J . Phys. Chem. 1985, 89, 153-163 Since chain long axes are disordered on a length scale always much shorter than A, the local director cannot be identified with any individual chain long axis (as Pace and Chan do) but must be identified with an average over the chain long axes within a length comparable to A. The most important points to emerge are these: The presence of an array of relatively long-wavelength modes which are uncorrelated with each other does not imply long-range cooperativity. (In a bulk hard-sphere liquid, a similar array of modes exists [density fluctuation modes]. Nevertheless, in this liquid, there is no long-range collectivity of deviations of the density from its bulk value.) All the continuum modes are superimposed in an uncorrelated way on each other and on a large-amplitude disorder between the tilt directions of neighboring chains. Thus, cooperative tilting of the chains exists only over length scales comparable to a chain diameter and even then only to a limited extent. In fact, a model which ignores tilt correlation between neighboring chains (the single-chain model) is capable of reproducing the simulation’s molecular tilt distribution with considerable fidelity (Figure 4). This suggests that cooperative tilting of the chains occurs hardly more often than would be expected from a collection of independent chains constrained, on average, to fill the available volume. The picture which emerges is very similar to an earlier picture” of the state of the chains in the bilayer interior. There it was suggested that as bilayer chains have only slightly fewer gauche bonds than random-coil chains, tilt correlation in the bilayer should be very short range, just as orientational correlation in liquid n-alkane is very short range.38 The molecular dynamics simulation strongly supports this conclusion. ( G r ~ e was n ~ primarily ~ concerned with dipalmitoyllecithin bilayers. The fraction of gauche bonds lost by palmitoyl chains going from a random-coil state to a bilayer seems to be only 12% while here the corresponding figure is 30%-35%.) Figures 5 and 6 show to what extent the distribution of gauche bonds in the chain is modified by the chain being incorporated into a bilayer. The bilayer curves are shifted substantially from the random-coil curves, but the general shapes remain similar. The chains in the bilayer exist as “perturbed random coils”.44
-
Summary 1. A simple model of the state of the chains in amphiphilic aggregates is described. The model involves generation of all the possible internal bond sequences in a single chain. After an
153
average area per chain for this model is chosen which gives a best fit to experimental order parameters down the chain, a large number of equilibrium properties of the model are generated, Without exception, these results show the same qualitative trends as results derived from a molecular dynamics simulation involving 128 chains. In most cases, the quantitative agreement between the two sets of results is very close indeed. 2. The precepts of statistical mechanics ensure the existence of an array of splay and twist distortion modes of the local director of the bilayer. After a detailed analysis, it is concluded that the available evidence suggests that the combined amplitude of all the (relatively long wavelength) independent splay and twist modes of the local director of the bilayer is very small. Further, they are superimposed in an uncorrelated way on each other and on a large-amplitude disorder between the tilt directions of neighboring chains. Cooperative tilting of the chains exists only over length scales comparable to a chain diameter and even then only to a limited extent. This is analogous to bulk n-alkane, where orientational correlation between chains is very short range.38 This accounts for the success of single-chain models.
Acknowledgment. I am most grateful to Bertil Halle for many enlightening discussions about the interpretation of NMR experiments and to Professor Berendsen for sending me a preprint of his molecular dynamics results. Appendix An Aspect of the Pace and Chan Model. Pace and Chan assume that “splay” and “twist” distortions of nearest-neighbor chain long axes introduce voids between the chains. They then calculate that collective fluctuating modes have a correlation length of -130 chain diameters. Applying their analysis to G(CH,),CH, chains (with an effective length in the bilayer (I,) of 10 A) leads to an estimate of the root-mean-square angular deviation between neighboring chain long axes ( (6n2)’I2)of -3.6O. This number may be used to estimate the spatial tilt correlation function defined by van der Ploeg and BerendsenZ3as C ( R ) . For nearest and next-nearest neighbors respectively the Pace and Chan model predicts C ( R ) values of -0.99 and -0.96 which may be compared to the values derived from the molecules dynamics simulation of -0.20 and -0.07. We may conclude that the Pace and Chan calculation substantially overestimates the free energy cost of misaligning neighboring chain long axes.
A Model for the Chains in Amphiphilic Aggregates. 2. Thermodynamic and Experlmental Comparisons for Aggregates of Different Shape and Size David W. R. Gruen Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, A.C. T. 2601, Australia (Received: March 15, 1984; In Final Form: August 24, 1984) Order parameter profiles are derived from the model and compared with experimentally determined order parameters. The comparisons are performed for bilayers, cylinders, and spheres, and with one exception, they are made at experimentally determined dimensions. In most cases, the model provides close agreement both in magnitude and trend with the experiments. A variety of thermodynamic and structural properties are generated for the three aggregate shapes over a wide range of hydrophobic core sizes. These include the extent of chain straightening and the free energy penalty incurred when chains are packed into any aggregate. A unifying idea, the volume-weighted mean area per chain, is introduced. For an aggregate of any shape, calculation of this mean area per chain gives a good measure of the extent of chain straightening required and the free energy penalty incurred when chains are packed into that aggregate. Alternative models for the chains in amphiphilic aggregates are critically examined. Introduction Having demonstrated in paper 1 the extent to which our model reproduces the static properties of a large molecular dynamics simulation,’S2it is time to confront the model with experimental 0022-3654/85/2089-Ol53$01.50/0
fact. However, first we must deal with some preliminaries. For all three aggregate shapes, spheres, cylinders, and bilayers, R ( 1 ) van der Ploeg, P.; Berendsen, H. J. C. Mol. Phys. 1983, 49, 233.
0 1985 American Chemical Society
154 The Journal of Physical Chemistry, Vol. 89, No. 1 , 1985
denotes the radius (or half-thickness) of the hydrophobic core, while t denotes the average radius (or half-thickness) of the hydrocarbon region (i.e. t includes the small amount of hydrocarbon sitting outside the hydrophobic core). Figure 1 in paper 1 makes this distinction clear. In the rest of this introduction, we describe extensions which are required to model amphiphiles with different head groups and to model aggregates containing more than one type of amphiphile. The introduction ends with an explanation of how the free energy of the system is calculated. The Results section contains two subsections. In the first subsection, order parameters derived from the model are compared with experimentally determined order parameters. The comparisons are performed for all three aggregate shapes and (with one exception) are made at experimentally determined dimensions. In the second subsection, several comparisons are presented between model calculations for aggregates of different shape and size. In the Discussion section several points which emerge from the model calculations are examined. Then, other models of chain packing in amphiphilic aggregates are discussed and compared with the present model. The paper ends with a summary in which the paper's most important contributions are listed. Head-Group Orientation. In the molecular dynamics simulation discussed in paper 1 no a priori probability density was applied to head-group orientations. Therefore, in the single-chain model, all possible orientations of the head group (within a hemisphere extending into the hydrophobic core of the aggregate) were chosen with equal probability. In this paper we model amphiphiles with a variety of head groups which (presumably) impose different orientational constraints on their chains. We introduce a single parameter to model this situation. As before, define head-group orientation by the vector joining the head group to the second carbon atom in the chain. Define p' as the angle between this vector in the ith chain conformation and the local normal to the hydrophobic core surface. We again generate all possible head-group orientations so that all solid angles (within a hemisphere extending into the hydrophobic core) are equally likely to be generated. But we now modify the probability of conformation i, pi, to read
pi = (sin p ' ) " ( f l a , k ) exp(-E'/kT)/Z J
Gruen chains do not reach the bilayer center. When applied to chains with n carbon atoms, the factor (flpJk) in eq 1 has n terms in it. Thus, the effect of the above akprofile is that the longer (decanol) chains will be more strongly biased than the shorter chains in favor of those conformations with many segments near the center. It is certainly clear that all the chains which reach the bilayer center are decanol chains, but without modification the model accentuates the difference between chains of different length. The accentuated difference is unphysical and undesirable. The problem is overcome by introducing a parameter x (x < 1) and modifying eq 1 for the longer chains to read
p' = (sin pl).(n~l/~)~ exp(-E'/kT)/Z I
(2)
Whatever value of x is used, the ensemble average of all conformations of both amphiphiles must pack each region of the core at liquid n-alkane density. Then, the parameter x changes the relative extent of straightening of the two different chains and hence their relative order parameters. A large value of x leads to relatively larger order parameters for the longer chain amphiphile, and smaller ones for the shorter amphiphile. This analysis is used only for the sodium octanoate-decanolwater system, and the results will be presented in Figures 5 and 6 (see later). Calculation of Free Energy. The large free energy cost associated with transferring short n-alkanes from bulk n-alkane to water is predominantly e n t r ~ p i c . The ~ internal energy changes involved in the transfer are small. So, when evaluating the internal energy of our system, we shall neglect the small internal energy changes involved when part of a chain sits outside the hydrophobic core. The ensemble average internal energy of our system is then U=( where Einlis the energy of gauche kinking for the ith chain conformation (eq 2, paper 1). Our definition for entropy of the chains must not depend on the number of repeats which we generate of each internal bond sequence. The entropy is defined as S = -kE,pl In pI where pI = E,,@and the sum over m denotes a sum over all conformations with the same internal bond sequence. p' is given by eq 1 or 2 depending on the system. The Helmholtz free energy per chain is then F = U - TS = (Elnt)+ kTx,p, In pi
(3)
where 2,the partition function, plays the role of a normalization constant. The formula is identical with eq 3 in paper 1 except for the head-group weighting provided by the (sin p')" term. In modeling a variety of aggregates with different head groups, we use values of a in the range Ia I0. In the absence of other implies ( 3 / 2 cos2 p' contributions to the chain energy, a = = 0. = 0.1 while a = 0 implies ( 3/2 cos2 0' When a given amphiphile is studied in a variety of aggregates (of different sizes or shapes), only a single value of a is used. This value is chosen to give a best fit to the order parameter profiles along the chain. The effect on order parameters along the chain of changing the value of a will be displayed in Figure 1 (see later). Aggregates Containing Two Different Amphiphiles. In order to illustrate a general principle, it is useful to discuss an example. When sodium octanoate, decanol, and water are mixed at respective weight fractions 0.2845:0.425:0.2905 at 20 O C , a lamellar phase is formed with an average area per amphiphile of 24.7 A2 (ref 3). Hence (from eq 1 of paper l), t = 10.96 8, and R = 10.6 8,while the fully extended chains have lengths 10.4 8, (octanoate) and 14.2 8, (decanol). In order to pack this structure at constant density throughout, it is necessary to use values of ak < 1 for shells near the bilayer surface and values ak> 1 near the bilayer center (this (Yk profile leads to a net straightening of the chains). There are many decanol conformations with several chain segments near the bilayer center but even fully extended octanoate
Results Results are generated for fully saturated amphiphiles with between 6 and 11 CH2 groups. For each system, a single sample of -(4-7) X lo4 amphiphile configurations is generated. As before, each sample contains a large fraction of the most likely configurations (those with 0, 1, or 2 gauche bonds). Since each sample is of a similar size to the samples used in paper 1, estimates based on a sample should have similar accuracy. Therefore, the error analysis developed in paper 1 was applied, remembering that standard deviations based on a single sample are 5lI2times larger than standard deviations based on five independent samples. When results are quoted, the single sample standard deviations will also be specified.
(2) Edholm, 0.;Berendsen, H. J. C.; van der Ploeg, P. Mol. Phys. 1983, 48, 319. (3) Fontell, K.; Mandell, L.; Lehtinen, H.; Ekwall, P. Acto Polyrech. Scand., Chem. Incl. Metall. Ser. 1968, No. 74, 111.
(4) Tanford, C. 'The Hydrophobic Effect: Formation of Micelles and Biological Membranes",2nd 4.; Wiley-Interscience: New York, 1980. (5) Flory, P. J. "Statistical Mechanics of Chain Molecules"; Wiley-Interscience: New York, 1969.
We wish to compare this free energy with the free energy of the chains in the rotational isomeric state or random-coil model (FRC), used to describe the behavior of bulk liquid n-alkane.5 In the random-coil model, all initial orientations of the chain are equally likely, and the probability of configuration i is pkc = exp(-E:,,/kT)/ZRc where ZRC= E, exp(-E:,,/kT). There are no constraints external to the chain which exclude any conformations, and hence the sum is over all t, g+, and g- conformers in the chain (excluding gig7 sequences). Then FRC
= -kT In
ZRC
(4)
A Model for the Chains in Amphiphilic Aggregates
The Journal of Physical Chemistry, Vol. 89, No. 1, 1985 155 -1
I
0.1 Oil_\
00';
I
4'
'
6 I 8I Carbon number
\i
, \ I
10 I
I
12 Carbon number
Figure 1. Model C-D order parameters along a -(CH2)&H3 chain in a bilayer at 31 OC: (-), order parameters derived assuming a = 0 in eq 1, i.e. assuming that all allowed head-group orientations are equally likely; (e), order parameter derived assuming a = in eq 1, i.e.
assuming that the probability distribution of head-group orientations is proportional to s i d 2 8. The order parameters are evaluated with respect to the bilayer normal. The calculations assume an ensemble average area per chain of 30.7 A2. Carbon 2 is the CH2 group bonded to the head group; carbon 12 is the terminal methyl group. The two sets of results are derived by using the same sample of chain conformations, and hence the relative error (of one set of results with respect to the other) is minimal. The standard deviation of either set of results is 0.007. 031
,
Theoretical
Figure 3. Comparison of experimental and model C-D order parameters in the hexagonal phase: (e), experiments performed by J. Charvolin (personal communication) on perdeuterated potassium laurate and 5 1.33 wt % water at 50 "C; (-), model calculation at 50 "C assuming a = and t = 14.0 A. The order parameters are displayed as 2%; where Ji CD is the order parameter with respect to the cylinder axis. The standard deviation of the model results is 0.007.
Experimental
I 00'
\
;
I
I
I
I
1
3
5
7
9
12
Carbon number
2
L
6
8 1 0 1 2
Carbon number
Figure 2. Comparison of experimental and model C-D order parameters in bilayers: (-), 31 "C at an area per chain of 30.7 A2; (---), 50 "C at an area per chain of 32.8 A2; 51 OC at an area per chain of 36.0 Az; (---), 110 "C at an area per chain of 41 A2. The experiments were performed by Mely et aL6 on perdeuterated potassium laurate [K+COO-(CDz)loCD,]-water lamellar phases at increasing levels of hydration (in wt % H20, respectively 21%, 24%, 30%,and 30%). The exper(-e),
imental order parameters were derived from the quoted quadrupolar splitting by using the value of 167 lcHz for the C-D quadrupolar coupling (in eq 1) for constant. The model values are derived assuming a = all profiles. The standard deviation of the model results is 0.007. Figure 1 displays the C-D order parameters derived from two different assumed probability distributions for head-group orientations. The curves are derived by assuming the two extreme values of the parameter a used in this paper to model a variety of head groups. It should be clear that varying a in the range -1/2 I a 5 0 changes the orientational probability distribution of the first three carbons in the chain. Further down the chain, order parameters are almost completely unaffected by the value of a. Comparison of Experimental and Model Order Parameters. Figures 2 and 3 show a comparison between the model and the experiments of Mely et aL6 and of Charvolin (personal communication). Since the same amphiphile (potassium laurate) was used in all the experiments, a single value of the parameter a (a = was used for all model calculations. For the four bilayer experiments, Mely et a1.6 quote values for the area per chain which are in close agreement with values which can be derived (in two cases by interpolation and in the other two cases by extrapolation) from the extensive X-ray diffraction work of Gallot and Skoulio~.~ (6) Mely, B.; Charvolin, J.; Keller, P. Chem. Phys. Lipids 1975,15, 161. (7) Gallot, B.; Skoulios, A. Kolloid Z.Z.Polym. 1966,208,37. (8) Klason, T.;Henriksson, U. In "Solution Behaviour of Surfactants: Theoretical and Applied Aspects";Mittal, K. L., Fendler, E. J., Eds.; Plenum Press: New York, 1982; Vol 1, p 417.
Figure 4. Comparison of experimental and model C-D order parameters (with respect to the local normal of the aggregate) for spherical micelles at 40 OC: (e), results derived from a "two-step" model (see text) to interpret T , NMR relaxation measurements on dodecyltrimethylammonium chloride (DOTAC) at 30 wt % by Walderhaug et al.;9(-), model results assuming a spherical hydrophobic core with R = 19.2 and the parameter a = 0 in eq 1. The standard deviation of the model results is 0.007.
The quoted areas per chain are used for the model calculations. For their hexagonal-phase measurement, the area per amphiphile quoted by Mely et al. is not in good agreement with the results of Gallot and Skoulios. Therefore, I am extremely grateful to Prof. J. Charvolin, who generated new results which are now described. At 51.33 wt % HzO, a perdeuterated potassium laurate-water mixture forms a hexagonal phase at 50 OC in which the potassium laurate forms cylinders with a radius of 14.76 A. Using the specific volumes for protonated potassium laurate quoted by Gallot and Skoulios' and our values for chain volume (eq 1, paper I), we may infer that, at 50 OC, the chain accounts for -0.895 of the volume of the amphiphile and hence t N (0.895)1/2 X 14.76 II 14.0 A. Figure 3 shows a comparison between the Charvolin experimental results and the model. In hexagonal phases, the symmetry axis is the axis of the cylinders. Diffusion of the molecules around this symmetry axis is sufficiently rapid that this motion contributes to the averaging of the quadrupole interaction. If there is a t least threefold symmetry around the cylinder axis and around the local normal to the aggregate surface, S$E = - 2 g ; (where LN means with respect to the local normal and C A means with respect to the cylinder axis). In this paper, order parameters derived from hexagonal phases (Figure 3 and the first profile of Figure 5) are displayed as 2%;. Figure 4 shows a comparison of order parameters for spherical micelles. The experimental values are derived assuming a "two-step" model for the motion of the amphiphile chain^.^ Because the head group is effectively anchored to the aggregate IO-" surface, on a fast time scale (with a correlation time 7 s) the chains are assumed to undergo orientationally restricted
-
(9) Walderhaug, H.;SMerman, 0.; Stilbs, P. J . Phys. Chem. 1984, 88,
1655.
(10) Mandell, L.; Fontell,
K.; Lehtinen, H.; Ekwall, P. Acta Polytech.
Scand., Chem. Incl. Metall. Ser. 1968, No. 7 4 , 11.
Gruen
156 The Journal of Physical Chemistry, Vol. 89. No. 1, 198
03
\
OlC I
:
01 -
\ I
b
Carbon number
Figure 5. Comparison of experimental and model C-D order parameters down the octanoate chain for three sodium octanoate-decanol-water mixtures at 20 OC. The systems studied (from left to right) were 50 wt % sodium octanoate:50 wt %water, 33 wt % sodium octanoate:32.6 wt %, decanok34.4 wt % water, and 28.45 wt % sodium octanoate:42.5 wt % decanob29.05 wt % water. The first mixture forms a hexagonal phase while the second and third form lamellar phases: (e), experiments performed by Klason and Henriksson;8(-), model results derived assuming a = (in eq 1) for all profiles and aggregate dimensions t = 10.16, 9.67, and 10.96 A, respectively (deduced from the density measurements of Mandell et al.'" and X-ray measurements of Fontell et aL3). The standard deviation of the model results is 0.007.
motion; i.e., on this time scale the C-H bond vector does not sample all directions with equal probability. There is thus a residual anisotropy and hence a nonzero order parameter may be defined. (S = ( 3 / 2 cos2 0;: - l / Z ) h where 0;: is the angle between the local normal to the aggregate and the C-D or C-H vector. The subscript f implies that the fast motion is averaged but the slow motion is not.) On a much slower time scale (with a cors) diffusion of the monomer around the relation time T micelle and/or rotational diffusion of the micelle eliminate the residual anisotropy. It is the order parameter S, defined above, which is displayed in Figure 4. Figures 5 and 6 display results derived from the sodium octanoate-decanol-water system. The dimensions of the aggregates have been derived by Fontell et aL3 The first profile in Figure 5 is for a hexagonal phase containing only water and sodium octanoate. The other two mixtures studied are lamellar phases containing all three components, and hence the hydrophobic cores contain chains of two lengths. As discussed in the previous section, the model requires modification to deal with such systems. A parameter x is introduced which determines the relative extent of straightening of the two different chains and hence the relative magnitude of their order parameters. For the aggregate containing only octanoate chains, the model gives quite close agreement (both in magnitude and trend) with the experimental order profile (Figure 5, first profile). For the mixed amphiphile systems, we therefore decided to set the parameter x (in eq 2) at a value for which the average model octanoate order parameters agreed with the average experimental order parameters (Figure 5). When this constraint was applied, the hydrophobic cores were packed at constant density at the observed aggregate dimensions. Although for both mixtures the shape of the resulting decanol order parameter profile quite closely resembles the experimental profile shape, the model results are -25% too large (Figure 6). The free energy cost of transferring an -OH group from an aqueous environment to a hydrophobic environment is very substantially less that the corresponding cost for a 400- head group. Therefore, we decided to do calculations assuming that as well as being able to exist outside the hydrophobic core, the -OH group could also sit inside core within 1 A of its surface. Of course, this must be regarded as a tentative and crude approach to dealing with the (limited) solubility of -OH groups in a hydrophobic environment. In these calculations, the parameter x was again set at a value for which the average model octanoate order parameters agreed with the average experimental order parameters. The model octanoate order profiles differed only slightly from those shown
-
(11) Franks, N.
P.J . Mol. Biol. 1976, 100, 345
A Model for the Chains in Amphiphilic Aggregates
The Journal of Physical Chemistry, Vol. 89, No. 1, 1985 157
F I kcal mol"
31e
F I k T per chain
2
16
t
4
4
1
'
5
F I kT per chain
I;1 110"
200°
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/
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,
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r
i'O
I
/
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05
Distance from center to core surface I
x
Figure 7. Free energy cost of packing -(CH2)&H3 chains into bilayers (-), cylinders (---), and spheres as a function of aggregate size and temperatures. The graphs display F - FRc (see eq 3 and 4), the free (e-)
energy cost of transferring the chains from a (model) bulk n-alkane environment to the aggregate. The curves are drawn through the results at 110 OC. The bottom (top) of the vertical bars marks the results at 31 OC (200 "C). Where no bars appear, the three sets of results are almost identical. For comparison, the free energy gained when a C,, chain is transferred from water to bulk n-alkane at room temperature is 1 8 l ~ T Simply .~ because of the changes of shape, some chain conformations which sit completely within a bilayer core will be partly outside a cylindrical core and hence almost suppressed by the hydrophobic effect. This effect is even more pronounced for a spherical core. It explains the different depths of the minima in the three free energy curves. The standard deviation of the results is 0.025.
-
center of the micelle. Our best estimate of the associated free energy cost is 1 137 cal/mol of holes each with a volume of one CH3 group,l* and this free energy cost has been included in the calculations for micelles with R > 15.4 A. From Figure 7 it should be clear that even after including this extra penalty, the free energy cost of packing chains into spherical micelles increases only slowly up to R = 19 A. Further comment on this interesting point will be found in the Discussion section. Polynomial fits were obtained for the results at 31 OC:
+
+
F / k T = 5.6067 - 25.0940t 42.8854t2 - 33.3869t3 10.1568t4 0.42 < t < 1.42 bilayers
+
+
= 4.7940 - 13.2634t 15.4157t2 - 8.7521t3 2.0508t4 0.62 < t < 1.58 = -0.7728
cylinders
+ 8.2959t - 12.5093t2 + 6.8953t3 1.2774t4
0.96
< t < 1.95
spheres (5)
where t (expressed in nm) is the average half-thickness of the hydrocarbon region. Aggregates with six different thicknesses are used for each fit, and the largest deviation in F / k T between a polynomial and the points used in its generation is 0.004. As F / k T appears to vary little with temperature up to 200 O C (see Figure 7), the polynomials may be regarded as reasonably accurate up to 200 OC. Further, a best guess is that both F / k T and t should scale linearly with chain length. Free energy curves for bilayers have previously been generated from a model involving a central chain and six surrounding chains J ~these calculations, all seven arranged in a hexagonal a r r a ~ . ' ~In chains were constrained to exist in the same conformation. Judged by the results of the van der Ploeg and Berendsen molecular dynamics simulation,1,2this constraint seems very unrealistic. W e now introduce a definition for the mean area per chain (AAV)which will prove useful in comparing aggregates of different shape. A bilayer with an area per (single chain) amphiphile of A has an average area A available to a chain at all depths in the bilayer. For curved aggregates however, the area available to a chain decreases as the center of the aggregate is approached. We therefore define the mean area per chain as the volume-weighted mean area throughout the aggregate. For a spherical aggregate (12) Gruen, D. W. R.; de Lacey, E. H. 8. In 'Surfactants in Solution"; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 1, pp 219-306. (13) Belle, J.; Bothorel, P.Noun J . Chim. 1977, 1 , 265. (14) Lemaire, B.; Bothorel, P. Macromolecules 1980, 13, 311.
OOL
1
,?W, I
30
LO
50
60
Mean area per chain Anv I
70
xz
Figure 8. Free energy of -(CH2)&H3 chains at 31 OC as a function of the mean area available to them, A," (see eq 6 and 7 ) : (-), bilayers; (---), cylinders; spheres. The free energy is plotted on the left-hand scale. Also shown is the surface pressure which the chains in a bilayer impose on each other (-.-) (right-handscale). The vertical arrow marks the area per molecule observed by Mely et aL6 for bilayers of KtCOO-(CD2),,,CD3 at 21 wt % water at 31 OC. At this area per molecule, the chains are exerting a surface pressure of ?r = 25 mN/m on each (-e),
other. containing N amphiphiles, with V denoting the volume per chain (NV = 4irt3/3), we have NAAV = L'4a$(47rr2 dr/[4rt3/3])
(6)
Evaluating this integral and the appropriate ones for cylinders and bilayers leads to AAV = 9V/5t spheres
= 4V/3t = V/t
cylinders bilayers
(7)
From eq 1 of paper 1, the volume of a -(CH2)&H3 chain at 31 OC is 326.4 A3. The polynomials of eq 5 may be transformed via eq 7 to give the variation of chain free energy with mean area per chain (AAV).Figure 8 displays these free energies as well as the surface pressure (T = -dF/aAAV) applied by the chains in a bilayer as a function of the area per chain. When plotted as a function of hydrophobic core radius (R), the free energies for different aggregate shapes have minima at distinctly different dimensions (Figure 7). As a function of mean area per chain, however, the curves are surprisingly similar (Figure 8). The free energy curves all have a minimum in the range 37 < AAv < 44 A2, and in each case the free energy climbs slowly for larger mean areas. For smaller mean areas, the free energy climbs more rapidly especially for AAV< 30 AZwhere the number of conformations available to the chains is substantially reduced. For spheres with no hole at their center (Le. R I 15.4 A), AAV > 37 A2. For cylinders, the corresponding figure is AAv > 27 A2. For bilayers, as the mean area per chain falls, the chains become increasingly all-trans. Eventually, their packing density must increase and the van der Waals attractions between them must become stronger than in bulk n-alkane. These enhanced van der Waals attractions (which are not included in our calculation) account for the gel phase of the bilayer. It should be emphasized that the free energy of the chains (Figures 7 and 8) is only one contribution to the free energy of the system. Other important contributions arise from interaction of the head groups and from the free energy cost associated with chain-water contact at the aggregate surface. Thermodynamics requires that, at equilibrium, the total free energy is a minimum, but it is clear from Figures 7 and 8 that the chains do not, in general, exist in their lowest free energy state. What is clear is that the chains are very flexible and can be packed into aggregates with different geometry and a wide range of sizes at little free energy cost. It is this property that makes possible the multitude of phases observed in amphiphile-water and amphiphile-oil-water systems. Figure 9 shows how much the chains must be straightened in order to pack into aggregates of different sizes. For spheres of
Gruen
158 The Journal of Physical Chemistry, Vol. 89, No. I , 1985 Fraction o f gauche bonds
I
011,
I
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O0
6
L
2
8
-
.
\
10
12
Carbon number 1
a
I
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1
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16
Distance from center to core surface /
I
8.
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20
Figure 9. Average number of gauche bonds in -(CH2)LOCH3 chains packed into bilayers (-), cylinders (---), and spheres (--) as a function
of aggregate size and temperature. The number of gauche bonds is expressed as a proportion of the number in a random coil at the same temperature. In a random coil, the average number of gauche bonds is respectively 3.297, 3.570, and 3.773 at 31, 110, and 200 OC. The curves have been drawn through the data at 110 OC. The bottom (top) of the vertical bars marks the results at 31 OC (200 "C). Where no bar appears, the three sets of results are almost identical. The standard deviation of the results is 0.005. any possible dimensions, the chains are only slightly straighter than in their random-coil state. For cylinders and bilayers, there is more straightening especially for aggregates with a larger hydrophobic core radius. At the dimensions observed by Mely et a1.6 and Charvolin, the ratios (number of gauche bonds in the aggregate)/(number in a random coil) are 0.86 for bilayers at 31 OC, 0.86 for cylinders at 50 "C, and 0.95 for bilayers at 110 OC. Examination of the extensive data for Gallot and Skoulios' at 86 "C reveals these ranges for the area per amphiphile (at the amphiphile-water surface): for bilayers 30.1 < A < 44.9 A2 (from Tables 2 and 4 in ref 7) and for cylinders 47.5 < A < 57.1 A2 (from Table 1 in ref 7). The observed area is dependent on the water content of the phase but independent of chain length of the a m ~ h i p h i l e , ~ so we may use these ranges to infer hydrophobic core thicknesses for our -(CH2)&H3 chains. Over these ranges, all the bilayers may be formed at a free energy cost of less than 0.4kT per chain, and all the cylinders may be formed at a cost of less than 0.5kT per chain. Further, the fractional loss of gauche bonds on transfer of the chains from a model bulk n-alkane environment to the aggregates is less than 0.20 for all bilayers and less than 0.22 for all cylinders. The chains have not lost much of their configurational freedom. This is also true for the chains in dipalmitoyllecithin bi1a~ers.l~ We turn now to a comparison of some structural features of aggregates of different geometry. For any given aggregate shape, it is clear from the simulations that increasing the hydrophobic core radius increases the magnitude of the order parameters ScD all along the chain. An interesting question intrudes: are there any conditions under which the order parameter profiles for aggregates of different geometry are similar? The answer appears to be yes, when the mean area per chain (eq 7) is the same. Figure 10 shows a comparison of order parameter profiles for a sphere, a cylinder, and a bilayer with the same mean area per chain (AAV). The profiles are quite similar. By contrast, there is roughly a factor of 2 difference between the order parameters from a sphere and a bilayer with the same hydrophobic core radius (see Figure 10). The volume-weighted mean area per chain can be evaluated for any conceivable aggregate shape. For very different shapes (bilayer, cylinder, and sphere), there is a close similarity both in the variation of free energy with mean area per chain and in the order parameter profiles for the same mean area per chain. So, it is tempting to speculate that whatever the aggregate shape (including, for example, vesicles and inverted structures), calculation of the mean area per chain should provide a good measure (15) Gruen,
D.W. R. Chem. Phys. Lipids 1982, 30, 105.
Figure 10. Comparison of model C-D order parameters for aggregates containing -(CH7)10CH3chains at a mean area per chain of 30.7 A*. The order parameters are all evaluated with respect to the local normal at the amphiphile's head group: (-), bilayer with t = 10.6 A; (---), cylinder with t = 10.6 X 4/3 = 14.1 A; (-), sphere with t = 10.6 X 9/5 = 19.1 A. For comparison (-*-), sphere with t = 10.6 A. All calculaas the head-group orientions were carried out at 31 OC using a = tational weighting. The standard deviation of the results is 0.007. Carbon number
2
O3I
sc 3
6
L
I
I
12
lo I
I
I
I
I
\f
I "12
aI
"
0
I
1
o\.
;O
d8
d6
dL
d2
JO
Fractional distance from center to core surface
Figure 11. Model C-D order parameters for two of the aggregates studied in Figure IO: the bilayer with t = 10.6 A (-) and the sphere with t = 10.6 X 9/5 = 19.1 A The two profiles with marked points are the average order parameters for chains whose head groups sit further than 2.5 A from the hydrophobic core (top scale). The other two profiles are the average order parameters as a function of position in the aggregate (bottom scale). (-a).
of the extent of chain straightening required and the free energy penalty incurred in packing chains into that structure. Figure 11 shows two other order parameter profiles for the bilayer with t = 10.6 A and the sphere with t = 19.1 A. They should be studied in conjunction with Figure 10. For both aggregate shapes, when the head group is some distance from the core surface, the first few segments of the chain are significantly straightened so that as many chain segments as possible sit inside the hydrophobic core. The hydrophobic effect is sufficiently strong that the probability distribution of head-group position decays approximately exponentially with a decay length of -0.6 A. Although not shown here, this probability distribution is very similar to one displayed for an earlier version of this model.16 Particularly for the sphere, the variation of order parameter with depth in the aggregate (Figure 11) is quite different from the variation with carbon number down the chain (Figure 10). Explanations for the shapes of the former curves may be found e l s e ~ h e r e . ' ~Essentially, ~'~ the effect arises because of the substantial freedom of movement of the groups in the chain (see later, Figure 14). Figures 12 and 13 show the distribution of gauche bonds in the chain for aggregates with the same mean area per chain and for the free random-coil chain. At 31 O C , the mean number of gauche bonds in the random-coil chain is 3.30 while the corresponding numbers for the bilayer, cylinder, and sphere are respectively 2.85, 2.75, and 2.62. The shape of the probability density of the number of gauche bonds remain similar to the free chain profile (Figure 12). The main loss of gauche bonds occurs in the first 2 / 3 of the (16) Gruen, D.W. R. J . Colloid Interface Sci. 1981, 84, 281.
A Model for the Chains in Amphiphilic Aggregates
The Journal of Physical Chemistry, Vol. 89, No. 1. 1985 159
Probability
I
:i
I
\-
O‘t ;I
I
\* I
I
I
1
I
6 8 Number of gauche bonds
Figure 12. The probability density of the number of gauche bonds in the chain for the three aggregates studied in Figure 10 with the same mean area per chain and for the random-coil chain (assuming E , = 500 cal/ sphere; (e),random-coil chain. mol): (-), bilayer; (---), cylinder; The standard deviation of the model results is 0.007. (-e),
.
. I
Probability
042
c2 , * * I C-.--i
I
-01
L
2
1
c
I
I
I
I
CL 06 a8 1.0 Fractional distance from center to core surface
a2
Figure 14. Distribution of chain segments in those aggregates studied in Figure 10 with the same mean area per chain. Each dot shows the ensemble average position of a segment. The length of the bar is one standard deviation either side of the mean: (-), bilayer with t = 10.6 A; (--), cylinder with t = 14.1 A; sphere with t = 19.1 A. (.e-),
Probability density I
i - 7
‘I t 012
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11
i
/
/
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41
0 2 4
T 1
1
2
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I
1
L
1
6
I
8
10
Position of gauche bond in chain
Figure 13. The probability density of the position of gauche bonds in the chain for the three aggregates studied in Figure 10 with the same mean area per chain and for the random-coil chain (with E , = 500 cal/mol): (-), bilayer; (---), cylinder; sphere; (e), random-coil chain. k = 2 is the C C bond nearest the head group. The standard deviation of the results is 0.01 1. (.e.),
chain while the loss becomes progressively smaller as the free end of the chain is approached (Figure 13). The two figures reflect the fact that those internal bond sequences which contribute substantially to the properties of the chain in a bulk n-alkane environment suffer only modest changes in their probabilities when the chain is incorporated into an aggregate. In an aggregate the chains exist as ‘slightly perturbed random coils”.15 Figure 14 shows the mean and standard deviation of position of the segments of model -(CH2)&H3 chains packed into aggregates of different shapes but with the same mean area per chain. For the bilayer, the 3rd, 8th, and 1 lth carbons in the chain have standard deviations of depth in the bilayer of 1.0, 1.7, and 2.4 A, respectively, while the mean differences of depth in the bilayer are 4.4 A (3rd to 8th) and 2.0 A (8th to 1 lth). These numbers may tentatively be compared with results of neutron diffraction experiments on the -(CH2)&H3 chains of dipalmitoyllecithin in the lamellar liquid crystalline phase at 70 OC.17 (The spread v, defined by Zaccai et al., satisfies v = 2Il2cr, where u is the standard deviation.) The ex erimentally derived mean distances between the groups are 4.2 (3rd and 8th) and 2.4 8, (8th and 1 1th) while the standard deviations are respectively 1.1, 2.2, and 2.4 8,. The reason for this trend in spatial delocalization is clear. There is a large free energy cost associated with moving the head group away from the hydrophobic core surface and so it has a narrow spatial distribution. Although the aggregate’s shape and size
1
(17) Zaccai, G.; Biildt, G.;Seelig, A.; Seelig, J. J . Mol. Biol. 1979, 134,
693.
, 4
8
12
20
16
Distance from micelle center I
1
24
Figure 15. Comparison of results derived from neutron diffraction experiments and from the model for the distribution of terminal methyl groups in spherical dodecyl sulfate micelles at 35 OC. The micelles are observed to have a mean aggregation number of N = 78, and hence t = 18.8 A. If the methyl group probability density is p ( r ) , where r is the distance from the micelle center, the plot shows the volume-weighted probability density P ( r ) = 4?rCrZp(r),where C is a normalization constant: (-), model results with a = 0 as the head-group orientational weighting. To improve resolution, 12 evenly packed regions of the core were used to generate these results: (---), from the neutron diffraction from a ‘simple model” (see Discussion section). The experiments;18 four points used to generate this profile are also shown: profile generated assuming that the CH3 groups are distributed uniformily in a hydrophobic core with R = t = 18.8 A. (.am),
(-e-),
impose a (limited) straightening on the chains (see Figure 9 and associated text), this effect is too weak to overcome the entropic advantages of delocalization. Thus, for all three shapes, spatial delocalization increases all along the chain. For the spherical micelle, the volume available to the chains increases sharply with distance from the center. On taking an ensemble average, although the C H 3 grou is closer to the center than any other group, it sits a full 11.1 from the center and only 7.0 A from the hydrophobic core surface (Figure 14). Figure 15 shows the complete spatial distribution of the terminal CH3 groups in spherical micelles containing 78 G-(CH2)1ICH3 amphiphiles. The model results are compared with results from a neutron diffraction study.’* The model profile shows a pronounced peak near the micelle surface (though it should be noted that if t h e CH, groups were distributed completely uniformly throughout the micelle, there would be -60% more of them at the surface than the model predicts (see Figure 15)). This peak occurs because volume increases sharply with distance from the center of the micelle and also because there are many conformations which go some way into the core and then return toward the surface. Thus, of the 14% of CH, groups which sit within
w
(18) Bendedouch, D.; Chen, S.-H.;Koehler, W. C. J . Phys. Chem. 1983, 87, 153.
Gruen
160 The Journal of Physical Chemistry, Vol. 89, No. 1, 1985
penetration into the hydrophobic core, all segments of the chain spend an appreciable proportion of time near the core surface and hence in close proximity to the hydrophilic exterior.
Figure 16. A scale drawing of a spherical dodecyl sulfate micelle. The hydrophobic core surface is drawn as a perfect sphere although there is, of course, an equilibrium distribution of core shapes. The hydrophobic core radius is equal to the fully extended length of the chain, R = 16.7 A. At 25 "C, 55 chains can fit inside the core while the aggregation number of the micelle is approximately 58. The hydrophobic core is divided into five regions, each of thickness 3.34 A. The model is used to dictate the number of chain segments belonging to each region. For clarity, only the C-C skeletons of the chains are drawn. It appears that the outer parts of the micelle are crowded with chains while the central regions are almost empty. However, the effective volume of a CH3group is about twice that of a CH, group and also the figure is a two-dimensional picture of an evenly packed three-dimensional structure. By way of illustration, the innermost region has a volume of 156 A3, room for 2.9 CH, groups. On average, 1 of the 14 amphiphiles shown should have its CH, group in this region only 70% of the time!
1.5 A of the model core surface, almost half of these (44%) belong to chains which somewhere along their length have ventured at least 3 A into the core. The peak is particularly sharp because, in the model, CH3 groups are completely confined to the micelle's hydrophobic core. Relaxing this restriction would broaden the peak and lower its height. For real micelles, polydispersity of size and fluctuations of shape must further broaden and lower the peak. Notwithstanding these qualificationsit is clear that the neutron diffraction results show no hint of the peak. However, these results were obtained only for scattering vectors with a magnitude Q 5 0.24 A-1 (and hence QR < 4.6 where R is the average radius of the micelles). Such measurements are only sensitive to features of the methyl group distribution with low spatial frequencies. This point is graphically illustrated by Cabane et al.,19 who generate the scattering curves for model spherical particles of the same average size but formed of concentric spherical shells with a variety of scattering densities. For the four different scattering density profiles considered by Cabane et al., the resulting scattering curves are all identical for QR < 4.6 (see Figures 2 and 4 in ref 19). Bendedouch et al.'s neutron diffraction results over this range of QR values are well fitted by assuming a Gaussian distribution for the CH3 groups.'* At this level of approximation, the model gives close agreement with the experiments for both the mean ( r ) and standard deviation (a) of the distance from the micelle center to the CH3 groups. Thus, the model gives ( r ) = 11.3 and u = 4.4 8, while the experiments yield ( r ) = 11.9 A and u = 5.0 A. It should be noted, however, that at this level of approximation, very simple models indeed give good agreement with this experiment (see the Discussion section). Figure 16 shows a scale drawing of a spherical dodecyl sulfate micelle. The drawing shows clearly that even without any water (19) Cabane, B.; Duplessix, R.; Zemb, T. In "Surfactants in Solution"; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 1, pp 313-404.
Discussion When modeling all lamellar and hexagonal phases containing only one type of amphiphile, the head-group orientational parameter a is the only free parameter in the model. As Figure l demonstrates, changes in this parameter lead to only limited changes in the order parameter profile down the chain. This should be remembered when model order parameters are compared with experimental ones. As should be expected, even with the parameter a the model is not capable of realistically treating the orientational constraints acting on the beginning of the chain. Figures 2 and 3 demonstrate that the model does not give a good prediction for the order of C-2. Apart from this, for the bilayer with the smallest area per chain (Figure 2; A = 30.7 A2), the model order parameter profile is in good agreement with experiment, both in order of magnitude and trend. As the area per chain increases, the experiments reveal a reduction in order all along the chain and a progressively steeper profile. Both these trends occur in the model profiles, but to a greater extent. Both Figure 3 and the first profile in Figure 5 suggest that the model gives a good description of the order parameter profiles in hexagonal phases. The impressive fit displayed in Figure 4 suggests that the model also gives a good description for spherical micelles. Here, however, it should be remembered that the fit required an extra parameter (R, the hydrophobic core radius). In studying aggregates containing chains of two different lengths, we introduced a parameter which effectively sets the relative values of the average order parameters of the two chains. Thus, the model can claim no credit for giving order parameters of the right magnitude in the last two profiles in Figure 5. Credit can be claimed for mirroring the change in shape of the order profile as the aggregate dimensions and shape are changed. When an ensemble average is taken, a chain in a bilayer or a sphere must display cylindrical symmetry around the aggregate's local normal. This is not necessarily so for a cylindrical aggregate. We have used the model to examine how the chains in cylinders are distributed with respect to their aggregate's local normal. We use the notation developed in paper 1, where the z axis is the local normal (the head group always sits on this axis), the y axis is the cylinder axis, and the x axis completes a Cartesian set. We have analyzed model calculations corresponding to the hexagonal phases for the potassium laurate-water system (see Figure 3) and for the sodium octanoate-water system (see Figure 5, first profile). In both cases we have examined the behavior of (i) the vector VCM connecting the head group to the center of mass of the chain and (ii) the sum of the C-H vectors (equivalent to C-D vectors), VCH, on all the CH2groups in the chain. Define $CM ($cH)as the acute angle between the cylinder axis and the projection of VCM (VcH) on the x-y plane. When the probability densities are expressed in terms of the normalized variables $cM/900 and $CH/9O0, perfect cylindrical symmetry of both vectors around the z axis requiresf(+cM/900) = g($CH/90°) = 1. In fact, the probability densities are approximately linear satisfying f($cM/900) = 1.388 - 0.776($cM/900) r = -0.989
+ 0.112($CH/9O0)
r = 0.626
for the model potassium laurate system and f($cM/900) = 1.317 - 0.633($cM/900)
r = -0.953
g($c~/90") = 0.944
g(+cH/900) = 0.948
+ 0.104($cH/900)
r = 0.823
for the model sodium octanoate system. For both systems, there is some preference for the model chains to have a component along the cylinder axis rather than along the x axis. By contrast, there is a very slight preference for C-H (or C-D) vectors to have a component along the x axis rather than along the cylinder axis. The former tendency is easy to understand. As is common, in both cases the chains must straighten to some extent to pack into the aggregate. Other things being equal, a chain conformation
A Model for the Chains in Amphiphilic Aggregates which reaches close to the central cylinder axis will have a higher probability than one which does not. Assume an all-trans chain is tilted at a fixed angle with respect to the local normal, and imagine this chain being rotated around the local normal. As &M is reduced from 90 to Oo, this chain conformation will reach nearer and nearer to the central cylinder axis and hence will become progressively more probable. This effect is a general one, and it explains the preference for small values of 4CM. It is not so clear why &H shows a slight preference for large values rather than small ones. Nevertheless, the effect is sufficiently small that the often used6,8 assumption of cylindrical symmetry of C-H groups around the local normal is a good one. It has long been asserted that the fully extended length of an amphiphile chain defines the upper limit of the radius of spherical micelles formed by those amphiphiles (seee.g. ref 4). The assertion is based on the very reasonable argument that liquids do not contain well-defined holes. However, as we have shown (Figure 7), a spherical micelle is a special case because a small increase in its linear dimensions produces a large change in its aggregation number but only a very small hole at its center. For example, increasing the radius of a spherical micelle containing -(CH2),oCH3 chains from the fully extended chain length ( I , = 15.4 A) to 17.74 A increases the micelle aggregation number by -50% but creates a hole at the micelle center with a volume of only one CH, group. The free energy cost of forming such a hole should be about 1140 cal/mol or 1.9kT at room temperature,12 clearly not a prohibitive cost. This is a dramatic observation, but it must be treated with caution. A valid criticism of our model is that it has only been applied to aggregates with symmetrical hydrophobic cores (spherical, cylindrical, and lamellar). Real aggregates exist in an equilibrium distribution of shapes, and all nonspherical micelle shapes have parts of the micelle surface nearer to the micelle center than a sphere of the same volume. It requires only a small deviation away from a perfect sphere to eliminate a hole with a volume of one CH, group. But it is also clear that the above free energy considerations imply that a spherical shape should form a nonnegligible part of the ensemble of possible shapes of micelles containing 50% more monomers than a spherical micelle with radius I,. Calculations have been performed assuming that the free energy changes involved in distorting a spherical micelle to a prolate or oblate spheroid arise from the cost of increasing the surface area (assumed to be 18 mJ/m2) and the reduced cost of charging up the micelle (Bengt Jonsson and Bertil Halle, private communication). An axial ratio of 1.5 may then be obtained at a free energy cost always less than 3 k T per micelle (and possibly at a cost very much less than 3kT, depending on the extent of counterion binding to the micelle). Of course, oblate and prolate spheroids form only a very limited subset of the possible shapes of small micelles. We have already seen that the order parameter profile down the chain is strongly dependent on the mean area available to the chains but almost independent of whether the aggregate is a bilayer, a cylinder, or a sphere. This strongly suggests that whatever the fluctuations in micellar shape, the order parameter profile down the chain should be sensitive to them only in sofar as they alter the volume-weighted mean area per chain. It seems interesting to ask, why is the hydrocarbon-water interface observed to be so sharp?IsJ9 A calculation of the distribution of head groups away from the hydrophobic core for conditions chosen to model sodium dodecyl sulfate just above its cmc in the absence of added salt20 gave an average protrusion of 1.96 A. We have previously suggested that there are at least two effects which were not considered in that calculation and which should reduce this value.I2 Although it is difficult to quantify these effects, they imply that, at the hydrophobic core surface, the average volume fraction of hydrocarbon should fall from a value near one to a value near zero over a distance of the order of the linear dimensions of a water molecule or a CH, group (Le. (20) Aniansson, G. E. A. J . Phys. Chem. 1978, 82, 2805.
The Journal of Physical Chemistry, Vol. 89, No. 1, 1985
161
1-3 A). Although our model does not very accurately treat the interfacial region of the micelle, the results are not without interest. The model produces a profile of hydrocarbon volume fraction outside the hydrophobic core which closely follows the form $HC(r) = A exp(-(r - R ) / I ) . For all the aggregates studied, 0.54 < A C 0.91 and 0.52 C I < 0.77 A. In all cases, therefore, the hydrocarbon-water interface is fairly sharp. It may seem extraordinary that a structure as dynamic as a micelle (with monomers being associated with the micelle for only 10-6-10-s S ) ~ O could have such a sharp hydrocarbon-water interface. However, the uast majority of time a monomer is associated with an aggregate; it sits with almost all of its chain inside the aggregate’s hydrophobic core and hence with its head group in close proximity to the core surface. It does this because of the prohibitive free energy cost of doing otherwise. Other Models. Perhaps surprisingly, progress can be made in explaining the results of many experiments on ionic micellar systems by keeping in mind these simple points: (i) On average, each amphiphile associated with a micelle has almost all of its hydrocarbon chain in the micelle core. (ii) The ionic head groups, ions, and water are almost completely excluded from the micelle core. (iii) The chains (which cannot separate from their respective head groups) are conformationally disordered (liquid-like) and fill the core at approximately liquid n-alkane density throughout. In the third point, the parenthetical clause is included because some interpretations of the “liquid hydrocarbon droplet” micelle mode12’q22seem to ignore it. The experimental evidence in favor of these assertions is very strong indeed (see Introduction to paper 1). They form a common starting point for three recent attempts at modeling the state of the chains in amphiphilic aggregates: Fromherz’ surfactant-block Dill and Flory’s lattice mode1,22,24,25 and the present mode1.I2J6 (By contrast, Haan and Pratt26have attempted a more fundamental calculation. Such an approach is commendable, but in its present form their model predicts very substantial water contact with the amphiphile tails which would appear to be inconsistent with the experimental data cited in the Introduction of paper 1.) The surfactant-block model2, assumes that in a fluid bilayer the chains pack in a parallel array with each chain being perturbed by only one or two gauche-trans-gauche kinks. In a micelle, it is argued that the chains exist in the same conformational state as in a bilayer, except that in a micelle they are packed at right angles to each other in order to form a compact core. As we have seen (in excruciating detail) in paper 1, both the single-chain model and the molecular dynamics simulation suggest that the chains in a bilayer exist in an enormous variety of conformations, many of which lie only very roughly parallel to the bilayer normal. In terms of gauche rotamers, the chains are closer to the random coil characteristic of bulk n-alkane than to chains with only one or two gauche-trans-gauche kinks. The conformations available to chains in micelles (and their probabilities) are quite similar to those available to chains in a bilayer with the same mean area per chain (see Figures 12 and 13). In arguing for the surfactant-block model, Fromherz dismisses the “coiled radial micelle” (which, from his description, seems roughly what is envisaged in the present model). He claims that the coiled radial micelle requires that most CH2 groups in the chain do not make contact with the aqueous phase, that the chain ends are fixed, not mobile, and that the chains have a high conformational energy and are more coiled than in the bilayer (see Figure 2 in ref 23), all of which are contrary to experiment. The chains in a micelle do have a high conformational energy compared to an all-trans chain. However, they appear to have (21) Dill, K. A,; Flory, P. J. Proc. Nafl.Acad. Sci. U.S.A. 1981, 78, 676. (22) Dill, K. A. J . Phys. Chem. 1982, 86, 1498. (23) Fromherz, P. Ber. Bunsenges. Phys. Chem. 1981, 85, 891. (24) Dill, K. A,; Flory, P. J. Proc. Nafl.Acad. Sci. U.S.A.1980, 77, 3115. (25) Dill, K. A.; Cantor, R. S. In “Physics of Amphiphiles: Micelles, Vesicles and Microemulsions”;Degiorgio, .~ V., Corti, M., Eds.; North-Holland: Amsterdam, 1984. (26) Haan, S . W.; Pratt, L. R. Chem. Phys. L e f f .1981, 79, 436.
162 The Journal of Physical Chemistry, Vol. 89, No. 1, 1985
only a very slightly higher conformational energy than in bulk n-alkane (see Figure 9) and need not have any higher conformational energy than a bilayer. The other claimed properties of the coiled radial micelle clearly do not apply to the present model. The Dill and Flory model is conceptually similar to the present model in several respects. In common with the present calculation, the Dill and Flory model explicitly considers a single chain and the constraint of constant density of chain packing in the hydrophobic core is imposed on the ensemble of conformations of this chain. The exact probability of a chain configuration on the Dill and Flory cubic latticez4requires exact enumeration of the number of configurations available to all the chains, a problem which has never been solved. The Dill and Flory model has the distinct advantage that it involves only a very small amount of computing, and so results can be generated quickly. It is however handicapped by the fact that the chains are treated very crudely. A lattice with the same symmetry as the hydrophobic core is introduced into the core, and the chains are constrained to fill the lattice with one “segment” per lattice site. As the geometry of the aggregate changes, the geometry of the Dill and Flory chains must also change in order to fit onto the lattice. The lattice sites are equidimensional, and so each Dill and Flory chain segment corresponds to =3.6 CH2 groups. It is also assumed that once chains have ventured some way into the core, they cannot move back toward the surface. [Apart from special groups of conformations (see text accompanying Figure 1 9 , this assumption seems a good one. Thus, even in the case of a spherical micelle, the present model suggests that 16% of conformations exhibit a reversal of more than 2 A while only 1% exhibit a reversal of more than 4 A. (The calculation was performed at room temperature on a spherical micelle containing -(CHZ)&H3 chains with a radius equal to a fully extended chain.)] The Dill and Flory model can only provide a modest amount of information about the state of the chains in amphiphilic aggregates. For example, if it were used to model the system studied in part 1 of this work, there should be 9/3.6, hence 2 or 3 “segments” in the chain, and thus 1 or 2 internal bonds. Examination of the figures in both parts of this work demonstrates that the Dill and Flory model gives either no information or very rough information about most of the quantities which may be derived from our single-chain model or from the molecular dynamics simulation. Predictions of the Dill and Flory model have been compared with Cabane’s N M R relaxation studiesz7on (roughly) spherical micelles, with neutron diffraction studies on (roughly) spherical micelles,28 and with order parameter profiles for the three aggregate shapes. Cabane’s N M R studiesz7provide strong experimental evidence for the simple points enunciated at the beginning of this section. Models consistent with these points give good agreement with Cabane’s data. Both the Dill and Flory model and the earlier version of our model give good agreement.l2sz2 But the data are also predicted by Cabane’s singly bend chain modelz7 and by the present model assuming no energy difference between gauche and trans bonds. It is possible to design a model which assumes a densely packed core and which gives bad agreement with the data. Cabane shows that forcing all the C H 3 groups to sit as close as possible to the micelle center gives just that, but this model is hardly consistent with conformational disorder of the chains and a liquid-like core. Any model consistent with our three simple points would be hard pressed to give bad agreement with Cabane’s data! We turn now to the neutron diffraction experiments of Bendedouch et al.’* As we have already remarked, their measurements are only sensitive to features of the methyl group distribution with low spatial frequencies. As we now show, a very simple model based on our three points is sufficient to produce a methyl group
--
(27) Cabane, B. J . Phys. (Orsoy, Fr.) 1981, 42, 847. (28) Dill, K. A.; Koppel, D. E.; Cantor, R. S.; Dill, J. D.; Bendedouch, D.; Chen, S.-H. Nature (London)1984, 309, 42.
Gruen distribution consistent with these experiments. We assume a spherical hydrophobic core which contains all 78 amphiliphic chains and hence has a radius R = 18.8 A. (We may assume either small fluctuations away from a spherical shape or a very small hole at the center of the micelle-neither will make much difference to the methyl group distribution.) We divide the core into four spherical annuli with boundaries at r = 5, 10, and 15 A. For geometrical reasons, the inner sphere (0 < r < 5 A) must be rich in CH3 groups and we assume that they account for 70% of the volume of this sphere. (The volume of this sphere is less than 2%of the core volume; hence, wide variations in its assumed CH3 group concentration produce only minimal changes to the final result.) Again for geometrical reasons, we assume that the first three C H z groups in the chain are confined to the outside annulus (15 < r < 18.8 A) and that the next four CH2 groups are distributed evenly in the (remaining) volume of the outside two annuli (10 < r < 18.8 A). Finally, we assume that those CH3 groups which are not in the inner sphere are distributed evenly in the (remaining) volume in the outer three annuli. The resulting CH3 group distribution is shown in Figure 15. From this simple model we deduce a mean and standard deviation of position of the CH3 groups (the experimental results are in parentheses): ( r ) = 11.9 A (11.9 A); u = 4.3 A (5.0 A). The upshot of these deliberations is this. Assuming that the chains form a compact liquid-like core which is almost completely inaccessible to head groups, water, and ions is sufficient to explain both the data of Cabane and of Bendedouch et al. The ability of the Dill and Flory model to give good agreement with these data is a result of the assumptions which are common to their model, to the Fromherz model, and the present model (namely, the three simple points set down at the beginning of this section). It provides strong evidence in support of the assumptions but does not distinguish between different models based on these assumptions. We turn now to a discussion of order parameters. Order parameters along the chain have been measured for many different amphiphiles in bilayers, cylinders, and spheres. Without exception, the order parameter is observed to fall as the free end of the chain is approached. Our single-chain model also exhibits this behavior in all cases. As we have seen, it also provides fairly close quantitative agreement with experiments both for the magnitude of the order parameters and for the profile shape. By contrast, the original version of the Dill and Flory model applied to curved aggregates2’ predicts that the order parameter slowly increases along the chain from the head group to the free end of the chain for both spheres and cylinders. This problem can be overcome if the average orientation of the first segment in the chain is changed and chain stiffness is introduced. Then the Dill and Flory model yields a decreasing order parameter profile along the Nevertheless, it should be remarked that this situation does not augur well for the predictive capacity of this model. For spherical aggregates, Dill and Flory are led by their model to conclude that the micelle center should exhibit a “degree of order approaching that in a crystal”.z’ Reflection on this conclusion suggests that it is erroneous. The one or two fully extended (and hence fully ordered) chains which sample the micelle center can diffuse rapidly around the micelle surface and can also exchange rapidly with other chains.’* The average order parameter in the innermost 2%of the core volume is small (for the spherical micelle studied in Figure 11, it is -0.18) and hence, even here, the micelle core is liquid-like. To conclude, some progress can be made by a thoughtful application of the three points listed at the beginning of this section. Neither the Fromherz model nor the Dill and Flory model seems to add much to this progress. In some cases, predictions based on these models contradict experimental findings or are not sustained by a more careful analysis. Summary 1. Our single-chain model is used to generate order parameter profiles down the chains in bilayers, cylinders, and spheres. Fully saturated chains containing between 6 and 11 CH2 groups are
J. Phys. Chem. 1985, 89, 163-171 studied in aggregates with a range of experimentally determined dimensions. In most cases, the model provides close quantitative agreement both in magnitude and trend with experimentally determined order parameters. 2. In common with other models, our model provides good agreement with both the N M R relaxation experiments of Cabane” and with the neutron diffraction experiments of Bendedouch et alelaon (roughly) spherical micelles. However, both experiments provide evidence supporting the assumptions inherent in the models, rather than distinguishing between different models based on those assumptions. 3. Defining t / l c as the ratio of the average half-thickness of the chain region to the fully extended chain length, the model suggests for Cll chains that at a free energy cost of less than 0.5kT per chain these aggregates may be formed: bilayers with ? / I c < 0.75, cylinders with ?/I, < 1.00, and spheres with t / I , < 1.25. The chains in any of these aggregates have, on average, more than 80% of the number of gauche bonds that they would have in the random-coil state characteristic of bulk n-alkane. For most of the experimentally observed aggregate dimensions which were examined, the above numbers (0.5kT free energy cost and 20% reduction in number of gauche bonds) represent upper bounds. Those internal bond sequences which contribute substantially to the properties of the chain in a bulk n-alkane environment suffer only modest changes in their probabilities when the chain is incorporated into an aggregate. In an aggregate, the chains exist as “slightly perturbed random coils”. 4. The fully extended chain length, I,, is almost universally used as the radius of the largest possible “spherical micelle”. Our model suggests that this is misleading, because micelles containing 50% more monomers should be only marginally more aspherical than micelles with an average radius equal to I,. Both exist in an equilibrium distribution of shapes, and the model suggests that for both the spherical shape makes a nonnegligible contribution to this equilibrium distribution.
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5. A unifying idea is introduced: the mean area available per chain in any aggregate. When results for the three aggregate shapes are compared, a close similarity is observed both between the order parameter profiles at the same mean area per chain and between the variations in free energy with mean area per chain. For any proposed aggregate (including irregularly shaped and inverted structures), calculation of the mean area per chain should provide a good measure of the extent of chain straightening required and the free energy penalty incurred when chains are packed into that structure. Note Added in Proof: In an elegant paper, Ben-Shad et al.29 have established an important result for single-chain calculations. They show that minimization of the chain free energy subject to the constraint of constant density in any number of distinct regions in the hydrophobic core leads directly to the condition imposed in eq 3 of paper 1. Ben-Shad et here, i.e. to the term (njajk) al. solve a Dill-Flory type model using this superior statistical mechanical approximation and derive results in semiquantitative agreement with those presented here.
Acknowledgment. I am indebted to Professor J. Charvolin for much informative correspondence about his past experiments and for doing the experiment which forms the basis of Figure 3. I am grateful to Olle Soderman and Bernard Cabane for sending me preprints of their papers and to Jacob Israelachvili for his artistic rendition of Figure 16. I have also benefited from discussions and comments from Bernard Cabane, Bertil Halle, Jacob Israelachvili, Roland Kjellander, Stjepan Marcelja, Professor Theo Overbeek, Thomas Zemb, and a most diligent reviewer. During the course of this work, I have been in receipt of a Queen Elizabeth I1 Fellowship. This research was not supported by any military agency. (29) Ben-Shaul, A.; Schleifer, I.; Gelbart, W. M. In “Physics of Amphiphiles: Micelles, Vesicles and Microemulsions”;Degiorgio, V., Corti, M., IUS.; North-Holland: Amsterdam, 1984.
Phase Behavior of Multicomponent Systems Water-Oil-Amphiphile-Electrolyte.
3
M. Kahlweit,* R. Strey, and D. Haase Max-Planck-Institut fuer biophysikalische Chemie, 0-3400 Goettingen, West Germany (Received: April 12, 1984; In Final Form: July 24, 1984)
In the two preceding papers of this series we studied the phase behavior of ternary systems of the type H20-oil-nonionic surfactant and of quaternary systems of the type H20-oil-nonionic surfactant-inorganic electrolyte. We showed that in applying inorganic electrolytes one has to distinguish between lyotropic salts, Le., salts which decrease the mutual solubility between H 2 0 and surfactant, and hydrotropic salts, which increase the mutual solubility. This difference leads to a qualitatively different phase behavior of the two types of the above quaternary systems. In part 2 we presented the influence of lyotropic salts, in particular of NaCl, on the phase behavior. In this third paper we compare the influence of hydrotropic salts, in particular of NaC104 and (C6HS).,PC1, which are not surface active, with that of NaDS as an ionic detergent. We find the influence of both types of salts on the phase behavior of the ternary system H20-oil-nonionic surfactant to be quite similar, which is of importance for the further discussion of the microstructure of so-called microemulsions. Finally, we suggest discussing the phase behavior of multicomponent systems of the type. H20-nonpolar liquids-nonionic surfactants-hydrotropic salts-lyotropic salts in terms of a pseudoquaternary phase tetrahedron. This leads to a transparent interpretation of the phase behavior of such systems on the basis of that of “simple” systems as treated in parts 1 and 2.
I. Introduction In section I1 of this paper we discuss the influence of the hydrotropic NaC104 on the phase behavior of ternary systems of the type H 2 0 (A)-oil (B)-nonionic surfactant (C). In section I11 we discuss the phase behavior of quaternary systems in a pseudoternary phase prism. In section V we compare the effect of ionic detergents with that of hydrotropic salts which are not surface active. In section VI we show that the effect of adding an electrolyte is equivalent to that of adding a second nonionic surfactant C’. In section VII, finally, we discuss the phase behavior 0022-3654/85/2089-0163$01.50/0
of multicomponent systems in a pseudoquaternary phase tetrahedron by placing pure H20 (A) in one comer of the basic triangle A-B-C, all nonpolar liquids as an oil (B) of effective hydrophobicity in the second corner, all nonionic surfactants and hydrotropic salts as an amphiphile (C) of effective hydrophilicity in the third corner, and all lyotropic salts (E) on top of the tetrahedron. 11. NaCIO4 as Hydrotropic Electrolyte
In the second paper of this series2 we showed that with respect 0 1985 American Chemical Society
