J . Phys. Chem. 1990, 94, 4728-4131
4728
Harmonic Shape Transitions In Lipid Monolayer Domains Harden M. McConnell Stauffer Laboratory for Physical Chemistry, Stanford University, Stanford, California 94305 (Received: November IO, 1989) Molecules in monolayer films at the air-water interface act on one another by long-rangeelectrostatic forces. These electrostatic forces compete with line tension in affecting the equilibrium shapes of lipid domains at the air-water interface. The present work gives a theoretical treatment of conditions for shape transitions from circular domains to two-dimensional quantized shapes, that is, to shapes with n-fold rotation symmetries, where n = 2, 3, .... It is shown that the critical radius R, at which a circular domain becomes unstable with respect to a transition to a shape with n-fold rotation symmetry is given by the equation, R, = (6/4)exp(2,) exp(h/p*), where 6 is the distance between neighboring dipoles, h is the line tension at the are shape transition domain boundary, p is the difference in the dipole density between the domain and its surround, and the Z,, exponents that arise from purely geometrical factors, independent of the physical variables. Our calculated values of the 2, agree with recent computer simulations of these shape transitions by Vanderlick and Mohwald.
Introduction A number of recent theoretical studies of lipid monolayers at the air-water interface have focused on the sizes, shapes, and shape transitions of finite lipid A separate but related subject concerns phase transitions between various infinite arrays of two-dimensional lipid domain^."^ Lipid domains can be visualized by epifluorescence micro~copy.~-'~ The domains of interest correspond to solid, liquid, or gas. The domain shapes and sizes can be. understood phenomenologically by a competition between line tension at the domain boundary, and electrostatic repulsions between molecules within and between domains.'-3 Line tension favors compact domain shapes that are circular for a fluid surrounded by a second fluid. The electrostatic repulsions favor noncircular domain shapes. A subject of special interest to us concerns shape transitions of lipid domains. This is because such shape transitions can be observed and characterized with epifluorescence microscopy. Moreover, near the critical conditions for shape transitions, lipid domain shapes are highly sensitive to physical chemical events at the air-water interface, such as, for example, photochemical changes. A quantitative understanding of the physical conditions for these shape transitions should facilitate the understanding of the underlying chemistry. Our earlier theoretical work focused on particularly simple shape transitions, such as the hypothetical square-to-rectangle2transition, and the experimentally relevant circle-to-ellipse transition.' The purpose of the present paper is to show how to extend our calculation to transitions from a circle to shapes with higher
harmonic symmetries, corresponding to rotation axes with n = 2, 3, .... A calculation of this sort has recently been carried out by Vanderlick and Mohwald using numerical (computer) metho d ~ Their . ~ calculation treats both small and large distortions from circular symmetry and also distortions with more complex symmetries. While the numerical methods are clearly more powerful, the analytical treatments provide physical insight and also provide a check on the numerical methods for small distortions. We anticipate that the analytical results will also be useful for treatments of fluctuations near the shape transitions, as well as transition kinetics. Our present calculation is related to an earlier theoretical treatment by Thiele of shape instabilities of circular "magnetic bubbles" in thin ferromagnetic films.'* Thiele's analytical treatment is different from ours, but similar integrals ultimately arise in both calculations, and we use his notation whenever possible to exhibit this similarity. The experimental relevance of our calculations is considered in the Discussion.
( I ) Keller, D. J.; Korb, J. P.; McConnell, H. M. J . Phys. Chem. 1987,91, 64176422. (2)McConnell, H.M.; Moy, V. T. J . Phys. Chem. 1988,92,4520-4525. (3)Vanderlick, T. K.; Mohwald, H. J . Phys. Chem. 1990, 94,886-890. (4)Andelman, D.;Brochard, F.; deGennes, P. G.; Joanny, J. C.R. Acad. Sci. Paris, Ser. C 1985,301, 675-678. (5) Andelman, D.; Brochard, F.; Joanny, J. J . Chem. Phys. 1987,86, 3673-368I. ( 6 ) Keller, D. J.; McConnell, H. M.; Moy, V. T. J . Phys. Chem. 1986,90, 231 1-2315. (7) McConnell, H.M. Prm. Natl. Acad. Sci. U.S.A. 1989.86.3452-3455, (8) Fischer, A.; Lbsche, M.; Mbhwald, H.; Sackmann, E. J . Phys. Lett. 1984. 5. L785-L791. (9) h;4cConnell, H. M.; Tamm, L.;Weiss, R. M. Proc. Natl. Acad. Sci. U.S.A. 1984,81,3249-3253. (IO) Lbche, M.; Sackmann, E.; Mbhwald, H. Ber. Bunsen-Ges. Phys. Chem. 1983.87. 848-852. ( 1 1 ) Peters, R.; Beck, K. Proc. Narl. Acad. Sci. U.S.A. 1983, 80, 71 . - x1-71 - - . .R7 - .. (12)Weiss, R. M.;McConnell, H. M. Nature 1984,310, 47-49. (13)Miller, A.; Mohwald, H. J . Chem. Phys. 1987,86,4235-4265. (14)Subramanian, S.;McConnell, H. M. J . Phys. Chem. 1987, 91, 17 15-171 8. (15) Moy, V. T.; Keller, D. J.; Gaub, H.; McConnell, H . M. J . Phys. Chem. 1986,90,3 198-3202. (16)Sed, M.; Subramaniam, S.; McConnell, H. M. J . Phys. Chem. 1985, 89,3592-3595. (17) Rice, P.;McConnell, H . M . Proc. Nutl. Acad. Sci. U.S.A.1989,86, 6445-6448.
In eq 1 6 is the distance of closest approach of neighboring dipoles, X is the line tension, and p is the difference in dipole densities in
0022-3654/90/2094-4128$02.50/0
Background Theories We consider a single lipid domain using the axis system sketched in Figure 1. The molecular dipoles within the domain and outside the domain are assumed to be oriented vertically. This approximation is most accurate for liquid domains surrounded by a second liquid. In our earlier study it was shown that, in a monolayer of fixed total area, the equilibrium radius of an isolated, circular lipid domain is2
the two phases. It was shown further that the circular shape is unstable with respect to distortion into an elliptical shape (C, symmetry) when the radius equals, or exceeds, R2, where'
The general problem is one of the variational calculus, where the shape-dependent part of the free energy is given by the equation's2 F = F:, + F,I + FA (3) (4)
The quantity FAis the line-tension free energy. The quantity F i l is simply the electrostatic energy of a uniform dipole array, (18) Thiele, A. A . Bell System Tech. J . 1969,48, No. IO, 3287-3335.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. I I , I990 4729
Shape Transitions in Lipid Monolayer Domains
+
identical, parallel, concentric, current loops, with radii Ro p, separated one from the other by the small distance 26. Clearly, the Taylor's perturbation series should be valid for both IpI > 6 and IpI 6 if the calculation is to be useful. The problem is solved by the following strategy. Let
G = 2HX
(14)
X = ( x + (1/2)(1 - cos 0))'/2
(15)
where
x = 62/P Figure 1. Cylindrical coordinates used for the calculations. The unit vectors u' and (and u" and 3') ar_e parallel and perpendicular to the radius vector R = u'(R,,+ p , ) and R' = u"(Ro+ p',,), where p , = r, cos n@ and p'" = r, cos n$'.
neglecting edge (finite size) effects. This term is constant for a given area and can be omitted from further consideration. The term Fd is the shape-dependent part of the electrostatic free energy. For a circular domain of area A = a R 2 expressions for FA and F,,are
Sometimes it is convenient to think of the energies in eqs 4 and 7 as corresponding to a negative contribution to the line-tension energy that depends on the size and shape of the domain.
To simplify the discussion, we consider only a single mode, with amplitude r,. (Coefficients of cross terms rmrnvanish in second order.) The problem of collecting all the terms in eq 17 that are quadratic in r, is tedious and sometimes tricky. To enable the reader to follow our steps, we use a special notation. When the sequence of terms NH-IX-' is used as reference, the free energy Fe1(200)arises from terms quadratic r,, in N , and zero order in H-I and XI;F,,(110) arises from terms linear in r,, in N and H-I and zero order in X , etc. The calculation of the terms in r: arising from NH-' alone is tedious, but straightforward:
Analysis The electrostatic energy in eq 4 is expressed as follows
-1 --F(,:)(200) P2
flo)= 2sXR,
(6)
(7)
+ Q ( l 1 0 ) + Fy(020) = ar'srdO 2R, o cos ( n
where
+ 1)0 +
I:
I-;
-(n)(n - 1)
cos 0
+
-cosnO)cosO/(l N = (dd/d$).(dd'/d$')
?I G = (2[Ro2+ Ro(p
= u'(Ro
(9)
+ p)
(10)
+ p') + pp'] [ 1 - COS 01 + (p
- p')2
e=$-$' p
= Cr,, cos (n$ n
+ 462)1/2
( 1 1)
The above integrals are evaluated by using Xo, that is, the value of X with r,, = 0. The various terms can be recombined into two types of integrals, V ( x ) and Ln(x),where V(x)= I r d O X 1 0
(12)
+ on)
(13)
In the above equations u' is a unit vector in the radial direction Formulas for these integrals are given by Thiele,18 and in the and the r,, are the amplitudes of the various possible harmonic Appendix. For the problem of interest to us, x
