Langmuir 2007, 23, 5267-5269
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Fast Dynamics of Wormlike Micellar Solutions Florian Nettesheim* and Norman J. Wagner Department of Chemical Engineering, UniVersity of Delaware, Newark, 150 Academy, Newark, Delaware 19716 ReceiVed December 11, 2006. In Final Form: February 14, 2007 We present the first measurements of the fast dynamics of cationic wormlike micelles (WLM) using neutron spin echo (NSE). The comparison with theory [Zilman, A.; Granek, R. Phys. ReV. Lett. 1996, 77, 4788. Granek, R. J. Phys. II 1997, 7, 1761]1,2 enables coarse grained parameters to be identified. We propose and validate a calibration procedure to extract the bending constant κ from NSE measurements.
Surfactant wormlike micelles (WLM)3 have similarities to polymers, and therefore, the description of their static as well as dynamic properties is largely borrowed from polymer physics. One key difference, however, is their equilibrium nature as they constantly break and reform, which in the fast breaking limit leads to a single relaxation time and Maxwellian behavior at low to intermediate frequencies.4 On time scales much shorter than the breakage time of the micelle, this dynamic process is essentially frozen and the micelles can be regarded as semiflexible chains. A dynamic structure factor for these fast dynamics of semiflexible chains has been derived by Zilman and Granek,1,2 which relaxes as a stretched exponential with an exponent of β ) 3/4. A recent review by Richter and co-workers5 demonstrates how these fast segmental dynamics can be probed by neutron spin echo (NSE) measurements. There is only one prior report of NSE measurements of the dynamics of wormlike micelles. Seto et al.6 studied the concentration dependence of the short time dynamics of a nonionic WLM solution. Although the semiflexible chain model was able to describe the observations they reported a discrepancy of 2 orders of magnitude between literature values for the bending modulus and that calculated from fitting the theory1,2 to their NSE data. This raises questions about the applicability of the theory and the use of NSE to determine properties of WLMs. In this communication the first NSE measurements of the short time dynamics of a cationic WLM system are presented. A model WLM solution of cetyltrimethylammonium bromide (CTAB) and sodium salycilate (NaSal) is used, as the rheology3 and rheo-optics7 of this system are well characterized. These systems are known to exhibit near Maxwellian behavior, for the concentration studied here. From the dynamic rheology, light and neutron scattering, and rheo-optics, length and time scales relevant for this study can be obtained (micelle diameter dcs, persistence length lp, mesh size ξm, entanglement le, and contour length Lc as well as the relaxation τr and the breakage time τbr as summarized in Table 1). In particular, rheo-birefringence studies provide an accurate value for lp of CTAB/NaSal WLM.7 * To whom correspondence should be addressed. E-mail:
[email protected]. Homepage: http://www.che.udel.edu/wagner. (1) Zilman, A.; Granek, R. Phys. ReV. Lett. 1996, 77, 4788. (2) Granek, R. J. Phys. II 1997, 7, 1761. (3) Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, 933. (4) Cates, M. E. J. Phys. Chem. 1990, 94, 371. (5) Richter, D.; Monkenbusch, M.; Arbe, A.; Colmenero, J. AdV. Polym. Sci. 2005, 174, 1. (6) Seto, H.; Kato, T.; Monkecnbusch, M.; Takeda, T.; Kawabata, Y.; Nagao, M.; Okuhara, D.; Imai, M.; Omura, S. J. Phys. Chem. Solids 1999, 60, 1371. (7) Shikata, T.; Dahman, S. J.; Pearson, D. S. Langmuir 1994, 10, 3470.
Table 1. Length Scales for Wormlike Micelles Considered Herea cD/cs, mM lp, nm dcs, nm ξm, nm Lc, nm le, nm η0, Pa s τr, s τbr, s DG × 10-2, nm8/3 ns-1
12.5/25 4.4 170 2500 580 1.37 1.51 0.04 4.60 ( 0.13
25/50 23.8 ( 0.5 4.4 100 1800 250 4.00 0.99 0.03 4.66 ( 0.13
50/100 25.1 ( 0.2 4.4 60 1600 110 26.1 1.41 0.04 4.30 ( 0.11
100/200 23.5 ( 0.1 4.4 40 1600 55 231 3.55 0.16 4.00 ( 0.10
a The persistence length lp was measured using the stress optic coefficient,7 dcs is the cross-sectional diameter determined from SANS, mesh size ξm, contour length Lc and entanglement length le were obtained from dynamic rheology.8,9
These values are used with the theory by Zilman and Granek to analyze the NSE results and evaluate the diffusion coefficient and the bending constant κ. WLM samples were prepared from the cationic surfactant cetyltrimethylammonium bromide (CTAB) cD, and sodium salicylate (NaSal) cs, obtained from Aldrich and used without further purification. A dilution series at a surfactant to salt molar ratio of cD/cs ) 0.5 with cD ) 12.5, 25, 50, and 100 mM, and a salt series at cD ) 25 mM and 16 mM e cs e 300 mM were prepared. All samples were prepared in D2O (Cambridge isotopes) to optimize contrast and minimize incoherent background. Neutron spin echo (NSE) was performed at the NG-5 beamline at NCNR at NIST, Gaithersburg, Maryland, with λ ) 0.8 and 1.1 nm, 0.13 e q e 1.5 nm-1, 0.5 e τ e 90 ns. Using dynamic rheology, small angle neutron scattering, flow birefringence, and quasielastic light scattering, the structure and dynamics of the wormlike micelles in solution were measured8,9 and the values are tabulated in Table 1. The persistence length measured by flow birefringence agrees with literature values.7 It was assumed that the value of the anisotropy in polarizability of CTAB does not change as a function of salt or surfactant concentration, so that the value of -8.5 × 10-22 cm3 from Shikata et al. was used. An example of the intermediate scattering function S(q,t)/ S(q,0) obtained by NSE is shown in Figure 1a. The intermediate scattering functions were fit to eq 1, based on the Zilman and (8) Schubert, B. A.; Kaler, E. W.; Wagner, N. J. Langmuir 2003, 19, 4079. (9) Liberatore, M. W.; Nettesheim, F.; Cook, L. P.; Vasquez, P.; Wagner, N. J.; Kaler, E. W.; Hu, Y. T.; Lips, A. 2006, in preparation.
10.1021/la0635855 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/03/2007
5268 Langmuir, Vol. 23, No. 10, 2007
Letters Table 2. Electrostatic Contribution lep to the Persistence Length lp Calculated from the OSF Theory for Varying Salt at Fixed Surfactant Concentration Compared to the Local Diffusion Coefficient DGa cD/cs, mM
lep (OSF), nm
lp, nm
DG × 10-2, nm8/3 ns-1
η0, Pa s
25/16 25/50 25/75 25/180 25/300
1 0.47 0.34 0.15 0.1
24.4 23.8* 23.7 23.5 23.4
4.49 ( 0.23 4.66 ( 0.17 5.06 ( 0.17 4.66 ( 0.17 4.43 ( 0.12
200 250 125 800 80
a The zero shear viscosities are also tabulated for comparison with DG. The values for lp are calculated based on the measured value indicated by the asterix.
Figure 1. (a) Normalized Intermediate scattering functions for 100 mM CTAB and 200 mM NaSal at selected q values in the range from 2 × 10-1 to 1.5 nm-1. Solid lines are fits to eq 1. (b) Data plotted according to semiflexible wormlike chain model for samples along the dilution line at a CTAB:NaSal ratio of 1:2.
Granek model.1 Its applicability to NSE was demonstrated previously by Seto et al.6
I(q,t) S(q,t) ) ) exp(-(Γ(q)t)β) I(q,0) S(q,0)
(1)
where β ) 3/4. Here, Γ(q) is the relaxation rate of the stretched exponential function
Γ ) DGq2/β
(2)
with DG is the segmental diffusion coefficient. Note that the time scale of measurement is significantly shorter than τbr, such that wormlike micelles are equivalent to semiflexible chains. The validation of a stretch exponent of 3/4 is shown in Figure 1b for the entire dilution series, where the normalized intermediate scattering functions nearly collapse onto a single master curve, independent of surfactant concentration. Interestingly, there is also no dependence on salt concentration (data not shown here for clarity, also see Table 2). The relaxation rates obtained by fitting eq 1 to the intermediate scattering function are presented in Figure 2, where at higher scattering vectors Γ(q) follows the 8/3 power law predicted for semiflexible chains.1 Deviations at low q are due to the influence of translational diffusion. A crossover from the 8/3 scaling observed in NSE for bending modes to the scaling expected for Rouse breathing modes (2), observable in dynamic light scattering (DLS) is expected for scattering vectors of q ≈ 2πlp-1. The first cumulant of the correlation functions measured by DLS is plotted in Figure 2 for comparison. The crossover between the Rouse scaling of the first cumulant from DLS and the relaxation rate from NSE is indeed observed in the range of 0.1 e q e 0.5 nm-1, in good agreement with a persistence length of lp ) 24 nm
Figure 2. Relaxation rate Γ(q) vs q for the dilution series of CTAB/ NaSal with a molar ratio of 1:2. Open symbols represent NSE and the filled DLS measurements (CTAB concentration: 12.5 mM (square), 25 mM (circle), 50 mM (up triangle), and 100 mM (down triangle)). The dashed line represents the scaling expected for bending modes of a semiflexible chain and the solid line represents the scaling expected for Rouse breathing modes. The crossover roughly corresponds to lp.
(qcross ≈ 2π/lp ) 0.26 nm-1). An additional contribution to the diffusivity at low q could originate from the sliding motion of branches as opposed to translational motion of a micellar segment. This motion is faster than translational diffusion10 and would thus lead to the observed trend in the relaxation rate from NSE at low q with surfactant concentration. This, however, remains inconclusive, since the data at low q are associated with rather large errors. The segmental diffusion coefficient DG was obtained by linear regression using eq 2. The resulting DG is shown in Table 1 as a function of surfactant concentration. The first important observation, which differs slightly from Seto et al.,6 is that the diffusion coefficient is nearly independent of surfactant concentration for the dilution series (see Table 1). Since all samples are viscoelastic and above the overlap concentration c*, this agrees with the observation and theory of Kato et al.11 The wormlike micelles used in these experiments are studied at high salt concentration, and therefore, the electrostatic interactions can be regarded as strongly screened. Thus, a close correspondence to the behavior of the nonionic wormlike micelles is expected. The dilution series presented above displays a continuous increase in viscosity (see Table 1). However, by varying salt, the viscosity passes two maxima (see Table 2). Thus, the dependence of DG on salt concentration at a fixed surfactant concentration of 25 mM was studied as well. Calculations of the change in lp based on the Odijk-Skolnick-Fixman (OSF) theory12,13 show (10) Constantin, D.; Freyssingeas, E.; Palierne, J. F.; Oswald, P. Langmuir 2003, 19, 2554. (11) Kato, T.; Taguchi, N.; Nozu, D. Prog. Colloid Polym. Sci. 1997, 106, 57.
Letters
Langmuir, Vol. 23, No. 10, 2007 5269
that lp is expected to only vary slightly (see Table 2). The persistence length can be written as the sum of an electrostatic and a steric contribution (lp ) l0p + lep). According to ref 6, the local diffusion coefficient can be calculated from lp using κ ) lpkBT, where κ is the bending modulus of the wormlike micelle and µ is the medium viscosity (i.e., water)
DG ) γ (kBT)4/3 κ-1/3 µ-1
(3)
As shown in Table 2, the measured DG’s are constant to within measurement accuracy for the salt series as expected, and independent of the zero shear viscosity of the samples (η0). The WLMs are transitioning from an entangled to a branched network as the salt concentration increases, as shown by the viscosity.14 The constant DG shows that network and junction formation does not significantly affect the segmental diffusion coefficient in agreement with reports for nonionic micelles by Kato and others.11 The latter is also expected since the length scales of observation are smaller than the mesh size of the system, and thus local dynamics probe the solvent viscosity. Equation 3 can also be used to obtain an estimate of the bending modulus κ of the wormlike micellar chain from the measured DG. The prefactor γ for semiflexible chains is obtained using the mean squared displacement for a semiflexible chain as given by Granek2 as
( [ ])
〈(∆h)2〉 ) 0.082 ln
( ) {( )
Lc t aπ 4πµa4
κ ln
3/4
kBT κ
1/3 k
BT
µ
}
3/4
t
(4)
A connection between the mean squared displacement and the diffusion coefficient is given by the Einstein relation
〈(∆h)2〉 ) 2d∆t
(5)
where d is the dimensionality of the problem. For the case of a semiflexible chain, segmental diffusion on short time scales is two-dimensional, hence
DG ) 0.0056 ln
( ) ( )
Lc t aπ (kBT)4/3 κ-1/3 µ-1 4πµa4
κ ln
(6)
By comparison with eq 3 the prefactor is
γ ) 0.0056 ln
( ) ( )
Lc t πa 4πµa4
κ ln
(7)
The prefactor is weakly dependent on κ and the experimental time t. As the NSE data is fit over the time window of 0.5-90 ns, an average time value is calculated as
tav ) exp
(∫
90
0.5
)
ln(t) dt
tmax - tmin
(8)
A sensitivity analysis shows that the results for the bending elasticity κ strongly depend on the choice of the lower cutoff length a, which is unspecified in terms of molecular parameters. Therefore, we use the measured persistence length of CTAB (12) Skolnik, J.; Fixman, M. Macromolecules 1977, 10, 944. (13) Odijk, T. Polym. Phys. B 1977, 15, 477. (14) (a) Lequeux, F. Europhys. Lett. 1992, 19, 675. (b) Lequeux, F Curr. Opin. Colloid Interface Sci. 1996, 1, 341.
Table 3. Lower Cutoff Length Calculated for All Samples Using the Experimental Values for DG, Lc ) 2000 nm, t ) 33 ns, µ ) 1 × 10-3 Pa s and lp ) 24 nm cD/cs, mM
a, nm
cD/cs, mM
a, nm
12.5/25 25/50 50/100 100/200
1.5 ( 0.1 1.5 ( 0.1 1.6 ( 0.1 1.8 ( 0.1
25/16 25/75 25/180 25/300
1.5 ( 0.1 1.3 ( 0.1 1.5 ( 0.1 1.6 ( 0.1
obtained from the stress-optic coefficient7 to calculate and identify the lower cutoff length a. Using κ ) kBTlp with lp ) 24 nm, DG ) 4.5 × 10-17 m8/3 s-1, µ ) 1 × 10-3 Pa s and t ) 33 ns leads to a ) 1.5 ( 0.1 nm. Table 3 summarizes the lower cutoff length calculated for all samples, showing it ranges from 1.3 to 1.8 nm. Hence, the lower cutoff length lies between the diameter of a surfactant headgroup (0.85 nm)7 and the cross-sectional diameter of the micelle (5 nm). This is reasonable, if one thinks of a wormlike chain composed of surfactant “discs” that can slide past each other. The same viewpoint is used to interpret the stress-optic coefficient.7 The shortest feasible wavelength is ≈ O(2 - 3) discs. Revisiting the calculation of lp of the nonionic system studied by Seto et al. now using a lower cutoff length of twice the headgroup size (a ) 1.5 nm) and the experimental time of 8.5 ns, yields a value of κ ) 3 × 10-28 N m2. This is a considerable improvement over the estimate reported by Seto et al., which are 2 orders of magnitude smaller, and is close to the value obtained from the measured persistence length. The discrepancy can easily be accounted for by a slightly higher value for the lower cutoff length, as for C16E7 the diameter of the headgroup is slightly larger than that for CTAB (for comparison, the headgroup diameter for C12E5 is 0.7 nm15). In conclusion, we show that cationic WLMs exhibit short time segmental dynamics in NSE experiments that are consistent with the predictions for semiflexible chains by Zilman and Granek over the concentrations of surfactant and salt studied here. Using independent measurements of lp, we identify the lower cutoff length a to be about twice the micellar headgroup diameter. Using this value for a, we reanalyze the experiments of Seto et al.6 and resolve their reported disagreement between NSE and light scattering measurements of κ. Therefore, we propose a calibration procedure for extracting κ for WLMs from NSE, which is similar to that of Richter, Monkenbusch, and co-workers for calibrating the bending elasticity of bicontinuous microemulsions against SANS measurements of the patch size ξ.16 Thus, NSE can be used to probe both the fast dynamics of wormlike micelles and, with the Zilman and Granek theory, becomes a method to measure κ. Acknowledgment. We acknowledge the support of the National Institute of Standards and Technology, U.S. Department of Commerce, in providing the neutron research facilities used in this work. This work utilized facilities supported in part by the National Science Foundation under Agreement No. DMR0454672. In particular we thank Antonio Faraone for helpful discussions. LA0635855 (15) Bagger-Jo¨rgensen, H.; Olsson, U.; Mortensen, K. Langmuir 1997, 13, 1413. (16) Monkenbusch, M.; Holderer, O.; Frielinghaus, H.; Byelov, D.; Allgaier, J.; Richter, D. J. Phys.-Condens. Matter 2005, 17, S2903.
