J . Phys. Chem. 1993,97, 2941-2945
2941
Dynamics of Domain Shape Relaxation in Langmuir Films M. Sed AT& T Bell Lobotatoties. Muttay Hill, New Jersey 07974 Received: October 21, 1992
The dynamics of domain shape relaxation exhibited by individual domains of finite size have been investigated in a Langmuir film containing the phospholipid dimyristoylphosphatidylcholine (DMPC) and cholesterol. The temporal evolution of shape distortions following a stepped expansion of the film has been monitored by examination of simple geometrical quantities, notably perimeter, enclosed area, and curvature. In addition, spectral shape analysis has been invoked to determine temporal autocorrelation functions for spectral densities representing the primary modes of boundary distortion. This analysis indicates that the mode relaxation is well approximated by an exponential time dependence and that the decay of shape distortions in dipolar fluids of the type represented by Langmuir films is mediated by a monotonic reduction of the domain perimeter, constrained by the condition that the enclosed area remain constant. While these findings would suggest a description of the shape relaxation in terms of an effective line tension, systematic deviations from such an approximation are revealed by quantitative comparison with a pertinent evolution equation describing area-conserving dissipative dynamics of plane curves. These deviations are attributed to the nonlocal action of dipolar forces.
Introduction The spontaneous formation of domain patterns in monomolecular films of amphiphilic molecules confined to an air-water interface has evoked widespread interest,IJ originating, to a significant extent, in the close analogy of this phenomenon with theappearanceof similar patterns in a widevarietyof realizations, including thin films of type I superconductor^,^ ferrofluid~,"-~ and ferrimagnetic garnets.6 Considerable evidence now supports the view that pattern formation in all these and similar systems arises in response to competing interactions. In the Langmuir films of particular interest here, this scenario invokes the presence of repulsive dipolar interactions of long range between identical amphiphilic molecules adsorbed at an air-water interface: the free energy contributed by this interaction is reduced by spatial modulations in the pertinent order parameter field, namely the lateral molecular density in single component films, or, in the case of binary mixtures, the (intralayer) concentration. Within this simple picture, domains are regions of uniform amplitude of theorder parameter, separated fromone another by sharpdomain boundaries or "walls". The preference of the repulsive interaction for continued subdivision of domains is balanced by the energetic cost incurred in the creation of these walls. A new characteristic length scale, the modulation period, is determined by balancing the strength of the repulsiveinteraction, Ap, and the finitedomain wall energy, or line tension, y, and may thus be tuned by altering the ratio Ap/y. This notion of competing interactions has been employed to construct a phasediagram for Langmuir films which contains uniaxially modulated "stripe" and triagonally modulated 'bubble" phases.' In addition tocontrolling the global phase behavior, the presence of competing interactions also affects the stability of the shapes adopted by inidividual domains. For magnetic systems such as thin, magnetic garnet films with uniaxial anisotropy6J as well as thin layers of ferrofluid,4.5 the limits of stability to harmonic distortions have been examined for many simple domain shapes, notably circular bubbles and linear stripes. For Langmuir films, closely approaching a truly two-dimensional dipolar sheet, requisiteanalysisI~~-~~ predicts circular shapes to become unstable to perturbations of 2-fold ("elliptic") and eventually to perturbations of n-fold symmetry at particular values of the ratio Ne 0 (&)2/r. Ne represents the analog of a 'bond number",4J2 defined here in terms of the differential of dipolar densities, A p , , of different regions of the film and in terms of the line tension,
y, associated with the interface between such regions. The existence of an elliptic domain shape instability in Langmuir films has indeed been verified experimentally.13 Whilestaticaspectsofdomain PatternsandshapesinLangmuir films and related systems have been the subject of much recent activity, comparatively little effort has been devoted to the study of dynamic phemomena. Early analysis of equilibrium shape fluctuations, involving small deviations from a circular ground state, indicated a quasi-capillary nature of the interface modes, thereby suggesting the dynamics to be controlled by an effective line tension, yew,and implying a renormalization of the 'bare" line tension, y, to represent the primary effect of dipolar interactions.13J4 Very recently, systematic experiments on the shape relaxation of branched droplets of ferrofluid in a magnetic field have been initiated in combination with theoretical analysis to investigate the evolution of complex interface configurations from simple initial states and to account for the relaxation to the circular state in quantitative detail.I2 A clever experiment probing dynamics in a Langmuir film has been recently reported in which a shear field was employed to distort an initially circular domain into a highly elongated ("bola"I5) shape in an effort to estimate the effective line tension acting to restore the circular state subsequent to eliminating the shear." The focus of the present article is the dissipative dynamics governing the decay of domain shape distortions. A mode analysis of distorted shapes is presented, and temporal autocorrelation functions are evaluated for the predominant modes in the shape spectrum. Of particular interest is the question to what extent the presence of nonlocal interactions affects the dynamics. It is shown here that the relaxation of distorted domains proceeds via the monotonic reduction of thecontour length, L, while conserving the area, A, enclosed by thecontour. In particular, the relaxation of the normalized ratio L2/4uA to its asymptotic value of unity is found to be well described by an exponential time dependence. This process exhibits qualitative features conistent with the evolution of plane contours to a circular shape as described by a 'curve-shortening equation",12 based on the predicate that the free energy functional governing the shape relaxation contains only a local, that is, a shape- and size-independent line tension. However, quantitative comparison with a pertinent equation of motion applicable to this type of area-conserving dynamics reveals systematicdeviationsfrom the simple local behavior. The nonlocal nature of dipolar interactions is examined as a possible source of
0022-3654/93/2091-2941s04.~0/~ Q 1993 American Chemical Society
2942 The Journal of Physical Chemistry, Vol. 97, No. 12, 1993
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VIDEO CAMERA
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KAPTON BARRIER TEFLON TROUGH
ISTAGE Figure 1. Simplified sketch of the experimental setup for video epifluorescence microscopy of monomolecular organic (Langmuir) films confined to an air-water interface. The area enclosed by the movable Kapton barrier is set under remote control via a dc servomotor, equipped with optical encoder, whose axis is aligned with one rhombus diagonal. The area is evaluated from the encoder output and subsequently stored on videotape along with the recorded image as well as additional experimental parameters. The lateral trough position, controlled via a pair of translation stages, omitted here for simplicity, is adjusted to center the rhombohedral barrier with respect to the microscope objective. Fine focusing is achieved by adjusting the subphase volume.
these deviations, and further measurements, suggested by the present analysis, are indicated. Microscopy and Image Analysis of Domain Shapes
Experiments were performed on monomolecular films, confined to an air-water interface and composed of a binary mixture of the phospholipid dimyristoylphosphatidylcholine (DMPC) and cholesterol of near-critical composition, 68:30 mol %, also containing 2 mol %of a fluorescent lipid analog whose preferential partitioning renders visible a domain pattern morphology typical of this type of Langmuir film.I6J7 Domain patterns were visualized via epifluorescence microscopy in an experimental arrangement sketched in Figure 1. Images were stored on videotape and disk for subsequent analysis. The analysis performed here to extract a set of shape descriptors of momentary domain boundary configurations followed the general prescription detailed elsewhere,18 employing a sequence of preprocessing steps to convert the original into a binary image on which the actual shape analysis was performed. The first level of processing entailed contrast enhancement via histogram manipulation, followed by low-pass filtering to remove highfrequency noise in the position of thedomain boundary to prepare the image for binarization. The subsequent shape analysis relied on the detection and efficient representation of the contour marking of domain boundary of interest. The evaluation of shape descriptors was based on an explicit polygonal representation of the contour in the form of a sequence of vertices, C=(vk ( x k ~ k ) 1; 5 k I N). Specifically,geometrical quantities were evaluated, including contour length, L,enclosed area, A, and local curvature, a. The enclosed area is given in form of the moment mm of order zero of the contour C. The
Seul curvature is defined as the inverse of the radius of curvature, &. I?&) represents the radius of the circle passing through three consecutive vertices vk-1, vk, and vk+l and thus coincides with the osculating circle19at vk of radius R = d(vk-l,vk+l)/(2sin 4); here, d(,) represents Euclidean distance and 4 denotes the angle subtended by the polygon edges e(vk-l,vk) and e(vk,vk+l). To determine a2 1/R2, sin2 4 is conveniently calculated from the cross product of the vectors v~vk-1and vk+l-vk. The total curvature energy, E,, stored in a contour of given configuration represents a global shape descriptori8which is of particular interest here. For a continuous contour, E, may be defined in the form E, f&sx2, yielding E, = 2 r / R for a circle of radius R. For a polygonal contour, E, is readily evaluated in the form E, = x;I:-' a2(vk)As,where As = I/2(d(vk,vk-1)+ d(vk,vk+l)) and N = 64 for the data reported here. In addition to the examination of geometrical attributes of a given contour, a spectral analysis was performed to obtain a set of spectral densities c,, la,,l2,representing the deviation of a momentary contour configurationfrom a circular reference shape. This reference shape is defined as the circle of radius po = (mm/a)*I2,enclosing the same area as the contour of interest. The mode analysis was based on a radial parametrization (6p*(v& 1 Ik I N), given in terms of the distance between the centroid c (xclyc)and vertices vk = (x&, corrected by the radius of the reference circle:I8
The time evolution of modes contributing significantly to shape distortionswas monitored and served as the basis for theevaluation of (normalized) temporal autocorrelation functions capturing details of the shape relaxation dynamics.'* Dynamics of Shape Relaxation
The room-temperature phase diagram of the DMPC4holesterol binary mixture, discussed in detail elsewhere," contains a critical consolute point which controls the balance of the local, bare line tension, 7, and the nonlocal, repulsive dipolar interactions, thus permitting the experimental adjustment of the bond number, NB.Dipolar interactions predominate as the consolute point is approached along the pressurearea isotherm of mixed films of near-critical composition. They act to destabilize the boundary of circular domains, eventually favoring the appearance of labyrinthine configurati~ns,~~J~ in close analogy to the instabilityexhibited by circular4ylindrical domains in thin films of ferrofluid in a magnetic field?v5J2 Highly distorted,"branched", initial shapes may thus be generated by carefully approaching the consolute region via film compression. Sudden subsequent expansion affords a stepped decrease in NBand induces a slow relaxation of the distorted domain to a circular state. Figure 2 illustrates the shape evolution induced by such a stepped expansion of a film which had been compressed beyond the point of a branching instability. As may be ascertained, the initial state is highly distorted, with a predominating 3-fold symmetric configuration. The relaxation of this distortion to a circular shape is conveniently characterized by tracking the temporal evolution of contour length, L,and enclosed area, A, shown in the upper two panels of Figure 3. This exercise reveals that A remains essentially constant, varying by no more than *2%, a margin limited by the accuracy of the preprocessing steps described above, while the perimeter, L, steadily contracts to approximately 80%of its initialvalue. As a result, the normalized ratio, S L2/4aA,of contour length to area exhibits a monotonic decay to its expected asymptotic value of S = 1.O, indicative of a circular contour. As the fit in the bottom panel of Figure 3 demonstrates, the relaxation of S is well approximated by an exponentialtime dependence, yielding a characteristic relaxation time TS 15 s. This observation implicates an effective line
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Dynamics of Domain Shape Relaxation
The Journal of Physical Chemistry, Vol. 97, No. 12, 1993 2943
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Figure 2. Sequence of snapshots illustrating the shape relaxation of a distorted droplet ('bubble") domain in a Langmuir film, composed of a DMPC-cholesterol binary mixture described in the text, following a steppedexpansionof approximately5%. Images were subjectedto contrast enhancement. Highly distorted shapes such as that displayed in panel A arise as a result of a 'branching" instability of the domain boundary inducing '1abyrinthine"configurations of the emerging branches.I3J4The depicted configurationscorrespond to the following values of the global shape parameter S = L2/4rA: 1.56 (A), 1.38 (B), 1.16 (C), and 1.05 (D). Theelapsed timebetween thesnapshotsis6s [(A)-(B)],9s [(B) (C)], and 21 s [(C) (D)]; the horizontal dimension of each panel is 88 mn.
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Figure 4. Temporal evolution of spectral densities c,, la,,l*,contributing to shape distortions of the domain depicted in Figure 2. The c,, for modes of 2-fold ("elliptic") (0),3-fold (A),and 4-fold (0)rotation symmetry are seen to dominate. 1.2,
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Figure 5. Temporal autocorrelationfunctions,&,(t)
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Figure 3. Temporal evolution of geometrical quantities describing the momentary configuration of the domain boundary in Figure 2 during shape relaxation. The monotonic decay of thecontour length, L (middle panel), under conditions of essentially constant enclosed area, A (top panel), expresses itself in the monotonic decay of the global shape parameter S = L2/4?rAto unity (bottom panel), as expected for shape relaxation to a circular final state. The solid line superimposed on the plot of Svs t is the result of a fit to an exponential,yieldinga characteristic decay time of 14.7 s. The mean area, ( A ) = 2485 pm2, is seen in the top panel to vary by no more than 1 2 % .
tension as the predominant source of restoring force driving the relaxation dynamics of domains in Langumir films.13-l5 A more detailed picture of the relaxation dynamicswhich lends itself to quantitative comparison with desirable theoretical predictions is provided by spectral shape analysis. The results of such an analysis are summarized in Figures 4 and 5. The monotonic decay of the most significant spectral densities in the spectrumof the evolving distorted domain shape is apparent from Figure 4. The predominant initial distortion of 3-fold symmetry ( n = 3), unmistakable in Figure 2A, exhibits a steep initial decay. At times exceeding approximately 30 s, the elliptic mode (n = 2) emerges as the predominant distortion. Its asymptoticapproach
(c,,(t)c,,(O))/(c,,(O)*
c,(O)), of modes of 2-fold and 3-fold rotation symmetry, computed from
the data in Figure 4. The inset shows the final stage of thedecay,covering the time interval beyond the initial 16 s of the relaxation, as indicated by arrows alongthe abscissa. Fits to an exponentialfunction,superimposed on all plots, yield a reasonable parametrization of the decay curves, the final portion of the decay, involving shapes close to the circular final state, being more faithfully reproduced. For the initial portion of the decay, the fit yields relaxation times of 7 2 = 9.3 s and 7 3 = 5.7 s; for the final stage of the decay the corresponding results are 7 2 = 21.5 s and 7 3 = 12.0 s.
to a finitevalue is consistent with continuing fluctuationsof small amplitude about the circular reference state. It is in this regime that equilibrium shape fluctuations of a quasi-capillary nature have been previously examined.13J4 Figure 5 contains autocorrelation functions, &(?) = (cn(t)c,,(O))/(c,,(O)c,,(O)), c,, la,,1*, computed from a sequence of spectral densities, {la,,(Tj)l*;1 5 i 5 T/ATj obtained for modes n = 2 and n = 3 at successive times Ti where AT E Tj+l- Tj and T denotes the total observation time. The initial portion of the decay was monitored adopting a time step of 2 s, while a time step of 5 s was chosen for a separate inspection of the relaxation at times exceeding 15 s, corresponding to a regime in which &,,(t) is reduced to values below 0.5. In the absence of theoretical predictions, fitsto an exponentialfunctional form were performed to parametrize the curves, yielding relaxation times 7 2 and 7 3 of
Seul
2944 The Journal of Physical Chemistry, Vol. 97, No. 12, I993 ( 2 ~ 1 - ~ E1 (i/pix) ~~ -0.24
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x suggested by the data. That is, the constant i,required for the strict validitr of ,a local curve-shortening dynamics, must be replaced by y = y(x). The linear fits diplayed in Figure 6 are based on a somewhat arbitrary, minimal segmentation of the data which does, however, provide a marked improvement over linear fits of the entire range. They are respectively based on the initial stage of the relaxation, corresponding to distorted shapes (S > 1.1) and the late stage approaching the circu!ar final state (S < 1.1): the respective slopes yield values for y of approximately 50 pix2l.s = 17 x 10-8 cm2/s and 15 pix2/s = 5 X 1 t 8cm2/s. Given a (bulk) viscosity, 7,these values for i.provide estimates of an effectiveline tension, yerr = y / v ; for example, using V H ~ O= 0.01 P,one finds Teff= 5 X 1 t I 2N for the late stage of near-circular shapes.
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Discussion The exponential time dependence approximating the decay of the global shape parameter S as well as the relaxation of the 2and 3-fold symmetric modes analyzed in Figure 5 would suggest that a singledecay time, 7 , accounts well for the overalldynamics. An expression for such a characteristic time, 7 , follows from a simple dimensional argument based the discussion of the dimensions of i. in the previous section, yielding 7 R2/? Rzq/y. This expression, which may also be obtained by considering the Laplace pressure associated with a curved interface, implies that the existence of a single characteristic decay time is predicated upon a constant value of the line tension. The implied scaling of T with R2 should be subjected to experimental tests. Similar checks of the size dependence of decay times, characterizing the relaxation of "bolas" formed after the coalescence of a pair of circular droplet domains in a surfaceadsorbed polymer film, were recently employed to conclude that in that system dissipation within the film predominates.21 In addition, it would be desirable to extend the investigation reported here to more highly distorted initial shapes to ascertain whether the deviations from the local dynamics discussed here will become more pronounced, a trend expected on the basis of the theoretical description12and suggested by the nonlinearity displayed by the data in Figure 6. This range of larger values of the parameter S is accessible to careful experimentation. The leading correction of magnetostatic and electrostatic dipolar forces to the line tension, y, associated with the onedimensional boundary of domains in magnetic systems or in Langmuir films is known8J2J2to be local, resulting merely in a downward renormalization of the "bare" value; that is, Teff< y. Linear stability analysis of circular-cylindrical domains of radius R and height h and thus of aspect ratio p 2R/h leads, in the limit of large p appropriate in view of the quasi-two-dimensional nature of Langmuir films, to the expression12 Teff= y{l - l / ~ ( l - C + Inp)NB}where Cdenotes Euler'sconstant. This expression implies that adjustment of the bond number, NB, may in fact produce negative values of Teff. However, it cannot account for the observed deviations from linearity in the curve-shortening dynamics. This conclusion emerges simply from the realization that In p, with p EC: lo4, remains essentially unaffected by an area-conserving distortion of a planar shape and from the assumption that the bond number, once set via a stepped expansion of the films, remains constant. Thevalidity of thelatter condition relies on the presumption that an abrupt density reduction of the order of 596, induced via layer expansion, is equilibrated within the region of the film containing the domain of interest on a time scale which is short compared with the typical shape relaxation times found here. These deliberations suggest that the observed deviation from linearity, discussed in connection with Figure 6, reflects a dependence of i.on the evolving shape itself. Such a dependence would constitute a manifestation of the nonlocal nature of dipolar interactions whose formulation does, after all, require integration
- -
where 4 denotes a normalized line tension of dimension length2/ time. Simple dimensional analysis relates i. to the physical line tension, y,of units dyn g cm/s2via multiplication bya viscosity, v, of units P = 10 g/(cm s): y = yv. A plot of the experimental data for y e dL/df vs those for x = ( ~ X ) ~-/ E, L is displayed in Figure 6. The time derivative was obtained by fitting the data for L = L ( t ) (Figure 3) to an exponential of the form L(t) = u exp(-bt) + L,, u and b denoting fit parameters and L, = L(t--) representing a constant. The resulting representation, not shown here, facilitates the evaluation of the derivative dL/dt = a(-b) exp(-bt) at the desired time points. The equation listed above requires a strictly linear relationship to hold between the quantities x and y, the slope being set by the normalized line tension, so that dy/dx = q. The plot in Figure 6 makes apparent that a linear approximation fails to describe the data over the full range. While a piecewise linear approximation may be constructed, this requires adjustments in the slope to accommmodate the superlinear dependence of y on
Dynamics of Domain Shape Relaxation over the bounding contour of the dipolar domain in question.i.4-6,8-12In the linear stability analysis quoted above, this shape dependecce is found to arise in the form of a logarithmic dependence of y on n, the mode of n-fold symmetry in the spectral decomposition of the contour. Specifically, this dependence implies an increase of the effective line tension with n, so that modes of higher symmetry experience an increasingly higher Teff. The notion of a local line tension loses its traditional meaning in such a situation and, as has been pointed out previou~ly,~~ should be employed with caution. A theory yielding predictions for the mode relaxation on the basis of incorporating the full dipolar kernel is currently not available. To accommodate deviations from strictly local behavior, it may be worthwhile to consider whether the essential features of the dynamics may be reproduced by an equation of motion constructed by a modification of the curve-shortening equation rather than by incorporation of the full nonlocality of the dipolar interaction. Such an approach, if tractable, would be particularly welcome in its application to the analysis of equilibrium shape fluctuations in the regime of large excursion from circularity, i.e., near the branching instability providing the highly distorted initial shapes in the current study. Shape analysis performed on fluctuatingsingledomainsin this regime18s24 suggests the existence of several low-lying distortion modes of similar free energy, favoring transitions between distorted states in a manner reminiscent of a ‘flickering” phenomenon observed in simulations of a simple model of a two-dimensional vesicle.25 The discussion of the linear fits of the data in Figure 6 as well as the quality of the exponential approximation to the mode relaxation suggests that a local effective line tension and the single characteristic decay time it implies represent useful, if approximate, concepts, as long as consideration is restricted to limited regimes in the degree of shape distortion, measured, for example, by the parameter S. That is, the notion of a strictly local energy functional to describe shape and dynamicsof domains in Langmuir films may be viable, as long as shapes sample only a limited set of distortions, such as those in the regime of nearcircular shapes identified in Figure 6, over which y remains essentially constant. In general, however, deviations from this behavior must be expected.
The Journal of Physical Chemistry, Vol. 97, No. 12, 1993 2945
Acknowledgment. The work described here has benefited from suggestions by R. Goldstein. It is a particular pleasure to acknowledge extended and continuing exchanges with H.M. McConnell concerning Langmuir films and other topics.
References rod Notes (1) McConnell, H. M. Annu. Reu. Phys. Chem. 1991,42, 171. ( 2 ) Mdhlwald, H. Annu. Rev. Phys. Chem. 1990, 41, 441. (3) Faber, T. E. Proc. R. Soc. London 1958, A248,460. (4) Rosensweig, R. E. Ferrohydrodynamics; Cambridge University Press: Cambridge, UK, 1987. Rosensweig. R. E.;Zahn, M.; Shumovich, R. J. J. Magn. Magn. Mater. 1983,39, 127. (5) Tsebers, A. 0.; Maiorov, M. M. Magnetohydrodynamics 1980, 16, 21. Tsebers, A. 0.Ibid. 1981, 17, 113. (6) Cape, J. A.; Lehman, G. W. J. Appl. Phys. 1971, 42, 5732. (7) Andelman, D.; Brochard, F.; Joanny, J.-F. J. Chem. Phys. 1987,86. 3673. (8) Thiele, A. A. J. Appl. Phys. 1970,41, 1139; Bell l a b . Tech. 1.1969, 48, 3287. (9) Kellcr, D. J.; Korb, J. P.; McConnell, H. M. J. Phys. Chem. 1987, 91,6417. McConnell, H. M.; Moy, V. T. J. Phys. Chem. 1988, 92, 5233. McConnell, H. M. J. Phys. Chem. 1990, 94,4728. (IO) Vanderlick, T. K.;Mdhlwald, H. J. Phys. Chem. 1990, 94, 886. (11) Deutch, J. M.; Low, F. E. J . Phys. Chem. 1992, 96, 7097. (12) Langer, S.A.; Goldstein, R. E.; Jackson, D. P. Phys. Reo. A 1992, 46, 4894. (13) Seul, M.; Sammmon, M. J. Phys. Rev. Lort. 1990, 64, 1903. (14) Sed, M. Physica 1990, A168, 198. Seul, M. In Macromolecular Liquids; Mater. Res. Soc. Symp. Proc. 177; Safinya, C. R., Safran. S.A., Pincus. P. A.. Eds.; Materials Research Society: Pittsburgh, 1990. (15) Benvegnu, D. J.; McConnell, H. M. J . Phys. Chem. 1992,96,6820. (16) Subramanian, S.;McConnell, H. M. J . Phys. Chem. 1987,91,1715. Rice, P.A.; McConnell, H. M. Proc. Narl. Acad. Sci. U.S.A. 1989,86,6445. (17) Hirshfeld, C. L.; Seul, M. J . Phys. (Paris) 1990, 51, 1537. (18) Scul, M.; Sammon, M. J.; Monar, L. R. Reo. Sci. Instrum. 1991.62, 784. (19) Struik, D. J. Lectures on Classical Differential Geomeiry; Addison-Wesley: Reading, MA, 1961; Chapter 1-4. (20) Seul, M.; McConnell, H. M. J . Phys. 1986,47, 1587. (21) Mann, E.K.; HCnon, S.; LangevhD.; Meunicr, J. J. Phys. (Paris) 1992,112, 1683. (22) Milner, S.A. Unpublished results. (23) Muller, P.; Gallet, F. J. Phys. Chem. 1991, 95, 3257. (24) Seul, M. Unpublished results. (25) Leibler, S.;Singh, R. R. P.; Fisher, M. E. Phys. Reu. &ti. 1987,59, 1989. Fisher, M.E. Physic0 1989, 038, 112. Maggs, A. C.; Leibler, S..; Fisher, M. E.; Camacho, C. J. Phys. Reu. 1990, A42, 691.
