fea-finn.qxp 7/1/99 9:56 AM Page 770

Capillary Surface Interfaces Robert Finn

nyone who has seen or felt a raindrop, under physical conditions, and failure of unique- or who has written with a pen, observed ness under conditions for which solutions exist. a spiderweb, dined by candlelight, or in- The predicted behavior is in some cases in strik- teracted in any of myriad other ways ing variance with predictions that come from lin- with the surrounding world, has en- earizations and formal expansions, sufficiently so counteredA capillarity phenomena. Most such oc- that it led to initial doubts as to the physical va- currences are so familiar as to escape special no- lidity of the theory. In part for that reason, exper- tice; others, such as the rise of liquid in a narrow iments were devised to determine what actually oc- tube, have dramatic impact and became scientific curs. Some of the experiments required challenges. Recorded observations of liquid rise in microgravity conditions and were conducted on thin tubes can be traced at least to medieval times; NASA Space Shuttle flights and in the Russian Mir the phenomenon initially defied explanation and Space Station. In what follows we outline the his- came to be described by the Latin word capillus, tory of the problems and describe some of the meaning hair. current theory and relevant experimental results. It became clearly understood during recent cen- The original attempts to explain liquid rise in a turies that many phenomena share a unifying fea- capillary tube were based on the notion that the ture of being something that happens whenever portion of the tube above the liquid was exerting two materials are situated adjacent to each other a pull on the liquid surface. That, however, cannot and do not mix. We will use the term capillary sur- be what is happening, as one sees simply by ob- face to describe the free interface that occurs when serving that the surface fails to recede (or to change one of the materials is a liquid and the other a liq- in any way) if the tube is cut off just above the in- uid or gas. In physical configurations such as the terface. Further, changing the thickness of the capillary tube, interfaces occur also between these walls has no effect on the surface, thus suggest- materials and rigid solids; these latter interfaces ing that the forces giving rise to the phenomenon yield in many cases the dominant influence for de- can be significant only at extremely small dis- termining the configuration. tances (more precise analysis indicates a large por- In this article we describe a number of such tion of these forces to be at most molecular in phenomena, notably some that were discovered range). Thus, for a vertical tube the net attractive very recently as formal consequences of the highly forces between liquid and wall must by symmetry nonlinear governing equations. These discoveries be horizontal. It is this horizontal attraction that include discontinuous dependence on the bound- causes the vertical rise. Molecules being pulled to- ary data, symmetry breaking, failure of existence ward the walls force other molecules aside in all Robert Finn is professor emeritus of mathematics at directions, resulting in a spread along the walls that Stanford University. His e-mail address is finn@gauss. is only partly compensated by gravity. Liquid is stanford.edu. forced upward along the walls, and cohesive forces

770 NOTICES OF THE AMS VOLUME 46, NUMBER 7 fea-finn.qxp 7/1/99 9:56 AM Page 771

carry the remaining liquid column with it. The low- g ered hydrostatic pressure at the top of the column is compensated by the curvature of the surface, in essentially the same manner as occurs with a soap S bubble. This basic observation as to the nature of the acting forces appears for the first time in the γ writings of John Leslie in 1802. V W An early attempt to explain capillarity phe- nomena was made by Aristotle, who wrote circa 350 B.C. that A broad flat body, even of heavy mater- ial, will float on a water surface, but a long thin one such as a needle will always sink. Any reader with Figure 1. Fluid interface S, support surface W ; γ is the angle access to a needle and a glass of water will have between the two surface normals. little difficulty refuting the statement. On the other hand, Leonardo da Vinci wrote in 1490 on the me- chanics of formation of liquid drops, using ideas very similar to current thinking. But in the ab- γ sence of the calculus, the theory could not be made quantitative, and there was no convincing way to test it against experiments. The achievements of the modern theory de- pend essentially on mathematical methods, and u0 specifically on the calculus, on the calculus of vari- ations, and on differential geometry. When one looks back on how that came about, one is struck by the irony that the initial mathematical insights were introduced by Thomas Young, a medical physician and natural philosopher who made no secret of his contempt for mathematics (and more specifically for particular mathematicians). But it was Young who in 1805 first introduced the math- Figure 2. Capillary tube configuration. ematical concept of mean curvature H of a surface and who showed its importance for capillarity by Theorem 1. The height u(x, y) of a capillary sur- relating it to the pressure change across the sur- face interface lying over a domain Ω in a vertical face: ∆p =2σH, with σ equal to surface tension. gravity field satisfies the differential equation Young also reasoned that if the liquid rests on a support surface W, then the fluid surface S meets (3) div Tu = κu + λ. W in an angle γ (contact angle) that depends only Here λ is a constant to be determined by phys- on the materials and not on the gravity field, the ical conditions (such as fluid volume) and bound- shape of the surface, or the shape or thickness of ary conditions; κ is positive when the denser fluid W; see Figure 1. lies below the interface; in the contrary case, the Young derived with these concepts and from the laws of hydrostatics the first correct approxima- sign of κ reverses. For the problem described tion for the rise height at the center of a circular above, considered by Young, λ =0and (3) becomes capillary tube of small radius a immersed vertically (4) div Tu = κu. in a large liquid bath: For a capillary surface in a cylindrical vertical 2 cos γ ρg (1) u0 ≈ ,κ= ; tube of homogeneous material and general hori- κa σ zontal section Ω, the Young condition on the con- here ρ is the density change across the free sur- tact angle yields the boundary condition face, g the magnitude of gravitational accelera- · tion. See Figure 2. (5) ν Tu = cos γ Young’s chief competitor in these developments on Σ = ∂Ω, with ν the unit exterior normal on Σ. was Laplace, who relied heavily on mathematics. Instead of a tube dipped into an infinite reser- Laplace derived a formal mathematical expression voir as considered by Young, one could imagine a vertical tube closed at the bottom and partially ≡ ≡ q Du (2) 2H div Tu, Tu filled with a prescribed volume of liquid covering 1+|Du|2 the base. In general in this case λ =06 , but addition for the mean curvature H of a surface u(x, y); he of a constant to u converts (3) to (4). From unique- was led to ness properties discussed below, it follows that in

AUGUST 1999 NOTICES OF THE AMS 771 fea-finn.qxp 7/1/99 9:56 AM Page 772

sult was obtained again independently by E. Mierse- mann in 1994. In 1830 Gauss used the Principle of Virtual Work, formulated by Johann Bernoulli in 1717, to unify the achievements of Young and of Laplace, and he δ obtained both the differential equation and the Γ boundary condition as consequences of the prin- α ciple. In the Gauss formulation the constant λ in O (3) appears as a Lagrange parameter arising from an eventual volume constraint. Ωα δ Capillarity attracted the attention of many of the leading mathematicians of the nineteenth and early twentieth centuries, and some striking results were obtained; however, the topic then suffered a hia- tus till the latter part of the present century. A great influence toward new discoveries was provided by the “BV theory”, developed originally for min- imal surfaces by E. de Giorgi and his co-workers. In the context of this theory M. Emmer provided Figure 3. Wedge domain. in 1973 the first existence theorem for the capil- lary tube of general section. For further references all cases, and independent of the volume of liquid, to these developments, see [1, 2]. Other directions the surfaces obtained are geometrically the same. were initiated by Almgren, Federer, Fleming, Si- This result holds also when κ =0 and a solution mons, and others and led to results of different exists; however, existence cannot in general then character; see, e.g., [10]. be expected, as we shall observe below. To some extent, these considerations extend to configura- The Wedge Phenomenon tions with κ<0, i.e., with the heavier fluid on top; It is unlikely that anyone reading this article will however, in general both existence and unique- be unfamiliar with the name of Brook Taylor, as ness may fail when κ<0. the Taylor series figures prominently in every cal- If γ is constant, we may normalize it to the culus sequence. It is less widely known that Tay- ≤ ≤ ≤ range 0 γ π. The range 0 γ<π/2 then in- lor made capillarity experiments, almost one hun- ≤ dicates capillary rise; π/2 <γ π yields capillary dred years prior to the work of Young and of fall. It suffices to consider the former case, as the Laplace. He formed a vertical wedge of small angle other can be reduced to it. If γ = π/2, the only so- 2α between two glass plates and observed that a ≡ lutions of (4), (5) with κ>0 are the surfaces u 0; drop of water placed into the corner would rise up if κ =0, the only such surfaces are u ≡ const. If into the wedge, forming contact lines on the plates κ<0, nontrivial such solutions appear. that tend upward in a manner “very near to the With the aid of (4) and (5), Laplace could improve common hyperbola”. Taylor had no theory to ex- the approximation (1) to plain the phenomenon, but in the course of the en- !! suing centuries a number of “proofs” of results 2 cos 1 2 1 sin3 leading to or at least suggesting a hyperbolic rise (6) u0 a. a cos 3 cos3 independent of opening angle appeared in the lit- erature. Laplace did not prove this formula completely, The actual behavior is quite different and varies nor did he provide any error estimates, and in fact dramatically depending on the contact angle and it was disputed in later literature. Note that the angle of opening. There is a discontinuous transi- right side of (6) becomes negative if the nondi- tion in behavior at the crossing point α + γ = π/2. mensional “Bond number” B = κa2 > 8, in which In the range α + γ<π/2, Taylor’s observation is case the estimate gives less information than does verified as a formal property that holds for any so- lution. Specifically, we consider a surface S given (1). The formula is, however, asymptotically cor- α rect for small B; that was proved for the first time by u(x, y) over the intersection Ωδ of a wedge do- almost two centuries later in 1980 by David Siegel, main with a disk of radius δ as in Figure 3. We ob- who also provided explicit error bounds; a proof tain: yielding improved bounds was given by the pre- α Theorem 2. Let u(x, y) satisfy (4) with κ>0 in Ωδ sent author in 1984, and further improvements and suppose S meets interior points of the bound- were obtained by Siegel in 1989. The expression ing wedge walls in an angle γ such that (6) was extended by P. Concus (1968) to the entire α + γ<π/2. Let k = sin α/ cos γ. Then in terms of traverse 0

772 NOTICES OF THE AMS VOLUME 46, NUMBER 7 fea-finn.qxp 7/1/99 9:56 AM Page 773

p cos θ − k2 − sin2 θ (7) u ≈ . kκr On the other hand, if α + γ ≥ π/2, then u(x, y) is bounded, depending only on κ and on the size of the domain covered by S near O: α Theorem 3. If u(x, y) satisfies (4) with κ>0 in Ωδ and if S meets the bounding wedge walls in an angle γ such that α + γ ≥ π/2, then

(8) |u|(2/)+ α throughout Ωδ . It is important to observe that the boundary is singular at O and the contact angle cannot be pre- scribed there. Despite this singularity, neither theorem requires any growth hypothesis on the solution near O. Such a statement would not be possible, for example, for harmonic functions under classical conditions of prescribed values or normal derivatives on the boundary. It is also note- worthy that no hypothesis is introduced with re- gard to behavior on Γ. The local behavior is com- pletely controlled by the opening angle and by the Figure 4. Water rise in wedge of acrylic plastic. (Left) contact angle over the wedge segments near O + </2. (Right) + /2. and is an essential consequence of the nonlinear- ity in the equation. (In another direction, it can be so that the surface has constant mean curvature. shown that if u(x, y) satisfies (4) in all space, then We find: u ≡ 0.) α Theorem 4. If u(x, y) satisfies (9) in Ωδ and defines If in an initial configuration there holds a surface S that meets interior points of the wedge + >/2 and α is then continuously decreased walls in the constant angle γ, then α + γ ≥ π/2 and until equality is attained, (8) provides a uniform u(x, y) is bounded at O. bound on height up to and including the crossing point. For any smaller α the estimate (7) prevails, Thus, the change is now from boundedness to and the surface is unbounded at O. nonexistence of a solution. This behavior was tested in the Stanford Uni- It may at first seem strange that this physical versity medical school in a “kitchen sink” experi- problem should fail to admit a solution; after all, ment by T. Coburn, using two acrylic plastic plates fluid placed into a container has to go someplace. and distilled water. Figure 4 shows the result of a Figure 5 shows the result of an experiment con- change of about 2◦ in the angle between the plates, ducted by W. Masica in the 132-meter drop tower leading in the smaller α case to a measured rise facility at NASA Glenn Research Center, which pro- height over ten times the predicted maximum of vides about five seconds of free fall. Two cylindrical Theorem 2. That result was confirmed under con- containers of regular hexagonal section were trolled laboratory conditions and for varying ma- dropped, both with acrylic plastic walls but with terials by M. Weislogel in 1992; it yields a contact differing liquids, providing configurations on both angle of water with acrylic plastic of 80, 2 . (See sides of the critical value. In this case, when ≥ the remarks in the next section.) α + γ π/2, the exact solution is known as a lower The relation (7) was extended by Miersemann spherical cap, and this surface is observed in the (1993) to a complete asymptotic expansion at O, experiment. When α + γ<π/2, the liquid fills out in powers of B = κr2 . It is remarkable that the co- the edges and climbs to the top of the container, efficients are completely characterized by and as indicated in Figure 6. Thus the physically ob- and are otherwise independent of the data and do- served surface folds back over itself and cannot be expressed as a graph over the prescribed do- main of definition for the solution. Thus again the main. The physical surface thus exists as it has to, behavior is strikingly different from that arising but not in the form of a solution to (9), (5). It in linear problems. should be noted that for regular polygonal sections, If gravity vanishes, the discontinuous depen- the change in character of the solution is discon- dence becomes still more striking. In this case (3) tinuous. That is clearly apparent, as (unique) so- becomes lutions exist as lower spherical caps throughout (9) div Tu = =2H, the closed range + /2.

AUGUST 1999 NOTICES OF THE AMS 773 fea-finn.qxp 7/1/99 9:56 AM Page 774

Existence and for example, for the water and acrylic plastic in- Nonexistence; the terface in the Coburn experiment described above, Canonical it is over 20◦. Such apparent anomalies of mea- Proboscis surement have led to some questioning of “con- The just-described con- tact angle” as a physical concept. The experimen- currence of experimen- tal confirmation of discontinuous dependence on tal results with predic- data at the critical opening in a wedge supports the tion from the formal view that contact angle does have intrinsic physi- theory provides a per- cal meaning; in addition it suggests a new method suasive indication that for measuring the angle in particular cases. One the Young-Laplace- need only place a drop of the liquid into a wedge Gauss theory, beyond that is formed by vertical plates of the solid and its aesthetic appeal, has initial opening sufficiently large that Theo- also correctly describes rem 3 applies. The opening angle is then slowly de- physical reality. The creased until the liquid jumps up in the corner to physical validity of that a height above the bound given by (8). The crite- theory has been ques- ria of Theorems 2 and 3 then determine γ. tioned on the basis of For values of γ close to π/2, remarkably good ambiguities that occur agreement has been obtained in this way with the in attempts to measure “equilibrium angle” measured under terrestial con- ditions. On the other hand, for values of γ close contact angle. There are ◦ experimental proce- to 0 , the physical changes occur in the immedi- O dures that give rise to ate neighborhood of with an opening close to reasonably repeatable π, and measurements become subject to experi- measurements of what mental error. We thus seek domains in which the may be described as an discontinuous behavior is manifested over a larger “equilibrium angle”. set. We can do so in the context of a general the- However, if one par- ory of independent mathematical interest, apply- tially fills a vertical cir- ing to zero gravity configurations and determin- cular cylinder with liq- ing criteria for existence and nonexistence of solutions of (9), (5) in tubes of general piecewise Figure 5. Liquids with different contact uid (symmetrically) and smooth section Ω. angle in acrylic plastic cylinders of then tries to move the Let Γ be a curve in Ω cutting off a subdomain hexagonal section during free fall. liquid upward by a pis- Ω∗ ⊂ Ω and subarc Σ∗ ⊂ Σ = ∂Ω, as in Figure 7. (Top) α + γ>π/2. (Bottom) ton at the bottom, the From the zero gravity equations (9) and (5) we ob- α + γ<π/2. contact line of liquid tain, for the area |Ω∗| and length |Σ∗|, with solid does not im- Z mediately move, but in- ∗ ∗ (10) 2H|Ω | = |Σ | cos γ + ν · Tuds. stead an increase in the Γ contact angle is ob- ∗ served. The maximum The same procedure with Ω = Ω yields possible such angle be- |Σ| cos γ (11) 2H = . fore motion sets in is |Ω| known as the “advanc- In (10) we observe that ing angle”. Corre- spondingly, the small- | | | | q Du est such angle ob- (12) Tu < 1 1+|Du|2 tainable before reverse motion occurs is the for any differentiable function u(x, y). It follows “receding angle”. The that whenever there exists a solution to the difference between ad- problem, the functional ( ; ) | | vancing and receding | | cos + 2H| | satisfies angles is the “hysteresis ∗ range”, which presum- (13) Φ(Ω ; γ) > 0 ably arises due to fric- tional resistance to mo- for all strict nonnull subsets Ω∗ ⊂ Ω cut off by tion at the interface, smooth curves , when H is determined by (11). but could conceivably This inequality provides a necessary condition for also reflect inadequacy existence of a solution. Figure 6. Sketch of side view at edge in of the theory. This To make the condition sufficient, we appeal to drop tower experiment, α + γ<π/2. range can be very large; the properties of “Caccioppoli sets” in the BV

774 NOTICES OF THE AMS VOLUME 46, NUMBER 7 fea-finn.qxp 7/1/99 9:56 AM Page 775

theory. Roughly speaking, these are subsets of Ω whose boundaries within Ω can be assigned a fi- nite length, in a variational sense. Details of gen- eral properties of these sets can be found in [1, 2]; reference [3] and references cited there give spe- Ω∗ Ω cific applications to capillarity. It was shown by Giusti in 1976 that if we rephrase (13) as a prop- ∗ erty of Caccioppoli sets, the condition also be- Σ comes sufficient; see also Theorem 7.10 of [3]. We obtain: Γ Theorem 5. A smooth solution u(x, y) exists in Ω if and only if the functional Φ(Ω∗; γ) is positive for Σ every Caccioppoli set Ω∗ ⊂ Ω, with Ω∗ not equal to ∅ or Ω. ∗ In order to use this result, one proves that if Ω Figure 7. Existence criterion; Ω is the portion of Ω cut off by an is piecewise smooth, then Φ can be minimized arbitrary curve Γ. among Caccioppoli sets, and the minimizing sets can be characterized geometrically. One obtains: mean curvature H (and accordingly the radius of the arcs) will, however, no longer be determined Theorem 6. For Ω piecewise smooth, there exists by (11). at least one minimizing set Ω0 for Φ in Ω. Any such The case N = N0 = ∞ can occur, and it is this set is bounded within Ω by a finite number N0 of case that provides the clue for obtaining domains nonintersecting subarcs of semicircles of radius in which there is abrupt transition in height of so- 1/(2H), each of which either meets Σ at a smooth lutions over large sets at the critical angle sepa- point with angle γ measured within Ω0 or meets Σ rating existence from nonexistence. Following joint at a corner point. The curvature vector of each work by Bruce Fischer, by Tanya Leise, and by subarc is directed exterior to Ω0. Jonathan Marek with the author, we look for a Thus, the existence question is reduced to eval- translational continuum of extremals, each of Φ =0 Ω uation of particular configurations, which often can which yields . We construct the domain by finding the curves that meet a given translational be determined explicitly by the geometrical re- family of circular arcs of common radius R in a quirements of prescribed radius and angle with Σ. fixed angle γ ; these curves can be written ex- In general, there will be a number N ≥ N of sub- 0 0 plicitly, in the form arcs of semicircles satisfying those requirements. We refer to these arcs as extremals. If N =0, then q since Φ vanishes on the null set, we find (14) x + c = R2 y2 ∗ ∗ q Φ(Ω ; γ) > 0 , all Ω ⊂ Ω with =6 ∅, , and R2 y2 cos y sin hence by Theorem 5 a smooth solution exists. That q 0 0 + R sin 0 ln . happens, for example, in a rectangle when 2 2 R + y cos 0 + R y sin 0 /4. (If γ<π/4, then by Theorem 4 there is no solution for that problem.) We choose upper and lower branches of the If N is finite and positive, then a finite number ∗ curves joining at a point on the x-axis as part of of subdomains Ω appears for examination. If one the boundary Σ, and then complete the domain by or more of these sets yields Φ ≤ 0, then no solu- a circular “bubble” whose radius is adjusted so that tion of (9), (5) can exist in Ω. If the value zero but the arcs become extremals with Φ =0, as indicated ∗ no smaller value is achieved by Φ on the sets Ω , in Figure 8. We refer to such a domain as a canon- then there exists a set Ω0 among them such that ical proboscis. It can be shown that when the con- \ there will be a solution u(x, y) in the set Ω Ω0 and tact angle exceeds γ0 , then a smooth solution of →∞ such that u as Ω0 is approached from within (9), (5) exists for that domain; however, for γ = γ0 Ω \ Ω0 . If we then set u ≡∞in Ω0, we obtain a gen- the surface becomes a generalized solution in the eralized solution in a sense introduced by M. Mi- sense of Miranda, infinite in the entire region cov- randa in connection with minimal surfaces in 1977. ered by the extremals. This solution will have mean curvature determined Since the relative size of the portion of such a by (11). domain that is covered by extremals can be made ∗ ∗ If a domain Ω exists with Φ(Ω ; γ) < 0 , then as large as desired, the canonical proboscides can that inequality holds also for each minimizer. be used for determining small contact angles based Again it can be shown [4] that there exists a solu- on rapid changes of fluid height over large sets (see tion u(x, y) in a subdomain such that u(x, y) tends the initial paragraph of this section). The theory to infinity on a finite number of circular subarcs was tested by Fred W. Leslie in the Space Shuttle of equal radius that intersect Σ in angle γ. The USML-2 using containers with two proboscides on

AUGUST 1999 NOTICES OF THE AMS 775 fea-finn.qxp 7/1/99 9:56 AM Page 776

normal vector continuous to O is that (B1,B2) lies in the closed ellipse

E 2 2 2 (15) : B1 + B2 +2B1B2 cos 2 sin 2.

See Figure 10. For data in the indicated domains D 1 bounded between the square and the ellipse, one can show the natural extension of Theorem 4 that solutions of (9), (5) are precluded without re- gard to growth hypotheses. D For data lying in 2 the situation is very dif- ferent. In this case solutions of (9), (5) have been shown to exist under general conditions, but their form must be very different from what occurs for E Figure 8. Proboscis domain with bubble showing extremals Γ. data in , in which case spherical surfaces yield D particular solutions; that cannot happen in 2 , as the normal is discontinuous for such data. The clas- sical Scherk minimal surface provides a particular example. But for data (B1,B2) that are not on the boundary of the square in Figure 10, the solution u(x, y) is bounded above and below. It has been conjectured by J-T. Chen, E. Miersemann, and this author that u(x, y) is itself discontinuous at O. The conjecture was examined numerically, initially in special cases by Concus and Finn (1994), and later by Mittelmann and Zhu (1996) in a compre- hensive survey of minimal surfaces achieving the data (B,−B) and (−B,B) on adjacent sides of a square. The calculations suggest that a disconti- D nuity does indeed appear in 2 , although at points close to the boundary with E it was so small as to create a numerical challenge to find it. The local problem we have been discussing can Figure 10. Ellipse of regularity. be realized either by a global surface in a capillary tube, with prescribed contact angle along the walls, opposite sides of a “bubble” corresponding to con- or alternatively by a drop of liquid sitting in a ◦ tact angles 30 and 34 . The fluid used had an equi- wedge with planar walls that meet in a line L and librium contact angle measured on Earth as 32◦, extend to infinity. In the latter case, if data come with a hysteresis range of 25 . Effects of resis- from E, a solution can be given explicitly for any tance of the contact line to motion (see beginning prescribed H>0, as the outer spherical surface S of this section) were observed in the experiment, of a portion of a ball of radius 1/H cut off by the but after the astronaut tapped the apparatus the wedge. This is effectively the only possible such successive configurations of Figure 9 (top row in surface. Specifically: cover montage) appeared. In an analogous con- tainer stored for some days, the fluid on the right Theorem 7. If (B1,B2) lies interior to E and if S is climbed over the top and descended on the other topologically a disk and locally a graph near its in- side. Thus, although time is required (and per- tersections with L, over a plane Π orthogonal to L, haps also thermal and mechanical fluctuations), the then S is metrically spherical, and is uniquely de- resistance effects are overcome and a clearly de- termined by its volume, up to rigid motion parallel termined contact angle is evidenced. to L. D Theorem 7 can be viewed as an extension of the Differing Contact Angles; the 2 Domains, and Edge Blobs H. Hopf theorem that characterizes genus zero We return to the discussion of wedges above and immersions of closed constant H surfaces as met- ask what happens when two distinct contact an- ric spheres. We now keep the volume of the drop constant gles γ1,γ2 are prescribed on the two sides of a wedge. Again, new differences in possible behav- and allow the data to approach the boundary of E. For simplicity we restrict attention to the two ior appear. Setting Bj = cos γj, one sees easily that  a necessary condition for existence of any surface symmetry lines B1 = B2 joining the D regions,  1 α − D over a wedge domain Ω of opening 2α and with and B1 = B2 joining the 2 regions. We find:

776 NOTICES OF THE AMS VOLUME 46, NUMBER 7 fea-finn.qxp 7/1/99 9:56 AM Page 777

(a) (b) (c)

Figure 11. Spherical blobs in a wedge of opening 2α =50◦ for three choices of data near boundary of E. (a) Data in E D− D− near 1 ; crossing the boundary into 1 yields continuous transition to a sphere with two caps removed. D+ (b) Data near 1 ; transition to boundary yields spread extending to infinity along edge. Transition across D boundary not possible. (c) Data near 2 ; transition across boundary yields fictitious drop realized physically as drop on single plane.

Theorem 8. As one increases B1 = B2 to move from the supplementary orientation condition that the the ellipse E to the region in the parameter space mean curvature vector is directed away from the D− 1 , the free surface of the drop changes in topo- vertex, the surface is spherical. logical character from a disk to a cylinder. The If N>3, the surface need not be spherical. The surface becomes a metric sphere with two caps re- surfaces described in the Appendix of [6] have moved at the places where the drop contacts the two four vertices. A spherical drop in a wedge has two plates. Figure 11a depicts a value of the parame- vertices. We are led to: ter shortly before the transition occurs. On ap- D+ E proaching 1 within on the same line, the radius Conjecture. There exists no blob in a wedge, with D of the drop grows unboundedly, with the drop cov- N =2vertices and data in 2 . ering very thinly a long segment of L; see Figure D− D+ E 11b. As either 2 or 2 are entered from on the Stability of Liquid Bridges I line B = −B , the drop becomes a spherical cap 1 2 A substantial literature has developed in recent lying on a single plate and not contacting the other years on stability criteria for liquid bridges join- plate; see Figure 11c. ing two parallel plates separated by a distance h, D− D+ in the absence of gravity. Earlier papers considered As already mentioned, for data in 1 or in 1 no drop in the wedge can exist even locally as a bridges joining prescribed circular rings, but more graph over a plane orthogonal to L. Theorem 8 sug- recently the problem was studied with contact D angle conditions on the two plates. The initial con- gests that the same result should hold for 2 data. The statement is here, however, less clear-cut; tributions to this latter problem, due to M. in fact, recent joint work of P. Concus, J. McCuan, Athanassenas and to T. I. Vogel, were for contact and the author [6] shows that capillary surfaces angles γ1 = γ2 = π/2 for which a bridge exists as D a circular cylinder, and established that every sta- with data in 2 can exist locally as graphs. The matter may be related to the presence of vertices, ble configuration is a cylinder with volume V≥ 3 in a sense introduced in [5]. There a blob of liquid h /π. A consequence is that instability occurs topologically a ball was considered, resting on a at half the value of h that is indicated in classical system of planes, each of which it meets in a pre- work of Plateau and of Rayleigh on stability of liq- scribed angle. Points at which intersection lines of uid columns. The discrepancy arises because of the the planes meet the surface of the blob are called use here of a contact angle condition, rather than vertices. This concept extends also to configura- the Dirichlet condition employed by those authors. tions in which the intersection point is not clearly It was later shown by Vogel and this author that D the same inequality holds regardless of contact an- defined, as can occur for data in 2 . The follow- ing results are proved, for a configuration with N gles. The most complete results on stability of vertices, and for which all data on intersecting such bridges are due to Lianmin Zhou in her 1996 planes that it contacts lie in E: Stanford University dissertation; Zhou established in the general case of equal contact angles γ on the Theorem 9. If N ≤ 2, then either N =0and the blob plates that regardless of γ there is a unique sta- consists either of a spherical ball or a spherical cap ble bridge if the volume exceeds a critical Vγ. In resting on a plane, or else N =2and the blob forms 1997 Zhou also obtained the striking result that a spherical drop in a wedge. If N =3, then the blob for unequal contact angles, the stability set in covers the corner point of a trihedral angle; under terms of a defining parameter can be disconnected.

AUGUST 1999 NOTICES OF THE AMS 777 fea-finn.qxp 7/1/99 9:56 AM Page 778

Figure 12. For given volume, uniqueness holds in cylindrical tube for any section Ω ; also drop on plane is unique. But uniqueness fails in tube with curved bottom. With regard to interpretation of these and also of data on any set of linear Hausdorff measure zero earlier results, see the editor’s note following her on ∂Ω have no effect on the solution within Ω; this 1997 paper and the later clarifying papers by behavior holds without growth condition on the so- Große-Brauckmann and by Vogel. The stability cri- lutions considered. Finn and Hwang showed if κ>0 teria just described should also be interpreted in (positive gravity) then the stated result can be ex- the context of the comments directly following: tended to unbounded domains of any form, with- out conditions at infinity. In the case of equations Stability of Liquid Bridges II (9), (5) (zero gravity), uniqueness can fail for un- Stable liquid bridges joining parallel plates are ro- bounded domains. tationally symmetric but are in general not spher- A liquid blob resting on a planar support sur- ical. The meridional sections (profiles) of these face is also uniquely determined by its volume surfaces were characterized by Delaunay in 1841 and contact angle, although the known proof pro- as the “roulades” of conic sections, and yield the ceeds along very different lines from the one for sphere as a limiting case. We obtain, however, the the statement above. We may ask whether unique- result [6]: Every nonspherical tubular bridge join- ness will persist during a continuous (convex) de- ing parallel plates is unstable, in the sense that it formation of the plane to a tube. The negative an- changes discontinuously with infinitesimal tilting of swer can be seen from Figure 12. If the tube on the either plate. In the earlier stability results, tilting right is filled from the bottom until nearly the of the plates was not contemplated. maximum height of the conical portion with a liq- From another point of view, there is the re- uid making contact angle 45, then a horizontal sur- markable 1997 result of McCuan: face will result. But if it is filled to a very large − height, then a curved meniscus appears. Removal Theorem 10. If the data (B ,B ) are not in D , then 1 2 1 of liquid can then lead to two distinct surfaces with there is no embedded tubular bridge joining inter- identical volumes and contact angles. secting plates. This line of thought can be carried further: D− It should be noted here that data in 1 are ex- Capillary surfaces are characterized as station- actly those for which tubular bridges that are met- ary configurations for the functional consisting rically spheres are possible (cf. Theorem 8 above). of the sum of their mechanical (gravitational and It is not known whether nonspherical embedded surface) energies. In any gravity field and for any bridges can occur in that case. H. C. Wente gave γ, “exotic” containers can be constructed that yield an example of a nonspherical immersed bridge, an entire continuum of stationary interfaces for fluid with contact angle π/2 on both plates (and thus configurations with the container as support sur- with data in E). face, all with the same volume and the same me- chanical energy. This can be done in a rotation- Exotic Containers; Symmetry Breaking ally symmetric container, so that all the surface For solutions of (3), (5) in a bounded domain Ω with interfaces have the same rotational symmetry, given κ ≥ 0, uniqueness for prescribed fluid vol- that no other symmetric capillary interfaces exist, ume and contact angle can be established under and in addition so that energy can be decreased much weaker conditions than are needed for lin- by a smooth asymmetric deformation of one of ear equations. In fact, arbitrary changes of the the surfaces. Thus, none of the interfaces in the

778 NOTICES OF THE AMS VOLUME 46, NUMBER 7 fea-finn.qxp 7/1/99 9:56 AM Page 779

continuum can minimize the energy. Using a the- orem due to Jean Taylor, we can prove that: A minimizer for energy will exist but is necessarily asymmetric. Local minimizers for this “symmetry breaking” phenomenon were obtained computationally by Callahan, Concus, and Finn, and then sought ex- perimentally, initially in drop tower experiments conducted by M. Weislogel. In the lower left of the cover montage are shown calculated meridian sec- tions for the continuum of stationary symmetric nonminimizing surfaces, corresponding to a con- tact angle of 80 and zero gravity. The second row of the montage shows the planar initial interface in the earth’s gravity field, and a “spoon” shaped configuration that appeared near the end of free fall in the 132-meter drop tower. The spoon sur- face has the general form of the global minimizer suggested by the calculations, which had also lo- cated two local minimizers of larger energy (“potato chip” and “lichen”). The time permitted by the drop tower was, however, insufficient to deter- mine whether the spoon configuration was in equi- librium or whether it might continue to evolve to some distinct further form. In view of the ambiguity in the drop tower re- sults, space experiments were later undertaken in order to obtain a more extended low gravity envi- Figure 14. Pendent liquid drop (Thomson, ronment. These were performed initially by 1886). Lawrence de Lucas on the NASA Space Shuttle USML-1 and then by Shannon Lucid in the Mir terior to E, u(x, y) has first derivatives Hölder con- Space Station. Figure 13 (the lower right corner of tinuous to O. Note that no growth condition is im- the cover montage) shows comparison of calcula- posed in any of this work. tion with the latter experiment. The upper row If 2α>πthen it can occur that a central fan of shows the calculated spoon and potato chip; the opening π appears, in addition to the two fans at lower row shows experimental observation. The ex- the sides. periment thus confirms both the occurrence and the qualitative structure of global and also of local Pendent Drops asymmetric minimizers, in a container for which We consider what happens when κ<0 in (3). an entire continuum of symmetric stationary sur- That occurs when the heavier fluid lies above the faces appears. interface. Examples are a liquid drop hanging from a “medicine dropper”, or a drop pendent Re-entrant Corners; Radial Limits from a horizontal plate. The behavior of such so- It was shown by Korevaar in 1980 that if the open- lutions is very different from what occurs in the ing 2α of a wedge domain exceeds π, then even cases discussed above and, in general, instabili- if B1 = B2, solutions u(x, y) of (3) can exist that are ties must be expected. This “pendent drop equa- O discontinuous at the vertex , in the sense that the tion” was much studied toward the end of the last tangential limits along the two sides can differ. Lan- century, and Kelvin calculated a remarkable par- caster and Siegel (1996) showed that under gen- ticular solution of a parametric form of the equa- eral conditions radial limits at the vertex exist tion; see Figure 14. from every direction. Further, if u(x, y) is discon- This surface is unstable; however its bottom tinuous, then there will exist “fans” of constant ra- tip could be observed as a stable drop hanging from dial limit adjacent to each side. If 2α<π and if a ceiling in a house with a leaking roof. The best (B1,B2) ∈E, then the fans overlap, yielding conti- nuity of u(x, y) up to O. Earlier work, initially for stability criteria were obtained in 1980 by Wente, the case B1 = B2, due originally to L. Simon and later who showed that in the development of a drop by extended by Tam, by Lieberman, and by Mierse- increasing volume, instability occurs after the ini- mann, can be used to extend the result to the parts tial appearance of an inflection in the profile, and E D of ∂ bordering 1 , yielding continuity of the prior to appearance of a second inflection. Wente unit normal up to O, and shows that for data in- showed existence of stable drops with both “neck”

AUGUST 1999 NOTICES OF THE AMS 779 fea-finn.qxp 7/1/99 9:56 AM Page 780

From the case R0.

Past and Future I have described only some of the many new re- sults that have appeared in recent years; choices had to be made as to what to cover, and they were based largely on the simple criterion of familiar- ity. Other directions of interest can be inferred from items in the bibliography. Let me close with the opening quotation from the 1851 Russian treatise on capillarity by A. Yu Davidov: The outstanding contributions made by Poisson and by Laplace to the math- ematical theory of capillary phenomena have completely exhausted the subject Figure 15. Stable pendent drop exhibiting inflection on and brought it to such a level of per- the profile, also neck and bulge. fection that there is hardly anything more to be gained by its further inves- and “bulge”. An example is the drop of colored tigation. water in a bath of castor oil, shown in Figure 15. For the later French and German translations of the Concus and Finn proved the existence of “Kelvin book, Davidov changed the quotation to: drops” that are formal solutions of the paramet- ric equation, and which have an arbitrarily large The outstanding contributions made number of bulges. In addition they proved existence by Poisson, by Laplace, and by Gauss to of a rotationally symmetric singular solution the mathematical theory of capillary U(r) 1/r of (4); this solution was shown by phenomena have brought the subject to M.-F. Bidaut-Veron to extend to a strict solution for a high level of perfection. all r>0. (If κ ≥ 0 then any isolated singularity of a solution of (3) is removable.) It was shown by Finn The subject has advanced in significant ways that in the limit as the number of bulges increases since the time of Davidov, but may nevertheless unboundedly, the Kelvin solutions tend, uniformly still be in early stages. The material I have de- in compacta, to such a singular solution. In a re- scribed should indicate the kind of behavior to be markable work now in preparation, R. Nickolov expected; some of it may serve as building blocks proves the uniqueness of that singular solution, toward creation of a cohesive and structured the- among all rotationally symmetric surfaces with a ory. nonremovable isolated singularity. The result holds Acknowlegments without growth hypotheses. The work described here was supported in part by Gradient Bounds grants from the NASA Microgravity Research Di- In addition to the height bounds indicated in The- vision and from the NSF Division of Mathematical orems 2, 3, and 4, gradient bounds can also be ob- Sciences. I am indebted to many colleagues and stu- tained. Again, some of these have an inimical char- dents for numerous conversations, extending over acter, reflecting the particular nonlinearity of the many years, that have deepened my knowledge and problems. We indicate two such results: understanding. I wish to thank the editors of the i) (Finn and Giusti, 1977). There exists Notices for careful reading of the manuscript and for perceptive comments from which the exposi- R0 =(0.5654064 ...)/H0 and a decreasing function tion has benefited greatly. The exposition has also G(RH0) with G(R0H0)=∞ and G(1) = 0, such that profited from comments by Paul Concus and by if u(x, y) satisfies (9) with H = H0 ≡ const. > 0 in a Mark Weislogel. —R. F. disk BR(0) and R>R0, then |Du(0)| < G(RH0). The condition R>R is necessary. There is no solution 0 Skeletal Bibliography of (9) in B (0) if RH > 1. R 0 [1] E. GIUSTI, Minimal Surfaces and Functions of Bounded ii) (Finn and Lu, 1998). Suppose H = H(u) with Variation, Birkhäuser-Verlag, Basel, Boston, 1984. 0 ≥ −∞ 6 ∞ H (u) 0 and H( ) = H(+ ). Then there exists [2] U. MASSARI and M. MIRANDA, Minimal Surfaces of F(R) < ∞ such that if u(x, y) satisfies (9) in BR(0) Codimension One, North-Holland, Amsterdam, New then |Du(0)| < F(R). York, 1984.

780 NOTICES OF THE AMS VOLUME 46, NUMBER 7 fea-finn.qxp 7/1/99 9:56 AM Page 781

[3] R. FINN, Equilibrium Capillary Surfaces, Springer- Verlag, New York, 1986; Russian translation (with ap- About the Cover pendix by H. C. Wente), Mir Publishers, 1988. The top row exhibits “nearly discontinuous” de- [4] R. FINN and R. NEEL, Singular solutions of the capil- pendence on data for fluid in a cylindrical lary equation, J. Reine Angew. Math., to appear. “canonical proboscis” container in zero grav- [5] R. FINN and J. MCCUAN, Vertex theorems for capillary ity; the container section has two noses with surfaces on support planes, Math. Nachr., to ap- pear. slightly different critical angles that encom- [6] P. CONCUS, R. FINN, and J. MCCUAN, Liquid bridges, edge pass the one for the fluid. Following successive blobs, and Scherk-type capillary surfaces, Stanford taps by the astronaut, the fluid remains University preprint, 1999. bounded in height below a predicted maximum [7] K. LANCASTER and D. SIEGEL, Existence and behavior of on one side, but rises well above that height on the radial limits of a bounded capillary surface at a the other side. corner, Pacific J. Math. 176 (1996), 165–194; edito- The second row shows symmetry breaking rial corrections in Pacific J. Math. 179 (1997), for fluid in an “exotic container” designed for 397–402. zero gravity, shown just prior to and near the [8] T. STÜRZENHECKER, Existence of local minima for cap- end of free fall in a drop tower. The curves on illarity problems, J. Geom. Anal. 6 (1996), 135–150. the lower left are calculated meridional sec- [9] A.D. MYSHKIS, V.G BABSKI˘,ı N.D. KOPACHEVSKI˘,ı L.A. SLOBOZHANIN, and A. D. TYUPTSOV, Low-Gravity Fluid tions of a continuum of symmetric equilibrium Mechanics. Mathematical Theory of Capillary Phe- surfaces bounding equal volumes in the con- nomena, translated from the Russian by R. S. Wad- tainer; all these surfaces are unstable. The fluid hwa, Springer-Verlag, Berlin, New York, 1987. volume for the drop tower experiment is cho- [10] F. ALMGREN, The geometric calculus of variations and sen so that the initial section in the earth’s modeling natural phenomena, Statistical Thermo- gravity field can be identical to the horizontal dynamics and Differential Geometry of Microstruc- section of the calculated zero gravity family. tural Materials (Minneapolis, MN, 1991), IMA Vol. The lower right shows comparison of cal- Math. Appl., vol. 51, Springer, New York, 1993, pp. culation and experiment for surfaces yielding 1–5. asymmetrical critical points for mechanical en- [11] K. GROSSE-BRAUCKMANN, Complete Embedded Constant Mean Curvature Surfaces, Habilitationsschrift, Uni- ergy in an exotic container during an experiment versität Bonn, 1998. on the Mir Space Station. The apparent global minimum (“spoon”) and a local minimum of larger energy (“potato chip”) are both observed by the astronaut. The cover montage was prepared by Gary Nolan of the NASA Glenn Research Center. —R. F.

AUGUST 1999 NOTICES OF THE AMS 781