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Surfactant Self-Diffusion in Wormlike Micelles Ve´ronique Schmitt*,† and Franc¸ ois Lequeux‡ LESOC, URA 406 CNRS Universite´ Henri Poincare´ Nancy I, Faculte´ des Sciences, BP 239, 54506 Vandoeuvre les Nancy, France, and LDFC URA 851 CNRS Universite´ Louis Pasteur Strasbourg I, Institut de physique 3, rue de l’Universite´ , 67084 Strasbourg Cedex, France Received June 20, 1997. In Final Form: October 24, 1997X Many self-diffusion measurements have been performed on giant micelles by either fluorescence recovery after photobleaching (FRAP) or Fourier transform pulsed-gradient spin echo (FT PGSE) and reported by many authors. As a general feature, one observes a minimum of the diffusion coefficient as a function of surfactant concentration, which is not clearly understood and has often been attributed to the existence of connections. In this paper, we explain that this minimum may result from the competition between two mechanisms: the diffusion of the micelle itself and the surfactant molecule diffusion on the micelle.
1. Introduction Some surfactant molecules are known to assemble in order to form long cylindrical micelles also called wormlike or giant micelles. For a complete and recent review of these systems the reader can refer to ref 1. Their static properties are very similar to those of polymer chains with some noticeable differences: in contrast with classical polymers the molecular weight distribution is not quenched by the synthesis but results from the thermodynamical equilibrium through reversible scissions. Because of these reversible scissions, their dynamical properties are modified compared to classical polymer dynamics,2,3 as described by Cates,4,5 and the micelles are therefore called living polymers. In this paper we focus on the diffusion process of the surfactant molecules themselves as it reveals how they are confined in the threadlike aggregates. We try to explain the self-diffusion coefficient D behavior generally observed: D surprisingly goes through a minimum as a function of the surfactant concentration Φ. The aim of this paper is not to give the exact power laws of D but to qualitatively describe its behaviour with Φ. The selfdiffusion coefficient of surfactant has been measured by different groups6-15on various systems: nonionic surfactants,6,10-12 ionic surfactants,7,9,13-15 or lecithin in organic solvent.8 The self-diffusion coefficient can be measured via two techniques: (i) By fluorescence recovery after photo* To whom correspondence should be addressed. E-mail:
[email protected]. † Universite ´ Henri Poincare´ Nancy I. ‡ Universite ´ Louis Pasteur Strasbourg I. X Abstract published in Advance ACS Abstracts, December 15, 1997. (1) Lequeux, F. Curr. Opin. Colloid Interface Sci. 1996, 1, 341 and references therein. (2) De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (3) Doı¨, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford Science Publication: Oxford, 1986. (4) Cates, M. E. Macromolecules 1987, 20, 2289. (5) Cates, M. E. J. Phys. (Paris) 1988, 49, 1593. (6) Nilsson, P. G.; Wennerstro¨m, H.; Lindman, B. J. Phys. Chem. 1983, 87, 1377. (7) Messager, R.; Ott, A.; Chatenay, D.; Urbach, W.; Langevin, D. Phys. Rev. Lett. 1988, 60, 1410. (8) Ott, A.; Urbach, W.; Langevin, D.; Schurtenberger, P.; Scartazzini, R.; Luisi, P. L. J. Phys. Condens. Matter. 1990, 2, 5907. (9) Khatory, A.; Kern, F.; Lequeux, F.; Appell, J.; Porte, G.; Morie, N.; Ott, A.; Urbach, W. Langmuir 1993, 9, 933.
bleaching (FRAP) as in refs 7-9, 13, and 14. In this technique some fluorescent molecules, called probes, with the same structure as the surfactant molecules are added in the sample. A strong laser beam illuminates the sample via a fringe pattern, the probes loose their fluorescence. As new probes diffuse in the studied volume, the fluorescence increases and is measured with a low-intensity laser beam. For a more detailed description of the experiment see for example ref 16. (ii) By Fourier transform pulsed-gradient spin echo (PGSE) as in refs 6, 10-12, and 15. In this technique one directly measures the self-diffusion coefficient of the surfactant molecule itself. The sample is located in a magnetic gradient field. A pulse sequence is applied to the sample, followed after a delay time ∆ by the reverse sequence. It produces a spin echo, and owing to diffusion, the echo amplitude is attenuated. From this attenuation, one can deduce the self-diffusion coefficient. For more details see for example ref 17. Let us point out that with both techniques the diffusion length scale is on the same order. With the FRAP technique, the diffusion coefficient is measured on a spatial length scale of the order of the fringe length, i.e., of few micrometers. This length is much larger than the typical radius of gyration of wormlike micelles. Similarly by PGSE technique, the diffusion occurs on a spatial length of the order of (D∆)1/2, which is also of the order of a few microns. Thus, by these two techniques, except for some very peculiar cases (as Levy flights18,19 ), one can assume that the self-diffusion coefficient is measured at infinite times. Below the critical micellar concentration (cmc) the diffusion coefficient is that of an isolated surfactant (10) Kato, T.; Terao, T.; Tsukada, M.; Seimiya, T. J. Phys. Chem. 1993, 97, 3910. (11) Kato, T.; Terao, T.; Seimiya, T. Langmuir 1994, 10, 4468. (12) Jonstro¨mer, M.; Jo¨nsson, B.; Lindman, B. J. Phys. Chem. 1991, 95, 3293. (13) Morie, N.; Urbach, W.; Langevin, D. Phys. Rev. E 1995, 51, 2150. (14) Narayanan, J.; Manohar, C.; Langevin, D.; Urbach, W. Langmuir 1997, 13, 398. (15) Nyde´n, M.; Schmitt V. Private communication. (16) Axelrod, D.; Koppel, D.; Schlessinger, J.; Elson, E.; Web, W. W. Biophys. J. 1976, 16, 1055. Lanni, F.; Ware, B. R. Rev. Sci. Instrum. 1982, 53, 905. Davoust, J.; Devaux, P. F.; Leger, L. Eur. Mol. Biol. Organ. J. 1982, 1, 1233. (17) Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1. (18) Ott, A.; Bouchaud, J. P.; Langevin, D.; Urbach, W. Phys. Rev. Lett. 1990, 65, 2201. (19) Bouchaud, J. P.; Ott, A.; Langevin, D.; Urbach, W. J. Phys. II 1991, 1, 1465.
S0743-7463(97)00654-9 CCC: $15.00 © 1998 American Chemical Society Published on Web 01/20/1998
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molecule in the solvent. Above the cmc, most of the surfactant molecules belong to micellar aggregates. In this paper we assume that the surfactant concentration is much larger than the cmc, so that the contribution of the isolated molecules in solution can be neglected in the measured diffusion coefficient (Dmeas ) PmonDmon + PmicDmic ≈ PmicDmic,6 where Pmon ) cmc/c and Pmic ) (c - cmc)/c are respectively the fractions of surfactant molecules in the monomer and micellar states for a surfactant concentration c, and where Dmon and Dmic are their respective diffusion coefficients). In general, as the cmc is about 10-4mol‚L-1, this assumption is correct. We will now discuss in the two coming sections, the two most important and relevant mechanisms that occur on our viewpoint: the micelle diffusion and the surfactant molecule diffusion on the micelle. Such an approach has already been introduced and qualitatively discussed in refs 10-12. A more quantitative analysis has been proposed in ref 13 but the mechanisms are so intricate that the authors did not conclude clearly on the nature of the minimum of the diffusion coefficient. Here we will present a dynamical phase diagram which clearly exhibits the various diffusion mechanisms. 2. Micellar Diffusion a. Theoretical Background. Let us first recall some useful properties of the wormlike micelles size distribution. In these systems, the length of the aggregates results from thermodynamical equilibrium and is widely distributed, with the probability p(L)
p(L) ∝ exp(-L/L h)
(1)
where L h is the average micellar length and is given by L h ∝ Φy0.20 This power law is valid for nonionic or screened ionic micelles. Electrostatic interactions largely modify both the micellar length distribution and the growth law as demonstrated theoretically by Safran et al.21 and MacIntosh et al.22 Indeed, electrostatics affects many properties as experimentally evidenced in ref 23. The micellar growth is particularly spectacular in a salt-free gemini anionic surfactant solution.24 Since all the following results are very sensitive to the growth exponent y0 which is actually not very well known experimentally, we will keep y0 as a parameter in the different following expressions. As indications we will make explicit the calculated exponents for the Flory approximation (ν ) 3/5 and y0 ) 1/2) and the scaling law exponents (ν ≈ 0.588 and y0 ≈ 0.6). All these results are summarized in Table 1. Let us now describe shortly the scission/recombination mechanism. The wormlike micelles are polymers that recombine and break continuously with time. Detailed mechanisms have been described in ref 4. Here we will just assume that the probability for a micelle to break on a given part of length λ is 1/(c1λ) (which simply means that the breaking is equiprobable along all the micelle, with a probability c1 per unit time and length). The lifetime of a micelle of average length is then given by (20) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. Tandford C. The hydrophobic effect; Wiley: New York, 1973. (21) Safran, S.; Pincus, P.; Cates, M. E.; Mackintosh, F. J. Phys. (Paris) 1990, 51, 503. (22) Mackintosh, F.; Safran, S.; Pincus, P. Europhys. Lett. 1990, 12, 697. (23) Candau, S. J.; Hebraud, P.; Schmitt, V.; Lequeux, F.; Kern, F.; Zana, R. Nuovo Cimento 1994, 16D, N.9, 1401. (24) Kern, F.; Lequeux, F.; Zana, R.; Candau, S. J. Langmuir 1994, 10, 1714.
Table 1. Recall and Sum-up of the Most Characteristic Functions and Their Power Laws with Surfactant Concentration Φ in the Flory and Scaling Law Theories Flory ν ) 3/5 y0 ) 1/2
scaling law ν ) 0.588 y0 ) 0.6
mean length L h ∝ Φy0 radius of gyration Rg ∝ Lν spatial displacement R2 ∝ Φy0+(2ν-1)/(1-νd) entanglement length le ∝ Φ1/(1-νd) mesh size ξ ∝ Φν/(1-νd) reptation time τrep ∝ Φ3y0+(νd-3)/(1-νd) reptation diffusion coefficient Drep ∝ Φ-2y0+(2+(2-d)ν)/(1-νd) lifetime τbreak ∝ Φ-y0 Cates parameter ζ ) τbreak/τrep ζ ∝ Φ-4y0+(3-νd)/(1-νd) new parameter m ) Dcτbreak/L2 m ∝ Φ-3y0 dilute regime Dmic ∝ Φ-νy0
Φ1/2
Φ0.6
Φ3/10
Φ0.35
Φ1/4
Φ0.37
Φ-5/4
Φ-1.31
Φ-3/4
Φ-0.77
Φ3
Φ3.42
Φ-11/4
Φ-3.05
Φ-1/2
Φ-0.6
Φ-7/2
Φ-4.02
Φ-3/2
Φ-1.8
Φ-3/10
Φ-0.35
semidilute regime: micellar reptation Drep semidilute regime: reptation + reversible scission Drepζ-1/3 surfactant without scission Drepζ-1 surfactant with scissions Drepζ-1/3m2/3 surfactant with connections DcΦ1/4 reptation of a branch Φ(3-ν(1+d))/(1-νd)
Φ-11/4
Φ-3.05
Φ-19/12
Φ-1.71
Φ3/4
Φ0.97
Φ-1/4
Φ-0.23
τbreak ∝
Φ1/4 Φ-3/4
Φ-0.85
1 ∝ Φ-y0 c1L h
b. Dilute Regime. In the dilute regime the diffusion coefficient of the micelle is given by Dmic ) kBT/6πηRG, where η is the solvent viscosity and RG is the gyration radius of the micelle, which is related to the micellar length through: RG ∝ Lν.2 Hence, the diffusion coefficient of a micelle of length L in the dilute regime behaves with surfactant concentration as
Dmic(L) ∝ L-ν The micellar diffusion coefficient is thus given by the average diffusion coefficient of the micelles
D h mic )
∫0∞L p(L) Dmic(L) dL ∝ dL ∝ Φ-νy ∫0∞L1-ν exp(-L L h )
0
(2)
This expression holds whether the probes belong successively or not to different aggregates. Hence, the diffusion behavior in the dilute regime does not depend on the scission-recombination kinetics, contrary to the semidilute case as we will see now. c. Semidilute Regime. In the semidilute regime, as shown by Cates4,5 the micellar motion is a combination between scission/recombination and reptation mechanisms. He has introduced the coefficient: ζ ) τbreak/τrep where τrep is the reptation time for a micelle of average length L h.
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Langmuir, Vol. 14, No. 2, 1998 285
We have to distinguish two cases depending on the relevant dynamics, i.e., on the relative values of τbreak and τrep. (i) Reptation. Let us first assume that the lifetime of the micelles is larger than the reptation time, that is to say ζ . 1. In this case the scission/recombination processes are not efficient and we can apply quite roughly the pure reptation mechanism to describe the micellar motions.2,3 Thus, the micellar coefficient of a given micelle of length L is equal to
Dmic(L) ) Drep(L) )
()
kBT le 6πηξ L
2
where ξ is the network mesh size, η the solvent viscosity, and le the curvilinear length between entanglements (L/le is actually the number of blobs of the chain). This expression is valid for entangled micelles, i.e., for micelles whose length is larger than le. It can also be written as 2
L h Drep(L) ) Drep(L h) 2 L
2
dL ∫l∞LhL exp(-L L h ) e
(3a)
where we have assumed a classical length distribution as in eq 1. We see that if le ) 0, the integral diverges logarithmically, and the diffusion coefficient is infinite. By taking into account the cutoff length le, one gets, up to a logarithmic prefactor, the diffusion law
h ) ∝ Φ-2y0+((2+(2-d)ν)/(1-dν)) D h mic ≈ Drep(L
D h mic ≈ Drepζ-1/3
(4a)
or as a function of the surfactant concentration
Let us now try to estimate the average diffusion coefficient
D h mic ) Drep(L h)
The reptation dynamics describes how a polymer chain gets out its initial tube resulting from the surrounding entangled chains. In the case of a wormlike micelle, the reptation of a given chain portion breaks down as soon as this portion has left its tube. This, as demonstrated by Cates, occurs owing to the contribution of both classical reptation and scission recombination mechanisms. When the two mechanisms are involved, a portion of chain diffuses over a length λ, before the chain breaks on this length and then the tube renews. Hence this distance λ is the distance over which the considered portion of chain diffuses before the scission occurs on the corresponding part of the tube. So λ2 ) Drep(L h ) τbreak(λ). This is well explained in the original paper of Cates.4 The micellar diffusion is Dmic ) R2(λ)/τbreak(λ) where R is the spatial length corresponding to the curvilinear length λ: R2(λ) ) h )λ/L h and τbreak(λ) ) τbreakL h /λ, (λ/le)ξ2. Since R2(λ) ) R2(L one deduces that the micellar diffusion in the presence of scission-recombination can be written as
(3b)
d being the space dimension. But, if the length distribution is such that the integral diverges strongly for le ) 0, then one may observe Levy flight.18 As explained and observed in refs 18 and 19, the diffusion coefficient may increase as a function of the measurement time if the probe explores successively micelles of different lengths. The exploration of a small micelle will give an enormous contribution to the spatial displacement. For increasing measurement times, each probe has an increasing probability to explore shorter and shorter chains. The diffusion of the shorter chain dominates all the other motions, and this leads to hyperdiffusion. This is the case only if the integral (3a) of the diffusion coefficient diverges for vanishing length of the aggregates. But, as there is a cutoff in expression 3a for L ) le, the hyperdiffusion acts only if the shorter chains explored by the probes are longer than le. In other words, the hyperdiffusion ceases when the probes begin to explore aggregates shorter than le. Actually, it can be observed only for specific length distributions, for measurement times of the order of several lifetimes, and for a small enough probe concentration (in order not to saturate the phenomenon). To sum up, in this regime, the diffusion coefficient is very sensitive to the length distribution, and Levy flights can be observed in some very peculiar conditions. (ii) Combination between Reptation and Scission Recombination Mechanisms. If, now, the lifetime τbreak is small such that ζ , 1, the reptation picture can no longer apply. For this case, Cates has introduced an original approach.4,5 Let us briefly recall these results.
D h mic ∝ Φ(-2y0/3)+(1+2ν(1-d/3))/(1-νd)
(4b)
Equation 4a shows how the micellar diffusion is accelerated (since ζ < 1) in the semidilute regime, by reversible scissions. Let us remark here that this rough description (there is no average over λ) gives the exact exponent as demonstrated in ref 25. For the same reasons, all the following descriptions will also be valid. The micellar diffusion is not the only one that will contribute to the measured self-diffusion coefficient of living polymers. The second important mechanism is the diffusion of the surfactant molecule itself inside the micellar aggregate, that we will now describe. 3. Diffusion of the Surfactant Molecules on the Micelle As surfactant molecules are very small objects, they can a priori diffuse extremely rapidly inside the aggregates. Let us now consider that the surfactant molecules diffuse much faster on the micelles than the micelles diffuse in space. In order to focus only on the surfactant diffusion, we will now completely neglect the micellar diffusion itself. This mechanism has also been proposed by Morie et al.13 Again two cases have to be considered, either the surfactant diffuses over the whole micelle before this last breaks or the micelle breaks before the surfactant molecule explores the whole micelle. We h 2, define a new dimensionless parameter m ≡ Dcτbreak/L where Dc is the curvilinear diffusion coefficient of the surfactant molecule. This parameter m gives the ratio between the time for the surfactant molecule to diffuse on a chain of mean length L h and the time before this chain breaks. (i) m > 1. The surfactant molecule diffuses over the whole micelle before it breaks. This means that the diffusion on the micelle is much faster than the scission process or, in other words, that the surfactant molecule is delocated on the micelle. The molecule has then to wait for a scission in order to move on long scales. The measured (spatial) diffusion coefficient is Dsurf ) R2/τbreak where τbreak ∝ 1/(c1L h ) and R is the spatial displacement of the surfactant molecule, corresponding to the curvilinear (25) Lequeux, F. J. Phys. II 1991, 1, 195.
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motion over the micelle of length L h . We have R2 ) (L h /le)ξ2 where L h /le is again the number of blobs in the mean chain and ξ is the blob size. This leads to
Dsurf ∝
L h ξ2 ∝ Drepζ-1 leτbreak
(5a)
or, as a function of the surfactant volume fraction
Dsurf ∝ Φ2y0+((2ν-1)/(1-νd))
(5b)
Let us notice that in this case the spatial surfactant diffusion coefficient depends neither on Dc, the curvilinear surfactant diffusion coefficient, nor on the reptation kinetics. It only depends on the micelle lifetime τbreak and on geometrical parameters (see eq 5a). (ii) If m < 1, breakage occurs faster than diffusion over the whole micelle and the surfactant molecule is located on a small part of the micelle. It only diffuses over a curvilinear length λ , L h , on the micelle before scission. This length λ is such that λ2 ) Dcτbreak(λ), i.e., λ3 ∝ Dc, where Dc is again the curvilinear diffusion coefficient of the surfactant molecule. The spatial diffusion coefficient is then Dsurf ) R2(λ)/τbreak(λ) where R2(λ) ∝ (λ/le)ξ2 is the mean displacement of the surfactant before the renewal of the explored part of the micelle and τbreak(λ) ∝ 1/(c1λ) is the survival time of it. Then
Dsurf ) Drepζ-1m2/3
(6a)
which gives the following power law for the surfactant diffusion:
Dsurf ∝ Dc2/3Φ(2ν-1)/(1-νd)
(6b)
Contrary to the previous case, the spatial surfactant diffusion coefficient here depends on the surfactant curvilinear diffusion Dc. 4. Combination of the Two Mechanisms a. Dilute Regime. In the dilute regime the scission process has no effect because even if breakage is very fast, the surfactant molecules are confined in the micelle and their diffusion is limited by the diffusion of the micelles. Thus in the dilute regime, whatever the value of τbreak, one always measures the micelle diffusion coefficient which is given by eq 2. This is not the case anymore in the semidilute regime. b. Semidilute Regime. In the two previous sections we have considered separately the micellar diffusion (eqs 3 and 4) and the surfactant diffusion (eqs 5 and 6) that actually occur simultaneously. The measured self-diffusion coefficient Dmeas is then the sum of the two diffusion coefficients Dmic and Dsurf. Nevertheless we can discuss which one of these two processes will dominate the diffusion, by comparing Dsurf and Dmic. This leads to the (m-ζ) diagram plotted in Figure 1, where only the dominating contribution has been indicated. As the surfactant concentration increases, one moves on a line parallel to the dashed line. Four different domains can be distinguished (noted I to IV). In the upper half plane of the diagram (domain I, ζ > 1), classical micellar reptation dominates the diffusion, and hence Dmeas is simply Drep (eq 3). In this case, because
Figure 1. Schematic diffusion diagram, presenting the dominant contributions to the measured self-diffusion coefficient. The contributions are given as a function of two important parameters: m (x-axis) and ζ (y-axis). ζ is the parameter defined by Cates ζ ) τbreak/τrep;4 depending on ζ, either reptation alone or reptation with scission recombinations is the relevant dynamics. The second important parameter is m ≡ Dcτbreak/L h 2; depending on m, either reversible scission or diffusion along the micelle is the fastest process. Note the four regimes: in regimes I and II the diffusion is dominated by micellar diffusion, while in regimes III and IV the diffusion is dominated by surfactant diffusion on the micelle. The two regimes differ by the fact that in regime III the diffusion coefficient decreases with increasing surfactant concentration and is Dc dependent, whereas it increases with Φ and is Dc independent in regime IV. Remark that by increasing Φ, one moves on a line parallel to the dashed line (whose slope is 7/3). The bold line has to be crossed in order to observe a minimum for Dmeas.
of both the reptation and large polydispersity, the viscoelastic spectrum is non-Maxwellian.26 For ζ < 1, there are three different regimes. In the regime noted II, the relevant diffusion mechanism stays the micellar diffusion, but scission-recombination processes modify the pure reptation (eq 4). In this regime, the rheological behavior becomes Maxwellian and is well characterized both theoretically27 and experimentally.28 In the two last regimes (III and IV) the diffusion is dominated by the surfactant diffusion along the micelle and modified by the reversible scissions. In regime III, the surfactant has no time to explore the micelle before scission-recombination (eq 6). At the opposite, in regime IV, the surfactant molecule explores the whole micelle before scission-recombination (eq 5). This last regime is very peculiar because it is the only one where the diffusion coefficient depends neither on the reptation dynamics nor on the surfactant motion along the chain (the measured diffusion coefficient is Dc independent). Moreover this is the only regime where the diffusion coefficient increases with surfactant concentration. It is then very probable that when an increase of Dmeas with surfactant concentration is observed at high surfactant volume fraction, this corresponds to case IV. The fact that the diffusion (26) The stress relaxation function µ(t), which is the average fraction of tube remaining after a time t has been calculated by Cates4 and scales as µ(t) ∝ exp(-t/τrep1/4). Hence the stress relaxation is not a pure exponential and thus leads to a non-Maxwellian behavior. (27) Cates showed4,5 that in the presence of scission recombination the stress relaxation becomes a single exponential: µ(t) ∝ exp(-t/τ) where τ ) (τbreakτrep)1/2, and hence one observes a Maxwellian behavior. Granek R. Langmuir 1994, 10, 1627. Granek, R.; Cates, M. E. J. Chem. Phys. 1992, 96, 4758. (28) Hoffmann, H.; Rehage, H.; Wunderlich, I. Prog. Colloid Polym. Sci. 1986, 72, 51. Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712 (regions noted III, IV, V). Kern, F.; Zana, R.; Candau, S. J. Langmuir 1991, 7, 1344. Berret, J. F.; Appell, J.; Porte, G. Langmuir 1993, 9, 1456.
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coefficient does not depend on the nature of the probe is consistent with this picture. Hence, the most probable explanation of the experimental results on diffusion coefficient is that ones measures successively the micellar reptation in domain I, and surfactant diffusion but limited only by scission/recombination in domain IV. One mechanism often invoked for the minimum of the diffusion coefficient is the presence of connections between the micelles. We will see now why this picture would lead to a very smooth minimum for Dmeas. 5. Role of Connections At higher salt concentrations, one expect intermicellar 3-fold connections to appear. These connections, which are negatively curved, originate in the screening of the electrostatic repulsion between polar heads by addition of salt.29 They are quite difficult to evidence experimentally, but recently they have been clearly observed by cryoTEM.30 Again, in the semidilute regime with numerous connections, two mechanisms compete: the surfactant diffusion over the large network and the reptation of a branch between two connections. i. Surfactant Diffusion. Let L3 be the mean distance between connections, it behaves as L3 ∝ Φ-1/2 in the mean field theory.31 The curvilinear diffusion time between two h 32/Dc and the spatial diffusion connections is given by τ3 ) L h 3/le)ξ2 becomes coefficient Dsurf ) R2/τ3 where R2 ) (L
Dsurf ∝ DcΦ1/4
(7)
This regime leads then to an increase of the diffusion coefficient as a function of surfactant concentration, but the exponent is quite small. Moreover, the diffusion coefficient must depend on the probe used. ii. Reptation of Branches. A second mechanism can occur, the reptation of the branches themselves between the connections. This mechanism has already been described earlier.31 Let us only recall the important result. The diffusion of this branch is equivalent to the diffusion of a single chain of the same length L3. Hence
Dbranch ) Drep(L3) ∝
R2(L3) τrep(L3)
∝ Φ(3-ν(1+d))/(1-νd) ∝ Φ-3/4 (8)
which is a decreasing function of the concentration! (29) Porte, G.; Gomati, R.; El Haitamy, O.; Appell, J.; Marignan, J. J. Phys. Chem. 1996, 90, 5746. Khatory, A.; Lequeux, F.; Kern, F.; Candau, S. J. Langmuir 1994, 10, 1714. (30) Danino, D.; Talmon, Y.; Levy, H.; Beinert, G.; Zana, R. Science 1995, 269, 1420. (31) Lequeux, F. Europhys. Lett. 1992, 19, 675.
Equations 7 and 8 show that the existence of connections leads, at best, to a small increase of the diffusion coefficient with surfactant concentration. Moreover one sees from eq 7 that in the best case (toward the surfactant power law), the apparent diffusion depends on the probe (since Dsurf is Dc dependent). Micellar diffusion always leads either to a decreasing function of the surfactant concentration (wormlike aggregates) or to a very smooth increase of the diffusion coefficient in the presence of connections which accelerate the diffusion. This shows that the connections are not responsible for the sharp minimum commonly observed. 6. Conclusion Experimentally, the diffusion coefficient of the surfactant or a fluorescent hydrophobic molecule exhibits a minimum as a function of surfactant concentration. Two hypotheses have been proposed to explain these observations: monomer exchange or existence of connections. Indeed molecular exchange is in contradiction with experimental results since the diffusion coefficient would depend on the probe molecule. Moreover, existence of connections leads to a less marked minimum (since the increase is smoother after the minimum) and the diffusion coefficient would be in that case, probe-dependent. It seems that only one possibility for the various mechanisms could account for experimental results. On both sides of the minimum, two different diffusion mechanisms are actually investigated corresponding to regimes I and IV. Below the minimum, the measured diffusion is dominated by micellar diffusion, either in the dilute regime or in the semidilute regime via pure reptation (with possible Levy flight). In this last case the rheological behavior should be non-Maxwellian. Above the minimum, the measured diffusion coefficient is dominated by surfactant diffusion on the micelle. In this regime, although the diffusion process is the surfactant diffusion, the measured diffusion coefficient should not depend on the probe diffusion along the micelle. Actually, one observes a quite wide range of exponents before (and after) the minimum of the diffusion coefficient. These variations are probably due to the extreme sensitivity of the diffusion coefficient with respect to length distribution, which depends itself for instance on electrostatic local interactions. We are currently studying the diffusion coefficient by PGSE of wormlike micelles in collaboration with Magnus Nyde´n. Acknowledgment. This work has been initiated in the department of Physical Chemistry 1 in Lund and V.S. would like to thank Professor Bjo¨rn Lindman for his hospitality and Magnus Nyde´n for fruitful discussions. LA970654U
