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 Acountered 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.