J. Phys. Chem. 1992, 96,60954102
6095
Ultrasonic Absorption Studles of Surfactant Exchange between Mlcelles and Bulk Phase in Aqueous Micellar Solutions of Nonlonlc Surfactants with a Short Alkyl Chain. 2. C&39 C&59 C8E4r and C8E8 M. Frindi, B. Michels, Laboratoire d'Ultrasons et de Dynamique des Fluides Complexes (LUDFC), URA 851 du CNRS, UniversitZ Louis Pasteur, 4, Rue Blaise Pascal, F-67070 Strasbourg Cedex, France
and R. Zana* Institut Charles Sadron (CRM-EAHP), CNRS-ULP Strasbourg, 6, Rue Boussingault, 67083 Strasbourg Cedex, France (Received: December 18, 1991; In Final Form: March 24, 1992)
Spectrofluorimetry (with pyrene as a probe), density and ultrasonic absorption measurements, and static and time-resolved fluorescencequenching have been used to obtain the critical micelle concentration (cmc), the isothermal volume change upon micellization and the micelle aggregation number N of the ethoxylated nonionic surfactants C6E3,C6Es,C&, and CsE8 (tri- and penta(ethy1ene glycol) monohexyl ethers and tetra- and octa(ethy1eneglycol) monooctyl ethers, respectively). The kinetics of the exchange of these surfactants between micelles and bulk phase has been investigated by the ultrasonic relaxation method. The results obey the predictions of Aniansson and Wall theory for micellization kinetics ( J . Phys. Chem. 1975, 79, 857). The full analysis of the relaxation data on the basis of this theory showed that the association of the surfactant to micelles is nearly diffusion-controlled, as for ionic surfactants. The rate constant k- for the dissociation of the C,E, surfactants from micelles decreases by a factor of about 10 for i increasing from 6 to 8 and can be taken as 5 X lo9 cmc, to within a factor of 2. The results also show that the effect of temperature on the rate constants is not large and that the micelle polydispersity is low. The volume changes determined from density measurements and from the ultrasonic relaxation amplitudes are in good agreement. Finally in the case of C6E3solutions where the measurements extend up to only 5 'C below the critical temperature, the results showed no contribution of the critical fluctuations of micelle concentration to the ultrasonic absorption.
Introduction In the past 20 years, a large number of studies have been reported, dealing with the dynamics of surfactant exchange between micelles and the bulk aqueous phase (intermicellar solut i ~ n ) . ' - Since ~ 1975, such studies have been usually interpreted in terms of Aniansson and Wall theoretical treatment4 or its improved version^^*^-' for the surfactant exchange. A survey of the literature until 1990 revealed only two ~ t u d i e sof ~ -the ~ dynamics of the exchange process in micellar solutions of nonionic surfactants. The first one dealt with Triton X-100, and the results were analyzed in terms of Aniansson and Wall theory.4 It was concluded that the rate constant k+ for Triton X-100 incorporation into its micelles is nearly diffusion-controlled (k+ N 2 X lo9 M-' s-l) and that the micelle polydispersity is small. Thex two results are very similar to those found for the large number of ionic surfactants investigated.' Also the rate constant for the exit of Triton X-100 from its micelles was reported to be about lo6 s-' at 25 OC? The second studygused ethoxylated surfactants, but the results were only of preliminary character. This paucity of results concerning the dynamics of surfactant exchange in solutions of nonionic surfactants led us to undertake the present studies which concern the effect of the surfactant chain length and nature of its head group (polyol, polyoxyethylene, sugar). Part 1 in this series reported ultrasonic relaxation data for the surfactant-like compounds: 1,2-hexanediol(1,2-HD) and 1,2,3octanetriol ( 1,2,3-OT).'O The results confirmed the validity of Aniansson and Wall treatment4 and yielded values of k+ close to the diffusionantrolled limit, whereas k- decreased in going from 1,2-HD to the longer chain 1,2,3-0T, as expected. Notice however that both the head group size and the hydrocarbon chain length are increased in geing from 1,2-HD to 1,2,3-OT. The present investigation deals with four poly(ethy1ene glycol) monoalkyl ether surfactants C,E, (where i is the number of carbon atoms in the alkyl chain and j the number of oxyethylene units), *To whom correspondence should be addressed.
0022-3654/92/2096-6095$03.00/0
namely, C&, C a 5 , c&4, and C a b . Short chain surfactants were selected in this series for the reasons given previously.1° These four surfactants permit a study of the effect of the size of the head group and of the hydrophobic moiety on the dynamics of the surfactant exchange. Besides, the critical temperatures of C6E3and CsE4are fairly close to room temperature (see below). This fact made it experimentally easy for us to check whether the approach of the critical temperature gave rise for these systems to an ultrasonic absorption contribution associated to the coupling between the ultrasonic waves and the fluctuations of micelle concentration. This problem has been addressed in recent papers dealing with the ultrasonic absorption of aqueous solutions of l-butanolll and various C,EP9J2 The bulk of the study reported below involved ultrasonic absorption measurements in order to obtain the relevant kinetic data. However, for a more complete interpretation of the results and/or independent checks, we have also determined the cmc (critical micellar concentration) of the surfactants by spectrofluorimetry'0J3J4and ultrasonic absorpti~n'~ and the isothermal volume change upon micellization from density measurements.16 Also the micelle aggregation numbers were obtained by using static and/or time-resolved fluorescence qu~nching.'~J'J~
Experimental Section Mntedak was purchased from Fluka with a purity higher than 98% and used as received. The samples of C6E3,c6E4,and C& were generous gifts from Dr. E. Platone (Enricerche, Milan, Italy). These surfactants have been purified by molecular distillation. Nevertheless we noted that the C6E3solutions became turbid close to the cmc. This surfactant was therefore further purified by washing with ether a submicellar aqueous solution of C6E3and separating the surfactant contained in the lower phase by distillation. Notice that from our experience a small content of impurities, to the extent of 1-2 mol %, d o a not appear to affect much the ultrasonic absorption or the micelle aggregation number at surfactant concentrations well above the cmc. On the contrary, 0 1992 American Chemical Society
6096 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992
surfactant
T ("C)
C6E3
15 20 25 40 25 40 15 20 25 40 25 40
C6ES
CBEB
b
C
105 94
97
109
103
12.4 10.4 8.0 9.7
Frindi et al.
d 100 96 92 72 94 68
AvO,
100,' 100-105,
9.6 f 1.5 10.0
115,
8.0 9.3
e-g
107c
1.5
8.49
10.4 8.2
10.2 f 1.5
'Values in cm3/mol at 25 OC. bThiswork, from the 11/13vs C plots. 'This work, from the fit of eq 2 to the aVvsC data. dThis work, from the a/f vs C plots at a frequency close to 0.7 MHz. CFromref 25 (surface tension). 'From ref 9 (a/yvs C plots at frequency above 5 MHz). XFrom ref 24 (light scattering). properties such as the cloud temperature can be very sensitive to small amounts of impuritie~.'~ Methods. Density Measurements. They were performed at 25 f 0.01 OC by means of an automatic densimeter (Anton Parr DMA 60)with an estimated accuracy of f5 X lo4 g/cm3. The densimeter was calibrated using air and water (do = 0.997 047 g/cm3). The apparent molal volume was calculated from
M is the surfactant molecular weight, d the density of the solution, and C the surfactant concentration. The isothermal volume change upon micellization A@ was obtained by fitting the equation16 =
+ AyOT(C - cmc)/C
v?
5
I
1.6
1.4
(2)
to the data, using a least-squares procedure. aV,- is the apparent molal volume at the cmc. This quantity is experimentally available by measurements below cmc and/or can be evaluated with good accuracy from reported results (see below). Spectrofluorimetry. In these measurements pyrene was used as a probe at a very low concentration of 1-2 pM, which ensured the absence of e ~ c i m e r . ' ~ . ' The ~ - ' ~fluorescence emission spectra of aerated surfactant solutions were obtained with a Hitachi F4010 spectrofluorimeter at an excitation wavelength of 335 nm. The spectra were used to determine the ratio 11/13 of the intensities of the first and third vibronic peaks of monomeric pyrene. This ratio is sensitive to the microenvironment of pyrene. Its variation upon micellization or aggregate formation reflects the change of polarity sensed by pyrene when it becomes partly or wholly solubilized within the aggregate^.'^^^^^^^ The changes of 11/13with the surfactant concentration were thus used to obtain the cmc of the surfactants. Spectrofluorimetry was also used to obtain the aggregation number of CbE3and CLESmicelles, using the static fluorescence quenching method.I8 In this method one solubilizes in the investigated micellar solution a fluorescence probe P and a fluorescence quencher Q which are partitioned in the micelles, at concentrations [PI and [Q] such that [P]/[M] cmc. However, the only data point below the cmc, determined for these two surfactants, is well above the extrapolation of the linear part. As previously emphasized,I0a similar behavior appears to characterize all short chain ionic and nonionic surfactants having a high cmc, as well as aqueous solutions of alcohols and related compounds where the self-associationprocess is much like a m i c e l l i i t i ~ n .In~ part ~ 1lo we have assigned this
Nonionic Surfactants with a Short Alkyl Chain
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 6099
TABLE HI: Vdw of A , B, and fa for the S o l ~ tInvestigated i~~ 3.1. C6EI
C(mM)
u
130 160 200 240 280
33 32 50 48 58
1019?(m-l s2) b C 29 32 35 34 40 39 46 38 53 38
d 19 24 27 38 44
iOi5A(m-I b 16550 14340 10040 7000 5070
U
8170 16840 13830 9 290 5 870
s2) C
9190 9580 7390 5180 4350
d 6660 5670 4480 3830 3060
U
0.58 0.76 1.13 1.90 2.72
lo4fR (Hz) b C 0.67 1.07 0.94 1.26 1.47 1.78 2.20 2.50 3.17 3.02
d 1.42 1.82 2.34 2.79 3.45
3.2. C6E5
C(mM) . . 130 160
109 (m-1 s 2) c d 27 22 32 28
1 0 9
10”A C
8520 8940
(ds2) d 5370 3800
(m-I s2)
104f~ (Hz) C d 1.23 2.11 1.63 2.89
C(mM) . . 180 200
c 34 37
d 35 32
104fR (Hz) C d 1.99 3.47 2.47 4.50
lO”A (m-’ s2) C d 8000 3500 6100 2550
3.3. C8E4
c (mM)
1015B(m-l s2) C
20 30 40
23 23 25
1015A (m-l s2) c 4000 4220 2700
10i5B(m-I s2)
lOI5A (m-l s2)
C
C (mM)
C
C
0.25 0.33 0.53
60 100
26 27
2420 1870
104fR (Hz)
104fR (Hz) c 0.71 1.16
3.4. CaEB 1015R (m-I s2)
C(mM)
c
20 25
26 27 25
30
d 15 17 17
1015~
10ISA (m-l s2) C d 3640 1720 2940 1710 3240 1570
104fR (Hz) c d 0.31 0.59 0.44 0.71 0.50 0.91
(m-I s2)
C(mM) 40 50
c 27 27
d 18 19
lOI5A (m-’ s2) C d 3140 1460 2710 1250
104fR (Hz) c d 0.68 1.15 0.88 1.52
“15 “C.*20 “C. ‘25 “C.“40 “C.
TABLE I V V d w s of k-/u2,k - / N , k-, k + , u, and AC for the Investigated Surfactants T (“c) CLEP . .
C6E5
C8E4 C8E8
15 20 25 40 25 40 25 25 40
1Odk-/a2(s-’) 2.01 1S O 2.70 4.01 3.40 2.60 0.73 0.96 1.33
lO-’k-/N (s-’) 0.50 0.70 0.79 0.60 1.03 1.16 0.57 1.19 1.58
submicellar process to the reversible formation of oligomers. At higher C, slightly above the cmc, the contribution of the oligomers vanishes as they are used up to form the micelles proper. It is noteworthy that the amplitude of the submicellar process is very small compared to that at C > cmc and decreases as the cmc decreases. For instance, for the c8Ej surfactants investigated in this work, the contribution of the submicellar process could hardly be evaluated owing to its small value. This process will not be discussed further in this paper. Table 111 shows that the values of fR are lower for the c&, than for the C6Ej surfactants, as expected on the basis of the effect of the surfactant chain length of the kinetics of the exchange process: the longer the chain, the slower the exchange.’ Note that the fR values for the C8Ej surfactants are close to the value of the lowest frequency investigated (0.55 MHz). The error on these values may reach 20?6 while that for the C6Ej surfactants is estimated to be of about 10%. The ultrasonic relaxation parameters (A, fR) have been analyzed as in part 1.lo Recall that AfRz is proportional to C - cmc 38 and that fR increases linearly with the reduced concentration (C cmc)/cmc,4 as indicated by eqs 6 and 7. Parts A-D of Figure 9 show that indeed the experimental points in the plots of AfR2 vs C and fR vs (C - cmc)/cmc fall on straight lines (the cmc values used were those obtained from the a/f vs C plots). The solid lines going through the results were obtained by least-squares fits of eqs 6 and 7 to the data. In the case of the AfR2vs C plots, the
1LY8k-( P I ) 2.6 3.9 4.5 6.0 5.5 6.4 0.84 0.86 1.10
104k+ (M-l s-I) 2.5 4.0 4.9 8.0 5.8 9.4 11.0 8.3 14.0
Q
Ae (cm3 mo1-l)
11
15 13 12 13 15
11 9 9
10.0 10.3 9.5 10.6 10.6 10.5 11.3 11.0 11.2
fits have been performed by forcing the straight lines to extrapolate to zero at C = cmc a2 C - cmc
du k-(Ac)’(C - cmc) (7) RTN In eqs 6 and 7,k- is the rate constant for the exit of a surfactant from the micelles proper, B is the standard deviation of the distribution of aggregation number, d is the density of the solution (g cm-j), v is the velocity of ultrasound in the solution (cm s-I; measured as part of this work), R is the gas constant (8.32 X lo7 erg mol-’ deg-I), T i s the absolute temperature, and AC is the isentropic volume change upon micellization (cm3 m ~ l - ’ ) . ~ , ~ ~ The fits of eq 6 to the fR vs (C- cmc)/cmc plots yielded the values of k - / d and k-/N listed in Table IV. The values of N in Table I1 permitted us to obtain the values of k-, 2, and k+ = k-/cmc,’ k+ being the rate constant for the association of a surfactant to a micelle proper. Finally the values of for the investigated surfactants were obtained from the values of k-/N and of the slope of the AfR2vs C plots. All numerical results are listed in Table IV. Several remarks must be made. (i) The values of k-/N are nearly 1 order of magnitude smaller for the c8Ej surfactants than for the C6Ej surfactants. Thus the residence time’ N/k- of a surfactant in a micelle greatly increases AfR2 = 0.05-
Ae
6100 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 0.5
C 50
(C-CMC)/CMC 1.0 1.5 I
2.0
:
Frindi et al. (C-CMC)/CMC
>
2.5
I
I
A 3.5
40 2.0 h
I
E
v
h
h
p
X
N
30
2.5
0:
N
B
v
L: 4
1.52
b.
20 1.5 10
0
0
100
200
0.5 300
C(mM)
3
0
(C-CMC)/CMC 5 10
15
D 1 .o
/.
0.8 h
7
h
!i? E
2
h
Nv
B
v
1
0.6
U
b.
2*!
0.5 1
0.4
0
Figure 9. Variations of AfR2 with C (0)andfR with (C - cmc)/cmc
(X)
with the surfactant chain length, roughly by a factor of 3 per additional methylene group. Of course the values of k- follow the same trend as the k-/N values since the N values for the two series of surfactants do not differ much. The value k- = 1.1 X lo6 s-l reported for Triton X-10O8is 10 times lower than for the C8Ej surfactants. In view of the change of k- with the length of the surfactant alkyl chain, Triton X-100thus behaves like a C& surfactant as far as surfactant exchange is concerned. This is also true for the cmc of this surfactant which is very close to 1 mM.48 The Triton X-100hydrophobic moiety is (CH3)3CH2C(CH3)2C6H47 A phenylene group (-C6H4-) is known to be equivalent to a linear alkyl chain with 3.54 carbon atoms.49*50Thus the branched octyl chain is equivalent to a linear alkyl chain with 6-6.5 carbon atoms. It is realized that Triton X-100is a commercial surfactant with a large polydispersity in the poly(oxyethy1ene) moiety. However our results show that the effect of the poly(oxyethylene) moiety is small. So this polydispersity may not much affect our conclusion. (ii) The values of k+ are all well above lo9 M-' s-l . This indicates that the process is nearly diffusioncontrolled, as reported for the nonionic surfactant Triton X-100." (i) The values of $/Nwhich provide a measure of the micelle polydispersity are small, between 0.02 and 0.06 for the C6E, surfactants and even smaller for the CsE, surfactants. Thus the micelles of the investigated surfactants show little polydispersity, and this polydispersity tends to decrease as the surfactant chain length is increased. A low polydispersity was also found for Triton x-100.8 (iv) The effect of temperature on k+, k-, u, and A e is generally
0.2
for CsE3 (A), C6ES(B), CsEl (C), and CsEs (D), at 25 OC.
weak. The large errors on k+ and k-, particularly for the c8Ej surfactants and the restricted T-range investigated, did not allow meaningful values of the activation energies to be evaluated from the data, except for C6E3. For this surfactant the activation energy for k+ was found to be 5.9 2 kcal/mol, that is, a value close to that characterizing a diffusion-controlled process in water. Results i-iv are very similar to those for ionic surfactants. Thus the short chain ethoxylated nonionic surfactants C,E, investigated here behave very much like ionic surfactants as far as kinetics of surfactant exchange is concerned. The fact that the rate constant for the association of ethoxylated surfactants to micella is nearly diffusion-controlled may look surprising. Indeed one would have expected the oncoming surfactant to experience some steric hindrance when crossing the layer of poly(oxyethy1ene) moieties E, coating the micelle hydrophobic core. Also this hindrance should have i n c r d with the number j of oxyethylene units. That these expectations are not borne out by the results reveals a remarkable mobility of the poly(oxyethy1ene) moieties in the micelle head group layer which permits it to be easily crossed by the alkyl chain of an oncoming surfactant. The conclusion that k+ is diffusion-controlled for the four ethoxylated surfactants investigated in this work and for Triton X-100s probably holds also for nonionic ethoxylated surfactants with longer alkyl chains, linear or branched. This allows one to obtain directly k- from the value of the cmc of the surfactant without having to undertake complicated and often tedious kinetic studies. Indeed, using the average value k+ 5 X lo9 M-'s-' (see Table IV and ref 8), one can readily obtain k- from cmc = k-/k+; that is
*
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 6101
Nonionic Surfactants with a Short Alkyl Chain
k- = 5 x 1O9cmc
(8)
where the cmc is expressed in mols/liter and k- in reciprocal seconds. The values of in Table IV are seen to differ only slightly from those of APTin Table 111. Recall that these two quantities are related as follows5'
Ae
AE
=a
e q - -AHQ
(9)
dCP where 8 is the coefficient of thermal expansion, Cp the specific heat of the solution, and A iP the enthalpy change upon micellization. can be evaluated from the change of cmc with T according to52
AH" = R[d In (cmc)/d(l/T)]
(10)
Equation 10 holds for surfactants characterized by N values which depend only a little on T.52This appears to be the case of C6E3 around 20 OC. For this surfactant the cmc data in Table I yielded AI+' = +2.3 f 0.5 kcal/mol. The enthalpic term in eq 9 thus amounts to less than 0.5 cm3/mol, Le., smaller than the experimental error on or AV$ This explains why the values of these two quantities are nearly equal. The last point to be discussed concerns the possible contribution of critical effects to the ultrasonic absorption of C6E3solutions. Indeed the measurements were performed up to 40 OC and C = 280 mM, whereas the critical temperature and concentration are 44.7 OC and 600 mM.26 Recall that the ultrasonic relaxation method has proved to be the most sensitive one for the detection of critical fluctuations in binary mixtures of liquids. A contribution arises from the coupling of concentration fluctuations and ultrasounds even at temperatures well below T,and concentrations not too close to the critical co~~centration.~*~~ For C6E3, the values of the relaxation amplitude, of the relaxation frequency, of the constant B, and of the aggregation number N show no special feature which would m e a l the approach of the critical conditions. Thus these results confirm previous ones for other aqueous solutions of surfactants and microemulsions where the approach of the critical conditions did not result in an additional excess ultrasonic a b s o r p t i ~ n . ~A. ~recent ~ studyI2 has shown that such a contribution arises only very close to T,, when T, - T < 1 K. This difference of behavior between liquid mixtures and amphiphilic systems at the approach of the critical conditions may be related to the fact that in the former the interactions take place between molecules whereas in the latter they involve supramolecular assemblies. As previously pointed such interactions do not affect the environment of the surfactant and therefore its thermodynamic properties, as it remains in the micelles or microemulsion droplets. For instance, the surfactant molar volume changes only a little at the approach of T, 57 (see also the values for C6E3in Table IV). The relaxation amplitude of the ultrasonic contribution of critical fluctuations involves the correlation lengtb of the fluctuations (which covers a broad spectrum) and also the changes of thermodynamic parameters associated with these fluctuations. It is because the latter are very small, as just discussed, that no contribution of the critical fluctuations was detected in the present study.
Ae
Ae
Conclusions The theory of Aniansson and Wall has been shown to apply to the kinetics of exchange of the ethoxylated nonionic surfactants C&, C6E5, C8E4, and C& between micelles and bulk phase, as in the case of nonionic diol and triol surfactants investigated in part 1 in this series. The association of these surfactants to the micelles is nearly diffusion-controlled. It results that the cmc allows a good estimate of the rate constant of surfactant dissociation from the micelles. The volume changes upon micellization obtained from direct density measurement and from the ultrasonic relaxation data are in good agreement. They are larger than those reported for ionic surfactants with the same chain length. Finally the results show no evidence of a contribution of critical fluctuations of micelle concentration to the excess ultrasonic absorption
even at temperatures only 5 degrees below the critical temperature. Re&* NO. C6E325961-89-1;C6E5, 86674-95-5;CUE,, 19327-39-0; CgEu, 19327-42-5.
References and Notes (1) Lang, J.; Zana, R. In Surfactant Solutions: New Methods of Investigation; Zana, R.; Ed.; Dekker: New York, 1987;Chapter 8. (2) Kato, S.;Nomura, H.; Zielinski, R.; Ikeda, S.J. Colloid Interface Sci. 1991, 146, 53 and references cited therein. (3) Wan-Badhi, W.; Palepu, R.; Bloor, D. M.; Hall, D. G.; Wyn-Jones, E. J. Phys. Chem. 1991, 95, 6642 and references cited therein. (4)Aniansson, E. A. G.; Wall, S.J. Phys. Chem. 1974, 78, 1024;1975, 79, 857. (5) Almgren, M.; Aniansson, E. A. G.; Holmaker, K. Chem. Phys. 1977, 19, 1. (6)Lessner, E.;Teubncr, M.; Kahlweit, M. J. Phys. Chem. 1981,85,3167. (7)Hall, D. G. J. Chem. Soc., Faraday Trans. I 1981, 77, 1973. (8)Hoffmann, H.; Kielman, H. S.;Pavlovic, D.; Platz, G.; Ulbricht, W. J . Colloid Interface Sci. 1981, 80, 237. (9)Borthakur, A.; Zana, R. J. Phys. Chem. 1987, 91, 5957. (10)Frindi, M.; Michels, B.; Zana, R. J. Phys. Chem. 1991, 95, 4832. (11) Zana, R.; Michels, B. J. Phys. Chem. 1989, 93, 2643. (12)Baaken, C.; Belkoura, L.; Fusenig, S.;Muller-Kirschbaum, Th.; Wcermann, D. Ber. Bunsen-Ges. Phys. Chem. 1990,94, 150. (13) Kalyanasundaram, K.; Thomas, J. K. J. Am. Chem. Soc. 1977,99,
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Bending Energy and Relatlve Stability of Micellar Forms G.Taddeit and L.Q. Amaral' Instituto de Ffsica, Uniuersidade de S ~ Paulo, O Caixa Postal 2051 6, Siio Paulo, Brazil (Received: March 13, 1992)
Calculationsof bending energy of two different micellar forms (prolate ellipsoid and spherocylinder) were performed using the Hyde's model in order to evaluate the relative stability of both kinds of micelles in a large range of micelle length. These calculations show that (i) prolate ellipsoids are more stable than spherocylinders at lower micelle lengths, and (ii) there is a "transition" point where both kinds of micelles have the same bending energies. These results are used in the interpretation of experimental results of X-ray scattering in concentrated micellar solutions near the isotropic-hexagonal phase transition.
Introduction In the last years a growing interest has been devoted to the calculation of elastic bending energy of various amphiphilic aggregates (lamellae, micelles, vesicles) of colloidal systems. The bending energy was considered as the leading term to the stability of these aggregates, if torques and tearing were excluded. There are two dif€crcnt phenomenological models for calculating the bending energy per unit area, . The former model was proposed by Canham' and Helfrich. They define g in terms of the two principal curvatures of the aggregate surface. In the latter model? g is defmtd in tenns of the surfactant parameter p = u/al of the aggregate, where a, u, and 1 are respectively the surface area of the polar head, the volume, and the average length of the hydrocarbon chain of the amphiphilic molecule. In a recent paper Fogden, Hyde, and Lundberg4 showed that the two phenomenological models are identical in the leading term, differing only in their treatment of the Gaussian curvature term: this term is linear in the Canham-Helfrich model, whereas it is quadratic in the Hyde model. When discussing micellar shapes deviating from sphericity, it is usual to consider ellipsoidals as well as spherOcylindricaF shapes. However the possibility of a specific change from ellipsoidal to spherocylindrical shape due to minimization of bending energy has not been 50 far discussed. The interest for such approach was rised by the recent study on the isotropic-hexagonal transition in the system sodium dodecyl (lauryl) sulfate (SLS)/H,O. It was found from analysis of X-ray data' the possibility of both forms, with different but low anisometries in concentrated micellar isotropic solutions. Since in the hexagonal phase micelles are expected to be cylindrical, the question arises whether the deviation from sphericity implies a direct growth from sphere to spherocylinder or the prolate ellip soidal shape occurs in between. Or, in more general terms, whether both prolate ellipsoids and spherocylinders are present in a distribution of sizes and shapes. The main goal of this paper is to understand in terms of the bending energy the structural information which was collected from the experimental data on this kind of transition.
f
Bending Energy and Forms of Micelles Hyde's mode13s4appear to us to be the more suitable for the calculations of bending energy in micelles. In fact g contains such quantities as the actual surfactant parameter and the spontaneous surfactant parameter of the micelles which have a simple and 'Resent address: Dipartimentodi Chimica, UniversitH di Firenze, Via G. Capponi 9, 50129 Firenze, Italy.
direct physical meaning. The bending energy per unit area is given by g = k b - Pol2
where k is a constant which has dimensions of an energy/surface, p = (u/al) is the actual value of the surfactant parameter of SLS molecule in the micelle, at that SLS concentration, and po 3: (ulal), is the "spontaneousn surfactant parameter, i.e. the surfactant parameter characteristic of the amphiphilic molecule in absence of external bending stress. po is a constant, within defined ranges of surfactant concentration. p depends on the amphiphilic concentration and on micelle size and shape. It is possible to calculate p in terms of the mean curvature H and of Gaussian curvature K of the micelle ~ u r f a c e . ~
p = l-Hl+K-
12 3
for a micellar monolayer (o/w). Throughout this paper the phrase micelle surface means the micelle interface between the polar heads and hydrocarbon chains of the surfactant molecules. H and K of eq 2 can be calculated in each point of the micelle surface from the parametric equations of the micelle surface by using the methods of differential geometry.* Equation 2 contains two assumptions: (i) the volume of the focal surface is vanishingly small9 and (ii) the effective chain length inside the ellipsoidal micelle varies-within certain limits imposed by the micelle geometryin order to fill the volume without voids and overlaps: The last requirement is due to the parallel surface formalism (in the calculation of H and K ) in which the hydrocarbon chains must be packed normal to the interface. The available structural data suggest that the hydrocarbon chains in the micelle behave as a liquid-like bodylobut at the same time ellipsoidal micelles in nematic phases orient due to the diamagnetic anisotropy of the hydrocarbon chains,I1showing that chains are in average normal to the interface. It appears therefore acceptable to use in eq 2 an averaged value of I for optimizing assumption ii: In this case H and K of eq 2 can be coupled9 so that (3)
The reliability of eqs 2 and 3 has been checked quantitatively considering the (correct) value of p averaged over the whole ellipsoid ptWc = Vchains/SI(where Vchpinr is the volume occupied by the hydrocarbon chains and S the interface area) against the corresponding (approximated) value p obtained from eq 3 with H averaged, upon the whole ellipsoid: The results of the calcu-
0022-3654/92/2096-6102$03.00/00 1992 American Chemical Society
