3188
Langmuir 1994,10,3188-3194
On Osmotic-Type Equations of State for Liquid-Expanded Monolayers of Lipids at the Air-Water Interface Si-shen Feng,*ytHoward L. Brockman,s and Robert C. MacDonaldf Department of Biochemistry, Molecular Biology and Cell Biology, Northwestern University, 2153 North Campus Drive, Evanston, Illinois 60208, and The Hormel Institute, University of Minnesota, Austin, Minnesota 55921 Received February 28, 1994. I n Final Form: June 2, 1994@ Wolfe and Brockman (Proc.Natl. Acad. Sci. U S A . 1988,85,4285-4289)derived a quasi-two-dimensional equation of state for liquid-expanded monolayers, which reproduces experimental data particularly well in comparison to other forms and can reveal mixing nonidealities in single phase, mixed-lipid monolayers (Smaby, J. M.; Brockman, H. L. Langmuir 1991,7 , 1031,and 1992,8, 563). To describe the surface pressure-area properties of monolayers,it utilizes three constant parameters: a lipid geometric parameter, a water activity parameter, and an osmotic coefficient-typeparameter. In this report, errors in the derivation of this equation of state are identified and corrected. It is shown that the simultaneous introduction of both constant activity and osmotic parameters, although necessary to accurately describe experimental data, leads to the loss of their original physical significance. Additionally, the true water activity and osmotic coefficients are calculated from the corresponding best-fit parameters and are shown to depend on the mole fraction of water in the interface and, hence, on lipid molecular area. In particular, the apparent water activity coefficient parameter of the Wolfe-Brockman equation is shown to be equal to the particular value of the true, area-dependent water activity coefficient evaluated at the lipid area at which the surface pressure extrapolates to zero. The lipid area dependence of the true water activity and osmoticcoefficientswas investigated for monolayers of oleic acid, 1-palmitoyl-2-oleoylphosphatidylcholine, and 1-palmitoyl-2-oleoylphosphatidylethanolamine; the quasi-two-dimensional mole fraction of water in the monolayer changes by as much as 110% over the area range of the lipid expanded state and for these three lipids, and this produces changes in the water activity coefficientofup to 21,43,and 106%, respectively. It is shown that although the two-dimensionalosmotic pressure due to the nonideal mixing entropy of the water and lipid head groups is a major factor in surface pressure, it cannot be the only one and that, contrary to earlier predictions ofthe asymptoticvalue ofthe surfacewater activitycoefficient,the difference between predictions of simple osmotictheory and experimental data for dilute lipid-water interface cannot be completely attributed to nonideal interactions of the type that are directly proportional to frequency of intermolecular contact.
Introduction The equation of state for a lipid monolayer a t an interface is a direct and quantitative description ofthe relationship among surface pressure, lipid molecular area, temperature, and other intensive parameters. It provides useful information for understanding the microscopic structure ofthe surface phase and the properties oflipid membranes. Various equations of state in the literature have been derived either from the kinetic theory of two-dimensional gases or from phenomenological treatments of a twodimensional Explicit recognition of water as a component of the interface leads to a two-dimensional structural model with water molecules associated with and intercalated between lipid molecules. Historically, it has been common to introduce a parameter called the surface water activity coefficient to cover the effects of the nonideality of mixing a t the interface. This parameter must be a function of the water or surfactant molecule concentration a t the interface; i.e., it must change when the monolayer is compressed. This parameter is, however, usually treated as a constant and there has been no analysis which could provide an evaluation of the error. Another modification to deal with the nonideal mixing is introduction of a n osmotic coefficient. Appearing in the Northwestern University. University of Minnesota. Abstract published inAdvanceACSAbstracts, August 1,1994. (1)Gaines, G. L., Jr. Insoluble Monolayers a t Liquid-Gas Interfaces; John Wiley and Sons: New York, 1966; pp 156-188. (2) Lucassen-Reynders, E. H. In Anionic Surfactants, Physical Chemistry of Surfactant Action; Lucassen-Reynders, E. H., Ed.; Marcel Dekker, Inc.; New York, 1981; Chapter 1. (3)Birdi, K. S. Lipid and biopolymer monolayers at liquid interfaces; Plenum Press: New York, 1989; pp 66, 72, 178. f
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equation of state as a power index of the water concentration, this parameter is sensitive to the surface water concentration; assuming it to be constant could produce serious error. Two-dimensional osmotic coefficientswere therefore rarely used in monolayer theory. Wolfe and Brockman4 introduced both water activity and two-dimensional osmotic coefficientsinto a n equation of state, assuming that the effects of the constant water activity coefficient approximation could be compensated by the simultaneous introduction of a constant osmotic coefficient. Smaby and Brockman5then evaluated various forms of equations of state by comparison with experimental measurement on monolayers of more than 14 chemical classes. They found that the equation of state which contains both a constant activity-type parameter and a constant osmotic-type parameter fits the experimental data best. This improved equation was found to be applicable to lipid mixtures as In spite of the experimental utility of the Wolfe and Brockman equation, its derivation from the LucassenReynders' excess free energy model7 appears flawed. A corrected derivation is given herein. Specifically, the original authors did not recognize that the introduction of two parameters to address nonideal mixing changes their physical meanings so that they no longer correspond to the surface water activity coefficient and the twodimensional osmotic coefficient. The surface water activity coefficient and the two-dimensional osmotic coefficient, (4) Wolfe, D. H.; Brockman, H. L. Proc. Nutl.Acad. Sci. U S A . 1988, 85, 4285. (5) Smaby, J. M.; Brockman, H. L. Langmuir 1991, 7, 1031. (6) Smaby, J. M.; Brockman, H. L. Langmuir 1992,8,563. (7) Lucassen-Reynders, E. H. J.ColloidInterface Sci. 1973,42,554.
0 1994 American Chemical Society
Osmotic-TypeEquations of State however, are important quantities for characterizing the interface. This paper addresses how they are related to and can be derived from the two constant best-fit parameters of Wolfe and Brockman, which are found to be only a special value of the true surface water activity coefficient and true osmotic coefficient at a specific lipid molecular area, for which the equation of state extrapolates to zero pressure. We then show, using monolayers of oleic acid, l-palmitoyl-2-oleoylphosphatidylethanolamine (POPE), and 1-palmitoyl-2-oleoylphosphatidylcholine (POPC) as examples, that the true surface water activity coefficient and the true osmotic coefficient are sensitive to the concentration of the water molecules a t the interface and should not be treated as constants. For the first time the two-dimensional water activity coefficient and osmotic coefficient are analytically given as a function of the lipid molecular area or, equivalently, of the surface water molecule concentration. Moreover, the sensitivity of these two important properties with respect to the lipid molecular area is quantitatively examined.
Theory We consider a surface layer composed of N components including the solvent water molecules. Each of the N components has ni molecules (i = 1, 2, 3, ..., N). For a nonideal two-dimensional mixing with surface tension effects, the chemical potential per molecule for component 1 (the water molecules), pis, is given by2t8
The symbols in eq 1 have the following meaning. p1O is the chemical potential of component 1 in the standard state (the pure air-water interface). The reference state for eq 1 has been chosen to be the clean air-water interface, for which ~d = 0, Xls = 1, and, consequently, flS = 1. k T is the Boltzmann constant multiplied by temperature. f15 is the surface water activity coefficient, which is introduced to include the nonideality effects of the N component interface mixing. XlSis the mole fraction of component 1 a t the interface. The term no1 is the surface force field contribution to the chemical potential, where w1 is the surface area occupied by a water molecule a t the interface, taken w1 = 9.65 A2.g Strictly, w1 should be given as 9.65 A21L,where L is the average number of layers of water molecules in the head group region. For this calculation we assume L = 1. See Discussion for additional commentary on this assumption. n = n(A,T) is the surface pressure of the lipid monolayer a t a lipid molecular area A and temperature T. The surface pressure is the average rate oflateral momentum exchange between lipid molecules per unit length on the surface.1° It is also the lateral, repulsive force per unit length on a barrier separating a clean water surface from the monolayer surface.l’ The (Langmuir) surface tension, y , is defined as the total mechanical effect of the clean water surface tension and the surface pressure
where ydo is the surface tension of a clean air-water interface. In the bulk phase, the chemical potential per water molecule is (8) Lucassen-Reynders, E. H.J . ColloidZnterface Sci. 1973,42,663. (9)Fowkes, F.M. J . Phys. Chem. 1962, 66,385. (10)Evans, E. A.; Skalak, R. Mechanics and Thermodynamics of Biomembranes; CRC Press, Inc.: Boca Raton, FL, 1980; p 85. (11) Cevc, G.;Marsh, D. Phospholipid Bilayers;John Wiley & Sons: New York, 1987; p 351.
Langmuir, Vol. 10, No. 9, 1994 3189
p 1 = plo
+ k T ln(flXl)
(3)
The symbols without the superscript s have the same meaning given above except that they apply to the bulk phase. For a bulk phase of pure water with a clean airwater interface, n = 0 andXlS=XI = 1and, consequently, flS = f1 = 1. Since the chemical potentials for water molecules a t the interface and in the bulk phase are equal a t equilibrium, p l S = p1. From eqs 1 and 3, we have
(4) For “insoluble surfactants”, like typical lipid molecules, the bulk concentration is very small and the bulk water activity coefficient is essentially 1. The term f1X1 in the denominator inside the logarithm in eq 4 can reasonably be equated to unity and eq 4 can then be approximated =2,12
kT
Jt = - -ln(flsXls) a1
If we assume that the water layer at the interface is one molecule thick, the surface concentration of water molecules in molecular fraction units can be approximated as
Here the Wolfe and Brockman notation is preserved, i.e., A represents the lipid molecular area. w , is the specific value of A at which the monolayer collapses. a, is the number of surface water molecules per lipid molecule in the collapse molecular area w,, i.e., a, = (0, - wo)/w1. The equation of state, containing the surface water activity coefficient,flS,to account for the nonideal mixing effect, is derived from eqs 5 and 6 as 0 1
A=wo+ fls
exp(nw,lkT) - 1
(7)
Correspondingly, if the two-dimensional osmotic coefficient, 4, is employed to represent the nonideal mixing effect, eq 1becomes
fils = pls + #kT ln(Xls)+ nwl
(8)
A parallel derivation leads to another form of equation of state, wherein 4 represents the nonideal mixing effect
According to the above definitions, the surface water activity coefficient and the two-dimensional osmotic coefficient depend on the surface concentration and the temperature, i.e.,flS=flS(A,T)and 4 = ~ A , T respectively. ), The assumption of constant fis or 4 is hence an approximation which may lead to significant error. It is then appropriate to ask whether or not the equation of state which contains both parameters can reproduce the surface pressure vs surface area relationship. Writing for A vs n (12) Gaines, G . L.,Jr. J . Chem. Phys. 1978, 69, 2627.
Feng et al.
3190 Langmuir, Vol. 10, No. 9, 1994 A
In eq 10, since both of the parameters f l S and 4 have been included, they no longer have their original physical meaning as the surface water activity coefficient and the two-dimensional osmotic coefficient. We therefore change the notation of f I s to f i b and that of 4 to q. Equation 10 can be recast as n vs A
Figure 1is a plot of eq 11with the surface pressure n as a function of the lipid molecular area A. Several various specific values of the surface pressure n and the lipid molecular area A are defined in this figure. 00 is the partial molecular area of a lipid molecule a t n = 05. we is the partial molecular area of lipid a t which the monolayer collapses. wp is the specific value of the lipid molecular area a t which the "lift-off of the n-a curve occurs. For w > wp eq 11 no longer has physical meaning and mathematically gives a very small value for the surface pressure (< 1dydcm). n,is the specificvalue ofthe surface pressure which causes the monolayer to collapse. n, has only a mathematical meaning and is the asymptotic value of n which is formally computed from the mathematical form ofthe equation of state, eq 11,asA becomes infinitely large. It was found that eq 11can fit experimental data for monolayers in the liquid-expanded state, i.e. in the surface pressure range from a near zero value to near ~ o l l a p s e .For ~ practical reasons, Smaby and Brockman limited their fitting to a point slightly under n,. This distinction is immaterial for present purposes. The equation of state in the form of eq 10 or eq 11 is thus applicable only in the surface pressure range of n,2 n > 0 or, correspondingly, only in the lipid molecular area range of w, 5 A < up.We know that the practical surface pressure of a lipid monolayer a t the air-water interface should always be positive and the tangent of an experimental n-a curve has a discontinuity due to phase transition a t a near zero pressure. Equation 11, however, mathematically becomes negative and gives a formal limiting value ofn, = n ( ~ -= - -(qltT/olIn )fib. In the case of oleic acid, for example, this value is -14.5 dydcm. Therefore, for A < wc or A > up,eq 10 or eq 11 does not reflect the thermodynamic behavior of the lipid monolayer and, therefore, no longer represents the equation of state of the monolayer a t the air-water interface. Smaby and Brockman showed that, among various equations of state, eq 10 or eq 11,with w , as a parameter in addition to fib and q, provided the best fit to experimental data for monolayers ofmore than 14different lipids, among which were several different chemical c l a s s e ~ .They ~ suggested that the introduction of the constant parameter q might have compensated for the effects caused by the constant f i b assumption. Since the symbol f l S in eq 7 and 4 in eq 8 have been reserved for the surface water activity coefficient and the two-dimensional osmotic coefficient, two new symbols of f i b and q have been used in eq 10 or eq 11 as the corresponding substitutions when both parameters are included simultaneously. The symbols f i b and q are introduced to emphasize that the physical significance of their origins has been lost due to their simultaneous introduction. Although the constant f i b is not the surface water activity coefficient, and the constant q is not the two-dimensional osmotic Coefficient, the relationships of
Figure 1. Graph ofthe equationof state, eq 10 or eq 11,written in terms of the constant parameters fib and q and plotted as surface pressure n vs lipid molecular area A. wo is the crosssectional area of a lipid molecule, w cis the lipid molecular area at which the monolayer collapses, and w Pis the specific value of the lipid molecular area at which the equation of state gives zero surface pressure. .n,is the collapse surface pressure and n- is the lower limit of n produced by the equation of state as A equals infinity. As the interface becomes infinitely dilute (A -), whereby the experimental surface pressure approaches zero, eq 10becomes negative. The limitingvalue ofn- is -(&Ti 01) In fib, which in the case of oleic acid, for example, is -14.5 dydcm.
-
the true water activity coefficient f i S and the twodimensional osmotic coefficient 4 to the corresponding substitutions f i b and q are easily derived. Since (flsX1s) in eq 1,(XIs)@in eq 8, and cfibXIsin )qeq 19 all represent the surface water activity a t the surface water molecule concentration XI, they must be equal, i.e.
from which we obtain that
and
Equations 13 and 14 show that, as expected from their defining equations, (1)and (8), the surface water activity coefficient,f I s , and the two-dimensional osmotic coefficient, 4, are both functions of the surface water molecule concentration Xis, even though the two parameters, f i b and q, are assumed to be constant in XIs. This is one of the advantages of the equation of state of the form eq 10 or eq 11. The assumption that the two parameters fis and q are constant is a matter of convenience in fitting eq 10 or eq 11to experimental data. The actual surface water activity coefficientand two-dimensional osmotic coefficient can then be evaluated from f i eand q using eqs 13 and 14 as functions of the water molecule concentration Xis. It should be pointed out that eqs 13 and 14 are valid only in the range of surface water concentrations X , < XIs < 1, or equivalently in the range of the lipid molecular area wc < A < wp or of the surface pressures 0 < ~d < n,. This is because eq 10 or 11can fit experimental data for monolayers in the liquid-expanded state in the pressure range between a near collapse pressure and a near zero ~ a l u e . ~We - ~note that eqs 13 and 14 do not approach the asymptotic value of 4 = 1and f I s = 1. This is contrary to our expectation and implies that the difference between predictions of simple osmotic theory and experimental data for dilute lipid-water interface cannot be completely
Osmotic-Type Equations of State
Langmuir, Vol. 10, No. 9, 1994 3191
attributed to nonideal interactions of the type that are directly proportional to frequency of intermolecular contact. The water activity coefficient f l S and the osmotic coefficient I$ can also be related to each other using eq 12
Wolfe and Brockman4did not recognize the loss in physical significance of f i b and q when they are both included in the equation of state. They, therefore, also did not appreciate the necessity of additional steps to extract the actual surface water activity coefficient f i s from their f i b and q . Smaby and B r ~ c k m a non , ~ the other hand, did recognize a possible correlation between the two parameters of f i b and q . On empirical evidence, they suggested a linear relation between f1 and ln(q), which reduced the total parameter number in the suggested equation of state from 3 to 2. Such a correlation between f i b and q is not surprising considering eqs 15. Moreover, since q appears in eq 19 a s the power index of the water concentrationXls, while f l S is only the proportionality coefficient of Xis, q is much more sensitive to XI than is fls. The derivation of the equation of state in the form of eq 10 or eq 11, which contains both constant parameters, fib and q , was first given by Wolfe and B r ~ c k m a n Although .~ their final form is the same as eq 10, their derivation is flawed and how it follows from Lucassen-Reynders' excess free energy model for nonideal two-dimensional mixing is obscure. A corrected and complete derivation follows. Lucassen-Reynders'J' assumed that the excess free energy, AG, for nonideal mixing of N components a t a n interface can be described by a linear function of the product of the mole fractions XXj for each pair of interacting components
A G / X n i s= XHiTisqs = HllXls2
+
Wl&sX,S
+
(16)
where Hu is the interaction coefficient for molecules of component i and componentj. For example, in the case of the two-component mixing a t the lipid-water interface, Hll, H I Z ,and HZZrepresent the interaction coefficients between two water molecules, between a water and a lipid molecule, and between two lipid molecules, respectively. The corresponding excess chemical potential per molecule of the component i due to the nonideal mixing at the interface is equal to
(17) Traditionally, this excess chemical potential due to nonideal mixing has been subsumed in the water activity coefficient,fls,Le., Lucassen-Reynders' assumption relates f i s to the sum of the mole fraction production for each pair of surface components:
kT In fls
=
XHiFFj= HllXlsz+
+
Wl.&lsX,S H;X,"'
(18)
from which it is clear that the water activity coefficient is definitely not a constant and it must depend on the surface water concentration XIs or on the lipid concentration Xzs (since XIs Xzs = 1). Alternatively, this excess chemical potential can also be formulated in the form of
+
qkT 1ncflbXls)as follows
where q denotes the quantity in the brackets and the last term n w l reflects the surface tension effect. Note that the result of this derivation, eq 19, does not depend on the specific form of the excess chemical potential aAGlanis. The Lucassen-Reynders' assumption, eq 16, is only one possible example. We can see from the preceding derivation that (1)the value of q depends on the concentration of the water molecules a t the interface, Le., q is a function ofXlS,and (2)q, as the quantity in brackets in eq 19, is closelyrelated to the parameter f l S by definition. Moreover, plSis the chemical potential of the water molecules a t the interface in a reference state, such state having been chosen to be the clean air-water interface, for which the surface pressure n = 0. The reference state of the clean airwater interface for the two-dimensional interface corresponds to the pure bulk phase of water for the threedimensional solution model. As shown above, eq 10 can be easily obtained from eqs 19 and 3.
Application Smaby and Brockman5 showed for data from more than 14classes of lipid molecules that, of five common equations of state, eq 10 with the three constant parameters wc,f l s , and q best reproduces the experimental data in the surface pressure range from near collapse to near zero. The use of two constant parameters, f i b and q , to account for nonidealities of lipid-water interactions was very satisfactory for monolayers in the liquid-expanded state, Le., for monolayers in the surface pressure range of n,> n > 0. Therefore, the equation of state (10) with f i b and q as constants is valid for the corresponding range of the lipid molecular area wc < A < wp. Here ncand wc are the specific values for the surface pressure and the lipid molecular area measured for a monolayer a t collapse and are related by the following equation, which is obtained by settingA = wc in eq 10 or eq 11
Given the experimental value of the collapse surface pressure nc,eq 20 can be used to compute a,, the number of surface water molecules which are associated with a lipid molecule at collapse. We have defined wp to be the "lift-off surface area of the lipid molecules. wp can be approximately calculated by extrapolating the equation of state to a zero value of the surface pressure. A slight error is incurred by assuming that A = opa t n = 0, since the pressure a t "lift-off" is actually the vapor pressure of the twodimensional gaseous phase. The latter is small (
