-
J. Phys. Chem. 1989, 93. 3809-3813
than the detection limit (estimated a t 10%). Simplifying the domain geometry to a cubic structure and using the known values for the lipid cross-sectional area and bilayer thickness (42 A2 and ) , can calculate 24 A, respectively, for anhydrous D L - D P P C ~ ~one in a straightforward manner that the domains must have a minimum size of -700 A, corresponding to lo5 lipids. This result compares favorably with the domain size in DSPC/DMPC hydrated bilayers for which a cooperative length of -500 A was found with neutron d i f f r a ~ t i o n .It ~ is possible in principle that domains of this order of magnitude cannot be detected separately with DSC. It has been shown for polymer blends that the detection limit may very well be greater than 1000 A.23 Therefore, we surmise that the domain sizes in “monohydrated” DL-DPPCare about 1000 A. The packing of anhydrous L-DPPC differs significantly from that in the anhydrous DL modification as is revealed by comparing the chemical shifts of specifically the sn-2 C=O, CHO, and w-methyl carbons. Given the fact that exactly the same sample preparation and drying techniques were used, it is concluded that the discrepancy in packing properties is caused partially by stereospecific interactions. Apparently, these interactions allow a more tightly packed arrangement for DL-DPPCthan for the L form as is demonstrated by the downfield shift of the internal methylene groups and the w-methyl groups of DL-DPPCcompared with the L form. This finding is supported by the DSC measurements showing a phase transition for the DL form some 20 OC higher than for the L enanti01ner.I~ Furthermore, additional evidence is found in a previous electron diffraction study on anhydrous L- and DL-DPPCfrom which it was concluded that the chains in L-DPPC are tilted with respect to the layer plane, whereas in the DL form essentially no tilting occurs.24 In bilayers of anhydrous L-DPPC 2 equiv of water can be accommodated without structural rearrangements in the hydrocarbon chains of a magnitude that they would impact the I3C N M R chemical shifts. Although the changes in anhydrous DL-DPPCby water are not accompanied by a substantial increase in gauche conformations in the acyl chains of DL-DPPC,as was suggested by Raman ~pectroscopy,~ there are certainly some pronounced differences in acyl chain and headgroup packings between the anhydrous and hydrated phases. Therefore, it is concluded that the discrepancy between the previous Raman9 and (23) DeLassus, P. T.; Wallace, K. L.; Townsend, H. J. Polym. Prepr. (Am. Chem. SOC.,Diu. Polym. Chem.) 1985, 26, 116. (24) Sakurai, 1.; Sakurai, S.; Sakurai, T.; Seto, T.; Ikegami, A,; Iwayanagi, S. Chem. Phys. Lipids 1980, 26, 41.
3809
the recently performed I3C N M R study,1° in which no changes in packing differences could be detected, can be largely ascribed to differences in optical purity of the used DPPC. It is interesting to compare the present results with a detailed study on chiral aggregation phenomena in bilayers of DL- and L-DPPC in Despite the high purity of the phospholipids, no significant differences could be demonstrated between racemic DPPC and its enantiomers by using various sensitive techniques. This finding is consistent with the present results on hydrated DPPC samples. However, our results also clearly point out that packing differences due to stereospecific interactions have to be taken into account in tightly packed anhydrous systems. Apparently, on hydrating anhydrous bilayers of DPPC, all chirally induced packing differences are overruled by the concomitant structural change in the lipid molecules. Vice versa, since dehydration of phospholipids is an important step in various membrane processes,8.26stereochemical control might be effected by different packing arrangements between phospholipids and other chiral molecules in the membrane bilayer. It was argued that in this way the membrane can act as a stereospecific screen toward molecules of unnatural chirality.25
Concluding Remarks The 13CNMR chemical shifts clearly indicate that the addition of 2 equiv of water to anhydrous DPPC bilayers results in a different packing of the hydrocarbon chains in the case of DLDPPC. This change in packing results in a very similar structure as was found for anhydrous L-DPPC on the basis of the I3C chemical shifts. On the other hand, the hydrocarbon chain arrangement of anhydrous L-DPPC is not affected by the addition of water. Apparently, in DL-DPPCa more tight packing is possible which is disturbed completely when at least 2 equiv of water is added. The structure obtained after the addition of only 1 equiv can be interpreted in terms of a coexistence of small domains of dihydrated and anhydrous DL-DPPC. In anhydrous L-DPPC such tight packing as in anhydrous DL-DPPCcannot be realized, and hence water can be accommodated without structural rearrangements in the hydrocarbon chains. Registry No. L-DPPC,63-89-8; DL-DPPC,2797-68-4. (25) Arnett, E. M.; Gold, J. M. J . Am. Chem. Soc. 1982, 104, 636. (26) Rossignol, M.; Uso,T.; Thomas, P. J . Membr. Eiol. 1985, 87, 269. (27) Stothers, J. B.; Tan, C. J. Can. J . Chem. 1976, 54, 917. (28) De Haan, J. W.; Van de Ven, L. J. M.; BuEinski, L. J . Phys. Chem. 1982, 86, 2517. (29) Yamanobe, T.; Sorita, T.; Komoto, T.; Ando, I. J . Mol. Struct. 1985, 131, 267.
Preferential Solvation in Two-Component Systems A. Ben-Naim Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel (Received: August 1 1 1988; In Final Form: November 14, 1988) ~
The theory of preferential solvation,originally developed for a three-component system,ls2is applied to two-componentsystems. The preferential solvation is computed for two sets of systems: the first includes mixtures of water with six organic liquids (methanol, ethanol, propanol, butanol, THF, and dioxane). The second includes mixtures of carbon tetrachloride with the same six organic liquids.
1. Introduction Recently a new approach to solvation thermodynamics has been proposed and applied to a variety of systems.’ On the basis of this approach we have suggested a new definition and a new way ( I ) Ben-Naim, A. Solvation Thermodynamics;Plenum: New York, 1987.
0022-3654/89/2093-3809$01.50/0
of measuring the preferential solvation (PS) of a solute in a two-component solvent.2 The typical question that has been asked in connection with is the following. Having a tWO-COmponent solvent, of A and B, with mole fraction XA, and a solute s very (2) Ben-Naim, A. Cell Eiophys. 1988, 12,
0 1989 American Chemical Society
255.
3810 The Journal of Physical Chemistry, Vol. 93, No. 9, 1989
dilute in the solvent mixture, what are the local concentrations of A and B in the immediate vicinity of S? Clearly, within the realm of the traditional concept of solvation thermodynamics, only very dilute solutions could be treated. Therefore, the PS question could have been dealt with only for a three (or more) component system: a solute and a two-component solvent. However, the question of PS can also be asked in a two-component system, say of A and B. At any composition xA, we may focus on one A molecule and ask what the PS of A is with respect to the two components, A and B, in its immediate vicinity. Likewise, we may focus on one B molecule and ask the same, but independent, question of the PS of B with respect to the two components A and B. Since our mixture may have an arbitrary composition, the last two questions concerning the PS of A and B could not be dealt with by the traditional concept of solvation. Fortunately the new concept of solvation’ can be easily applied to these systems, a feature that opens the way for investigating two-component systems in the entire range of compositions. In the next section we present the definition of PS in a twocomponent system, in terms of the Kirkwood-Buff integrak3s4 The latter may be computed through an inversion of the Kirkwood-Buff theory of solution. This leads to a simple relation between the PS of either A or B and measurable quantities. In section 3 we present a sample of results based on data available in the literature.
2. Theoretical Background We present here only the theoretical aspects of the PS problem that are relevant to a system of two components. The more general case of a three-component system has been discussed elsewhere.2 Consider an A molecule, placed at the center of a spherical volume of radius R,. For any arbitrary radius R,, the average number of A and B molecules in this sphere is given by
where pA and pB are the number densities of A and B, respectively, and g,, is the angular averaged pair correlation function for a pair of species (Y and p. In the following treatment we focus on a single A molecule to which we refer to as an A solvaton.’ A similar treatment applies to a B solvaton. For any radius R,, we define the local mole fraction of A molecules around an A solvaton by
Next we define the PS of an A solvaton with respect to A molecules, simply by the deviation of the local from the bulk composition, Le.,
Ben-Naim of the sphere (the solvaton) on its local environment could extend beyond the “first coordination shell”. Therefore, it is desirable to have a definition of PS that is free of the arbitrary choice of Rc, yet conveys information on the immediate environment of a molecule. Also, of course, we require that such a quantity be measurable. To achieve this goal we exploit the well-known fact that (except near critical points) all the pair correlation functions gapdecay to unity for distances of the order of a few molecular diameters. Thus, if we tentatively choose a radius R, (still arbitrary) for which g,,(R) = 1 for R > R, and for all pairs of species a,& then we can rewrite N A , A in (2.1) (and similarly N B , A in (2.2)) as
PALR’[gu(R) - 114aR2 dR E
pALm[gAA(R) -
1i4s2
dR
+ PAV,
=
PAGAA
+ PAV,
+ PAVC (2.5)
where we have introduced the so-called Kirkwood-Buff integral^,^ defined in (2.5). This approximation is rendered possible by the choice of R, sufficiently large, but still undefined, such that for R > R, the integrand in (2.5) is negligibly small. (In the actual calculation of G,, through an inversion of the Kirkwood-Buff theory: we require that all g, be defined in an open system, where g 1 at R m.) We now introduce (2.5) into our definition of the PS of A, (2.4), to obtain
- -
Similarly for the PS of B in the same system we have
Note that G A B = GBA and that 6A,A and 6A,B are, in general, independent quantities (see also the Appendix), but 6A,A = -hBb and 6B,B = -dkB, which follows from the definitions (2.6) and (2.7). Since all the G,, are computable from thermodynamic quantities, using the inversion of the Kirkwood-Buff theory of solution: one can compute 6A,A and 6A.B for any choice of V,. Clearly, if V, is chosen too small, then the approximation made in (2.5) becomes invalid. On the other hand, for V, a,both 6A,A and 6A,B become zero, as expected. Indeed, as we show in the next section, it is instructive to find that 6, and 6A.B become negligibly small in the entire range of compositions, whenever R, is chosen to be of the order of five or six (average) molecular diameters. This behavior suggests the expansion of 6,,+ and 6A.B in power series about y = V’; to linear terms to obtain
-
+ ~ x A x B ( G A A - GAB) + ... 6A,B = 0 + ~ x A x B ( G A B - GBB) + ...
6A.A
=0
(2.8) (2.9)
The linear coefficient of the PS can now be defined by Such a definition of PS is implied in the literature, normally for R, of the order of R , N (uAA + (rBB)/2, or roughly the first coordination shell about A (uAA being the molecular diameter of A). If a lattice-type model is used, then the first coordination shell includes all nearest neighbors to A.5 Unfortunately, since g,,(R) are not known individually for any two-component system, we cannot compute by using the above definition. Furthermore, there is no clear-cut definition of the “first coordination shell” for real molecules (say, water and ethanol). In most cases the effect of our chosen molecule at the center (3) Kirkwood, J . G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774. (4) Ben-Naim, A. Water nnd Aqueous Solutions; Plenum: New York, 1974; Chapter 4. ( 5 ) Marcus, Y. Ausr. J. Chew. 1983, 36, 1719.
~‘A,A
= XAXB(GAA - GAB)
(2.10)
~‘A.B
= X A X ~ G A-BGBB)
(2.1 1)
Thus, besides the product XAXB the difference G A A - GAB characterizes the linear coefficients of the PS of A, and likewise GAB - GBB characterizes the linear coefficient of the PS of B. By using these quantities as our measures of the PS, we have eliminated the need to make a choice of R,. In fact these quantities include the effect of the solvaton molecule on the entire range for which the correlation between the various molecules extends. Also the difference G , - GAB is a simple measure of the difference in the relative affinities4 of A toward A and A toward B. (6) Ben-Naim, A. J. Chem. Phys. 1977, 67, 4884.
:F
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3811
Preferential Solvation in Two-Component Systems
WATERMETHANOL
WATER METHANOL
X U
x
A
n
:
v
0
0 0
0
h
x,
h
0
I A
'i.
1
W
-5
-55 XA
x,
XA
WATER.PROPAHOL
X U
'h
h
0
XA
1
X U
'h
0
-
h
I
0
1
h
1
A
XA
WATER THF
XAB
*
XA
WATER D O U N E
.** H
,,::
0 0
h
1
XA
Figure 1. Preferential solvation in six systems of water (component A) and an organic liquid (component B) (all at 25 OC and atmospheric pressure). For each system (as indicated in the figures) we present the local mole fraction of A around A, the local mole fraction of A around B, and the linear coefficients in the preferential solvation of both A and B. The various curves of xA,Aand x ~correspond , ~ to different values of n, defined in (3.2). The larger the value of n the closer the corresponding curve to the diagonal line. In this figure, butanol stands for tert-butyl alcohol.
3812
The Journal ofPhysica1 Chemistry, Vol. 93, No. 9, 1989
Ben-Naim
As we have noted above, the main advantage of using either and 6A,Bin (2.6) and (2.7) or h O A , A and SoA,B in (2.10) and (2.11) is that all the G,, may be obtained from measurable quantities such as partial molar volumes, partial pressures, and conipressibilities. 6A,A
3. Results and Discussion In the following we have used as our main source of data, tables of G,, that were computed by Matteoly and L e p ~ r i . ’ ~These tables were obtained by the inversion procedure of the Kirkwood-Buff theory,6 using experimental data for mixtures of water and an organic liquid, and similar mixtures of carbon tetrachlorides with the same organic l i q ~ i d s . ’ ~ In order to gain some feeling for the order of magnitude of the correlation distance, we took the values of the effective molecular diameters as used in the scaled particle theory.8 The following values of the diameters were used in our calculations: water, 2.8 A; methanol, 3.3 A; ethanol, 4.2 A; propanol, 4.6 A; tert-butyl alcohol, 5.1 A; 1,4-dioxane, 5.3 A; tetrahydrofuran (THF), 5.3 A. We also define the distance of closest approach between two different molecules as AB = ( ~ A +A U B B ) / ~
(3.1)
where urn and uBB are the effective molecular diameters of A and B, respectively. With the above definition of uAB we define a series of correlation distances Rc = gAB(n
+ 1)/2
(3.2)
Thus, n = 1 corresponds roughly to the first coordination sphere. We used values n = 2-5 to compute xA,A(R,) as defined in (2.3). These were plotted in Figure 1 as a function of the bulk mole fraction of water, xA, in the various mixtures. It should be noted that the values, cited above for the effective diameters, are used here merely as a rough measure of the size of the solvent molecules. Nonspherical molecules have no “diameter”, and therefore there exists no clear-cut definition of the distance of closest approach between such a pair of molecules. For spherical molecules, the series of values of R, defined in (3.2) correspond to a series of coordination spheres around a given molecule. For nonspherical molecules, we still use the values of R, merely as a series of spheres around a given molecule, in which we calculate the solvent composition. These spheres clearly do not correspond to “coordination spheres” in a conventional sense. Although we have assigned no particular significance to either the effective diameters listed above or to the correlation distances R , computed from (3.2), it is quite clear from Figure 1 that as R, becomes of the order of 5 or 6, uAB, all the curves of xA,A(R,), and xA,B(R,) approach the diagonal line. In all cases we found that for n = 6 the relative deviation (xA,A(Rc) - xA)/xA becomes less than 0.01, which we consider to be an effective limit of the correlation distance. In other words, for R, > 6aABthe local composition approaches the bulk composition. For each of the systems reporeted in Figure 1 we have also calculated the linear coefficient of the PS of both A and B. The case of water-methanol is outstanding in the sense that the PS of both water (component A) and of methanol (component B) are positive in the entire range of compositions. This means that, at a n y composition, water molecules are preferred by both water and by methanol as solvaton. The absolute magnitude is clearly larger for the PS of water around water as compared with water around methanol. In the case of ethanol we observe a still positive PS of water around water (this is actually the same behavior for all the systems studied in this report). However, the PS of water around ethanol changes sign as the composition becomes more and more rich in water. This is not unexpected. Since 6 O A , B = -6’8.8 it follows that the PS of ethanol with respect to ethanol becomes positive
.-
0
0.2
0.6
0.8
IO
XCCl,
Figure 2. Values of G I I- GI, for CCI, (component 1) and an organic liquid (compound 2). The curves correspond to 1, methanol; 2, ethanol; 3, propanol; 4, 1-butanol; 5 , THF; 6 , p-dioxane. All curves are for 25 O C and atmospheric pressure.
in the water-rich region. This trend becomes even more pronounced for the cases of propanol and tert-butyl alcohol. It is worthwhile noting that although the PS does include, in a qualitative way, the effect of solute-solute interaction, which in the water-rich region may be referred to as hydrophobic interaction, such an interpretation is not straightforward, first, because 6OB,B, for instance, depends on the difference G B B - GAB and not only GBB. Second, the quantities of GAB and GBB do not reflect only the solvent-induced forces between the two molecules. These quantities depend also on the direct intermolecular potential, which in turn includes a repulsive contribution. The latter is also a qualitative measure of the size of the molecules A and B. In the case of T H F and dioxane we still observe that the PS of water by water is positive, whereas the PS of the organic molecules by water is almost always negative (except for a very small region near the limit of xA 0). In Figures 2 and 3 we report similar data of GI1- GI, (1 being CCI4 and 2 the second organic liquid as indicated in the captions) and of G12- GZ2. In all of the CC14-alcohol curves we observe a maximum of the PS of CCI, around CC14 in the region of 0.7 < xccl < 0.9 and a minimum of the PS of CCI, around the alcohol. It should be noted that the values of GI2- GZ2are almost an order of magnitude larger than the corresponding values of GI1- G12. In contrast, the values of both GI2- GZ2and GI1- GI?are nearly zero for the CCI,-THF and CC1,-dioxane systems. This is distinctly different behavior from the water-THF and waterdioxane systems shown in Figure 1. A small value of both GI2 - G22 and GI, - GI, in the entire range of concentration is indicative of a symmetrical ideal behavior of the mixture; Le., from
-
it follows GI1
( 7 ) (a) Matteoli, E.; Lepori, L. J . Chem. Phys. 1984, 80, 2856. (b) Matteoli, E. Private communication. (8) Reiss, H. Adu. Chem. Phys. 1966,9, 1.
0.4
+ G22 - 2G12 = 0
i.e., the systems CC14-THF and CCI,-dioxane behave very nearly as symmetrical ideal solutions. This is in sharp contrast to the
Preferential Solvation in Two-Component Systems
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3813 1. The first is deviation from the dilute ideal solution at constant T and M ~ .The relevant first-order deviation from dilute ideal solution of A in B is MA(TIMB.PA) = MOA(T,MB)
+ k T In PA - ~TGOAAPA + ...
(A. 1) 2. Another is deviation from dilute ideal solutions at constant 7' and P. In this case the relevant expansion is MA( 7
'2,~= ~) MOA( T,P)
+ kT In PA - k T ( G o U - GoAB)pA + ... (A.2)
we see that in (A.2) we have the difference GoAA- GOA6 which is the linear coefficient of the PS of A in the limit pA 0. Thus only in this limit we may extract information on PS from studies of deviations from dilute ideal solution. 3. The more important and relevant case is the deviation from the symmetrical ideal solutions. Here the necessary and sufficient condition for symmetrical ideality is4 A = GAA+ GBB- GAB = O ('4.3) -+
XCCI,
Figure 3. Values of G,2 - G22for the same systems as in Figure 2.
corresponding water-THF and water-dioxane systems.
Acknowledgment. I am grateful to Professors Matteoli and Lepori for providing the tables of their computed values of Gap. Also, thanks are due to Dr. K. L. Ting and M. Figura for their kind assistance in processing the data.
Appendix. Relation between PS and Deviations from Ideality of the Solutions There are a few deviations from ideal solutions that could be expressed in terms of the Ga,+ The most common ones are the following:
in the entire range of composition. We see that if both 6A,A = 0 and 6A,B = 0, then it follows that A in (A.3) is alm zero. In other words, lack of PS for both A and B leads to symmetrical ideal solution. The inverse of this is in general not true as can be seen already from the linear coefficients in (2.10) and (2.1 1). That is, for a system that obeys the condition A = 0, it follows that bAA = 6A,B; Le., the PS of A and B are equal to each other, but not necessarily zero. Clearly for nonideal solutions A # 0 and the sign of A (which determines the direction of the deviations from Raoul's law) does not imply anything regarding the sign of either 6A,A or bA,B. Finally, it should be noted that all the GI, used in this article are defined in an open system, in which case the limiting behavior of g,, is gl,(R) 1 at R a (A.4)
- -
The limiting behavior of g,,(R) is different in a close system. This follows from the different normalization of the pair correlation function in open and closed systems. More details on this can be found in ref 3, 4, and 9. Registry No. THF, 109-99-9; H 2 0 , 7732-18-5; CC14, 56-23-5; MeOH, 67-56-1; EtOH, 64-17-5; PrOH, 71-23-8; BuOH, 71-36-3; dioxane, 123-91-1. (9) Hill, T L. Statistical Mechanics; McGraw-Hill: New York, 1956.
