5389
J. Phys. Chem. 1994, 98, 5389-5393
Shape Transitions of Lipid Monolayer Domains in an External Field Rudi de Koker and Harden M. McConnell' Department of Chemistry, Stanford University, Stanford, California 94305 Received: January 31, 1994"
The equilibrium shapes of lipid monolayer domains are normally determined by the balance between the interface (line tension) energy and the long-range electrostatic repulsion between lipid dipoles. When an external electric field acts on the monolayer, a third energy term affects the domain shapes: the energy of the dipoles in the external field. We study the effects of a static field on harmonic shape transitions and of a periodically varying field on the dynamics of a distorting lipid domain.
Introduction
Lipid monolayers at the air-water interface can be prepared in a two-phase regime, where separate domains of one liquid phase are surrounded by a coexisting, continuous background liquid phase. In equilibrium, the shapes and sizes of the domains are determined by a fine balance between two energy The interface energy between the two phases, with a line tension A, favors compact, circular domains. When thedomains are large, however, or when the line tension is small compared to the difference in dipole density m between domains and background, the electrostatic repulsion between the lipid dipoles dominates and leads to distorted, noncircular domains. Transitions between equilibrium shapes occur when the ratio of X/mZ reaches certain critical values. For example, it is known6 that a circular domain is unstable with respect to a harmonic distortion of 2-fold symmetry when the radius R of the domain exceeds a critical value R::
For coexisting liquid phases, both X and m depend on the lateral pressure p ; shape transitions are therefore normally triggered experimentally by changes in the pressure. Recently, monolayer domains have been studied in the presence of externally applied electrostatic fields.',* In an inhomogeneous field, each lipid dipole experiences a force due to the field gradient. The total force on all the lipid dipoles in a domain will make the domain move until it finds an equilibrium position. Thus, external fields can be used to study domain motion and drag forces. At the same time, however, external fields can have an effect on domain shapes. In an inhomogeneous field, forces on different parts of a domain are generally different and can facilitate or suppress distortions of the domain. The equilibrium shape is now the shape that minimizes the total free energy, including the electrostatic energy of the dipoles in the external field. Thus, externally applied fields can be used to influence equilibrium shapes and to trigger shape transitions in a controlled way. In this paper, we investigate the effects and possible uses of a particular, rotationally symmetric electrostatic field. Theory
The total free energy of a monolayer domain can be written as the sum of three terms:
+ F d d + F,,,
F = FA
Fk=
w
(3)
respectively, where Pis the perimeter of the domain, A is a suitable cutoff parameter that keeps the electrostatic self-energy finite, and the double line integral is along the contour Cof the domain. In the presence of an external field, ther_e is a Lhird term, FeXt. The energy of a dipole$ in an electric field E is-pE. Accordingly, for a domain with dipole density k per unit area, the energy Fext can be written as
A
In fluid monolayers, the dipole densities in the plane of the monolayer (the xy-plane) can be taken zero. That leaves a scalar dipole density m (in the z-direction, perpendicular to the monolayer plane; the z-axis is taken to point into the subphase). As discussed in ref 8, in a two-component system, where the dipole density m is a function of the composition, an external field gradient will affect the composition and dipole density. We assume here that such composition changes are small and that m can be taken constant (this will be true when one stays far away from the critical point of the two-phase system). The energy contribution due to the external field then becomes
F,,, = - m J J E z ( i ) dA
(6)
A
For a given external field, this energy can be readily calculated. The calculations in this paper will be restricted to one particular field configuration, which has suitable theoretical properties, and can be easily generated experimentally. A straight wire with a circular cross-section (radius a ) is lowered into the monolayer, perpendicular to the monolayer plane, and held a t potential V, against the grounded subphase (Figure 1). As shown in ref 7, this generates a field in the monolayer plane given by
2 = (O,O,E,) (7)
(2)
As discussed in previous papers,9J0 the line tension energy FAand Abstract published in Advance ACS Absrracts, April 15, 1994.
the domain shape and size-dependent dipole4ipole electrostatic energy F d d can be written as
where r is the distance from the center of the wire, and KOis the zero-order modified Bessel function of the second kind. Note that this field is infinite for r = a. Since the wire is held in a glass
0022-3654/94/2098-5389$04.50/0 0 1994 American Chemical Society
De Koker and McConnell
5390 The Journal of Physical Chemistry. Vol. 98, No. 20, 1994
TABLE 1: Slum h i t i o n Exwmats and Slam n
z.
~ ( a / ~ / (-n1)‘ z
2 3 4
1013 1113 11.7613 12.3713 12.8813
0.1389 0.0476
5
6
0.0238 0.0142
0.0094
.The values for the slopcs given are for @ = 4on.
2.5 I
111
I
R y n I. Origin of the external electrostaticfield. A thin wire of radius a. held perpendicular to the monolayer plane, is kept at potential V . with respct tothesubphaseandshielded bya glasscapillaryofradiusr..Thc thin lines are the field lines. In the monolayer, the field is perpendicular to the monolayer plane. capillaryofradiusr,> a,however,thefieldiskeptbelowdischarge everywhere,and the area betweenr = a a n d r = ?,can beexcluded from the calculations. Note also that the field is positive everywhere when V. is positive. In that case, a domain will be attracted to the wire. and position itself around it, when its dipole density m is positive. We take V, > 0. m > 0 in the rest of this article. Lowest Order Stability Analysis
First, we investigate how the external field (eq 7) affects the stability of a circular domain with respect to small harmonic distortions. Take, therefore, a monolayer domain in its equilibrium position, centered around the wire. For small enough values of the potential (and high enough values of Aim2), the domain is circular, and we study shape distortions away from the circular reference shape. Note that the lipid molecules are excluded from the region where r < r., which is occupied by the capillary that carries the wire. The correction to the self-energy due to this small perforation is ignored in the following calculations. Assumethat theshapeofa domain isgivenin polar coordinates by r(0) = r,
+ r, cos(n0)
-
2
Le
1.5
0
N
5 c
-
1 0.5
0 0
5
10
15 11
20
25
30
Rgun 2. Critical radius for the harmonic distortion with n = 2. for two different values of @/a.
The second-order contribution to Fa, can be calculated analytically (see the Appendix) and is
with Corr(x) given by
where K , is the first-order modified Bessel function ofthesecond kind. Thus, one gets for the total free energy
(8)
In order for the area of the domain to remain fixed at rrR2, r. must depend on r. according to
:r = R2- 1/2r:
(9)
The energy of a domain with a shape given by eq 8 depends on r., and can be expanded in a power series:
where only even terms in r. are included because the sign of r. does not affect the energy. The sign of the second-order term determines whether the circle is stable or unstable with respect to the harmonic distortion. The second-order terms in the line-tension and dipole-dipole energies have been calculated in ref 11 and are given by
The Z. are numerical constants and are given in Table 1 for n between 2 and 6.
The sign of FC0 changes for R such that
R = -Ae 8
[
ynttez. exp zvoarr(a/R)]
mr(n2- 1)
(la)
Equation 16 implicitly defines a critical radius R. above which the domain becomes unstable with respect to an nth-order harmonic distortion. This critical radius depends as before on the ratio Xlmz. and now also on the wire-potential V., and on the wire radius a. Once a is fixed, R. can be expressed as a function of two independent, dimensionless parameters: Xlmz and 7 = 2V./mrr. The second parameter will be called the potential for simplicity. The functional dependence of the critical radius on q is sketched in Figure 2 and Figure 3. Figure 2 shows R2 as a function of 7 for two different values of a. One sees that the influence of a is slight. Figure 3 compares the critical radii R2 and R3 for one value of a. Note that, although R2 < R3 in the absence ofan external field and both radii increase with growing V, the critical radii eventually cross when the applied field is strongenough. Notealso that thecrossover valueof the field 7 ~ 3 can bedetermined fairly accurately from thelinear approximation for small fields, given by
Lipid Monolayer Domains in an External Field
7ke Journal of Physical Chemistry. Vol. 98, No. 20, 1994 5391
3
1.4
2.5
1 0.5 0 0
5
10
15
20
25
30
35
rl
Figure 3. Critical radii for harmonic distortions n = 2 and n = 3. with e / a = 40.
Figure 4. Critical radii (linear approximation)for harmonic distortions n = 2-5. Domains in the region under the thick line are circular and stable. In the triangular regions. domains are unstable with rcspct to
exactly one harmonic distortion.
where
is the critical radius in the absence of an external field. T h e approximate critical radii suffice for our p u r p e s and are sketched in Figure 4. Because the R-axis is logarithmic, eq 17 gives straight lines with the slopes given in Table 1. Figure 4 illustrates the following interesting properties. First of all, the external field stabilizes the circle: all critical radii grow with increasing potential 7. Second, the external field can be used to induce shape transitions. For example, if the domain radius is slightly below the critical value R2(point A in Figure 4). a drop in the potential (toward point B in Figure 4) induces a shape transition, as will alsoa decreasein X/m2(toward point C;adecreasein h/mZshifts all points of fixed R vertically upward in Figure 4, but leaves the instability curves unchanged). Thus, there are two independent ways to bring about a shape transition. Finally, and most interestingly, low-order harmonics are stabilized more than higher-order harmonics. It is this property which leads to the crossover behavior of the successive critical radii and has important experimental consequences. In the absence of an external field, the critical radii follow each other in order:
@ < R:< R! < ...
(19)
WhenX/mlisgradually decreased, thedomain becomesunstable first for a 2-fold distortion and only later for higher-order distortions. Iftbedomainradiusisonlyslightlyabove thecritical value R2, the domain assumes a distorted shape with 2-fold symmetry, and the distortion process is determined solely by equilibrium considerations. This is not true for higher-order distortions: when a domain is unstable with respect to distortions of order n > 2, harmonic distortions of lower order are also unstable. In that case, the domain will generally not assume the shape of lowest energy but will go into a shape that is kinetically favored, i.e. a shape determined by the fastest growing harmonic (see refs 4 and 12). In the presence of an external field gradient, however, the order of the critical radii changes. Thus, e.&, for a potential just above the crossover value for R2 and R, R,
< R, < R, < R, < ...
(20)
This means that the 3-fold distortion becomes unstable first.
Starting from a domain represented by point D in Figure 4, a dropinv(1owardpointE) willnowinducea transitiontoashape of3-foldsy"etry.and this transition tooisdetermined by energy considerations,notby kinetics. This transitioncanalso be brought about by a suitable change in X/m2 (point F). As is seen in Figure4, transitionstoshapesofanygivenorderncanbeinduced ina similar manner by makingappropriatechangcs intheexternal field (or in X/m2),starting from the stability region below the thick line.
Thermodynamic Order of tbe Transition Asis knownfrompreviouswork,'Ointheabsenceofanexternal field the transition from a circle to a shape of 2-fold symmetry at R2 is a first-order transition in the thermodynamic sense. The harmonic distortion which becomes unstable at R > Rz will therefore "run away", higher harmonics mix in, and the domain assumes a highly distorted, nonharmonic shape (approximated by an oval of Cassini in ref IO). One may wonder whether this transitionremainsof first order when theexternal field is switched on. Clearly, very small fields cannot change the order of the transition. The contribution of the external field to the total energy of a domain can be made arbitrarily small for sufficiently small potential V,. Since the thermodynamic order of the transition depends on the presence of a low-energy distorted shape at R = R2, and the energy of this shape must be lower than that of the circle by a finite amount for the transition to be first order, arbitrarily small energy changes due to the external field do not affect the order of the transition. This may be different in strong external fields. Indeed. shape transitionsmay be continuous, second-order ones when the applied field becomes strong enough, in particular when the potential 7 > vZ3, Le. for transitions to shapes of 3-fold. 4-fold, etc. symmetry. We have at present no theoretical arguments one way or the other. However, experimental observation of fieldtriggered shape transitions may provide evidence about theorder of the transition. The external potential V, can be controlled, and changed, more easily than the ratio X/m'. By decreasing V, slowlyuntilashape transitionoccurs,onemaybeabletoobserve whether the harmonic distortion stabilizes (second-order transition) or runs away (first-order) and to what extent the transition is obscured by thermodynamic fluctuations.
Observing Harmonic Distortions: Periodically Varying Fields One way to observe harmonic distortions of a given n-fold symmetryexperimentally is tovary theexternal field periodically around the critical value at which the transition occurs. Thus, while keeping the pressure (and thus h/m') fixed, one decreases V. until the harmonic distortion develops and then increases V,
5392
De Koker and McConnell
The Journal of Physical Chemistry, Vol. 98, No. 20, 1994
again before thedistortion hasa chance to runaway. We therefore investigate the dynamics of a monolayer domain in a time-varying external field. Let the external potential Vo(t) be time-dependent and given by
+ uac cos(w?)
v,(?) = Vd,
(21)
where the time constant
7, now
depends on
vdc:
-1= - -1 8mn2Corr(a/R)d'c 7, 7; *pR2(4n2- 1) Equation 31 can be solved exactly:
Assume a domain shape described by a time-dependent harmonic distortion r(0,t) = ro(t)
+ r,(t)
cos(n0)
(22)
with again, to keep the area fixed,
+ 1/2r:(t)
r:(t)
= R2
(23)
According to Stone and McConnell,13 the dynamical equation for the amplitude r,(t) of a harmonic domain shape is
n2
),=-- 4fn
W 4n2 - 1
1 8mn2~orr(a/~)vdC -= when TI] ?rpR2(4n2- 1)
Ar(s)
= -?rRf,Ar,
1 aF TR ar,
and thus, with (24) and (1 5 )
kR2(4n2- 1)
n'
-
8mV0
n2
z o r r ( a / R ) r , , (28) ?r2kR24n - 1
or ),--r 1 7;
8m V,
E--
,
"2orr(o/R)r, ?r2pR24n2 - 1
(29)
with the field-independent time constant
Separating the constant potential one gets i.,
- -r,1 7,
=-
vdc
does depend on the ac-frequency, however, and is determined by the ratio of the time period T of the external potential and the dynamic time constant 7::
T / r : = 2 ~ / w 70 ,
The expansion (eq 10) to second order gives
4 m2(n2- l)n2 ln(R/R:)
from the ac potential u,,,
8mua,n2Corr(a/R) 7rpR2(4n2- 1)
(34)
y / r p= exp [2ua,/ vd,w7:] (25)
Thus,
i., = -
= 03
Note that A, can have any value, independent of the frequency w . The shape oscillation is not a resonance effect, but reflects the periodic instability of the domain shape, brought about by the periodic change in the external field. The amplitude ratio of the shape oscillation,
= -Rrf,Ar,, cos2(ne)
f,=---
7,
One then gets a purely periodic oscillation given by
Here, p is the (dynamical) viscosity of the subphase under the monolayer, andf, is interpreted as the amplitude of a harmonically varying (in 0) force acting a t the boundary of the domain. This force can be derived from the total energy F as follows. For a small distortion &(e) = Ar, cos(n0) and a force per unit length f(0) = f, cos(n0), the energy changes by hF = -$dsAs)
i.e. an exponentially growing or decaying oscillation, depending on the sign of 1/T,. The growth or decay rates become smaller as one gets closer to the critical line in Figure 4. If one is exactly on this line, i.e. vdc is the critical field, 7, = m and thus
(37)
The larger this ratio, i.e. the more time the domain has to distort during one field oscillation, the larger the amplitude ratio. Note however that the amplitude of the harmonic distortion must at all times remain small enough for the approximations in the derivation of eq 35 to be valid. Since u,, will at most be of the order of vdc, a sufficient condition is that the ac-period T be small compared to 7;. One may wonder what happens when uaEis such that, for part of the cycle, more than one harmonic distortion is unstable. For small enough distortions, one can then simply superimpose two or more solutions of the form given by eq 33. If vdc is adjusted so that the domain is near the (thick) instability line in Figure 4, all distortions other than the one of lowest energy, will decay rapidly, and one is left with a single oscillating, harmonic distortion. Suitable values of uac and w can probably best be determined empirically. Equation 35 offers an approximate and qualitative description of the expected behavior in an external field with an ac component. It remains to be seen whether such harmonic amplitude oscillations can be observed experimentally. Acknowledgment. This work was supported by the National Science Foundation (Grant N S F DMB 9005556-01). R.d.K. is a Howard Hughes Medical Institute Predoctoral Fellow. Appendix
r, cos(wt)
(31)
can be calculated analytically as follows. The energy of a domain shape given by eq 8 in the electrostatic field of eq 7 is, according to eq 6,
The Journal of Physical Chemistry, Vol. 98, No. 20, 1994 5393
Lipid Monolayer Domains in an External Field
Substitution in (A3) gives
2m V, - - -Jo2“d8Jm a
dk J y d u u K o ( u ) ( A l ) k2Ko(ka)
where r, is the radius of the capillary, and u = kr‘ has been substituted. Here and throughout this paper, K,(x) stands for the nth-order modified Bessel function of the second kind. Using the property of these functions (see ref 14)
Using property A2 and also the fact that
d -du [ u K , ( u ) ] = -uKO(U) one obtains finally the integration over u can be performed easily, and one gets
F,,, = - 4 m r~
dkKi(kr~) J0- k Ko(ka)
+
References and Notes (1) McConnell, H.M. Annu. Rev. Phys. Chem. 1991,42,171-195, and references cited therein. (2) Vanderlick, T. K.; Mohwald, H. J . Phys. Chem. 1990, 94, 886-
The first integral does not depend on the amplitude of the distortion. The second integral contains the function rKl(kr), which, since 2
r(8) = R
‘n + r, cos(n8) - + O(r;) 4R
(‘44)
can be expanded as
r(B)K,(kr(B))= R K , ( v )
+ r, cos(n8) K , ( v ) +
r: kr, cos(n8) K,’(Y)- Z K , ( v ) + kr: cos2(n8) K,’(v) kr: Rk2r: ~ K , ’ ( u+)-cos2(n8) K,”(v) + O(r;) ( A 5 ) 2 where all functions K 1are evaluated at v = kR.
890. (3) Deutch, J. M.; Low, F. E. J . Phys. Chem. 1992, 96, 7097-7101. (4) Langer, S.A.; Goldstein, R. E.; Jackson, D. P. Phys. Reu. A 1992, 46 (8), 4894-4904. (5) Tsebers, A. 0.;Maiorov, M. M. Magnetohydrodynamics 1980,16, 21-28. (6) Keller, D.; Korb, J. P.; McConnell, H. M. J. Phys. Chem. 1987,91, 6417-6422. (7) Klingler, J. F.; McConnell, H. M. J . Phys. Chem. 1993, 97, 2962,2966. (8) Lee, K. Y. C.; Klingler, J. F.; McConnell, H.M. Science 1994,263, 65 5-6 58. (9) McConnell, H. M.; De Koker, R. J . Phys. Chem. 1992, 96, 71017103. (10) De Koker, R.; McConnell, H.M. J. Phys. Chem. 1993,97, 1341913424. (11) McConnell, H. M. J . Phys. Chem. 1990, 94,4728-4731. (12) Lee, K. Y. C.; McConnell, H. M. J . Phys. Chem. 1993,97, 95329539. (13) Stone, H.; McConnell, H.M. Proc. R. SOC.London, in press. (14) Arfken, G. B. Mathematical Methods for Physicists, 3rd ed.; Academic Press: Orlando, FL, 1985.
