J. Phys. Chem. 1987, 91, 137-145
and 12-CH3 is the same, within error, for dodecane, premicellar SDS, and micellar SDS. This is consistent with the low degree of order at the CH3ends of the chains and the fact that all C H 3 ~ c values f are similar. The similarity between N O E values for premicellar and micellar SDS is somewhat surprising. It suggests that segmental order does not change much when one goes from the premicellar to the micellar state. This observation is not entirely unexpected in light of the substantial evidence indicating that 5 mM SDS consists mainly of dimers and/or higher oligomer~.~~,~~ Conclusions At 361 MHz, the TI relaxation rates for protons in SDS micelles are determined almost exclusively by the rates of fast local motion (bond torsions, isomerizations, and librations). The spin-lattice relaxation for protons is dominated by intrasegmental dipolar interactions. The dependence of the rates of fast local motion upon segment position can be calculated and are found to be greatest at either end of the amphiphile. These rates, obtained from 'H T I data, are in quantitative agreement with the rates obtained from I3C T I data. Information on the local molecular order in SDS micelles is obtained when I3C T Ior 'H TI data are combined with values for I3C T2)s or 'H NOES,respectively. The order (38) Mukerjee, P. Adv. Colloid Interface Sci. 1967, 1, 241. (39) Shchipunov, Y. A. Colloid Interface Sci. 1984, 102,36.
137
profiles obtained by using either carbon or proton N M R are qualitatively consistent and indicate that the molecular order for SDS micelles increases slightly from segment position 1 to 2, remains rather constant over most of the chain, and decreases at the methyl end. The high sensitivity of 'H NMR allows us to compare the dynamics and ordering of premicellar as well as micellar SDS. The rates for the fast local motions, which are similar in premicellar SDS and neat dodecane, decrease by a factor of only 1.5 in SDS micelles. Thus, the rates for fast local motion are relatively independent of the statc of aggregation. Segmental order appears to be similar in premicellar and micellar SDS. Much recent work has focused on the precise structure of ionic micelles and the influence of the headgroup in micellar struct ~ r e . ' ~Clearly, * ~ further N M R studies, similar to those presented above, could help characterize micelle structure and clarify apparent differences among micelles composed of different amphiphiles. Acknowledgment. This work was supported by a grant from the National Institutes of Health (GM35215), a grant from the Jeffress Trust (5-61), and a Dreyfus Foundation grant for young faculty in Chemistry (all to D.S.C.). Registry No. SDS, 151-21-3. (40) Jones, R. R. M.; Maldonado, R.; Szajdzinska-Pietek, E.; Kevan, L. J . Phys. Chem. 1986, 90, 1126, and references therein.
Solubilization in Lyotropic Llquld Crystals: The Concept of Partial Molecular Surface Area Neville Boden,* Stephen A. Jones, and Frank Six1 Department of Physical Chemistry, The University, Leeds LS2 9JT, England (Received: June 25, 1986)
A case is made for the need to measure the partial molecular properties of the constituents in the lyotropic liquid crystal phases formed in amphiphile/water/solubilizate mixtures. By way of illustration, the concept of the partial molecular surface
area of a constituent in such a mixture is formulated. The procedures employed for the calculation of this quantity from X-ray data are outlined. Results are presented for the two-component soap/water and lecithin/water mixtures and for the three-component potassium oleate/water/decanol and lecithin/water/cholesterol mixtures. They provide a new insight into the relationship between structure and composition: in particular, the effects of varying the concentration of either water or solubilizate on the structure are manifest in a quantitative manner. For the soap/water systems, the partial molecular surface areas provide an insight into the mechanisms by which the counterion affects the structure of the aggregate; this is not apparent from direct studies of the structure. For these systems it is also demonstrated that, contrary to current theories of amphiphile aggregation, the chain-packing free energy significantly influences the optimal surface area per amphiphile in an apparently chain-length-independentway. The transition from the lamellar to hexagonal phase with increasing concentration of water is seen to be governed by the interplay of the interfacial forces, which want to increase the surface area of the aggregate, and the chain-packing free energy, which wants to maintain the volume-weighted mean area per chain close to its optimal value of around 0.40 nm2. In the lamellar phases of the three-component systems, the partial molecular surface areas of the amphiphile and solubilizate behave similarly with respect to variation of the concentration of solubilizate. The addition of solubilizate enhances the degree of order of the bilayer core at a rate that rapidly attenuates with increasing concentration. On the other hand, the partial molecular surface area of the water behaves quite differently, yet in a manner consistent with the different chemical structures of the bilayer-water interface. In the case of the potassium oleate/water/decanol system, the results show that incorporation of decanol into the bilayer stabilizes the lamellar phase by increasing the average separation between neighboring carboxyl groups: this effects a reduction in both the electrostatic repulsion between these groups and also the steric repulsion of the hydrated potassium ion. The net effect is to substantially reduce the total repulsive force at the interface so that the surface area per amphiphile (soap + decanol) is maintained well below the upper limit of stability (g0.45 nm2) irrespective of the concentration of water.
Introduction The lyotropic liquid crystals obtained on dissolution of amphiphilic substances in water may structurally be regarded as ordered arrays of either discrete micelles or more extensive aggregates of amphiphilic molecule^.^-^ The principal phases are (1) Ekwall, P. Adu. Liq. Cryst. 1975, 1 , 1-142. (2) Tiddy, G. J. T. Phys. Rep. 1980, 57, 1-46.
0022-3654/87/2091-0137$01.50/0
nematic (orientationally ordered solutions of columnar Nc or diskoid N, micelle^),^ columnar (two-dimensional arrays Of COh m aggregates), ~ k"-Ilar (alternating layers of bimolecular a€%regates and water), and cubic (Phases of cubic symmetry with (3) Luzzati, V. In Biological Membranes; Chapman, D., Ed.; Academic: London, 1968, pp 71-123. (4) Boden. N.; Radley, K.; Holmes, M. C. Mol. Phys. 1981, 42, 493.
0 1987 American Chemical Society
138 The Journal of Physical Chemistry, Vol. 91, No. 1 , 1987
quite complex s t r ~ c t u r e s ) .The ~ manner in which a solubilizate interacts with and perturbs the structures and properties of the amphiphilic aggregates is fundamental to many of their applications, as, for example, in cellular biology, detergency, emulsification, or enhanced oil recovery. It also provides an insight into the factors that determine the relative stabilities of the different phases, knowledge essential for the development of a theoretical understanding of aggregation in amphiphile-water mixtures. Studies of solubilization in liquid crystals have, to date, focused on the structures3 of the aggregates and phase diagrams.’ It is, however, of vital importance to understand the underlying thermodynamics of the process. To achieve this we propose to apply the classical thermodynamic techniques and methodologies that have, hitherto, been used to study multicomponent isotropic liquid mixture^.^ The results have been couched in terms of partial molar quantities such as partial molar volumes, heat capacities, chemical potentials, etc. The aim of this paper is to demonstrate the utility of adapting and applying the methodology of partial molar quantities to lyotropic liquid crystals. In particular, we shall describe the measurement and application of a new quantity that we call the partial molecular surface area. This concept was first described6 at a NATO summer school in 1983. Here, we present a more complete formulation of the concept and delineate the procedures employed for the calculation of this quantity from X-ray diffraction data. Results are presented for the two-component soaplwater and lecithinlwater mixtures and the threecomponent potassium oleate/water/decanol and lecithin/ water/cholesterol mixtures. Concept of Partial Molecular Surface Area Consider a volume element of lyotropic mesophase of arbitrary structure containing NA, Nw, and Ns molecules of, respectively, amphiphile, water, and solubilizate. The total surface area A of the amphiphilic aggregate can be related to the molecular composition in several ways. The customary approach is to express the total area A in terms of the area per amphiphile S A
= A/NA
(1)
This is useful for a simple two-component amphiphilelwater system but has little practical significance when applied to systems with three or more components. This is because the surface area of the aggregate is a communal or cooperative property of all of the constituents of the mesophase. A precise prescription is, therefore, required for apportioning the surface area among the various constituents. To do this we must recognize that the total surface area of the aggregate is an extensive property of the phase and can be expressed as a function of temperature T, pressure p , and the amount ni of each component, i.e., A(T,p,nA,nw,ns). Now, as this is a homogeneous function of degree one of the variables ni, it is easy to show (Euler’s theorem) that = CNiAi T.p,n,fn,
(i = A, W, S) ( 2 )
i
where Ni = Lnj
and (3) L is the Avogadro constant and the quantity Ai defines the partial molecular surface area of component i. In a three-component system the mean area per molecule at the aggregate-water interface is A A, = = CxiAi (4) N A + N W + Ns j ( 5 ) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworth Scientific: London, 1982. ( 6 ) Boden, N.; Jones, S . A . NATO ASI Ser., C 1985, 141, 497.
Boden et al. where x i is the mole fraction of component i. Since xA + xw + xs = 1, there are two independent composition variables. But when the ratio of the amounts of any two of the three components, say nW/nA= xW/xA = RWIA,is fixed, only the mole fraction xs is independent. Differentiating A, with respect to xs at constant T , p , and RWIAand recognizing the constraint
we obtain
AA
-
-
RW/AAW
+ RWJA
+ RW/A
+As (5)
since
and RWJA
xw =
(1 - x s )
-k RWJA
Multiplying eq 5 by xs and combining the result with eq 4 give A,=xs(2)
T,P.Rw/A
+
1 + RW/A (AA + RW/AAW)
(6)
Similarly, multiplying eq 5 by (1 - xs) and combining the result with eq 4 give
Equations 6 and 7 tell us that the tangent to the curve of A, vs. xs at xs = xs’ has intercepts As(xS’) at xs = 1 and [ ( l / ( l RWIA))(AA+ RWIAAw)](xs’) at xs = 0 as illustrated in Figure 1. Alternatively, choosing xw as the independent variable and fixing the ratio XSIXA = RSIA,we obtain
+
A,=xw(”.)
axW
+-
T,P,RSIA
1
+ RS/A
(AA + RS/AAS) (8)
Proceeding as above, values for both Aw(xw’) and (AA + Rs/A As)(xw’) corresponding to any value xw’ of xw can be obtained. Combining the results of the two experiments yields values for AA’, Aw’, and As’ corresponding to the composition xA’, xw‘, xs’. To obtain the partial molecular surface areas of the various components it is necessary to calculate the mean molecular surface area A, as a function of composition. The way in which values for A,,, are obtained from X-ray measurements for both lamellar and columnar-hexagonal phases is described below. Lamellar Phase. The classic model of the lamellar phase is one of alternating layers of indefinitely extending bimolecular aggregates and water (Figure 2 ) . The hydrophilic-hydrophobic interface is considered to be quite sharp with negligible water penetration into the The bilayer thickness db is related to the bilayer repeat distance do by
4 E &do
(10)
where &, is the volume fraction of the bilayer. Values for do can (7) Buldt, G.; Gally, H. U.; Seelig, A,; Seelig, J.; Zaccai, G. Nature (London) 1978, 271, 182. (8) Dilger, J. P.; Fisher, L. R.; Haydon, H. A. Chem. Phys. Lipids 1982,
30, 159.
The Journal of Physical Chemistry, Vol. 91, No. 1, 1987 139
Partial Molecular Surface Area I
Am I
Figure 3. (a) Cross-section of a hexagonal liquid crystal phase showing the honeycomb-like arrangement and the two repeat distances dl and dz wit': their relationship to the center-to-center separation s of the cylindrical aggregates. (b) Prismatic section of one element of the honeycomb of length I . Ahexand A,] are the cross-sectional areas of the hexagon and the cylinder, respectively.
I
Am I
Figure 1. Schematic illustration of the procedure used to calculate the partial molecular surface areas of all of the components in amphiphile/water/solubilizate mesophases. (a) The mean molecular surface area A , is plotted against the mole fraction of solubilizate xs at constant mole ratio RWIA'of water to amphiphile. (b) A , is plotted against the mole fraction of water xw at constant mole ratio RSIA' of solubilizate to amphiphile. area a
Columnar-Hexagonal Phase. The classic model is that of a two-dimensional hexagonal arrangement of cylindrical aggregates as illustrated in Figure 3a. It is also considered that the water-aggregate interface is well-defined and that there is negligible water penetration into the core of the aggregate. The X-ray diffraction pattern of a sample with a random distribution in the orientation of the optic axis consists of a series of concentric rings with spacings in the ratios: 1:1/3Il2:1/4lI2: 1/7lI2:1/ 12lI2:.... The innermost ring arises from the spacing dl (see Figure 3a) which is related to the center-to-center separation s of the cylindrical aggregates by d , = (31/2/2)s
(14)
To calculate the radius r of the aggregate we assume that all of the solubilizate is incorporated within the core of the aggregate. The volume fraction $cyl of aggregates, which can be calculated with eq 12, is, therefore, given by
where Acyland Ahexare respectively the cross-sectional areas of the cylindrical aggregates and the hexagonal prisms in Figure 3a. Hence, from eq 15, we obtain
Figure 2. Element of a lamellar phase showing the bilayer repeat distance do, the bilayer thickness db, the water layer thickness dw,and the surface area a.
be obtained by measuring the diffraction angle 20 and using the Bragg equation nX 5: 2do sin 8 (11) With the assumption that all of the solubilizate is within the bilayer XAVA+ xsVs b'
= xA VA
+ xw VW + xs VS
(12)
where is the partial molar volume of component i. Now, if the volume element ado of bilayer in Figure 2 contains N A , Nw, and Ns molecules of respectively amphiphile, water, and solubilizate, then the mean area per molecule at the bilayer-water interface is 2a A, = NA + Nw + Ns But, since Ni =
2 A, = -ExiVj Ldo i
LadO9i -
v,
( i = A, W, S )
(13)
We now consider the prismatic element in Figure 3b, which has volume Ahcxland contains water and solubilizate. The mean area per molecule at the aggregate-water interface is 27rl A,,, = NA + Nw + Ns But, since
4 rr A, = - -XxiVi (i = A, W , S) (18) 3112 Ls2 i Comments on Calculation Procedures. The first step in the calculation of partial molecular surface areas is the calculation of values for the mean molecular surface area A , using either eq 13 or 18, according to the structure of the mesophase. This requires values for the partial molar volumes of the constituents. The measurement of these quantities is, in principle, straightforward-it simply involves measurement of the mass density as a function of composition. However, this is a major undertaking, especially for three-component systems, because of the difficulty of removing air bubbles from the very viscous samples. Nevertheless, this kind of data is important as it is needed for the interpretation of all diffraction experiments. For the few lyotropic liquid crystal systems for which partial molar volumes have been measured?-l3 the values have been found to be quite (9) Ekwall, P.; Eikrom, H.; Mandell, L. Acra Chem. S c a d . 1963,17,1 1 1.
140 The Journal of Physical Chemistry, Vol. 91, No. 1, 1987
Boden et al.
TABLE I: Hydration Energies and Ionic Radii for the Alkali Metal Ions” and Values for q and SA’(See Eq 20) for Alkali Metal Soaps in Lamellar and Hexagonal Phases at 359 K1‘ lamellar phase cation Na’
hexagonal phase
-AH/(kJ mol-’)
ionic radius/nm
4
SA’/nm2
4
SA’/nm2
405 321 300 217
0.102 0.130 0.149 0.170
0.24 0.21 0.20 0.15
0.241 0.256 0.259 0.286
0.09
0.397 0.396 0.395 0.396
K+
Rb’
cs+
0.10 0.11 0.11
N
E
\ C
cia N
0.7
0.8
0.9
1.0
-
XW
Figure 4. Variation of the partial molecular surface area of the amphiphile AA with mole fraction of water xw for various alkali metal soaps in lamellar phases at 359 K.
close to those of the molar volumes of the pure components at the same temperature. For this reason the molar volumes of the pure components are often used instead of the partial molar volumes in the calculation of aggregate dimensions from X-ray diffraction measurements. For the calculation of the partial molecular surface areas presented herein we have used existing surface area data that has been obtained on this basis. The values of the partial molecular surface areas will also be subject to errors arising from the calculation of the intercepts of the tangents to the A,,, vs. composition curves. In practice, these are calculated analytically from polynomial functions obtained by fitting these data. The errors in the resulting partial molecular surface areas will, therefore, depend on the extent and precision of the original X-ray measurements. The data we have used is not ideal for our purposes and the errors are undoubtedly greater than they would be for properly designed experiments. A rough estimate for the errors in the results presented in Figures 4-12 are A, = 10-15% Aw i= 5-lo%, and As = 10-50% with the greater error being associated with the smaller values of As. Application to Two-Component Systems Soap-Water Systems. For a two-component system, eq 8 reduces to
A , = X W ( ” >axW
7,p
+A,
and eq 9 is still applicable. Thus, the tangent of the plot of A, vs. xw has intercepts Aw and AA on the xw = 1 and xw = 0 axes, respectively. To illustrate the procedure we shall make use of the extensive X-ray diffraction data for systems of saturated alkali metal soaps and water, as reported by Gallot and S k o ~ 1 i o s . l ~ These authors have studied the effect of composition, chain length, counterion, and temperature on the aggregate dimensions in both lamellar and columnar-hexagonal phases. The original data has been reanalyzed by Ekwall,’ who found that in both mesophases the relation between the surface area per amphiphile SAand the water content could be empirically represented by S A = SA’RWJA‘ (20) (IO) Mandell, L.; Fontell, K.;Lehtinen, N.; Ekwall, P. Acra Polytech. Scand., Chem. Incl. Metall. Ser. 1968, 74, 11. (11) Tardieu, A.; Luzzati, V.;Reman, F. C . J . Mol. Biol. 1973, 75, 71 1. (12) Kita, Y.;Bennett, L. J.; Miller, K. W. Biochim. Biophys. Acta 1981, 647,130. (13) Laggner, P.; Stabinger, H. J . Colloid Interface Sci. 1976, 5 , 91. (14) Gatlot, B.;Skoulios, A. Kolloid Z.U.Z. Polym. 1966, 208, 37.
0
0.0
0.7
0.9
1.0
XW
Figure 5. Variation of the partial molecular surface area of the water Aw with mole fraction of water xw for various alkali metal soaps in lamellar phases at 359 K.
0.8
0.9
1.0
XW Figure 6. Variation of the partial molecular surface area of the amphiphile AA with mole fraction of water xw for various alkali metal soaps in hexagonal phases at 359 K.
where RWIA is the mole ratio of water to amphiphile. SArand q are constants whose values are summarized in Table I and are seen to depend on counterion and temperature, but, significantly, not on the alkyl chain length. To obtain the partial molecular surface area of each component we could reanalyze the data to evaluate the mean molecular surface areas, plot these against the mole fraction of water, and take tangents of the resulting curve. It is, however, far simpler to obtain analytic expressions for A , and Aw from eq 20. To do this we recognize that the total surface area is A=
NASA
(21)
from which it follows that A = NASArRW/AP or, since RWiA= Nw/NA
From eq 3 we then obtain and
(22)
The Journal of Physical Chemistry, Vol. 91, No. 1, 1987 141
Partial Molecular Surface Area
0.8
0.9 XW
1.o
Figure 7. Variation of the partial molecular surface area of the water Aw with mole fraction of water xw for various alkali metal soaps in hexagonal phases at 359 K.
Equations 24 and 25 have been used to calculate the partial molecular surface areas of both soap and water using the values of q and SA’given in Table I. The results are summarized in Figures 4-7. Figure 4 shows the variation of AA with water content in the lamellar phases of sodium, potassium, rubidium, and cesium soaps at 359 K (86 “C). Figure 5 shows the variation of Aw for the same systems. The corresponding results for hexagonal phases are summarized in Figures 6 and 7. The hexagonal phase, which occurs at higher water content, is characterized by higher values of AA and lower values of Aw. In both phases, increasing the water content causes an increase in AA and a decrease in Aw. This behavior indicates that the addition of water causes an expansion of the interface, presumably a consequence of the hydration of the ionic head groups. The behavior of Aw shows that as the water content increases the hydration of the surface tends to saturate. The mechanism by which hydration leads to swelling of the aggregates is revealed by the dependence of the data on the counterion. This is most significant in the lamellar phase: the partial molecular surface area of the amphiphile is seen to be largest when the counterion is cesium and smallest when it is sodium (Figure 4). Conversely, the partial molecular surface area of water is largest for the sodium soaps and smallest for the cesium ones (Figure 5). A plausible explanation for these observations is as follows. At lower water concentrations the counterions are intercalated between neighboring carboxylate groups. However, on addition of water they become hydrated. But the fully hydrated counterions are too large to occupy the “intercalation” sites and are forced to migrate to new binding sites atop the carboxylate layer. (It is quite possible that in these sites the counterions bridge carboxylate groups in opposing bilayers.) As a consequence, there will be a stronger lateral electrostatic repulsion between the carboxylate groups within the interface, resulting in an expansion of the surface area. This process is expected to be more favorable for the smaller cations, which have the higher hydration energies (Table I). It is precisely for this reason that the partial molecular surface area of water increases as the size of the counterion decreases. The assertion that the counterions are bound at either intercalation sites between neighboring carboxylate groups or at interbilayer sites is supported by the results of recent N M R exp e r i m e n t ~ . ’ ~39KN M R studies of the lamellar phase of the potassium oleate/water system are consistent with the exchange of K+ ion between two such binding sites, with the fraction at the interlayer site increasing with water content and temperature. It is important to recognize that the effect of the counterion on the interaction of water with the aggregate is only apparent in the partial molecular surface area and not in the dimension of the aggregate. To illustrate this point we have plotted in Figure 8 values for the surface area per amphiphile SAas a function of composition in the lamellar phase, as calculated with eq 20. It can be seen that in the concentration range R W I A between 5 and (15) Boden, N.; Jones,
S.A. Zsr. J . Chem. 1983, 23,
356.
I P+ 30 0
5
10
15
20
RWA
Figure 8. Variation of the surface area per molecule of amphiphile SA with the mole ratio RWIA of water to amphiphile for various alkali metal soaps in lamellar Dhases at 359 K. Curves were calculated with the relationship S A = S,’RWI~~(eq 20) and the values for SA’and q given in Table I.
10, where the lamellar phase is stable, the structures of the bilayer aggregates are seemingly very similar with nearly identical values of SA,irrespective of the counterion. This observation would seem to indicate that the underlying interactions that determine the surface area are independent of the counterion. However, the values of the partial molecular surface areas show that the similarities in the values of SAare a coincidence and are the result of two opposing effects. At a given water content one expects the fraction of cations in the intercalation sites between the carboxylate groups to be greater for the larger cations. As a consequence there will be a smaller electrostatic repulsion between the carboxylate groups. This will tend to cause SAto become smaller as the size of the counterion increases. but this effect is largely compensated by the greater fraction of bound larger counterions which have a greater steric interaction and tend to push the carboxylate groups apart. Thus, while the SAvalues are essentially independent of the counterion, the ion distribution at the bilayer-water interface is not. At higher water concentrations, the differences in the values of the partial molecular surface areas for the various soaps become smaller (Figures 4 and 5) and, furthermore, are much less pronounced in the hexagonal phase (Figures 6 and 7). The latter result suggests that in hexagonal phases, where RWIA covers the range 6-40, the counterions are essentially fully hydrated and there is little tendency for them to reside within the carboxylate layer. Effect of Chain Length. For a soap with a particular chain length the mesophases exist only over narrow ranges of concentration, but for different soaps these ranges overlap. At a given temperature, the mesophases occur at higher water contents for longer hydrocarbon chain lengths. Consequently, although in each phase the data for all soaps with a particular counterion lie on a smooth curve, it is not clear whether this arises as a result of a truly continuous change in the values of the partial molecular surface area or as a result of the superposition of the data for each different soap. In particular, it is possible that the smooth curves shown in Figures 4-7 represent the average behavior of all of the soaps but not the individual behavior of any one of them. To clarify this uncertainty we have analyzed the original data of Gallot and S k o ~ l i o sfor ’ ~ the individual soaps. Figure 9 shows the results for the lamellar phases of the potassium soaps with four different chain lengths. The behavior for the individual soaps is within processing errors seen to be identical with the average behavior represented by the dashed curves. Phospholipid/ Water System. Phospholipids and water form lamellar phases which often exist over a much wider concentration range than the ordinary soap/water systems.I6 This property makes them more suitable for reliable measurements of partial molecular quantities. X-ray data of the lamellar phase formed (16) Reiss-Husson, F.; Luzzati, V. Adu. Bioi. Med. Phys. 1967, 11, 87.
Boden et al.
142 The Journal of Physical Chemistry, Vol. 91, No. 1, 1987
N
E
\
a P
N
consequence of the hydration of the interface. Discussion. The most striking single feature of the above results is that the dependence of the values of A, and Aw on water concentration is qualitatively the same for all of the systems examined, irrespective of the chemical structure of the amphiphile and the geometry of the amphiphilic aggregate. In all cases, the surface areas of the aggregates expand with increasing water concentration. This is a direct consequence of the hydration of the hydrophilic groups in the interface, which is the driving force for changes in the size and shapes of the aggregates. The hydration of the interface causes the aggregates to relax to a state of lower free energy. But the extent to which the structure may change is constrained by interaggregate interactions. These can be repulsive double-layer forces’*or indirect solvent “structural” Both tend to maximize the separation and minimize the surface areas of the aggregates. Thus, in a mesophase the surface area of an aggregate will always be smaller than that of the fully relaxed structure which would obtain at infinite dilution where the interaggregate forces vanish. This 0 as xw 1.0 (see Figures 4, 6, and 10). is the reason Aw It also follows that AA (xw 1.0) = SAo,the surface area per amphiphilic molecule in the fully relaxed aggregate. For xw < 1, the value of A, is always less than the average surface area per amphiphilic molecule SA. The reason for this is readily understood by combining eq 1 and 2 to give
- -
-+
0.7
0.9
0.8
1.o
xw
Figure 9. Variation of the partial molecular surface areas of amphiphile A A and water Aw with mole fraction of water xw for potassium soaps in lamellar phases at 359 K. Dashed curves are from Figures 4 and 5 and represent the average behavior of a variety of soaps with different chain lengths. Continuous curves show the results for individual soaps with particular chain length C,, as indicated by the values of n.
I
0.8
1.0
0.9 XW
Figure 10. Variation of the partial molecular surface area A iwith mole fraction of water xw for both components in the lamellar phase of the dimyristoylphosphatidylcholine/watersystem at 3 10 K. For values of xw > 0.96, the lamellar phase is in equilibrium with excess bulk water (“swelling limit”).
by the phospholipid dimyristoylphosphatidylcholine (DMPC) and water at 310 K (37 “C) and for various concentrations has been published by Janiak et a].’’ The values for the partial molecular surface areas AA and Aw were calculated from a polynomial function obtained by fitting the A,,, vs. xw data. The results are shown in Figure 10. As the amount of water is increased, AA is Seen to increase, whereas Aw decreases. This behavior is similar to that for the lamellar phases of the soap/water systems. In both cases the addition of water causes the bilayer surface to expand rapidly at low water concentrations, but the effect saturates at high concentrations. The values of Aw are quite similar at cor0 as xw 1. The responding xw and, in both cases, Aw similarity in the behavior of these structurally quite different systems is a direct consequence of the common origin of the swelling behavior of bilayers. In both cases the latter is a direct
-
-
(17) Janiak, M. J.; Small, D. M.; Shipley, G. G. J . Biol. Chem. 1979, 254, 6068.
= S A - RW/AAW Thus, we see that AA < SAat finite concentrations whenever Aw > 0. Equation 26 tells us that a value for SAo can be obtained by extrapolating AA to xw = 1.O. The values thus obtained are in the range 0.40-0.45nm2 and 0.55-0.60 nm2 for respectively the lamellar and hexagonal phases of the soap/water systems, apparently irrespective of the counterion. These values are quite consistent with those calculated for C12surfactants in various geometries:22it was concluded that for SA< 0.47 nm2 only bilayer aggregates can exist, while for 0.70 > SA> 0.47 nm2 cylindrical aggregates are predicted. The limiting values of the area per amphiphilic molecule in the unstressed, isolated aggregates are, significantly, independent of the length of the alkyl chain (see Figure 9) and thus appear to be exclusively determined by the “opposing forces” that act at the i n t e r f a ~ e . ~These ~ - ~ ~are, notionally, the repulsive forces (electrostatic, steric, and hydration) between adjacent head groups and the hydrophobic attractive “force” arising from the interaction between water and the hydrocarbon chains which tend respectively to maximize and minimize the average area per amphiphilic molecule at the surface of the aggregate. But the limiting values for the isolated planar and cylindrical aggregates are quite different and both are smaller than the estimated22lower limit (0.70 nm2) for a spherical micelle. There must, therefore, be some additional interaction, dependent on the geometrical shape of the aggregate, which constrains the expansion of the surface. The van der Waals attraction between chains is far too weak at the distances in question to be important. Rather, the conformational free energy of the chains and the constraints imposed on it by packing the chains into aggregates of different g e o m e t r i e ~ is ~ ~the - ~controlling factor. Gruen?’ using ~~~~~~~
~~~~~~~
~~~
(18) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (19) Parsegian, V. A.; Fuller, N.; Rand, R. P. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 2750. (20) Marklja, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (21) Israelachvili, J. N.; Sornette, D. J . Phys. (Les Ulis, Fr.) 1985, 46, 2125. (22) Mitchell, D. J.; Tiddy, G. J. T.; Waring, L.; Bostock, T.; McDonald, M. P. J . Chem. SOC.,Faraday Trans. I 1983, 79, 915. (23) Tanford, C. The Hydrophobic Effecr; Wiley: New York, 1980. (24) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J . Chem. SOC., Faraday Trans. 2 1976, 72, 1525. (25) Israelachvili, J. N.; MarEelja, S.; Horn, R. G. Q. Reu. Eiophys. 1980, 13, 121. (26) Jonsson, B.; Wennerstrom, H. J . Colloid Interface Sci. 1981, 80, 482.
Partial Molecular Surface Area a mean field model, has calculated the chain-packing free energy (for -(CH2)&H3 chains at 31 “C) as a function of the volumeweighted mean area per chain A,, (=9V/5t for spheres, 4V13t for cylinders, and V l t for bilayers, where Vis the volume of the hydrocarbon chain and t the half-thickness of the hydrocarbon core). The free energy curves (Figure 8 in ref 27) all have a minimum in the range 0.37 C A,, C 0.44 nm2 with the free energy at the minimum increasing in the order bilayer C cylinder C sphere. Using in the above expressions for A,, our values of SAo ( = V / t for bilayers and 2Vlt for cylinders), we obtain for fully relaxed isolated bilayers A,,,“ = 0.40-0.45 nm2 and for cylinders A,,,“ = 0.37-0.40 nm2, values that compare very closely with those at the free energy minima (0.44 nm2 for bilayers and 0.37 nm2 for cylinders) as calculated by Gruen. This is a remarkable result. It implies that the chain-packing free energy is an important factor in determining the equilibrium structure of an aggregate. This has hitherto been n e g l e ~ t e d because ~ ~ - ~ ~ of the apparent chainlength invariance of the surface area per am~hiphi1e.l~ While A,, will be independent of chain length for aggregates of given geometry, we do not know how the chain-packing free energy will vary. However, the experimental behavior suggests that the shapes of the free energy vs. A,, curves in Figure 8 of ref 27 are independent of chain length. It is not clear to us why this should be so. Nevertheless, it is clear that the transition from the lamellar to hexagonal phase with increasing concentration of water as observed for the soaps is govemed by the interplay of the interfacial forces, which want to increase the surface area of the aggregate, and the chain-packing free energy, which wants to maintain the value of A,, close to its optimal value of around 0.40 nm2. The observation that the value of Aw is always positive, provided of course that xw C 1, is quite important. It tells us that in a mesophase an aggregate is always in a compressed state, Le., in a state of high free energy. This internal stress is partially relaxed a t the transitions from lamellar to hexagonal phases and the hexagonal phase to the isotropic micellar solution where the aggregates are, in general, sperical. Clearly, the formation of a large interfacial area exposed to water is energetically favorable. But it is not easy to distinguish between the roles of electrostatic repulsion and hydration of the head groups. The decrease in free energy that accompanies the change in size and shape of the aggregates with increase in water concentration must be a significant source of free energy for the spontaneous uptake of water by lyotropic mesophases. This is of particular significance for the behavior of the lamellar phases formed by zwitterionic phospholipids such as DMPC3 These amphiphiles possess two hydrocarbon chains per molecule. This enhances the hydrophobic interaction to such an extent that they are unable to form aggregates with cylindrical or spherical geometries. Instead, they form lamellar phases that spontaneously take up water with a concomitant swelling of the water layers and a lateral expansion of the bilayers until a critical concentration, the so-called “swelling limit”, is attained.3 At greater water concentrations the lamellar phase coexists in equilibrium with bulk water. Now, the value of Aw at the swelling limit for DMPC (Figure 10) is finite and positive. This means that the bilayers are under lateral compression. The fact that the lamellar phase does not spontaneously take up water until the fully relaxed bilayer structure is attained implies the existence of a net attractive interbilayer force. Put in other words, the decrease in free energy, associated with the lateral expansion of the bilayers, is the source of the free energy expended in pulling the surfaces apart against this attractive force. The hydration limit is attained when the free energy available becomes insufficient to pull the surfaces further apart. We have recently argued31that the dominant forces (27) Gruen, D.W.R. J . Phys. Chem. 1985, 89, 146, 153. (28) Ben-Shaul, A.;Szleifer, I.; Gelbart, W. M. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 4601. (29) Ben-Shaul, A,; Gelbart, W. M. Annu. Rev. Phys. Chem. 1985, 36,
. ,.
170 I
(30) Ben-Shaul, A.;Szleifer, I.; Gelbart, W. M. J. Chem. Phys. 1985, 83, 3591,3612. (31) Boden, N.;Sixl, F. Faraday Discuss. Chem. Soc., in press.
The Journal of Physical Chemistry, Vol. 91, No. 1, 1987 143
0
0
0.5
1.0
RSh
Figure 11. Variation of the partial molecular surface areas of all three components in the lamellar phase of the potassium oleate/water/decanol system as a function of the mole ratio of solubilizate to amphiphile RSIA at 293 K. Results are shown for water/amphiphile mole ratios of RWjA = 5.3 and RWiA = 1.6.
between zwitterionic phospholipid bilayers are indeed attractive and probably have their origin in surface-induced perturbations in the structure of the water layers. Application to Three-Component Systems Three-component systems are of more practical importance than the simple two-component mixtures. To illustrate the procedure outlined in the theoretical section we shall consider two systems for which adequate X-ray data have already been reported. These are the lamellar phases of the potassium oleate/water/decanol and the egg lecithin/water/cholesterol systems. Potassium Oleatel WaterlDecanol System. Ekwall et aL3*have reported measurements of the bilayer surface area per surfactant molecule as a function of water content at several fixed mole ratios of decanol to potassium oleate. The data are not very extensive but sufficient to extract values for the partial molecular surface areas of all three components in order to illustrate the utility of the approach and the nature of the information attainable. Values of the partial molecular surface areas were obtained by fitting the A , vs. concentration curves to a polynomial function and calculating the intercepts of the tangents analytically. The variation of the partial molecular surface areas of all three components with the mole ratio of solute to soap R s / A at two fixed mole ratios of water to soap RWiAis shown in Figure 11. Because of the paucity of the data, there being only three data points at each value of R W / A , there is some uncertainty in the behavior of both A, and As. However, a more detailed study of this and similar systems has recently been completed in our laboratory and although the results obtained showed quantitative differences with those presented here, the qualitative behavior is the same. There are a number of notable features in the results: First, the partial molecular surface area of the water Aw falls from approximately 0.16 nm2 in the two-component system to almost zero at high decanol concentrations. This corresponds to a transition from a situation in which water interacts with and causes an increase in the interfacial area of the bilayer to one in which all added water is intercalated between the bilayers, causing an increase in the bilayer separation without any lateral expansion, i.e., a transition from a “nonswelling” to a one-dimensional “swelling” system. This behavior has its origin in the effect of the decanol on the interactions at the bilayer-water interface. It is generally accepted3335that C, alcohols with n > 6 are solubilized (32) Ekwall, P.;Mandell, L.; Fontell, K. J . Colloid Interface Sci. 1969, 31, 508, 530. (33) Klason, T.;Henriksson, U. Solution Behauiour of Surfactants: Theoretical and Applied Aspects; Mittal, K. L., Fendler, E. J., Eds.; Plenum: New York, 1982,Vol. 1, p 417. (34) Stilbs, P.J . Colloid Interface Sci. 1983,94, 643.
Boden et al.
144 The Journal of Physical Chemistry, Vol. 91, No. 1, 1987
with their hydroxyl groups located between neighboring carboxylates at the interface and with the hydrocarbon chain penetrating into the core of the bilayer. 39KNMR studies15have shown that in the presence of decanol the counterions are preferentially located between the charged carboxylate groups at low water concentrations. However, as the concentration of water is increased, the ions are displaced into outer layer sites atop of the carboxylate groups, as for the two-component system, but in this case the expansion of the surface is substantially smaller. This is because of the increase in the average separation of the charged carboxylate groups due to the presence of the decanol, which relaxes the repulsive electrostatic interactions. This, in turn, relaxes the lateral compressive stress and supresses the bilayer to hexagonal phase transition and the tendency to form micellar solutions at high water concentrations. Instead, the lamellar phase remains stable and takes up large amounts of water with concomitant onedimensional swelling, presumably as a result of repulsive double-layer forces between the bilayers. Second, the partial molecular surface area of decanol (As in Figure 11) at infinite dilution, Le., when R S j A= 0, typifies the solubilizate-solvent (bilayer) interaction. The value in this case is approximately 0.1 1 nm2 and is considerably smaller than the cross-sectional area of the decanol molecule (=0.37 nm2).6 The explanation is that the expansion of the bilayer brought about by the addition of decanol is largely offset by an enhancement in the order of the hydrocarbon chains with a commensurate reduction in their cross-sectional area. The value of As is seen to increase with the concentration of decanol, indicating a saturation of this ordering effect. The value of As is smaller at higher water concentrations, because the oleate chains are now disordered and consequently more responsive to the addition of decanol. Third, the behavior of the partial molecular surface area of the amphiphile AA is more complex than that of either of the other two components. The decrease in AA a t high decanol concentrations is quite simply a result of the induced ordering of the hydrocarbon chains within the core of the bilayer. On the other hand, the initial increase at low concentrations is more difficult to comprehend. It is a consequence of the effect of the decanol on the interaction of water with the surface of the bilayer. This can be understood by combining eq 1 and 2, which gives
where Ss is the area per solubilizate molecule at the bilayer surface. At low concentrations, RS/A is small and the third term in eq 27 can be neglected. The behavior of AA is now seen to be mainly determined by the way in which Aw varies with RS/* The initial increase in the value of AA is, therefore, quite simply a reflection of the decrease in the value of Aw with increasing concentration of decanol. At high decanol concentrations, Aw 0 and the values of both AA and As tend toward a limiting value of 0.25 nm2, which is close to the limiting value of 0.25-0.26 nm2 for the mean interfacial area per amphiphilic molecule in the swelling systems. This limiting behavior is again explicable by eq 27: at high decanol concentrations, Aw = 0, Ss = As, and A, i= SA.These limiting surface areas are approaching the value for the cross-sectional area of an unsaturated hydrocarbon chain in a densely packed monolayer.23 Egg Lecithin/ Water/Cholesterol System. Egg lecithin is a complex mixture of phospholipids with the same head group (choline) but a variety of alkyl chains and is extracted from egg yolks. Despite the heterogeneity of its composition it can be treated as a single component.j6 The egg lecithin/water/cholesterol system is, therefore, quite analogous to the potassium oleate/ water/decanol system and the partial molecular surface areas A , can be obtained by the same procedures. Suitable X-ray data have been published by Lecuyer and Deri~hian.~'These authors
-
(35) Stilbs, P.; Lindman, B. Prog. Colloid Polym. Sci. 1984, 64, 39. (36) Bourges, M.; Small, D. M.; Dervichian, D. G. Bioehim. Biophys. Acra 1967, 137, 157. (37) Lecuyer, H.; Dervichian, D. G. J . Mol. Biol. 1969, 45, 39.
0
0.5
1.0
bA Figure 12. Variation of the partial molecular surface areas of all three components of the egg lecithin/water/cholesterol system with the mole ratio of solubilizateto amphiphile RSIA.Numbers indicate mole ratios of water to amphiphile RW,*. The temperature of the X-ray measurements used for these calculations is not quoted by the author^.^'
have measured repeat distances do for egg lecithin/water/ cholesterol mixtures over a wide range of concentrations. The variation of the partial molecular surface areas A, of all three components with the mole ratio of cholesterol to lipid Rs,A at two fixed mole ratios of water to lipid RW,A is shown in Figure 12. Aw is seen to be, within experimental error, independent of thqcomposition and to have the value 0.006 f 0.001 nm2. This behavior is different from that of the potassium oleate/water/ decanol system (Figure 11) and indicates that the interaction of water with the aggregate surface is not significantly affected by the presence of cholesterol. The reason might be that the cholesterol O H group is located too far in the interior of the bilayers to have a significant effect on the structure of the interface, which is characterized by the rather bulky choline head groups. Extensive X-ray and neutron scattering experiment^^^.^^ have shown that the OH group is located at, and possibly hydrogen bonded to, the carbonyl group of the sn-2 chain of the lecithin molecules, leaving the polar head groups unaffected. The behavior of the partial molecular surface area of cholesterol As is similar to that for decanol in the potassium oleate/ water/decanol system and a similar interpretation can be attached to it. At RS/A = 0, As has the values of =O and -0.02 nmz at RW,A values of 20 and 35, respectively; the corresponding values at Rs/A = 0.98 are 0.28 nm2 (RwiA = 20) and 0.32 nm2 (RWIA = 35). Comparing these results with the cross-sectional area of 0.36 nm2 of the cholesterol molecule40tells us that its incorporation into the bilayers induces an ordering of the lecithin chains. This so called "condensing effect" of cholesterol has been studied in great detail (for a review, see ref 41) by a variety of techniques. It is most apparent in the negative values of As for small cholesterol-amphiphile mole ratios Rs,A at the higher water to amphiphile mole ratio of RWIA= 35 (Figure 12), where the total surface area of the bilayer is smaller in the presence than in the absence of cholesterol. Again, as the amount of cholesterol increases, the extent to which the lecithin chains become ordered decreases and As increases. Since the partial molecular surface area of the water Aw is essentially independent of RSIA,the variation in the partial molecular surface area of the amphiphile at constant hydration must solely be a result of the ordering of the lecithin chains (see eq 27). Conclusions The concept of partial molecular surface area has been shown to provide a new insight into the relationship between the structure and composition of lyotropic liquid crystals. In particular, the (38) Franks, N. P. J . Mol. Biol. 1976, 100, 345. (39) Worcester, D. L.; Franks, N. P. J . Mol. Biol. 1976, 100, 359. (40) Joos, P.; Demel, R. A. Biochim. Biophys. Acta 1969, 183, 447. (41) Yeagle, P. C. Biochim. Biophys. Acra 1985, 822, 267. (42) Atkins, P. W. Physical Chemistry, 2nd ed.; Oxford University Press: Oxford, 1982.
J . Phys. Chem. 1987, 91, 145-148 effects of varying the concentration of water or solubilizate on the structure are manifest in a quantitative manner. The data should be useful for testing theoretical models of aggregation and phase behavior in amphiphile/water/solubilizate mixtures. More generally, the measurements should be useful in investigative applications of solubilizate-aggregate interactions. For example, combining such data with complementary N M R measurements on counterions, water, amphiphile, and solubilizate is expected to provide an insight into the molecular mechanism of the solubilization process and into the manner in which the solubilizate affects the mesophase stability. To demonstrate this strategy we have carried out a detailed study of solubilization in the bilayers of the lamellar phase of the potassium oleate/water system; the results will be published in the near future. A more detailed study
145
of the interaction of cholesterol with lecithin bilayers will also be published elsewhere. Finally, we emphasize that the partial molecular surface area is just one example of a whole range of partial molecular properties that could be measured. We believe that measurements of these properties will provide a wealth of information about the relationship between molecular and interaggregate interactions and the structure and properties of lyotropic mesophases. Acknowledgment. We thank the Science and Engineering Research Council for research studentship to S.A.J. and the University of Leeds for a research fellowship to F.S. Registry No. Potassium oleate, 143-18-0;decyl alcohol, 112-30-1; cholesterol, 57-88-5.
Quantum Mechanical Calculations on Molecular Sieves. 1. Properties of the Si-0-T (T = Si, AI, B) Bridge in Zeolites E. G. Derouane*?and J. G. Fripiat? Central Research Laboratories, Mobil Research and Development Corporation, Princeton, New Jersey 08540 (Received: June 4, 1986)
Ab initio (nonempirical) molecular orbital calculations have been performed to investigate the behavior and the stability of Si-GT bridges such as found in zeolites. Dimeric model clusters of the type (OH),Si4M-T(OH),, where M is a monovalent charge compensatingcation and T is Si, Al, or B, were considered. It is shown that elongation of the T-O bond, upon replacement of Si by Al, occurs preferentially by a local deformation of the Si-O-A1 bridges, Le., a rearrangement of the 0 atoms directly coordinated to AI. The stability of the Si-0-T bridge increases with the electropositive character of the charge compensating cation M. The H form of the boron-containing cluster appears particularly unstable, suggesting that tetrahedrally coordinated framework boron in the hydrogen form of crystalline borosilicates has poor stability.
Introduction
Quantum chemical a b initio calculations may be a help to investigate the stability and the behavior of Si-0-T bridges ( T = Si, Al, or B) in zeolite frameworks, in particular because such properties are not always easily quantitated by experimental means.’ Quantum chemical calculations become reasonable when the cluster model is adopted. However, for computational reasons, the size of the clusters is determined by the choice of the theoretical method. The restricted Hartree-Fock-Roothaan (LCAO-SCFMO) approach has proved to be successful in zeolite and silica chemistry as shown, for example, by the works of Sauer et a1.,24 Hass et a1.,5.6Geisinger et al.,’ Mortier et al.,s and Derouane et aL9-I2 Generally, in these idealized geometries were used for the model clusters. In contrast, the models chosen in our previous investigation^^-'^ reflected more intimately the structure of the zeolites under study as they were derived from crystallographic structural data. The latter remark holds also for the geometry of the bridge clusters considered in the present investigation. This paper examines the effect of various cations, Le., H+, Li+, and Na+, on the charge distribution and stability of the Si-0-A1 and Si-0-B bridges in zeolites, and discusses possible mechanisms by which the T-0 bond can be elongated by relaxation of the framework upon substitution of Si by Al. Theoretical Method and Models Restricted Hartree-Fock-Roothaan results were obtained by using the STO-3G basis et'^,'^ and the GAUSSIAN-80 program Present address: Center for Advanced Material Research, Facultds Universitaires Notre-Dame de la Paix, 61, rue de Bruxelles, B-5000 Namur, Belgium.
0022-3654/87/2091-0145$01.50/0
TABLE I: Interatomic DistancesU bond distance, nm 0-H 0.0975 Si-0 0.160 T-0 0.160 s-O(1) 0.160 AkO(1) 0.176
bond B-O(1) H-O(1) Li-O(1)
Na-O( 1)
distance, nm 0.136 0.0975 0.200 0.229
package,I5 adapted for the 36-bit computerword of the Digital DEC 20/60 computer. All bielectronic integrals larger than 10” Sauer, J.; Zahradnik, R.Int. J. Quantum Chem. 1984, 26, 793. Sauer, J.; Hozba, P.; Zahradnik, R. J. Phys. Chem. 1980, 84, 3318. Sauer, J.; Engelhardt, G. Z . Naturforsch. 1982, 37a, 277. (4) Sauer, J. Acta Phys. Chem. 1985, 31, 19. (5) Hass, E. C.; Mezey, P. G.; Plath, P.J. J. Mol. Struct.: THEOCHEM 1981. 76. 389. ( 6 ) Hass, E. C.; Mezey, P. G.; Plath, P. J. J. Mol. Struct.: THEOCHEM 1982, 87, 26 1. (7) Geisinger, K. L.; Gibbs, G. V.; Natrotsky, A. Phys. Chem. Miner. 1985. 11. 266. (8) Mortier, W. J.; Sauer, J.; Lercher, J. A,; Noller, H. J. Phys. Chem., 1984, 88, 905. (9) Fripiat, J. G.; Berger-Andre, F.; Andre, J. M.; Derouane E. G. Zeolites 1983, 3, 306. (10) Derouane, E. G.; Fripiat, J. G. In Proceedings of the 6th International
Zeolite Conference; Olson, D. H., Bisio, A., Eds.; Butterworths: Guildford, UK, 1984; p 717. (1 1) Derouane, E. G.; Fripiat, J. G. Zeolites 1985, 5, 165. (12) Fripiat, J. G.; Galet, P.; Delhalle, J.; Andre, J. M.; Nagy, J. B.; Derouane, E. G. J . Phys. Chem. 1985,89, 1932. (13) Hehre, W. J.; Stewart, R. F.; Pople, J. A. J . Chem. Phys. 1969.51, 2657. (14) Hehre, W. J.; Ditchfield R.; Stewart, R. F.; Pople, J. A. J . Chem. Phys. 1970, 64, 5142.
0 1987 American Chemical Society
