4520
J . Phys. Chem. 1988, 92, 4520-4525
Shapes of Finite Two-Dimensional Lipid Domains Harden M. McConnell* and Vincent T. Moy Stauffer Laboratory for Physical Chemistry, Stanford University, Stanford, California 94305 (Received: December 10, 1987)
Distinct lipid domains with different two-dimensional density can be visualized in monomolecular films at the air-water interface by using fluorescence microscopy, when the monolayers are doped with a low concentration of fluorescent lipid probe. The shapes of these domains are determined to a large extent by a competition between line tension and long-range electrostatic dipolar repulsions. The line tension favors compact, often circular shapes, whereas the long-range electrostatic dipolar repulsions favor other shapes, such as thin stripes. A line integral technique is described for calculating the dipolar energies of these two-dimensional domains. Calculations are given for various domain shapes and shape transitions. It is shown that as domains grow in area they tend to thin in one dimension because of long-range dipolar forces. It is also shown how these long-range dipolar forces contribute to the formation of chiral domain shapes.
Introduction Lipid monolayers at the air-water interface can be observed by using epifluorescence microscopy, when the monolayers are doped with a low concentration of a fluorescent pr0be.l This technique has permitted the observation and study of two-dimensional solid and fluid phases a t the air-water interface, and transitions between them.z4 In the “two-phase” region, where the solid domains are surrounded by fluid (melted solid), the domains exhibit a remarkable variety of shapes and arrangements, depending in part on the lipid involved and the extent of compression of the m o n ~ l a y e r . ~For . ~ certain lipid compositions and experimental conditions, the domain shapes are equilibrium shapes.’ These equilibrium shapes are determined in part by a competition between line tension (that tends to stabilize isotropic circular domains) and long-range dipole-dipole repulsion between lipid molecules (that tends to stabilize elongated domains).8-10 The chirality of the domain shapes is related to anisotropic crystal forces, as discussed later. The epifluorescence microscope-fluorescent lipid probe technique also permits the study of domains formed by two fluid systems (e.g., liquid-gas,” or liquid-liquid12). The theory of domain shapes that involves only isotropic line tension and twodimensional isotropic electrostatic forces is particularly appropriate to the shapes and arrangements of such domains. Hexagonal arrays of circular domains and linear arrays of domain stripes are the theoretically predicted thermodynamic equilibrium states of these two-fluid systems.*JO Note that a hexagonal arrangement of circular domains is one thermodynamic phase, for example, when circular domains of one liquid are surrounded by continuous regions of a second liquid. A phase transition can take place between an hexagonal array of circular domains and a linear array of parallel domain stripes.sJO In the case of monolayers consisting of regions of solid lipid and fluid lipid, there are clearly cases where the shapes and sizes (1) von Tscharner, V.; McConnell, H. M. Biophys. J. 1981,36,409-419. (2) Peter, R.; Beck, K. Proc. Natl. Acad. Sci. U.S.A. 1983,80, 7183-7187. (3) B c h e , M.; Sackmann, E.; Mohwald, H. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 848-852. (4) McConnell, H. M.; Tamm, L. K.; Weis, R. M. Proc. Natl. Acad. Sci. U.S.A. 1984,81, 3249-3253. (5) Weis, R. M.; McConnell, H. M. Nature (London) 1984, 310,47-49. (6) Miller, A.; Knoll, W.; Mohwald, H. Phys. Reu. Lett. 1986, 56, 2633-2636. (7) Gaub, H. E.; Moy, V. T.; McConnell, H. M. J. Phys. Chem. 1986,90, 1721-1725. (8) Andelman, D.; Brochard, F.; de Gennes, P. G.; Joanny, J. F. C . R. Acad. Sci. Paris. Ser. C 1985, 301, 675-678. (9) Keller, D. J.; McConnell, H. M.; Moy, V. T. J . Phys. Chem. 1986,90, 231 1-231 5. (10) Andelman, D.; Brochard, F.; Joanny, J. F. J . Chem. Phys. 1987,86, 3673-3681. (1 1) Losche, H.; Mohwald, H. Colloids Surf. 1984, 10, 217-224. (12) Subramaniam, S . ; McConnell, H. M. J . Phys. Chem. 1987, 91, 17 15-1 7 18.
0022-3654/88/2092-4520$01.50/0
of solid domains are determined by kinetic factors; thus some solid domains have fractal shapes.6 In the present paper we are concerned with “equilibrium” states of the monolayer. In this case there is a hierarchy of equilibria. Consider a lipid monolayer composed of liquid and solid regions, and for simplicity assume that the pure fluid and pure solid phases are incompressible. For a given area of trough, the fractions of the area covered by fluid and solid are then fmed. A state of global equilibrium is achieved if the number and shape of the solid domains represent the state of minimum free energy. In the present paper we are primarily concerned with a simpler equilibrium problem. We assume that we are given a solid domain of a certain area, and we are to calculate the shape that minimizes the free energy (neglecting interactions between different solid domains). We suspect that this is a realistic experimental condition; on first compressing a monolayer a certain number of solid domains appears, either by nucleation or by spinodal decomposition. On further compression, these domains grow in size but not in number. Thus, the experimental system can be in equilibrium with respect to domain shape but not domain number. In fact, as will be seen, the analysis of domain shapes provides one of the important clues for understanding the physical properties of lipid monolayers. Background Theory and Model Assumptions Here we review briefly a simple model for the shapes of finite two-dimensional crystal domains of phospholipid at the air-water interface. Again it is assumed that the pure solid and pure fluid phases are incompressible, so that when the monolayer is compressed to convert fluid to solid, in this intermediate region, the area of the trough and the quantity of lipid present together define the area of the lipid that is solid. It is observed experimentally that, for a number of lipids, the solid is divided into a large number of small, finite-sized domains, surrounded by fluid. The solid domains repel one another due to electrostatic forces.2-10 In the simplest model the shape of an individual solid domain is determined by the interplay of the molecular dipolar electrostatic forces FC,and the line tension X of the boundary between fluid and solid regions. F = Xp 4- F,, (1) Here p is the perimeter of the solid domain. The free energy is minimized with respect to the two-dimensional shapes of the solid domains. At low solid area fractions, the forces between different solid domains can be neglected, permitting the minimization to be carried out for a single isolated domain. Comparisons of the predictions of this model and experiment have been limited to simple geometrical shapes (parallel stripes) because of the difficulty of calculating the electrostatic dipolar energy in eq 1.8-10 However, in recent work it has been shown that this electrostatic energy for dipole-dipole interactions can be calculated by using a line integral around the perimeter of a domain, greatly simplifying the mathematical problem, and also 0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4521
Shapes of Finite Two-Dimensional Lipid Domains
where P and Q are functions of x and y, P = P(x,y) and Q = Q(x,y). This form of Green’s theorem for one surface S can be generalized to two surfaces i and j , as follows.
/7
s 1( S,
S,
axfax
+ ay‘ay -”-
)
dx dy dx’ dy ’ =
If one takes P = Q = l / r , where
then eq 5 becomes
where Ci and C . refer to the line integrals around the perimeter of surfaces i and j . The dipolar energy in eq 3 is therefore reduced to
Figure 1. Calculation of dipolar interactions in a two-dimensional finite crystal. Line integrals are along paths around the little squares, in the directions indicated.
permitting a conceptual ~imp1ification.I~The electrostatic terms act like a negative contribution to the line tension, this contribution depending on the size and shape of the domains. The present paper gives a different and somewhat more general derivation of this line integral representation of the electrostatic dipolar energy. We first give this derivation and then give examples of domain shape calculations. Dipolar Sum in Two Dimensions We calculate the shape dependence of the electrostatic dipole-dipole energy of a two-dimensional lattice array of dipoles on a planar surface S. The symmetry of this lattice is not critical; we take it to be square planar, with one dipole per unit cell. To begin with, only the component of the dipole moment perpendicular to the plane is considered. The dipole density is p = p L . We approximate the lattice sum for the dipolar energy
by a line integral around the perimeter. Here a2 is the area occupied by a given molecular dipole, and k, k’enumerate lattice sites. The dipole array in the surface S is divided into an array of a large number N of small square surfaces si, i = 1, 2, 3, ... each containing n dipoles. See Figure 1. The sides of the squares are separated from adjacent squares by the lattice spacing 6. The electrostatic energy Fe, is
(3)
The similarity of these line integrals to those that appear in calculations of self-inductance and mutual inductance brings to mind electromagnetic interactions of currents.Is By reference to Figure 1 we see that there are three types of terms. Adjacent sides of adjacent squares (adjacent “currents”) give large integrals (the lines are separated by 6); the number of these contributions is proportional to N a n d can be combined with the first term on the right in eq 8, replacing Nne‘, by Nne,. The interactions of pairs of adjacent currents with other currents can be neglected, as each member of an adjacent pair cancels the contribution of the other. The remaining integrals are those along the perimeter of the surface S. The dipolar energy can therefore be written
--
F,, = Nne, - P2 -$$-1 dbdl‘ 2 r
where the line integrals are around the outer perimeter of the surface S, and it is understood that the distance r between points on the perimeter is not allowed to become less than 6n1/2. Since 6 is only a parameter in our calculations we replace 6n112by 6. If the integrals in eq 9 are not restricted as they are in eq 8, then a third term should be added to the right-hand side of eq 9. See Appendix. The result in eq 9 was obtained previously by using a different method of c a l ~ u l a t i o n . The ~ ~ method used here is convenient because it also permits the calculation of the energy due to in-plane dipole-dipole interaction, for dipole density pli. If the in-plane dipole component points in the x direction, the energy can be calculated as above, but by using Q = l / r , P = 0 in eq 4, which yields
1 1(1 - 3 cos2 s1 s,
Here e’, is the electrostatic energy per dipole due to electrostatic interaction between dipoles within each square. The second term on the right-hand side of eq 3 includes all dipolar interactions between dipoles in different squares and thus includes all longrange dipolar interactions. In calculating dipolar interactions between different squares, the dipoles are replaced by a continuous uniform dipole density. In eq 3 the distance r is between a point i in square i and a point j in square j , dA, and dA, are elements of area in square i and j . The integral on the right-hand side of eq 3 can be evaluated by using Green’s theorem14 dxdy = #(Pdx
J(z-dy> aQ
+ Qdy)
(9)
cp)-
1 dAi dA, = r3
1 $$dyi dyj r
(10)
Here cp is the angle between 7 and the x direction. In this case the electrostatic energy is l.LL2 1 Nne, - y $ $ -d ? d p + Nneli L r
h2 1 + -$$dy dy’ (11) 2 r
This equation shows how a lengthening of the perimeter reduces the energy of the dipole-dipole repulsions of the perpendicular dipoles. On the other hand the contribution to the energy from in-plane dipolar interactions can be reduced by reducing the length
(4)
(13) Keller, D. J.; Korb, J. P.; McConnell, H. M. J . Phys. Chem. 1987, 91, 641 7-6422.
(14) Kaplan, W. Advanced Calculus, 3rd. Ed.; Addison-Wesley: Menlo Park, CA, 1984; pp 287-303. (15) Landau, L. D.; Lifshitz, E. M. Electrodynamics of Continuous Media; Pergamon: New York, 1960; Vol. 8, pp 132-137.
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The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
of the perimeter in one direction. Note that the factors of ‘I2that appear in eq 9 and 1 1 apply when the double line integrals are around the same path. When each line integral is around a different path, the factor of 1 / 2 is omitted. As noted in the Appendix, if the integrals in eq 1 1 are not restricted as they are in eq 8, there is an additional contribution to FeIin eq 1 1 from the in-plane dipole moment that scales like an anisotropic line tension.
Global Equilibrium By using eq 9 we obtain the following expression for the perpendicular electrostatic dipolar energy for a solid domain of radius R, using Fel = Nne,
+fCl
McConnell and Moy
n
i
n
U
Figure 2. Shape transition from squares to rectangles. As the area of the solid domain increases, the shape of minimum free energy changes suddenly from a square to a rectangle at the critical area A,.
(12)
where fel
= 2.rrRpL2In
e26 4R
Imagine a trough of fixed large area, and fixed total area of solid, A,. When interactions between solid domains are neglected, the free energy that depends upon the number of domains is 414
The number of solid domains is n, = A,/(.rrR2). When the free energy F is minimized with respect to R, dF/dR = 0, one obtains
This result describes the state of global equilibrium, neglecting interactions between solid domains, and assuming the shape of minimum free energy is a circle. Similar calculations for other specific shapes yield higher free energies (see below); the circular disk is the shape of lowest free energy for an isotropic solid, for global equilibrium. This calculation may be relevant to solid domains formed in the early stages of monolayer compression where solid domains are first formed. As indicated in the Introduction, further monolayer compression may lead to the growth of solid domains rather than the formation of new solid domains, thus departing from global equilibrium. We have carried out calculations similar to the one described above for a square and for a long thin rectangle. The free energies per unit area, pmin, for the three shapes are listed below. circular disk
square
Figure 3. Normalized width vs normalized area for rectangular domains. The widths and areas are normalized to their values at the transition point (w, and A,). For areas below A,, the shape of the domain is a square. At A,, the domain undergoes a shape transition from a square to a rectangle. Above A,, the width of the domain decreases and approaches an asymptoticvalue w, = [2/( 1 + 21/2)]e~1/(’+21’2))wc = 0 . 5 4 7 for ~ ~ large areas.
that, if a circular domain is considered to grow in area, then the shape transition takes place at the radius13
That is, for R > Rshapcthe shape of lowest energy is an ellipse, the shape of lowest free energy is circular. whereas for R < kpe Since Rshapc> R&, we can understand how, on monolayer compression, one might first observe circular domains and then on further compression observe elliptical domains. Our earlier shape transition calculation is limited to the immediate vicinity of the critical point. It is also not certain that the distorted shapes are precisely elliptical. The difficulty in extending such analytic calculations beyond the neighborhood of the shape transition point has prompted us to consider transitions involving simpler shapes, where the shapes can be calculated both at the transition point and far from the transition point. With this idea in mind we describe a second-order transition from a square to a rectangular domain, analogous to the circle-to-ellipse transition. The shape-dependent part of the free energy of a domain in the form of a square (sides w = I ) or rectangle (sides w and 1) is given by the sum of the line tension term 2(w I)X plus the dipolar electrostatic term (the double line integral in eq 9). One obtains
+
long, thin rectangle
+ In ( I + (w2 + I 2 ) ’ I 2 ) Shape Transitions In recent work the idea of a two-dimensional lipid domain shape transition was ir~troduced.’~In the isotropic line tension-perpendicular dipole model it was shown that domains can undergo a sharp (second-order) transition from a circular shape to an elliptical shape, as the area of the domain increases. It was found
Z - 2 (w2 + 12)1/2 (20)
For a given area, the shape of minimum energy is obtained by setting dF/dw = 0 with 1 = ( A / w ) . The critical length of the side of the square at the transition point is w, = (1 + 21/2)~-21”/(1+2”2)e2~~A1e2 (21) These shape changes are illustrated schematically in Figure 2 , and plots of the width of the rectangle (or square) as a function
Shapes of Finite Two-Dimensional Lipid Domains
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4523
0-0
CIRCLE-CIRCULAR
i
TORUS TRANSITION
Figure 4. Schematic representation of circle to torus transition.
0
1
1
0
2
I
I
I
4
I
6
1
I
8
Areal Area,
Figure 6. Plot of width vs area for circle-torus transition. At A,, the circular domain undergoes a transition to the torus shape. The width is the outer radius minus the inner radius. As shown, the width of the torus increases with increasing area and asymptotically approaches w, = 26eh/P**1.The asymptotic widths for the torus and the rectangle are equal.
I
I
1
5
0
I
10
,-0
1
15
INNER RADIUS (arbitrary units)
Figure 5. Energy diagrams of a circular torus of different areas. For areas less than A,, the equilibrium shape is a circle. At A,, the domain undergoes a first-order shape transition from a circle to a circular torus.
of domain area are given: in Figure 3. It is interesting to note that, after the transition point, the solid domain thins as it becomes larger in area. As discussed later, this effect has been seen experimentally.16 As noted in Figure 3 the asymptotic width w, is approximately 0.547 of the maximum width w,, and this is of the order of magnitude of the ratio observed experimentally for more complex shapes.I6 There are other shape transitions that are useful to consider in connection with the experimental data. For example, torusshaped domains have been observed.I6 Theoretically, there is a first-order circle to circular torus transition. See Figure 4. The electrostatic energy of a circular torus with radii r, and r, is
F = 27rX(ri + r,,)
+
8ap2(rir0)
[(l - 0 . 5 k 2 ) K - E ]
+
+
where k2 = 4r,ro/(ri r0)*,and K and E are elliptic integrals of the first and second kind, respectively. Figure 5 illustrates the energies calculated for the circle and circular torus, for various areas. Note again that, for the smaller areas, the circle is the most stable shape. Figure 6 describes the dimensional changes of the circle to torus transition. We anticipate that it may be possible to study this transition experimentally for monolayers having two coexisting fluid domains. Torus-shaped domains, once formed, should be stable, or metastable, according to Figure 5 . This appears to be the case for the solid torus structures observed previously.16 A torus was observed following rather rapid monolayer expansion.16 The calculations show that the critical domain area for the circle to ellipse transition is approximately one half of the circle to torus transition, so on slow compression we do not (16) Weis, R.M.; McConnell, H. M. J . Phys. Chem. 1985,89,4453-4459.
125
O 0 2
0 25
0 5 Area/ Area,0 7 5
Figure 7. Effects of i,,-plane polarization on the shape transition of rectangular solid domains. In these calculations, the vertical component of the dipole moment was kept constant and the in-plane component was varied. q is the square of the ratio of the components of the dipole moments, 7 = ( P , , / P ~ * .
anticipate the formation of torus structures. Effect of In-Plane Dipole Components on Shapes and Shape Transitions It is plausible that in-plane dipole components can be neglected in calculating the shapes of fluid domains, but it is dubious that these components can be neglected for solid domains known to have long range vector orientational order (tilt order).” We therefore discuss briefly the effects of the in-plane dipole moments on the shape transitions. As noted previously, the perpendicular dipole moments tend to make solid domains long and thin. It can be seen from eq 11 that in-plane dipoles can have a similar effect, if they are all oriented in one direction. If the molecular dipoles are tilted in the x direction, then the solid domain will tend to elongate in that direction. Dipole tilt has a large effect on the shape transitions, as illustrated in Figure 7 . It will be seen that a fixed tilt of only a few degrees (9 = 0.005) has a pronounced effect on the sharpness of the transition, and a fixed tilt of 17O (7 = 0.1) obliterates the transition completely. Note that this calculation does not allow for a simultaneous change of tilt and shape. In general one expects a change in tilt with a change in shape, and vice versa. Thus, in the latter case a transition that is nominally a change in tilt should give a change in shape. A major question with respect t o the (17) Moy, V. T.; Keller, D. J.; Gaub, H. E.; McConnell, H. M. J . Phys. Chem. 1986, 90, 3198-3202.
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The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
in-plane dipole moment is the extent to which this component is screened by the aqueous phase. This screening may be much stronger for the in-plane component than for the perpendicular component. It will be shown elsewhere that, theoretically, in-plane components of the dipole moment can give rise to bound pairs of solid domains. An expermental search for such bound pairs may therefore be the best approach to study this problem. Origin of Chiral Shapes Solid domains of some phosphatidylcholines, when composed of one or the other of the two optical isomers, form highly chiral shapes.5*7*9~'6~'7~19 These shapes include chiral bananas, cloverleaves, and chiral spirals. While one can readily understand qualitatively the appearance of chiral shapes for solid lipid domains when the molecules are chiral and show long-range orientational order, since this is expected from Wulfs theorem?O it has not been so obvious why the observed chirality is so pronounced. Here we suggest an interpretation of these chiral shapes. Consider the straight line segment of solid-lipid fluid-lipid interface of length L. The contribution of this interface (alone) to the shape-dependent free energy is easily shown from eq 9 to be
f, = (A, - l . In~ (~ L / e s ) ) L
(23)
McConnell and Moy
r-f
!
t
Figure 8. Model for spiral domain growth. Spiral domains are formed by successive addition of unit square blocks to the existing structure. Associated with each block is a director (tilt). The (n + 1)th block adds onto the nth block either (1) with the directors of both blocks pointing in the same direction (no bend) or (2) with the director of the bend block rotating by YOo within the block. In the first scheme (1) the addition of the new block increases the free energy of the structure by the electrostatic interaction energy between the new block and the existing blocks plus one unit of line tension free energy (outside 0, inside 1). In the second scheme (2) the free energy increases only by the electrostatic interaction energy. The electrostatic interaction energy in case 2 is larger due to the compactness of the bend structure. The defect energy associated with the 90' bend is absorbed into the difference in the line tension energy in the calculation. The direction of growth is determined by the path of minimum increase in free energy.
Note that the effective line tension due to these terms is A', = AI
- lUz In ( L / e h ) and this line tension decreases with increasing size of the solid domain. For a two-dimensional crystal of chiral molecules, all of which have a common orientation, the line tension will be different on different line elements of the crystal. For example, the line tension will be different on the left- and right-hand sides of a crystal domain (A, # A,). The electrostatic contributions can thus exaggerate differences in line tension, since ratios such as (A'l/A'r) can become very large, or small. This then can lead to strongly chiral shapes, as observed. As noted in the Appendix, in-plane dipole-dipole interactions can also contribute to the anisotropy of the line tension. However, the situation is more complicated since it is known experimentally that there is a twist (in-plane rotation) of molecular orientation in solid domains with pronounced chiral shape.' This twist of molecular orientation can be thought of as arising in various ways. Imagine we start with a solid domain with uniform molecular orientation, a uniform tilt in one direction. (i) If the line tension forces exceed the crystal's resistance to deformation, the domain will deform, resulting in a twist in the molecular orientational order. (ii) The molecules at the solid-fluid interface are subject to local forces that might twist them away from a uniform orientation, and this twist propagates into the bulk of the solid. (iii) There is an intrinsic twist to the packing of chiral molecules, which leads to a twist in molecular orientation of all the molecules in the solid domains. These three effects are interrelated, as discussed below. The work of Langer and Sethnal" on thin chiral smectic liquid crystal films shows that the tendency of chiral molecules to twig can be_represent_edby a free energy density of the form -2qK(V X &)-k, where k is a unit vector perpendicular to the air-water interface in the presentproblem. The bulk elastic resistance to The net curvature elastic energy twisting has the form K(V X is F, = L [ - 2 q K ( 9
X
(24) ,ill).z + K ( 9 X ,ill)*] dA
Note that the first-order "bulk" curvature term under the integral sign in eq 24 is represented by a line integral in eq 25, which scales like a line tension. Thus, the tendency of molecules within a solid domain to twist can be thought of as being equivalent to (18) Lapger, S . A.; Sethna, J. P. Phys. Reu. A 1986, 34, 5035-5046. (19) Heck], W. M.; Lbsche, M.; Cadenhead, D. A,; Mohwald, H. Eur. Biophys. J. 1986, 14, 11-17. (20) Burton, W. K.:Cabrera, N.; Frank, F. C. Prm. R. SOC.London 1951, A 243, 299-358.
Figure 9. Square spiral. Computer-generated domain structure using the algorithm described in text and the legend to Figure 8. The line tensions are 0 (outside) and 1 (inside).
an anisotropic contribution to the line tension. The combined effects of anisotropic line tension, and the electrostatic dipolar repulsions, on domain shapes can be illustrated with the following simple model. Imagine that the electrostatic forces are large, so that the domain shape (in the absence of anisotropic line tension) is a long, thin rectangle, as in the right-hand side of Figure 2. We now add an anisotropic line tension, chop the rectangle into little square blocks, and reassemble the domain. When the blocks are reassembled the electrostatic forces favor the retention of the long, thin shape. The anisotropic line tension favors a twist of the shape so that the length of the side with the largest line tension is as short as possible. To simplify our calculations we allow only 90° bends. This scheme is illustrated in Figure 8. The self-energy of the square introduced at each bend involves the outer line tension and the energy of the defect line. The defect energy is analogous to the bulk elastic resistance term on the far right side of eq 24. In our simple model, the net energy gained by producing a right-angle bend is one parameter that competes with the long-range dipole repulsions. In Figure 8, we have assumed that there is an in-plane ordering with order parameter wII that is parallel to the long axes of the solid domains. The anisotropy in the line tension can then be described by the first term on the right-hand side of eq 25. Other assumptions regarding boundary conditions are equally plausible; for a general theoretical discussion of the anisotropic two-dimensional molecular film, see Langer and Sethna.18 The calculated shape of the square spirals in Figure 10 is more open than the square spiral in Figure 9. Both calculations use the same dipole density, and differ in the strength of bending energy. This calculation of the shapes of the square spirals is not
Shapes of Finite Two-Dimensional Lipid Domains
P Figure 10. Square spiral. Same as Figure 9 except that the line tensions are 0 (outside) and 0.8 (inside).
a true free energy minimization. The crystal shape is frozen after the addition of each elementary square. No allowance is made for further thinning of the long, thin sections but this effect should be small for domains already long and thin. See Figure 3. In spite of these several simplifications the present calculation does illustrate the important forces responsible for the formation of strongly chiral shapes.
Discussion Lipid monolayers a t the air-water interface confront the experimentalist using epifluorescence microscopy with a wide variety of domain shapes and patterns. The present work was undertaken to help interpret current data and to guide future experiments, not to attempt a rigorous, comprehensive theoretical treatment of the subject. The principal conclusions from the present work are the following. Consider a lipid monolayer compressed to the point where solid domains are first formed, and assume for the moment that the solid is isotropic. The solid domains first formed are expected to be circular, as frequently observed.24 On further monolayer compression, more solid is formed, either by the formation of new circular domains that must be all the same size at equilibrium, or by growth of the circular domains first formed. In the latter case, which may represent a common growth condition, the solid domains grow in size and undergo a change of shape. For the isotropic case, this change of shape may be a sharp shape transition. For an anisotropic solid with long-range molecular tilt, there may or may not be a sharp transition but shape changes are nonetheless certain to take place due to the increase of electrostatic forces with increasing solid domain size. Shape transitions are expected to be accompanied by changes of tilt, and vice versa. A coupled tilt-shape transition may have been observed by Heckl et al. in monolayers of phosphatic acid.19 Our calculations (Figure 2) show how a single solid domain can grow in area but undergoes a thinning in one direction at the same time. This effect has been seen experimentally and was previously attributed to an effect of cholesterol in reducing line tension.16 At present it appears most likely that the observed thinning is due both to an effect of cholesterol on line tension and to the electrostatic effects described here. (Compare Figure 2 of the present paper with Figure 3 of ref 16. The maximum in the plot of the experimental values of width vs monolayer compression in Figure 3 of ref 16 may be analogous to the theoretical maximum in Figure 2.)
Heckl et al. have described the formation of chiral spirals of dipalmitoylphosphatidic acid on monolayer^.'^ In one region of their pressure-area curve they report solid domains that remain constant in number, grow in area, and become thinner in the manner discussed in the present work. These results are similar to the observations made in this Heckl et al. also report a lower pressure region where domains associate with one We have not yet observed another on monolayer compres~ion.'~ this effect in monolayers composed of dipalmitoylphosphatidylcholine.
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4525 The present work also shows how, in principle, the dipolar electrostatic forces operate in concert with molecular asymmetry to produce what otherwise might seem like highly exaggerated chiral domain shapes. If we were to remove these electrostatic forces, then the solid domains in Figures 9 and 10 would collapse to two-dimensional compact solids, which would still be chiral by Wulfs theoremem Two other proposed explanations of these chiral domain shapes have also included long-range electrostatic dipolar forces.21~22We suggest that our treatment of the dipolar forces may be the more accurate for general domain shapes and anticipate that it will be possible to extend these calculations to more realistic representations of chiral spirals as well as other chiral shapes. The present paper has completely neglected the "interaction" between different solid domains, and the problem of how these domains are ordered relative to one another. The problem of supercrystals and associated thermodynamic phases has been treated by Andelman and collaborators.10 In the terminology of the present paper these studies by Andelman assume global equilibrium, and thus they have treated hexagonal arrays of circular domains and parallel arrays of linear strips. In a separate work we shall report on calculations that include interactions between different domains, with geometries more complicated than the parallel strips that have been treated previously.8-10 Acknowledgment. This work was supported by N S F Grant DMB 8619320.
Appendix Strictly speaking, a third term should be included on the right-hand side of eq 9 in the main text. This is because the line integral, as written, also includes some of the energy contained in e,. It will be seen that
where the last term on the right-hand side is the correction term. The limits of integration in the third term are taken along the outer edge of one small block in Figure 1, p is the perimeter of the domain, and p/n11Z6is the number of blocks around the perimeter of the domain. The third term is evaluated simply as follows:
U2
D
= -ph2, with n = 1
('44)
Hence,
Note that p p 2 scales like the line tension energy, pX. An in-plane component of the dipole moment makes an anisotropic contribution to the line tension; since domains with uniformly tilted dipoles are generally anisotropic, such line tensions are already anisotropic. Thus, as long as the line tension X is treated as an unknown adjustable parameter, we can neglect the third term in eq A l , as we have in the present paper. However, this third term must be included in the calculations where a change in dipole tilt is considered. (21) Gabay, M.; Garel, T.; Botel, R. J . Phys. (Les Ulis, Fr.) in press. See also Pomeau, Y.Europhys. L e r r . 1987, 3, 1201-1204. (22) Heckl, W. M.; MBhwald, H. Ber. Bunsen-Ges. Phys. Chem. 1986,90, 1159-1163.
