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Temperature-jump study of elongated micelles ofcetyltrimethylammonium bromide

S.J. Candau, F. Merikhi, G. Waton, P. Lemarechal

To cite this version:S.J. Candau, F. Merikhi, G. Waton, P. Lemarechal. Temperature-jump study of elongated mi-celles of cetyltrimethylammonium bromide. Journal de Physique, 1990, 51 (10), pp.977-989.�10.1051/jphys:019900051010097700�. �jpa-00212426�

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Temperature-jump study of elongated micelles ofcetyltrimethylammonium bromide

S. J. Candau, F. Merikhi, G. Waton and P. Lemarechal

Laboratoire de Spectrométrie et d’Imagerie Ultrasonores (*), Universite Louis Pasteur, 4 rueBlaise Pascal, 67070 Strasbourg Cedex, France

(Reçu le 20 novembre 1989, accepté le 22 janvier 1990)

Résumé. 2014 Une étude de la cinétique de solutions aqueuses de micelles allongées de bromure decetyltrimethylammonium en présence de bromure de potassium a été réalisée au moyen d’unetechnique de saut de température utilisant la lumière diffusée comme moyen de detection. Letemps de relaxation a été mesuré en fonction de la concentration en tensio-actif et en sel et de latempérature. Les résultats obtenus, combinés à des mesures rhéologiques fournissent desinformations sur le mécanisme de relaxation des contraintes dans ces systèmes.

Abstract. 2014 A study of the kinetics of elongated micelles of cetyltrimethylammonium bromide inaqueous solutions containing potassium bromide has been performed by means of a T-Jumptechnique, using light scattering to probe the relaxation. The relaxation time has been measuredas a function of the surfactant concentration, salt concentration and temperature. The T-Jumpresults combined with rheological measurements provide information on the stress-relaxationmechanism of these systems.

J. Phys. France 51 (1990) 977-989 15 MAI 1990,

Classification

Physics Abstracts82.35 - 82.70 - 61.10D

Introduction.

The temperature jump (T-Jump) technique has been widely used for the study of the micellarkinetics [1, 2]. A rapid and small change of temperature is generated in the solution. As aresult, the micellar system shifts to a new state of equilibrium determined by the final value ofthe temperature. The evolution with time of the system is usually monitored by a change in anoptical property such as absorption, scattering or fluorescence emission.Most of the studies performed up to now have dealt with highly dilute solutions, in the

vicinity of the critical micellar concentration (CMC). The first successful attempt to describetheoretically the kinetics of micelle formation was published by Aniansson and Wall [3]. Itwas based on the assumption that the micelle formation-breakdown takes place through aseries of stepwise reactions, the micelles growing by incorporation of monomer only. Thismodel which was restricted to nonionic surfactants was subsequently extended by Kahlweitet al. [2, 4-5] to the case of ionic surfactants. However, the experimental results for ionic

(*) Unité de Recherche Associée au CNRS n° 851.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019900051010097700

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surfactants could not be fitted by the theory except for a restricted range of concentrationabove the CMC. More specifically, in the high concentration range, the relaxation time wasfound to decrease as the surfactant concentration was increased, in contradiction with thetheoretical predictions [1-2]. To interpret these results, Kahlweit et al. suggested that forsystems where the intermicellar interactions are not too repulsive the stepwise reaction path isbypassed by a reversible coagulation process of submicellar aggregates [4, 5].

This process is likely to be the dominant one for the systems in which very long, flexible,wormlike micelles exist [6-17]. Recently, Turner and Cates [18] have studied theoretically therelaxation of a system of polymers or wormlike micelles that can break and recombinereversibly after a small perturbation. An important prediction of this model concerns the caseof an initial perturbation corresponding to a shift in the mean chain length, as caused by a T-Jump. For this case, the entire perturbation is expected to decay exponentially with thecharacteristic time equal to twice the breaking time T b (the mean-waiting time for a chain ofthe average length to undergo a scission somewhere along its length). To monitor therelaxation, one needs to measure a physical quantity sensitive to a change in the mean lengthof the wormlike micelle. An obvious candidate is the light scattering which probes the weightaverage molecular weight in the dilute regime.

In this paper we present results obtained by T-Jump in aqueous solutions of cetyl-trimethylammonium bromide (CTAB) in the presence of KBr. The results are compared withthe theoretical predictions of the model of Turner and Cates. Furthermore, the T-Jump dataare used in conjunction with the results of previous rheological experiments to provideinformation on the mechanism of stress-relaxation and give an estimate of the scission energyof wormlike micelles.

Theory.

We recall below the main results of the theoretical models derived by Cates for the stressrelaxation [16, 17] and the relaxation spectrum of micellar length distributions [18].

Static equilibrium properties.

Mean-field models [16-20] predict that Co(L), the number density of wormlike micelles of

length L is exponential with some mean L (L being expressed in monomer units)

with

where 0 is the surfactant volume fraction and Escis is the scission energy of the micelle thatrepresents the excess free energy for a pair of spherical end-caps relative to the cylindricalinterior regions [20, 21]. The above relationships have been derived for non ionic micelles orionic micelles at large ionic strength.

Indeed, dynamic measurements in semi-dilute CTAB solutions at high ionic strength(0.25 M KBr) gave results supporting the predicted 03A61/2 micellar growth [12, 13]. However,at low salt (0.1 M KBr) the strong dependence of the viscosity and self-diffusion constant on03A6 could not be fitted by equations (2) [12, 13]. To explain these anomalies, a modelcalculation for semi-dilute elongated micelles solutions with no added salt was recentlyproposed [22]. This model suggests that Coulomb interactions result in an additional

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contribution to the free energy of an end-cap that depends logarithmically on 03A6. This modifies

the growth-law for L which now varies approximately, if we consider a relatively narrowrange of 0 (about one decade), as 03A6 (1/2)(1 + A) where A> 0 depends on the renormalizedCoulomb charge of an end-cap.The physical origin of the increased growth exponent is that the electrostatic free energy

contributions favor the spherical end-caps over the cylindrical regions. This same effect leadsto smaller micelle for a fixed 0. This was discussed by Porte [23] and more recently by Odijk[24].

Relaxation spectrum of micellar length distributions.

Turner and Cates [18] have considered two types of chemical kinetics.

i) The reversible unimolecular scission, characterized by a temperature dependent rateconstant k per unit time per unit arc length, which is the same for all wormlike micelles and isindependent of time and of the volume fraction. Such assumptions are strictly valid in theentangled regime when reaction rates are determined by the local motion of subsections ofchain and not the diffusion of polymers over distances large compared to their gyration radii.

ii) The end-interchange with conservation of micelle number.Within the framework of a model for reaction kinetics taking these two mechanisms into

account, Turner and Cates found that the end-interchange reactions play no part in therelaxation after T-Jump whereas reversible scission leads to a single exponential decay withthe characteristic time

Equations (2) and (3) show that ’r T - J should decrease upon increasing the volume fraction as03A6 -1 /2 in the limit of high ionic strength

The effect of added salt is more difficult to analyze. An increase of salt leads to an increaseof Land therefore to a decrease of ’r T - j but k is likely to be also modified.

Stress relaxation.

In the semi-dilute range, i.e., at surfactant concentration large enough so that the elongatedmicelles overlap, the systems exhibit a viscoelastic behavior very reminiscent of that oftransient polymeric networks [8-12, 14-17]. In the latter systems, the viscoelastic propertiesare described by a model based on the reptation theory [25]. However, the « living » characterof the micelles provides additional pathways for disentanglement. Several regimes of behaviorare predicted depending on the relative values of two characteristic times : (a) T r, the

reptation time of a polymeric micelle with a length equal to the average micellar lengthL and (b) T b the breaking time. When T b is long compared to T r, the theory of reptation of

polydisperse polymers should apply, leading to a strongly non exponential form of the stressrelaxation function.

In the opposite case where Tb T r, the micelle breaking plays an important role in theviscous flow process as stressed first by Hoffmann et al. [8-10]. The Cates model predicts analmost pure exponential form of the stress relaxation function with the terminal time [ 16-17] :

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A single-exponential stress decay has been observed in solutions of several ionic surfactants[10-12, 14, 15] including the CTAB solutions studied here [10-12].An interesting consequence of the relation (5) concerns the temperature effect. All three

relaxation times are expected to follow arrhenian laws of the form exp (E/kB T), which leadsto the following relationship between the activations energies ER, Eb and Er correspondingrespectively to the temperature dependences of TR, Tb and T r

According to the reptation theory [25] Tr ~ L3. This result combined to equation (2) yields :

The above relation shows that independent measurements of ER by stress relaxation and ofEb by T-Jump (for the case of reversible scission) provide a determination of Esis.

Materials and methods.

The sample of CTAB was the same as in previous investigations [10-12]. The solutions werecarefully degassed in order to avoid resonances of bubbles that would generate an importantstray modulation of the light.The schematic diagram of the set-up is given in figure 1. Basically the set-up is similar to

those described by Hoffmann et al. [26] and Strey and Pakusch [27]. The T-Jump is producedby Joule heating by means of the discharge of a capacitor. The rise time of the T-Jump is- 1 ps and the amplitude of the T-Jump calculated from the values of the stored electricalenergy and the heat capacity of the sample can be varied between 0.1 and 2 ° C.The scattering cell is illuminated by an intense light beam obtained from a powerful

(100 W) mercury source (OSRAM HBO) in conjunction with a large aperture condenser(ORIEL Aspherab). To avoid heating the solution, a shutter which is controlled electricallyby computer, only permits the passage of light during the measurement. The cell is

Fig. 1. - Schematic diagram of the set-up.

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thermostated at ± 0.1 ° C. The calibration of the temperature in the cell as a function of thetemperature of the thermostat was realized using a thermistor. Interference filters allow toselect the following wavelengths : 313, 365, 405, 436, 546, 577 nm. A fraction of the incidentlight beam is sent into a photodiode, which thus delivers a reference signal. An electronicdevice under control of the computer uses this signal to compensate the instabilities arisingfrom the lamp power supply and the arc position. This device and the use of intense lightbeam allow one to improve the signal/noise ratio by a factor of the order of 100.The scattered signal is digitized and logged in the computer. For each measurement, about

ten relaxation functions are added up in order to get an averaged curve which results in afurther improvement of the signal/noise ratio.

Figure 2 gives’ typical experimental curves.

Fig. 2. - Typical relaxation curves for a CTAB solution at concentration C = 10- 2 M in presence of0.25 M KBr. The full lines are the best single exponential fits to the experimental curves. The relativevariation of the scattered intensity is of the order of 1 %.

Results and discussion.

SHAPE OF THE T-JUMP RELAXATION FUNCTION. - Figure 2 shows that single exponentialcurves fit the experimental recordings of the T-Jump relaxation functions. In fact, for mostmeasurements the best fits and the experimental curves are quite undistinguishable. This is inagreement with the prediction of equation (3) and allows one to determine Tb.

EFFECT OF SURFACTANT CONCENTRATION. - Figure 3 shows the variations of 7-b as a

function of CTAB concentration for 0.25 M KBr solutions at temperatures T = 30 ° C andT = 35 ° C. It is seen that Tb decreases steadily as the CTAB concentration is increased.However the results are not well fitted by the l/J - 1/2 dependence resulting from equations (2)and (3), but rather describe a sigmoïdal curve. This deviation can be understood from thefollowing considerations.

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Fig. 3. - Variations of Tb and TR as a function of surfactant concentration in 0.25 M KBr solutions.The dotted lines have the theoretical slope - 1/2.

i) The change of mean length L resulting from a T-Jump can be detected through thescattered intensity I, only in the dilute regime. In this range and if one neglects the

intermicellar interactions, Is is given by :

where II is the osmotic pressure, C the surfactant concentration, and P (K) the form factorthat varies only slightly during the T-Jump. However, in the highly dilute range the rateconstant k for reversible scission is likely to become concentration dependent as the micelleshave to diffuse through the medium in order to recombine. In this case, equation (4) would nolonger be valid. Moreover the contribution from the step-wise reaction path discussed in theintroduction becomes non-negligible. This is presumably at the origin of the peak observed inthe variation of the amplitude of the relaxation curve as a function of surfactant concentration(cf. Fig. 4).

ii) On the opposite, in the high concentration range corresponding to semi-dilute solutions,the micelles are entangled and the scattered intensity does not probe any longer the length ofthe micelle. In this regime I, is given by [ 11 ]

where e is the correlation length.As C increases, I, decreases. The cross-over between the regimes described by

equations (8) and (9) respectively is quite broad because of the polydispersity of the micelles.At a scattering angle of 90° the maximum of 7g (C) occurs at C = 6 x 10- 2 M for

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Fig. 4. - Amplitudes (per K) of the relaxation curves as a function of surfactant concentration in0.25 M KBr solutions. The lines are guides for the eye.

T = 30 ° C. Beyond this concentration, the change of intensity is related to that of e and notthat of L which might introduce corrections to the law given by equation (4). Atconcentrations higher than those investigated here, 1 becomes nearly temperature indepen-dent. This explains the large decrease observed in figure (4) for the amplitude of therelaxation. This results in a quite poor accuracy for the data taken in the highest concentrationrange.An important conclusion to be drawn from the results of figure 3 is that the 7b(C) curves

extrapolate in the high concentration range to values smaller than those of TR, the terminaltime determined from stress-relaxation experiments [12]. Such crossing of the curves of7-b(C) and TR(C) has also been observed by Hoffmann et al. [26].This is in fact the very condition required for obtaining a single exponential stress-

relaxation function, which is actually the experimental observation [12]. However it should benoted that in the low concentration side of the TR(C) curve the ratio Tb/TR is close to unitywithin experimental accuracy. Therefore, according to the theoretical calculation, the stressrelaxation in this concentration range should show deviations from single exponentialbehavior in the short time range [16, 17]. Such deviations have not been observed in theexperiments performed using the magneto-rheometer. On the other hand, they have beenobserved in experiments performed on other surfactant systems by means of dynamicviscoelastic measurements [28, 15].The same result is found for aqueous solutions in presence of 0.1 M KBr as shown in

figure 5. However, in that case, the ratio Tb/TR is much larger with an excess of salt. Oneobserves also in figure 5 a maximum in the curve of T b ( C ) . This maximum can be associatedwith the occurence at high dilution of the stepwise mechanism for the chemical kinetics. Italso appears in the amplitude of the relaxation curves (cf. Fig. 6).

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Fig. 5. - Variations Of’rb and TR as a function of surfactant concentration in 0.1 1 M KBr solutions. Thelines are guides for the eye.

Fig. 6. - Amplitude (per K) of the relaxation curves as a function of surfactant concentration in0.1 M KBr solutions. The lines are guides for the eye.

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EFFECT OF ADDED SALT. - According to the recent model of Safran, Cates and Pincus [22],the micellar growth is enhanced if Coulombic interactions are present (cf. theoretical section).As a result, one expects a larger variation of Tb with C when the amount of salt is decreased.This is what is observed in figure 5. However, it is difficult to draw quantitative conclusions,considering the different factors liable to affect the determination of Tb, as mentioned above.The effect of added salt at fixed volume fraction is illustrated in figures 7 and 8. The values ofTh become smaller at high salt concentration. This behavior corresponds to a transition fromthe dilute regime at low salt to the semi-dilute one at high salt, the micelles growing underaddition of salt. As expected, the larger the CTAB concentration, the smaller the salt contentnecessary to induce the transition. The decrease of T b upon increasing the concentration inKBr is accompanied by a large decrease of the amplitude of relaxation as shown in figure 8.

Fig. 7. - Variations of Tb as a function of KBr concentration. The lines are guides for the eye.

EFFECT OF TEMPERATURE. - The variations of log (rb) as a function of 11T are given infigures 9-11 for three KBr concentrations (0.1 M, 0.175 M and 0.25) and different CTABconcentrations. The data are fitted by straight lines, indicative of an Arrhenian behavior. Fora given salt concentration, the straight lines run parallel to each other within the experimentalaccuracy. From the slopes of these straight lines, one obtains the following values of theactivation energy

Increasing the salt content leads to larger micelles and therefore to a larger scission energy.This in turn would tend to increase Eb (cf. Eqs. (2) and (3)). Therefore the observed decreaseof Eb when [KBr] is increased must be associated with the effect of added salt on k. The

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Fig. 8. - Amplitude (per K) of the relaxation curves as a function of KBr concentration. The lines areguides for the eye.

Fig. 9. - Variations of Tb as a function of 103/ T for CTAB solutions in 0.1 M KBr solutions.

activation energies relative to the terminal time were previously measured for [KBr] ] =0.1 M and [KBr] ] = 0.25 M [12]

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Fig. 10. Variations of Tb as a function of 103/ T for CTAB solutions in 0.175 M KBr solutions.

Fig. 11. - Variations of Tb as a function of 103/ T for CTAB solutions in 0.25 M KBr solutions.

Combining the values of ER and Eb in equation (7) yields the following values of the scissionenergy

The above values of the scission energy correspond to an energy of the order of

kB T per surfactant molecule which appears reasonable.

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Because of the micellar growth induced by an increase of the added salt, one would expect asmaller value of Escis for the sample with 0.1 M KBr. The experimental accuracy doesn’t allowto draw such a conclusion.

Conclusion.

The results presented in this paper provide an experimental support to the theoretical modelof Turner and Cates [18]. More specifically, we have verified that the perturbation of thedistribution of lengths of wormlike micelles decays exponentially. The correspondingrelaxation time, associated with the reversible scission kinetics, is a decreasing function of thesurfactant concentration and becomes smaller than the terminal time of the stress relaxation

function in the entangled regime. This is consistent with the observation of a singleexponential decay of the stress relaxation. Combined T-Jump and stress relaxation

experiments allowed us to determine the scission energy of the wormlike-micelles.

Acknowledgments.

The authors acknowledge the precious advices from M. Cates. They are also very grateful toP. Pincus, S. Safran and R. Zana for very useful discussions.This work was funded in part under EEC Grant Number SC 1 * 0288-C(EDB).

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