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Suprafroth in type-I superconductors

RUSLAN PROZOROV*, ANDREW F. FIDLER†, JACOB R. HOBERG AND PAUL C. CANFIELDAmes Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA†Permanent address: Department of Physics, Albion College, Albion, Michigan 59224, USA*e-mail: prozorov@ameslab.gov

Published online: 2 March 2008; doi:10.1038/nphys888

The structure and dynamics of froths have been subjects of intense interest owing to the desire to understand the behaviour of complexsystems where topological intricacy prohibits exact evaluation of the ground state. The dynamics of a traditional froth involvesdrainage and drying at the cell boundaries; thus, it is irreversible. Here, we report a new member of the froth family: suprafroth,in which the cell boundaries are superconducting and the cell interior is normal, or non-superconducting. Despite having a verydifferent microscopic origin, topological analysis of the structure of the suprafroth shows that the same statistical laws, such asthose of von Neumann and of Lewis apply to a suprafroth. Furthermore, for the first time in the analysis of froths, there is a globalmeasurable property, the magnetic moment, which can be directly related to the suprafroth structure. We propose that this suprafrothis a model system for the analysis of the complex physics of two-dimensional froths—with magnetic field and temperature as external(reversible) control parameters.

The fundamental physics and chemistry of complex systems alongwith practical issues related to the stability of froths (for example,insulating foams, ultralight metallic foams and fire extinguishers)have made them a topic of broad interest1,2. Although precisemicroscopic analysis of froth dynamics is difficult, general laws,that take into account both the topological constraints and physicsand chemistry of the froth matter, have been developed. Thisgives rise to the hope that, whereas the behaviour of individualcells may be unpredictable, the overall system can be describedby relatively simple rules. Various froths have been studied,and in most cases the coarsening parameter has been timeand the microscopic mechanism has been diffusion of vapourmolecules between the cells as well as drainage of the liquidfrom the cell walls. In the case of a magnetic froth, however,coarsening is promoted by the application of a magnetic field.Two types of magnetic froth have been known: (1) a ferrofluid,in an immiscible liquid, stimulated by an applied alternatingmagnetic field2,3 and (2) the magnetic domain structure observed intransparent ferromagnetic garnets1,4. However, in both cases, onlylimited analysis exploring the similarities between time-dependentcoarsening of the conventional froth and magnetic-field-inducedcoarsening in the magnetic froth was carried out.

Here, we used low-temperature, magneto-optical imaging ofsuperconducting lead to add an entirely new type of magnetic froth,formed in clean, type-I superconductors. Specifically, in this froth,the cell boundaries consist of superconducting phase, whereas cellinteriors are normal-state metal filled with magnetic flux. Unlikeany froth studied before, this superconducting froth involves onlyelectrons—normal and paired in Cooper pairs and a magneticfield. We propose to abbreviate this new topological phase as‘suprafroth’ in the spirit of early discoveries when superconductorswere called ‘supraconductors’. The coarsening of the suprafroth ispromoted either by increasing applied magnetic field or increasingtemperature and does not involve mass transport, thus enabling thestudy of the topological hysteresis decoupled from the hysteresisof the ‘ageing’ of the froth matter. In addition, it is important tonote that, unlike all previous cases, here we know macroscopic

H (O

e)

4 5 6 7

T (K)

Meissner state, B = 0

Normal metal, B = HHc(T )

Suprafroth

(1–N )Hc(T )Intermediate

state

0

200

400

600

Figure 1 Phase diagram of a superconducting lead disc. Below the (1−N )Hc (T )line, there is a Meissner state in which no magnetic flux exists. Above the Hc (T ) line,there is no superconductivity. The shaded region between these lines is theintermediate state. Solid colour inside this region shows where the suprafroth isstable, albeit without well-defined boundaries. The inset magneto-optical imagesillustrate pattern evolution for magnetic field increasing after cooling in zero field(4.8 K path shown). Vertical lines at 4.0, 4.8 and 6.0 K as well as the horizontal lineat 320 Oe show where the images and data used in this work were acquired.

laws governing the behaviour of clean superconductors. Thus,we are able to check the compatibility of the statistical laws ofcellular evolution with the macroscopic behaviour as a function ofmagnetic field and temperature.

Lead is a type-I superconductor in which the interface energybetween the normal and superconducting phases is positive, so thesystem wants to avoid or minimize the number of such boundaries.

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c

d

b

e f

g h i

a190 Oe 207 Oe 220 Oe

4.80 K 5.07 K 5.22 K

0.2 mm

1 mm

n = 7

Figure 2 Suprafroth coarsening and cell analysis. a–c, Coarsening of the suprafroth with increasing magnetic field at T= 6.0 K showing the entire sample (5 mm disc).d–f, Coarsening with increasing temperature at H= 320 Oe shown in a 1×1mm2 region of the disc. g–i, 2×2mm2 image measured at 4.8 K and 390 Oe: originalimage (g), superimposed trace of the boundaries (h), counting the polygons (i), n= 7 is shown (see Supplementary Information, Fig. S1 for other n).

On the other hand, owing to the non-zero demagnetization factor,N , the magnetic field on the edge of a disc-shaped lead crystal,He, is larger than the applied field by He = H/(1− N ). When He

exceeds the critical field, Hc, magnetic flux penetrates the specimenand an inhomogeneous, intermediate state is formed. It has recentlybeen shown that in pinning-free bulk type-I superconductors,the intermediate state consists of tubes5,6 rather than a textbook,laminar structure7. At high enough densities, the tubes evolveinto a well-defined superconducting froth as shown in Fig. 1 (seeSupplementary Information Video S1 and S2). It should be notedthat tubular structure was observed and studied before6,8,9, butmostly in thin films8 where tubes do not evolve into the suprafroth,most likely because of the relatively low values of the magnetic Bondnumber, 0.3–350 (ref. 8), and pinning. In our case of clean bulk

crystals, the magnetic Bond number is >2,800, which means thatdipolar interaction between normal domains can be neglected andtubes interact in the interior repelling each other by supercurrentsflowing around them enabling formation of the suprafroth.

The magneto-optical images show a clear pattern evolutionon increase of the applied magnetic field. Initially, separate fluxtubes are injected into the sample (Fig. 1, lower inset). As theirnumber increases, they coalesce and grow. As the field grows, therepulsive forces between the tubes ultimately lead to formationof the suprafroth with well-defined polygonal cellular structure(Fig. 1, top right inset). For magnetic fields that approach the upperlimit of superconductivity, Hc, this structure ultimately degradesand forms extended concave cells with rounded boundaries. Thesuprafroth structure is very different from that of the mixed state

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seen in the more common, bulk type-II, superconductors wherethe interface energy is negative and the magnetic field inside thematerial exists in the form of Abrikosov vortices each carrying asingle flux quanta and no suprafroth is possible.

The coarsening of a suprafroth with applied magnetic fieldis shown in Fig. 2a–c. The images were obtained at T = 6 Kwhen, after zero field cooling, the indicated magnetic fields wereapplied (also see Supplementary Information Videos S1 and S2).Figure 2d–f shows coarsening with increasing temperature sampledat H = 320 Oe. The observed patterns were traced and convertedinto black and white images as shown in Fig. 2g–i. Traced imageswere analysed in terms of cell statistics; n = 7 is shown in Fig. 2iand other polygons are shown in more detail in SupplementaryInformation, Fig. S1. Our typical region of interest was 2 × 2 mmand contained up to 160 cells.

Before we proceed to the discussion of the statisticallaws, however, it is important to examine the questions ofreversibility and structural evolution and identify the elementarytransformation processes in a suprafroth. In conventional froths,the parameter that is varied to study coarsening is time. Thephysical processes of drainage and diffusion responsible forcoarsening are inherently irreversible. However, the irreversibilityof coarsening can be twofold: changes in the froth matter (ageingeffects such as drying and drainage) and/or changes in the frothtopology. Although it is impossible to reverse time, in the caseof the suprafroth, however, the equivalent controlling parameter,magnetic field can be increased and decreased so as to examine thetopological elasticity of the structure, while the physical propertiesof cell walls remain perfectly reversible.

Figure 3a–f shows what, in physics of conventional froths, areknown as T1 (side swapping, Fig. 3a,b) and T2 (cell disappearance,Fig. 3c–f) processes. These images are exactly like those observedin coarsening of the conventional soap froth. Therefore, themesoscopic cellular dynamics in suprafroth seems to be quitesimilar to that of conventional froths.

Figure 3g–i examines the effect of a minor loop in applied fieldon the suprafroth. In the experiment, H = 466 Oe was appliedafter zero field cooling to T = 4.0 K and an image was taken (anunderlying black trace in each of the three panels of Fig. 3g–i).Then the magnetic field was increased by the amount shown andreturned back to 466 Oe and a new image (shown in red) wasacquired. An immediate (and experimentally new compared withregular froth) conclusion is that the regions with the most uniformdistributions of cell sizes (shown by the grey square in Fig. 3g–i) aremore robust compared with the regions with broader distributionsof cell size. This result is directly related to the empirical Aboav–Weaire law1 as applied to the dynamics of cells whose neighbourshave a certain number of sides. Moreover, after the initial cycle,H → H + 1H → H , the structure remains perfectly elastic andreproducible (for small 1H ∼ 5 Oe) on several subsequent cycles.Similar elasticity is observed if the magnetic field is decreased by asmall amount and returned back to the base field.

Having examined the data in a qualitative manner, we nowproceed to a quantitative analysis of the cellular structure of thesuprafroth using a topological analysis similar to that shown inFig. 2 and Supplementary Information, Fig S1. Figure 4 shows thedistribution of the number of cells with n sides for suprafrothcoarsened by an applied magnetic field at 4.8 K (Fig. 4a) orcoarsened by temperature at 320 Oe (Fig. 4b). In both cases, itis clear that n = 6 is the most probable polygon, consistent withEuler’s theorem that states that the average number of sides ina continuous two-dimensional tiling with n-sided cells that havethree-fold vertices is n = 6 (ref. 1). If C is the total number ofcells in a studied area A and Cn is the number of n-sided cells,then the distribution function pn = Cn/C. As shown by the solid

T1

T2

T2

6 Oe 9 Oe

17 Oe

a b

g

c d

h

e f

i

Figure 3 Structural evolution of the suprafroth. a–f, T1 process (a,b) measured at4.8 K and in fields changing from 290 Oe (a) to 300 Oe (b) and two T2 processes forn= 4 (c,d) and n= 5 (e,f) imaged at 6.0 K in fields changing from 200 to 207 Oe.g–i, Response of a suprafroth to a small variation of a magnetic field at T= 4.0 K.Black lines correspond to the state obtained after cooling in zero field and applying466 Oe. Red lines show the structure obtained after the field was increased by 1Hindicated in the figure and decreased back to 466 Oe. Grey squares show regions ofmost reversible behaviour.

line, a simple triangular distribution, pn = 0.5(1 − 0.5|n − 6|)describes the observation quite well. Importantly, pn does notdepend significantly on either field or temperature.

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0

10

20

30

40

50

Dist

ribut

ion

of p

olyg

ons

(%)

0

10

20

30

40

50

Dist

ribut

ion

of p

olyg

ons

(%)

2 3 4 5 6

n

7 8 9 10 11n

H = 320 Oe

4.80 K5.07 K4.90 K5.22 K

328 Oe358 Oe342 Oe378 Oe389 Oe

T = 4.8 K

2 3 4 5 6 7 8 9 10 11

a b

Figure 4 Distribution of the number of n-sided cells. a,b, For froth coarsened by magnetic field (a) and temperature (b).

0

0.01

0.02

0.03

A n (m

m2 )

A n (m

m2 )

328 Oe342 Oe358 Oe378 Oe389 Oe

H =

T = 4.8 K

00 1 2 3 4

n – 35 6

n – 3

7 80 1 2 3 4n – 3

5 6 7 8

0.01

0.02

0.03

H = 322 Oe

T =

4.80 K 4.90 K 5.07 K 5.22 K

0

0.02

0.04

dAn/

dT (m

m2

K–

1 )

dAn/

dH (m

m2

T–1 )

0 1 2 3 4 5 6n – 3

0 1 2 3 4 5 60

1

2

3

dAn /dH = 0.6(n – 3)dAn /dT = 0.01(n – 3)

a b

c d

Figure 5 Lewis’ and generalized von Neumann’s laws in suprafroth. a–d, n-dependence of an average cell area (a,b) and its rate of change (c,d) at different fields (a,c)and temperatures (b,d), respectively. dAn/dH was calculated at T= 4.8 K and dAn/dT was calculated for H= 330 Oe. The error bars in c and d are the standard error ofthe linear regression.

We can also examine how the average area of an n-sidedcell depends on n for different magnetic fields and temperatures,Fig. 5a,b. This statistical correlation was first studied in biological,cellular structures by Lewis and is now known as Lewis’ law10,11,

An =A

Cl

(n−

[6−

1

l

]),

where the empirical constant l is typically between 1/3 and 1, butits microscopic meaning is not well understood. It is clear fromFig. 5a,b, that to make our observation compatible with Lewis’ lawwe need to set l = 1/3, so that An = (A/3C)(n−3).

Having deduced a form for the coarsening of a suprafroth,we should recall that the physics of a type-I superconductorin the intermediate state dictates a certain dependence of the

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0

100

200

300

400

C(H

)

C(T

)

0

50

100

150

0.002

320 340 360

H (Oe)

380 400

320 340 360

H (Oe)

380 400

4.80

0.003

0.006

0.009

0.012

4.9 5.0

T (K)

5.1 5.2 5.3

4.8 4.9 5.0

T (K)

5.1 5.2 5.3

0.004

0.006

0.008

Cells break

α=

dA n/

dn (m

m2 )

α=

dA n/

dn (m

m2 )

T = 4.8 K

λ = 1/3

λ A/C (H )

λ A/C (T )

H = 322 Oe

λ = 1/3

~ 1– HHc

2

a b

c d

Figure 6 Parameters of Lewis’ law. a,b, Field and temperature dependence of the total number of cells, C (H, T ). Solid line in a is a prediction on the basis of M (H ) of asuperconductor. c,d, The coefficient in Lewis’ law. Symbols: experiment, solid line: calculations using C from a,b.

total volume of the superconducting phase, Vs, on temperatureand magnetic field. Specifically, in the intermediate state, themagnetic moment of a type-I superconductor of volume V is4πM = V (H − Hc)/N = −VsHc (ref. 12). Therefore, the totalperimeter of all cell boundaries, P = Vs/δt = A(Hc − H)/δN Hc

where A is the total area, t is sample thickness and δ is thewidth of the superconducting walls in the suprafroth. It was found,by direct measurements, that δ ' 14 µm and is practically fieldindependent in our range of fields and temperatures. On the otherhand, the total perimeter is expressed through the distributionpn and the average length of a side of an n-sided cell, sn, as,P = C

∑npn sn/2 = 1.85

√AC. Therefore, we expect for the total

number of cells, C ∼ (Hc − H)2, which is, indeed what we observein Fig. 6a. Similar behaviour can be predicted for the temperaturedependence of C (Fig. 6b). However, the most striking result isthat the l = 1/3 coefficient of Lewis’ law obtained from thelinear fits shown in Fig. 5a,b is in excellent agreement with thedirect, parameter-free calculation of dAn/dn = lA/C with theexperimental values of A and C as shown by the solid (B-spline)line in Fig. 6c,d. The deviations at higher temperatures are relatedto breaking the cellular structure apart.

One of the most studied and discussed statistical laws in thephysics of froth coarsening is the von Neumann law1,13,14, whichpredicts a linear dependence of the rate of change of the averagearea of an n-sided cell to its number of sides, n. In conventionalfroths, it has been shown both experimentally and theoretically thatdAn/dt = γ(n − 6). Historically, the fact that the offset to n wasthe number six was associated with the Euler tiling theorem: sixbeing the most probable polygon. In the case of the suprafroth, wecan generalize von Neumann’s law and consider the temperature

and magnetic field derivatives. By direct differentiation of Lewis’law, An = (A/3C)(n − 3), with C = A(Hc − H)2/(1.85δN Hc)

2,we expect

dAn

dH=

2.28

Hc

(Nδ)2

(1−H/Hc)3(n−3) = β(n−3).

If the derivative is calculated for the small variation of themagnetic field (∼5% in our case), so that H/Hc ≈ constant, theprefactor β does not vary by much. Therefore, despite the fact thatwe still have hexagons as most probable polygons, the rate of changeis predicted to increase for any n > 3. This is, indeed, observed inthe experiment as shown in Fig. 5c,d where the derivatives wereevaluated in the small range of fields (or temperatures). Clearly, a(n−3) dependence is seen in both graphs. We can even estimate thecoefficient β by substituting the experimental values, H = 0.8Hc,Hc = 0.044 T, δ = 0.014 mm and N = 0.63, so that we estimateβ ≈ 0.5, which is quite close to the observed value of 0.6 giventhe uncertainty in the demagnetization factor and superconductingwall width, δ. It is more difficult to estimate the coefficient fordAn/dT (Fig. 5d), because M(T) is a more complex functionof temperature.

The ultimate explanation for the observed (n −3) dependenceof von Neumann’s law in the suprafroth rather than (n − 6)behaviour of the conventional froths has to be associated withthe differences between the microscopic mechanisms of thecoarsening. Whereas in conventional froths the amount of frothmaterial remains constant or decreases through drainage andthe process is controlled by the surface tension and diffusionof the boundaries, in suprafroth the coarsening is controlled

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by the magnetic field that exerts magnetic pressure inside thesuperconducting cells. Therefore, every polygon is inflated andthe minimum planar object has n = 3 sides, so all polygonsgrow. In addition, the exact formula for the area of a regularpolygon with side length sn, An = (n/4)s2

n cot(π/n), can be roughlyapproximated by the (n − 3) behaviour. On the other hand, thelinearity of the growth rate with n is always observed in variousfroths during coarsening. To this end, this work on suprafrothformally separates the most probable number of sides (6) andthe offset in von Neumann’s law (3 for the suprafroth). It shouldbe noted that a reversed von Neumann’s law due to specificsof the interfacial interaction with the substrate was reported inferrofluids2,15. This shows that dimensionality and non-localityare important parameters determining the statistical behaviourof froths.

METHODS

Direct visualization of the magnetic induction on the sample surface wascarried out in a flow-type, 4He, close-cycle cryostat using Bi-doped iron garnetswith in-plane magnetization, serving as indicators5,6. The sample was in avacuum and placed on a copper cold stage. The indicator was placed on topof the sample. In the experiment, linearly polarized light was passed throughthe indicator, reflected from a mirror sputtered on its bottom to the analyserthat was turned 90◦ with respect to the polarizer. The distribution of themagnetic moments in the indicator mimics the distribution of the magneticinduction on the surface of a studied sample, thus enabling direct, real-time,visualization of the magnetic flux. In all images, dark regions correspond to thesuperconducting, and bright regions to the normal, phase.

Bulk single-crystal lead samples with (001) orientation were grown, cutand polished by Mateck GmbH (http://www.mateck.de/). The samples were inthe form of discs, 5 mm in diameter and 1 mm thick.

Received 10 December 2007; accepted 28 January 2008; published 2 March 2008.

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AcknowledgementsWe thank J. Clem, N. Goldenfeld, R. Huebener, V. Kogan, R. Mints and J. Schmalian for helpfuldiscussions. Work at the Ames Laboratory was supported by the Department of Energy, Basic EnergySciences under Contract No. DE-AC02-07CH11358. R.P. acknowledges support from NSF GrantNumber DMR-05-53285 and the Alfred P. Sloan Foundation.Correspondence and requests for materials should be addressed to R.P.Supplementary Information accompanies this paper on www.nature.com/naturephysics.

Author contributionsR.P. idea, measurements, data analysis; A.F.F. measurements, data analysis; J.R.H. measurements;P.C.C. idea, data analysis.

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