Langmuir 1986,2, 331-337 seems unsuitable for the higher members of the given homologous series. For the amphiphile electrolyte such as dodecylammonium chloride there appears to be a lower exposure of the hydrocarbon chain, which may be explained by its high solubility in water resulted from the dissociation of molecule into ions.
331
As pointed out earlier, the method used in this paper implies that the surface (interfacial)tensions are applicable to individual molecules, this is still a controversial question though a similar assumption has been used by some authors.’~~ Accordingly, the results on the configuration of the adsorbed film for amphiphile molecules can only be qualitatively significant.
Theory for the Equation of State of Phospholipid Monolayers Robert S. Cantor* Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755
Ken A. Dill Departments of Pharmaceutical Chemistry and Pharmacy, University of California, San Francisco, California 94143 Received October 17, 1985. In Final Form: January 21, 1986 We present a simple theory for the interactions among the phospholipids in bilayers and monolayers over a wide range of surface densities, temperatures, and chain lengths. This work draws from three recent efforts: (i) lattice-based interphase theory which explicitly treats the chain conformations in three-dimensional interfacial phases of short-chain molecules, (ii) a recent “equation of state” treatment of van der Waals interactions which predicts volume changes, enthalpies, and temperatures of melting of the “rotator” phases of n-alkanes, and (iii) detailed study of the extensive measurements of Mingins et al. (Mingins, J.; Taylor, 3. A. G.; Pethica, B.; Jackson, C. M.; Yue, B. Y. T. J . Chem. SOC.,Faraday Trans. 1 1982, 78,323) of pressure-area isotherms of phosphatidylcholinesat the heptanefwater interface which we have used to assess empirical characteristics of the head group interactions and of the “solid state”. This approach rationalizes pressure-area isotherms and phase transitions of monolayers of lecithins of chain lengths from 16 to 22 and predicts temperatures, areas, volume changes, and enthalpies of melting of the corresponding bilayers. Thermodynamic measurements of equations of state have long provided the principal source of information on intermolecular interactions within the various phases of matter. For amphiphilic molecules, equations of state have been measured principally through use of film balance methods developed by Pockels, Langmuir, Adam, Wilson, McBain, and others.’-3 The resultant pressure-area isotherms often resemble those of pressure-volume measurements of van der Waals gases and liquids. Hence the earliest surfactant equations of state were premised on the assumption that amphiphilic molecules at an interface act like two-dimensional ideal gases2v3or gases with excluded area4* or that they could be modeled by the two-dimensional analogue of the van der Waals equation7 or virial expansions.6v8 These equations only apply to low surface densities for which interactions among the surfactants are weak. For greater surface densities, intermolecular in(1) Adameon, A. W. ‘Physical Chemistry of Surfaces”, 4th ed.; Wiley Interscience: New York, 1976. (2) Adam, N. K. ‘The Physics and Chemistry of Surfaces”, 3rd ed.; Oxford Univ Press: Oxford, UK, 1941. (3) Adam, N. K.; Jessop, G. R o c . R. SOC.London, Ser. A 1926,110, 423. (4) Volmer, M. Z . Phys. Chem., Abt. A 1925,115, 253. (5) Mingins, J.; Taylor, J. A. G.; Owens, N. F.; Brooks, J. H. Adu. Chem. Ser. 1975, 144,28. (6) Langmuir, I. J. Chem. Phys. 1933, 1, 756. (7) Magnus, A. Z . Phys. Chem. 1929,142,401. (8) Mitchell, J. S. Trans. Faraday SOC.1935,31,980.
teractions among the amphiphilic molecules become important. There have been many recent attempts to model the molecular interactions a t high surface densities, particularly the phase transitions (reviewed in ref 9-12). Some of these approaches are phenomenological, based on Landau-de Gennes theory13 or on the Ising lattice gas model.l’J4 More rigorous treatments have attempted to treat the chain interactions more e ~ p l i c i t 1 y . l ~In ’ ~ one such approach, it is assumed that the alignment of chains at high densities is the consequence of anisotropic attractive forces whereby chain segments have lower energy when they are aligned than when they are not. However, it is well-known that anisotropic attractive forces are much too weak to cause chain alignment in hydrocarbons: forces of attraction are largely isotropic. Steric repulsive forces are primarily responsible for the anisotropy and molecular organization in condensed phases of hydrocarbons.’“20 (9) Wiegel, F. W.; Cox, A. J. Adv. Chem. Phys. 1980, 41, 195. (10) Baret, J. F. B o g . Surf. Membr. Sci. 1981, 14, 291. (11) Bell, G. M.; Combs, L. L.; Donne, L. J. Chem. Rev. 1981,81, 15. (12) Nagle, J. F. Annu. Rev. Phys. Chem. 1980, 31, 157. (13) Priest, R. G. Mol. Cryst. Liq. Cryst. 1980, 60, 167. (14) Caill6, A.; Pink, D.; de Verteuil, F.; Zuckermann, M. J. Can. J. Phys. 1980,58, 581. (15) Marcelja, S. Biochim. Biophys. Acta 1974, 367, 165. (16) Gruen, D. W. R. Biochim. Biophys. Acta 1980,595, 161. (17) Nagle, J. F. J . Chem. Phys. 1973,58, 252. (18) Wulf, A. J. Chem. Phys. 1975, 64, 104.
0743-7463/86/2402-0331$01.50/00 1986 American Chemical Society
332 Langrnuir, Vol. 2, No. 3, 1986
Cantor and Dill
Other treatments have taken into account the steric reinto two steps; the all-trans chains are mixed with solvent pulsions, principally through use of exact lattice methods,17 chains, followed by chain disordering with no change in but mathematical intractability has prohibited their apmixing. The former can be represented by a distribution plication to real surfactants in three d i m e n s i ~ n s . ~ J ~ function, Qt, for the number of arrangements of a system of N amphiphilic molecules, each of which occupies area The purpose of the present work is to develop a statistical mechanical theory for the equation of state for all A,, in a total area NA: surface densities of monolayers of amphiphilic molecules (NA)! at the oil/water interface. It is based on recent theory for Qt = (1) [ N ( A- l)]!N! the chain conformations in interfacial phases of chain molecules”Vz2and on theory for the van der Waals interwhere A = A / A o is the reduced area per molecule and A, actions among chain molecules in semiordered s y s t e m ~ . ~ ~ = 40 A2 is the hard-core minimum area of individual The present approach is simple, is intended to comprehend phosphatidylcholine molecules.26 Through use of Stirling’s the full range of surface densities in monolayers and biapproximation, we obtain the translational free energy per layers, and is intended to be applicable to geometric molecule structures which comprise other surfactant phases. The Ft = (kBT/N)In Qt = recent development of surface balance methods by which -kBT[AIn A - ( A - 1) In ( A - l)] (2) surfactants can be compressed at the oil/water interface, and with which careful and extensive experiments have where kBT is Boltzmann’s constant multiplied by absolute been p e r f o r ~ n e d ,permits ~ ~ , ~ ~ us to extract additional intemperature. The translational contribution to the lateral formation about intermolecular interactions for which a pressure is statistical mechanical theory is not yet available. Theory We first consider a surfactant monolayer at an oil/ water interface. In the standard experiment, a movable barrier occupies a position at the interface, with surfactant at the oil/water interface on one side and pure oil/water interface on the other. A lateral pressure is applied to the barrier toward the side containing surfactant in order to sustain equilibrium. The lateral pressure exerted by the surfactants is due to a difference in interfacial free energies on each side of the barrier. With increasing pressure, a phase transition is induced from a “fluidlike” to a “solidlike” state. In this experimental arrangement, the free energy due to hydrocarbon/water contact, per se, does not contribute to the free energy difference, in first approximation, inasmuch as moving the barrier does not change the sum of the areas of hydrocarbon/water contact on the two sides of the barrier; this applies only when the surfactant chains are in a fluid state, however, and is discussed further below for chains that are in the solid state. (Note that the same cancellation should not apply to the air/water interface or in bilayers or other aggregates dissolved in a bulk solvent, where this hydrophobic interaction is an important driving force.) We assume then that the interfacial free energy per molecule of the liquid state of the system is the sum of four area-dependent contributions, due to two-dimensional translational freedom of the surfactants in the plane of the interface, head group interactions, chain configurations, and the volume dependence of chain packing. Each of the terms is described in detail below.
I. Translational Free Energy When the area per chain is large, the dominant contribution to the lateral pressure is due to the two-dimensional entropy of mixing of surfactants with solvent. Translational freedom may be taken into account through the assumption that the mixing process can be decomposed (19)Israelachvili, J. N.; Marcelja, S.; Horn, R. G. Q.Reu. Biophys. l!
This expression is valid for all densities. A low-density approximation to this expression, valid to second order in A is
(4) (Through use of a simple excluded volume a factor of 1is sometimes substituted for ‘/2 in eq 4,but this is only valid to first order.) 11. Head Group Interactions In the present work, we restrict our attention to lecithins, surfactants containing zwitterionic phosphatidylcholine head groups. They provide two principal advantages for present purposes: lateral pressures of diacylphosphatidylcholine monolayers are observed to be nearly independent of the salt concentration in the solvent,24and extensive data are available over a range of temperatures and chain lengths which permits experimental determination of the area dependence of the energy and entropy of the ~ y s t e m . ~We * , ~have ~ deduced the head group contribution to the pressure by subtracting the translational pressure, described above, from the experimental isotherms of Mingins et for areas/molecule larger than 200 A2. The remaining lateral pressure is observed to be independent of the length of the alkyl tails of the phospholipids; thus we attribute the excess pressure to purely head group interactions at large areas/molecule. We have found that the excess pressure determined in this manner, taken from the data of Mingins et al.,” can be well characterized by the following area and temperature dependence:
where b = 2140 X (dyn cm3)/K and To = -32 OC are the empirical constants appropriate for phosphatidylcholine head groups. We assume that eq 5 is valid at all surface densities, although it will be clear from subsequent discussion that other interactions among head groups and water also contribute when chains are in the solid state. The contribution of the head groups to the free energy, (26) Tardieu, A.; Luzzati, V.; Rzman, F. C.J. Mol. Biol. 1973,75,711. (27)Lleneras, E.;Mingins, J. Biochim. Biophys. Acta 1976,419,381.
Langmuir, Vol. 2, No. 3, 1986 333
Equation of State of Phospholipid Monolayers relative to the state of infinite dilution, is
O%
111. Chain Conformational Free Energy The physical properties of polymeric materials are, in part, determined by the many degrees of intramolecular freedom of rotation about single bonds along the chains. With increasing surface density, the number of accessible states of the chain molecules in amphiphilic aggregates is increasingly restricted by the constraints imposed by the packing of the chain among its neighbors. These constraints may be taken into account through use of lattice statistical mechanical methods developed to describe the conformational properties of chain molecule “interphases”.21~22~2s The lattice permits the enumeration of the various chain conformations subject to the requirements that chain segments do not have steric overlap with neighboring segments. Many of the conformational properties of interphases can be predicted through these methods.2WP-31 The partition function for the chain conformations in interphases has been described in detail elsewhere.21v22It is readily computed through a matrix method which can be applied to aggregates of any geometry and for chains of any surface density or length. For present purposes, we consider only planar arrays of surfactants such as those encountered in monolayers or bilayer membranes. For chain molecule interphases, two different levels of approximation for the states of internal energies of the chains have been described. in simplest approximation, all chain configurations are assumed to have the same intramolecular energy; in the absence of intermolecular constraints, all chain conformations are taken to be a priori equally likely.28This approximation that chains are freely flexible may be termed “Bernoullian” inasmuch as the probability of orientation of any “bond” (i.e., two adjacent segments in a chain) is independent of that of neighboring bonds along the chain. A better approximation is that which accounts for the dependence of the orientation of one bond on that of immediately adjacent neighbors along the chain.21p22In this approximation, bond conformations are given by conditional, or “Markoviann, probabilities. Approximate intramolecular energy differences between trans and gauche bonds and the degeneracies of each may be taken into account through use of a statistical weight, a b , for a bent pair of adjacent bonds relative to an unbent one. For freely flexible chains, wb = 1, for rigid rods, wb = 0, and for real polymethylenic chains, wb 0.5.29In the present treatment, we adopt the Markovian approximation with wb = 0.5. The theory permits the prediction of the conformational free energy as a function of temperature, chain length, and surface density. The matrix procedure does not directly yield a closed form expression for this contribution to the free energy except in the dilute (low surface density) limit, but we have found that it can be approximated with less than 2% error by a simple exponential function of area (see Figure 1): Fc(n,,A,T) = 2Fo(nC,T)[1- exp[-kf(A - U11 ~~~
~~~
~
(7)
~~~
(28)Dill, K.A.;Flory, P. J. Proc. Nutl. Acad. Sci. U.S.A. 1980,77, 3115. (29)Dill, K. A.;Koppel, D. E.; Cantor, R. S.; Dill, J. D.; Bendedouch, D.; Chen, S.-H. Nature (London) 1984,309,42. (30)Dill, K. A. J. Phys. Chem. 1982,86,1498. (31)Dill, K.A. In "Surfactants in Solution”; Mittal, K. L., Lindman, B.,Eds.;Plenum Press: New York, 1983;Vol. 1, p 307.
,-. -I 2
1
2
3
4
REDUCED AREA
5
(i)
Figure 1. Calculated chain configurational contribution to the free energy for n, = 18, T = 15 “C (discretepoints). Continuous curve is exponential best fit to calculated points.
where n, is the number of lattice segments per chain and kf = 2.1 is approximately constant with relatively small dependence on temperature (i.e., on wb) and on chain length, both of which we neglect here. Fo represents the configurational free energy difference between the disordyed, amorphous state (A a) and the totally ordered (A = 1)state of a chain; it is predicted by the lattice theory to vary nearly linearly with n,. The factor of 2 accounts for the two acyl chains per phospholipid molecule. The principal function of the lattice treatment is to establish the dependence of free energy on area per molecule. It is obvious, however, that lattice segments are not realistic models of methylene groups; thus predicted values of the factor Fo from the lattice treatment cannot be used directly. Fo represents, approximately, the conformational free energy difference between the crystalline and amorphous bulk states of n-alkane molecules of corresponding length. We compute this factor through use of the rotational isomeric state theory,32 in the notation of Jernigan and F10ry:~~ Fo(T,nc)= -kBT In 2, (8) where
-
Zc(n,,T)= [1,0>Ol
:
aw
“1 [;I (nc-nco) 1
(9)
w
and (r = exp(-E%/RT), w = exp(-E,/RT) are the statistical weights and E , = 900 cal/mol, E,, = 2000 cal/mol are the energies of trans/gauche and gauche (+)/gauche (-) bond pairs, respectively, which have been used to predict the thermal properties of alkane melting.23 By this approach, Fois found to vary nearly linearly with n,. The parameter n, represents the number of carbons which do not contribute to the conformational free energy. In free n-alkane chains, nco = 3,32,33however, the carbonyl carbons in phospholipids are not freely flexible, and the sn-2 chains are bent near the glycerol backbone. Thus, in the absence of more complete information, we have arbitrarily assigned the value nco= 4; use of slightly different values does not significantly alter the predictions of the calculations. The conformational contribution to the lateral pressure may be obtained by taking the areal derivative of F, (eq 7)
That the lattice treatment is a reliable predictor of chain conformations in amphiphilic aggregates has previously been established through independent comparison with a variety of experiments. The orientational order of the (32)Flory, P. J. “Statistical Mechanics of Chain Molecules”; Interscience: New York, 1969. (33)Jernigan, R. L.;Flory, P. J. J. Chem. Phys. 1969,50,4165.
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334 Langmuir, Vol. 2,No. 3, 1986
termined through fits of predicted and experimental various bonds along the chains has been measured by 2H pressures in the fluid phase at high surface densities; as NMR in bilayer membranes of pure phospholipids, fatty with the related constant for the chain configurational acids, and biological membranes.3P36 Predictions of the interactions, use of slightly different values does not alter interphase theory are in good agreement with these rethe results significantly. For the present studies of lecsults22,28 and with small-angle neutron scattering experiithins, n,, = 5. ments which measure the “positional” disorder of specifThe factors x and Q are determined as functions of ically deuterated segments of bilayer chains.37 Predictions surface density in the following manner. The factor x of the theory are also in good agreement with neutron accounts for the degree to which the range of the repulsive scattering, 2H, and 13CNMR experiments on chain conformations in spherical and cylindrical micelles.29~34~38*39part of the chain pair potential is reduced with the better steric packing that accompanies increased chain alignment. IV. Volume-Dependent Chain Interactions For the melting of alkanes, this factor is a function of the The van der Waals attractions among chain molecules relative numbers of methylene and methyl end groups in are responsible for the cohesion within alkane liquids and the chains and is established through experimental measolids; they must be a principal source of attraction among surements of volume changes upon melting of n-alkanes the alkyl tails of surfactant molecules. At temperatures of different lengthsaZ3For monolayers and bilayers, of a few degrees above their crystal transitions, short-chain interest here, we assume that x depends, in addition, on n-alkane liquids freeze to a “rotator” state in which the the degree of chain alignment in the fluid state; greater chains have a high degree of conformational order and alignment of the chains should permit better packing. alignment, but in which there is rotational disorder. This When there is no alignment, x = 1; when there is total state serves as a useful model for the intermolecular inalignment, x adopts a value, I,,corresponding to that of teractions among the alkyl tails in monolayer films when alkane “rotator” phases, and x is assumed to be a linear the chains are relatively compressed and highly aligned. function of the degree of orientational order between those Inasmuch as attractive dispersion forces among hydroextremes. From the lattice treatment of chain conforcarbons are almost completely isotropic,18,20they alone mations,21-22 the degree of order is found to be an expocannot be responsible for the chain alignment that acnential function of reduced area, as for the conformational companies freezing. The anisotropy of short-range steric free energy, but with exponential constant k, = 2.26. This repulsive forces must be largely responsible for the anconstant varies to a small degree with chain length, but isotropy in rotator phases of alkanes and ordered phases that variation is neglected here. Thus within amphiphilic aggregates. The van der Waals atx(A) = x , + (1 - x,)[l - exp[-k,(A - I)]] (12) tractive and repulsive forces and the free volume within where n-alkane liquid and “rotator” phases have been taken into (n, - ncl)- 1 account through a simple statistical mechanical theory 1 (13) x , = (n, - ncJ XCH* + (n, - ncl)xCHa developed recently for the melting of alkanes.23 That treatment provides a basis for the following approach. Provided the chains are distributed with constant density, and XCH, = 0.9765 and XCH, = 1.0342 are the values previously established for alkanes.23 The coefficients multithe disptrsion energy will depend linearly on reduced plying xCH, and x a , reflect the relative number of CH2 and density, V1.Following treatments that date back to the CH3 groups, respectively, in the molecule. For a given work of Eyring and H i r s ~ h f e l d e r ,we ~ ~assume ~~ the degree of ordering, corresponding to a particular surface principal intermolecular entropy is due to free volume. censity, x is thus specified by eq 12 and 13. To compute These contributions sum to give a volume-dependent free V(A),the atmospheric pressure isobar 0, = 0) is obtained energy per moleculez3 from the volume-dependent free energy expression (eq 11) EO F,(V,x) = -[2(n, - n,,) ne]- x
+
N A
where NA is Avogad_ro’snumber and the constants are Eo = 1.57 Kcal/mol, VI = 1.23 (the reduced volume of the disordered liquid), and y takes the values 1.078 and 1 in the solid and fluid states, re~pectively.~~ The constant ne = 1.1derives from the Flory, Orwoll, and Vrij equation of state for alkane l i q ~ i d s , 4and ~ ~n,,~ ~is required to account for chain units which do not contribute to changes in volume-dependent interactions. The value of n,, is de(34) Mely, B.; Charvolin, J.; Keller, P. Chem. Phys. Lipids 1975, 15, 161. (35) Seelig, J. Q. Reo. Biophys. 1977, 10, 353. (36) Seelig, J.; Seelig, A. Quart. Reu. Biophys. 1980, 13, 19. (37) Zaccai, L.G.;Buldt, G.; Seelig, A,; Seelig, J. J. Mol. Bid. 1979, 134, 693. (38)Bendedouch, D.; Chen, S.-H.; Koehler, W. C. J.Phys. Chem. 1983, 87, 153.
(39) Cabane, B. J. Phys. 1981, 42, 847. (40) Eyring, H.; Hirschfelder, J. 0. J. Phys. Chem. 1937, 41, 249. (41) Prigogine. 1. “The Molecular Theory of Solutions”; Interscience: New York,-1557. (42) Flory, P. J.; Orwoll, R. A,; Vrij, A. J. Am. Chem. SOC.1964, 86, 3507. (43) Orwoll, R. A,; Flory, P. J. J. Am. Chem. SOC.1967, 89, 6814.
For the fluid state (y = i),eq 14 can be solved for V(A,nJ. These values of x and V, having thus been computed as functions of chain length and reduced area, are substituted into eq 11 in order to obtain the volume-dependent contribution of the chains to the free energy. Since the parameter x deviates only very slightly from unity, we can expand eq 14 in a Taylor series-around x = 1 to obtain an approximate expression for V(A)for the fluid phase (y = 1). If we define 7 = 1 - x , then eq 12 becomes (15) s(A) = 7, exp[-k,(A - 0 1 where 7,=1-
Expanding eq 14 to first order in 7,we obtain V ( A )=
(16)
Equation of State of Phospholipid Monolayers
Langmuir, Vol. 2, No. 3, 1986 335
V. Contributions to the Free Energy of t h e Solid loo50
L
-100 -5040
80
120
160
200
240
A (A2/molecule)
Figure 2. Predicted contributions to monolayer pressure-area isotherm of the fluid phase of (CIS) 1,2-distearoylphosphatidylcholine at 15 O C : (--) translational;(---) head group interactions; (-) chain configurational interactions; (- --) volume-dependent chain interactions; (-) total.
Substituting eq 17 into eq 11 and expanding around q = 0 gives the explicit area dependence of the free energy to first order in q
F,(A)
%
F,"
-
where F," is obtained from eq 11 with y = 1:
F,"
E
F,(A
+
m)
= -[2(n, - nc,)
6 = VI, x = 1, and
+ ne]
The volume-dependentcontribution to the lateral pressure, to first order in 7, is obtained by taking the areal derivative of eq 18
n.,=
The total free energy of the fluid state is obtained as the sum of the terms given in eq 2,6,7, and 11. Contributions to the lateral pressure are obtained from corresponding areal derivatives: eq 3, 5, and, in first approximation, eq 10 and 20. Figure 2 shows the magnitudes of the four terms for a typical case. Equations 10 and 20 reveal that to good approximation, the chain configurational (repulsive) and volume-dependent (attractive) contributions to the lateral pressure both depend exponentially on area and nearly linearly on chain length and are of similarly large magnitude. Their sum is thus predicted to be small, except at high surface densities, where in most cases the fluid phase is no longer experimentally observed. The chainlength-independent contributions to the lateral pressure are expected to predominate at low densities, resulting in pressure-area isotherms predicted to be nearly independent of chain length in thq,fluid phase, as is observed e ~ p e r i m e n t a l l y This . ~ ~ ~predicted ~~ cancellation of chainlength-dependent contributions depends on, and thus supports, our assumption (eq 12) that the factor x, representing the range of the repulsive part of the chain pair potential, depends linearly on the degree of orientational order.
State The interactions described in the preceding sections, I-IV, are those that are expected to contribute to the surfactant pressure when the surfactants are in their fluid state. At high lateral pressures, lecithin monolayers at the oil/water interface undergo a conformational phase ttansition to a solidlike state. The state of lecithins in lipid bilayer membranes below the main phase transition (melting) temperature is one in which the chains are tilted with respect to the bilayer normal, and there is little conformational disorder. In the absence of evidence to the contrary, we assume that the solid state of the monolayer is similar, Le., that there is little translational or conformational freedom, but that the free energy of the molecules can be reduced by tilting which increases the head group separation but may decrease the van der Waals attraction. Thus the free energy of the solid state will have no contribution from either the translational or conformational terms described above. In addition to contributions from head group and volume-dependent chain interactions, we assume further that the solid-state free energy is comprised of a term reflecting differences in hydration of the head group in the fluid and solid phases and a contribution due to the difference in interfacial tension of hydrocarbon/ water contact between the solid phase and the clean oil/water interface. Each of these contributions is described in turn below. The empirical expression for the head group repulsion, given by eq 5, was deduced from experimental isotherms at large areas. Predicted pressures of the fluid phase are in good agreement with experiment even at the smallest areas for which the fluid phase is observed; for example, for DPPC (n, = 16) at 20 "C (see Figure 7), agreement is good at all areas down to the transition point ( A = 65 A2). Although the head group interactions may be significantly different at the high surface densities characteristic of the solid phase, we assume that the form given by eq 5 remains valid in the solid state. The volume-dependent free energy for the solid state at the minimum area is obtained through evaluation_of eq 11 and 1,4 for x = x, and y = 1.078, i.e., F,(solid, A = 1) = Fu(xr,Vr,y = 1.078). As the chains tilt with increasing area, we expect the efficiency of packing of the chains to decrease slightly, resulting in an increase of the free energy (expected to be small). This free energy increase should vary linearly with the number of chain segments which contribute to the volume-dependent interactions. To first order in area, the total volume-dependent contribution is assumed to take the form F,(solid,A) = F,(xr,6,)+ 2(n, -nc,)Ftilt ( A - 1)
(21)
where Ftiltis a constant, the determination of which is discussed below. We have assumed that at the oil/water interface, the interfacial tension due to hydrocarbon/water contact is equal on the monolayer and Yclean"sides of the barrier, when the monolayer is in the fluid phase. For the solid state at high surface densities, we can no longer make this assumption, principally because the bulky head groups preclude hydrocarbon/water contact to a large degree. It may also be argued that with increasing area, the increasing degree of tilt of the chains does not bring additional chain segments close to the interfacial plane, as occurs with increasing area in the fluid phase. Thus we expect the interfacial tension due to hydrocarbon/water contact of the monolayer in the solid state to be much less than that of the pure oil/water interface, resulting in a constant
Cantor and Dill
336 Langmuir, Vol. 2, No. 3, 1986
3 n
\
E
Y -
e
A
0 40
60
EO
100
120
E
140
E:
A (A’/molecule)
-i
Figure 3. Predicted free energy-area isotherms (continuous curves) of fluid and solid monolayer phases of (CIS) 1,2-distearoylphosphatidylcholine at 15 “C. Dotted line represents the two-phase equilibrium region.
I
= -Yin+AO(A - 1)
15
I 01
1
50
0
(22)
In aqueous dispersions of lecithins, the hydration of the head groups is known to be reduced upon we assume that the release of ‘bound” water molecules occurs as well in the freezing of monolayers at the oil/water interface. This may be represented by a free energy that is independent of chain length and area
I1
I
positive lateral pressure Tint. The resulting molecular free energy can be written as Fint
1
0’
100
150
1
250
200
A (P/molecule)
Figure 4. Predicted monolayer isotherms of (Cz2) 1,2-dibehenoylphosphatidylcholine (continuous curves) at the temperatures (“C) indicated above the curves. Experimental data
points are taken from ref 25.
where Ehyd and s h y d are constants, determined as indicated below. The total molecular free energy of the solid state is obtained as the sum of the four contributions given by eq 6, 21, 22, and 23.
(24) The lateral pressure is obtained from the areal derivative of eq 24 aF(solid,A) n(solid,d) = aA b(TlrT0) Ftilt + 2(nc - ncJ -- Tint (25) A2 NAAO Unlike the expression derived above for the free energy of the fluid phase, that for the solid phase (eq 24) is not based on molecular theory, for the molecular organization in the solid state is unknown. We consider eq 24 to represent simply a reasonable conjecture, premised on the assumption that the solid state is comprised of all-trans, tilted chains. Values of the parameters Fdt,Tint, E h y & and Shdare determined by fitting to the solid-state areas and transition pressures of all the experimental isotherms of Mingins et al.,% for n, = 16,18,20, and 22. The best values are found to be Ftilt= 0.159 Kcal/mol, yint = 58 dyn/cm, E h y d = 6.95 Kcal/mol, and S h y d = 15.6 cal/(mol K). The (44) Ruocco, M. J.; Shipley, G. G. Biochim. Biophys. Acta 1982,684,
59. (45) Ruocco, M. J.; Shipley, G. G. Biochim. Biophys. Acta 1982,691, 309. (46) Janiak, M. J.; Small, D. M.; Shipley, G. G. J.Bid. Chem. 1979, 254, 6068.
OA
0
I
50
100
150
200
250
A (A2/moIecuIe)
Figure 5. Predicted monolayer isotherms of (Czo) 1,2-diarachidoylphosphatidylcholine (continuous curves) at the temperatures (“C) indicated above the curves. Experimental data points are taken from ref 25. positive values of Ehyd, shy,, and F h y d are consistent with the release of ‘bound” water molecules from the phosphatidylcholine head groups upon freezing of the chains. Phase transitions are predicted by constructing tie lines that are mutually tangent to the fluid and solid free energy curves, an example of which is shown in Figure 3. Combination of the expressions for the fluid and solid-state pressure-area isotherms with the calculated transitions permits prediction of equilibrium isotherms over the full range of surface densities. In Figures 4-7 are shown comparisons of the predictions with the experimental data of Mingins et aLZ5for lecithins of acyl chain lengths n, = 16-22, each over a range of temperatures. VI. Bilayer Membranes Provided that the two monolayers in a bilayer membrane can be considered relatively independent (evidence for this assumption is given in ref 47), then it is possible to adapt (47) Ohki, S.; Ohki, C. B. J. Theor. Biol. 1976, 62, 389.
Langmuir, Vol. 2, No. 3, 1986 337
Equation of State of Phospholipid Monolayers
Table I. Predicted Properties of the Main Transition of Phosphatidylcholine Bilayers" A,: A2/ AL,CA2/ AH: kcall AV,d cm3/ nc
l"ainYb
16 18 20 22 24
molecule 60.9 [65] 71.1 76.4 80.3 83.4
molecule 47.5 [48] 49.2 48.5 47.3 46.0
O C
25.1 [41.50] 42.2 [54.54] 52.6 [64.81] 60.4 [72.96] 66.6 [80.07]
mol of lecithin 7.35 [7.6] 12.5 [9.5] 16.7 [11.5] 20.6 [13.5] 24.4 [14.2]
mol of lecithin 28.2 [27] 40.1 [36] 49.6 58.7 67.6
a Experimental data (in brackets) are included for comparison. *Experimentaltemperatures and enthalpies (in brackets) from ref 48. eExperimental areas from ref 50. dExperimentalvolume changes from ref 51.
A
E
Ya
=e5. v
0
50
100
150
200.
250
A (A2/moIecuIe)
Figure 6. Predicted monolayer isotherms of (CIS) 1,2-distearoylphosphatidylcholine (continuouscurves) at the temperatures ("C) indicated above the curves. Experimental data points are taken from ref 25. the foregoing theory to predict the thermal properties of bilayers. In a bilayer at equilibrium, there is no external applied pressure. Unlike the oil/water interface paradigm, however, in which total hydrocarbon/water contact is approximately independent of position of the movable barrier, the hydrocarbon/water contact at the bilayer/ solvent interface depends on surface density of the chains. Thus the hydrophobic effect contributes an additional free energy to the balance of forces in a bilayer, an effect that is negligible for the interfacial monolayer. This provides an additional cohesive tension in a bilayer which tends to decrease the area/chain; it may be thought of as resulting in the same state of molecular organization as a monolayer at an interface under an effective applied lateral pressure. Provided this free energy is linear in hydrocarbon/water contact area, then this tension, the derivative of the free energy with area, is a constant whose value is best estimated as the interfacial tension at the simple hydrocarbon/water interface, approximately 50 dyn/cm. Using the theoretical approach described above, thermal properties of melting transitions for lecithins of length n, = 16-24 were calculated at the temperature at which the transition pressure takes the value 50 dyn/cm; the results are summarized in Table I, along with representative experimental data on the bilayer main transition from the literature. Predicted melting temperatures are seen to be consistently 12-15 K below the experimental values.48
I
"I
0.
50.
100.
150.
200.
250.
A (A2/moIecuIe)
Figure 7. Predicted monolayer isotherms of (C1&1,2-dipalmitoylphosphatidylcholine(continuous curves) at the temperatures ("C) indicated above the curves. Experimental data points are taken from ref 25. Enthalpies of melting are calculated to increase with chain length approximately twice as rapidly as is observed in experiment, although there is some discrepancy among the experimental values.49 The molecular area of the lowtemperature phase at the transition is predicted to be nearly independent of chain length, whereas the transition areas of the fluid phase increase with chain length. Areas of the n, = 16 (DPPC) transition, as measured by Albon and Baret,5oare in reasonable agreement with the calculated values. Nagle and Wilkinson51have measured volume changes for lecithins of length n, = 14-18. If we adopt the assumption of Nagle and Wilkinson that the heat groups do not contribute to the volume change and suppose, as we have done throughout, that only nc-ncl carbons (ncl= 5) contribute to the volume-dependent interactions, then the predicted volume changes are in reasonable agreement with the experimental values. (48)Lipka, G.;Chowdhry, B. Z.; Sturtevant, J. M. J. Phys. Chem. 1984,88,5401. (49)Silvius, J. R.In "Lipid-Protein Interactions";Jost, P. C., Griffiths, 0.H.,Eds.; Wiley: New York, 1982;Vol. 2,p 239. (50)Albon, N.;Baret, J. F. J . Colloid Interface Sci. 1983,92, 545. (51)Nagle, J. F.;Wilkinson, D. A. Biophys. J. 1978.23, 159.
