3167
J. Phys. Chem. 1992,96, 3167-3169
Electrohydrodynamic Instabilities in Lipid Monolayers at the Air-Water Interface Harden M. McConnell Department of Chemistry, Stanford University, Stanford, California 94305 (Received: November 14, 1991) The shape stability of a large rectangular monolayer lipid domain is analyzed. A straight edge of this domain is stabilized by the line tension A and is destabilized with respect to harmonic shape distortions by long-range dipole forces. It is shown that this straight edge is unstable with respect to harmonic distortions for all spatial frequencies k that are less than a critical frequency k,, where kW-l = (6/v‘2) exp([l + 2C/2]) exp(A/p2). Here 6 is the distance between neighboring dipoles in the film, Cis Euler’s constant, and p is the difference in dipole density in the two phases on either side of the edge. If the two phases on either side of the domain boundary are fluid and a simple hydrodynamic model is used to describe the motion of this fluid, it is found that the harmonic distortion that grows most rapidly has a spatial frequency k,, = (k,/e’/j). The instability of the edge of the long rectangular domain is removed if the width w of the domain is less than 2e1/’kW-l.
Introduction Recent work has shown that monolayers of certain mixtures of lipids can give immiscible liquid phases a t the air-water interface.I4 For example, a binary mixture of cholesterol and dimyristoylphosphatidylcholineyields two immiscible phases at certain temperatures and pressures, one phase being rich in cholesterol, and the second phase rich in phosphatidylcholine.I4 Theoretical models of the sizes and shapes of lipid domains have been developed in terms of a competition between line tension and long-range dipolar interaction^.^-^ The line tension favors large circular domains, and the long-range dipolar interactions favor domains that are circular and small or else those that are elongated. The present work is concerned with theoretical aspects of transient or unstable shapes of large domains in monolayers of immiscible fluids at the air-water interface. Such time-dependent domain shapes have been observed when monolayers are affected by photochemical events,2pressure changes near a critical point,” and hydrodynamic shear. Transient structures brought about by shear will be described elsewhere. It is important to distinguish between the instability of “small” domains, and the instabilities of “large” lipid domains. A small circular liquid d o m a i n s u c h as a domain of equilibrium s i z e i s stable. As the radius of the domain is increased, a point is reached where it becomes unstable with respect to distortion to an elliptical shape (2-fold symmetry).E As the size is increased still further, the circular domain becomes unstable with respect to higher harmonic distortion^.^*'^ The instabilities of small lipid domains thus have distortion wavelengths of the same order of magnitude as the size (radius) of the domains. In the present paper we are concerned with large domains that are unstable for a wide range of spatial frequencies, and we wish to calculate the spatial frequency of the instability that grows most rapidly with respect to time. For related discussions of magnetohydrodynamic instabilities, refer to Chandrasekhar” and to Ci5bers and Maiorov.I2 Instability of a Straight Edge Consider a large, rectangular lipid domain of length = 2L and width w. As in previous work, the shapedependent free energy (1) Subramaniam, S.; McConnell, H. M. J . Phys. Chem. 1987, 91, 17 15-1 7 18. (2) Rice, P. A.; McConnell, H. M. Proc. Narl. Acad. Sci. U.S.A. 1989, 86,6445-6448. (3) Hirshfeld, C. L.; S e d , M. J . Phys. (France) 1990, 51, 1537-1552. (4) Seul, M.; Sammon, M. J. Phys. Rev. Lerr. 1990, 64, 1903-1906. (5) Andelman, D.; Brochard, F.; Joanny, J.-F. J . Chem. Phys. 1987,86, 3673-3681. (6) McConnell, H. M.; Moy, V. T. J . Phys. Chem. 1988,92,4520-4525. (7) McConnell, H. M. Annu. Rev. Phys. Chem. 1991,42, 171-195. (8) Keller, D. J.; Korb, J. P.; McConnell, H. M.J . Phys. Chem. 1987, 91, 6417. (9) Vanderlick, T. K.; Mohwald, H. J . Phys. Chem. 1990, 94, 886-890. (IO) McConnell, H. M. J . Phys. Chem. 1990, 94,4728-4731. (1 1) Chandrasekhar, S.Hydrodynamic & Hydromagnetic Stability; Dover: New York, 1961. (12) Cebers, A. 0.;Maiorov, M. M. Magnetohydrodynamics 1980, 16, 21-28.
of this domain is written as a sum of an electrostatic free energy F,, and a line-tension free energy FA: F = F,I FA (1)
+
Here FeI = -y*’$
$dFdi’/r
(2)
dr
(3)
FA = $A
In eqs 2 and 3, p is the difference in dipole density between the domain and the surrounding region, and A is the line tension, assumed constant. For a single straight edge along the x axis the contribution of the integrations in eqs 2 and 3 to the total shape-dependent free energy is F’, where
e6 = Lc~’ In - LOA LO The quantity A = 26 is a cutoff parameter introduced in eq 4 and previously to avoid di~ergence.6,~ This parameter is taken to be of the order of magnitude of a neighboring dipoledipole distance. The above free energy Fr becomes negative when L,,is larger than e6 exp(X/p2), suggesting that this shape is unstable with respect to distortions if the domain length is large enough. As discussed below, this conclusion is correct if the width of the rectangular domain is also large enough. In earlier work it has been shown that the equilibrium radius of an isolated circular domain is6
+
R, = (e36/4) exp(h/p2)
(6)
Circular domains with larger radii are unstable with respect to J ~the following, we carry out distortions to harmonic ~ h a p e s . ~In a related calculation, seeking harmonic instabilities in the straight edge of a lipid domain. As sketched in Figure 1, we consider a harmonic distortion of a domain edge ranging from x = -L to x = L, as follows: y = Ac sin kx (7) Here e is a dimensionless amplitude, and k is the spatial frequency of the distortion. The electrostatic energy with this distortion is then 1 Ffcl - - p 2 X 2 (1 + k2A2c2cos kx cos kx’) [ ( x - x ’ ) ~+ A2e2(sinkx - sin kxq2 + A’]‘/’ (8)
and the line tension energy is FrA
XJ(1
+ k2A2t2COS’
kx)II2 d x
The onset of instability is determined by the condition alae* ( ~ + ~f ’ ~ ~= ~)o l ~
0022-3654/92/2096-3167%03.00/0 0 1992 American Chemical Society
(9) (10)
3168 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 Y
McConnell corresponding to instability of a straight edge. A point of experimental interest is the question of which harmonic distortion grows most rapidly. This is then a problem in fluid mechanics, since the drag on these distortions is due to viscosity. We model this by assuming that the fluid motion is two-dimensional, with a coefficient of friction y due to coupling to the subphase. We assume that the fluid motion in the monolayer is irrotational, with a velocity potential 4:
t W
o = - -U
Figure 1. Schematic representation of edge harmonic instability of a large rectangular domain, of width w. A harmonic distortion is depicted for the edge parallel to x , for which the undisturbed straight edge corresponds to y = 0. The right-hand side of the drawing shows the directions of liquid flow on the two sides of the domain, at the beginning of the time-dependent distortion.
It is shown in the Appendix that the following expression can be derived for the first term: aFfCl/at2= ( p / f i ) p 2 L o k 2 s 2In ks exp[(l
+ 2~9121
(11) where p is between 1 / 4 2 and 4 2 , doubtless very close to 4 2 . Also
aFfA/ac210= k2s2L0A
(sin kx)e-kv
(19)
With this velocity potential, the fluid in_ each lipid monolayer phase. is incompressible,that is v' = V4, and 8-v'= V24 = 0. The velocity v' corresponding to 4 in eq 19 satisfies the boundary condition at the straight edge when t = 0. That is, u = A de/dt. The potential in eq I 9 has been used to describe waves at the interface between two 1iq~ids.l~The time dependence of the instability amplitude c can be obtained by equating the rate of loss of domain shape energy to the rate of dissipation of energy by viscous fluid motion. Per unit length the rate of loss of domain shape energy is
The loss of energy due to dissipation is attributed to the viscous drag of the subphase. If the viscous drag force is simply proportional to the monolayer velocity, the loss of energy due to subphase viscosity is (per unit length)
(12)
Taking p = 4 2 , we obtain from eqs 10-12 the critical value k , of the spatial frequency:
Here y is the coefficient of friction. By eqs 20 and 21, we obtain
According to this calculation, the straight edge of the domain is unstable for all spatial frequencies k less than k,, k < k,. The above instability is suppressed for rectangular domains of small width w. The repulsive electrostatic "interaction" between the two opposite sides of the rectangular domain is given by the integral
From this equation we see that when w = a,t increases exponentially with time for all k < k,. The rate of increase is maximal for the spatial frequency, kmo:
F",, = !/*2x:x:dx
k,, = k,,/e1J3
(23)
The most rapidly growing mode for a domain of finite width w is obtained from the solution of the equation dx' [ ( x - x ' ) ~+ A2(N - e sin k ~ ) ~ ] - l / ~ (24) In [e-'k,/km3] = 2km2w2 (14)
where the width w is written as w = NA. The integrals arising from the second derivative t32Fff,l/at2(o can be evaluated when L is very large, with the result
From this equation it is seen that the spatial frequency k , is smaller than k,,, since In (k,,3/km3) = 2/km2w2
(25)
Note that eq 19 does not include effects related to a domain of finite width. The condition for instability is then
and this gives the equation (k,o/kc)-2In (k,/k,)2 L 4(w2k,,2)-' (17) Since the left-hand member of this relation is always less than e-l, it follows that the instability vanishes when the right-hand member of eq 17 is greater than e-I, that is w < 2e1/2kc,-1 (18) Thus, when w is less than the right-hand member of eq 18, the straight edge of the rectangle is stable. Kinetics of Instability The foregoing calculations give the range of values for the critical spatial frequencies k , and k , for harmonic distortions
Discussion The present work was motivated by observations of transient domain shapes in lipid monolayers at the air-water interface. Such transient structures can be produced for example by photochemical effects that result in abrupt changes in the line tension and/or dipole density: as well as by transient hydrodynamic shear f0r~es.l~ The structural changes of domain shapes are readily observed by eye in the fluorescence microscope, thus providing a strong motivation for the present calculation. Our most significant conclusion concerns the magnitude of the critical spatial frequency k, (or wavelength, 2r/k,) of the harmonic instability of a straight edge of a large domain. It will be seen that the wave length of ~ ,the same order of the dominant instability, 2 ~ d / ~ / kis , of ( 1 3 ) Lamb, H. Hydrodynamics; Dover: New York, 1945; p 456. (14) Benvegnu, N.; McConnell, H. M., to be published. (15) Gradshteyn, I. S.; Ryzhik, I . M. Tables of Integrals, Series and Products; Academic Press: New York, 1980.
The Journal of Physical Chemistry, Vol. 96, NO. 7, 1992 3169
Instabilities in Lipid Monolayers
X‘
t
the region of (x,x’)space over which this integration is carried out, -L Ix IL, -L 5 x’ IL. The integrand is everywhere positive in this region. Introduce the change of variables z = ( l / d 2 ) ( x ’ - x ) and z ’ = ( l d 2 ) ( x ’ + x ) , for which the Jacobian is unity. Figure 2 depicts two regions of (z,z’)space overlaid on the (x,x’)space. The inner region corresponds to - L / d 2 I z I L / d 2 , - L / d 2 Iz‘ IL l d 2 , and the outer region corresponds to - d 2 L Iz Id 2 L and - d 2 L Iz‘ Id 2 L . The desired integral in eq 8 clearly has a value somewhere between the integral over the inner region and the integral over the outer region. In terms of the variables z and z’, the integral in eq 8 can be iterated - ~ z p 2 S-DL P L d z ’ S P L d( 1z -DL
+ k2A2t2A)/(2z2+ A2t2B+ A2)-1/2 (28)
where
+ t/,(cos f i k z )
(29)
B = 4 sin2 ( k z / f i ) cos2 ( k z ’ / f i )
(30)
A = !lZ(cos f i k z ’ ) I
I
-Lo-
Figure 2. Areas of integration. There are three squares in the figure, small, intermediate, and large. The intermediate square, of dimensions Lo X Lo is the area of integration of the desired double integral in eq 8. The sides of this square are parallel and perpendicularto the axes x and x’. The sides of the smaller and larger squares are parallel and perpendicular to the axes z and z’. The smaller and larger squares have dimensions L J 4 2 X LJv’2 and 4 2 L 0 X 4 2 L , , respectively. Since the integrands in eqs 8 and 28 are everywhere positive, the integrals over the smaller and larger squares are necessarily smaller and larger than the desired integral over the intermediate square. Thus, a new square of dimensions pL, X pLo can be chosen so that the integral in eq 28 is equal to the integral in eq 8, where 1 / 4 2 C p C 4 2 . The integrands take on large values along the line x = x’, or I = 0, and thus the square of dimension pL, X pLo is expected to be close to the larger square. As discussed in the text, p 4 2 as Lo in one special case, when both integrations can be carried out analytically.
-
--
magnitude as the equilibrium radius of a lipid domain, R , in eq 6. It is of interest to note that in the stripe phase of lipid monolayers, when the relative area of each of two liquid phases is a previously calculated16 equilibrium width is wq = seb exp(X/p2)
and where p is between l / d 2 and d 2 . From eq 28 one obtains
-
Here d2 = 2b2. These integrals do not depend on the value of L when L is large, so the above limits can be taken between 0 and a. The integrals can then be expressed in terms of Struve functions, K0(2kb):15 aF’,l/at210 =
I
- ( p 2 p L o / f i ) 2b2K0(2kb) -
a X - (kb)’ a(kb)2
K0(2kb) = -In kb Zo(2kb) +
(26)
m
(2k6)”Q(n m=O
+ 1)/22n(n!)2 (33)
We may calculate the stability of the edges in this phase by extending the calculation of eq 15 to an infinite array of edges, each separated from its neighbor by a width w. In this case the right-hand side of eq 15 is simply ( p 2 / W ) L O ( 2 ) (1 2-2 + 3-’ - 5-2 + ...) = ( ~ ~ / 6 ) ( p ~ / N 2 ) L , (27) The equations analogous to eqs 17 and 18 show that the stripe phase is stable with respect to these harmonic distortions. This result is consistent with experimental observations4 if the observed high-frequency irregularities in the stripe pattern are not due to this instability but are due to thermal excitations. The major uncertainty in our calculations involves the assumptions regarding the hydrodynamics. Equation 19 embodies the conservation of monolayer fluid and satisfies the boundary conditions. The expression for the dissipation of energy2’ assumthat the subphase exerts a viscous drag on monolayer motion that is proportional to monolayer velocity. These are the simplest approximations that can be made and may be quantitatively accurate under special experimental conditions, especially very shallow troughs.
By setting the sum of dF’,,/a& and aF’X/at210equal to 0, one obtains the condition for instability of the straight edge given in eq 13. The value of p is not known, but it is doubtless very close to 4 2 . This can be seen by reference to Figure 2. The integrand in eq 8 is very large close to the line x = x’; the central and outer square regions have this line in common, indicating that p should be closer to d 2 . By way of illustration, the integral in eq 4 has the exact value L&2 In (eS/L,). If instead we use the method of integration employed above, with the variables z and z’, we obtain (~/42)L& In ~( 4 2 6 / p L 0 ) . An accurate value of p is obtained by equating these two results. If we let q = p / d 2 , we obtain
Appendix The value of the integral in eq 8 can be bracketed rather narrowly without resort to numerical methods. Figure 2 depicts
In (LO/b) = (1 + 4 In q)/(l - 4) From this it can be seen that as ( L o / @ a,q 42.
(16) McConnell, H. M . Proc. Nurl. Acad. Sci. U.S.A. 1989, 86, 3452-3455.
m
Z0(2kb) = C ( k b ) 2 n / ( n ! ) 2
(34)
n-0
n
s(n aF’ at2
+ 1 ) = -C + pE= p - l
(35)
1
= ( p L , / f i ) p 2 k Z b 2In kb exp[(l
+ 2C)/2]
- -
(36)
-
(37) 1 , and p
Acknowledgment. This work was supported by the National Science Foundation, Grant NSF DMB 90-05556.
