2308
Langmuir 1996, 12, 2308-2315
A Simple Surface Equation of State for the Phase Transition in Phospholipid Monolayers Eli Ruckenstein* and Buqiang Li† Chemical Engineering Department, State University of New York at Buffalo, Amherst, New York 14260 Received August 2, 1995. In Final Form: February 5, 1996X Phospholipid monolayers at an oil/water interface are treated as two-dimensional regular solutions made up of three components, namely, singly dispersed phospholipid molecules, clusters of phospholipid molecules, and “empty” sites occupied by water and oil molecules. A simple surface equation of state is derived and used to explain the phase transition that occurs in the monolayer. The equation contains two terms: one is due to the two-dimensional mixing entropy and the other one arises from intermolecular interactions calculated in the nearest-neighbor approximation. The equation reduces to the two-dimensional van der Waals equation of state when no clustering occurs. The temperature-dependent interaction parameter is obtained by fitting the experimental data for 1,2-dimyristoyl phosphatidylcholine at relatively large surface areas per molecule. Good agreement with experiment is obtained for 1,2-dimyristoyl (C14), 1,2-dipalmitoyl (C16), 1,2-distearoyl (C18), 1,2-diarachidoyl (C20), and 1,2-dibehenoyl (C22) phosphatidylcholines (lecithins) over the entire range of surface areas and temperatures.
Introduction At low surface densities (large molecular areas), the surfactant molecules at an air/water interface behave like a two-dimensional gas. A first-order phase transition may occur on compression of the gaseous film. The phase transition is similar to the three-dimensional vapor-liquid condensation and can be predicted via a two-dimensional van der Waals equation of state.1 However, deviations from the first-order phase transition occur near the critical temperature, which were attributed to clustering in the gaseous monolayer.2 At high surface densities (low molecular areas) at an air/water or oil/water interface, a phase transition may occur, but generally, the phase transition is not first-order. A first-order phase transition implies a horizontal transition region with discontinuity in the slope of the π-A isotherm at the two end points of the phase transition. Most experimental π-A isotherms exhibit, however, a nonhorizontal region and continuity at the two end points.3,4 Even when a horizontal region was experimentally observed,5 the phase transition could not be considered completely first-order, since at the high-density end (close to molecular close-packing), the transition was in the form of a smoothly rising curve. It was generally recognized that the phase transition in the monolayers does not conform to a first-order transition in a strict thermodynamic sense.6-10 * To whom the correspondence should be addressed. † Permanent address: Chemical Engineering Department, Beijing University of Chemical Technology, Beijing 100029, PRC. X Abstract published in Advance ACS Abstracts, April 15, 1996. (1) Adamson, A. W. Physical Chemistry of Surfaces, 4th ed.; Wiley: New York, 1990; p 129. (2) Stoeckly, B. Phys. Rev. A 1977, 15, 2558. (3) Birdi, K. S. Lipid and Biopolymer Monolayers at Liquid Interfaces; Plenum: New York and London, 1989. (4) Mingotaud, A.-F.; Mingotaud, C.; Patterson, L. K. Handbook of Monolayer; Academic: New York, 1993; Vols. 1 and 2. (5) Pallas, N. R.; Pethica, B. A. Langmuir 1985, 1, 509. (6) Philips, M. C.; Chapman, D. Biochim. Biophys. Acta 1968, 163, 301. (7) Adam, N. K. The Physics and Chemistry of Surface, 3rd ed.; Oxford University: London, 1941. (8) Scott, H. L., Jr. Biochim. Biophys. Acta 1975, 406, 324. (9) Marcelja, S. Biochim. Biophys. Acta 1974, 367, 165. (10) Nagle, J. F. J. Membrane Biol. 1976, 27, 233; J. Chem. Phys. 1973, 58, 252; 1975, 63, 1255.
S0743-7463(95)00653-6 CCC: $12.00
Many attempts8-22 to model the phase transition at high surface densities have been made. Most9-19 lead to a first-order phase transition and few to a second-order phase transition.8,20-22 In the first set of models, some investigators9-15 assumed that the phase transition is due to the change in the number of gauche conformations of the hydrocarbon chains, and others16,17 assumed that the excluded volume interactions are more important than the intramolecular chain interactions. Molecular dynamics18 and Monte Carlo simulations19 also provided a firstorder phase transition. It is worth noting the theory of Cantor and Dill15 for phospholipid monolayers, which was concerned with the same surfactants as those examined in the present investigation. Their theory involved four area-dependent contributions to the surface pressure: the two-dimensional translation, the headgroup interactions, the intramolecular interactions due to the chain configurations, and the volume-dependent interactions between the hydrocarbon chains. First-order phase transitions were obtained by constructing tie lines mutually tangent to the fluid and solid free-energy curves. Among the models which exhibit a second-order phase transition,20-22 one may note the statistical mechanical model of Bell et al.22 which fits, however, the experimental data only qualitatively.3 One possibility to obtain a second-order phase transition is to account for the clustering of the surfactant molecules. Very simple theoretical treatments in this direction were (11) Caille, A.; Pink, D. A.; De Verteuil, F.; Zuckermann, M. J. Can. J. Phys. 1980, 58, 581. (12) Doniach, S. J. Chem. Phys. 1978, 68, 4912. (13) Caille, A.; Rapini, A.; Zuckermann, M. J.; Cros, A.; Doniach, S. Can. J. Phys. 1978, 56, 348. (14) Jahnig, F. J. Chem. Phys. 1979, 70, 3279. (15) Cantor, R. S.; Dill, K. A. Langmuir 1986, 2, 231. (16) Firpo, J.-L.; Dupin, J. J.; Albinet, G.; Bois, A.; Casalta, L.; Baret, J. F. J. Chem. Phys. 1978, 68, 1369. (17) Dupin, J. J.; Firpo, J.-L.; Albinet, G.; Bois, A.; Casalta, L.; Baret, J. F. J. Chem. Phys. 1979, 70, 2357. (18) Kox, A. J.; Michels, J. P. J.; Wiegel, F. W. Nature (London) 1980, 287, 317. (19) Georgallas, A.; Pink D. A. J. Colloid Interface Sci. 1982, 89, 107. (20) Scott, H. L., Jr.; Cheng, W.-H. J. Colloid Interface Sci. 1977, 62, 125. (21) Kaye, R. D.; Burley, D. M. J. Phys. A 1974, 7, 1303. (22) Bell, G. M.; Mingins, J.; Taylor, J. A. G. J. Chem. Soc., Faraday Trans. 2 1978, 74, 223. Bell, G. M.; Combs, L. L.; Dunne, L. J. Chem. Rev. 1981, 81, 15.
© 1996 American Chemical Society
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Langmuir, Vol. 12, No. 9, 1996 2309
suggested by Langmuir23 and Smith.24 Recently, assuming a size distribution of clusters, Ruckenstein and Bhakta25 developed a theoretical treatment for the adsorption isotherms. Ruckenstein and Li extended the treatment to the phase transition from a gas to a liquid monolayer26 and to the phase transition from a liquidexpanded to a liquid-condensed monolayer.27 Depending upon the extent of clustering, first- or second-order phase transitions were predicted. A simple surface equation of state, which involves a single cluster size, was also proposed by Israelachvili.28 In the present paper, we consider phospholipid monolayers at a nonpolar oil/NaCl aqueous solution interface. Extensive experimental data are available29 for lecithin surfactants regarding the chain length and temperature dependencies of the phase transition occurring in the monolayers. A simple surface equation of state based on clustering is developed and compared with experiment. Basic Thermodynamic Relation The monolayer of surfactant molecules at the oil/water interface is treated as a ternary two-dimensional solution made up of singly dispersed surfactant molecules, clusters of a single size and “empty” sites, containing several water and oil molecules, which are considered as fictitious species. Each surfactant molecule is assumed to occupy the surface area A0, regardless of whether it is in a singly dispersed state or in a cluster. A cluster is assumed to contain n surfactant molecules and therefore occupies the surface area nA0. Each empty site is regarded as a fictitious species that occupies the surface area A0 and consists of several water molecules in the water side of the interface and oil molecules in the oil side of the interface. For the two-dimensional solution, the chemical potentials (µ) are expressed as30
µ1 ) µ01 + kT ln(a1) - A0σ
(1)
µc ) µ0c + kT ln(ac) - nA0σ
(2)
µ0 ) µ00 + kT ln(a0) - A0σ
(3)
where the subscripts 1, c, and 0 stand for the singly dispersed surfactant molecules, clusters, and empty sites, respectively, a is the activity, σ is the interfacial tension, and µ0 is the standard chemical potential. Since the singly dispersed surfactant molecules are free to rotate, the standard state for the singly dispersed molecules is considered to be the solid monolayer in which each molecule has complete freedom of rotation about its long axis. The standard state for the clusters is considered to be the solid monolayer in which the surfactant molecules are completely frozen. The standard state for the empty sites is assumed to be the pure oil/water system. Each empty site is considered to contain nw water molecules in the water side of the interface, noil oil molecules in the oil side of the interface, and the oil/water interface of area (23) Langmuir, I. J. Chem. Phys. 1933, 1, 756. (24) Smith, T. Adv. Colloid Interface Sci. 1972, 3, 161. (25) Ruckenstein, E.; Bhakta, A. Langmuir 1994, 10, 2694. (26) Ruckenstein, E.; Li, B. Langmuir 1995, 11, 3510. (27) Ruckenstein, E.; Li, B. J. Phys. Chem. 1996, 100, 3108. (28) Israelachvili, J. N. Langmuir 1994, 10, 3774. (29) Mingins, J.; Taylor, J. A. G.; Pethica, B.; Jackson, C. M.; Yue, B. Y. T. J. Chem. Soc., Faraday Trans. 1 1982, 78, 323. (30) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans, Green and Co.: London, 1966. (31) Taylor, J. A. G.; Mingins, M.; Pethica, B. J. Chem. Soc., Faraday Trans. 1 1976, 172, 2694.
A0 which separates them. Consequently, its standard chemical potential can be written as
µ00 ) nwµw + noilµoil + A0σo/w
(4)
where µw and µoil are the chemical potentials of the pure water and oil molecules, respectively, and σo/w is the oil/ water interfacial tension. Assuming that the surfactant molecules are present only in the surface phase and denoting with NB,w and NB,oil the number of water and oil molecules in the bulk water and oil phases, respectively, the total Gibbs free energy (G) of the whole system is given by
G ) NB,wµw + NB,oilµoil + N0µ0 + Nµ1 + Atσ
(5)
where At stands for the total surface area and N0 and N stand for the number of empty sites and the total number of surfactant molecules, respectively. By combining eqs 1 and 3-5, one obtains t G ) Ntwµw + Noil µoil + Nµ01 + N0A0σo/w + kT[N0 ln(a0) + N ln(a1)] (6)
where Ntw ) NB,w + N0nw and Ntoil ) NB,oil + N0noil represent the total number of water and oil molecules in the oil/ water system, respectively. The thermodynamic relation
σ)
( ) ∂G ∂At
)
T,P,N
[
]
∂ (G/N) ∂A
(7)
T,P,N
yields the following surface equation of state
[
πA0 ∂ N0 )A ln(a0) + A0 ln(a1) kT ∂A N 0
]
T,P,N
(8)
In the above equations, A ) At/N is the average surface area occupied by a surfactant molecule (molecular area) and π ) σo/w - σ is the surface pressure at the oil/water interface. Regarding the derivation of eq 8, one should note that the first three terms in the right-hand side of eq 6 provide no contribution to the surface pressure (π), since the total numbers of water and oil molecules, Ntw and Ntoil, and their chemical potentials, µw and µoil, as well as Nµ01, are constant. Since an empty site is assumed to occupy the same surface area as a surfactant molecule,
At ) (N0 + N)A0
(9)
Combining with eq 9, eq 8 becomes
{
πA0 ∂ )[(A - A0) ln(a0) + A0 ln(a1)] kT ∂A
}
T,P,N
(10)
Activity Coefficient The activities of component j (0, 1, or c) can be expressed as
aj ) xjγj
(11)
where xj is the mole fraction and γj is the activity coefficient of component j, which will be calculated as for a regular solution. The activity coefficient can be considered to be a result of the interactions in the side of the headgroups and the interactions in the side of the hydrocarbon tails. Because of the similarity between the hydrocarbon tails and the oil molecules, the contribution of the exchange energy in the side of the hydrocarbon tails is negligible
2310 Langmuir, Vol. 12, No. 9, 1996
Ruckenstein and Li
and γj can be calculated on the basis of the mixing enthalpy in the side of the headgroups (∆Hmix), using the expression
kT ln(γj) )
(
∂ ∆Hmix ∂Nj
)
(12)
T,P,Ni*j
The nearest-neighbor interactions will be the only ones employed in the calculation of ∆Hmix. The zwitterionic headgroup hydrated with some water molecules is assumed to occupy the volume of a cylinder of cross-sectional area A0, which is equal to the crosssectional area of the two hydrocarbon tails of the lecithin molecule. In the headgroup side, the empty sites are made up of water molecules and are assumed to occupy the surface area A0. An empty site or a surfactant molecule occupies a lattice site surrounded by z0 nearest neighbors. A cluster which contains n surfactant molecules occupies n lattice sites and has a larger number of nearest neighbors. Assuming that the number of nearest neighbors of a species is proportional to its perimeter (pj), one obtains zc/z0 ) pc/p0. Because pc ∝ (nA0)1/2 and p0 ∝ A01/2, the number of nearest neighbors of a cluster can be calculated from
zc ) n1/2z0
(13)
Since only the interactions with the nearest neighbors are taken into account, the interaction exchange energy between the singly dispersed molecules and the clusters is zero because molecules of the same kind are involved and the mixing enthalpy (∆Hmix) depends only on the interactions between the empty sites and the headgroups of the surfactant molecules in contact with the empty sites in the side of the headgroups. Consequently,
∆Hmix ) N0z0(y1∆W10 + yc∆Wc0)
(14)
where yj, the fraction of nearest neighbors of component j, is given by
yj )
Njzj (N0 + N1)z0 + Nczc
(16)
where χ ) z0∆W/kT represents the interaction parameter between the headgroups and the water molecules of the empty sites. Inserting eq 16 into eq 12, one obtains
ln(a0) ) ln(x0) + (1 - y0)2χ
(17)
ln(a1) ) ln(x1) + y02χ
(18)
ln(ac) ) ln(xc) + n1/2y02χ
(19)
Surface Equation of State Introducing eqs 17 and 18 into eq 10, the following surface equation of state is obtained
π)-
kT kT ln(x0) (1 - y0)2χ A0 A0
π)-
kT [ln(1 - x1) + x12χ] A0
(21)
Since x1 ) A0/A, the above equation becomes
π)
(
) ( )
kT A0 2 A kT ln χ A0 A - A0 A0 A
(22)
When the second term in the right-hand side is omitted, eq 22 reduces to the Frumkin and Volmer equation. A low-density approximation of the first term leads to the two-dimensional van der Waals equation of state
(
π+
)
A0z0∆W A2
(A - A0) ) kT
(23)
where A0z0∆W ) kTA0χ. Effect of Clustering
(15)
and ∆Wj0 (j ) 1, c) is the interaction exchange energy between component j and the empty sites. Because only the nearest-neighbor interactions are taken into account, ∆Wc0 ) ∆W10 ) ∆W. Since y0 + y1 + yc ) 1, eq 14 becomes
∆Hmix ∆W ) N0z0 (1 - y0) ≡ N0 (1 - y0)χ kT kT
Equation 20 contains two terms: the first is due to the two-dimensional entropy and the second to the lateral interactions between the headgroups of the surfactant molecules and the water molecules of the empty sites. For a monolayer at an air/water interface, a similar derivation leads again to eq 20. However, in the latter case, χ involves the interactions both between the hydrocarbon chains and air and between the headgroups and water molecules. For a monolayer at an oil/water interface, only the interaction between the headgroup and the water molecules provides an important contribution, because the hydrocarbon chain of the surfactant and the oil molecules are very similar. Equation 20 can be reduced to a simple form when no clusters are generated in the monolayer. Indeed, in this case, the two-dimensional solution contains only two components: the singly dispersed surfactant molecules and the empty sites. Since the two components are assumed to have the same molecular area, x0 ) y0 ) 1 x1 and eq 20 reduces to
(20)
The chemical potential per surfactant molecule at equilibrium must be the same regardless of whether it is in a singly dispersed state or in a cluster. Consequently, µc ) nµ1. By combining eqs 1, 2, 18, and 19, one obtains
xc ) x1nK
(24)
where K is an equilibrium constant given by
[
K ) exp (n - n1/2)y02χ -
]
n∆µ0 kT
(25)
In the above equation, ∆µ0 ) (µ0c/n) - µ01 stands for the standard chemical potential change per surfactant molecule for the transfer from the singly dispersed to the cluster state. The total number of surfactant molecules is equal to the sum of the singly dispersed surfactant molecules (N1) and those included in the clusters (nNc); that is,
N ) N1 + nNc
(26)
Dividing the two sides of eqs 9 and 26 by the total number (Nt ) N0 + N1 + Nc) of chemical species of the two-dimensional solution and using eq 24 lead to
x0 + x1 + nx1nK )
1 A0Γt
(27)
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Langmuir, Vol. 12, No. 9, 1996 2311
and
x1 + nx1nK )
1 AΓt
(28)
where Γt ) Nt/At stands for the total number of chemical species per unit area. By combining eqs 27 and 28 and taking into account that x0 + x1 + xc ) 1, one obtains
[
x1 + n +
]
A0 A0 (1 - n) x1nK ) A A
(29)
The above equation contains x1 and y0 as variables (y0 via the equilibrium constant K). Equations 15 and 24 lead to
y0 )
1 - x1 - x1nK 1 + (n1/2 - 1)x1nK
(30)
For given values of A and n, eqs 24, 29, and 30 allow us to calculate x1, y0, and xc. Finally, the surface pressure can be obtained from eq 20. Interaction Parameter (χ) and Standard Chemical Potential Change (∆µ0) Equation 20 contains three molecular parameters: the surface area of a surfactant molecule (A0), the interaction parameter between the headgroups and water molecules (χ), and the standard chemical potential change (∆µ0). The molecular area is considered to be equal to the cross sectional area of the surfactant tail. Since the lecithin surfactants have two saturated hydrocarbon chains, we use the value of 40 Å2 for A0.15 The other two parameters were calculated as follows: Interaction Parameter (χ). χ represents the interaction parameter between the empty sites and the headgroups. Since the headgroups are the same for all the surfactant molecules considered in the present paper, its value will be determined from the experimental data available for one of them. 1,2-Dimyristoyl phosphatidylcholine (C14) in the range of relatively large surface areas per molecule is selected for the determination of χ. By fitting eq 22 with the π-A isotherms for the C14 at four temperatures, we obtained (Figure 1)
χ ) -0.1566(T - T0)
(31)
where T0 ) 239 K. This expression for χ is used to predict all the phase transitions occurring in the lecithin monolayers over a range of temperatures. Standard Chemical Potential Change (∆µ0). The standard chemical potential change includes a bulk term proportional to n and a line contribution proportional to n1/2.25,26 The latter contribution appears because the lecithin molecules lying at the boundary of a cluster provide a different contribution to ∆µ0 than those in the interior of the cluster, due to a different environment. Assuming only nearest-neighbor interactions, the following expression was obtained (see Appendix):
n∆µ0 ) (n - n1/2)∆µ0B
(32)
where ∆µ0B stands for the standard free-energy change of a lecithin molecule from singly dispersed molecules to molecules located in the bulk of the cluster. It contains
Figure 1. π-A isotherms for 1,2-dimyristoyl phosphatidylcholine.
two contributions: one due to van der Waals interactions 0 (∆µB,vdW ) and the other to the loss of rotational freedom 0 (∆µB,rot ). 0 is evaluated as follows: It is well-known32,33 ∆µB,vdW that the liquid n-alkanes having an odd number of carbons from 9 to 43 or an even number from 22 to 42 can generate by freezing a solid “rotator” phase in which the chains form layered arrays and exhibit a high degree of rotational freedom. They can also generate an orthorhombic solid phase in which the chains are fully ordered and the molecules have lost almost completely the rotational freedom. In both cases, the chains are oriented perpendicular to the plane of the terminal methyl groups. The former solid phase is denoted RH and the latter βO.33 Their resemblance to the standard states of the singly dispersed and clustered molecules suggests we consider that 0 ∆µB,vdW is equal to the enthalpy change from the RH to the βO phase. It is calculated from the group contributions of two terminal methyl groups and 2(nc - 1) methylene groups using the expression 0 ∆µB,vdW ) 2β[∆HCH3 + (nc - 1)∆HCH2]
(33)
In the above equation, ∆HCH3 and ∆HCH2 represent the contributions of a methyl and a methylene group, respectively. The factor β is included to account for the fact that the interaction between the two hydrocarbon chains belonging to a given molecule provides no contribution to 0 ∆µB,vdW . Considering that a hydrocarbon chain has four sides, only three of them are in contact with the hydrocarbon chains of the neighboring molecules; consequently, β ) 0.75. Experimental data33 for the enthalpy change from the RH and to the βO phase are available for the n-alkanes with an odd number of carbon atoms from nc (32) Cantor, R. S.; Dill, K. A. Macromolecules 1985, 18, 1875. (33) Broadhurst, M. G. J. Res. Natl. Bur. Stand., Sect. A 1962, 66A, 241; J. Chem. Phys. 1962, 36, 2578.
2312 Langmuir, Vol. 12, No. 9, 1996
Ruckenstein and Li
) 11 to 35. However, with the exception of the lower alkane lengths, there is a nonlinear variation of the enthalpy change with the chain length, which does not allow us to evaluate constant contributions for the methyl and methylene groups. For odd n-alkanes of relatively low length, which are of particular interest in the present context because the lecithin surfactants considered here involve hydrocarbon chains with a relatively low odd number of carbon atoms, a linear relationship could be established. By using the experimental data for the n-alkanes with 15, 17, 19, and 21 carbon atoms, we obtained ∆HCH3 ) 0.60 kcal/mol and ∆HCH2 ) -0.26 kcal/ mol. The two group parameters will be used at all temperatures, since no experimental data are available regarding their temperature dependence. The loss of rotational freedom (an entropic contribution), which provides a positive contribution to ∆µ0B, is given by34
[(
0 ∆µB,rot ) kT ln 2π
)]
2πISkT 2
h
1/2
(34)
where h is the Planck constant and IS is the moment of inertia of a single lecithin molecule about its long axis (IS ) mv/(2πl, where m, v, and l are the mass, volume, and length of the lecithin molecule). Since v ) A0l, the above equation can be rewritten as 0 ∆µB,rot )
(
)
4π2A0mkT kT ln 2 h2
(35)
Comparison with Experimental Data In this section, eq 20 is used to interpret the π-A isotherms for a homologous series of 1,2-dicyl phosphatidylcholines (lecithins) (with two identical hydrocarbon chains) at the n-heptane/NaCl aqueous solution interface. They include 1,2-dimyristoyl (C14), 1,2-dipalmitoyl (C16), 1,2-distearoyl (C18), 1,2-diarachidoyl (C20), and 1,2-dibehenoyl (C22) phosphatidylcholines. Their general formula is RCO2CH2CH(O2CR)CH2OPO3(CH2)2N(CH3)3, and the number of carbon atoms in R is nc ) 13, 15, 17, 19, and 21. The notation Cm is used for the lecithin with m ) nc + 1. The experimental π-A isotherms of these surfactants show that the surface pressure is nearly independent of the salt concentration in the aqueous phase and that at a given temperature, the phase transition, which depends upon the length of the hydrocarbon chain, occurs as a break from a master curve. The latter observation indicates that the isotherm is independent of the chain length for areas larger than that at the onset of the phase transition, hence, that at sufficiently large values of A there is a common π-A isotherm, which depends only on the nature of the headgroup and temperature.29,31 Effect of the Cluster Size. Figure 2 show the effect of the cluster size (the number of lecithin molecules included in a cluster). The theoretical π-A isotherms strongly depend on the size of the clusters. When n ) 1, eq 20 reduces to eq 22 and the theoretical π-A isotherm exhibits a continuous increase of π with the surface density, corresponding to a one-phase region. When n tends to infinity, a phase transition involving a horizontal line (first-order phase transition) occurs. Corresponding to this first-order transition, µ∞c - µ01 ) kT ln(a1) ) ∆µ0B, where µ∞c stands for the chemical potential of a surfactant molecule in a frozen solid monolayer in equilibrium with the fluid monolayer phase. By using eq (34) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1979, 71, 580.
Figure 2. Effect of cluster’s size on the theoretical π-A isotherm for 1,2-distearoyl phosphatidylcholine (C18) at 20 °C.
18 for a1, one obtains
∆µ0B ) kT ln(x∞1 ) + kT(y∞0 )2χ
(36)
where x∞1 and y∞0 represent the mole fraction of the singly dispersed surfactant molecules and the fraction of the nearest neighbors of the empty sites in the fluid monolayer phase in equilibrium with the frozen solid monolayer. Since only singly dispersed surfactant molecules and empty sites are present in the fluid monolayer phase and they have the same molecular areas, y∞0 ) 1 - x∞1 . From the above equation, one can obtain x∞1 for a given value of ∆µ0B. The surface pressure corresponding to the first-order phase transition can then be calculated using eq 20. Compared with the two extreme π-A isotherms (for n ) 1 and n ) ∞), a finite value of n leads to a phase transition with a nonhorizontal line in the π-A isotherm. The slope of the nonhorizontal line decreases with increasing values of n. Furthermore, the surface pressure changes continuously in the transition region with the surface density, as shown also by experiment. Figure 2 compares the theoretical curves with experimental data for the 1,2distearoyl phosphatidylcholine (C18) at 20 °C. Equation 20 fits well the experimental data for n ) 150. Since, for 50 < n < 150, the calculated π-A isotherm is insensitive to the value of n, we take n ) 150 for the prediction of the phase transitions for all the lecithins considered in this paper. Effect of the Chain Length. Besides the value of n, the phase transition strongly depends on the value of ∆µ0B, which constitutes the driving force for the formation of a cluster from the singly dispersed molecules. The more negative is ∆µ0B, the greater is the stimulation for cluster formation and, hence, the lower the transition surface pressure and the smaller the slope of the isotherm in the transition region. Table 1 contains the calculated values of ∆µ0B for the lecithins at different temperatures. One
Phase Transition in Phospholipid Monolayers
Langmuir, Vol. 12, No. 9, 1996 2313
Table 1. Standard Free-Energy Change (∆µ0B, kcal/mol) for Five Lecithins t, °C 0.3 1.1 5.0 8.0 10.0 11.0 15.0 15.2 19.8 20.0 23.0 25.0 28.6 30.0 32.4 40.0 40.3
C14
C16
C18
C20
C22
-5.20 -0.97 -3.62
-6.38
-0.79 -2.09
-3.44
-6.15 -4.75
-1.94
-3.27
-5.93 -4.59 -4.41
-1.79
-3.10
-5.71 -4.29
-2.94
-5.50 -4.09
-2.78
-5.30 -3.96 -4.92 -3.69
Figure 4. π-A isotherms for five lecithins at 20 °C.
Figure 3. π-A isotherms for five lecithins at 15 °C.
can see that ∆µ0B becomes more negative with increasing length of the hydrocarbon chain and decreasing temperature. Figures 3 and 4 present a comparison between predicted and experimental π-A isotherms for the five lecithins at 15 and 20 °C, respectively. They show that with increasing chain length from nc ) 13 to 21, the transition surface pressure at the onset of the phase transition (πtr) decreases and the π-A isotherm in the transition region gets closer to a horizontal line, hence to a first-order phase transition. With increasing temperature, πtr increases. Very good agreement is obtained for 1,2-dimyristoyl (C14), 1,2-distearoyl (C18), 1,2-diarachidoyl (C20), and 1,2dibehenoyl (C22) phosphatidylcholines. For 1,2-dipalmitoyl phosphatidylcholine (C16), the predicted π-A isotherms in the transition region exhibit a somewhat smaller slope than the experimental one. The π-A isotherms presented in Figures 3 and 4 show that at a given temperature, the phase transitions appear as a break from a master curve for all the lecithins. This
behavior is a result of the fact that at relatively high values of A, the π-A isotherms are independent of the chain length. In the present model, the chain length affects only the value of ∆µ0B, which is the driving force for the formation of a cluster. Since the clustering leads to phase transition, the chain length is expected to influence the phase transition only, and this is, indeed, shown by experiment. No phase transitions were found experimentally29 for temperatures near 0 °C for 1,2-dilauroyl phosphatidylcholine (C12). The present model predicts that for ∆µ0B > 0, there should be no phase transition. For the C12 lecithin, the present approach predicts ∆µ0B ) 0.21 kcal/mol and, hence that, indeed, no phase transition should occur. It is obvious that ∆µ0B ) 0 provides a critical temperature below which a phase transition can occur. For the C12 lecithin, the critical temperature corresponding to ∆µ0B ) 0 is -16.5 °C. No experimental data are available for comparison. It is of interest to examine the nature of the phase transition. Experiment indicates that the phase transition is second order,29 even though for lecithins with long hydrocarbon chains an approximate horizontal line in the transition region is observed at low temperatures. The experimental π-A isotherm in the transition region clearly deviates from a first-order phase transition, even when the transition region is nearly horizontal over a large range of surface densities. Indeed, at least at the high surface density end of the phase transition, both at the oil/water and at the air/water interface, the experimental isotherms always show a smoothly rising continuous curve. Many experimental π-A isotherms indicate, however, a break at the lower surface density end (onset) of the phase transition, but this may be due to the uncertainty of the experiment. The theory predicts that at both surface density ends of the phase transition, the π-A isotherm is continuous in slope and density and is nonhorizontal between the two. Effect of Temperature. A comparison with experimental data for the five lecithins at different temperatures
2314 Langmuir, Vol. 12, No. 9, 1996
Ruckenstein and Li
Figure 5. π-A isotherms for 1,2-dimyristoyl phosphatidylcholine (C14) at various temperatures.
Figure 7. π-A isotherms for 1,2-distearoyl phosphatidylcholine (C18) at various temperature.
Figure 6. π-A isotherms for 1,2-dipalmitoyl phosphatidylcholine (C16) at various temperatures.
Figure 8. π-A isotherms for 1,2-diarachidoyl phosphatidylcholine (C20) at various temperatures.
is presented in Figures 5-9. The predicted π-A isotherms for 1,2-distearoyl (C18), 1,2-diarachidoyl (C20), and 1,2dibehenoyl (C22) phosphatidylcholines are in good agreement with experiment both in the one-phase and in the transition region. The theoretical isotherms exhibit a somewhat smaller slope in the phase transition region for 1,2-dimyristoyl (C14) and 1,2-dipalmitoyl (C16) phosphatidylcholines. However, if instead of n ) 150 a smaller
value, namely n ) 35, is employed, the agreement becomes very good. Furthermore, the theoretical and experimental π-A isotherms exhibit continuity at both surface density ends of the phase transition region. The temperature dependence of the onset transition surface pressures (πtr) for the five lecithins is presented in Figure 10. Good agreement between the predicted and the experimental values is obtained.
Phase Transition in Phospholipid Monolayers
Langmuir, Vol. 12, No. 9, 1996 2315
is derived and used to interpret the phase transition occurring in the phospholipid monolayers at the nonpolar oil/NaCl aqueous solution interface. The model considers the monolayer as a two-dimensional solution composed of three components: singly dispersed molecules, clusters, and fictitious molecules (formed of water and oil molecules). The surface equation of state contains two terms: one of them is due to the two-dimensional mixing entropy, and the other arises from the intermolecular interactions in the monolayer. The model leads to a second-order phase transition. The standard state of the singly dispersed surfactant molecules is considered to correspond to a solid monolayer in which each molecule has rotational freedom about its long axis and the standard state of the clusters to a solid monolayer in which the molecules are completely frozen. The standard chemical potential change is calculated by taking into account two contributions: one is due to the change of van der Waals interaction energies among the hydrocarbon chains and the other to the loss of the rotational freedom of surfactant molecules. The former is assumed to be equal to the enthalpy change from a hexagonal “rotator” solid monolayer to an orthorhombic frozen solid monolayer. The model provides predictions for the phase transition occurring in the phospholipid monolayers in good agreement with experiment. Figure 9. π-A isotherms for 1,2-dibehenoyl phosphatidylcholine (C22) at various temperatures.
Appendix: Derivation of Equation 32 Denoting np as the number of boundary molecules of a cluster, the standard chemical potential change from the singly dispersed to the cluster state can be written as
n∆µ0 ) (n - np)∆µ0B + np∆µ0p
(A)
where ∆µ0B and ∆µ0p stand for the changes when a surfactant molecules is transferred from the standard state of singly dispersed molecules to the bulk and the boundary of the cluster at their standard states, respectively. Considering only nearest-neighbor interactions, ∆µ0B ) z0/2, where and z0 stand for the interaction energy between two surfactant molecules and the number of nearest neighbors of a surfactant molecule. Since, for large clusters, a boundary molecule interacts approximately with 2z0/3 nearest neighbors belonging to the cluster, ∆µ0p ) 2∆µ0B/3 ) z0/3, and eq A becomes
(
n∆µ0 ) n -
Figure 10. Comparison between theoretical and experimental transition surface pressure (πtr).
Conclusions On the basis of a model that involves the clustering of surfactant molecules, a simple surface equation of state
)
np ∆µ0B 3
(B)
The number of boundary molecules of a cluster is calculated by dividing the perimeter of the cluster by the diameter of a surfactant molecule. For circular clusters, np ) πn1/2. Consequently,
π n∆µ0 ) n - n1/2 ∆µ0B ≈ (n - n1/2)∆µ0B 3
(
LA9506537
)
(C)
