Dynamic Effect In Surfactant's Shape.pdf

6897

J . Phys. Chem. 1989, 93, 6897-6901

is possible for a given equilibrium behavior, since the rates are independently assigned. A well-known relation due to Adam and Gibbs20describes the temperature (7') dependence of average relaxation time ( 7 )in glasses in terms of the configurational entropy S,: (7)= 70

'0.

Ob

0.b5

O.;O

0.\5

*.

OL5

(6.1)

Here T~ and A are temperature-independent positive constants, which can be expected to vary from substance to substance. While the Adam-Gibbs relation appears to describe the glass-transition-region behavior of many substances quite satisfactorily?' both experimentalzz and theoretical* exceptions have been noted. In order to understand in principle how such exceptions could arise, it is worthwhile inquiring what circumstances in a kinetic tiling model should produce substantial violations of the AdamGibbs relation. One qualifying scenario has x / e = j~ 1)

-..... .....

0.20

exp[A/TSc(T)]

O!O

B

Figure 7. Configurational entropy per lattice site, divided by ke, for the case D = 3. 0 = 0.01.

+

properties. Furthermore the results indicate a shift of the smeared transition to larger 0 (lower temperature): the heat capacity maximum in Figure 4 occurs at 0 z 0.416. Figures 6 and 7 respectively present heat capacity and entropy curves versus 0 for another case, namely D = 3, B = 0.01. The qualitative pattern agrees with the preceding two-dimensional example, with the B = 0 sharp transition now smeared out and shifted to larger ,13. The heat capacity maximum in this case occurs at 0 0.182. The entropy curves in Figures 5 and 7 are relevant to the "ideal glass transition" controversy. Both display regions of rapid decline toward zero. If kinetic sluggishness at lower temperatures were to make observation of the remainder of the heat capacity curves impossible, it would seem natural to extrapolate the entropy curves to zero at a finite 0 (approximately 0.44 and 0.19 for D = 2 and D = 3 ) . However, we see that the entropy curves smoothly turn over and approach zero only as 0 approaches infinity. Consequently, tiling model behavior with frustration present seems to be inconsistent with the existence of an ideal glass transition. Nevertheless, the presence of well-developed heat capacity peaks in the tiling models with aDDroDriatelv chosen couDling . -.Darameters make the& useful in atiiining a qualitative understanding of enhanced configurational heat capacities in fragile glass formers."

where j is a large integer, so that the ground state consists of a degenerate mixture of size-j and size-(j 1) tiles. In this event lim Sc(7') = Sc(0) > 0 (6.3)

+

T-0

as a result of the degeneracy. At very low (but positive) temperatures the system would be tiled almost exclusively with these two large sizes, and relaxation rates would be controlled just by the basic transition rates K(C-4') within this special tiling set. If these rates have a common Arrhenius form, then (7)in this very low temperature regime can be expected to adopt the Adam-Gibbs form (5.1) with appropriate T~ and A values. At somewhat higher temperatures the mean tile size will be subwill be negligibly stantially smaller, and concentrations pj and small. Consequently a different set of basic transition rates K will be involved that in principle can be assigned independently. In particular these rates could be given Arrhenius form with a very different activation energy. If the result for (7)were forced into Adam-Gibbs form the constant A so obtained would differ from that appropriate to the very low temperature regime.

=

Acknowledgment. F.H.S. acknowledgesbeneficial conversations with Dr. S. Bhattacharjee concerning ground-state structures of frustrated tiling models. We are also grateful to Dr. J. McKenna for discussions involving material in section IV.

VI. Discussion Time-deoendent DroDerties of the tiling models reauire sDecification of {he basic iraisition rates betwe& configuraiions. Even after accounting for the constraints on these rates mentioned in section 11, it appears that a wide range of relaxation behaviors

(20) Adam, G.; Gibbs, J. H.J . Chem. Phys. 1965, 43, 139. (21) Angell, C. A.; Tucker, J. C. J . Phys. Chem. 1974, 78, 278. (22) Laughlin, W. T.; Uhlmann, D. R. J . Phys. Chem. 1972, 76, 2317.

Shape Fluctuations in Ionic Micelles Kyoko Watanabe and Michael L. Klein* Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323 (Received: February 3, 1989)

Molecular dynamics calculations are reported for a microscopic model of a sodium octanoate micelle in aqueous solution. The micelle, whose equilibrium structure is found to be a prolate spheroid, undergoes shape fluctuations with a characteristic time scale of order 30 ps. The results are discussed in the light of recent theoretical and experimental studies on related systems.

introduction The rich variety of phase behavior exhibited by ionic surfactant solutions' can ultimately be traced back to the competing interactions between the constituent species: the repulsive interactions (1) Ekwall, p. I,, Adoances in ~ i ~ ,-,ystuls; ~ i dB demic: New York, 1975; Vol. 1, p 1.

~ G.H,, ~ Ed.;~

0022-3654/89/2093-6897$01.50/0

~

between polar head groups, the attractive interactions between the surfactant tails (hydrophobic effect), and the solvation forces associated with the hydrophilic head groups. Considerable theoretical progress has recently been made in understanding the overall behavior of micelles and related microemulsion systems ,in terms of mean-field theories and simplified or idealized models.2-'z However, the bulk of experimental and theoretical work 0 1989 American Chemical Society

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The Journal of Physical Chemistry, Vol. 93, No. 19, 1989

has centered on microemulsion Here, the equilibrium structures and phase diagrams have been rationalized in terms of macroscopic quantities such as surface tension and the bending energy of the i n t e r f a ~ e .Indeed, ~ the theory has been extended beyond equilibrium properties to include hydrodynamic fluctuations of an ensemble of microemulsion droplets that take place subject to the constraint that the total volume of surfactant and hence the total droplet area are c o n ~ e r v e d . ~Relatively little theoretical work has been devoted to ionic surfactant micelles.I0 Recent elegant neutron spin-echo experiments7 have demonstrated the existence of such shape fluctuations in microemulsion droplets, in this case reverse micelles, of radius ro = 27-70 A, with a labeled layer of sodium bis(2-ethylhexy1)sulfosuccinate surfactant. The wave vector (Q) dependence of the scattering spectrum from these droplets, whose internal (water rich) phase had the same neutron-scattering-length density as the external continuous phase (decane), exhibited a sharp peak in the relaxation frequency spectrum at a wave vector Q,. Neutron spin-echo measurements yield an effective diffusion constant for the droplets that contains two contributions: the real center-of-mass diffusion which obtains when Q 0 plus a contribution from shape fluctuations that give rise to the peak at Q,. In the case of the experiment in ref 7, one might expect the shape change to be dominated by the 1 = 2 spherical harmonic. However, because microemulsion droplets can exchange surfactant through collisions, the scattering experiment will also sample an ensemble distribution of droplet radii which, in effect, gives a nonvanishing 1 = 0 contribution, even for incompressible droplet^.^ Fluctuation modes in droplets are customarily regarded as being driven by surface tension or so-called capillary waves? Theoretical arguments lead to the conclusion that fluctuation modes, driven by surface tension y, will have a relaxation rate T.,-I = y / v r o , whereas for modes driven by the bending or curvature modulus K,, the rate is given by T;' = K,/vr:, where v is the viscosity.' By measuring the variation in peak position as a function of droplet radius and the ratio of peak height to line width (diffusion constant), it was concluded that the neutron spin-echo data are consistent with a mode driven by bending elasticity and not surface tension.' The above considerations for swollen microemulsion droplets need not apply to ionic surfactant micelles. Indeed, a recent articlelo has been quite critical of the above approach that neglects electrostatic interactions. The picture that emerged from this theoretical studylo was that ionic micelles could adopt a wide range of structures, including prolate and oblate spheroids as well as quasi-spherical. Collective dynamical shape fluctuations were postulated with time scales T 1 100 ps. The theory on which these conclusions were based was, of necessity, highly approximate due to the obvious complexity of the problem. It is natural therefore to seek an alternative complementary approach to the problem of shape fluctuations in micelles. One such approach is offered by computer The present article reports the results

-

(2) De Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982, 86, 2294. (3) Safran, S.A. J. Chem. Phys. 1983,78,2073. Safran, S.A.; Turkevich, L. A. Phys. Rev. Lett. 1983, 1930. (4) Milner, S. T.; Safran, S.A. Phys. Rev. A 1987, 36, 4371.

(5) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. Proc. Nutl. Acad. Sci. U.S.A. 1984, 81, 4601: J . Chem. Phys. 1985, 83, 3597, 3612. (6) Ben-Shaul, A.; Gelbart, W. M. Annu. Reo. Phys. Chem. 1985,36, 179. Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Chem. Phys. 1986, 85, 5345;

Watanabe and Klein TABLE I: Molecular Dynamics Results for a Sodium Octanoate Micelle in Aqueous Solution time.

(PE),

(P),

PS

kJ/mol

kbar

(RHG),A

(&),A

0-8 8-16 16-24 24-32 32-40 40-48 48-56 56-64 64-72 72-80 80-88 88-96

-57.25 -57.21 -57.23 -57.28 -57.24 -57.25 -57.16 -57.26 -57.29 -57.13 -57.25 -57.25

-0.12

10.2 10.4 10.2 10.3 10.6 10.8 10.7 10.6 10.4 10.8 10.6 10.4

5.8 5.8 5.8 5.9 5.9 6.2 6.4 6.0 5.9 5.9 6.2 5.9

0.00 -0.15 0.05 -0.1 1 -0.12 -0.06 -0.05 -0.12 -0.05 -0.11 -0.10

(ntrana) 3.4 3.6 3.7 3.6 3.7 3.9 3.8 3.9 3.9 4.1 3.8 3.6

of a molecular dynamics study, based on a completely microscopic model, of a single sodium octanoate micelle in aqueous solution. From this study, we are able to demonstrate explicitly the nature of fluctuations that can arise in small ionic micelles. Our principal finding is that such micelles undergo dramatic fluctuations in shape, a result that is in excellent agreement with predictions of a considerably simpler model introduced by Pratt and co-workers." The presence of these fluctuations calls into question the utility of model calculations that employ constrained head groups and that ignore the solvent.I2 Our results also reiterate the view that the current methods of analyzing small-angle neutron scattering (SANS) data from micelles are incomplete" because they invariably ignore shape f l u c t u a t i o n ~ . l ~ ~ ~ ~

The Model The interaction potentials and the basic model we employ for the simulation are very similar to those that we used recently to study structural aspects of the sodium octanoate mi~e1le.l~In particular, we employ effective pair potentials for the water,17 sodium ions,I8 and the polar carboxylate head groups.19 The methylene (>CH2) and methyl ( X H J groups were treated as fused pseudoatoms with no explicit treatment of the hydrogen atoms. These groups interacted via pair potentialsI9 that yield good thermodynamic properties for liquid butane.20 All the cross-interactions were constructed by using combining rules for the Lennard-Jones parameters. Hydrogen and oxygen atoms of the SPC water model carry fractional charges 0.41 and -0.82 e, respectively, and the sodium ions have their full charge. The carbon and oxygen atoms of the head groups carry charges 0.4 and -0.7 e, respectively. The relative positions of the head-group atoms plus the first methylene grou were held rigid by constraining the C-0 distance at 1.26 and the 0-C-0 and 0C-CH, angles at 122O and 119O, respectively. The hydrocarbon chains were treated as flexible with a simple C-C-C-C dihedral angle torsional potential and a C-C-C angle bending force constant.,' There are five dihedral angles for each hydrocarbon tail so that, in an all-trans configuration of an octanoate ion, the quantity ntrans = 5. The C-C distance was held fixed at 1.53 h; by using the method of constraint^.^^*^^

K

Details of the Simulation The molecular dynamics cell, which contained 718 water molecules plus 15 sodium and octanoate ions, was a periodically

1987, 86, 7094.

(7) Huang, J. S.; Milner, S. T.; Farago, B.; Richter, D. Phys. Rev. Left. 1987, 59, 2600. (8) Brown, D.; Clarke, J. H. R. J. Phys. Chem. 1988, 92, 2881. (9) Lunggren, S.;Eriksson, J. C. J . Chem. SOC.,Faruduy Trans. 2 1984, 80, 489. (10) Halle, B.; Landgren, M.; Jonsson, B. J . Phys. (Les Ulis, Fr.) 1988, 49, 1235. ( 1 1) Owenson, B.; Pratt, L. R. J . Phys. Chem. 1984,88, 2905. Pratt, L. R.; Owenson, B.; Sun, Z. Adv. Colloid Interface Sci. 1986, 26, 69. (12) Haile. J. M.; O'Connell, J. P. J. Phys. Chem. 1984,88, 6363. Woods, M. C.; Haile, J. M.; OConnell, J. P. J . Phys. Chem. 1986, 90, 1875. (13) Jonsson, B.; Edholm, 0.;Teleman, 0. J . Chem. Phys. 1986,85,2259. (14) Watanabe, K.; Ferrario, M.; Klein, M. L. J . Phys. Chem. 1988, 92, 819.

(15) Hayter, J. B.; Zemb, T. Chem. Phys. Lett. 1982, 93, 91. (16) Sheu, E. Y.; Chen, S.-H.;Huang, J. S. Preprint, 1988. (17) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces; Pullman, B., Ed.; Reidel: Dordrecht, 1981. (18) Chandrsekhar, J.; Spellmeyer, D. C.; Jorgensen, W. L. J. Am. Chem. SOC.1984, 106, 903. (19) Jorgensen, W. L. J . Am. Chem. SOC.1981, 103, 335. (20) Jorgensen, W. L. J. Am. Chem. SOC.1981, 103, 4721. (21) van der Ploeg, P.; Berendsen, H. J. C. J. Chem. Phys. 1982,76, 3271. (22) Ryckaert, J. P.; Bellemans, A. Chem. Phys. Lett. 1975, 30, 123. Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J . Comput. Phys. 1977.23, 327. (23) Allen, M. P.; Tildesley, D. Computer Simulation of Liquids; Clarendon: Oxford, 1987.

The Journal of Physical Chemistry, Vol. 93, No. 29, 1989 6899

Shape Fluctuations in Ionic Micelles 0.04

I

I w

0.03 -

c-

0.4

h 0

h

0.3

r

*'&

0.02 -

9.

0.01 -

0.1

A,

0.00 0

0.2

0.0 2

4

6

8

10

12

14

0

16

TIA Figure 1. Density distributions for carbon atoms (squares), water molecules (triangles), and counterions (circles) measured with respect to the micelle center of mass.

2

4

6

8

10

12

14

16

TIA Figure 2. Probability distributions for carbon atoms measured with respect to the micelle center of mass. The curves labeled by squares, open circles, triangles, and dots refer to carbon atoms 1, 3, 5 , and 8 of the hydrocarbon chains, starting at the head group, respectively.

replicated truncated octahedron24with a volume V = 0.5 X (37 = 25326.5 A3.The simulation was carried out at constant volume and constant temperature (298 K) using an Ewald method to handle the long-range electrostatic interactions.23 The equations of motion were integrated by using a Verlet algorithm and a time step of 2.5 fs. As in our previous work, the simulation was carried out in three stages.I4 First, the micelle core, Le., the tails of the 15 octanoate ions, was equilibrated by carrying out a long molecular dynamics run with the head-group carbon atoms constrained to lie on the surface of a sphere whose radius was gradually decreased from 15 8,to the final value of 10.8 8,. Next, the micelle core was inserted into an equilibrated box of water along with the 15 counterions, and a run of 35 ps was then carried out with the head groups still constrained to lie on the surface of a sphere. This was followed by a further equilibration run of 23 ps, with the head-group constraint removed. Finally, a production run of 96 ps was used to generate the structural and dynamical results that are described below and listed in Table I.

Structural Results The variation of the potential energy ( P E ) and pressure ( p ) of the micelle-water system, throughout the 96-ps production run, is given in Table I. For the purpose of analysis, the total run has been partitioned into 12 equal segments for which subaverages are recorded. (Each segment is actually a convenient overnight run on our IBM 3090/200 VF computer.) The overall structure of the micelle can be characterized through various probability distribution functions. Figure 1 shows the density distribution of carbon atoms in the micelle core, the sodium counterions, and the solvent water with respect to the center of mass of the micelle, calculated during the last 64 ps of the run. The extent of water penetration into the core region of the micelle is somewhat greater than we found in our previous calculation on this system. On the average, the number of water molecules found within a 4-8, radius from the methylene and methyl groups is, from head to tail, 5.0, 4.6, 3.9, 3.7, 3.3, 3.2, and 3.4. The total number of water molecules within 4 8, of any methyl or methylene group in the micelle is 127. Thus, on average, there are 8.5 water molecules in contact with the hydrophobic tail of each monomer comprising the micelle. This number is significantly larger than the result found in our previous simulation which gave 4.6 water molecules in contact with each hydrocarbon chain. In the latter case, the average pressure was actually quite negative, being (p) = -1.1 kbar compared with the present value of (p) = -0.07 kbar. In the present near-zero-pressure calculation, unlike in the previous one, the carbon atom density in the micelle core region is close to that the water content of a liquid hydrocarbon. As found previ~usly,'~*'~ of the micelle core is large and disagrees with a recent analysis of new SANS data. This new work estimates the number of water

molecules in the core to be less than one per monomer.16 In previous molecular dynamics calculations, the sodium ions were seen to localize in the region of the head groups (the Stern r e g i ~ n ) . ' ~In ? ' ~the present calculation, about four of the ions are actually in contact with the head groups. The net charge on the micelle is thus about -1 1 e, which is a little larger than our previous value of -9 e.I4 The value derived from a fit to the original SANS data is -10 f 2 e.Is The distribution functions for carbon atoms at positions 1, 3, 5, and 8 (counting from the head-group carbon atom along the chain backbone) are shown in Figure 2. The average distances of the carbon atoms, C1 to C8, from the micelle center of mass are 10.6 f 1.6,9.4 f 1.7, 8.6 f 1.7, 7.6 f 1.7, 7.1 f 1.7, 6.4 f 1.8, 6.2 1.9, and 6.0 2.2 8,. Overall, the carbon atom distributions are rather similar to those reported previously by other but the peak positions occur at distances that are 12-19% smaller than in our previous sim~1ation.l~ The main point to note in the carbon atom distributions is that the terminal methyl group distribution is very broad.2s*26 The mean location, with respect to the micelle center of mass, of the terminal methyl group ( R T )is given in Table I for each segment of the run. This value of ( R T )appears to be rather constant as is the analogous quantity for the carbon atom of the head groups (RHG).The values of (nb,,) listed in Table I indicate that, on average, there is at least one gauche defect in each chain.

*

*

Micelle Shape A natural interpretation of the data in Table I is that the micelle is indeed stable for the duration of the simulation. Accordingly, it seems worthwhile to examine the shape of the micelle in a more quantitative fashion, particularly in view of the remarks in the Introduction.lo Figure 3a shows the variation of R, the micelle volume, throughout the run, where the quantity R = (4/3)ir( R H G ~ ) is defined in terms of the head-group positions at each time step. The volume defined in this fashion varies between extremes of about 4200 and about 6000 A3with a mean value of roughly 5200 A3over the last 64 ps of the run. An analogous calculation for the surface area yields a mean value for each head group of about 94 A2. As mentioned in the Introduction, micelles need not necessarily be spherical. Indeed, the calculations of Pratt and co-workers on an idealized model suggest dramatic departures from sphericity." We have examined representative micelle configurations generated in the molecular dynamics run in some detail using an IBM 5080 work station. The inescapable conclusion from these visualizations is not only that the micelle is undergoing large volume fluctuations but also that it is very (25) Cabane, B.; Duplessix, R.; Zemb, T. J . Phys. (La.Ulis, Fr.) 1985, 46, 2161.

(26) Dill, K. A.; Naghizadeh, J.; Marqusee, J. A. Annu. Reu. Phys. Chem. (24) Adams, D. J. Chem. Phys. Lett 1979,62, 329.

1988, 39, 425.

6900 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989

Watanabe and Klein 20

1.6

10 n

&

1.4

5

0

2. 1.2

-10

1

-20

6000

10

n n

in

5

96 ps

0

4000 -10 ) 1

2000

, '

I 1

20

0

1 1

,

'

40

/ '

60

1 1

80

, '

1

tips Figure 3. (a) Time evolution of the micelle volume, defined in terms of the head-group carbon atom positions (see text). (b) Time evolution of the ratios of principal moments of inertia of the micelle.

TABLE II: Spherical Harmonic Expansion Coefficients Characterizing the Micelle Shape set 1 set 2

4 3

(0.8)

00 10

36.85 0.00 0.00 7.40 -0.13 -0.15 -2.02 -1.68 1.07 0.75 -1.94 0.43 -1.15 1.38 1.75

11 20 21 22 30 31 32 33 40 41 42 43 44

(s(a,,)2)1'2 0.79 3.17 3.40 2.58 8.05 5.83 5.48 4.32 8.84 5.28 5.37 5.41 5.40

(a,$) 36.85 0.00 0.00 2.73 0.59 0.98 -0.78 1.49 -0.38 -0.62 1.22 -1.72 1.33 0.95 -1.05

(6(0,@)2)1'*

0.79 6.26 3.32 3.44 7.22 5.06 5.44 5.29 1.43 5.91 5.24 5.90 5.80

nonspherical. In order to better characterize the micelle shape, we first attempted a spherical harmonic expansion of the radial function R(t,8,$) which defines the surface of the head-group carbon atoms:* R(t9894) = Ca/m(t) Y/m(e*4) /m

The expansion coefficients aIm(t)were calculated from the equation a/m(t)

= ( 4 ~ / n ? C R ( t , @ i , 4Yi )d e i , A ) 1

where i runs over the N = 15 head-group carbon atoms at positions R(t,8,&) with respect to the center of mass of these atoms. Table I1 lists two sets of values for the quantities (al,(t)) and their rms fluctuations; the brackets denote an average over the run. The first set of coefficients are calculated by defining 8 with respect to the axis corresponding to the smallest principal moment of inertia. The second choice of coordinate frame sets the z axis along the principal axis with the largest a20coefficient. In both cases, ( a20)gives the largest nonspherical contribution which suggests a prolate micelle shape. However, contributions from higher order spherical harmonics are appreciable and indicate the complexity of the actual micelle shape.

I -20 -, -20

-10

10

20

x ;A) Figure 4. Instantaneous view of the micelle after 32 and 96 ps. The head groups are shown as circles and the tails as squares. The simulation cell boundaries are indicated by lines and the sodium ions by crosses. Views from orthogonal directions (not shown) confirm that at 32 ps the micelle is spherical.

Since there are relatively few monomers in the micelle, characterization in terms of the spherical harmonic expansion yields only qualitative information. We therefore also used an alternative scheme based on variations in the moment of inertia tensor of the micelle throughout the run. We focused on the moment of inertia tensor because it is dominated by the contributions from the head groups, and hence this quantity provides a simple way to characterize the micelle shape. The values of the moments Zl, Z2, and 13,corresponding to the principal axes, leave no doubt that, on average, the micelle is roughly a prolate spheroid, with Z, C Z2 = 13.The variation throughout the run of the ratios of the principal moments of inertial R = Z2/11 or Z3/Z1 is shown in Figure 3b. The values of the two ratios vary quite widely but, nevertheless, track each other quite closely as the run proceeds. At 32 ps, R = 1 which implies that the micelle is spherical. Figure 4 shows a view of the micelle at exactly this point in the trajectory. As expected, the picture confirms that the micelle shape is quite spherical. The mean value of R over the latter part of the run is about 1.33, which indicates a significant spheroidal distortion. A second configuration shown in Figure 4, which is the last one of the run, clearly shows that at 96 ps the micelle is nonspherical. The fact that, on average, we observe substantial deviation from a spherical shape, even for such a small micelle, is in excellent accord with the results of Pratt and co-workers." However, the actual numerical agreement for the predicted shape of the octanoate system is only semiquantitative. Shape Fluctuations The apparent success of the analysis used to interpret the structural features of the octanoate micelle makes it worthwhile to explore the time evolution of the system in more detail. In the Introduction, we mentioned some current theories of shape fluctuations of microemulsion droplets4v9and micelles10and the recent elegant neutron spin-echo experiments that confirmed the presence of shape fluctuations in the former case.' The results shown in Figures 3 and 4 already strongly suggest that periodic volume and shape fluctuations are occurring. In order to highlight this phenomenon more clearly, we have examined two different autocorrelation functions, Cn(t)= (6Q(t) sQ(O))/(sQ(O) sQ(0)) and CR(t)= ( 6 % ( t ) b R ( 0 ) ) / ( 6 2 ( 06R(O)), ) where the brackets

The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6901

Shape Fluctuations in Ionic Micelles 1

0

n

0

es v

n

-1

\

0

0

-1

0

20

40

60

80

t/ps Figure 5. (a) Autocorrelation function of fluctuations in the ratios of moments of inertia for the principal axes. The bold and dashed curves have the same meaning as those in Figure 3b (see text). (b) Autocorrelation function of fluctuations in the micelle volume (see text).

denote an average over the run and time origins and 6n(t) = n(t)

- ( Q ( t ) ) .Figure 5 gives rudimentary information on the shape

and volume fluctuations that are occurring in an octanoate micelle in aqueous solution. Although the statistics on the time correlation functions are inevitably rather poor, Figure Sa suggests that the time scale for the shape fluctuations is T R = 30 ps. On the other hand, it is evident from Figure 5b that the time scale for the volume fluctuations is considerably longer and is of the same order of magnitude as the run length, namely, 70 ir 100 ps. One of the questions that naturally comes to mind when one attempts to characterize the micelle shape is the overall translation and rotational motion of the entity. The diffusion coefficient for the micelle center of mass was calculated from the trajectory and was estimated to be 0.3, in units of cmz s-', which is to be compared to the values 3.3 and 1.2 for water molecules and sodium ions, respectively, in the same units. The diffusion coefficient for individual monomers is about 0.6 in these units. Diffusion of water molecules in the hydration shells of the hydrophobic methyl and methylene groups and solvent associated with the head groups and sodium ions is somewhat slowed down compared to bulk water, the respective diffusion coefficients being 2.9 and 2.2 in the above-mentioned units. Finally, the diffusion of monomers on the micelle surface was examined by calculating the autocorrelation function (u(t).u(O)),where u(t) is a unit vector in the direction of the radial vector connecting the center of mass to the head-group carbon atom. The characteristic time for the decay of this function was found to be in the region of 500 ps, assuming an exponential decay.

Discussion A number of questions immediately come to mind concerning the observations reported above. For example, to what extent is the calculation influenced by the periodic boundaries and the fact that there is only one micelle? Moreover, the run is rather short on the time scale that is appropriate to micelle diffusion, rotation, and intermicelle collisions, etc. Also, there are the usual concerns about the dependence of the results on the choice of potential parameters, especially the use of combining rules for waterhydrocarbon interactions and the neglect of hydrogen atoms in the construction of the model. Also, to what extent are the fluctuations we observed in Figure 5 a consequence of our initial conditions? It is obviously difficult to rule out the influence of any of these factors. However, arguments similar to those used to rationalize Onsager's regression hypothesis suggest that imposing a constrained starting configuration and observing the resulting relaxation toward equilibrium is not an unreasonable way to probe the spontaneous fluctuations of the system of interest. This is in effect what we have done. In so doing, we have explicitly demonstrated that shape fluctuations and volume fluctuations occur on a time scale less than that of the exchange of octanoate particles. That is, a run of 100 ps is apparently long enough to sample shape changes that involve considerable monomer motion but no actual loss of monomers from the micelle. The actual time scales we seem to find have implications for larger more commonly studied micelles such as those formed by lithium and sodium dodecyl sulfate. Here, the time required to follow shape fluctuations will likely be much longer than r0 = 30 ps. Indeed, a study of such systems will likely present a real challenge to simulations. The situation may be saved, however, because the magnitude of the fluctuations in a larger micelle might be smaller than in the case considered herein. In summary, we have used molecular dynamics calculations to demonstrate the nature of fluctuations that can arise in a small ionic surfactant micelle. We find that a sodium octanoate micelle in aqueous solution is best described as being a prolate spheroid that undergoes shape fluctuations on a time scale of 30 ps. It remains to be seen whether any of these findings can be carried over to larger micelles of wider interest. Our results appear to be at odds with a recent analysis of new SANS data on sodium octanoate.I6 Here, the usual approximations were invoked to analyze the data.2s These include an assumption of sphericity and total neglect of shape fluctuations plus effects of polydispersity. With these rather drastic assumptions, it is perhaps not too surprising that theory" and experimentI6 are found to differ somewhat on the nature of ionic micelles in aqueous solution. Acknowledgment. We have benefited from discussions with John Huang, Sam Safran, John Shelley, and Eric Sheu. The research described herein was supported by the National Institutes of Health under Grant GM-40712-02 and is dedicated to Bob Zwanzig on the occasion of his 60th birthday. The calculations were carried out on an IBM 3090-200/VF supported by the School of Arts and Sciences at Penn and the National Science Foundation under CHE-8815130. Registry No. Sodium octanoate, 1984-06-1.