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Equilibrium Thermodynamics of Lipid Monolayer Domains Harden M. McConnell* and Rudi De Koker Department of Chemistry, Stanford University, Stanford, California 94305 Received April 26, 1996. In Final Form: June 27, 1996X Coexisting thermodynamic phases of lipid molecules at the air-water interface exhibit domains with a variety of sizes and shapes. Both the sizes and shapes of these domains are affected by a competition between a line tension and long-range dipole-dipole repulsion. In the case of coexisting liquid phases, line tension favors large, circular domains, whereas the dipolar repulsions favor small and/or highly elongated domains. The present work discusses the thermodynamics of the equilibrium size and shape of these domains. It is shown that whereas there is only one stable equilibrium domain size, there is an infinite number of sizes that represent metastable equilibrium at different monolayer pressures. Circular domains that are far enough apart as to not interact with one another electrostatically have the same radii under this condition of metastable equilibrium. It is shown that, in principle, the metastable equilibrium radius reached in a monolayer can depend on the initial distribution of nonequilibrium domain sizes. Domains formed in multicomponent lipid mixtures are shown to have lipid compositions in the vicinity of the domain boundaries that are different from the bulk lipid compositions. These composition variations tend to stabilize domains with different radii under conditions of metastable equilibrium.
Introduction Both pure lipids and mixtures of lipids exhibit coexisting thermodynamic phases at the air-water interface. These phases are often observed using fluorescence microscopy, employing low concentrations of fluorescent lipid probes that partition preferentially into one of the phases.1 -3 In general one of the coexisting phases forms domains that exhibit a wide variety of shapes and sizes, depending on the lipids involved and the experimental conditions. The equilibrium size and shape of lipid domains depends on line tension and on long-range dipole-dipole forces.4,5 From a theoretical point of view, a particularly simple situation arises when both phases are liquid (or liquid and gas), since in such cases the line tension is independent of the domain size and shape and the dipole forces are due to dipoles that are on average perpendicular to the surface of the monolayer. There have been a number of experimental and theoretical studies of the shapes of liquid lipid domains.6,7 A simple theoretical model of shapes and shape transitions appears to be in good agreement with experiment.2,5,6 One reason for this is that domain shape equilibrium is reached rapidly, often within minutes or less.1-3 The experimental situation is quite different with domain size equilibrium. In many experiments one observes a dependence of domain size on monolayer pressure and in some cases a wide distribution of sizes. The purpose of the present paper is X Abstract published in Advance ACS Abstracts, September 1, 1996.
(1) Mo¨hwald, H. Annu. Rev. Phys. Chem. 1990, 41, 441. (2) McConnell, H. Annu. Rev. Phys. Chem. 1991, 42, 171. (3) Knobler, C. M. Science 1990, 24a, 870. (4) McConnell, H. M.; Moy, V. T. J. Phys. Chem. 1988, 92, 4520. (5) McConnell, H. M.; De Koker, R. J. Phys. Chem. 1992, 96, 7101. This reference and ref 4 above deal with the energetics of isolated domains. Earlier work on the role of dipolar forces and line tension in monolayers was concerned with strongly interacting domains that give rise to stripe and hexagonal superstructure phases. See: Andelman, D.; Brochard, F.; deGennes, P. G.; Joanny, J. C. C. R. Acad. Sci. Paris, Ser. C 1985, 301, 675-678. Keller, D. J.; McConnell, H. M.; Moy, V. T. J. Phys. Chem. 1986, 90, 2311. Andelman, D.; Brochard, F.; Joanny, J. J. Chem. Phys. 1987, 86, 3673. The thermodynamics of the superstructure phases is complicated by the thermal excitations of longwavelength distortions. See: De Koker, R.; McConnell, H. M. J. Phys. Chem. 1996, 100 , 7722 and references therein. (6) Lee, K. Y. C.; McConnell, H. M. J. Phys. Chem. 1993, 97, 9532. (7) Stone, H. A.; McConnell, H. M. Proc. R. Soc. London, Ser. A 1995, 448, 97.
S0743-7463(96)00411-8 CCC: $12.00
to describe some of the thermodynamic aspects of domain size equilibrium. Background As in previous work, we first consider two coexisting incompressible phases, a black (B) solid phase and a white (W) liquid phase. For simplicity assume that the solid phase is isotropic in two dimensions and that the molecular dipoles are perpendicular to the plane of the monolayer. This black solid phase forms circular domains each with radius R. These circular domains are imagined to be sufficiently far apart from one another that they do not interact electrostatically. Nevertheless, they may “communicate” with one another by means of monolayer pressure, as discussed later. In previous work it is shown that the energy of an isolated circular domain is4
E ) 2πR(λ + m2 ln(e2∆/8R))
(1)
where λ is the line tension and
m ) mB - mW
(2)
The quantity ∆ is a closest approach cutoff distance used in the calculation of the dipole-dipole interaction, and can be taken to be of the order of magnitude of an intermolecular distance, 5-10 Å. Here mB and mW are the electric dipole densities in the two phases. The derivation of eq 1 is discussed extensively elsewhere.5 We refer to E as an edge energy. Both the line tension and the electrostatic contribution to E arise from the fact that the domain is finite in extent and has a boundary. The following calculations employ a material parameter Rq,
Rq ) (e3∆/8) exp(λ/m2)
(3)
It turns out that Rq is an equilibrium radius under conditions to be defined later. In some of the monolayers studied most extensively it is believed that Rq is of the order of magnitude of 10 µm. Using this material parameter, eq 1 is written in the convenient form
E ) 2πRm2 ln(Rq/eR)
(4)
It will be seen that the line tension and electrostatic forces at the domain boundary are exactly balanced when R ) © 1996 American Chemical Society
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Rq/e2, since then ∂E/∂R ) 0. The energy at this radius is a maximum not a minimum, since ∂2E/∂R2 ) -2πm2/R. Gibbs Free Energy It is instructive to consider the problem of the equilibrium size of lipid domains using the Gibbs free energy, G. The procedure is to define the conditions under which G is a minimum for all possible variations of domain size and shape, while maintaining the monolayer pressure constant. In this analysis let the chemical potentials of the black and white phases be µ°B and µ°W in the absence of the edge energy term in eq 1. In other words, the two phases are in equilibrium at pressure p0 in the absence of the edge energy. Then, when the edge energy is “turned on", the system comes to equilibrium at a different pressure p∞. Here the subscript infinity indicates that this pressure is measured at a point far away from all black domains. The Gibbs free energy is then
G ) nWµ°W + nBµ°B + (nWAW + nBAB)(p∞ - p0) + NBE (5) Here nB and nW are the number of moles in the black and white phases, AW and AB are the molar areas, and NB is the number of black domains
NB ) nBAB/πR2
(6)
In minimizing the Gibbs free energy at fixed pressure and total number of moles of lipid, one considers three variables, NB, nB, and R, of which only two are independent. In the minimization we fix one of these variables and vary another. The total number of moles of lipid is maintained constant, nW + nB ) constant. In the immediately following calculations we make no attempt to incorporate the energy NBE into the chemical potentials but simply regard this term as part of the system energy. It is shown later that this term can be incorporated into the chemical potential so as to yield the same final results, but such modified chemical potentials must be used with caution, as the system free energy is not a homogeneous linear function of the composition variables. Condition I. Minimize the free energy with respect to a process in which the number of moles of lipid in the black phase is kept constant and the domain radius is changed by changing the number of black domains.
|
∂G/∂NB p,nB ) 0
(7)
The solution to eq 7 gives an equilibrium radius equal to Rq. This calculation is equivalent to one given earlier.4 The calculation is also generically related to the energy minimizations used to calculate domain shapes.2 Condition II. Minimize the free energy with respect to a process in which the domain radius is kept constant and the number of domains is changed.
|
∂G/∂NB p,R ) 0
(8)
The solution of eq 8 yields for the equilibrium pressure
p∞ ) p0 + (η - 1)-1(2m2/R) ln(Rq/eR)
(9)
In this equation η ) AW/AB. In order to simplify the discussion, we assume that η is greater than 1, except for the black gas phase discussed in the Appendix. The several conclusions reached later concerning domain size equilibration do not depend on whether η is larger or smaller than 1.
Condition III. Minimize the free energy with respect to a process in which the number of black domains is kept constant and the domain radius is changed by changing the amount of black phase.
|
∂G/∂nB p,NB ) 0
(10)
The solution of eq 10 yields for the equilibrium pressure
p∞ ) p0 + (η - 1)-1(m2/R) ln(Rq/e2R)
(11)
It will be seen that the pressures obtained from conditions II and III are only mutually consistent for R ) Rq, the same equilibrium radius obtained from condition I. Note that in eq 11 the equilibrium pressure p∞ is equal to the equilibrium pressure p0 in the absence of edge energy when R ) Rq/e2. As discussed above in connection with eq 4, at this value of the domain radius the electrostatic and line tension forces exactly balance, and the edge energy is a maximum. The equilibrium pressure is a minimum or maximum when R ) Rq/e, depending on whether η is larger or smaller than 1. In connection with the following discussion of domain sizes it is helpful to note here that the equilibrium pressure p∞ in eq 11 increases with decreasing domain size as long as Rq/eR > 1 and η > 1. More generally,
dp∞/dR ) -(η - 1)-1m2/R2 ln(Rq/eR) At the pressure in eq 11 isolated domains are unstable when η > 1 and Rq/eR > 1, since there the calculated equilibrium pressure increases with decreasing radius, a process leading to domain annihilation when the external pressure acting on the system is maintained at a constant value. Compressible Phases As far as we can tell the assumption of incompressible liquid and solid phases in the above calculations has no significant qualitative effect on the conclusions, as long as the compressibilities are small. If the black and white phases are compressible, the areas need to be replaced by A°W(1 - κW(p - p0)) and A°B(1 - κB(p - p0)) where the compressibilities are κB ) - A-1 B ∂AB/∂p and κW ) -A-1 ∂A /∂p. The pressure-area terms in eq 5 then W W become
(nBA°B + nWA°W)(p∞ - p0) 1
/2(nBA°BκB + nWA°WκW)(p∞ - p0)2
If eq 11 is rewritten so that the factor (η - 1)(p∞ - p0) is on the left hand side, then we see that when monolayer compressibility is included in the calculation, this factor is simply replaced by (η - 1)(p∞ - p0) - 1/2(ηκW - κB)(p∞ - p0)2 and one needs to solve a quadratic equation for the equilibrium pressure. For highly compressible phases a problem arises because of the issue of the pressure dependence of the dipole density. Gas phase domains represent a special problem in this connection and are discussed in the Appendix. Metastable Equilibrium In experimental monolayer studies it is frequently observed that monolayer compression results in an increase of domain size rather than the formation of more domains of a given size. Thus conditions I and II above do not represent common experimental observations. However condition III may be valid in many situations, corresponding to metastable equilibrium. Circular do-
Equilibrium Thermodynamics of Lipid Monolayer Domains
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mains with radii that conform to eq 11 are stable, since
|
∂2G/∂n2B p,NB ) -(ABm2/2πRnB) ln(Rq/eR) g 0 when Rq/eR < 1
(12)
The domains are metastable rather than stable in that the system free energy is lower when the domain radii are given by Rq in eq 3. That is, the domains are stable with respect to small transfers of lipid between the black and white phases, but when R * Rq, they are not stable with respect to processes that change the number of domains. Processes that change the number of domains include domain fission, fusion, creation, and annihilation. All of these processes except annihilation are expected to have large activation energies.2 Note that a typical order-ofmagnitude for a domain energy corresponds to 105 K. The stability condition in eq 12 can also be understood by reference to the condition for the equilibrium pressure in eq 11. It will be seen that for η > 1, ∂p∞/∂R|NB > 0 when Rq/eR < 1, which is an equivalent condition for stability. Approach to Metastable Equilibrium The approach to metastable equilibrium under typical experimental conditions can be viewed as follows. We start with a monolayer having an arbitrary distribution of domain sizes, each domain well separated from its neighbors. Consider a pair of domains in this monolayer after it has been compressed to a fixed total monolayer area. Let the domains have radii equal to R0(1 + χ)1/2 and R0(1 - χ)1/2, so that the total area of the two domains is constant, independent of χ. Minimization of the Helmholtz free energy of the system at constant temperature and monolayer area reduces to minimization of the energy of all pairs of domains. During the process of equilibration the monolayer pressure may change. We assume the processes of domain fission and fusion to be too slow to be significant. The energy of any given pair is
E2 ) 2πR0m2[(1 + χ)1/2 ln(Rq/eR0(1 + χ)1/2) + (1 - χ)1/2 ln(Rq/eR0(1 - χ)1/2)] (13) The following derivative is of interest.
- p(1-χ) ) dE2/dχ ) πR20(η - 1)(p(1+χ) ∞ ∞
(14)
and When χ is restricted to being a positive number, p(1+χ) ∞ p(1-χ) are the equilibrium pressures of the larger and the ∞ smaller of the two domains. With respect to the energy function in eq 13, three cases are of interest. (i) Rq/eR0 > 1. In the representative plot in Figure 1a for this case it will be seen that for positive values of χ the slope dE2/dχ is always negative so that the equilibrium pressure of the smaller domain is larger than that of the larger domain. This is a condition for instability: the smaller domain annihilates and the larger domain reaches the radius x2R0. (ii) 1 > Rq/eR0 > 1/e. Here the energy function has two minima, as illustrated in Figure 1b. In this case a pair of domains reaches a state of metastable equilibrium with equal radii (χ ) 0) for an initial χ < 0.7. For an initial χ < 0.7 the smaller domain annihilates and the surviving domain has the radius x2R0. This dependence of metastable equilibrium size and number on initial conditions is discussed later. (iii) Rq/eR0 < 1. As illustrated in Figure 1c, when χ is taken to be positive, the slope dE2/dχ > 0, so that by eq 14 the larger domain has the higher
Figure 1. Energy of a pair of lipid domains, one with radius R0(1 + χ)1/2 and the other with radius R0(1 - χ)1/2 in units of 2πR0m2. See eq 13. Values of Rq/eR0 are (a) 1.05, (b) 0.80, and (c) 0.3679. Energy minima at χ ) 1, -1 reflect the spontaneous annihilation of one domain whereas minima at χ ) 0 reflect equilibration of pairs of domains to the same radius. Equilibration of domain size occurs for values of the domain radii such that dE2/dχ > 0 for χ > 0.
equilibrium pressure, and the pair of domains reaches the same metastable equilibrium size. Domains with Noncircular Shapes In previous work it has been shown that the edge energy of a domain of arbitrary shape can be expressed as follows.8
E ) PR′(λ + m2 ln ∆/R′ + m2F/P)
(15)
Here R′ is defined so that the area of the domain is equal to π(R′)2 irrespective of its shape. In this equation P and F are functions that depend only on the shape of a domain and not on the domain size. Using the definition, γ ) (8/e2) exp(F/P), one may repeat all of the above thermodynamic calculations using an energy expression similar to that in eq 4.
E ) PR′m2 ln(γRq/eR′)
(16)
From this equation it can be seen how the domain energy depends on both the domain size (through R′) and shape (through γ and P). Minimization of the free energy with respect to size yields a pressure dependence of R′ similar to that given in eq 11. Minimization of the free energy with respect to shape usually leads to a single preferred shape although shape degeneracy can occur. In at least one known case two distinct shapes may have the same (8) De Koker, R. E.; McConnell, H. M. J. Phys. Chem. 1993, 97, 13419.
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pB ) pW + λ/R
(21)
to obtain eq 11. More generally, eqs 17-19 and 21 can be combined to yield
p∞ ) p0 + (η - 1)-1(λ/R - mΣ)
Figure 2. Sketch of a black lipid domain surrounded by a white lipid domain, where the domain boundary has a noncircular shape. The pressures just inside and outside the domain boundary are designated pB and pW, and the radius of curvature at a point on the domain boundary is R. These quantities and the electrical field strength at the domain boundary Σ are related to the monolayer pressure p∞ as described in eq 22 in the text. The pressures pB and pW vary from point to point along the boundary due in part to the electrical field associated with the domain boundary. The pressure jump across the domain boundary (pB - pW) at a given point on the boundary is determined by the line tension λ and the radius of curvature R.
energy, separated by an activation barrier.8,9 Thus in principle domains with different shapes and sizes may coexist in monolayers in metastable equilibrium at certain pressures. The next section gives a relation between monolayer pressure and the properties of the domain perimeter that are related to domain size and shape. Electric Field at the Domain Edge It is sometimes convenient to consider local equilibrium in terms of the thermodynamic conditions at the domain edge. Three chemical potentials are of interest,
µB ) µ°B + AB((pB - p0) - mBΣ)
(17)
µW ) µ°W + AW((pW - p0) - mWΣ)
(18)
µ∞W
) µ°W + AW(p∞ - p0)
Equation 22 is general and holds for domains with circular as well as noncircular shapes, where in the latter case both the local radius of curvature R and the local electric field Σ vary from point to point along the perimeter. This situation is sketched in Figure 2. For general domain shapes the electric field can be obtained as described in the Appendix. The condition that (λ/R - mΣ) is a constant along the perimeter of a domain can also be derived simply from conditions of mechanical stability of a domain. That is, under conditions of mechanical stability the hydrodynamic “body forces” acting inside and outside the ˆ Σ and ∇ ˆ pW - mW∇ ˆ Σ, must be zero,7 domain, ∇ ˆ pB - mB∇ and thus pB - mBΣ and pW - mWΣ are constant inside and outside the domain and along the perimeter. Since for stability of the domain boundary the pressure jump at the boundary is equal to λ/R, it follows that (λ/R - mΣ) is a constant along the domain perimeter. This same general approach can be used to treat the problem of gas phase domains. The calculation is outlined in the Appendix. Note that (λ/R - mΣ) ) 0 when the electrostatic and line tension forces exactly balance, which is the case when R ) Rq/e2, as discussed previously in connection with eq 4. Immiscible Liquids with Two Chemical Components Consider a binary mixture forming black circular liquid domains surrounded by white liquid. In the absence of edge energy the mole fraction composition of component 1 is X°1B and this changes to X1B in the presence of the edge energy. The ambient temperature is taken to be lower than the critical temperature Tc. The three chemical potentials for component 1 are
µ1B ) µ°1B + δµ°1B + A1B((pB - p0) - m1BΣ) (23) µ1W ) µ°1W + δµ°1B + A1W((pW - p0) - m1WΣ) (24)
(19)
By definition, the equilibrium condition without the edge energy is µ°B ) µ°W, and that with the edge energy is µB ) µW at the domain boundary. In these equations pB and pW refer to pressures just inside and outside a domain boundary, and Σ is the electric field strength at the domain boundary. At equilibrium the chemical potential of the white phase at infinity µ∞W is equal to the chemical potenital of the white phase just outside the domain boundary µW. By using the electrostatic model described in the Appendix (see also ref 10), one obtains the following expression for Σ for a circular domain,
Σ ) (m/R) ln(8R/e∆)
(22)
(20)
Equations 17-20 can be combined with the YoungLaplace relation (9) Mayer, M. A.; Vanderlick, T. K. J. Phys. Chem. 1993, 100, 83998407. (10) Keller, D. J.; Korb, J. P.; McConnell, H. M. J. Phys. Chem. 1987, 91, 6417.
∞ µ1W ) µ°1W + A1W(p∞ - p0)
(25)
The quantities µ°1B, µ°2B, µ°1W, and µ°2W refer to the equilibrium chemical potentials of the two components in the two phases in the absence of the edge energy. The dipole densities used in eqs 23 and 24 are defined in terms of partial molar dipole moments. That is, for example m1B -1 ) A1B ∂M1B/∂n1B, where M1B is the total electric dipole moment of component 1 in the black phase. To shorten the discussion, we use the simplest possible expression for the chemical potentials that lead to a liquid-liquid phase separation: o2 µ°1B ) µ°° 1 + RT ln X° 1B + 2RTcX2B
(26)
Here µ°° 1 is the chemical potential of pure component 1. Similar expressions hold for component 2 in the black phase and for the two components in the white phase. In the absence of edge energy the compositions of the phases are X°1B, X°2B and X°1W, X°2W. In the presence of the edge energy the compositions are X1B ) X°1B + δX1B, and the chemical potentials are µ°1B + δµ°1B, and so forth. To simplify the calculations further, we assume A1B ) A2B
Equilibrium Thermodynamics of Lipid Monolayer Domains
and A1W ) A2W. As the composition changes will prove to be small under the conditions of interest, we neglect the effect of these changes on the field strength Σ. When eqs 23-26 are combined, retaining only linear terms in δXij, one obtains the folllowing equations.
p∞ - p0 ) pW - p0 - mWΣ
(27)
η(p∞ - p0) ) pB - p0 - mBΣ
(28)
δX1W ) X°1WX°2WA1W(m1W - m2W)Σ/Rg(T 4TcX°1WX°2W) (29) δX1B ) X°1BX°2BA1B(m1B - m2B)Σ/Rg(T - 4TcX°1BX°2B) (30) In eqs 27 and 28, mW ) X1Wm1W + X2Wm2W, mB ) X1Bm1B + X2Bm2B, and Rg is the gas constant. Equations 27 and 28 can be combined to yield eq 22, which now applies to a system with two chemical components. As long as the system is well removed from the critical point, the composition changes are estimated to the small δXij ≈ 10-4 for binary mixtures of phosphatidylcholine and cholesterol. Note that for the simple chemical potential in eq 26, the critical compositions are X°1B ) X°2B ) X°1W ) X°2W ) 1/2, so that near the critical point the denominators on the right hand side of eqs 29 and 30 approach zero. None of the simplifying assumptions made in the present discussion should obviate our conclusion concerning the magnitude of the composition changes. We think of the approach to metastable equilibrium in this two-component system as follows. Starting with the black and white liquid phases with their unperturbed compositions, and with a fixed number NB of black domains, we turn on the edge energy and let the system reach a state of metastable size equilibrium by having the domains all reach the same radius, as discussed earlier in connection with eq 13. With this domain radius we then let chemical equilibration be achieved by lipid composition changes so that eqs 23-26 are satisfied. This analysis is self-consistent when the composition changes are so small as to produce negligible effects on the dipole densities, line tension, and domain area. Discussion In the present work we have considered the problem of thermodynamic equilibrium involving coexisting phases of lipid at the air-water interface. In these systems one commonly observes domains of one phase surrounded by a second phase. The sizes and shapes of such domains can be strongly affected by energies associated with domain boundariessthe “edge energy” due to line tension and dipolar electrostatic forces. It is observed that the shapes of lipid domains often respond quickly to changes in experimental conditions, such as monolayer pressure. Thus the assumption of equilibrium of domain shape is plausible, and the agreement between theory and experiment is generally good.2,5,6 On the other hand difficulties arise in comparing calculated equilibrium sizes with experiment, due to the fact that often these domains do not have sizes that correspond to the lowest possible free energy minima. In previous work4 the equilibrium size of a circular lipid domain was calculated to be equal to Rq, as given in eq 3. The calculation used to obtain this result is equivalent to that given in the present paper in connection with equilibrium thermodynamic condition I. This condition corresponds to a process in which domains in a monolayer change their radii by changing the number of domains,
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maintaining constant total domain area. Given a random distribution of domain sizes, this equilibrium condition could be reached by processes involving domain fission, fusion, creation, and annihilation. Fission, fusion, and creation are all estimated to involve activation energies so large are to render them unlikely as mechanisms for reaching equilibrium. On the other hand, domain annihilation can be thermodynamically “downhill” at every step, with no activation energy. Thus the domain radii observed experimentally may differ substantially from Rq, as discussed below. The domain radius given in eq 11 corresponds to thermodynamic process III, in which one transfers molecules back and forth across the domain boundary at fixed monolayer pressure. As discussed in connection with eq 12, the radius-pressure relationship in eq 11 represents metastable equilibrium as long as R is greater than Rq/e. It is important to contrast this metastable equilibrium with an unstable equilibrium. For example, a small drop of water may be in equilibrium with surrounding water vapor at an appropriate pressure, but this equilibrium condition is unstable. That is, a small variation in drop radius leads to a drop that either evaporates or grows in size. In contrast the lipid domain is metastable for a given radius and pressure. At constant monolayer pressure variations in domain radius tend to return to the metastable equilibrium value, in accord with eq 12. For non-interacting single-component domains the metastable equilibrium discussed thus far does require that the domain sizes be all the same, a situation sometimes observed experimentally, especially with gas domains. One expects that at equilibrium there can be large thermal fluctuations in domain radii, of the order of
|
(δR)2∂2G/∂R2 p,NB ≈ kT
(31)
so that for a typical dipole density difference, m ≈ 10-4 (cgs units), Rq ) 10-3 cm, and R ) eRq, one obtains
(δR/R)2 ≈ 1/4
(32)
It is important to note that the metastable equilibrium radius of domains in a monolayer can depend on the history of the sample. Refer to Figure 1 for the energy of a pair of domains, one with the radius (1 + χ)1/2R0 and the other with the radius (1 - χ)1/2R0, so the total area of the pair of domains is 2πR02. Assume that these domain sizes are such that Rq/eR0 ) 0.80, as in Figure 1b. It will be seen that when the initial condition is χ < 0.7, the two domains will reach metastable equilibrium with the same radius, R0. On the other hand, it the initial sizes of the domains are such that χ > 0.7, then the smaller domain annihilates and the larger domain grows to the metastable equilibrium radius x2R0. This simple example is easily generalized to the conclusion that the metastable equilibrium radius reached by a monolayer can depend on the monolayer history, namely the original size distribution of the domains. Two monolayers each with different metastable equilibrium radii, brought into contact with one another, would then sense the difference in pressure in the two monolayers, leading ultimately to a combined monolayer with a common equilibrium radius and pressure. However the rate of this process may be low. In this way one can see how a long lived heterogeneity of domain radii in monolayers might arise. The pressure changes associated with edge energies are small, order of magnitude 10-4 dyn/cm, so that equilibration rates dependent on differences in pressure may also be small.
4902 Langmuir, Vol. 12, No. 20, 1996
Monolayers composed of cholesterol and phosphatidylcholine form two immiscible liquid phases at lower applied pressures.2 The domains in these monolayers often show wide variations in domain sizes. In the above calculations we have shown that for the metastable domain radius the changes in lipid composition due to edge energy effects are quite small for this binary mixture of lipids. Nevertheless these composition variations tend to stabilize lipid domains with different radii, even when these domains should all have the same radii under conditions of metastable equilibrium. The observed large variation in domain sizes seen in these monolayers may be due to this effect. These systems should ultimately approach the condition of metastable equilibrium where all domains have the same radii. We have not attempted to discuss equilibrium domain sizes for domains that interact with one another electrostatically. This is a much more complex problem. However it can be seen that variations in domain size under conditions of metastable equilibrium can be brought about by variations in boundary electrical strength Σ due to neighboring domains. In this case eq 22 is also a condition on domain translational stability. Strongly interacting domains are present in the stripe and hexagonal phases of lipid monolayers, and the size uniformity often seen in such systems11,12 suggests that they are in stable or metastable equilibrium. Domains in thin films of ferrofluids can develop various shapes in the presence of applied magnetic fields perpendicular to the plane of the film. These shapes are sometimes remarkably similar to those seen in binary liquids composed of cholesterol and phosphatidylcholine. There are significant similarities as well as differences in these systems from a theoretical point of view. In the magnetic systems shape changes are brought about by changes in the applied field strength, which changes the magnetic dipole density.13-16 In the monolayer systems the shape changes can be brought about by applied electric fields but are usually brought about by monolayer compression or expansion, which changes domain area and sometimes composition. An approach used in calculations of ferrofluid domain shapes seeks a constant value of (λ/R - mΣ) for all points on the domain perimeter subject to the constraint of fixed domain area, where mΣ refers to a magnetic energy.14,15 In the monolayer systems this quantity must satisfy eq 21 for metastable equilibrium for domain size and shape at a given pressure. Many of the remarkable shape and size variations seen in binary mixtures of phosphatidylcholines and cholesterol are related to the fact that in these mixtures both the line tension and dipole densities change as monolayer pressure is changed, especially near the consolute critical point.2 The complex shapes seen in the monolayers as well as ferrofluid films doubtless represent still another type of metastability, where two shapes of the same or nearly the (11) Seul, M.; Morgan, N. Y.; Sire, C. Phys. Rev. Lett. 1994, 73, 2284. (12) Seul, M.; Chen. V. S. Phys. Rev. Lett. 1993, 70, 1658. See also: To, K.; Akamatsu, S.; Rondelez, F. Europhys. Lett. 1993, 21, 343. (13) Rosensweig, R. E. Ferrohydrodynamics; (Cambridge University Press: Cambridge, 1985; and references therein. (14) Tsebers, A. O.; Maiorov, M. M. Magnetohydrodynamics 1980, 16, 21. (15) Langer, S. A.; Goldstein, R. E.; Jackson, D. P. Phys. Rev. A 1992, 46, 4894. (16) Dickstein, A. J.; Erramilli, S.; Goldstein, R. E.;Jackson, D. P.; Langer, S. A. Science 1993, 261, 1012. (17) Deutch, J. M.; Low, F. E. J. Phys. Chem. 1992, 96, 7097. (18) McConnell, H. M.; Bazaliy, Ya. B. Proc. Natl. Acad. Sci. U.S.A. 1995, 92, 8823. (19) De Koker, R. E. Domain Structures and Hydrodynamics in Lipid Monolayers. Thesis, Stanford University, 1996. (20) Lee, K. Y. C.; Klingler, J. F.; McConnell, H. M. Science 1994, 263, 655-658.
McConnell and De Koker
same energy are separated by an activation barrier. A simple example of this has been demonstrated in connection with the “circle-to-dogbone” transition.8,9 In the case of monolayers, there is clearly a multiplicity of metastabilities, including domain size, number of domains, domain shape, and, in the case of interacting domains, domain arrangement as well. To our knowledge there has been no systematic study of domain size equilibration under the conditions of low domain number density appropriate to the present work. Exploratory work by Lee and McConnell has show a slow (hours) disappearance of very small domains and the growth of larger domains in binary mixtures of cholesterol and phosphatidylcholines under these conditions of low domain density.21 Since domains with radii less than Rq/e are expected to undergo annihilation and those with radii greater than e1/3Rq are expected to undergo a shape transition, even qualitative observations of domain size maturation and shape can serve to bracket the value of the equilibrium radius Rq. In the exploratory work mentioned this radius is of the order of magnitude 10 microns.21 Acknowledgment. This work was supported by the National Science Foundation (Grant NSFG 9005556). R.D. is a Howard Hughes Medical Institute Predoctoral Fellow. Appendix Gas Phase Domains. Two special factors play a role in the case of gas phase domains. First, the gas phase is compressible. Second, it is necessary to divide the dipole density mB into two componenents, mBg and mBa, so that mB ) mBg + mBR. Here mBg is the dipole density due to the gas molecules, which is presumably very small due to their low molecular density. The major contribution to the dipole density of the black (gas) phase then arises from the water molecules at the air-water interface, mBa. The dipole density difference m appearing in eq 20 for the electrical field strength Σ is thus m = mBR - mW whereas the chemical potential of the gas phase lipid molecules involves the dipole density mBg. The thermodynamic calculation is then straightforward. The pressure just inside the boundary of a circular gas phase domain is pB, where pB ) p∞ + mWΣ + λ/R. The chemical potential of the gas phase just inside the boundary is pB
µB ) µ°B +
∫AB dp - ABmBgΣ
(33)
p0
This equation can be simplified by assuming the gas is a perfect gas. The combination of eq 33 and eqs 17 and 18 is then a condition on metastable equilibrium for a monolayer composed of non-interacting gas phase domains, each of which has radius R. The initial pressure p0 is the twodimensional vapor pressure of the liquid phase. At present very little is known about the magnitude of this vapor pressure, except that it is too low to measure for phospholipid monolayers. If it is assumed that the pressure change pB - p0 is small (an assumption of uncertain validity) and that η , 1 and |mBg| , |mW| (both doubtless true), then one obtains for the equilibrium pressure.
p∞ ) p0 - (λ/R - mWΣ)
(34)
It is of interest to contrast the electrostatic term in eq 34 (21) Lee, K. Y. C.; McConnell, H. M. Unpublished.
Equilibrium Thermodynamics of Lipid Monolayer Domains
Langmuir, Vol. 12, No. 20, 1996 4903
and
∫
∫
ΣW(x0,y0) ) 2πmW - m Q dx dy + 3m∆2 F-5 dx dy B
B
(41)
The first integral on the right hand side of eq 41 can be converted into a line integral around the perimeter of the black domain using Stokes theorem in two dimensions. The corresponding electrical field strength Σ ˜ W(x0,y0) is conveniently represented in terms of a vector potential V ˆ W(x0,y0):
ˆ0 ∧ V ˆ W) Σ ˜ W(x0,y0) ) kˆ ‚(∇
Figure 3. Coordinate system used for the calculation of the electrical field Σ acting at the boundary of a circular domain. See eqs 20 and 46.
with that in eq 22. In eq 22 the electric field depends on the difference in the dipole density in the two lipid phases, whereas in eq 34 this electric field is determined by the difference in the dipole density in the lipid liquid phase and the dipole density due to water molecules at the airwater interface. The (dipole density) coefficients of Σ are likewise different in the two equations. This difference in the two results stems from the fact that in the liquid lipid phases we combine the contributions of the lipid molecules, the water molecules, and the image dipoles into one effective dipole density. In the gas phase, the molecules that produce the dipole density are the water molecules, whereas the molecules that produce the pressure are the lipid gas molecules. In spite of this complication, the thermodynamic condition I is valid with m ) mB - mW; the lowest energy equilibrium radius remains Rq with this definition of the dipole density difference. Local Electric Field Strength. In the above discussion of the equilibrium thermodynamics of lipid domains we have used a quantity Σ to represent the electric field strength at the boundary of a lipid domain. Here we give a derivation of Σ that is closely related to electrostatic energy calculations given earlier.5 Consider a finite black domain surrounded by an infinite white domain with dipole density mW. The electrical field at a point (x0,y0) in domain W is,
ΣW(x0,y0) ) 2πmW/∆ +
∫F-3 dx dy
(35)
B
where
F ) [(x - x0)2 + (y - y0)2 + ∆2]1/2
(36)
A similar expression holds for the electrical field strength at a point in domain B.
∫
ΣB(x0,y0) ) 2πmB/∆ - m F-3 dx dy
(37)
W
The homogeneous monolayer (infinite domain) electrical fields 2πmW/∆ and 2πmB/∆ are easily deduced from energies given previously.17,18 The area integrals in eqs 35 and 37 can be written in terms of line integrals, as follows. Define a vector P ˆ and the quantity Q by the equations
P ˆ ) [-(y - y0)ıˆ + (x - x0)jˆ]F-3
(38)
∇ ˆ∧ P ˆ ) Qkˆ
(39)
Q ) -F-3 + 3∆2F-5
∫
V ˆ W(x0,y0) ) m F-1 drˆ
(43)
B
∇ ˆ 0 ) ıˆ∂/∂x0 + ˆj∂/∂y0
(44)
The electrical fields Σ˜ W(x0,y0) and Σ˜ B(x0,y0) are equal to one another when points (x0,y0) are on the domain boundary; this is the “edge field” Σ used in the thermodynamic calculations in the present paper. The energies associated with uniform infinite domain electrical fields, 2πmW/∆ and 2πmB/∆, are of course incorporated into the chemical potentials of the infinite phases, µ°W and µ°B. The second integral on the right hand side of eq 41 also represents an electrical field associated with the domain edge, but this field is of short range and the effects of this field can be incorporated into the line tension. The simplest way to see this is to relate the present calculation to earlier work.5 The total electrical energy associated with domains B and W is
∫
∫
Etot ) (1/2)mW ΣW dx dy + (1/2)mB ΣB dx dy W
)
(π/∆)(m2WAW
B
+
m2BAB)
∫∫F-3 dx dy
2
- 1/2m
B W
(45) The double integral on the right hand side of eq 45 is a linear superposition of contributions from both edge fields. When F-3 is expressed in terms of Q and 3∆2F-5 according to eq 40, this double area integral can be written as a double line integral derived from Q and an energy equal to -m2 times the domain perimeter, derived from the term 3∆2F-5. See McConnell and De Koker.5 (A direct calculation of these electical fields at domain boundaries is given by De Koker.19) Thus the effect of this short-range electrical field is incorporated into the line tension energy. This electrical field strength is equal to -πm/∆, independent of domain size and shape. We now calculate the electric field at the domain boundary of a circular domain, as given earlier in eq 20. Consider the cylindrical coordinate system given in Figure 3. In this system,
F ) (R2 + r20 - 2Rr0 cos φ + ∆2)1/2
(46)
and the only nonzero component of the vector potential is in the φ-direction. 2π
∫
Vφ ) mR F-1 cos φ dφ
It then follows that
(42)
(47)
0
(40)
The calculated electrical field strength at the point r0 is
4904 Langmuir, Vol. 12, No. 20, 1996
[
Σ ˜ B(r0) ) 2m/[(R + r0)2 + ∆2]1/2 K +
McConnell and De Koker
R2 - r20 - ∆2
E (R - r0)2 + ∆2
]
(48)
Here K(k) and E(k) are elliptic integrals of the first and second kind and
k2 ) 4Rr0/[(R + r0)2 + ∆2]
(49)
When we set r0 ) R, we obtain the electrical field at the circular domain boundary given in eq 20. For typical material parameters, m of the order of 1 Debye per 100 square angstroms, R equal to 10 µm, and ∆ equal to 10 Å, one obtains a field Σ equal to 300 V/cm. It may be noted that externally applied electrical fields can be made orders of magnitude larger than this edge field, with corresponding larger effects on domain size and composition.20 LA960411M
