6251
J. Phys. Chem. 1995, 99, 6251-6257
Instabilities of the Stripe Phase in Lipid Monolayers Rudi de Koker, Weina Jiang, and Harden M. McConnell" Department of Chemistry, Stanford University, Stanford, Califomia 94305 Received: November 30, 1994@
In lipid monolayers consisting of two phases, the competition between line tension and electrostatic repulsion can give rise to a stripe phase. The stability of this phase with respect to small harmonic distortions is analyzed. It is shown that the stripe phase is just marginally stable when the stripes are of equilibrium width. As soon as the stripes exceed this width, coupled long wavelength distortions are energetically favored. Intermediate results concerning the stability of a single edge and a single stripe are also presented.
Introduction Phospholipid monolayers at the air-water interface can be readily prepared and observed in a two-phase regime.' The coexisting phases, which appear as dark and light phases under the fluorescence microscope, form a variety of spatial structures, such as circular or near-circular domains, parallel stripes, or more irregular, wormlike structures. Such structures are thought to arise from a competition between line tension and electrostatic r e p ~ l s i o n . ~In - ~the simplest possible macroscopic model, the total free energy of the monolayer contains a line tension term, due to the interfacial energy between the two phases, and an electrostatic term, due to the long range repulsion forces between surface dipoles:6
E = AP -
'/+'ff
Instability of One Edge
d-AT '
-4
(1)
where A is the line tension, ,u is the difference in the dipole densities of the two phases, and A is a cutoff parameter that keeps the double line integral from diverging. Note that P is the total length of the interface between the two phases. The double line integral, which has the form of the Biot-Savart law between current loops, is taken over the interface. Equilibrium structures are those which minimize the free energy, usually under the constraint that the area of each phase is fixed. Mathematically, this translates into a challenging problem in variational calculus. Minimization of the total free energy leads to an integrodifferential equation, which in the general case is intractable. A more promising approach has been to postulate certain simple geometries for the two phases. The geometric parameters that characterize such geometries are then varied until the free energy is minimized. For example, if one assumes that one phase exists as circular islands of equal radius in a sea of the majority phase, it is straightforward to determine the radius which minimizes the total free energy:7 Reg = (e3/8)A eAtp2
(2)
Similarly, for a phase with alternating black and white stripes of equal width w ,the equilibrium stripe width is found to be8
(3) The next logical step in this approach is to subject these @
Abstract published in Advance ACS Abstracts, April 1, 1995.
geometries to various distortions and see whether the total free energy is thereby increased or decreased. Accordingly, in previous work, we have studied both the stability of circular domains with respect to small harmonic distortions9 and large nonharmonic distortions1° and the stability of a straight edge with respect to small deformations." We now apply the same method of (lowest order) stability analysis to a stripe phase, consisting of alternating parallel stripes of the two phases. For this purpose, we first recapitulate the stability analysis of a single straight edge. We then look at the stability of a stripe of small width with respect to coupled deformations of the two edges. Finally, we discuss a restricted class of deformations of an infinitely extended stripe phase.
In the study of small wavelength deformations of a domain edge, the edge can be approximated as initially straight and infinitely long. Imagine the edge to lie along the x-axis, given by the equation y = 0. The upper half-plane (y > 0) is then occupied by one phase, the lower half-plane 0, < 0) by the other phase. A harmonic deformation is superimposed on the edge, giving
y = A sin kx
(4)
The line tension energy of the deformed edge is simply proportional to the length of the edge and is given by
We have introduced an integration limit L so that the integral remains finite. The total length of the undeformed edge is thus & = 2L. For sufficiently small A , this energy can be approximated by the lowest order terms. The excess energy per unit length is then approximately
E,' = -J A 2L
-L
dr(J1-I- A2k2cos2 kx - 1)
Several remarks are in order. First, note that the approximation made here is valid only for very small A , such that Ak In z
+ 2/z2 + C - In 2 -
This condition is shown graphically in Figure 1. Note that a stripe with a given width is represented by a horizontal line in this plot. A deformation of a particular wavelength is then given by a single point on this line. Instabilities occur when this point lies in the region above the thick line. With this in mind, we note several properties of physical interest. First, for a sufficiently narrow stripe, with
w < w , = e Cwo
Instability occurs when this energy change is negative, Le., when the deformed edge has a lower energy than the straight edge. The instability condition is thus
kA eCf1/2 ekip2 < 2
(9)
Le., the wavelength of the deformation exceeds a critical value
A = nec+1/2 A eklp2 C
(10)
Note that this critical wavelength is of the order of the characteristic length
which fixes the length scale of the structures that one expects to see in a monolayer in equilibrium (compare eqs 2 and 3). In short, a single infinite edge is always unstable, irrespective of Alpz, and deformations will be seen on a scale of the order of wo.
(17)
the unconstrained edge is stable for all wavelengths. Second, at the critical stripe width w,, the unconstrained edge becomes unstable for deformations with z = 2, i.e., with finite wavelength
A = m, = ne Cwo
(18)
Third and last, when wc > w , the unconstrained edge is unstable for deformations with a range of wavelengths around A. We now lift the artificial condition that the second edge be kept straight and look at what happens when both edges are allowed to deform. In particular, we are interested to see if a deformation of one edge can facilitate the deformation of the other edge. This will occur if coupling between the two edges lowers the free energy. As mentioned in Appendix A, such coupling will occur only if both edge deformations have the same wavelength. We therefore write for the edge equations: y1 = A sin kx y 2 = B sin (kx
+ 0) + w
(19)
where the amplitudes A and B can be taken as positive. The line tension energy of the stripe is simply the sum of the contributions of both edges:
Instability of One Stripe
The next step is to consider how the stability of an edge is affected by the presence of a second, parallel edge a distance w away, Le., to study the stability of an infinite stripe of width w. First, to simplify the calculations and gain some physical insight, we artificially constrain the second edge to be straight. The excess electrostatic energy is then given by
(16)
Ak2 E,’ = -(A2 4
+ B2)
The electrostatic energy contains self and cross terms given by eqs A.26 and A.27 in Appendix A, respectively. Thus, the electrostatic deformation energy is
‘e5
=
+
p2k2(A2 B2){ kw, In 4 2
+-}2 k2w2 p2AB(k/w)cos 0 K,(kw) (21)
This will be positive and the stripe stable if which, using eqs A.25-27 of Appendix A, gives, to lowest order,
+ e2 - AB > o
~ V I , B=) d2
(22)
where for clarity we have introduced the notation The straight edge at y = w always increases the energy, and therefore stabilizes the edge at y = 0. The total energy
a = In
b o eC-ll2 2
y=4coso-
gives the instability condition In
b o
2
2 k2w2
+- 0. A necessary condition is clearly that a > 0, Le., precisely the condition we found when one edge was kept fixed. If y is negative, this is a sufficient condition: the coupling between the edges increases the energy and cannot induce instabilities. The more interesting case is when y > 0.
Stripe Phase Instabilities in Lipid Monolayers
J. Phys. Chem., Vol. 99, No. 16, 1995 6253 1
I
-
y
X
a
0
0
1
2
3
4
5
1
2
Figure 1. Critical line for the stripe with one deforming and one straight edge. The deformation is unstable in the region above the line.
I
-
y
X
2.5 n
3" \
g
2 1.5 -
-2 -1
0
+2 +3
+1
+4 +5
W
c
b
1 .
Figure 3. Stripe phase: (a) undeformed and (b) deformed.
0.5 . I
0
1
2
I
I
3
4
I s
2
Figure 2. Critical lines for the stipe with two deforming edges, for five different values of the phase difference 8: 0 (a), n/8(b), n/4 (c), 3n/8(d), and n/2(e).
In that case, the quadratic form will be positive definite and the stripe stable only if two conditions are met:
must have equal amplitude; we therefore call this a buckling instability. With increasing stripe width, shorter wavelength deformations become unstable and can be increasingly out of phase. Finally, when w exceeds w, = eCwo,both edges become separately unstable, and no simple amplitude or phase relationship between the deformations is to be expected.
Instability of the Stripe Phase We now turn to the stripe phase, which can be thought of as an infinite array of equally spaced, equally wide stripes (Figure 3). The ratio of the stripe width w over the spacing D is simply related to the area ratio of the two phases
w/D = q5 = A,/(A, The second condition is stricter than the first and is sufficient. The coupling between both edges therefore induces an instability as soon as
This condition is shown graphically in Figure 2, where the axes are labeled exactly as in Figure 1. For a given 8, the instability region is the region above the critical curve corresponding to that 8. The global stability region is the region below the lowest critical curve. As before, this plot demonstrates important physical properties. Instability sets in at the lowest point in this graph, Le., the point z = 0 on the curve for 8 = 0. Using the expansion
1/2zIn 2
+ O(z2) (26)
we see that at this point, w = WO. Thin stripes, with width w < WO,are stable. When the stripe width exceeds WO,the stripe becomes unstable for deformations with k = 0. Since the critical point for this instability lies on the curve for 8 = 0, this instability sets in when both deformations are completely in phase. A simple analysis shows that both deformations also
+ A2)
(27)
where A1 and A2 are the total areas of the first and second phases, respectively. The total energy of the stripe phase is a minimum when*
For C#J = l/2, this gives the equilibrium width quoted in eq 3. The stability analysis of this stripe phase will be limited to the special case when all edge deformations have the same wavelength, the same amplitude, and the same phase, as shown in Figure 3b. This is the easiest case to describe and analyze mathematically. Plausibly, it is also the case of greatest physical interest: as we saw in the previous paragraph, it is precisely for this kind of coupled deformation that a single stripe first becomes unstable. To make the mathematical analysis more definite, we choose one edge as our reference and number the other edges starting from zero for the reference edge. Edges on one side of the reference edge will therefore have positive numbers, and edges on the other side will have negative numbers (see Figure 3b). The profiles of the deforming edges are then given by
yh = A sin kx for even-numbered edges, and
+ nD
(29)
6254 J. Phys. Chem., Vol. 99, No. 16, I995
y2n+l= A sin kx
de Koker et al.
+ nD + w
(30)
for odd-numbered edges. Note that n goes through the entire range of negative and positive integers. The total free energy is calculated as follows. The reference edge contributes a line tension energy
In w/wo> g(z,4) or, using weqinstead of
(39)
WO,
+
In w/weq > g(z,@) In (sin z@/m$)
(40)
g(z,@)can be derived directly from eqs 36 and 37 above. As
shown in Appendix B, substitution of the integral representations of the Bessel functions leads to more convenient expressions for g(z,@),namely, and an electrostatic self-energy g(z9
I
12)
=
C - 1/2 - In 2
as before. Furthermore, the edge interacts with all other edges. The interaction energy with an even-numbered edge, 2n, is given by
E
1
' = - /+2Z'(A,A,k,B = 0,lnlD)
(33)
where I' is the integral calculated in Appendix A. The 1, gets an interaction with an odd-numbered edge, 2n additional minus sign because of the different orientation of such edges (see Figure 3):
n? - 4 h m d t s i h 2 t + lnz + 7 exp(z cosh t) + 1 32 (41)
when 4 =
lI2,
and
g(z,4) = c - 1/2 - In 2
+ In z - 7Z2 +
+
E :ir+l ' = 1/#2Z'(A+4,k,B = 0,lnD
+ wl)
(34)
In total, therefore, edge 0 contributes an interaction energy +m
m
n=l m
Z'(nD - w)- 2Z'(nD)]} (35) where some of the arguments of integral I ' have been omitted. Since there is nothing special about the reference edge, this is simply the interaction energy per edge. Substituting I ' from eq A.25 and adding the different energy contribution gives a total energy
E,,' = '/&'A2k2[ In
kA eCf1/2 eAlp2
+- n? +
2
3k2w2
4i(-1)nF] n=l nkw when 4 =
(36)
(black and white stripes equally wide), and
P2A2k2[1nkA Eto: = 4
e
2
Kl(kw)
k2w2
kw
+-2f#J2 k2w2
- 411 2K1(nkD)]j + K,[kD(n (37) n-4 n
in the general case (see Appendix B for details). The stability condition 4 0 :
'0
can again be expressed in the form
cosh(z cash t ) - 1 4Jm dt sinh2 t (42)
(38)
in the general case. The critical lines for these instabilities are shown in Figure 4 for various values of 4. Note that, again, instabilities set in first for long wavelength deformations (k = 0) and, remarkably enough, for a stripe phase where the stripes have their equilibrium width wq. When w = wq,the deformation energy is given by
Eta: = '/41U2A2k2[g(z,4)+ ln(sin n4/n4)]
when w = weq (43)
As can be seen from Figure 4, for very long wavelengths, when
z = kw is very small, we have Etot)a k4
when w = weq
(44)
The absence, at equilibrium, of an energy term proportional to k2 corresponds to what Cebers calls zero effective line ten~i0n.l~ Stripes of width smaller than weqare stable with respect to harmonic deformations. Stripes of width greater than wq are unstable for deformations with a range of wavelengths. In principle, the minimum wavelength for deformation for a given w > weqand a given 4 can be determined from this plot and from eqs 40-42.
Discussion The present work was motivated by the observation in this and other laboratories that the stripe phase in lipid monolayers is never a series of strictly parallel straight stripes. The stripes are invariably serpentine. However, it has not been obvious to us whether this serpentine pattern is due to artifacts of sample preparation or to thermal excitations or is fundamental to the energetics of the system. The early work of Cebers and Maiorov5on the instabilities of a single isolated stripe suggested the last possibility, namely that the idealized stripe phase is fundamentally unstable. The present work shows that this is indeed the case, but of course does not go beyond a treatment of small distortions from parallel straight stripes. On the other hand, the observed deviations from parallel straight stripes are often not large. Comparison between theory and experiment
J. Phys. Chem., Vol. 99, No. 16, 1995 6255
Stripe Phase Instabilities in Lipid Monolayers 0.7 I
I = Io
I
0.6
+ I , + I, + z3 + z4
with
1
0.5
Io = f 3-
h 2
L
+ w2 + A2 A,A,k,k, cos k,x, cos(kg, + e) I/(xl - xJ2 + w 2+ A2 I/(xl - x2),
0.4
\
3 v
-c
0.3
I, =
0.2 0.1
s-”,
h 2
hlJ-:
1, =
[A, sin k,x, - A, sin(kg2
0 0
0.5
1
2.5
2
1.5
3
-92S_LL h l fL
3.5
h 2
Figure 4. Critical lines for the stripe phase, for different values of $J = WID.
then requires that one determine the dynamics of shape changes in the stripe phases, so as to show that a distorted stripe shape would return to parallel straight edges in a reasonably short time if the straight stripes represented the more stable phase. This comparison will be made in a subsequent publication. Acknowledgment. We are indebted to Andrejs Cebers for correspondence and for preprints of his work related to stripe phase instabilities. This work was supported by the National Science Foundation, Grant MCB93 16256. W.J. was supported by the Tonya Foundation. R.D.K. is a Howard Hughes Medical Institute Predoctoral Fellow.
13 = - w f L
y 1 = A, sin k,x y , = A, sin(kg
+ 6)+w
(-4.6)
+ w 2+ A2
+ e)
A, sin k,x, - A, sin(kg2 h, 3 4(xl - x,), w 2 A2
h1fL
z4 = 3/2~2s_L,h,f
+ +
+
[A, sin k,x, - A, sin(kg, @I2 h2 5 I/(xl - x2)2 w2 A2
L
+ +
In these integrals, L is introduced as a length cutoff. The integrals of physical interest are the values per unit length when L goes to infinity. Since l o is just the integral when the edges are undeformed, we begin with I,; much of wat we learn from the evaluation of this integral can then be readily applied to the other integrals. A change of variables
Appendix A
In this appendix, we calculate the basic double line integral that underlies all the calculations.12 Consider two parallel edges, each of length h = 2L, and a distance w apart. Let them undergo harmonic distortions with amplitudes A1 and A?, wavenumbers kl and k2, and a relative phase difference 8. The equations for these edges are then given by
+ @I2 3
J(xl - x2),
Z
y=x1+x2 u = x l -x,
(A.7)
with Jacobian
(A. 1) gives
The cross integral
I , = ‘ / ~ , A , k , k 2dy~ d~u x
+
COS(K1l;l
then becomes
+ e) + c o s ( K , +~ J K Z 2
K2U
K 2 ~
e)
(A.9)
The integration is over the region in Figure 5,and the new wave numbers ~1 and ~2 are
+ A,A,k,k, cos k,x, cos(kg2 + 0) 4(xl - x2), + [A, sin k,x, - A, sin(kg, + 0) - wI2 + A’
K,
1
(A.3) where both edges have been given the same orientation. Obviously, this integral cannot be expressed in standard analytical functions. We therefore expand the integrahd in a power series in A1 and A2, using
z/l+a This gives
(-4.5)
1 - ‘t2a
+
3/8
a2
6 4 )
K,
+
= ‘t2(kl k2) = ‘/,(kl - k2)
(A. 10)
Because of the symmetry of the integration region, this integral further reduces to
I , = A,A,k,k, cos
OS,”
duh2L2L-u dy x
+ COS JZ-E-2
K I U COS K 2 y
COS K1y COS K2U
(A. 11)
To evaluate this, we first set
k, = k2 = K, = k,
K,
=0
(A.12)
de Koker et al.
6256 J. Phys. Chem., Vol. 99, No. 16, 1995
Thus, we have for the sum:
x2
t
I ‘= A,A2k2 K o ( k f ) (
+ k
m
I
)
2 w2- A2 (A. 18) l/2(A12 +A, ) (w2 A2)2
+
which can be further simplified using the property of the Bessel functions (A.19)
Ko(x) - K2(x) = - (2/x)Kl(x)
0=x,-x2
giving Figure 5. Integration region for the double line integral.
J i l l ( k J w ‘ k w2+ A’
In that case, the integration over 7 can be readily performed, and gives 2~ d o sin[k(2L-
I, = A,A2k2cos
OIL 1
*
*
a)] -I
+ A’)
+
kdd+w”+A‘ dacos ka
1+
(A.20)
Note that in all of these integrals, the term in k appears only when Since we are not interest in ZIitself, but rather in the contribution per unit length,
limL-2L
I*
I,’
(A. 14)
k, = k2 = k
Two cases of this integral Z ’are of special interest. First, when both edges coincide, w=O, A , = A 2 = A ,
the calculations are simplified enormously. Indeed, only one term in eq A.13 contributes to the limit, giving
I,’ = AlA2k2J d a
cos k a
47T7T-Z = A,A2k2KO(kJw2 + A2) when k, = k2 = k
#
k2
(A.15)
=
w2
+ A2
- =,A2
COS
0 Jw2
+
K , ( k f ) }
13’ = 0
14)= w2{ A12 +A;
Since the cutoff A is very small, we can use the series expansions for the Bessel functions, and find
J ’ = -‘/fi2k2 In
(w2
+ A2)2
w2+A2
kA 2
(A.24)
-
where C is Euler’s constant, C = 0.577. A second special case is when w >> A. Letting A 0 simplifies the integral to
I‘ =
( k m ) }
A12 +A: 2w2
- A1A2k2COS 0 ~1K , ( k w )
(A.25)
Equations A.24 and A.25 are the integrals we need for the rest of our discussion. The change in self-energy per unit length of a deforming edge is simply
E Elf ’(A,k) = -‘/p2J‘
(A.17)
- A1A2cos OLK
+ K2(kA) - -} (A.23)
(A.16)
The other integrals can be calculated by the same methods:
;z
(A.22)
(W2
*
when k,
O=O
we get J ’ = A2k2{ - GK,(kA) 1
where KO is the modified Bessel function of the second kind and zero order. When ~2 0, I,’ is more complicated. However, it can be seen easily that in this case, 11’ does not contain a term proportional to L. Therefore,
I,’ = 0
(A.21)
(A.26)
The interaction integral between two deforming edges is
E
:y’(a,A1,A2,0,k,w)= -11p2aI’
(A.27)
J. Phys. Chem., Vol. 99, No. 16, 1995 6257
Stripe Phase Instabilities in Lipid Monolayers where u = 4-1 when both edges are oriented parallel, and u = -1 when they are antiparallel.
to calculate numerically. The integral representation
Kl(z)/z= Appendix B
hm e-zcoshrsinh2 t dt
(B.7)
allows one to do the summation at the expense of introducing an integration:
The calculation of the energy of the stripe phase is completed as follows. The cross electrostatic energy follows directly from eq 35 upon substitution of the integral A.25, giving
(B.8) and similarly for the other sums. Thus, the three summations in (B.4) reduce to
hmdr
'
kw cosh f + e-kw cosh f
ekD cosh f
-1
-
2
(B.9)
giving a total cross energy
Using the summations m
l/n2 = d l 6 n=l m
n=O
where is Riemann's function. This can be somewhat simplified to
Analogously, when 9 = l I 2 ,
which, substituted into B.6, gives a cross energy sinh2 t exp(z cosh t )
+1 (B.12)
2K1(nkD)]) n (B.3) In the special case when 4 = I/*, this can be further reduced. Indeed, since
ly '('/,I
+ ly '(-V2)
=d
+4
(B.4)
and
kw(2n - 1) Kl(nz) -2C(-1)"n= 1
nz
Kl(z)
- - (B.5) Z
the cross energy then becomes
""')
(B.6) nz Equations B.3 and B.6, combined with eqs 31 and 32, give the expressions for the total energy quoted before (eqs 36 and 37). The sum of Bessel functions is awkward to write and difficult
These forms for the cross energy B.10 and B.12 can then be substituted into eqs 36 and 37 to give the expressions for g(z,$) quoted above (eqs 41 and 42). 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, 886890. (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. Rev. A 1992, 46 (8). 4894-4904. (5) Cebers, A. 0.;Maiorov, M. M. Magnefohydrodynamics 1980,16, 21-28. Blum, E. Y.; Maiorov, M. M.; Cebers, A. 0. Magnetic Fluids; Riga: Zinatne, 1989 (in Russian). (6) McConnell, H. M.; De Koker, R. J. Phys. Chem. 1992,96,71017103. (7) McConnell, H. M.; Moy, V. T. J. Phys. Chem. 1988. 92, 4520. (8) McConnell, H. M. Proc. Natl. Acad. Sci. U S A . 1989, 86, 34523455. (9) McConnell, H. M. J . Phys. Chem. 1990, 94, 4728. (10) De Koker, R.; McConnell, H. M. J. Phys. Chem. 1993,97, 1341913424. (11) McConnell, H. M. J . Phys. Chem. 1992, 96, 3167. (12) Compare with the approximate treatment in ref 11 above. (13) Cebers, A. 0. On the Elastic Properties of Stripe Structures in Magnetic Fluids, preprint (in Russian). JF943 174+
