J. Phys. Chem. 1987, 91, 6417-6422 The theoretical results for the relaxation time and for the dielectric increment are plotted as functions of the conductivity of the electrolyte in Figures 2 and 3, together with all the available experimental results for suspensions of polystyrene spheres in KCl electrolyte. The plots correspond to the value D,,, = 0.5 X m2/s (chosen in order to fit the experimental data of Schwan) and two different values of the surface conductivity. The scatter of the experimental points might arise in part from experimental uncertainties. It more likely reflects that the relaxation parameters do not simply depend on R and u, but rather on combinations of these variables.” Figure 2 shows that in the condensed counterion model the surface diffusion only determines the relaxation time in the limit of very high electrolyte conductivities. On the contrary, for low conductivities, the relaxation time is mainly determined by the surface conductivity of the counterion layer. The original results of Schwarz can be made to fit any experimental result for the relaxation time and the dielectric increment by a proper choice of the parameters X and D,,,. Figure 3 shows, on the contrary, that due to bulk diffusion the dielectric increment has an upper bound given by eq 8. Therefore, Schwarz’s model fundamentally cannot account for a large part of the experimental data, 5 ~ 1
Conclusion We have shown that the results originally obtained by Schwarz should be corrected to take into account the diffusion of ions in (17) Grosse, C.; Foster, K. R. J. Phys. Chem. 1987, 91, 3073. (18) Springer, M. M.; Korteweg, A.; Lyklema, J. J . Electroanal. Chem. 1983, 153, 55. (19) Lim, K.-H.; Franses, E. I. J. Colloid Interface Sci. 1986, 110, 201.
6417
the bulk electrolyte. When this is done, it appears that the interpretation of the polarization as a mechanism solely controlled by the surface diffusion of counterions only applies to large particles in high conductivity electrolytes. In many other cases of interest, the tightly bound counterions essentially behave as a conducting layer so that the counterion polarization reduces to a simple capacitive effect. The relaxation time is then mainly dependent on the surface conductivity while the dielectric increment is determined by the conductivity of the electrolyte. Our results formally resemble those of Shilov and Dukhin: our eq 6 is identical with eq 26 in ref 3, except for the factor ,E which is missing in this last expression. Because of this difference, the dielectric increment calculated in ref 3 was more than an order of magnitude lower than the experimental results of Schwan.l Figure 3 shows, on the contrary, reasonably good agreement between the corrected Schwarz theory and part of the experimental value^,^*^**'^ showing that a layer of tightly bound counterions which do not exchange with the bulk electrolyte can produce a significant contribution to the low-frequency relaxation. However, the dielectric increments predicted by the theory are substantially below other measured value^.^^^'^*'^ This difference is particularly significant since eq 8, which represents the upper limit of the corrected Schwarz result, has no adjustable parameters. It follows that, a t least for these systems, Schwarz’s model does not adequately represent the behavior of the counterions. A more appropriate simple model is presented in ref 17 that might be adequate to interpret experimental data. More rigorous (and far more complex) models are reviewed in ref 11.
Acknowledgment. We thank Professor H. P. Schwan for numerous discussions and suggestions about this work. This work was partially supported by Office of Naval Research Contract NO0014-86-K-0240.
Theory of Shape Transitions in Two-Dlmenslonal Phospholipid Domains D. J. Keller, J. P. Korb; and H. M. McConnell* Stauffer Laboratory for Physical Chemistry, Stanford University, Stanford, California 94305 (Received: January 26, 1987; In Final Form: June 4, 1987)
A theory is presented in which the noncircular shapes of two-dimensional solid domains of phospholipid are determined by a competition between repulsive electrostatic forces (which favor elongation of the domains) and interfacial line tension (which favors round domains). A general argument is given that, in the absence of domain fission, the crossover from circular to noncircular shapes must occur in a sharp, second-order “shape transition”. An explicit calculation is given for the special case where the noncircular shape is elliptical. When the electrostatic forces are due entirely to free charges, or perpendicular polarization, it is found that the shape transition can be described quantitatively.
I. Introduction Pure, single-component, phospholipid monolayers a t the airwater interface can exist in (at least) three two-dimensional phases: a highly expanded “gas” phase at low pressure, an intermediate fluid phase at higher pressures, and finally a dense solid phase. In passing from the fluid to the solid phase, there is a region of two-phase coexistence where domains of solid lipid emerge and grow from the fluid Perhaps the most striking feature of the solid domains is their unusual ~ h a p e . ~ When -~ a small amount of cholesterol is added to a monolayer of dipalmitoylphosphatidylcholine (DPPC) or dimyristoylphosphatidic acid (DMPA) under certain conditions of temperature, pH, and ionic strength, it is found that the solid domains form long thin strips of uniform width. The width of the strips changes reversibly as Laboratoire de Physique de la Matiere Condensee, Ecole Polytechnique, 91 128-Palaiseau. France.
0022-365418712091-6417$01 SO10
the monolayer is compressed or the temperature is lowered and the domains increase in size. When the lipid molecules are chiral the striplike domains curl and form spirals, with the sense of the curling determined by the handedness of the lipid.5 (1) Losche, M.; Sackmann, E.; Mohwald, H. Eer. Bunsen-Ges. Phys. Chem. 1983, 87, 848-852. (2) Peters, R.; Beck, K. Proc. Natl. Acad. Sci. U.S.A. 1983, 80,
-
71R1-71R7 . - -- . - . .
(3) McConnell, H. M.; Tamm, L. K.; Weis, R. M. Proc. Narl. Acad. Sei. U.S.A. 1984 81, 3249-3253. (4) Weis, R. M.; McConnell, H. M. Nature (London) 1984,310.47-49, (5) Weis, R. M.; McConnell, H. M. J . Phys. Chem. 1985,89,4453-4459. ( 6 ) Gaub, H. E.; Moy, V. T.; McConnell, H. M. J . Phys. Chem. 1986,90, 1721-1725. (7) Moy, V. T.; Keller, D. J.; Gaub, H. E.; McConnell, H. M. J . Phys. Chem. 1986, 90, 3198-3202. (8) Moy, V. T.; Keller, D. J.; McConnell, H. M. J . Phys. Chem., to be published.
0 1987 American Chemical Society
6418 The Journal of Physical Chemistry, Vol. 91, No. 25, 1987
There is evidence suggesting an electrostatic origin for the elongation of the domains. When viewed through a microscope, the domains visibly repel each other over distances of tens of micrometers suggesting that the domains have either a net charge or a perpendicular dipole m ~ m e n t Second, .~ if a conducting needle is brought very close to the monolayer surface and an electric field applied, the solid domains can be either attracted to or repelled from the area under the needle tip.' Finally, it has been found that when a phospholipid with a charged head group like DMPA is used, the elongation of the domains depends on the charge of the head group and the ionic strength of the aqueous subphase.I0 In a previous paper we showed that the thermodynamically controlled widths of highly elongated striplike domains could be accounted for by a combination of electrostatic repulsions (which favor elongation and narrowing) and interfacial line tension along the boundaries (which favors shortening and widening)." Related studies have been camed out by Andelman et al., who have treated the striplike phases as well as other phases of polarized monol a y e r ~ . ' ~In, ~the ~ present paper we consider domains which are nearly round, Le., at the early stages of elongation into strips. The domains are taken to have either a net charge or a permanent polarization. As before, the electrostatic forces are opposed by line tension at the boundaries of the domain. Domains with a net free charge can occur in two situations: (1) when the system as a whole has a net charge, Le., has a nonzero potential with respect to ground; and (2) when the domains are small compared to the Debye screening length for the subphase, so that countercharges in the subphase are far away. The first case can be achieved experimentally, and the second case may occur in the very early stages of solidification, when solid nuclei are very small. We find that when in-plane components of the polarization are absent the domain will either break into pieces or there will be a second-order "shape transition" from perfectly circular domains to noncircular shapes. The transition occurs as a function of the area of the domains, with only circular domains occurring below the critical area and noncircular domains occurring above the critical area. In the present work we neglect electrostatic interactions between separate solid domains. Such interactions should be taken into account when the fraction of solid is large.I1J2 11. The Shape Transition in the General Case
In the absence of any charge or polarization the minimum energy domain shape is determined entirely by line tension and is circular for an isotropic solid. In the other extreme, when the charge or polarization is large and the line tension is weak, and if the domain does not break into pieces, the minimum energy shape is highly extended and striplike. Somewhere between these two extremes the domain crosses over from a circular to a noncircular shape. In this section we present a general argument that in the absence of in-plane components of the polarization the crossover from circular to noncircular shapes occurs in a sharp, second-order transition. The electrostatic energy of a uniformly charged two-dimensional plate of any shape can be written E,,, = j / , p 2 J 1 ( 1 / R )
da da'
(1)
where p is the area charge density, R = 17 - 71 is the distance between two points inside the plate, and the integrations are over the area of the plate. By rescaling all the variables this can be written in the form E, = )/zp2A3I2f (2) where A is the domain area and f is a function which depends on the shape of the domain. Likewise the electrostatic energy of (9) Miller, A.; Mohwald, H. Europhys. Lett. 1986, 2, 67-74. (10) HecW, W. M.; Mohwald, H. Ber. Bumen-Ges. Phys. Chem. 1986,90, 1159-1 163. (11) Keller, D. J.; McConnell, H. M.; Moy, V. T. J . Phys. Chem. 1986, 90, 2311-231s. (12) Andelman, D.; Brochard, F.; de Gennes, P. G.; Joanny, J. F. C. R. Acad. Sci. Paris, Ser. C 1985, 301, 615-618. (13) Andelman, D.; Brochard, F.; Joanny, J. F. J . Chem. Phys. 1987,86, 3673-3681.
Keller et al.
r=Fb
A = In b l a
Figure 1. Elliptical domain showing definition of parameters. The same quantities are used to parametrize any shape with two orthogonal planes of symmetry.
a plate with uniform vertical polarization can be written in the form Ed =
[
y2p2s
R3 + 4"63(7)] 3 da da' (3)
where p is the polarization (dipole per unit area) and t is a small distance (essentially the thickness of the monolayer). Next we assume that the shape of the domain as it first deviates from a circle is ovoid, that is, has two planes of symmetry at right angles to each other like an ellipse. Let the family of minimum energy shapes be parameterized by b/a, the ratio of the width of the ovoid to the length (see Figure 1). The functions f and g are then functions of b/a. Now let A = In ( b l a ) . Since the ratios b / a and a / b describe the same shape (except for a 90° rotation), f and g must be even functions of A: f(A) =f(-A) g ( 4 = g(-A) (4) Finally, we takefand g to be monotonic decreasing in A for small A. This can be shown to be true for elliptical shapes (see section 111), and so must also be true for the minimum energy shape, whether or not it is elliptical. For A near-zero, therefore f(A) fo -f2A2 + f4A4 + ... g(A)
N
go - g2A2
+ g4A4 + ...
(5)
wheref,, f2, go, and g2are positive constants, but f4 and g4may be either positive or negative. These constants may in general depend on the domain area, A. We now turn to the boundary contribution to the energy Eb = X p (6) where X is the line tension and p is the perimeter length. If the domain increases in area without changing shape, the perimeter, being a length, must scale like A'/2. Also p must be even and monotonic increasing in A. Therefore p N A'12(po + p2A2 p4A4 ...) (7)
+
+
where poand p2are positive and p4may be of either sign. In what follows we assume that po and p2 are positive (since a circle has positive perimeter and is a shape of minimum perimeter for given area), and p4 may be of either sign. The total energy of the domain is now the sum of the boundary energy, Eb, and either the monopole or dipole electrostatic interaction energy E = eo e2A2+ e4A4 + ... (8) where eo = )/zp2A3/2f,+ XA'12po or '/zw2A1I2go+ XA1/2po
+
e2 = e4 =
+ XA1I2p2 or y2p2A3/2f,+ XA'12p4 or
-!/212A'/2g2
+ XA'l2p2
+
' / 2 ~ ~ A ' / ~ g 4XA'12p4
The equilibrium shape is determined by minimizing E: dE/dA = 0 = 2e2A + 4e4A3
(9)
(10)
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 6419
Shape Transitions in 2-D Phospholipid Domains
V
There are two physically unique roots: A=O A = &[-e2/(2e4)]1/2
I (11)
Which of these two roots applies for a given choice of A , p , p, and X depends on the sign of e2 and e4. We note that as A goes to infinity, the perimeter and hence the line tension energy also goes to infinity. On the other hand, the electrostatic energy, whether due to dipoles or monopoles, goes asymptotically to zero as A goes to infinity. Therefore, the line tension energy must always eventually overcome the electrostatic energy, and the total E(A) will always be an increasing function at large enough A. It is therefore reasonable to take e4 greater than zero. If e4 is negative, it will always be possible to extend the expansion to higher powers of A2 until a positive highest order coefficient is obtained. For A near zero, however, whether E(A) increases or decreases depends on the sign of the coefficient of the lowest order term, e2. For those values of A , p , or p where ez is positive, E(A) is increasing near zero. The physically significant root is then A = 0; Le., the domain is a perfect circle. But as A, p , or p increases and e2 goes negative, E(A) becomes a decreasing function near zero. This means that the minimum energy occurs for nonzero values of A, and the physical root is A = *[-e2/(2e4)]1/2 (A = 0 is a local maximum in this case) (see Figure 3). Since both solutions are equal at the transition point, the transition is second order. From the definition of e2 it can be seen that the transition point, where e2 crosses from positive to negative and the domain changes from circular to noncircular, occurs when P2A -
= -2P2
U
Figure 2. Definition of variables used in calculating the electrostatic energy of a charged ellipse. The coordinates ro and &, define the position of a point within the ellipse which interacts with all other points.
to points on the boundary along the direction given by 4. By the law of cosines, these distances are the roots of the equation 1 = ro2 r2 2rro cos (4 - +o) (17)
+ +
from which we find rl r2 = 2[1 - ro2sin2 (4 - 40)]1/2
+
(18) The total electrostatic interaction energy for a charged ellipse is therefore 1 E,,, = jp2(ab)’ X
f2
[ I - ro2sinZ(4 - 40)]1/2
for a charged domain and
[a2sin2 4
+ b2 cos2 4]1/2
for a polarized domain. For small A , small p , small p, or large X, the domain is circular. For large A , p , or p, or small A, the domain becomes noncircular. 111. The Shape Transition for Elliptical Domains In the preceding section we argued that a charged or polarized domain undergoes a second-order shape transition from circular to noncircular shapes, without specifying the nature of the minimum-energy shape. In this section we consider the noncircular shapes to be elliptical and calculate the electrostatic energy as a function of A. For a charged elliptical plate, we need to find the integral
E,,, = Y $ 2 L L ( 1 / R ) da da’ where the region of integration is an ellipse. An ellipse may be formed by starting with a circle and then “stretching” the circle along one direction. Accordingly, we choose the variables u = x/a
v =y / b
(15) where a and b are the primary radii of the ellipse. In the u,v coordinates the ellipse appears as a unit circle. It will therefore be convenient to use the polar coordinates, u = r cos 4, u = r sin 4. Consider one point located at ro,40 inside the circle, which interacts with the rest of the circle. (See Figure 2.) This interaction energy is
16 1 = Tp2(ab)2i;K(k)
where K ( k ) is the complete elliptical integral of the first kind and k = ( b 2 / a 2 )- 1 The electrostatic energy of a polarized ellipse could in principle be calculated by an integration similar to the one above with l/R3 in place of 1/R, but we have found this to be difficult. Instead we have exploited an analogy between the electric field of a dipole layer and the magnetic field of a plane current loop. Let A be the area of the polarized domain and C be the contour bounding the domain. Then the electrostatic energy of a vertically polarized 2-D region is (see Appendix)
where t is the thickness of the dipole layer, p is the dipole density (dipole moment per unit area), and X and X‘are points on the boundary. For an ellipse the boundary points are given by 2 = a cos 42 + b sin 4 9 where the angle 4 has the same interpretation as in the case of the charged ellipse above. After some algebra the integral in (20) can be written
cosh A cos A 4 sin A4/2[cosh A
where rl and r2are the radial distances from the point of interest
(19)
+ sinh A cos 26 + sinh A cos 2$]’12 d$ dA4 (21)
6420
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987
Keller et al.
+
where r = (ab)'I2,A+ = - +', 6 = Il2(4+ #), and A = In bla. In terms of r and A the principal radii are a = re-A/2and b = real2. The quantity 6 is a small angle which represents the fact that no two dipoles actually touch each other. Let d be the distance of closest approach between dipoles (Le., approximately the distance between neighboring molecules). Then 6 is related to d by
= r[cosh A
+ sinh A cos 2$I1l26
(22)
The integral over A+ can be performed analytically. The energy is then
+ p2Izr[(cosh
Ed = 2 a t
sinh A cos 24)'/* In 6 / 4
A
+
cosh A + (cosh A +2sinh A cos 2$)'12
1
d+ (23)
We now expand the integrals in (19) and (23) in powers of A
+ -A411 2* Ed = -Ap2 - 2arp2 t
-!21' In where r = by p =
(
+ ...
")A4 e46/3d
+
...I
(25)
Figure 3. Energies of a polarized elliptical domain. The parameters have been chosen so that the domain is past the transition point, Le., the electrostatic energy is dominant near In ( b / a ) = 0, and the total energy has a minimum at nonzero values of A. These curves were calculated by numerical integration to test the validity of the expansions in powers of A.
jl
dipoles,,'
A u
The perimeter length of the ellipse is given
1 161 21
d3
d+ = r J
2%
(cosh A
+ sinh A cos 24)'"
d$
The energy due to line tension is then 0 0
I
1
2 0
AIA,
We now proceed as in section 11. For the charged ellipse the two roots of the minimized energy equation are
A=O A = f[-e2/(2e4)I1I2
[ (,; )/(
= f 96 - - 1
1:
+ 1)]1'2
(28)
where A , = 9aX/(4p2). For the polarized ellipse the roots are A=O
1
In r / r c 'I2 12 - In r / r c where
8 E, = 7 p 2 r 3
In Figure 3 the electrostatic, line tension, and total energies are shown as a function of A for a value of r, slightly past the critical point. These curves were calculated by direct numerical integration of the integrals in eq 19, 23, and 26 to test the validity of the expansions in powers of A. The dashed curves for the separate electrostatic and line tension energies have been shifted along t h e y axis so that they go to zero as A goes to zero. (The
(30)
and for a polarized circular domain is 2iT 4r Ed = -Ap2 - 2ap2r In t e2d Now consider two circular domains of area A , and A2, such that A , + A, = A , at infinite separation. For two charged circular
The Journal of Physical Chemistry, Vol. 91, No. 2.5. 1987 6421
Shape Transitions in 2-D Phospholipid Domains domains the energy (including line tension energy) is E / ( 2 r r X ) e(a,y) = 7[a3/'
+ (1 - u ) ~ /+~ ] + (1 -
(32)
where a = A I / A and y = 4p2?/(3X) By plotting e as a function of a it is easy to see that for small y the minimum energy occurs at a = 0. But as y increases a new local minimum develops at a = The value of y for which this minimum becomes the global minimum is given by e(0,y) = e(1/2,y) which implies A, =
3 ~x --
r
P2
(33)
For A larger than A, the circle will break into two smaller circles of equal size. For two polarized circular domains the energy is E / ( m 2 ) = e(a,r) = -a1l2In (y2a)- ( I
- a)Il2In (y2(1 - a ) )
(34)
where (35) As in the case above, for small values of y the minimum energy is at a = 0, but for larger values a new minimum occurs at a = The fission point is given by
fide2
Yc =
2
where r, is the critical radius, related to the critical area by A, = rr?. The critical areas for these fission processes are smaller than the critical areas for elongation, so in the absence of domain-domain repulsion the physically important transition is fission, not elongation. In experimental monolayers, however, there will be unfavorable repulsion energy between domains. A rough calculation in which the disks are replaced by two point charges or point dipoles indicates that when the domains are close to each other (as they would be just after fission) the unfavorable repulsion energy is large enough to make elongation favored over fission. This suggests that when the domains are confined to a small area (i.e., when the fraction of solid material in the monolayer is high) the repulsion energy should stabilize elongated domains. Also, even when the domains are not confined and could in principle separate infinitely far apart, the process of breaking a single domain in half requires an activation energy related to the line tension.
Figure 5. Elongated gas-phase "bubble" domain in coexistence with smaller circular bubbles in a condensed monolayer of the liquid crystal molecule K18. The bright streak in the upper left is a reflection on the television screen from which this photograph was taken, and is not a feature of the monolayer.
thinning of the domains as the temperature is lowered is then explained as being due to increasing in-plane polarization. We have not included in-plane components of the polarization in our calculations here, but our preliminary calculations indicate that the primary effect of in-plane dipoles is to add an anisotropic term to the effective line tension. This does cause extension of the domain and also smooths out the sharp shape transition described above. If their mechanism is correct, then the nature of the transition from round to extended shapes is quite different from the one calculated in the present paper. Recent experiments with DPPC monolayers have shown that certain fluorescent dyes can give information on changes in molecular orientation when the monolayers are illuminated with polarized light.* If the mechanism suggested by Heckl and Mohwald is correct we would expect the shape transition to be accompanied by a change in the tilt orientation of the lipid molecules, while no such change would be required if the mechanism is of the sort calculated here. It may therefore be possible to distinguish between the two mechanisms by observing DMPA monolayers under polarized illumination during the transition. For polarized domains we found that the critical radius r, for either elongation or breakup depends exponentially on the ratio of the line tension to the squared dipole density; X / p 2 . This dependence is very similar to an earlier result for striplike domains,"*'*where we found that the equilibrium width of a long striplike polarized domain was determined by a combination of line tension and electrostatic repulsion according to the formula
w a eX/P2
V. Conclusions The shape transition predicted here is mechanical in nature. That is, entropy plays no role comparable to that in common thermodynamic phase transitions. In DPPC monolayers no sharp transition from circular to noncircular shapes has yet been observed for solid domains. (Such shape transitions may have been observed in fluid domains of DPPC near a classical thermodynamic critical point. See below.) In most experiments done to date the domain shapes in DPPC monolayers are already noncircular when they first become visible under the microscope. However, in monolayers of DMPA Heckl and Mohwald have observed a sharp shape transition which depends strongly on the pH and ionic strength of the subphase, and on the temperature.'O The head group of DMPA is acidic, and the transition occurs under conditions of high pH, where the acid should be ionized. This suggests that the transition may be of the sort predicted here, with increasing pH or decreasing ionic strength having the effect of increasing the charge density or polarization until the critical point is reached. Heckl and Mohwald have suggested an alternative mechanism in which the shape transition is brought on by the sudden appearance of in-plane components of the polarization, i.e., by an underlying ferroelectric phase transition.'O The elongation and
The chief significanceof these results is that the strong exponential dependence on X explains the high sensitivity of domain shapes to factors which affect X such as the presence of impurities in the monolayer. In recent work Leone et aI.l4 have observed elongated elliptical domains in monolayers formed from the liquid crystal molecule K18 (see Figure 5 ) . These domains are apparently low-density two-dimensional gas-phase "bubbles" in a higher density twodimensional fluid. In these monolayers the elongated domains coexist with many smaller circular domains, and it may be that the elongated domains are metastable structures, prevented from breaking into smaller, lower energy pieces by the need to go through high-energy shapes during the fission process. Also Subramaniam and M~Connell'~ have recently described a binary monolayer mixture of lipids (phosphatidylcholine and cholesterol) that shows liquid-liquid immiscibility and a critical composition such that on compression the two liquid phases merge to a single (14) Leone, A.; Keller, D.; McConnell, H. M., to be published. (15) Subramaniam, S.; McConnell, H. M. J . Phys. Chem. 1987, 91, 1715-1 718.
6422
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987
phase by passing through a critical point. The shape transition discussed in the present work may be useful in describing such systems, where the fluid domains differ in perpendicular dipole density, and where the liquid-liquid line tension approaches zero as the critical point is approached. In some experiments elongated ovoid fluid domains are observed near the consolute critical point whereas circular fluid domains are observed further from the critical point.I5
Acknowledgment. We thank David Andelman for his help and suggestions with respect to the calculations presented here. This work was supported by NSF Grant DMB 83-13770-A1, and by an N I H postdoctoral fellowship to D.J.K. Appendix A domain of arbitrary shape, uniformly filled with vertical electric dipoles, can be thought of as a very thin parallel-plate capacitor of the same shape. The electrostatic energy of this capacitor can be calculated if we know the electric field produced by it everywhere in space:
The task is therefore to find the electric field due to a very thin plane capacitor of arbitrary shape. If the capacitor were infinite in spatial extent, the field would be confined entirely between the plates and would be given by
-
47rP
E, = --
t
i
where p is the dipole density (dipole moment per unit area), f is the thickness of the capacitor, and 2 is a unit vector parallel to the dipoles (and perpendicular to the plane of the capacitor). A finite capacitor will have nonzero fields outside of the region between the plates near its edges. We call these fringe fields. We take the total fie_ld everywhere in space to be a superposition of t_he fringe field Ef and the "internal" field, Ein,which is equal to E, in the space between the plates and zero everywhere else: E = 8, El, ('43)
+
The task is now to-find 8,. The total field E must satisfy the Maxwell equations v.E = 4*p
vxE=o
Keller et al. It is not hard to see that the second Maxwell equation implies
v x E, = 4s/.ld2(2)i
(-47)
E.,=VX;i
With an appropriate choice of gauge (A5) and (A6) then become with solution
where Cis the contour bounding the capacitor. d j is a line element along the contour, and R is the distance from the point on the contour to the point of observation. The fringe field is then
where R is a unit vector X / R . The electrostatic energy may now be computed by using A 1 : 1 U= [18in1222i,.8f l8,1*]dV (A1 1) 87r
-1
+
+
There are three terms in the integrand of (A1 l), corresponding to three contributions to the total energy. The first and third terms are the self-energies of the internal and fringe fields, respectively, and are always positive. The second term arises from the overlap of internal and fringe fields. It is not hard to see [from (AlO), for example] that in the space between the plates of the capacitor the fringe field always points in the direction opposite to the internal field. The second term, therefore, always gives rise to a negative contribution to the energy and is responsible for the elongation effects calculated in t_he m a i t text. By use of the relationships for Ei, and Ef derived above, U can be reduced to
('44)
The first of these equations is satisfied by &, alone, so we have
v.8, = 0
('45)
(A61
where 2 is a point on the boundary of the caeacitor and i is a unit vector tangent to the boundary curve a t X . Equations A5 and A6 show that the fringe electric field, Ef, is identical with the magnetic field caused by a current-carrying wire along the boundary of the capacitor with current Z = c p (where c is the speed o,f light in vacuum). Accordingly, we define a vector potential A such that
Substitution of A9 then gives eq 20 in the main text.
