys. II 1992, 2, 1045.

10.1021/la0105275 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/02/2001

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Figure 1. Pictorial representation of the relationship between (2Dct)1/2, the experimental space resolution (diameter of the gray circle), and the density of branch points (small circles). (A) The branch points are most likely relatively rare and randomly distributed in the low concentration regime of the oil dilution line at W0 ) 2. For short NMR time scales, the experimental space resolution (2Dct)1/2 allows us to measure directly the curvilinear motion between two consecutive branch points. Here, Lb > (2Dct)1/2 where Lb is the mean curvilinear distance between two consecutive connections i and j. (B) At intermediate concentrations, only a few micellar junctions, connected by the distance R, fall in the circle of diameter (2Dct)1/2. Here, Lb e (2Dct)1/2. (C) High concentration regime. The surfactant molecules can visit many branch points inside the space resolution, making the observed diffusion typically Gaussian. Here, Lb , (2Dct)1/2.

well as a structural rod-to-sphere micellar transition does as found recently in a reverse micellar system.11 Furthermore, a multiconnected branched micellar network and an entangled micellar network cannot be distinguished by scattering techniques. Although cryo-TEM is a technique suitable to recognize branch points in micellar networks for aqueous systems,12 for organic solvent based systems it is much more difficult, if not impossible, to perform. One way of getting information on the microscopic topology is to measure a parameter the value of which is sensitive to the microstructure. As Anderson and Wennerstro¨m have treated it,13 such a parameter is constituted by the surfactant self-diffusion coefficient D, which can be conveniently determined by the Fourier Transform Pulsed field Gradient Spin-Echo NMR (FT PGSE NMR) technique.14 The most important feature of the NMR selfdiffusion approach is that it monitors movements over macroscopic distances (order of micrometers) characterized by the mean square displacement (from here forward, msd) 〈z2〉 (in one dimension). For the micellar system lecithin-water-cyclohexane previously investigated,15 we argued from the treatment of the NMR self-diffusion data that the transport processes would be constituted by curvilinear surfactant diffusion along giant wormlike micelles and concluded that such a system behaved like real living polymers (entangled but not branched micellar network). Indeed, in that system has been verified, for the first time in a nonconventional polymer system, the scaling law 〈z2〉 ∝ t1/2, that is, the expected time-dependence behavior of the anisotropic motion of a polymer segment during its reptative movement.16 To test the curvilinear diffusion model in a system with “true” branch points, we have performed for the present study a further self-diffusion investigation on a slightly different system, namely, lecithin-water-isooctane, but characterized by different phase equilibria. Indeed, water (11) Angelico, R.; Ceglie, A.; Cirkel, P. A.; Colafemmina, A.; Giustini, M.; Palazzo, G. J. Phys. Chem. B 1998, 102, 2883. (12) (a) Danino, D.; Talmon, Y.; Levy, H.; Beinert, G.; Zana, R. Science 1995, 269, 1420. (b) Lin, Z. Langmuir 1996, 12, 1729. (c) Strey, R. Colloid Polym. Sci. 1994, 272, 1005. (13) (a) Anderson, D. M.; Wennerstro¨m, H. J. Phys. Chem. 1990, 94, 8683. (b) Jo¨nsson, B.; Wennerstro¨m, H.; Nilsson, P.; Linse, P. Colloid Polym. Sci. 1986, 264, 77. (14) Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1986, 19, 1. (15) Angelico, R.; Olsson, U.; Palazzo, G.; Ceglie, A. Phys. Rev. Lett. 1998, 81, 2823. (16) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986.

dilution of lecithin viscoelastic solutions in apolar media results in a Winsor II phase separation if cyclohexane is the organic solvent,11 while with isooctane we have a highly viscous isotropic solution in equilibrium to almost neat isooctane17 at the phase separation. Furthermore, unlike the cyclohexane system, dynamic rheology18 and dielectric spectroscopy19 measurements strongly support the presence of a connected branched network for the micellar system in isooctane. Also, the high diffusion coefficients obtained by Ott et al.20 suggest the presence of branches. In this paper, we will prove the presence of the crosslink points inside the microstructure of the network in the isooctane system and make conclusions about the microstructural properties in comparison with the unbranched network in the analogue cyclohexane system. By tuning opportunely the time scale of the self-diffusion experiments, we are able to measure a continuous transition from a surfactant one-dimensional diffusion along the curvilinear micellar pathway connecting two branched points (see Figure 1A) to a three-dimensional Gaussian diffusion in a fully connected network (Figure 1C), passing through an intermediate configuration (Figure 1B). The experimental spatial resolution 〈z2〉1/2 ) (2Dct)1/2 dictates the dynamical character of the observed surfactant diffusion, Dc being the local curvilinear lecithin self-diffusion coefficient and t being the time scale of the FT PGSE NMR experiment. 2. Experimental Section Materials. Soybean lecithin Epikuron 200 was a generous gift from Lucas Meyer A. G., with a purity of 95% and an average molecular weight of 772 Da. The soybean lecithin used in this study is of the same quality as in our previous studies.21 Isooctane was of analytical grade and purchased from Carlo Erba. Water was Millipore filtered. All these chemicals were used as received, without any further purification. Sample Preparation. Two series of samples were prepared as oil dilution lines at fixed W0, at W0 ) 2 and at W0 ) 3, over the extension of the L2 isotropic solution region. For the line at (17) Schurtenberger, P.; Scartazzini, R.; Magid, L. J.; Leser, M. E.; Luisi, P. L. J. Phys. Chem. 1990, 94, 3695. (18) Cavaco, C. Ph.D. Thesis, ETH Zu¨rich, Nr. 10897, 1994. (19) Cirkel, P. A.; van der Ploeg, J. P. M.; Koper, G. J. M. Phys. Rev. E 1998, 57, 6875. (20) Ott, A.; Urbach, W.; Langevin, D.; Schurtenberger, P.; Scartazzini, P.; Luisi, P. L. J. Phys.: Condens. Matter 1990, 2, 5907. (21) Angelico, R.; Ceglie, A.; Olsson, U.; Palazzo, G. Langmuir, 2000, 16, 2124 and references therein.

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W0 ) 2, samples were obtained by dilution, adding weighed quantities of isooctane to a “stock-gel” with a micellar volume fraction Φ ) φlec + φw ) 0.46 (taking the value 1.0198 kg/L for the density of lecithin)22 where φlec and φw are the lecithin and water volume fractions, respectively. The corresponding samples were obtained with Φ in the range 0.036 e Φ e 0.36. For the second dilution line at W0 ) 3, samples were prepared singularly by dissolving weighed amounts of water to lecithin solutions in isooctane as previously described11 in the case of the cyclohexanebased system. The resulting concentrations were in the range 0.11 e Φ e 0.38. All the samples were let to equilibrate at 25 °C for 2 weeks before running the self-diffusion experiments. The lecithin used always contained strongly bound hydration water of the surfactant headgroups. Depending on the lot and on the storing procedure, such water content has been estimated between 0.3 and 1 water molecule per lecithin molecule.23 Owing to the large error in the evaluation of the real water-to-lecithin molar ratio occurring at low W0 values, it might be misleading to compare experimental results from other studies about the same micellar system, obtained using different lecithin batches. Self-Diffusion Measurements. All proton self-diffusion measurements were performed on a 200 MHz Bruker DMX spectrometer equipped with a Bruker field gradient probe, capable of giving field gradients of 9 T/m at a current of 40 A. The temperature was kept at 25 °C and was controlled with a thermocouple with an accuracy of (0.1 °C. A typical experiment was performed by keeping the gradient pulse length δ constant and gradually increasing the gradient strength G. For water self-diffusion experiments, a normal Hahn echo sequence was used with δ typically 4-5 ms and the maximum G in the range 1-2.5 T/m; for all experiments, four scans were accumulated for each value of the gradient strength. The echo intensities E(q,t) were reduced by at least 2 orders of magnitude. To ascertain that the water echo decays were single exponential and time scale independent, the experiments were carried out at different time scales in the range 50-100 ms. Since the lecithin wormlike reverse micelles are very large, the slow reorientation of the aggregates together with the slow diffusion of lecithin on the micellar surface made the T2 relaxation of the surfactant quite rapid. For this reason, self-diffusion measurements were carried out with a stimulated echo sequence14 by varying the gradient strength up to 8 T/m, while both the time between the first two 90° pulses and the gradient pulse duration δ were kept constant and equal to 5 ms. For each gradient strength, 16 phase cycling steps were accumulated in order to remove unwanted echoes. The resonance intensity of the methyl groups at 3.5 ppm of the -N+-(CH3)3 of the lecithin headgroup was used in the evaluation of the surfactant self-diffusion experiments. To check for either any deviation from the expected single-exponential echo decay or any time-scale dependence, the total diffusion time t in the stimulated echo sequence was varied in the range 50-1000 ms.

3. Diffusion in Wormlike Micelles: Living Polymers versus Living Networks Within the short pulse limit (δ , t), the normalized echo attenuation, E(q,t), corresponds to the Fourier transform of the one-dimensional diffusion average propagator parallel to the applied magnetic field gradient (here, the z direction). For a homogeneous and isotropic solution, we may write24

E(q,t) )

∫-∞ dzP(z;t) exp{iqz} ∞

(1)

Here, q ) γgδ, where γ is the gyromagnetic ratio of the nucleus (here, 1H). The average diffusion propagator P(z;t) describes the probability that the molecule has diffused for a distance z during a time t. In the PGSE experiment, the observation or diffusion time t is given by the time (22) Angelico, R.; Ambrosone, L.; Ceglie, A.; Olsson, U.; Palazzo, G. Prog. Colloid Polym. Sci. 1999, 112, 1. (23) Scartazzini, R. Ph.D. Thesis, ETH Zu¨rich, Nr. 9186, 1990. (24) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Clarendon: Oxford, 1991.

separation between the two gradient pulses. In a simple liquid, P(z;t) is Gaussian,

P(z;t) )

exp{-z2/(4Dt)} (4πDt)1/2

(2)

where D is the self-diffusion coefficient of the species under investigation, resulting in a Gaussian echo attenuation:

E(q,t) ) exp{-Dtq2}

(3)

This situation we refer to as Gaussian diffusion, where 〈z2〉 ) 2Dt. When the average diffusion propagator which enters eq 1 is unknown, one can still determine the msd 〈z2〉 from evaluating the initial slope of E(q,t) when plotted as a function of q2. In the limit of small q, we have

E(q,t) )

(

∫-∞∞ dzP(z;t) 1 + iqz -

)

(qz)2 〈z2〉q2 (4) )12 2

Curvilinear Diffusion. In a system of giant wormlike micelles, the surfactant molecules are bound to the micelles, with a characteristic lifetime, τ, where they are free to perform one-dimensional curvilinear diffusion along the micellar contour. If this lateral diffusion is the dominating mode of long-range material transport, the average diffusion propagator will be given by the (Gaussian) one-dimensional lateral diffusion propagator, averaged by a distribution function ψ(z;l), describing the probability that the projection of a lateral displacement l corresponds to a displacement z along the gradient direction. This we refer to as curvilinear diffusion that resembles the polymer segment diffusion in the case of reptation.16 For Gaussian chain statistics (Flory regime), the corresponding echo attenuation is given by15,25

E(q,t) ) exp{x2} erfc{x}

(5)

(Dct)1/2λq2 x) 3

(6)

where

Here, Dc is the lateral diffusion coefficient and λ is the persistence length related to the chain stiffness. For a continuous, wormlike chain, the concept of Kuhn segment length that is twice the persistence length is also used.26 This peculiar anomalous diffusion can be evidenced more conveniently by plotting the experimental E(q,t) versus q2t1/2 measured at different time scales t, resulting in an almost complete overlap of all the echo decays in this type of plot.15 For curvilinear diffusion, the msd is given by

〈z2〉 )

( )

4 Dct λ 3 π

1/2

(7)

From the t1/2 scaling, we again see the resemblance to reptation. Migration between Living Polymers. If the average time that a molecule resides on a micelle is short compared to the lifetime of micelles, in general, they can exchange molecules in essentially three different ways: (i) Molecules on the aggregates are in equilibrium with a small concentration of monomers in the continuous oil solvent. (25) Fatkullin, N.; Kimmich, R. Phys. Rev. E 1995, 52, 3273. (26) Fujita, H. Polymer Solutions; Elsevier Science Publishers: Amsterdam, The Netherlands, 1990.

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In this case, τres denotes the average residence time on the aggregates before leaving an aggregate for the bulk solvent.27 From the organic bulk, the molecule joins up with another micelle. (ii) Micelles can possibly exchange molecules, also by means of direct exchange through collisions. Such a mechanism has been suggested by Kato et al.28 to be significant for the surfactant self-diffusion in solutions of wormlike micelles formed by nonionic surfactants. For the wormlike micelles, Kato considered micellar exchange of molecules at entanglement points with a certain probability. In the case of the nonionic surfactant C12E5, Nilsson et al.29 found an increase in the observed surfactant self-diffusion coefficient at higher surfactant concentrations and higher temperatures, approaching the cloud point, similar to the observation in the present system. They considered that the increase in D comes from the fact that the exchange of surfactant monomers between different micelles is enhanced. For this exchange process, there is another residence time on the micelles that we call the exchange time, τexch, to distinguish it from τres defined above. (iii) In the two exchange processes discussed above, material is exchanged as individual molecules. A third mechanism involves fragments of micelles that are broken off and then join up (coalesce) with another micelle. For wormlike micelles, this is Cates’ breaking-recombination process2 that also is important for the rheology. If the breaking event has the same probability to occur anywhere in the micelle, the average lifetime of a micelle, τbreak, before it breaks is inversely proportional to the micellar contour length L, that is, τbreak ∼ 1/L. When considering processes that are important for the surfactant self-diffusion, the three characteristic times τres, τexch, and τbreak have to be compared one to each other and to the experimental observation time, t, that here was varied in the range from 50 ms to 1 s. If all three exchange processes are slow compared to the observation time, they will not influence the self-diffusion on this time scale. In the case of very long micelles, for which the micellar diffusion is negligible, it is only the curvilinear diffusion along the micellar contour that contributes to the displacement of molecules. This behavior was observed for a broad range of time scales and concentrations for lecithin reverse micelles in cyclohexane,15 and as will be discussed below, we also find it here in the isooctane system at W0 ) 2 and low Φ. The condition is that the observation time is shorter than the diffusion time along a micelle, τL, which for a contour length, L, is given by

τL )

L2 2Dc

(8)

When an exchange process is fast compared to the diffusion rate in the experimental time scale, the diffusion behavior is altered (we still neglect micelle diffusion) and the msd shifts from the t1/2 scaling to a linear time dependence, that we also refer to as Gaussian diffusion (see Figure 2). In an entangled solution with very long micelles so that the exchange process is fast compared to τL, the diffusion may be modeled as a three-dimensional random walk with a step length ξ and a step time τ so that (27) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80, 905. (28) (a) Kato, T.; Terao, T.; Tsukada, M.; Seimiya, T. J. Phys. Chem. 1993, 97, 3910. (b) Kato, T.; Terao, T.; Seimiya, T. Langmuir 1994, 10, 4468. (29) Nilsson, P. G.; Wennerstro¨m, H.; Lindman, B. J. Phys. Chem. 1983, 87, 1377.

Figure 2. Log-log representation of the predicted msd 〈z2〉 vs experimental time scale t: for t e τ (characteristic step time), 〈z2〉 ∼ t1/2 according to the curvilinear diffusion model, while for t . τ, 〈z2〉 scales linearly with time (Gaussian diffusion).

the self-diffusion coefficient is given by

D)

ξ2 6τ

(9)

The step time, τ, can be identified with the characteristic time for the exchange process (i.e., τres, τexch, or τbreak), and ξ2 is the (three-dimensional) msd due to curvilinear diffusion during a time τ. For a time τ, the average curvilinear displacement is l ) (2Dcτ)1/2, associated to the end-to-end distance ξ by

ξ2 ) 2λl

(10)

(in the limit l . λ), and the diffusion coefficient becomes

D)

λl 3τ

(11)

that with l ) (2Dcτ)1/2 can be also written as

D)

λ(2Dc)1/2 3τ1/2

(12)

Thus, we have D ∼ τ-1/2. If τ is longer than τL but still shorter than t, we are in the so-called “waiting regime”30 where the molecules can explore the whole micelle before the subsequent exchange. In this case, l can be replaced by L in eqs 10 and 11 and the diffusion coefficient becomes31

D)

λL 3τ

(13)

Now, depending on which is the fastest and therefore dominating exchange process, the concentration dependence of the self-diffusion coefficient can be different. Considering exchange via the solvent, τres increases with increasing concentration and therefore the diffusion coefficient decreases with increasing concentration. For the case of exchange at entanglements, we expect the diffusion coefficient to increase with the concentration because of an increasing entanglement density. However, for the specific surfactant in object, the characteristic time for the lecithin flip-flop motion is of the order of hours (if (30) Morie, N.; Urbach, W.; Langevin, D. Phys. Rev. E 1995, 51, 2150. (31) Schmitt, V.; Lequeux, F. Langmuir 1998, 14, 283.

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not days).32 This suggests that the transfer of a lecithin polar head in a hydrophobic environment, that is, through an entanglement point, is severely hindered from a kinetic point of view. Moreover, we do not observe any effect on the observed self-diffusion coefficients due to the surfactant migration mechanism, for the analogous system lecithin-water-cyclohexane, in the whole range of volume fractions explored. In addition, the breaking and recombination exchange leads to increase of the diffusion coefficient with increasing concentration. This because τbreak ∼ L-1 and L is expected to increase with the concentration.1 In the discussion above, we have assumed that l . λ. If the opposite holds, that is, l , λ, either because of very stiff micelles or because of very rapid exchange, then the diffusion process will be as along randomly oriented straight stiff pipes and the diffusion coefficient is given by13

Dc D) 3

D)

This corresponds to unrestricted lateral diffusion where the factor 1/3 comes from the fact that the diffusion is locally one-dimensional. For reasonable values of the persistence length (λ ≈ 100 Å), this limit is not realistic in semidilute solutions since it essentially means that there is free passage between micelles at any contact. Free passage rather implies a branched network. Diffusion in Living Networks. In recent years, one has found that in certain systems of wormlike micelles the micelles are not only entangled but also branched, forming a three-dimensional infinite network.8 Obviously, the presence of branches has the same effect on the surfactant diffusion as the exchange between micelles discussed above. If the surfactant molecule passes many branch points during the observation time t, the diffusion process becomes Gaussian. In this case, the diffusion coefficient will depend on the average number of branch points sampled during the time t. We define Lb as the curvilinear contour distance along the cylindrical micelle between two branch points. If we sketch a connected network, like in Figure 1A, where the cross-link points are marked by the circles, we see that a lecithin molecule localized in the position i must move along a curvilinear one-dimensional path of length Lb before reaching the first neighboring branch j. The diffusion time, τb, between two branch points is given by

(17)

Lb 2λ

(18)

where

N)

is the number of Kuhn lengths between two branch points. In the other limit when Lb < λ, the diffusion coefficient is given by eq 14. Mean field theories foretell that Lb ∼ N should decrease8 with increasing Φ, and indeed, from eq 17 we expect an increase of D with increasing the volume fraction of the dispersed phase. Finally, the density of micellar branch points Fz is related to the average curvilinear length (Lb ) N2λ) between two junctions by

Fz ) (14)

Dc ξ2 ) 6τ 3N

2Φ zπrc2Lb

(19)

where z is the connectivity of junctions and rc denotes the cross-sectional radius of the cylindrical micelles. In the next sections, we shall present the analyzed data and discuss them in the framework of the previous diffusion mechanisms. 4. Results

and the step time is τb. Analogous to eq 11, the diffusion coefficient can be written as

Lecithin Diffusion: W0 ) 3. At this W0 value, we found that all the amphiphile self-diffusion measurements were independent of time scale, in the range 25-500 ms explored (see Figure 3). Moreover, for W0 ) 3, the echo decays are exponential with a surfactant self-diffusion coefficient D which is essentially independent of the micellar volume fraction Φ in the range 0.1 < Φ < 0.38 investigated resulting, thus, in an average diffusion coefficient D ) 4.2 × 10-13 m2 s-1 (see Figure 4). We also note that in another connected network micellar system, Monduzzi et al.33 reported a very weak surfactant self-diffusion dependence on the micellar volume fraction. The Φ-independence of D and its average value, which is approximately one-third of the lateral diffusion of lecithin in bilayers at 25 °C,34 (≈10-12 m2 s-1), are consistent with a fully branched network (N e 1 in eq 17). Lecithin Diffusion: W0 ) 2. The self-diffusion experiments performed for all the samples belonging to the oil dilution line at fixed W0 ) 2 (0.036 e Φ e 0.36) show very peculiar and quite singular features. The initial slopes of the echo decays are very low, the apparent lecithin diffusion coefficients evaluated as 〈z2〉/2t, lying between 10-14 and 10-13 m2 s-1. In Figure 5C, the experimental echo attenuations at Φ ) 0.36 (the highest volume fraction explored) can also be fitted by eq 3, with an observed surfactant diffusion coefficient D which is constant over the time window investigated, 50 ms e t e 1 s. On the contrary, at the lowest volume fraction Φ ) 0.036, a significant dependence on the experimental time scale has been detected, as is evident from Figure 5A. Here, more interestingly, the initial slope slightly reduces with time. For intermediate concentrations (Figure 5B), a smooth decrease of the dependence on the time scale of the initial slope of E(q,t) occurs with increasing the volume fraction Φ. Note that at low Φ, by plotting the echo decays versus q2t1/2 the apparent time dependence on the experimental

(32) Ben-Shaul, A. Structure and Dynamics of Membranes; Lipowsky, R., Sackmann, E., Eds.; Elsevier: Amsterdam, 1995; Vol. 1A, Chapter 7 and references therein.

(33) Monduzzi, M.; Olsson, U.; So¨derman, O. Langmuir 1993, 9, 2914. (34) Lindblom, G.; Ora¨dd, G. Prog. Nucl. Magn. Reson. Spectrosc. 1994, 26, 483.

τb )

Lb2 2Dc

(15)

Under the condition t e τb, we observe the onedimensional curvilinear diffusion of surfactant, while for τb , t the diffusion is Gaussian. The diffusion coefficient may still be very low if the density of branch points is low, so that Lb . λ. Analogous to the exchange case, we can model the diffusion in this limit as a random walk where the characteristic step length now obeys

ξ2 ) 2λLb

(16)

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Figure 4. The time-independent lecithin self-diffusion coefficients at W0 ) 3, calculated by fitting eq 3, plotted vs the micellar volume fraction Φ. The hatched line corresponds to the mean value D ) 4.2 × 10-13 m2 s-1.

Figure 3. Semilog plots of experimental normalized lecithin echo decays vs q2t at W0 ) 3 for (A) micellar volume fraction Φ ) 0.11 and t ) 50-200 ms, (B) Φ ) 0.16 and t ) 100-400 ms, and (C) Φ ) 0.33 and t ) 25-500 ms. All the graphs are plotted in the same coordinate system.

echoes disappears for E(q,t) measured up to t ) 250 ms, while deviations from the curvilinear behavior occur for t’s higher than 250 ms (see Figure 6A), indicating that the curvilinear model does not hold above a certain time scale. Experiments performed on samples at Φ ) 0.051 and Φ ) 0.073 reveal that also at these compositions the echo attenuations fan out for t > 250 ms (not shown). At Φ g 0.11, all the experimental echo attenuations are focused in the usual Stejskal-Tanner plot, E(q,t) versus q2t (see Figure 5B,C), and spread out when plotted versus q2t1/2 (Figure 6B,C). For Φ ) 0.036 and t e 250 ms, the fit of experimental decays to eq 5 is shown as a solid line in Figure 6A. Further, using the lateral diffusion coefficient of lecithin, Dc ) 1 × 10-12 m2 s-1, measured for bilayers of lamellar phase at room temperature,34 we estimate λ ≈ 150 Å from the calculated fitting parameter. From accurate small-angle neutron scattering (SANS) studies on the same lecithin

system in isooctane, Jerke et al.5 had persistence lengths of λ ≈ 150 Å and λ ≈ 160 Å at W0 ) 1.5 and W0 ) 2.5, respectively. The time dependence on the experimental surfactant msd is strongly affected by the micellar volume fractions as shown in Figure 7. Here, we represent only four measurements of 〈z2〉 (at t ) 50, 250, and 500 ms and 1 s) per sample for selected volume fractions Φ in a wide concentration range, but this has no consequence for the basically qualitative discussion presented below. At low values of Φ, we observe 〈z2〉 ∝ t1/2 for time scales up to ≈250 ms and a linear dependence of 〈z2〉 with t for longer time scales. The overall picture is very similar to Figure 2 with an estimated crossover time t*, between the two scaling behaviors, of about 0.4 s. The higher the volume fraction, the shorter the crossover time. Indeed, at high Φ values we observe only the direct proportionality between 〈z2〉 and t (Figure 7). Summarizing, for W0 ) 2 and Φ e 0.11, we have curvilinear diffusion for t e t* and Gaussian diffusion at longer times. For higher micellar volume fractions, we find always Gaussian diffusion with long time D (measured at t ) 1 s) that increases with Φ. The experimental surfactant self-diffusion data, together with the number of Kuhn lengths N between two junctions, calculated from eq 17 under the assumption of 3-fold branches,35 are listed in Table 1 for different Φ values. The N values are also plotted versus Φ in Figure 8. Finally, from eq 19 we can evaluate the density of branch points Fz obtained for z ) 3 and using the rc value known from SANS measurements.5 The values, reported in Table 1, indicate that the branch density increases with increasing the micellar volume fraction. Water Diffusion. The echo decay of the water peak for lecithin organogels in isooctane was recorded at different time scales t, and for all the samples examined the decay was found to be monoexponential and time independent. At the two W0 values investigated, the water self-diffusion coefficient decreases with Φ along oil dilution lines. A similar situation was found for water in the lecithinwater-cyclohexane system and was proven to be due to a fast exchange of water molecules secluded in the polar (35) Systems with negative saddle-slay modulus, κj, prefer a minimal number of arms, z ) 3. This is in accordance with the observation of “Y-like” structures by Danino et al. (ref 12a) which suggests 3-fold junctions.

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Figure 5. Semilog plots of normalized lecithin echo decays vs q2t at W0 ) 2, measured in the range t ) 50 ms to 1 s for (A) micellar volume fraction Φ ) 0.036, (B) Φ ) 0.11, and (C) Φ ) 0.36. All the graphs are plotted in the same coordinate system.

core of reverse micelles with water monomers dissolved in the organic bulk.6 In the case of fast water exchange, the value of Dw is a population-weighted average of the diffusion coefficients of water solubilized in the bulk oil, Dw,o, and of the water molecules confined in the aqueous core of the aggregates, Dw,aq, and one can write

Dw )

φo KDw,o + Dw,aq φw

(20)

where Dw is the observed diffusion coefficient of water, K is the water solubility in the organic solvent (on a volume fraction basis), and φo and φw are the oil and water volume fractions. Equation 20 predicts a linear dependence of Dw on the ratio φo/φw, as experimentally found (Figure 9); moreover, the two straight lines have the same slope KDw,o

Figure 6. The same normalized lecithin echo decays shown in Figure 5 are plotted here vs the new abscissa q2t1/2: only at low Φ (Φ ) 0.036) and at short times (t ) 50-250 ms), the decays overlap each other. The solid line is the best fit according to eq 5 (curvilinear diffusion).

) (2.1 ( 0.1) × 10-13 m2 s-1 (a value very close to that found in cyclohexane)6 as expected for samples sharing the same diffusion and solubility of water in isooctane. On the other hand, the values of the intercepts depend on W0, Dw,aq being equal to (1.24 ( 0.06) × 10-11 m2 s-1 and (2.64 ( 0.08) × 10-11 m2 s-1 at W0 ) 2 and 3, respectively. Summarizing, we have studied the lecithin and water diffusion along two different oil dilution lines, corresponding to different water-to-lecithin molar ratios, W0. The time window in the surfactant diffusion experiments was varied in the range from 50 ms to 1 s. At W0 ) 3, a Gaussian diffusion, characterized by a msd varying linearly with time, was observed for all concentrations and all observation times investigated. Furthermore, the self-diffusion coefficient was found to be independent of the concentration in the concentration regime studied from Φ ) 0.1 to Φ ) 0.38. The value of the

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Figure 7. Log-log plot of surfactant msd 〈z2〉 vs experimental time scale t for selected volume fractions in the range 0.036 e Φ e 0.36 and W0 ) 2. The slope of the hatched line is 0.5 in accord with the correct exponent for the curvilinear diffusion observed, here, only at low Φ and up to t ) 250 ms. The solid lines have unitary slope identifying the regimes where the diffusion is Gaussian. Table 1. Surfactant Self-Diffusion Coefficients D Measured at the Longest Time Scale of t ) 1 s, for Nine Different Volume Fractions Φ at Fixed W0 ) 2a F 0.036 0.036 0.051 0.073 0.11 0.15 0.19 0.23 0.32 0.36

D (10-14 m2 s-1)

N

F3 (µmol/L)

1.68 1.60 2.13 2.25 3.00 4.13 4.88 5.88 6.40

23-33b 20 21 16 15 11 8.1 6.8 5.7 5.2

0.4-0.5 0.6 0.8 1.5 2.4 4.4 7.6 11 18 23

a For each sample, both the mean number of Kuhn lengths between two branches N (eq 17) and the densities of branch (3-fold) points F3 (eq 19) are presented. b Lower and upper limit for N estimated identifying the curvilinear-Gaussian crossover time to t ) 0.25 and 0.5 s, respectively (see section 5).

Figure 8. Log-log plot of the number of Kuhn lengths, N, connecting two consecutive branch points on the micellar network (calculated from eq 17), plotted vs Φ. The hatched line is the best fit to the power law N ∝ ΦR, yielding R ) -0.55 ( 0.07. Closed square and closed circle symbols represent the lower and upper limit for N, estimated identifying the curvilinear-Gaussian crossover time to t* ) 0.25 and 0.5 s, respectively (see section 5).

diffusion coefficient is approximately 1/3 of the value of the lateral diffusion coefficient, Dc. At the other dilution line, W0 ) 2, the situation is

Figure 9. Experimental water self-diffusion coefficients Dw along oil dilution lines for two different W0 values. The lines are the best fits to eq 20.

markedly different. At lower concentrations (Φ < 0.11), we found at shorter times the msd 〈z2〉 scaling as t1/2, consistent with curvilinear diffusion. For longer times, there was a crossover to a Gaussian diffusion with 〈z2〉 ∝ t. The observation time t* where there is a crossover from curvilinear to a Gaussian diffusion shifts to shorter times with increasing Φ. At higher concentrations, only a Gaussian diffusion was observed within the experimental time window. The diffusion coefficient evaluated from the Gaussian regime increases with Φ, the value varying from Dc/100 to Dc/20. Water diffusion reveals a different mechanism governed by time-independent Gaussian diffusion with water selfdiffusion coefficients decreasing with Φ along both oil dilution lines. 5. Discussion Microstructure in the Lecithin-Water-Isooctane System. We now turn to discuss the experimental results obtained in the present study. Starting with W0 ) 3, we found for this dilution line a concentration-independent surfactant diffusion coefficient with a value of approximately Dc/3. This very high diffusion coefficient strongly implies that the structure here is a branched micellar network. The branching density Fz where z is the connectivity of junctions, which is expected to increase with increasing concentration,8 is so high that already for the lowest concentration investigated (Φ ) 0.1) Lb < λ. At the other dilution line (W0 ) 2), we found different surfactant diffusion behavior. At lower concentrations and shorter observation times, we found a curvilinear surfactant diffusion coefficient, from which we could estimate a micellar persistence length λ ≈ 150 Å. This value is in quantitative agreement with the value found by Jerke et al.5 when analyzing SANS data from dilute solutions. Increasing the observation time, a Gaussian diffusion was found for all concentrations. Explanations compatible with a microstructure based on living polymers are unlikely. The concentration dependence of the diffusion coefficients rules out exchange of molecules via bulk and through breaking-recombination processes faster than the diffusion over the whole micelle. For disconnected micelles, the only regime where the observed diffusion coefficient increases with surfactant concentration is the so-called waiting regime.30,31 Here, the condition is L < (2Dcτ)1/2, that is, a surfactant molecule diffuses over the whole

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micelle before the exchange of material takes place (either as molecular hopping30 τ ≡ τbreak or as micellar coalescence28 τ ≡ τexch). However, in this case the lecithin diffusion coefficient should obey eq 13 identifying τ as the crossover time t* between the curvilinear and Gaussian diffusion (see section 4). The estimated t* at low Φ is 0.4 s, while it has to be less than 50 ms at high Φ. On this ground, we can reject the presence of living polymers since, for Φ < 0.11 and calculating L from eq 13 using the experimental D values in Table 1 and τ ) t*, the inequality L < (2Dcτ)1/2 is not verified. Moreover, at higher Φ we have t* < 50 ms, and considering that the average contour length of living polymer increases with Φ, the inequality L < (2Dcτ)1/2 breaks to a larger extent. Possibly, we have a branched network also at W0 ) 2 and this is the rationale for the observed Gaussian diffusion at long observation times. Here, however, the diffusion coefficients are low implying that the branching density Fz is much lower at W0 ) 2 compared to that at W0 ) 3. Since D is inversely proportional to N (see eq 17), this implies that the Fz increases strongly with W0. In Figure 8, we have plotted how N varies with Φ, calculated from eq 17 with Dc ) 1 × 10-12 m2 s-1 and using the experimental surfactant self-diffusion coefficients D measured at the longest experimental time scale of 1 s (see Table 1), that is, beyond the crossover time t* to Gaussian diffusion, where eq 17 holds. As can be seen, it decreases from a value of about 20 at the lowest concentration to a value of about 5 at the highest concentration studied (see also Table 1). Corresponding values for Lb are 6000 and 1500 Å, respectively, in the two limits, using eq 18 and λ ) 150 Å. From mean field theory, one expects a weak concentration dependence in the curvilinear distance between connections, for example, Lb ∝ Φ-1/2 for noninteracting wormlike micelles with 3-fold junctions,9,37b and the (approximate) theories are in rather good agreement with our data. Indeed, the hatched line in Figure 8 represents a power law fit to the data (open symbols), N ∝ ΦR, yielding R ) -0.55 ( 0.07. Values of N can also be estimated, in the case of the less concentrated samples, from the observed crossover between curvilinear and Gaussian diffusion when varying the observation time scale. For the lowest concentration, Φ ) 0.036, we observe the crossover somewhere between 0.25 and 0.5 s (see Figure 7). Identifying the crossover time with τb, we obtain with eq 15 Lb between 0.7 and 1 µm. With λ ) 150 Å, the corresponding N values are 23 and 33 (closed symbols in Figure 8) that closely follow the power law dependence. Furthermore, we expect for our shortest observation time, 50 ms, to observe curvilinear diffusion only for N > 10. Indeed, experimental signs of curvilinear diffusion were observed only for Φ < 0.11 (Figure 5). From this, we conclude that the analysis of the long time self-diffusion coefficient is consistent with the observed crossover from curvilinear to a Gaussian diffusion. A branched network structure for the micellar system in isooctane has also been suggested from dynamic rheology18 and dielectric spectroscopy19 measurements. Water Diffusion. In the present (reverse) micellar system, the water dynamics is very different from the surfactant dynamical behavior. Unlike lecithin molecules, water can escape easily from the micelles; the points where the water exchange takes place act as virtual branches (for further explanations see ref 6). Let Dc,w be the local (36) Wassel, S. R. Biophys. J. 1996, 71, 2724. (37) The case of microemulsions is discussed in: (a) Tlusty, T.; Safran, S. A.; Menes, R.; Strey, R. Phys. Rev. Lett. 1997, 78, 13. (b) Tlusty, T.; Safran, S. A.; Strey, R. Phys. Rev. Lett. 2000, 84, 1244. (c) Tlusty, T.; Safran, S. A. J. Phys.: Condens. Matter 2000, 12, A253.

Ambrosone et al.

(curvilinear) self-diffusion coefficient of water inside a micelle and τres be its residence time, as in the case of surfactant monomers discussed above. Thus, water molecules experience a virtual network with a choice of paths distributed every Lw along the micelles where

Lw ) (2Dc,wτres)1/2 ) Nw2λ ) Dc,w2λ/(3Dw,aq) (21) Here, Nw is the number of Kuhn lengths explored by a water molecule before escaping from the interfacial layer, and Dw,aq was defined in section 4. Using as Dc,w the values, dependent on W0, measured in the lecithin/water LR phase36 and the Dw,aq values evaluated in section 4, we have Nw ) 0.5 and 1 for W0 ) 3 and 2, respectively. The corresponding τres values are 1.3 and 6 µs, confirming a residence time of the order of microseconds previously found in the lecithin-water-cyclohexane system. Relation between Structure and Phase Behavior. Several micellar systems exist for which branching rather than end-cap formation is the favored structure.33,37,38 The analysis of the self-diffusion data above outlined employs few assumptions on which there is a large agreement, viz., for living polymers31 L is an increasing function of Φ and for living networks8 Lb decreases with Φ (specific power laws are not assumed). This is strong evidence supporting the existence of living branched networks in the lecithin-water-isooctane system but not a true proof since we compare the predictions for different models and these cannot describe the physical system in all the details. Therefore, it is useful (and encouraging) to list evidence coming from other studies (with other techniques) that fully supports the description of the L2 phase in the system lecithin-water-isooctane as a multiconnected network. As stated above, dielectric spectroscopy19 and Kerr effect39 studies on the same system are fully consistent with our interpretation. Rheological investigations40 on the system lecithin-water-decane give the same conclusions (this last, together with the isooctane-based system, shares the same rheological behavior and type of phase separation). Micellar branching has significant influence on the rheology:9 as entanglements are replaced by branched points we approach a network structure that resembles the structures of bicontinuous microemulsions and sponge phases, that are Newtonian, low-viscosity fluids.41 Finally, the phase behavior of the lecithin-water-isooctane system42 closely follows the predictions for a threecomponent system of living networks,37 that is, the phase sequence viscoelastic L2 phase f finite swelling of the network and phase separation with excess solvent f spherical reverse micelles in equilibrium with excess water, with increasing the water content (emulsification failure). Above, we have presented an approach to experimentally determine the branching density in a micellar network. The next approach will be to incorporate this information in the interpretation of rheology data. Such work is in progress. Acknowledgment. This work was supported by the Swedish Research Council and MURST (Italy) PRIN 1998 Sistemi a Grande Interfase. LA0105275 (38) Kato, T.; Tagushi, N.; Terao, T.; Seimiya, T. Langmuir 1995, 11, 4661. (39) (a) Cirkel, P. A.; Stam, D. D. P.; Koper, G. J. M. Colloids Surf., A 1998, 140, 151. (b) Cirkel, P. A.; Koper, G. J. M. Langmuir 1998, 14, 7095. (40) Shchipunov, Y. A.; Hoffmann, H. Langmuir 1998, 14, 6350. (41) Snabre, P.; Porte, G. Europhys. Lett. 1990, 13, 641. (42) Angelico, R.; Ambrosone, L.; Ceglie, A.; Olsson, U.; Palazzo, G. Manuscript in preparation.