4712
J . Phys. Chem. 1988, 92. 4712-4719
particle size4bindicates that the trapped electron senses the rest of the particle, i.e., it is not completely localized, its wavefunction extending to a certain degree over the whole particle. The fact that the absorption of eaq-and bleaching occur within less than 100 ps shows that the reaction between the excited states must be very fast. As a consequence, one has to conclude that such states cannot be accumulated to a large extent on a small colloidal particle. At very low light intensities, where klcl >> Uc0, c1 and c2 are much smaller than c., Ordinary stationary-state treatment of the reactions of eq 2 leads to the following rate of electron emission: (3)
where C is the overall concentration of colloidal particles. This explains the shape of the rate versus dose curves at low doses (Figure 6 ) . At medium light intensities, where co is still greater than c1 and c2, one obtains (4) Le., the rate of e,- formation is proportional to a power in intensity between 1 and 2. Finally, at high intensities where co > 1 G’approaches a constant limiting plateau value. Under such experimental conditions the solutions behave as an elastic body. At low frequencies u7
I
l 1 / 1 1 1 1
1bZ
I
I IyIT
I
I l'"lI,bo
1
10 '
creases again (Figure 8). Beyond that maximum, the dynamic properties of the surfactant solutions are strongly altered. Typical properties of these solutions are summarized in Figure 10. The dynamic properties of the surfactant solutions at high excess salt concentrations can be represented by the theoretical equations of the Maxwell model. The drawn symbols represent experimental values, and the lines are calculated from eq 5 and 6. It is noteworthy to mention that for salt concentrations above the first maximum, the flow behavior of the viscoelastic surfactant solutions can be explained by such a simple mechanical model. According to our knowledge, such behavior was observed only once before in another surfactant system and explained by the formation of a three-dimensional network built up from rodlike micelle^.'^ The theoretical equations of the Maxwell material hold in all concentration regions beyond the first maximum. At high shear rates, however, severe deviations occur that have already been described in Figure 4. We can concfude from our measurements that the stress relaxation process depends upon the average orientation of the micellar particles. Only in the limit of small deformations or shear rates do we get a single stress relaxation time constant. When the shear viscosity is investigated as a function of the surfactant concentration, we observe characteristic curves which are summarized in Figure 11. In these experiments, the NaSal concentration is kept constant. We observe a single maximum, which is shifted to higher surfactant concentrations with increasing electrolyte concentration. The experimental data indicate that there exists a well-defined value of the ionic strength where maximum viscosities occur. The exact relationship between surfactant and salt concentration is given in Figure 12. In the log-log plot we observe a single straight line, indicating that there exists a simple relation between these two concentrations. From the fit of the data we obtain in the concentration range between 1 and 100 mmol log (cNaSal/mmol)= 0.23 + 0.8 log (ccpycl/mmol) (14)
I
lO*~,,P
10.'
I l l l l l l
Figure 10. Storage modulus G'and the loss modulus G"as a function of the angular frequency w for a solution of 15 mmol CPyCl and 12.5 mmol Nasal at T = 20 "C.
' I
r
w /s-l
lo-* 10' 1oo 10' Figure 9. Storage modulus G'and the loss modulus G"as a function of the angular frequency w for a solution of 15 mmol CPyCl and 11 mmol Nasal at T = 20 O C .
When the electrolyte concentration is further increased, the viscosity decreases again and attains rather low values at high salt concentrations. In region I1 we observe a striking analogy to entangled polymer solutions. A typical example is illustrated in Figure 9, where the dynamic properties of these solutions are expressed in the form of the storage modulus and the loss modulus. The storage modulus and the loss modulus are strongly increasing as a function of the angular frequency. A quantitative comparison of the experimental results and eq 5 and 6 shows that the surfactant solution cannot be described by-the simple Maxwell equations. Instead of one well-defined stress relaxation time we observe a continuous spectrum of relaxation processes. This is similar to polymer mol&ules, where many configurational motions of the flexible chains contribute to the decay of the shear stress. When the salt concentration of the surfactant solution is increased, the viscosity passes through a maximum and then de-
Similar results are obtained for the minimum and the second maximum of the viscosity. From the analysis of our experimental data we get: minimum of the zero shear viscosity (Figure 8): cNaSal/mmol= 30 mmol + 0.85ccbc,/mmol (15) second maximum of the zero shear viscosity (Figure 8): CNaSal/mmol = 140 mmol + 1.2ccbcL/mmol (1 6) (13) Janeschitz-Kriegl, H.; Papenhuijzen, J. M. P. Rheol. Acta 1971, 10, 461.
The Journal of Physical Chemistry, Vol. 92, No. 16, 1988 4717
Rheology of Viscoelastic Surfactant Systems
Figure 11. Zero shear viscosity T(m,O) as a function of the surfactant concentration at T = 20 ‘ C . Figure 13. Stress relaxation time T as a function of the surfactant concentration ccpYcland the salt concentration cNaSalat T = 20 “C. 7
1 -
-.
I
I
I
I
p
1 13’
CCPycl’mmol I
l1/11,’/
10’
I
I
, I 1
i 3’
Figure 12. Salt concentration cNaSa,as a function of the surfactant
concentrationccpyclat conditions where maximum viscosities appear ( T = 20 “C). Equations 14-16 can be used to design viscoelastic surfactant solutions with desired rheological properties. As we will discuss in the next paragraph, these relations describe different regions of micellar kinetics. Relaxation Behavior. The dynamic properties of the viscoelastic surfactant solutions can be characterized by measuring the stress relaxation time after cessation of steady-state flow. Typical results of these experiments are summarized in Figure 13. We obtain a curve that is very similar to the zero shear viscosity in Figure 7. The positions of the extreme values are in excellent agreement with the corresponding maxima and minima of the viscous resistance. At salt concentrations before the first maximum we observe a multiexponential decay of the shear stress, and the values given in this regime correspond to the longest relaxation process. After passing the maximum, there is only one single relaxation time, which depends strongly upon the ionic strength. In former investigations we already pointed out that in this region the stress relaxation time coincides with the slow relaxation time of kinetic processes that are present in micellar solution^.'^^^^ We (14) Lobl, M.; Thurn, 1984,88, 1102.
H.; Hoffmann, H. Ber. Bunsen-Ges. Phys. Chem.
can conclude, therefore, that the viscoelastic behavior of the solution in this regime is controlled by micellar kinetics. In situations where the average lifetime of the micellar structures becomes comparable to their orientational relaxation time, the micellar kinetics control the macroscopic flow behavior of the solutions. The average lifetime of micellar systems is known to vary from milliseconds to hours, depending on the surfactant concentration and the electrolyte concentration of the solution^.'^^^^ The structural relaxation time can therefore change by many orders of magnitude. It is for this reason that surfactant solutions with identical structures can be so different in their rheological properties. There are two different mechanisms by which micellar aggregates can be formed or destroyed: at low salt concentrations, ionic micelles change their aggregation number only in a stepwise fashion, while at high electrolyte concentrations the micelles can also coalescence and break into pieces. The molecular details of these fundamental processes are not so important for the understanding of the different relaxation processes. In a first approximation it is necessary to know only that the stress can relax by disentanglement or by kinetic processes and that the mechanisms are changed as a function of the salt concentration. According to these ideas, the complicated shape of the stress relaxation curve can easily be traced back to the kinetic properties of micellar aggregates. Elastic Properties. The elastic properties of the surfactant solutions are characterized by the shear modulus. This quantity can be investigated by measuring the time-dependent relaxation modulus C ( t , y )after a step function shear strain or by measuring the storage modulus G’(w) at sinusoidal flow conditions. The plateau values of these two functions coincide, and they are denoted as equilibrium shear modulus G(0,O). This quantity describes the (15) Hoffmann, H.; Lobl, M.; Rehage, H. “Progress and Trends in Rheology 11”, Proceedings of the Second Conference of European Rheologists, Prague, June 17-20, 1986; Giesehus, H., Hibberd, M. F., Eds. (supplement to Rheol. Acta 1988, 26, 246). (16) Baumiiller, W.; Hoffmann, H.; Ulbricht, W.; Tondre, C.; Zana, R. J . Colloid Interface Sci. 1978, 64, 418. (17) Hoffmann, H.; Nagel, R.; Platz, G.; Ulbricht, W. Colloid Polym. Sci. 1976, 254, 821.
4718 The Journal of Physical Chemistry, Vol. 92, No. 16, 1988
Rehage and Hoffmann
,02i
/
i
G ( 0 , O )1 P a
/l
/
i 7
/
1
i
1
0
1
1
M
l
1
LO
l
1
60
1
1
80
l
I
I
100
I
120
l
I
-
110
Reduced concentration , , 1 1 , :
,
I
'
Figure 15. Shear modulus G(0,O) as a function of the reduced surfactant concentration at T = 20 O C .
sites. In the vicinity of the percolation threshold the shear modulus follows a power law:1s-23
where peeldenotes the gelation threshold of the surfactant solution. The exponent x has been calculated by several authors.*23 Recent advanced studies seem to x = 1.98
If only the surfactant concentration is changed, eq 20 can be replaced by
where cBelis the gelation concentration of the surfactant system. The exact experimental determination of cgel is very different because the elastic effects in the vicinity of the gel point are very small. From measurements of the zero shear viscosity we can conclude that cgelis of the order of a few millimoles (1 mmol < cgel< 5 mmol). A more accurate value can be obtained by a two-parameter least-squares analysis of the experimental data. From the nonlinear least-squares fit we obtain x = 1.9 f 0.1,
cgel= 2 mmol
The solid line in Figure 15 represents the best-fit curve according to eq 2. It is evident that the exponent x thus determined is in excellent agreement with the theoretical predictions. Percolation theories imply the formation of chemical or permanent bonds between the molecules, and in micellar systems such a cross-linking mechanism seems to be doubtful. On the other hand the exponent 2 is rather close to the value predicted for entangled polymer systems either in the mean field approach (2) or in the excluded volume region (2.25). We can hence conclude that the viscoelastic surfactant solutions have "entanglement properties" and they seem to undergo real sol-gel transitions. Investigations of dodecyl-, tetradecyl-, and hexadecyltrimethylammonium bromides that have recently been carried out by Candau et al. gave results that are very similar to our own c o n ~ l u s i o n s .From ~ ~ ~ ~extensive ~ measurements of the diffusion coefficient these authors found an interesting analogy to systems (18) Stauffer, D. Phys. Rep. 1979, 54, 1. (19) Stauffer, D.; Coniglio, A.; Adam, A. Adu. Polym. Sci. 1982, 44, 103. (20) de Gennes, P.-G. J . Phys. Lett. 1976, 37, 1. (21) de Gennes, P.G. Scaling Concepts in Polymer Physics; Cornel1 University Press: Ithaca, NY, 1979; Chapter V. (22) Alexander, S.; Orbach, R. J . Phys. Lett. 1982, 43, 625. (23) Rammal, R.; Toulouse, G. J . Phys. Lett. 1983, 44, 13. (24) Candau, S. J.; Hirsch, E.; Zana, R. Prog. Colloid Polym. Sci. 1987, 73, 189. (25) Candau, S. J.; Hirsch, E.; Zana, R. J . Colloid Interface Sci. 1985, 105, 521.
Rheology of Viscoelastic Surfactant Systems of semidilute polymer solutions. We may conclude, therefore, that the elastic properties of the surfactant solutions are due to the presence of a three-dimensional network that is formed from flexible micelles. The molecular details of such a supermolecular structure are not known in the present state and further studies are necessary to evaluate the complicated structure of these surfactant systems.
Discussion Recently Olsson et al. have made extensive N M R studies on cetyltrimethylammonium salicylate and on cetylpyridinium salicylate systems.26 In the framework of these studies, the authors have investigated the influence of excess Nasal on the correlation times of C H 2 groups.26 It is interesting to compare these measurements with the results of our rheological measurements. From N M R investigations carried out on the cetyltrimethylammonium system Olsson et al. proposed a reverse of the electric charges of the rodlike micelles with increasing salicylate/surfactant ratio.26 In N M R experiments, the correlation times decrease with increasing salicylate concentration.26 This experimental result was explained by an increasing flexibility of the rodlike micelles.26 At very high excess salicylate concentrations the observed N M R lines became very narrow, which was explained by a rod-sphere transition of the micellar particles.26 In the CPySal system, the situation should be very similar. However, judged from rheological measurements, such a rod-sphere transition cannot be observed. The shear modulus does not depend upon the salicylate concentration, and this implies that the micellar structures are not changed as a function of the excess salt concentration. The stress relaxation time constant, however, is strongly influenced by the ionic strength, and we guess that there exists a certain relationship between the N M R signals of CH2 groups and the structural relaxation times. From previous measurements we know that the structural relaxation time above the first viscosity maximum is determined by kinetic processes, in which the micellar particles break and re-form again.l4>l5Under these conditions the mechanical stress relaxation proceeds through chemical pathways. The occurrence of the second viscosity maximum is still an unsolved problem. In analogy to the results of Olsson et al. we assume a change from a positive to a negative electric charge with increasing salicylate/surfactant ratio. An excess negative charge of the micellar particles must lead to a decrease of collision rates and thus results in an increase of the structural relaxation time, which is experimentally observed. It is also conceivable that the excess charge leads again to a stiffening of the micelles and an increase in the persistence lengths. At very high values of the salicylate concentration the micellar charges are completely shielded by the normal salt effect of the ionic strength, and we obtain the typical properties of uncharged systems. The results of our rheological measurements indicate that the longest structural relaxation times occurs at a salicylate/surfactant ratio of less than unity. This is consistent with conclusions of Olsson et al., which are based on NMR measurements on a similar surfactant system.26 These authors propose a stiffening of the rodlike micelles when some of the salicylate counterions are replaced by bromide ions. The increase of the structural relaxation time is then a simple consequence of the decreasing flexibility. In principle, the stress relaxation time constant of rodlike micelles should strongly depend upon the dimensions of the anisometric particles. Quantitative theories, however, are available only for simple, ideal systems of stiff rods without interactions. In micellar solutions, the influence of polydispersity, flexibility, and interaction (26) Olsson, U.;Soderman, 0.;Guering, P. J . Phys. Chem. 1986,90,5223. (27) Sakaiguchi, Y.; Shikata, T.; Urakami, H.; Tamura, A.; Hirata, H. Colloid Polym. Sci. 1987, 265, 750.
The Journal of Physical Chemistry, Vol. 92, No, 16, 1988 4719 forces is still under debate, and it is therefore difficult to determine molecular dimensions from stress relaxation time constants. In a system of stiff rods there is only the dynamic orientation of these particles during flow, which corresponds to an entropy elasticity of the sheared solutions. After cessation of flow, the quiescent and isotropic state is re-forming again, and the stress relaxation process is determined by the rotational diffusion constant of the anisometric particles. Micellar rods have, even when they are completely stiff, a dynamic character that is controlled by kinetic processes, and as a consequence we observe only one relaxation time. It is interesting to note that such a pure exponential stress decay has recently been treated theoretically by Cates for concentrated micellar solutions and microemulsions.28~29The theoretical predictions of this model are in encouraging agreement with our experimental results, and the incorporation of chemical reactions into rheological properties seems to be a special feature of surfactant solutions that can quantitatively be described in the framework of this new t h e ~ r y . Our ~ ~ ,results ~ ~ clearly indicate that these kinetic processes are important especially at high Nasal concentrations. If the rodlike micelles are flexible, which can be expected in analogy to polymer systems, at least at high values of the ionic strength, we have to assume that there are entanglement and disentanglement processes, which can also contribute to the stress relaxation of these systems. In situations where the molecular dimensions of the flexible aggregates are much larger than their mean distance of separation, we obtain overlap and the formation of a three-dimensional network with physical or temporary contacts. Recently, Sakaiguchi et al. exhibited the occurrence and existence of such dynamic networks in a very similar surfactant system by electron m i c r o ~ c o p y .The ~ ~ measurements were performed on the cetyltrimethylammonium bromide/salicylate system, which has qualitatively the same rheological properties as the cetylpyridinium chloride/NaSal system. With eq 19 it is possible to calculate the number of elastically effective chains per unit volume from measurements of the shear modulus. The dimensions of the flexible particles can be determined by
cMM = w 2 p v L
(22)
where cM is the concentration of the aggregated surfactant, M is the molecular weight, r is the radius of the micelles, p denotes the density of the aggregates, and L is the length of the rodlike particles. All quantities except L are known, and it is therefore possible to calculate the dimensions of the flexible aggregates from eq 19 and 22. For our solutions we get dimensions of several thousand angstroms, which is in fairly good agreement with the results of Sakaiguchi et al. revealed by electron microscopy on a similar surfactant system.27 We can conclude, therefore, that the striking phenomenon of viscoelasticity is due to the presence of a temporary, three-dimensional network that is formed by very long, entangled rodlike micelles. According to these ideas, the strong analogy to the flow behavior of polymer systems and the validity of percolation theories, which have originally been developed for polymer solutions, is not very surprising.
Acknowledgment. Financial support of this work by a grant of the AIF is gratefully acknowledged. Thanks are due to C. Wagner for technical assistance. We would like to thank the Deutsche Forschungsgemeinschaft for a grant and the experimental equipment (fluids rheometer RFR 7800). This work is part of the research project of the SFB 213. Registry No. CPyCI, 123-03-5; CpySal, 23647-43-0; Nasal, 54-21-7. (28) Cates, M. E. Macromolecules 1987, 20, 2289. (29) Cates, M. E. ACS Symposium on Polymeric Microemulsions and Polymer-Microemulsion Interactions, New Orleans, August 1987.
