132. D. Weigent, J. Blalock, R. LeBoeuf, Endocrinol-ogy 128, 2053 (1991).

133. A. Brunetti et al., J. Biol. Chem. 265, 13435(1 990).

134. F. Cope, J. Wille, L. D. Tomei, in (77), pp.125-142.

135. F. Cope and J. Wille, Proc. Nati. Acad. Sci.U.S.A. 86, 5590 (1989).

136. A. Ao et al., Antisense Res. Dev. 1, 1 (1991).137. A. Caceres, S. Potrebic, K. Kosik, J. Neurosci.

11, 1515 (1991).138. E. Sorscher et al., Proc. Nati. Acad. Sci. U.S.A.

88, 7759 (1991).139. M. Listerud, A. Brussaard, P. Devay, D. Colman,

L. Role, Science 254, 1518 (1991).

140. M. Clark et al., Proc. NatI. Acad. Sci. U.S.A. 88,5418 (1991).

141. A. Teichman-Weinber, U. Z. Littauer, I. GinzburgGene 72, 297 (1988).

142. M. Shoemaker et al., J. Cell Biol. 111, 1107(1 990) .

143. G. Thinakaran and J. Bag, Exp. Cell Res. 192,227 (1991).

144. R. Pepperkok et al., ibid. 197, 245 (1991).145. L. Bavisotto et al., J. Exp. Med. 174,1097 (1991).146. P. Porcu et al., MoL. Cell. Biol. 12, 5069 (1992).147. L. Neyses, J. Nouskas, H. Vetter, Biochem. Bid-

phys. Res. Commun. 181, 22 (1991).148. M. Kanbe et al., Anti-Cancer Drug Des. 7, 341

(1992).

149. J. Cook et al., Antisense Res. Dev. 2, 199(1992).

150. A. West and B. Cooke, Mol. Cell. Endocrinol. 79,R9 (1991).

151. A. Witsell and L. Schook, Proc. Nati. Acad. Sci.U.S.A. 89, 4754 (1992).

152. C.A.S. is the Irving Assistant Professor ofMedicine and Pharmacology at Columbia Uni-versity and is grateful to the Mathieson Foun-dation and the Dubose Hayward Foun-dation for support. A. Krieg, J. Reed, and M.Caruthers participated in helpful conversa-tions. The research of Y.-C.C. is supported by agrant from the National Cancer Institute (CA44538).

R RESEARCH AR.T.I.CLE.......

Labyrinthine Pattern Formationin Magnetic Fluids

Akiva J. Dickstein, Shyamsunder Erramilli, Raymond E. Goldstein,*David P. Jackson, Stephen A. Langer

A quasi two-dimensional drop of a magnetic fluid (ferrofluid) in a magnetic field is oneexample of the many systems, including amphiphilic monolayers, thin magnetic films, andtype I superconductors, that form labyrinthine patterns. The formation of the ferrofluidlabyrinth was examined both experimentally and theoretically. Labyrinth formation wasfound to be sensitively dependent on initial conditions, indicative of a space of configu-rations having a vast number of local energy minima. Certain geometric characteristics ofthe labyrinths suggest that these multiple minima have nearly equivalent energies. Kineticeffects on pattern selection were found in studies of fingering in the presence of time-dependent magnetic fields. The dynamics of this pattern formation was studied within asimple model that yields shape evolutions in qualitative agreement with experiment.

Several distinct physical systems formstrikingly similar labyrinthine structures.These include thin magnetic films (1, 2),amphiphilic "Langmuir" monolayers (3-5),and type I superconductors in magneticfields (6). Similarities between the energet-ics of these systems suggest a commonmechanism for pattern formation. In eachcase, the labyrinth is formed by the bound-ary between two thermodynamic phases(oppositely magnetized domains, expandedand condensed dipolar phases, or normaland superconducting regions), and has anassociated surface tension favoring a mini-mum interface length. Each also has long-range dipolar interactions. These may be

A. J. Dickstein, S. Erramilli, R. E. Goldstein, and D. P.Jackson are in the Department of Physics, JosephHenry Laboratories, Princeton University, Princeton,NJ 08544. S. Erramilli and R. E. Goldstein are alsoaffiliated with the Princeton Materials Institute, BowenHall, Princeton University, Princeton, NJ 08540. S. A.Langer is in the Department of Physics, Simon FraserUniversity, Burnaby, British Columbia, V5A 1S6, Can-ada.

*To whom correspondence should be addressed.

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electrostatic, as in amphiphilic m(the dipolar molecules of which andicular to the air-water interface)be due to bulk magnetization (perninduced, as in superconductors). 1teractions are repulsive, favoringtended interface. From a dynamicaview, the evolution of each of thefrom some unstable initial state (the presence of a global constrainprescribed magnetization, constanarea, or fixed magnetic flux, respeceach case, the shape evolution is ainated by dissipation.

It has been recognized for solboth in the context of amphiphilif(7) and superconductors (8), thatpetition between long-range forcesface tension can result in a varietypatterns such as lamellar stripe donhexagonal arrays. The more widelytered irregular, or disordered, patpoorly understood. In analyzirshapes, a number of general questurally arise. (i) Is an observed ti

cnolayerse perpen-1, or mayianent orThese in-g an ex-1 point ofpatterns

occurs init such as,t domaintively. Inalso dom-

me time,c systemsthe com-s and sur-of regularnains andyencoun-tterns areng these:ions nat-ime-inde-

pendent shape a unique energetic groundstate or does the energy functional containmultiple minima? (ii) If the latter, are theminima roughly equivalent in energy? (iii)Might kinetic considerations force a relax-ing system into a metastable minimum in-stead of the true ground state? Such ques-tions are of course not confined to theseparticular examples of pattern formation,but also arise in systems such as spin glasses(9) and protein folding (10).

Motivated by the above-mentioned sim-ilarities among labyrinthine pattern form-ing systems, we have investigated the fin-gering instabilities of macroscopic domainsof magnetic fluids (also known as "ferroflu-ids"), which are colloidal suspensions ofmicroscopic magnetic particles in a hydro-carbon medium (1 1) . Ferrofluids are knownto produce complex labyrinthine patternswhen trapped between closely spaced glassplates (a "Hele-Shaw cell") and subjectedto a magnetic field normal to the plates(1 1-13). Here, as in the systems describedabove, there is a competition between theferrofluid-water surface tension and bulkinduced magnetic dipole interactions. Themotion satisfies a global constraint (fixedfluid volume) and is dominated by viscosity.The macroscopic nature of this system af-fords distinct experimental advantages, in-cluding ease of visualization and direct con-

0 0 X 1

Fig. 1. Stages in the fingering instability of amagnetic fluid drop of initial diameter 2.1 cm, ina field of 87 gauss, as seen from above.

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trol of the strenOur focus is

circular domainplication of a rIiments are comrof dipolar shapemonolayers (14tal answers toabove and desci(15, 16) that reitative features

Existence ofperimental appthat already dethe time evolutfingering instabisee the initialshape evolvebranched strucspreads withoutnectivity. Ram[zero restored ticircular form,sometimes resulThe variability (to the next is jshows shapes prhigh fields. TIaccomplished irthe results of enimately 60 to 1tion of the fieldpatterns was tesdc magnetic fieWhile this pertshape, the tree ]when the oscill

The final trlengths, and co(19). All observn - 2 threef

0ec

0

Fig. 2. Final stateand field strengt&= 75 gauss; (lovgauss. Circles inin the space ofgyration (RG), noinitial circle, shov

igth of dipolar interactions. which branches met at nearly 120°. Theyon the shape evolution of may be classified according to their "topol-

is made unstable by the ap- ogy," the particular connectivity of theiagnetic field. These exper- nodes. We have observed all possible dis-plementary to recent studies tinct topologies for n ' 7, and many topol-e relaxation in amphiphilic ogies for larger n.). We obtained experimen- Equivalence of multiple minima. As isthe three questions raised apparent from the branched patterns in Fig.ribe here theoretical models 2, even within a given topology there was,produce the essential qual- variability of the details of the shape. Nev-of the experiments. ertheless, the trees formed under givenFmultiple minima. The ex- experimental conditions (ramp rate andaratus (17) was similar to final field value) yielded similar values forscribed (18). Snapshots of two very basic geometric quantities: the:ion of a drop undergoing a perimeter L and radius of gyration RG,ility are shown in Fig. 1. We defined as R'=(l/L) i ds[r(s) - rJ2. Hereinstability of the circular r(s) is the boundary of the pattern, s isquickly to a well-defined arc length, and r, (i/L) ; ds r(s) is theture (a "tree") that then center of mass of the boundary. A conve-:any further change in con- nient method for summarizing the shapeping the field slowly back to evolution is to consider its trajectory in thehe shapes to their original L - RG plane, with time varying paramet-although rapid quenching rically (Fig. 2). Evidently, patterns whichIted in fission into droplets. differ in the details of their branching mayof the patterns from one run nevertheless exhibit a high degree of over-illustrated in Fig. 2, which lap, both during the evolution and in theiroduced by ramps to low and final states. We suggest below that thishese ramps generally were overlap indicates that these patterns havei 1 second. The patterns are nearly equivalent energies.volutions lasting for approx- Kinetic effects. We found that the de-50 second after the applica- gree of branching of the initially circularThe local stability of these shape depended not only on the magnitude

ted by superimposing on the of the applied magnetic field, but also onAd a small sinusoidal field. the rate at which that field was ramped tourbed the arm positions and its final value. This pattern selection mayreturned to its original shape be quantified by determining the wave-rating field was removed. length X of the initial instability by exam-ees differed in the shapes, ining videotaped images just after the onsetinnectivity of their branches of the ramp. Although, as a result of tip-ied trees had n free ends and splitting or competition between branches,old coordinated nodes, at the stationary long-time tree and the initial

instability may differ in the number of theirbranches, the initial instability is more

I*' 'I,.' l likely to be understandable within a linearor weakly nonlinear analysis. The initialmode number n = L/X depended on thefield ramp rate a dH/dt, as ax varied overthree orders of magnitude (Fig. 3). An

r t _ ~~~~approximate power-law behavior, n = (dHIdt)* is shown in the inset, along with theprediction i = 0.25, arising from the the-ory described below. The selection occurredlong before the ramping was complete.

Energetics. A theory for this patternformation must address both the energeticsand the dynamics of labyrinthine struc-

I I tures. In designing the simplest model, we5 10 15 assume that the magnetization M = M2

LILO within the domain is uniform (20), butaccount for the energy associated with the

hs (upper left) RO = 0.55 cm, H demagnetizing field, so the magnetic energyver right) RO = 1.1 cm, H = 87 is just that of two sheets of opposite chargeidicate radii of gyration. Paths density a = M z, separated by a height h inperimeter (L) and radius of tne z direction, and boundei by the curve

)rmalized by the values for the r(s) in the xy plane. It is thus equivalent inwing high degree of overlap. form to the energy of a parallel plate capac-

itor, a model used to describe dipolar am-phiphilic domains. The field energy may beexpressed (15) as a pair interaction betweenthe local unit tangent vectors t(s) of thecurve. Apart from a term proportional tothe slab volume, the total energy is,9o[r]=

yL - M2h;dsids't(s) . t(s')(D(R/h) (1)where y is the line tension, R = IRI = Ir(s)- r s' I, and ¢(e) = sinh-'(1/e) + g -

+ . The somewhat complicated func-tion cP(R/h), arising from integrations overthe thickness of the slab, decays like 1/R atlarge R, but crosses over to -InR at small R,preventing divergences in the integral andobviating the need for additional smallscale cutoffs (2 1). The form of the magneticinteraction in Eq. 1 is to be anticipatedfrom the usual association between magnet-ic moments and current loops, the scalarproduct reflecting the attraction (repulsion)between parallel (antiparallel) current-car-rying wires.

Just as the existence of long-range forcesin fluids justifies mean or molecular fieldtheories, we expect the magnetic energy tobe less sensitive to details of the spatialarrangement of the interface than to theoverall scale of the pattern, given by RG.The surface energy naturally scales with theperimeter L, so different experimentallyobserved shapes with similar L and RG arelikely to be similar in energy.

Turning now to the details of thebranched patterns, we find a convenientdimensionless measure of the relativestrength of dipolar and surface energies isthe Bond number (11)

NB,= 2M2h2/,y (2)When NBo is small, line tension dominates,and the droplets will be circular. When NBo

40

30

5=.520 So ~ ~ ~ ~20

10e 10

100 101 102 103 104dH/dt (gaussas)

0 1, A0 1000 2000

dH/dt (gaussls)

Fig. 3. Initial mode of instability n as a functionof magnetic field ramp rate, for quenches of a4.9-cm (diameter) circular drop to 265 gauss.Inset shows data on a log-log plot. The slope ofthe line is 1/4, as shown in Eq. 14.

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liskaffi............

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is large, dipole forces dominate, and laby-rinthine patterns develop. The Bond num-ber for electric dipole systems with dipoledensity p, per unit area is obtained bysubstituting p, = Mh in Eq. 2.

To understand the well-defined arm widthseen in the experimental patterns, consider along, thin, rectangular stripe of width w > h,length ( > w, and conserved area A = wl. Ifwe ignore end contributions, the surface en-ergy of the stripe is 2ye. To estimate itsmagnetic energy, we focus on the scalingproperties of the double integral in Eq. 1,approximating the function (D¢(e) by 1/2t for e> 1 and 1/2 for 0 < e < 1. Recognizing thatw replaces h as the cutoff for interactions onopposite sides, we obtain the estimate

-0- 2^y( + 2M2h2f41n(h/w) - 1] (3)Minimizing with respect to w at fixed A,and considering the limit h2/A < 1, we findthe equilibrium width

w ~ he2'/NB (4)Thus the width decreases with increasingfield, as is indeed the case in experiment(Fig. 2). The observed threefold coordina-tion of the nodes in the pattern arises fromthe interplay of the repulsion of the armsand surface tension effects. A fourfold nodeis unstable to the creation of two threefoldnodes with internal angles of 1200.

Dynamics. At the ferrofluid-water inter-face the force that leads to shape relaxationis the pressure difference All arising fromthe line tension and magnetic interactions.That pressure is related to the energy % via

A~~~l=-6~~(5)brfi is the unit outward normal to the inter-face. The simplest dynamical model (15)for the pattern formation is that the shapefollows the path of steepest descent of itsenergy: a "gradient flow" in configurationspace. Here, the dissipation is taken to belocal (as one might find for domain wallmotion in solid-state systems), and

ar - - 8%ia-=raHl=-rfi- (6)at 8r

where F is a kinetic coefficient. The con-straint of constant enclosed area is enforcedwith a Lagrange multiplier p by using aLegendre-transformed energy % = O- pAin Eq. 6. The resulting equation of motionfor this "dissipative dynamics" is

arf. =-YK+2M2+ds' X t(s')

at

X[Vil+ (h/R) - 1 + p[r(s)] (7)where K(S) is the curvature, A = R/R, andplr] is determined by dA/dt = O. The firstterm in Eq. 7 is the familiar Laplace pres-

1014

sure across a curved interface, while thesecond is the Biot-Savart contribution froma current ribbon of finite height h.A more realistic description of the hydro-

dynamics in the Hele-Shaw cell accounts forthe entire flow field through Darcy's law(22), which gives the fluid velocity v interms of gradients of the total pressure HI,

v=-FVIH (8)Here, F = h2/12iq, and q is the fluidviscosity. This law of motion, also applica-ble to flow in porous media, may be derivedby neglecting the inertial terms in theNavier Stokes equation and assuming thatthe dissipation is dominated by viscousinteractions with the plates. The velocity vin Eq. 8 is a two-dimensional, z-averagedquantity, and the pressure jump in Eq. 5becomes a boundary condition applied atthe ferrofluid-water interface. Area conser-vation is automatically obtained from theincompressibility of the flow (23).

These two approaches to the interfaceevolution are complementary; the first, infocusing on the forces and ignoring thenonlocality associated with the fluid incom-pressibility, is computationally simpler,while the second is more faithful to thehydrodynamics but computationally inten-sive. We have studied both in detail andfind their predictions to be qualitativelysimilar, although quantitatively distinct.

Insight into the basic mechanism of thefingering instability comes from a linear sta-bility analysis of a circle of radius Ro (12,15), particularly in the limit of large aspectratio, Ro/h > 1, which is the case for ourexperiments. A perturbation of the radius ofthe circle in the form ~n(t)cos(n6) evolves asadi/at = cni,, where the growth rate con inHele-Shaw flow is given for small n by

rrn [i3n'1 - n2) - M2h2n2ln(n)

(9)The result from dissipative dynamics dif-fers only in the absence of an overall factorof n/RO, reflecting the additional gradientin Eq. 8. Here, the effective line tension is

y = --M2h2[1 -C + In(2Ro/h)]

selection in the ramping experiments followsfrom these results. Let us suppose that the fieldis linear in time, as in our experiments: H =At. Then, for fields below the saturationmagnetization, M will also be linear: M =Xat, where X is the susceptibility. From Eq.11, at different times different modes will begrowing fastest, and the mode which is ob-served at long times will be the mode whichwas growing fast enough and long enough toacquire a sufficient amplitude to suppress theother modes. If the ramp is slow, low-ordermodes grow large before higher ones becomeunstable, and the pattern has few branches. Ina fast ramp, the low modes have not had timeto grow before higher modes, with fastergrowth rates, become unstable and overtakethem.When the wavelength of a mode is small

compared to the perimeter of the circle,and the magnetization has the above-men-tioned time dependence, we find that thedispersion relation is well approximated bythe form

(7n(t) = i3 (X0,)R,~-7Ro)(12)

If there were no explicit time dependence toa, it would be plausible to postulate that thefastest-growing mode would be selected.With time dependence, however, we claimthat mode selection occurs through nonlin-ear effects when the amplitudes reach somethreshold value. The time Tn required for thenth mode to reach the threshold is given by

L(0) [e Jt (t)= constant(13)

--8

a-

-9

(10) Wlwhere C is Euler's constant. Equations 9and 10 show that, as the magnetic field (oraspect ratio) is increased, the circle be-comes unstable (cra > 0) due to an effec-tive negative line tension while the mag-netic forces stabilize the short-wavelengthmodes. Within this approximation, thefastest growing mode n* is given by

n* -R

e - 2/NBO,lhincreasing with increasing field.A qualitative explanation for the mode

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-10

0 5 10 15 20Time

Fig. 4. Shape evolution and energy relaxationfrom the dissipative dynamics formalism, forNBO = 1.12 and Rolh = 10.0. Dashed lineindicates the energy of a perfect circle, solidlines are for three initial conditions weakly per-turbed from a circle. The unbranched structurelies lower in energy.

00,111,1

.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. .............

The selected mode n* will be the one withthe smallest Tn. From Eqs. 12 and 13, wefind that n* scales as a power of the ramprate a,

(14)in fair agreement with the data (see Fig. 3).

The linear analysis is of limited predic-tive ability regarding the final state. Atpresent, an understanding of the nonlinearregime requires numerical study. The mod-els described above allow us to study branchcompetition near the linear regime, as wellas to compare the energies of the patternsobtained at long times. Figure 4 shows threeevolutions from nearby initial conditionsobtained with the use of the dissipativedynamics in Eq. 7. We see that the un-branched tree lies lowest in energy, andthat there is apparently a "vertex energy"associated with each threefold coordinatednode.

The fingering of several branched struc-tures obtained from a conformal mappingapproach (16, 24) to the Hele-Shaw dy-namics is shown in Fig. 5. The overallstructure is in good qualitative accord withthe experimental patterns, displaying awell-defined finger width and the properinternal node structure. We do not expectnonuniformities in the magnetization tomake qualitative changes to the patternformation although there may be quantita-tive effects, such as a lowering of the energybelow that estimated in Eq. 1. An under-standing of these effects remains, however,an open problem. The dynamics also dis-plays sensitive dependence on initial con-ditions, in the sense that nearly identicalstarting configurations (circles perturbedslightly by different sets of modes Q canlead to vastly different final states. Theseinitial conditions can arise experimentallyas a consequence of thermal fluctuations ormaterial inhomogeneities. The precise con-nection between these initial perturbationsand the final branched structures is notcompletely understood. In accord with ex-periment we find that fourfold vertices areunstable toward vertex fission, creating twothreefold vertices. This process bears a re-semblance to topology transitions in soapfroths (25). In numerical simulations, it isoften extremely difficult to distinguish be-tween true local minima and very shallowslopes of the energy functional in the con-figuration space. Thus, we cannot rule outthe possibility that the experimentally ob-

oFig. 5. The theoretical shape evolutions ob-tained from conformal mapping solution to Dar-cy's law dynamics for magnetic fluids. Bothsequences are for R1/h = 10.0 and NBO = 1.20.The top resulted from a ramp in NBO., while thebottom followed an instantaneous jump of theBond number.

served patterns are not true local minima,but are made locally stable by stick-slipfriction against the plates in the Hele-Shawcell or other effects not accounted for with-in Darcy's law.

In conclusion, the branched patternsseen in experiment can be understood with-in perhaps the simplest dynamical modelsincorporating the competition between sur-face and dipolar energies. Taken together,the experimental and theoretical resultsindicate that an enormously complex ener-gy landscape in the space of shapes can arisefrom a competition between short-rangeforces and long-range dipolar interactionsin systems subject to a geometric con-straint. A static theory of labyrinths wouldfind only the minimum energy configura-tions, whereas the dynamic theories reflectthe complexity of the landscape in thecomplexity of the labyrinths.

1.

2.

3.

4.

5.

6.

7.

8.9.

10.

11.

REFERENCES AND NOTES

M. Seul, L. R. Monar, L. O'Gorman, R. Wolfe,Science 254, 1616 (1991).P. Molho, J. L. Porteseil, Y. Souche, J. Gouzerh, J.C. S. Levy, J. Appl. Phys. 61, 4188 (1987).H. Mohwald, Annu. Rev. Phys. Chem. 41, 441(1990), and references therein.H. M. McConnell, ibid. 42,171 (1991), and refer-ences therein.M. Seul and M. J. Sammon, Phys. Rev. Lett. 64,1903 (1990); M. Seul, Physica A 168, 198 (1990).F. Haenssler and L. Rinderer, Helv. Phys. Acta 40,659 (1967); R. P. Huebner, Magnetic Flux Struc-tures in Superconductors (Springer-Verlag, NewYork, 1979).D. Andelman, F. Brochard, J.-F. Joanny, J. Chem.Phys. 86, 3673 (1987).L. D. Landau, Zh. Eksp. Teor. Fiz. 7, 371 (1937).K. Binder and A. Young, Rev. Mod. Phys. 58, 846(1986).H. Frauenfelder, S. G. Sligar, P. G. Wolynes, Sci-ence 254, 1598 (1991), and references therein.R. E. Rosensweig, Ferrohydrodynamics (Cam-bridge Univ. Press, Cambridge, 1985).

12. A. 0. Tsebers and M. M. Maiorov, Magnetohydro-dynamics 16, 21 (1980).

13. A. G. Boudouvis, J. L. Puchalla, L. E. Scriven, J.Coil. Int. Sci. 124, 688 (1988).

14. M. Seul, J. Phys. Chem. 97, 2941 (1993).15. S. A. Langer, R. E. Goldstein, D. P. Jackson, Phys.

Rev. A 46, 4894 (1992).16. D. P. Jackson et al., unpublished data.17. The experimental apparatus consisted of a Hele-

Shaw cell constructed from two plates of opticallyflat glass, each 5.78 + 0.01 mm thick, and sepa-rated by a rubber gasket with thicknesses of 1.94+ 0.01 mm, creating a circular working volume ofradius 5.8 cm. The ferrofluid was placed in thecenter of this volume and surrounded by a dilutesolution of surfactant (Tween) in water. The ferro-fluid (EMG 901) was obtained from Ferrofluidics(Nashua, NH) with a saturation magnetization of600 gauss and a viscosity of 10 cp. Once filled,the cell was placed in a water-cooled Helmholtzcoil with an outer radius of 18.0 cm. and boreradius of 7.9 cm., whose field uniformity was ±5percent from bore center to periphery. The timecourse of the magnetic field was controlled by aPCL-818 digital to analog converter card andsoftware written for this purpose, and the intrinsicresponse time of the apparatus was less than 0.1second. Videotaped images were obtained with aSony XC-39 CCD video camera collinear with thebore of the Helmholtz coil, and controlled by anNEC PC-VCR (PV-S98A). Recorded images weredigitized and processed with edge-detection soft-ware.

18. R. E. Rosensweig, M. Zahn, R. Shumovich, J.Magnetism Magnetic Mater. 39, 127 (1983).

19. There are also indications of an important roleplayed by a ferrofluid-glass wetting phenomenonin that two successive patterns often have strongsimilarities, suggesting the presence of a thin filmon the glass surface which biases the arm evolu-tion. This effect can be reduced significantly bymanipulating the ferrofluid drop back and forthacross the glass prior to the field quench.

20. The magnetization is most uniform for highly reg-ular shapes like the initial condition, which closelyapproximates an oblate ellipsoid, a shape forwhich the demagnetizing field is uniform. Forhighly branched patterns this approximationbreaks down as fringing fields come to dominate.

21. Our calculation differs from the important work ofD. J. Keller, J. P. Korb, H. McConnell, [J. Phys.Chem. 91, 6417 (1987)], in which the cutoff wasadded by hand.

22. G. K. Batchelor, An Introduction to Fluid Dynam-ics (Cambridge Univ. Press, Cambridge, 1967),pp. 222-224.

23. For a related hydrodynamic calculation, see H. M.McConnell, J. Phys. Chem. 96, 3167 (1992).

24. See, for example, D. Bensimon, L. P. Kadanoff, S.Liang, B. I. Shraiman, C. Tang, Rev. Mod. Phys.58, 977 (1986).

25. D. Weaire and N. Rivier, Contemp. Phys. 25, 59(1984).

26. We thank N. P. Ong, M. Shelley, M. Seul, and R.Rosensweig for their long-standing interest andadvice; P. W. Anderson, R. H. Austin, S. M.Gruner, H. M. McConnell, and A. 0. Cebers forimportant discussions; and K. D. Bonin and M.Kadar-Kallin for experimental assistance at anearly stage of this work. Funded in part by Nation-al Science Foundation grant CHE91-06240, aDepartment of Energy Graduate Fellowship(A.J.D.), the Alfred P. Sloan Foundation (R.E.G),and the NSERC of Canada (S.A.L.).

6 May 1993; accepted 17 July 1993

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n* - Cl1/4

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