J . Phys. Chem. 1994, 98, 5481-5486
548 1
Excluded-Volume Effects and Internal Chain Dynamics in Giant Polymer-like Lecithin Reverse Micelles Peter Schurtenberger’ and Carolina Cavaco Institut f u r Polymere, ETH Zurich, CH-8092 Zurich, Switzerland Received: October 28, 1993; In Final Form: March 9, 1994”
W e report a detailed analysis of data from static and dynamic light scattering experiments with polymer-like lecithin water-in-oil microemulsions. W e show that a theoretical treatment of the static structure factor using concepts from polymer physics, which include chain statistics appropriate for excluded-volume interactions in semiflexible classical polymers, yields a quantitative description of the Q dependence of the scattering intensity I ( Q ) from the giant polymer-like lecithin reverse micelles present in these solutions. We demonstrate that we can study “internal” micellar chain dynamics and that the dynamic structure factor exhibits a universal behavior similar to classical polymer solutions.
Introduction It has previously been shown for a number of aqueous surfactant solutions that it is possible to find conditions where the micelles grow dramatically with increasing surfactant concentration into giant and flexible rodlike aggregates.’“ These wormlike micelles can then entangle and form a transient network above a crossover concentration c* with static properties that are comparable to those of semidilute solutions of flexible polymers. These solutions were then used as model systems for an experimental investigation of the static and dynamic properties of “equilibrium polymers”, where the term equilibrium (or “living”) polymer is used for linear macromolecules that can break and recombine. It was demonstrated that scaling theory for polymers could successfully be applied to semidilute micellar solutions and that a number of experimentally accessible quantities such as the osmotic compressibility, (XI/a@)-l, or the static correlation length, &,directly obey the same simple universal scaling laws as do classical polymers. It was also clearly shown that the understanding of the concentration dependence of dynamic properties was much more complicated due to the transient nature of the aggregates.( Lecithin water-in-oil (w/o) microemulsions represent a unique system which shows a characteristic sphere-to-rod transition normally observed in aqueous solutions We have previously been able to show that the addition of trace amounts of water to lecithin reverse micellar solutions induces onedimensional growth and the formation of giant flexible cylindrical aggregates. For lecithin/cyclohexane solutions at a given surfactant concentration, the micellar size increases with increasing water-to-surfactant molar ratio, W O ,and reaches a plateau at wo 1 14. At wo 1 10,the resulting micellar particles areextremely large even at very low surfactant concentrations. We have been able to demonstrate the close analogy between the structural properties of lecithin reverse micelles and classical semiflexible polymers using different scattering techniques such as static and dynamic light scattering and small-angle neutron ~cattering.~-l* The results from these scattering experiments were quantitatively interpreted using the wormlike chain model, and we were thus able to obtain estimates on the micellar flexibility (Le., persistence length), contour length, polydispersity, and local packing as a function of the solution composition. However, we recently found evidence that the wormlike chain model may not appropriately describe the dependence of the radius of gyration or the hydrodynamic radius as a function of the molecular weight or contour length for solutions where the water-induced micellar growth is particularly pronounced and that (intramicellar) Abstract published in Advance ACS Abstracts, May 1, 1994.
0022-365419412098-5481$04.50/0
excluded-volume effects should be taken into account.13J4 Here we now present a detailed analysis of the static and dynamic structure factor of lecithin reverse micelles at low concentrations c C c* and show how excluded-volume effects can be included in the micellar form factor. Since the lecithin system is oil-continuous and no complicating effects arise due to additional contributions from electrostatic interactions or salt effects, it serves as a good model system for structural and dynamic studies of equilibrium polymer solutions. Moreover, nonaqueous solutions are much easier to investigate with SLS and DLS at low scattering angles due to their relatively high index of refraction (good index matching with the scattering cell) and the fact that they can easily be made and remain dustfree (mainly due to their low dielectric constant), and we were thus able to obtain highly precise SLS and DLS data over an extended range of Qvalues. Since conditions can be found where giant micellar aggregates are formed even at low values of c, we can cover the regimes where QEs 1 and Q E h >> 1, and we can thus study overall coil conformation and dynamics as well as contributions from local (internal) dynamics. For this reason we have extended the experimental work to lecithin/isooctane solutions, where the water-induced micellar growth is even more dramatic and giant polymer-like micelles form at wo I1.5. While for this system we cannot work under conditions c 1. Typical examples for the Q dependence of D(Q) = (I’)/QZ, where (r)is the first cumulant obtained from a second-order cumulant analysis of the intensity autocorrelation functions, are shown in Figure 2 for the same samples already characterized by means of SLS (Figure 1). The dynamic light scattering
9
10.0-
I
9. 0
5.0-
0.04 0.000
0.002
0.1 04
Q [A”] Figure2. D(Q) = (r(Q))/@asa functionof Qfor lecithin/cyclohexane w/o microemulsions at wo = 8.0 and c = 4.57 mg/mL (A) and wo = 14.0 and c = 5.1 1 mg/mL (0)and for lecithin/isooctane w/o microemulsions at wo = 3.0 and c = 2.47 mg/mL (0).
experiments also confirm the water-induced formation of giant aggregates. For the lecithin/cyclohexane solutions, an extrapolation to Q 0 leads to values for the hydrodynamic correlation length of t h = 501 8, for wo = 8.0 and c = 4.57 mg/mL and of (h = 605 8, for wo = 14.0 and c = 5.11 mg/mL. For lecithin/ isooctane we obtain t h = 943 8,for wo = 3.0 and c = 2.47 mg/mL. We find two characteristic regimes for the Q dependence of (r)/ Q2. At Q[h 1 and QI, - - JN d2 n ( N - n ) J o m d r P , , ( rsin( ) ~ Q r ) (4)
I(0)
N2
where P,(r) is the probability density that elements separated by n segments along the chain contour are a distance r apart from each other in space.I9 Equation 4 assumes that P,(r) is a function of n alone. The exact distribution function Pn(r)is unknown, but several approximations have thoroughly been discussed in the literature. A frequently used model for the chain conformation of “realistic” polymer chains is the wormlikechain, for which reliable equations exist for P ( Q ) , ( R : ) , and Rh as a function of the persistence length I,, the contour length L, and the chain diameter d.20-22 In particular, the use of the first Daniels approximation for P,,(r)results in a quite accurate form for P(Q) which is valid for LII, I 20 and QI, I3.1 and which has been evaluated by Sharp and B l ~ o m f i e l d : ~ ~ , ~ ~
(5) where x = Ll,Q2/3 = (Rg2)Q*.A particularly sensitive way to test the agreement between theoretical and experimental P(Q) for semiflexiblechains is the so-called “Holtzer plot” (or “bending rod” plot), in which QP(Q) is plotted versus QR,.24 In such a plot, a horizontal asymptote is reached a t high Q values (Qlp> 1.9), whose magnitude is a measure of the mass per unit length M Lof the polymer independent of contour length polydispersity. At lower values of Q, the curve passes through a maximum and then decreases and approaches 0 a t Q = 0 due to the finite length of the polymer. The ratio of the height of the maximum to the height of the plateau depends on the ratio L/I, and polydispersity, and the location of the maximum is mainly a function of the p~lydistpersity.~~ An example of a Holtzer plot is given for lecithin/cyclohexane at wo = 14.0 and c = 5.11 mg/mL (see Figure 1) in Figure 3 . We do see the characteristic features of semiflexible chains, although the maximum is relatively weak and the Q range is too limited to reach the asymptotic plateau at Ql, > 1.9. We can now quantitatively compare the experimental data with the theoretical form of P(Q) given in eq 5. First, we have to take into account polydispersity. For polydisperse solutions, the experimentally determined values of ( P ( Q )) correspond to a z average given by
where w(L) is the weight fraction distribution and PL(Q) the particle form factor of a polymer of contour length L. We now have to chose an explicit form for the micellar size distribution. Based on multiple chemical equilibrium models or scaling arguments, an exponential size distribution of the form n(L) exp(-L/L,,,) which results in L,,,/Ln= 2, where L,, is the numberaverage and L,,, is the weight-average contour length, has been
-
0.0 0.0
1.0
2.0
3.0
4.0
Q,Rg
Figure 3. QP(Q) versus QRg for lecithin/cyclohexane, wo = 14.0 and c = 5.11 mg/mL. Also shown are theoretical curves for polydisperse wormlike chains with 1, = 110 A using a Schulz-Flory distribution (&/ L = 2.0) (--) and a Schulz-Zimm distribution with z = 0.5 and z = 0.175 (- - -) and for polydisperse semiflexible chains with excludedvolume interactions (-). See text for details. (-e)
predicted for micelles.’ Therefore, we have included polydispersity in our analysis using a Schulz-Flory “most probable distribution”, for which &/L, = 2.0.25 The Schulz-Flory distribution is a one-parameter distribution, and w(L) is uniquely determined by the corresponding value of &. Having chosen the form of the micellar size distribution, we can now calculate ( P ( Q ) )for polydisperse wormlike chains with a given R,. The mean-square radius of gyration of a wormlike chain of contour length L is given by20
where r = L / l p ,and the corresponding z average is then given by
[
R, = ~ w ( W R , ( L ) 2d L ) ]If2
L m w ( L ) Ld L
(8)
We have previously obtained an estimate of the persistence length of 1, = 110 f 30 8, from small-angle neutron scattering experiments.10 We can thus calculate ( P ( Q ) ) for the sample shown in Figure 3 using R , = 1 142 8, and eqs 5-8. A comparison between the experimental data points and the theoretical curve for the polydisperse wormlike chain model shown in Figure 3 reveals clear deviations. Whereas the low Q data are in good agreement with the model calculation due to the fact that it depends on the coil size (Le., R g ) only, the high Q part which mainly reflects the chain conformation is in clear disagreement. The theoretical model leads to a much more pronounced maximum and a faster decay at high Q values. It is known that an increase in polydispersity leads to a shift of the maximum and a broadening of the curve Q ( P ( Q ) )versus QRgnZ4Therefore, we have also used the more general Schulz-Zimm distribution
where y := ( z + l)/L,,,, &/L, = 1 + l/z, and r is the gamma function. The Schulz-Zimm distribution permits us to further study the effect of polydispersity on ( P ( Q ) ) ,and two additional curves for z = 0.5 and z = 0.175 (L,.,/L= 3.0 and L,.,/L,, = 6.7, respectively) are included in Figure 3 . It is clear from Figure 3 that even a very high polydispersity cannot explain the scattering data at QR, L 2 within the wormlike chain model.
5484 The Journal of Physical Chemistry, Vol. 98, No. 21, 1994
However, this is not surprising looking at the value of & required to obtain RB = 1142 A. For the Schulz-Flory distribution, we have to use a value of & = 24 000 A, Le., r, = &/Ip = 218. It is known that excluded-volume effects become important for r, > 100,24 which means that the first Daniels approximation which provides the basis for eq 5 is not a good description of the chain distribution function P,(r) in eq 4. There have been attempts to use a Gaussian distribution functionZ6 in eq 4 or to modify the first Daniels approximation in order to take intoaccount excluded-volume effects.23 However, we believe that these choices for the chain distribution function are not well founded on theoretical gro~nds.2~,~8 In our attempt to quantitatively describe the scattering pattern, we have therefore used thedistribution function for a self-avoiding random walk proposed by Domb, Gillis, and Wilmers on the basis of direct enumeration dataZ7
Schurtenberger and Cavaco
0.01 0.0
2.0
4.0
10.0
where C, is a normalization factor and u,, is a scaling factor.*9 Furthermore, we have used 6 = t = 2/(1 - e) = 2.5, where e is the excluded-volume exponent that describes the influence of excluded-volume effects on the molecular weight dependence of the mean-square end-to-end distance ( R 2 ) nI+' of a coil with n segments.28 At small values of Qr, the sine function in eq 4 can be expanded in a Taylor's series, and integration of eq 4 then leads t0*~+28
-
(6
P(Q) = 2 2 ( - 1 ) ' ( i=O
r([3
+ 2i-
(2i
I /---
(1 1)
in x = (R,2)Q2 is now given b~2~-30
I
""
I
\O
5.0
0.0i 0.0
+ l)t]/2)x'
r [ ( 3 - ~ ) / 2 ] [ ( 1 + t ) i + 1][(1 + t ) i + 2 ] r ( 2 i + 2 ) where (R:)
I
7
+ 5 t + t2)r[(3 - c)/2] r[(5- 34/21
6.0
2.0
4.0
6.0
Q.Rg
Figure 4. QP(Q) versus QRg for lecithin w/o microemulsions: (A) lecithin/cyclohexane,wo = 8.0 and c = 4.57 mg/mL (A)and wo 12.0 and c = 4.93 mg/mL (0); (B) lecithin/isooctane, wo = 2.5, c = 2.45 mg/mL (A)and c = 4.08 mg/mL (0) and wo = 3.0, c = 2.47 mg/mL ( O ) , respectively. Also shown are theoretical curves for polydisperse semiflexible chains with excluded-volume interactions as solid lines. See text for details.
Equation 11 is useful for numerical evaluations of P(Q) up to x I10. At higher values of x, we have used an approximation in the form of an inverse power series of x given in eq 20 of ref 19. We now have an analytical form for P,(Q) which we can use in the theoretical calculation of (P(Q)). The starting point is again eq 8, which we use in conjunction with eq 12 and the (one parameter) Schulz-Flory distribution in order to determine w(L) from R, = 1142 A and I, = 110 A. Having fixed w(L), we can now integrate PL(Q) given by eq 11 or eq 20 of ref 19 over the size distribution w ( L ) (eq 6) and obtain (P(Q)). The result of such a calculation with t = 0.2 is shown in Figure 3 as the solid line. We obtain excellent agreement between the experimental data points and the theoretical calculation for polydisperse semiflexible chains with excluded-volume effects and e = 0.2. The explicit form of P(Q) resulting from the Domb, Gillis, and Wilmers distribution function is further tested with scattering curves obtained for lecithin/cyclohexane and lecithin/isooctane solutions at different values of wo and c. The corresponding Holtzer plots and the theoretical curves are shown in Figure 4. Again we observe very good agreement between the experimental data and the theoretical curves. This indicates that the data for lecithin/cyclohexane and lecithin/isooctane solutions a t high values of wo and low values of c, where the static correlation length is largest, are consistent with the scattering pattern from polydisperse semiflexible chains with excluded-volume effects and an exponential size distribution with &I&,= 2. The Domb, Gillis, and Wilmers distribution function and a value o f t = 0.2
appear to yield a correct description of the micellar conformation, in close agreement with previous findings for synthetic polymers.lg However, it is important to point out that we have made no attempt to search for an 'optimal combination" of e and a more general choice of w(L) or use independent values of 6 and t in eq 10, since we believe that such atask would not be justified by the available data. Nevertheless, we can say that we now have an analytical form for the scattering curve over an extended range of QRg which permits the incorporation of polydispersity in a straightforward way and which goes well beyond simple asymptotic relationships or an empirical expression such as the one by Fisher and Burford previously used in order to analyze the intermediate scattering behavior of giant polymer-like aqueous micelles.5 Dynamic Light Scattering. In our discussion of the dynamic light scattering results, we shall concentrate on the Qdependence oftheinitialdecay ( r ( Q ) ) oftheintensity autocorrelation function as obtained for example using a second-order cumulant analysis of the data. For this quantity, theoretical expressions have been derived for various concentration and solvent regimes.18 At QI > 1 we primarily see the internal motion of the polymer chains. The dynamic structure factor is then characterized by a decay rate (I'(Q)) Q3 in the so-called Zimm model. The theoretical calculations predict an asymptotic behavior of (I'(Q)) = 0 . 0 6 2 5 ( k ~ T / ~ o ) for Q ~ theta solvents (v = 0.5) and (I'(Q)) = 0.0788(k~T/qo)Q~ for good solvents (v = 0.6).1* Generally, experimental values are -25% smaller than the theoretical predictions, which is believed to be due to the fact that the correct scaling regime is reached at very high values of M only.18 The data shown in Figure 2 do indeed demonstrate that we can examine both regions for polymer-like lecithin reverse micelles at low concentrations c < c*. At low values of Q, we observe a relatively weak Q dependence which is primarily a result of the micellar polydispersity. Under these conditions, the observed apparent diffusion coefficient corresponds to a z average of the form
-
where DL is the apparent translational diffusion coefficient of a polymer of contour length L. Under these conditions, an extrapolation to Q = 0 can be made with good accuracy, and the corresponding value of [ h can be determined. At higher values of Q, the Q dependence becomes more pronounced due to the increasing contributions from internal modes, and we finally observe a crossover to a Q3dependence typical for the regime QSh >> 1. We can now try to look more quantitatively at the Qdependence of ( I'(Q))/Q2. Dynamic scaling calculations predict that ( I'(Q))/Q2Dappshould be a function of Q&, only. For Qlh 0, this function F(&) should approach 1, whereas for Q& >> 1 it should approach the asymptotic regime for internal motion given by the Zimm model. We have thus replotted in Figure 5 the experimentally determined values of ( r(Q))/Q2 for various solutions in a plot of ( I'(Q))/Q2Dappversus Q&, where Dappand (h have been determined from extrapolations to Q 0. We indeed find that the experimental data fall onto a "universal"
-
-
mastercurveirrespectiveofthenatureofthesolventor thelecithin concentration. A similar behavior was previously reported for classical polymer solutions or aqueous solutions of ionic wormlike micelles at high ionic strength and c > c*.32-34 It is interesting to note that the experimentally determined master curve exhibits a perfect agreement with the theoretical expression of F(Q&), for theta solvents, Le., ideal Gaussian chain statistics (see dashed line in Figure 5).18 At first this is not consistent with the findings from the static light scattering experiments, where we observe good agreement between the static structure factor and the theoretical predictions for polymer chains in good solvents with excluded-volume effects and t = 0.2 (or v = 0.6). However, this could be another indication for the fact that the correct scaling behavior for good solvents is obtained a t very highvalues of Monly and is in agreement with theobservation made with synthetic polymers that the experimental values of (I'(Q)) are generally too low when compared to the theoretical predictions for the Zimm model, We also find no indication that the invariance of the prefactor in the Q3 dependence resulting from internal modes could beviolated and exhibit a concentration dependence as indicated in a previous publication by Appell and Porte for aqueous ionic micelles.5 Conclusions A detailed analysis of the osmotic compressibility measured in an extensive SLS study for lecithin cyclohexane solutions over a broad range of volume fractions using polymer renormalization group theory had previously indicated that for very large micellar sizes excluded-volume effects need to be taken into account.13 Our current static light scattering study provides now a firm basis for the application of the relation R, M O . 6 to these s o l ~ t i o n s . ~ 3ItJ demonstrates ~ that a theoretical treatment of the static structure factor which includes chain statistics appropriate for excluded-volume interactions yields a quantitative description of the Q dependence of the scattering intensity Z(Q) from giant polymer-like lecithin reverse micelles. The present work also shows that the dynamic structure factor exhibits a universal behavior when plotted as ( l?(Q))/Q2Dap, versus QSh. However, an analysis of the thus obtained master curve shows that, while the data are in agreement with an asymptotic Q3 dependence expected for internal dynamics in the Zimm model, the quantitative values are not in agreement with the good solvent prediction but correspond to the case of Gaussian chains. Our study thus shows that the use of the wormlike chain model could underestimate the dependence of the radius of gyration on the molecular weight and lead to errors in the determination of the persistence length from a combination of static and dynamic light scattering if the model is applied to data from samples with very pronounced micellar growth.
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Acknowledgment. This work was supported in part by the Swiss National Science Foundation (Grant 21-37'274.93) and by the Portuguese National Fund for Technological and Scientific Research (Grant BD/1256/91-IC). References and Notes (1) Cates, M. E.; Candau, S. J. J . Phys.: Condens. Marter 1990, 2, 68694892. (2) Candau, S. J.; Hirsch, E.; Zana, R. J . Colloid Znterface Sci. 1985, 105, 521-528. (3) Candau, S. J.; Hirsch, E.; Zana, R.; Adam, M. J . Colloid Interface Sci. 1988, 122, 430-440. (4) Marignan, J.; Appell, J.; Bassereau, P.;Porte, G.; May, R. P. J . Phys. (Paris) 1989, 50, 3553-3566. ( 5 ) Appell, J., Porte, G. Europhys. Lett. 1990, 22, 185. (6) Imae, T. Colloid Polym. Sci. 1989, 267, 707-713. (7) Schurtenberger, P.;Scartazzini, R.; Luisi, P.L. Rheol. Acta 1989, 28, 372-381. (8) Schurtenberger,P.;Scartazzini, R.; Magid, L. J.; Leser, M. E.; Luisi, P.L. J . Phys. Chem. 1990, 94, 3695-3701.
5486 The Journal of Physical Chemistry, Vol. 98, No. 21,.1994 (9) Schurtenberger, P.; Magid, L. J.; Penfold, J.; Heenan, R. Langmuir 1990,6, 1800-1803. (10) Schurtenberger, P.; Magid, L. J.; King, S.;Lindner, P. J. Phys. Chem. 1991, 95,41734176. 1 1) Schurtenberner. P.: Maaid. L. J.: Lindner, P.; Luisi, P. L. Pron. - Colloid Pol)m.. Sci. 1992, 8G, 274-277: (12) Schurtenberger, P.; Peng, Q.;Leser, M.; Luisi, P. L. J. Colloid Interface Sci. 1993, 156, 43-51. (13) Schurtenberger, P.; Cavaco, C. J. Phys. II 1993, 3, 1279-1288. (14) Schurtenberger, P.; Cavaco, C. Lungmuir 1994, IO, 100-108. (1 5) The original phase diagram studies were made at 20 O C , and we can
slightly shift the phase boundary for liquid-liquid phase separation in lecithin/ isooctane solution when working at T = 25.0 OC. Under these conditions, we can work at concentrations c 2 1.6 mg/mL for wo = 2.5 and c 2 2.5 mg/mL for wo = 3.0. (16) Schurtenberger, P.; Augusteyn, R. C. Biopolymers 1991,31, 12291240. (17) Koppel, D. E. J . Chem. Phys. 1972,57,4814-4820. (18) Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, 1986. (19) Utiyama, H.; Tsunashima, Y.; Kurata, M. J. Chem. Phys. 1971,55, 3 133-3 145. (20) Yamakawa, H. Modern Theory of Polymer Solutions; Harper & Row: New York, 1971.
Schurtenberger and Cavaco (21) Yamakawa, H.; Fujii, M. Macromolecules 1973. 6, 407-415. (22) Yoshizaki,T.;Yamakawa,H.Macromolecules 1980,13,15 18-1525. (23) Sharp, P.; Bloomfield, V. A. Biopolymers 1968,6, 1201-1211. (24) Denkinger, P.; Burchard, W. J. Polym. Sci. B, Polym. Phys. 1991, 29, 589-600. (25) Peebles, J. L. H. Molecular Weight Distributions in Polymers; Interscience Publishers: New York, 1971. (26) Loucheux, C.; Weill, G.; Benoit, H. J . Chim. Phys. 1958,55,540546. (27) Domb, C.; Gillis, J.; Wilmers, G. Proc. Phys. SOC.(London) 1965, 85, 625. (28) Sharp, P. A.; Bloomfield, V. A. J . Chem. Phys. 1968.49,4564-4566. (29) Domb, C. J. Chem. Phys. 1963.38, 2957-2963. (30) Peterlin, A. J . Chem. Phys. 1955, 23, 2464-2465. (31) Brown, W.; Nicolai, T.Colloid Polym. Sci. 1990, 268, 977-990. (32) Nemoto, N.; Kuwahara, M. Langmuir 1993, 9, 419-423. (33) Wiltzius, P.; Cannell, D. S.Phys. Rev. Lett. 1986, 56, 61-64. (34) However, a close comparison of our Figure 5 and Figure 7 from ref 32 clearly reveals the advantage of reverse micellar solutions as model systems
for a quantitative test of polymer theories, since the organic solventsare ideally suited for very precise DLS and SLS measurements at low scattering angles.
