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Langmuir 1997, 13, 7012-7020
Rheological Master Curves of Viscoelastic Surfactant Solutions by Varying the Solvent Viscosity and Temperature Peter Fischer*,† and Heinz Rehage‡ Department of Chemical Engineering, Stanford University, Stanford, California 94305-5025, and Institut fu¨ r Umweltanalytik, Universita¨ t-GHS Essen, 45141 Essen, Germany Received May 30, 1997. In Final Form: October 17, 1997X In this paper we present a study of the rheological properties of aqueous cetylpyridinium chloride/ sodium salicylate solutions that have been measured as a function of both sucrose concentration and temperature. By using three different superposition methods we evaluated the rheological master curves for these solutions. In the low-frequency regime the rheological behavior can be described by a simple Maxwell material while in the high-frequency regime the sample is characterized by a Rouse-like behavior. Increasing the sucrose concentration causes some structural changes of the micelles to occur. In the Maxwellian part of the spectrum we always obtain a uniform master curve independent of the amount of added sucrose and the temperature. Therefore in the limit of monoexponential stress relaxation a complete linear rheological description becomes possible by just knowing the relaxation time and the zero shear modulus of the solution. However, some deviations from this picture are observed in the regime of high frequencies and extremely high sucrose concentrations. Here we observe a Rouse-like spectrum and structural changes in the elongated micelles due to the added sucrose.
Introduction It is well-known that surfactant molecules in aqueous media tend to form elongated rod-shaped aggregates.1-9 For sufficiently long wormlike micelles, an analogy with solutions of flexible polymers in good solvent is experimentally found. One important difference between network structures, formed in solutions of pure macromolecules and micellar aggregates is that micelles are in thermal equilibrium with single monomers. The dynamic properties of these solutions are characterized by a permanent exchange of material. Some viscoelastic surfactant solutions behave like of an ideal Maxwell material with a single relaxation time, λ, for the whole system. However, the wormlike micelles can break and recombine on an individual time scale, λBREAK.5,10 An ideal Maxwell behavior is predicted in the limit where the micellar breaking time is short compared to the diffusion time, λREP, of the whole aggregate. Therefore, the rheological properties of these solutions depend on the relative magnitude of λBREAK and λREP.11 Due to the technical limitations of almost all rheometers it is often not possible to obtain detailed information on the behavior of viscoelastic materials. This holds, espe* To whom correspondence should be addressed: e-mail,
[email protected]. † Stanford University. ‡ Universita ¨ t-GHS Essen. X Abstract published in Advance ACS Abstracts, December 1, 1997. (1) Appell, J.; Porte, G. J. Phys. Lett. 1983, 44, L 689. (2) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081; Langmuir 1988, 4, 354. (3) Rehage, H.; Wunderlich, I.; Hoffmann, H. Prog. Colloid Polym. Sci. 1986, 72, 51. (4) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712. (5) Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, 933. (6) Schurtenberger, P.; Scartazzini, R.; Luisi, P. L. Rheol. Acta 1989, 28, 372. (7) Schurtenberger, P.; Scartazzini, R.; Margrid, L. J.; Leser, M. E.; Luisi, P. L. J. Phys. Chem. 1990, 94, 3695. (8) Candau, S. J.; Hirsch, E.; Zana, R.; Adam, M. J. Colloid Interface Sci. 1988, 122, 430. (9) Candau, S. J.; Hirsch, E.; Zana, R.; Delsati, M. Langmuir 1989, 5, 1225. (10) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Mater. 1990, 2, 6869. (11) Cates, M. E. Macromolecules 1987, 20, 2289.
S0743-7463(97)00571-4 CCC: $14.00
cially, for short relaxation times where the stress relaxation process is very fast. As a consequence, the zero shear viscosity, the plateau modulus, and the relaxation time are just insufficiently detectable. Unfortunately, these data are essential to describe the dynamic properties of the sample. In addition, it is often of interest to obtain the rheological behavior of those systems in the highfrequency regime. Besides using special rheometers to extend the accessible regime of frequency, one may also use superposition methods to obtain information in the high-frequency regime.12-14 When parameters such as temperature or solvent viscosity are changed, it is often observed that the rheological properties are shifted to virtually lower or higher frequencies. Superposition of all these different experiments leads to a master curve that covers a wide range of viscoelastic properties. To extend the accessible frequency regime we performed not only time-temperature superposition (TTS) and time-concentration superposition (TCS) but also time-temperature-concentration superposition (TTCS) by systematically changing the temperature and solvent viscosity. In this way it was possible to obtain master curves, covering a large frequency range of the investigated solution of cetylpyridinium chloride/sodium salicylate (100-250 mmol/L) with and without sucrose (0-200 g/100 mL). Theory Maxwell Model. The linear viscoelastic properties of fluids can be described by the Maxwell element. It is convenient to express the rheological properties in terms of the relaxation modulus, G*(ω)
G*(ω) ) G′(ω) + iG′′(ω)
(1)
Here the storage modulus, G′(ω), describes the elastic properties of the sample, while the loss modulus, G′′(ω), is proportional to the viscous resistance. The behavior of (12) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley & Sons: New York, 1980. (13) Schmiedl, P. PhD Thesis, Universita¨t Bayreuth, 1995. (14) Zeegers, J.; Ende, D. v. d.; Blom, C.; Altena, E. G.; Beukema, G. J.; Mellema, J. Rheol. Acta 1995, 34, 606.
© 1997 American Chemical Society
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log dc ) e1
Figure 1. Dynamic moduli as a function of oscillation frequency and different solvent viscosities ηSV (0.001, 0.005, and 0.01Pa s).
a Maxwell material under harmonic oscillations can be described by
G′(ω) ) G0
ω2λ2 1 + ω2λ2
(2)
G′′(ω) ) G0
ωλ 1 + ω2λ2
(3)
where λ denotes the relaxation time. The behavior of both dynamic moduli as a function of angular frequency is shown in Figure 1. A deeper introduction is given elsewhere.5 The simplicity of comparing the rheological data with the Maxwell model is rather attractive. For example, the Maxwell model is in qualitative agreement with the dynamic properties of viscoelastic surfactant solutions as shown in one of the following sections. Superposition Methods. Superposition principles for the dynamic moduli can be used to obtain master curves. This technique was originally described by William, Landel, and Ferry.12 The principle of the equivalency of time and temperature is basically described by timetemperature superposition (TTS) and is shown by
log aT ) b1
T - T0 b2 + T - T0
(4)
The shift factor aT is defined as the ratio of the relaxation time, λT, at a given temperature, T, and the relaxation time, λTo, at the reference temperature, T0. The empirical constants b1 and b2 are obtained from the graph (T - T0) versus log aT. Varying the temperature from T0 to T results in multiplying G0 by bT/T0 and in each T being multiplied by aT. This is the same as if ω were multiplied by aT, which leads to an extended frequency regime. As a consequence, the storage modulus, G′(ω), measured at frequency, ω, and temperature, T, is equivalent to the storage modulus measured at a frequency ωaT and temperature T0. Thus, if measurements over a range of temperatures are plotted in the form of G′T0/T versus ωaT, a composite curve will be formed. This technique holds if the supermolecular structure remains constant upon changing the temperature. A similar technique is used in time-concentration superposition (TCS). Here the solvent viscosity is changed. An example of how this may be accomplished is by the addition of sucrose. The corresponding equivalence for time and concentration, c, is given by
c - c0 e2 + c - c0
(5)
The shift factor, dc, as well as the empirical constants e1 and e2 are of the same meaning as in the timetemperature superposition. This method is only applicable when the increase in the solvent viscosity does not affect the molecular structure of the surfactant solution. For example, the electrostatic interactions among surfactants and counterions have to remain unaffected by the addition of sucrose, otherwise the rheological properties would change and a comparison would no longer be possible. This requirement may be tested by analyzing both dynamic moduli. In general the rheological properties have to be constant at small and intermediate angular frequencies. It is only at high frequencies that a second upturn of both moduli is desired. Here, the rheological properties cannot be described by the Maxwell model any longer and a transition to a glassy state is expected. Any deviation from the obtained master curve would indicate some structural rearrangements in the wormlike micelles. As long as the superposition principle holds, one should just obtain a horizontal shift of the storage modulus and a Rouse motion in the high-frequency regime.15 Nevertheless, changes in solvent viscosity due the adding of sucrose and/or a decrease temperature affect the slope of the loss modulus in the high-frequency regime. This influence is given by
G′′(ω) ) ηSVω + G0
ωλ 1 + ω2λ2
(6)
where the solvent viscosity, ηSV, is added to the original Maxwell equation in order to include friction caused by the solvent. From this equation one expects a second upturn of the loss modulus in the high-frequency regime while the low-frequency region remains unaffected. This effect is schematically demonstrated in Figure 1 for three different values of the solvent viscosity. Last but not least a combination of both techniques is practical. This time-temperature-concentration superposition provides a wider frequency range than both single superposition methods. Viscoelastic Surfactant Solutions. The phenomenon of viscoelasticity can be observed by adding certain counterions to cationic surfactant solutions. These systems may be used as simple model systems for rheological studies because of their model-like flow properties.4,16-21 In comparison to solutions of macromolecules, there is, however, one important difference. The anisometric aggregates of surfactants are in thermal equilibrium with individual monomers. The permanent exchange of surfactant molecules leads to breaking and re-formation processes of the rod-shaped aggregates. This has important consequences on the stress relaxation mechanism. Cates proposed a model for the dynamics of these entangled micelles, which involves the conventional reptation model for polymers and the reversible scission and recombination of chains.11,22,23 In situations where the elongated micelles are breaking within the time scale (15) Rouse, P. E. J. Chem. Phys. 1953, 21, 1272. (16) Hoffmann, H.; Ulbricht, W. Chem. Z. 1995, 29, 76. (17) Appell, J.; Porte, G. Europhys. Lett. 1990, 12, 185. (18) Berret, J.-F.; Appell, J.; Porte, G. Langmuir 1993, 9, 2851. (19) Fischer, P.; Rehage, H. Prog. Colloid Polym. Sci. 1995, 98, 94. (20) Fischer, P. Nicht-lineare rheologische Pha¨ nomene in viskoelastischen Tensidlo¨ sungen; Wissenschaftlicher Buchverlag Dr. Fleck: Giessen, 1995. (21) Fischer, P.; Rehage, H. Rheol. Acta 1997, 36, 13. (22) Turner, M. S.; Cates, M. E. J. Phys. II 1992, 2, 503.
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shape are observed. For example in the high-frequency regime when Rouse-like motion is present, an upturn of G′′ is seen as a function of G′. To estimate the different time constants from the experimental Cole-Cole plots Turner and Cates simulated Cole-Cole plots for various values of the reptation time and the breaking time and the Rouse-like motion.24,25 The calculated ratios ζ and ζh are given by
ζ ) λBREAK/λREP
(8)
ζh ) λBREAK/λ
(9)
One may obtain both ratios directly from a normalized Cole-Cole plot as shown schematically in Figure 2b. An increasing ratio, ζ, manifests itself by a modification of the semicircular Cole-Cole plot. As a consequence, the time constants of the solution under investigation become available. But, these predictions only hold for the fast breaking regime assuming that a tail in the Cole-Cole plot is correlated to λBREAK . λREP (ζ , 1).25 A departure from the semicircular shape for an increasing storage modulus is observed in the high-frequency regime and one will obtain a tail and in some cases a dip indicating a Rouse-like behavior as shown in Figure 2c.26 The actual sizes of the rodlike micelles now can be estimated. The ratio of the minimum of the loss modulus in the high-frequency regime to the relaxation modulus is correlated to the ratio of the entanglement length, le, and the average contour length of the micelle, L. The ratios are given in the following equation
G′′min le ) G0 L
(10)
The entanglement length, le, can be estimated by using the correlation length, ξ, and the zero shear modulus Figure 2. Cole-Cole plots for a ideal Maxwell fluid (a), for different values of the ratio ζ (b), and for Rouse-like motion (c).
of observation, the rheological properties are controlled by kinetic processes. It is only under these circumstances, that we obtain very simple scaling laws and linear relations between all rheological quantities. As long as the average lifetime, λBREAK, of the micelles is much smaller than the reptation time, λREP (λBREAK , λREP), there are numerous breaking and re-formation processes within the time scale of observation. This leads to a pure, monoexponential stress decay and the solution can be described by the Maxwell model. The relaxation time, λ, now is the geometrical average of λBREAK and λREP
λ ) xλBREAKλREP
(7)
It is often convenient to represent the storage and loss moduli in the form of a Cole-Cole plot. In these diagrams, the loss modulus is plotted as a function of the storage modulus. The Cole-Cole plot provides a more precise determination of the relaxation behavior of samples than simple frequency sweeps. In the case of pure monoexponential stress relaxation (λBREAK , λREP) a semicircular shape is characteristic for the Cole-Cole plot as shown in Figure 2a. When the fluid can no longer be described as a simple Maxwell fluid, deviations from the semicircular (23) Cates, M. E. In Structure and Flow in Surfactant Solutions; Herb, C. A., Prud’homme, R. K., Eds.; American Chemical Society: Washington, DC, 1994; Vol. 578; p 32.
G0 )
kBT ξ
3
≈
kBT le1.8
(11)
As a consequence, one can obtain detailed information about the different time constants and the sizes of the micellar aggregates from the viscoelastic properties of the solution. This becomes very important when evaluating the possible changes of the micellar structure during different superposition methods. Significant changes, except those effects discussed before, would indicate a structural change of the micelles that might be induced by adding sucrose or changing the temperature. Sample. To obtain the master curve of 100 mmol/L cetylpyridinium chloride to 250 mmol/L sodium salicylate two techniques were performed. In the first set of experiments, the viscous resistance of the surfactant solution was increased by decreasing the temperature. In the second technique the solvent viscosity, ηSV, was increased by adding sucrose. In Figure 9 the viscosities of the corresponding aqueous sucrose solutions, ηSS, are plotted. All these aqueous sucrose solutions are of Newtonian character. Experimental Section The rheological experiments were performed using a Rheometrics RFS II rheometer fitted with an environmental chamber. The Couette device consisted of a 34 mm diameter aluminum(24) Turner, M. S.; Cates, M. E. Langmuir 1991, 7, 1590. (25) Granek, R.; Cates, M. E. J. Chem. Phys. 1992, 96, 4758. (26) Khatory, A.; Lequeux, F.; Kern, R.; Candau, S. J. Langmuir 1993, 9, 1456.
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coated cup and a titanium bob with a diameter of 32 mm and a height of 33.35 mm. A small amplitude oscillatory shear flow with a constant shear strain of 20% is applied to the investigated viscoelastic solution. The magnitude of the complex viscosity, |η*| (ω), the storage modulus, G′(ω), and the loss modulus, G′′(ω), are measured in periodical experiments in the frequency range of 0.01-100 rad/s. The relaxation time was obtained from the intersection point of the dynamic moduli. The zero shear modulus was obtained from the plateau value of the storage modulus G′(ω). In addition both parameters were checked by fitting to a Maxwell model as given in eqs 2 and 3. The zero shear viscosity was calculated from the magnitude of the complex viscosity, |η*|(ω), at very small oscillation frequencies.
Data Analysis Superposition Methods. The time-temperature and time-concentration superposition as well as the combination of both techniques were performed by using the “Rhecalc” Software.27,28 The overlay procedure and determination of the shift factor were performed by using eqs 6 and 7. Cole-Cole Plots. To obtain comparable results the Cole-Cole plot must be normalized. In one procedure the results were presented in the form of G′′(ω)/G0 versus G′(ω)/G0. Experimentally it is often difficult to obtain an accurate value of G0. In a slightly different normalization procedure both dynamic data are divided by G′′max(ω), the maximum of the semicircle in the classical Cole-Cole plot. This method is identical to normalizing by the maximum of the semicircle, Gosc, as mentioned in several papers.26 In both methods G0 can be estimated by an extrapolation of the tail of the Cole-Cole plot down to the horizontal axis.
Figure 3. Normalized Cole-Cole plots of pure cetylpyridinium chloride-sodium salicylate solutions as a function of the temperature.
Results and Discussion Time-Temperature Superposition. In the timetemperature superposition experiments, the viscosity is increased by decreasing the sample temperature until the Krafft point of the pure surfactant solution (0 g of sucrose) is reached. Typical experiments were performed at many temperatures between 14 and 24 °C. In this temperature range the solution still behaves as an ideal Maxwell material without any Rouse-like behavior. This is indicated by the semicircular shape of the Cole-Cole plots in Figure 3. By time-temperature superposition it is possible to increase the frequency regime up to 1000 rad/s without changing the viscoelastic properties of an almost ideal Maxwell material. The master curve in Figure 4 shows perfect superposition for the investigated temperature regime. From an Arrhenius plot the activation energy, Ea, can be calculated by
η0 ) AeEa/RT
Figure 4. Master curve of pure cetylpyridinium chloridesodium salicylate solution using data from Figure 3.
(12)
We obtain Ea ) 3.1 × 105 J/(mol K). The activation energy describes the energy that is necessary to move individual micelles in an environment of surrounding micelles. The activation energy is therefore given by the interactions between individual aggregates.12 With increasing temperature we observe a decrease in the relaxation time, λ, as shown in Figure 5. The scaling behavior for the relaxation time with temperature is given by λ ∼ T-4.3. Similar values are reported in literature.29 It is interesting to note that the zero shear modulus, G0, is not affected by varying the temperature as shown in Figure 5. This is somewhat unusual as the zero shear modulus is normally affected by temperature.29 According (27) Rheometrics Owners Handbook (1991). (28) Honerkamp, J.; Weese, J. Rheol. Acta 1993, 32, 57. (29) Buhler, E.; Munch, J. P.; Candau, S. J. J. Phys. II 1995, 5, 765.
Figure 5. Relaxation time, λ, zero shear modulus, G0, and zero shear viscosity, η0, of pure cetylpyridinium chloridesodium salicylate solution as a function of the temperature as obtained from dynamic moduli. The solid lines are obtained according to eq 13.
to the theory of rubber-elasticity the zero shear modulus depends upon the temperature and the relative amount of intersections in the molecular network structure as
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Figure 7. Master curve of cetylpyridinium chloride-sodium salicylate-sucrose solution for data from Figure 6 (T ) 20 °C).
Figure 6. Normalized Cole-Cole plots of cetylpyridinium chloride-sodium salicylate-sucrose solution as a function of the sucrose concentration (T ) 20 °C).
given by eq 11. In the TTS experiment the entanglement length increases by increasing the temperature. Further, Figure 5 shows the effect of the temperature on the zero shear viscosity, η0. The scaling behavior is approximately η0 ∼ T-4.0. A cross-checking of all three properties according to the following equation
η0 ) λG0
(13)
shows a very good agreement between both measured parameters and the calculated one. The calculated properties are given in Figure 5 as the solid lines. Time-Concentration Superposition. In the timeconcentration superposition experiments the viscosity of the surfactant solution is changed by increasing the solvent viscosity, ηSV. For this purpose the sucrose concentration, cSucrose, is varied from 0 to 200 g of sucrose in 100 mL of cetylpyridinium chloride-sodium salicylate solution. For this set of experiments the temperature was maintained at 20 °C. One obtains changed rheological parameters and different behavior of the dynamic moduli in the highfrequency regime as a function of the sucrose concentration. Indeed all frequency sweeps clearly show the typical behavior of a Maxwell material in the low-frequency regime but some deviations are present at high frequencies. In the normalized Cole-Cole plots, Figure 6, this is visible as both a dip in G′′/G′′max and the appearance of a tail as increasing amounts of sucrose. Both dynamic moduli slightly increase a second time at high frequencies and therefore denote a Rouse-like motion as mentioned previously in Figure 2. Simultaneously the typical frequency regime where a plateau of the storage modulus appears becomes much shorter. The Rouse-like part of the spectrum leads to a strongly increasing loss modulus at high frequencies. By shifting several frequency sweeps it is possible to map out the high-frequency regime up to 1000 rad/s as shown in Figure 7. The increasing effect of the solvent viscosity is shown in the upturn of the loss modulus in the high-frequency regime similar to Figure 1. The master curve shows good superposition for the investigated concentration regime except for the highest sucrose concentration of 200 g. In comparison to Figure 4 it turns out that even between TTS and TCS a superposition is possible. In the next section we therefore
Figure 8. Relaxation time, λ, and zero shear modulus, G0, of cetylpyridinium chloride-sodium salicylate-sucrose solution as a function of the sucrose concentration. The solid lines are obtained according to eq 13 (T ) 20 °C).
discuss the TTCS more intensively. The activation energy is calculated to be Ea ) 2.84 × 105 J/(mol K). The lower value of the activation energy in comparison to the pure surfactant solution implies that the added sucrose decreases the interaction between the micelles. On a molecular level the addition of sucrose screens the individual micelles from each other and as a consequence lesser energy is necessary to move the micelles. The shift of the relaxation time as a function of sucrose concentration, cSucrose, is shown in Figure 8. Clearly as the amount of sucrose increases so does the relaxation time. The relaxation time scales with sucrose concentration as λ ∼ cSucrose0.7. The zero shear modulus, G0, decreased slightly with sucrose concentration as shown in Figure 8. The scaling behavior is given by G0 ∼ cSucrose-0.6. Actually, if the idea of changing only the solvent viscosity holds, the plateau modulus must not depend on the sucrose concentration. This is shown in Figure 8 and Figure 13. But as seen in Figure 8 (G0 ∼ cSucrose-0.6) there is a small influence of the sucrose concentration to the rheological properties. This result suggests structural changes of the material. The analysis of the solvent viscosity as shown in Figure 9 confirms this observation. The zero shear viscosity, η0, of the surfactant-sucrose solution as a function of sucrose concentration, cSucrose, is plotted together with the viscosity of the corresponding pure sucrose solution, ηSS. One clearly observes no influence of ηSS in the regime of low concentrations. Only in the regime of high sucrose concentration can one detect a small influence on the zero shear viscosity. It is also interesting to see the relation between the viscosity ηSS of the pure
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Figure 9. Zero shear viscosity, η0, of cetylpyridinium chloridesodium salicylate-sucrose solution as a function of the sucrose concentration. The solid line is from eq 13 using data from Figure 8. Comparison to the viscosity, ηSS, of the pure sucrose solution and the solvent viscosity, ηSV, obtained from eq 8 (T ) 20 °C).
sucrose solution and fitted solvent viscosity, ηSV, from eq 6. It is difficult to see any correlation between these properties that were expected to be equal. It becomes clear that the sucrose is altering the molecular properties of the micelles and therefore influences the scaling behavior of the zero shear modulus. Due to eq 11 we expect that the entanglement length is increased as a function of the sucrose concentration. But, the interesting point is that for moderate amounts of sucrose a superposition with the pure surfactant solution is possible. This is shown in Figure 7 as well as comparing Figure 4 and later on in Figure 11. A cross-checking of the relaxation time, the zero shear modulus, and the zero shear viscosity according to eq 13 again gives good agreement between these basic rheological parameters (solid lines in Figures 8 and 9). Only the solution with 200 g of sucrose behaves differently compared to all the other solutions. Here the scaling behavior deviates and the frequency sweeps do not superimpose accurately. This implies that at this concentration more dramatic changes in the micellar structure occur and that it is no longer possible to obtain a uniform master curve. A Rouse-like behavior is observed in the regime of very high amounts of sucrose and high oscillation frequencies. The frequency range of the plateau of the storage modulus decreases and a flattening of the curves around the maximum of the loss modulus occurs. This type of behavior is observed in special cases for classical polymers when the ratio of chain length to persistence length becomes small. The theory of living polymers provides a relation between the minimum in the loss modulus and the average contour length of the micelle, L, in the case that Rouse modes govern the relaxation process at high frequencies.25 Experimentally it is found that the minimum depends upon temperature, so that a relation between the minimum and the micelle length can be established. This suggests that Rouse modes are appropriate for describing the observed relaxation process at high frequencies. Time-Temperature-Concentration Superposition. In the time-temperature-concentration superposition experiments the motion of the micelles is reduced by decreasing the temperature of the surfactant-sucrose solution until the Krafft point is reached. This has been performed for three surfactant solutions with discrete amounts of 61, 101, and 200 g of sucrose in 100 mL of cetylpyridinium chloride-sodium salicylate solution in a temperature range between 14 and 28 °C.
Figure 10. Normalized Cole-Cole plots of three cetylpyridinium chloride-sodium salicylate-sucrose solutions as a function of temperature (cSucrose ) 63 g/100 mL (A), 101g/100 mL (B), and 200 g/100 mL (C)).
With decreasing temperature it is possible to adequately reach the glassy state as shown below. The Maxwell behavior of all samples is clearly shown but a Rouse part is also observed in the high-frequency regime. Figure 10 summarizes these results in normalized Cole-Cole plots. In all cases the Maxwell behavior is clearly shown through variation of the temperature in the low-frequency regime (ωλ , 1). In the high-frequency regime (ωλ . 1) the values increase creating a second intersection of the dynamic moduli. The shifted frequency sweeps are shown in Figure 11 for all investigated surfactant-sucrose solutions. Good superposition is seen for all concentrations of sucrose at the different temperatures except for the solution with 200 g of sucrose as reported before. The activation energies
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Figure 12. Relaxation time, λ, of three cetylpyridinium chloride-sodium salicylate-sucrose solutions as a function of temperature. The solid lines are obtained from data in Figures 13 and 14 according to eq 13 (cSucrose ) 63 g/100 mL, 101 g/100 mL, and 200 g/100 mL).
Figure 13. Zero shear modulus, G0, of three cetylpyridinium chloride-sodium salicylate solution-sucrose solutions as functions of temperature. The solid lines are obtained from data in Figures 12 and 14 according to eq 13 (cSucrose ) 63 g/100 mL, 101 g/100 mL, and 200 g/100 mL).
Figure 11. Master curves of three cetylpyridinium chloridesodium salicylate-sucrose solution for data in Figure 7 (cSucrose ) 63 g/100 mL (A), 101g/100 mL (B), and 200 g/100 mL(C)).
are Ea ) 1.47 × 105 J/(mol K) for 61 g of sucrose, Ea ) 1.42 × 105 J/(mol K) for 101 g of sucrose, and Ea ) 1.7 × 105 J/(mol K) for 200 g of sucrose. All the activation energies are lower than those for the pure surfactant solution determined in the previous section. We assume from this observation that both temperature and sucrose concentration decrease the interaction between the micelles as observed before. The decrease in relaxation times as a function of temperature is shown in Figure 12. The scaling behavior of the relaxation time as a function of the temperature is approximately the same for all amounts of added sucrose and behaves like λ ∼ T-5.0 for 61 g and for 101 g of sucrose, and λ ∼ T-5.5 for 200 g of sucrose. The dependence of the zero shear modulus and the zero shear viscosity on temperature is summarized in Figures 13 and 14, respectively. Again, the zero shear modulus is independent of temperature. The scaling behavior of the zero shear viscosity as a function of temperature depending on the amount of added sucrose is η0 ∼ T-4.5 for 61 g and 101 g sucrose and η0 ∼ T-3.6 for 200 g of sucrose. Except
Figure 14. Zero shear viscosity, η0, of three cetylpyridinium chloride-sodium salicylate-sucrose solutions as functions of temperature. The solid lines are obtained using data in Figures 12 and 13 with eq 13 (cSucrose ) 63 g/100 mL, 101 g/100 mL, and 200 g/100 mL).
for the solution with 200 g of sucrose the scaling exponents agree. Additionally a cross-check according to eq 13 (lines in Figures 12 to 14) shows a tremendous discrepancy in the scaling behavior of the relaxation time and the zero shear viscosity (both exponents should be equal) for the solution with 200 g of sucrose. Therefore the cross-check
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Figure 15. Ratio G′′min/G0 of cetylpyridinium chloride-sodium salicylate solution-sucrose solution as a function of the sucrose concentration. The solid line is obtained by a power law fit (T ) 20 °C).
Figure 16. Ratio G′′min/G0 of three cetylpyridinium chloridesodium salicylate solution-sucrose solutions as a function of temperature. The solid lines are obtained by a power law fit (cSucrose ) 63 g/100 mL, 101 g/100 mL, and 200 g/100 mL).
produces large deviations especially in the zero shear modulus as shown in Figure 13. As noted before the sucrose seems to alter drastically the molecular structure and superposition is no longer applicable. But, in the regime of sucrose concentration up to 101 g/100 mL, one can note that the micellar structure is not changed with temperature and, as a consequence, superposition with all previous data is possible. In general it is interesting to understand what happens to the viscoelastic surfactant solution with the addition of sucrose or a change in the temperature. Based on the behavior of the zero shear modulus both mechanisms act very differently on the molecular structure. As shown previously, Figures 5 and 13, the zero shear modulus is not a function of temperature and we can conclude that the entanglement length is altered. However, in most of the investigated solutions the superposition method is applicable. This means that the stress relaxation (Maxwell type) is identical for all samples independent of temperature and/or sucrose concentration. In contrast, the zero shear modulus is dependent on the concentration of sucrose; hence, the micellar structure is changed, too. A structural change occurs and it is obvious at the lowfrequency part of the spectrum superposition but not the high-frequency regime. As a consequence, one can describe the rheological properties completely by the relaxation time and the zero shear modulus. This very interesting result can simplify the rheological investigation of numerous viscoelastic surfactant solutions since the obtained master curve would be valid for different surfactant concentrations and even counterion concentrations. Of course, one always has to keep in mind that the Maxwell relaxation type must be valid and be careful not to move into concentration regimes where stress relaxation or strong molecular changes due to the sucrose concentrations occur. Further it is interesting to understand which molecular properties are altered by increasing the sucrose concentration. According to Khatory et al. and Turner and Cates, we analyzed the macroscopic and molecular properties as mentioned in eqs 8-11.22,24,26 In the first case where we added sucrose at a constant temperature the decreasing zero shear modulus (Figure 8) implies an increasing entanglement length, le, with increasing sucrose concentration. Simultaneously G′′min and the ratio G′′min/G0 are increasing as a function of the sucrose concentration as shown in Figure 15. The scaling behavior of the ratio as a function of the sucrose concentration is G′′min/G0 ∼ cSucrose0.5. This implies that the average contour length of
the micelle, L, is constant as a function of the sucrose concentration. Introducing the previous trends into eqs 8 and 9, one obtains an increasing ratio ζ. Together with the assumption that λBREAK is approximately a linear function of L, we obtain a shorter reptation time for the solutions.26 In temperature superposition the situation is somewhat different. Here the zero shear modulus is constant with temperature, while the relaxation time and the zero shear viscosity decrease with increasing temperature. According to eq 11 this means that the entanglement length is not decreasing as rapidly as in the concentration governed experiments. We are not able to check the ratio in the TTS experiment (Figure 3) because a perfect semicircle is obtained and therefore no Rouse-like motion occurs. In the TTCS experiment (Figure 10), however, G′′min increases with temperature at constant sucrose concentrations as does the ratio G′′min/G0 shown in Figure 16. The scaling behavior of the ratio as a function of temperature is G′′min/G0 ∼ T1.2 for 61 g of sucrose, G′′min/G0 ∼ T0.7 for 101 g of sucrose, and G′′min/G0 ∼ T0.4 for 200 g of sucrose. This implies that the average contour length of the micelle is slightly decreasing or even constant as a function of the temperature. According to eqs 8 and 9 and assuming that λBREAK is a linear function of L we obtain increasing reptation times. However, in the end it turns out that in both cases the change of the rheological properties is governed by the reptation of the wormlike micelles. The next step is to investigate in more detail the structural changes to the micelles by the addition of sucrose. In this paper we are unable to further probe this question but the obtained results indicate that this is strongly recommended. Here SALS and SANS experiments could be helpful. Further it would be useful to determine if the obtained uniform master curve is of a more general character as mentioned earlier. All of these ideas and experiments are in the process of being performed. Summary In conclusion we have shown that the time-temperature superposition method is applicable to a viscoelastic surfactant solution of cetylpyridinium chloride/sodium salicylate. We also have added sucrose to the sample to perform a time-concentration superposition that gives the same results as altering the temperature. For all methods we obtain a superposition in Maxwellian part of the spectrum along the frequency axis. Depending on the
7020 Langmuir, Vol. 13, No. 26, 1997
temperature and on the amount of sucrose, a Rouse-like motion occurs in the high-frequency regime of the spectra. In addition, structural changes occur when the sucrose is added, but these phenomena did not change the stress relaxation processes in the regime of low and mediate concentration. In conclusion, it turns out that the obtained master curves are valid for different surfactant and counterion concentrations as long as the stress relaxation is of the form of a Maxwell fluid and we can describe the structural changes simply by the relaxation time and the zero shear modulus. This very interesting result simplifies rheological investigations for viscoelastic surfactant solutions such as cetylpyridinium chloride/sodium salicylate.
Fischer and Rehage
However, for the time being we should be careful to extend our analysis to various kinds of viscoelastic surfactant solutions. Acknowledgment. The authors thank Elizabeth K. Wheeler for the helpful discussion and the critical reading of the manuscript. P.F. acknowledges the “Deutsche Forschungsgemeinschaft” (Fi 665/1-2) for financial support that enabled us to finish this work. H.R. acknowledges grants from the “Deutsche Forschungsgemeinschaft” (Re 681/4-1) and the European Community (CHRX-CT940696). LA970571D
