Dynamics and Structure of Micelles
into account by employing the "effective Kihara potential parameters". The presented results show that the perturbation theory of convex molecule systems represents a powerful tool for the prediction of equilibrium behavior of nonpolar mixtures, yielding values that could be used in chemical engineering design. I t must be stressed that the method is completely consistent in the sense that all of the properties, including the molar volumes, compressibilities, and expansion coefficients of the solvents, are calcukted from the perturbation theory employing the given sets of the Kihara parameters. In comparison with the previously known methods, a comparatively broader class of systems can be considered. Wider practical applications of the described approach depend mainly on the increase of our knowledge of the parameters of the Kihara core pair potential.
Acknowledgment. The authors are indebted to the National Research Council of Canada for financial support. References and Notes (1) H. Reiss, H. L. Frisch, and J. L. Lebowitz, J . Chem. Phys., 31, 369 (1959). (2) R. A. Pierotti, J . Phys. Chem., 67, 1840 (1963).
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E. Wilhelm and R. Battino, J . Chem. Thermodyn., 3, 379 (1971). E. Wilhelm, R. Battino, and R. J. Wilcock, Chem. Rev., 17, 219 (1977). N. S.Snider and T. M. Herrington, J. Chem. Phys., 47, 2248 (1967). R. 0. Neff and D. A. McQuarrie, J . Phys. Chem., 77, 413 (1973). P. J. Leonard, D. Henderson, and J. A. Barker, Trans. Faraday Soc., 66, 2439 (1970). S. Goidman, J . Phys. Chem., 81, 608 (1977). S. Goldman, J . Solution Chem., 6, 461 (1977). S. Goldman, J . Chem. Phys., 67, 727 (1970). J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys., 54, 5237 (1971). L. L. Lee and D. Levesque, Mol. Phys., 26, 1351 (1973). S. Goldman, J . Phys. Chem., 81, 1428, (1977). T. Boublik, Mol. Phys., 27, 1415 (1974). T. Boublik, Collect. Czech. Chem. Commun., 39, 2333 (1974). T. Boubiik, J . Chem. Phys., 63, 4084 (1975). T. Kihara, Adv. Chem. Phys., 5, 147, (1963). T. Boublik, Mol. Phys., 32, 1737 (1976). T. Boubiik, presented to CHISA Conference, Prague, 1978. T. M. Reed and K. E. Gubbins, "Applied Statistical Mechanics", McGraw-Hill, New York, N.Y., 1973, p 250. T. Boublik, I.Nezbeda, and 0. Trnka, Czech. J . Phys., 826, 1081 (1976). I.Nezbeda and T. Boublik, Czech. J . Phys., 828, 353 (1978). J. A. Barker and D. Henderson, Ann. Rev. Phys. Chem., 23, 439 (1972). A. Koide and T. Kihara, Chem. Phys., 5, 34 (1974). E. Wilhelm and R. Battino, Chem. Rev., 73, 1 (1973). H. L. Clever and R. Battino in "Solutions and Solubilities", Part I, "Techniques of Chemistry", A. Weisberger and M. R. Dack, Ed., Wiley, New York, N.Y., 1975.
Dynamics and Structure of Micelles and Other Amphiphile Structures Gunnar E. A. Aniansson Department of Physical Chemistry, University of Goteborg and Chalmers University of Technology, Fack, S-402 20 Goteborg, Sweden (Received June 21, 1978)
A previous calculation of the rates of exit and entrance of tenside molecules and ions out from and into micelles is extended to the dynamics and extent of partial exits. The results indicate a very pronounced and varying head group protusion from the hydrophobic core. Several implications for the structure and properties of micelles are discussed and extended to amphiphile mono- and bilayers.
Introduction Within the context of the kinetics of stepwise micelle formation and dissolution1 the elementary process whereby one tenside molecule or ion (hereafter called "monomer") enters or leaves the micelle was considered in a quantitative way.2 A precursor to this latter treatment developed by the author and based on a treatment3 of reaction rates when the mean free path along the reaction coordinate is smaller than the width of the activation battier kT under its top was included in a later, joint paper of three groupsS4 The simple picture emerging was that of a largely uncurved monomer moving diffusively essentially a t a right angle to the micellar surface into and out of the micelle in a free energy field composed of the hydrophobic bonding energy of about one kT per exposed CH2groupI3 and the potential energy of the hydrophile end group, when charged, in the electric double layer. The average time required for a monomer with a 12 carbon atom chain to leave the micelle will be of the order of s when the electric field is absent and otherwise shorter in essential agreement with experimental result^.^ In this paper the dynamics and extent of partial motions of the monomers out into the aqueous environment and back is explored. Some important consequences are discussed but confined mainly to qualitative statements in anticipation of a fuller treatment taking into account, 0022-3654/78/2082-2805$0 1.OO/O
inter alia, the finer details of the electric double layer.
Dynamics Since the motion is a diffusive one there will be partial movements out from and back into the micelle without the monomer leaving the micelle. The time scale of these motions is easily calculated using the methods of Smol u c h ~ w s k i .For ~ example, the average time ( t ) required for the protrusion x to occur can be obtained by solving the following diffusion problem. J monomers are added per unit time at x = 0 and removed when they reach x = p . At steady state there will hold, simply ( t )= n / J (1) where n is the number of monomers between x = 0 and x = p. The flow equation to be used is dc D d V J = -D- - _dx kT dx where D is the diffusion constant, c(x) is the number of monomers per unit length at distance x from the core, V(n) is the free energy potential, and kT has the usual meaning. An analytical solution in terms of definite integrals is obtained in a straightforward way. It is particularly simple when, for example, the charge effects are absent or neutralized by a high ionic strength. (l/kT)(dV/dx) will then 0 1978 American
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Thus about every third monomer would protrude more than one CH2 group, about every seventh more than two CH2groups, etc. The average protrusion would be one CH2 group and the hydrophobic core proper would have a correspondingly smaller volume and radius. With a view to the discussion below it is of interest to examine the effect on the distribution of the electric field for charged monomers in the absence of added salt. For a plane surface, singly charged monomers, and monovalent counterions, the Poisson-Boltzmann equation takes the form
'OIW t
t -5-6-
-7. 8-
-9-
-10-
d2$
_Ilt
2
4
6
8
11
1_
12
Figure 1. Mean first arrival time, ( t ) , for a monomer with protrusion p , to reach p > p , as a function of p , / / , : ( 0 )p , = 0; ( X ) p , / l , = 3;(0) PI//, = 6; and (A) p , / / , = 9.
simply be the inverse of the length 1, of a CH2 group projected into the extended chain and one finds 11,
( t ) = D[exp(p/h)
-
1 - p/lll
(3)
The results are shown in Figure 1 where for D has been taken as 5 x cm2 s-' which would hold for a monomer with a 12 carbon atom chain. For p > I,, the last two terms in (3) are negligible compared to the first so that ( t ) essentially increases exponentially with p / l l . The time scale for this process extends from the low picosecond range, protrusion of one CH, group, to the microsecond range, protrusion of 12 CH2groups. With a 16 carbon atom chain it would extend to the millisecond range, etc. In a very similar way one can obtain ( t ) for a monomer with protrusion p1to reach p , p > pl. The results are also shown in Figure 1 for three choices of p1 and differ little from the previous ones. For example, the time needed to go from 9 to 12 CH, groups protrusion is almost the same as that needed to go from 0 to 12 CH2groups protrusion. The reason is that there is such a large propensity for inward motion that in most cases the monomer quickly moves down to the very lowest protrusion before finally reaching a larger one. The average time to go from p1 to p , p1 > p , under the same assumptions and the additional one that p1 is not closer to the full chain length but that the chance of the monomer to fully leave the micelle is negligible, is found to be 11
t = -(PI D
-
P)
(4)
The time scale is here wholly within the picosecond region. When there is a nonnegligible electric field in the double layer the free energy to overcome the protrusion is lowered. The mean first arrival times in outward going will be shortened, those for inward going lengthened. They could be obtained by numerical quadrature using the potential values obtained below but are not required in the present study. Structure Still under the same assumptions the equilibrium protrusion distribution would be c ( p ) = 4 0 ) exp(p/lJ (5)
dr2
4nq = -[m
exp(-q$/kT) - m exp(q$/kT)
f
€
where $ is the electric potential at the distance x from the hydrophobic core, q is the elementary charge, t the dielectric constant of water, and m the bulk concentration of positive and negative ions. c(0) is the number of charged head groups per unit area of the micelle surface and per unit of x at x = 0. The intergral of the term within brackets containing c(0) which is the distribution of the charged head groups over x should equal the total number of charged head groups per unit area of the hydrophobic core. The ordinary Gouy form will result when 1, 0 and c(0) m so that the term containing these terms becomes proportional to a delta function, the charged monomer groups all lying in one plane. No analytical solution was found so that numerical integration had to be used. This is complicated by the fact that c(0) and $(O) are unknown at the outset. On the other hand, a relation obtained by multiplying (6) with d$ldx and integrating from zero to infinity as well as the translational property of the equation greatly diminishes the labor involved. Since we are here interested only in the region close to the core the effect of curvature in micellar surfaces can, for the present purposes, be neglected. In Figure 2 is shown the result for a case chosen so as to correspond to sodium dodecyl sulfate micelles in the absence of added salt. A mean aggregation number of 62 was used.6 Data given by Reiss-Husson and Luzzati7J2 were used to calculate the volume and surface area of the hydrophobic core and from that the number of charged head groups per unit area of the core. For m a value corresponding to 10 mM was chosen allowing for the fact that in micellar solution the concentration of free counterions would be larger than the cmc = 8.1 mM a t 25 O C . The characteristic quantity of the Gouy double layer, (kTt/41~q2m)1/2,takes the value of 43.0 A a t 25 "C. For comparison the Gouy potential for the same total surface density of charged end groups is shown. It is seen that there is a considerable effect in allowing for the distribution of head groups away from the hydrophobic core. That the slope of the curve tends to zero when x goes to zero is evident since the distribution of charges is continuous and the potential is constant for x < 0. This has the effect of bringing the potential a t the core surface down to below the Gouy value. That the potential for large K values is larger than the Gouy POtential must be a result of this decrease of the potential close to the core bringing about a decrease in the number of neutralizing counterions in this region. The distribution of charged head groups away from the surface is shown in Figure 3. In a logarithmic plot a graph with some positive curvature is obtained. From this distribution one finds that now about one third of the head groups protrude more than 2 A 1.6 CH2 groups from
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-
The Journal of Physical Chemistry, Vol. 82, No. 26, 1978 2007
Dynamics and Structure of Micelles
q
3
5
10
15
I(
x
lo2
Figure 2. 9 $ l k T as a function of K X obtained through numerical integration of eq 6, curve 1: 1/x = ( k T d 4 ~ q * m ) ” *= 43.0 A. For comparison curve 2 shows the Gouy form, Le., the limit form when I, 0.
-
Diatributlon o f
head group protrusion
\ \ ‘ I ,, /
x i
2
4
Figure 3. The (non-normalized) distribution of head groups as a function of x in the electrostatic potential shown in Figure 2, curve 1.
the core. The average protrusion is 1.96 A. The correspondingly decreased hydrophobic core will have a radius of 16.59 A, somewhat shorter than the length of the fully extended chain, 16.68 A.
Discussion and Further Comments a. Micelles. The picture emerging from the above calculations is thus one of a very dynamic micellar surface with frequent and rather large protrusions of monomers out from the hydrophobic core. The time scale, 20 ps and longer, is today accessible with a variety of experimental methods, e.g., photophysical ones, and should constitute a fruitful field of study. The picture would also have important bearing on kinetic studies where for example a group attached to one of the CH2 groups of the chain is reacted on by a species only present in the bulk phase of the solution. The monomer will not need to leave the micelle for a reaction to occur but only protrude so much that the group considered is exposed to the bulk. This is of a much more frequent occurrence than the full dissociation of the monomer and for a group situated close to the hydrophilic head the difference is many orders of magnitude. The sizable protrusion of monomers out from the hydrophobic core has many interesting connotations. As seen above it has direct consequences for the electric double layer even rather far out from the core. In more refined calculations such as those of Stigters taking into account the finite size of the charges and ions and the penetration
of the electric field into the hydrophobic core a much less well ordered structure, reminiscent of a very concentrated solution, would seem relevant. It is interesting to note in this connection that Clifford and Pethicag in the interpretation of NMR results treated the inner part of the double layer as an approximately 5 A thick concentrated solution. The first few CH, groups from the hydrophilic end group will be very frequently exposed to the surrounding water. Since the motion in and out a distance of one or a few CH, groups is so fast NMR and similar measurements will not distinguish between states when these groups are wholly within the hydrophobic core or out in the water but give a signal corresponding to an average state. The very dynamic state of the micellar surface thus explains in a simple and straightforward way the evidence obtained’O that the first and probably more of the CH2groups appear to be surrounded by water. It was noted by Tartar’] and more recently emphasized and elaborated in a most interesting treatment by Tanfordl, that for most micelles even in the absence of added salt the total volume of the hydrophobic chains would, if spherical, have a radius larger than the length of a fully extended chain. For sodium dodecyl sulfate in the absence of added salt this radius would have been 17.31 A, incompatible with the 16.68-A length of the fully extended chain. An ellipsoid-like form would be necessary to account for this. The varying protrusion of the monomers relieves this contradiction in that the protruding monomers decrease the volume of the hydrophobic core so that those not protruding can, with more or less fully extended chains, reach closer toward the innermost space of the micelles. As noted above, for sodium dodecyl sulfate in the absence of added salt, the radius becomes even a little smaller than the length of the chain so that in this case there is no compulsion for a nonspherical micelle. With the addition of salt the electric field will decrease. The average protrusion will shrink and in the limit reach the value given above of only one CH2group length. Even with an unchanged aggregation number this would be insufficient, the radius of the core becomes 16.85 A, Le., larger than the chain length. In addition the decrease of head group repulsion leads to larger aggregation numbers and for these an ellipsoid-like form would seem to occur. However the deviation from spherical form would clearly be smaller than calculated without taking the varying protrusion into account. The protrusion of monomers as well as the resulting increase of the electric potential at larger distances from the micelle surface will lead to an increased effective size of the micelle. From Figure 3 one can deduce that for the micelle with aggregation number 62 considered above on the average 8 monomers will protrude more than 4 A and some 3 more than 6 A. As regards the resistance to migration, as studied for example in diffusion and electrophoretic mobility measurements, these large protrusions would give a much more than proportional contribution to the resistance. The total effect on the Stokes radius would be considerable and it is noteworthy that calculations of micellar aggregation numbers from migration measurements have given much larger values than light scattering measurement^.^^ The surface of the micelle is thus characterized by a very high degree of roughness. This roughness should, however, not be confused with that proposed by Stigter and Mysels14 who introduced the term rough to describe a micelle surface where the hydrophilic groups only were considered to protrude from the hydrophobic core rather than being
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buried in it, in which the latter case the surface would be very smooth. b. Amphiphile Monolayers and Bilayer Membranes. For monolayers at interfaces between water solution and air, oil, or a hydrophobic solid surface a picture very similar to that given above for micelles would exist. In these cases the density of charged head groups is mostly much larger and so the electrostatic increase of the protrusion would be even larger. For close-packed layers where an essentially parallel packing of straightened chains is expected the varying protrusion brings about an interesting effect on the chain configuration in that the varying protrusion would cause an increase of disorder in the hydrophobic tail end region. One would estimate that the concomitant increase in entropy per monomer would be of the order of k and the free energy decrease of the order of kT. Since the hydrophobic bonding which limits the protrusion is of the same magnitude per CH2 group this entropy gain, which would grow further with increased protrusion, would, reasonably, magnify the protrusion further. In lipid bilayers the protrusion would be lowered by the fact that two chains are normally attached to one head group. The second chain increases the hydrophobic bonding per CH2 group length on the order of13 2/3kTso a total free energy of 5/3kTper CH2group would have to be overcome in this case. If this were the only factor influencing the protrusion its average value would be 3 / 5 of a CH2 group length. However just as found for amphiphile monolayers an increased disorder in the region of hydropobic chain ends would result and lower considerably the free energy hindrance for the protrusion. Such a strongly increased disorder toward the chain ends fits in very nicely with the magnetic resonance results14 showing that the average angle between the chain direction and the normal to the layer increased as the chain ends were approached. Another factor that may influence the protrusion is the arrangement of the zwitterionic hydrophilic groups. If they
tend to be oriented out from the surface a considerable electrostatic gain would result from a varying protrusion compared to the case of zero protrusion. Instead of having like charges as near neighbors the charges would then to some extent have opposite charges or no charges as near neighbors. EPR studies'j on spin-labeled amphiphiles in lipid membranes showed that the spin label was in contact with water to an extent that decreased as the spin label was attached farther from the hydrophilic end. The result is very similar to thoselO referred to above for micelles and the explanation might be the same. Due to the fact that the spin label is rather bulky the state with the spin label in the hydrophobic part is energetically less favorable. This would enhance such protrusions that the spin label comes into contact with water and may explain that water appeared to penetrate so far into the layer. References and Notes (1) G. Aniansson and S. Wall, J . f h y s . Chem., 78, 1024 (1974); 79, 857 (1975). (2) G.Aniansson, M. Aimgren, and S. Wall, "Chemical and Biological Applications of Relaxation Spectrometry", E. WynJones, Ed., D. Reidel, Dordrecht, 1975, p 239. (3) G. Aniansson, ref 2, p 245. (4) G. Aniansson, S. Wall, M. Almgren, H. Hoffman, I. Kielmann, W. Ulbricht, R. Zana, J. Lang, and C. Tondre, J. fhys. Chem., 80, 905 (1976). (5) M. v. Smoluchowski, "Abhandlungen uber die Bronwsche Bewegung und verwandte Erscheinungen", Ostwald Klassiker Nr 207, Leipzig, 1923. (6) K. Mysels and L. Princen, J . Phys. Chem., 63,1696 (1959); M. F. Emerson, and A. Holtzer, ibid., 71, 1898 (1967). (7) F. Reiss-Husson and V. Luzzati, J . Phys. Chem., 68, 3504 (1964). (8) D. Stigter, J . f h y s . Chem., 68, 3603 (1964). (9) J, Clifford and B. A. Pethica, Trans. Faraday Soc., 60, 1483 (1964). (IO) J. B. Rosenholm, P. Stenius, and I, Danielsson, J . Colloid. Interface Sei., 57, 551 (1976). (111 H. V. Tartar. J . f h v s . Chem.. 59. 1195 (19551. (12j C. Tanford, f h y s ; Chem., 76, 3020 (
