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Capillary Engineering for Zero Gravity. Critical Wetting on Axisymmetric Solid Surfaces Jacob Lucassen* Mathenesselaan 11,2343HA Oegstgeest, The Netherlands
Emmie H. Lucassen-Reynders Unilever Research Laboratorium, Olivier van Noortlaan 120, 3133 A T Vlaardingen, The Netherlands
Albert Prins Wageningen Agricultural University, Department of Food Science, Section Dairy and Food Physics, Bomenweg 2, 6703 HD Wageningen, The Netherlands
Philip J. Sams Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, Merseyside L63 3JW, United Kingdom Received May 27,1992. I n Final Form: September 18,1992 Fluid surfaces of revolution contained in two coasial rings, tend to adopt a catenoidal shape because this minimizes their surface area. With a variation of the distance between the rings, the surfaces paas through a family of catenoids which are all tangent to a common envelope, the Lindeliif envelope. We discuss the physical significance of two mathematical loci in a catenoid family. The first locus is found by terminating all catenoids at the point where they all have an area equal to that of the generating circle of the family. A solid body shaped according to this locus gives indifferent or critical wetting equilibrium for fluids which approach it with a 90° contact angle. This includes the case of thin f i i drawn from soap solutions. We propose the term ‘Sinclair locus” for the curve which generates such conditions for a 90° angle. The physical significance of the second locus, the Lindel6f envelope, is found by combining ita mathematical properties with Young’s equation for contact angle equilibrium. It is found that a solid body with a shape generated by rotating a Lindeliif envelope gives critical wetting conditions for liquids with a zero contact angle. Both the Sinclair locus and the Lindeliif envelope are cusps, with the latter being more sharplypointed. The three-dimensionalbodiesobtained by their rotation are therefore cuspoids. For contact angles between zero and 90°, solid bodies of intermediate shape are required for indifferent wetting equilibrium to be possible. It can be predicted that they should all be cuspoids, but no simple way to calculate their shape is available as yet. The effect of “capillary engineered” surfaces on wetting and dewettingkinetics is discussed. It is arguedthat solid surfacescovered with cuspoids giving indifferent wetting equilibrium should lead to easy wetting reversal. Rupturing of foam lamellae based on capillary instability could be optimized by engineeringthe shape of the particles which trigger the instability. An experiment with soap f i i supported by a Sinclair cuspoid machined from PVC confirmed the theoretical expectation about indifferent equilibrium.
Introduction The explanation or prediction of fluid boundary shapes is a field of physica (and of surfacechemistry)which, during its early stages, ran parallel with the development of calculus of variation and its application to the search for surfaces of minimum area. During the 19th century there was a frequent exchange of ideas concerning these problems between physicists and mathematicians, and often their object of common interest was soap films. Soap films are good models for the behavior of surfaces in general. As they are relatively unaffected by gravity, the mathematics required for their description is greatly simplified. They are of direct relevance for understanding the behavior of fluid surfaces in cases when gravity can be neglected, due to small size or to a location outside the field of gravity. As they consistently adopt a minimal area, as far as boundary conditions permit, they also provide mathematicians with an analog model which can be used for solving complex variational problems.
* To whom all correspondence should be directed. 0743-746319212408-3093$03.00/0
Half a century before Perrin’ made his observations on the molecular architecture of soap films, Plateau2 published a very thorough study of the physical and mathematical background of their shape and changes in shape. His work is still an essential part of the foundations of present day physics and chemistry of fluid surfacee. He was the first to use the word catenoid for catenaries of revolution, the axisymmetric surfaces of zero average curvature. His discussion of shape and stability of soap films stretched between two coaxial rings ran parallel with the work of Linde16f,3v4a Finnish mathematician. Lindel6f, who worked in Helsingfors (as Helsinki was called when Finland was part of the Russian empire), published a number of papers on catenoids between 1860 and 1870. One objective of the present paper is to discuss an aspect of this work which is relevant for the behavior of fluids on solid surfaces. ~~
~
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(1) Perrin, J. B. Ann. Phys. 1918, 10, 160. (2) Plateau, J. A. F. Statique experimentale et thkorique de8 liquides soumis aux seules forces molkculaires; G a u t h i e r - V i : Paris, 1873. ( 3 ) Lindelbf, L. L. Mathematische Annalen 1870,2, 170. (4) LindelQ, L. L.; Moigno,F. Legons de calcul des uariations; Paris, 1861.
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In the early part of this century, a group of American mathematicians further developed the understanding of the properties of catenoids. Prominent among them was Gilbert Ames Bliss: who treated the theory of catenoids as a major example in his classical book on calculus of variation. Mary Emily Sinclair6 solved the problem of stability of minimal surfaces of revolution terminating on a fixed surface. More recently, Isenberg’ wrote an extremely readable review on soap films, and it is through his extensive bibliography that we were able to trace much of the relevant old literature. But in general, the subject of minimal surfaces and of catenoids seems to be very much at the periphery in present day surface and colloid chemistry, as witnessed by the scarcity of references to it in most textbooks.
Survey of Catenoid Properties The area of the body of revolution, obtained by rotating a line between two points around the z-axis, is given by
(-0)
(r.0)
Figure 1. Projection of a family of catenoids passing through a circle with radius r1 according to eq 2 for different values of a and b given by eq 3. Each curve starting at (r1,O)proceeds to a point of tangency with the envelope (along the line shown in bold), and proceeds continuously away from this point (along the thm limes). Only fully bold face curves between (r1,O) and (r,O),i.e. those which do not extend beyond the point of tangency, represent minimal surfaces. 2.5
where r is the distance to the axis of symmetry. To find the minimizing arc, i.e. that line which upon rotation gives the minimum surface area, Euler’s equation, applied to the integrand of eq 1 should be solved. The solution gives the general equation of a catenary 2-a
r = b cosh (2) b where the constants a and b have to be obtained from boundary conditions. Although the discussion in the following will be based on properties of (two-dimensional) catenaries, all conclusions apply to catenoids, the bodies of revolution of which they are the describing line. If one point of the describing catenary is kept fixed at (zl,rl),a and b of all the catenaries passing through that point are interdependent and can be expressed in terms of a new parameter A
b = - rl (3) cosh A and all catenaries passing through the fixed point depend on one parameter only. They are called a one-parameter family of catenaries. Figure 1 shows such a family of catenaries through (r1,O). To illustrate the point that they can be considered to be the projection of a family of catenoids, the catenaries through (-rl,O) are shown as well. The line connecting the two points is the projection of the circle through which all catenoids of the family must pass.
a = zl-rl
A cosh h
LindelBPs Envelope and LindelBf s Envelope Theorem As can be seen from Figure 1, there is a common envelope, tangent to all catenoids of the family. This surface which is called the Lindehf envelope, divides space into two parts. Through each point outside the envelope two catenoids pass. Of those two, only one represents a true minimal surface. According to Jacobi’s condition, (5) (a)Bliss, G. A. Calculus of Variations;The Open Court. La Salle, IL,1925. (b) Bliss, G. A. Calculus of Variations;The Open Court. La
Salle, IL, 1925; p 93, e~ 8. (c) Bliss, G. A. Calculus of Variations; The Open Court: La Salle, IL,1925; p 102. (6) Sinclair, M . E. Ann. Math. 1907,8, 177; 1908, 9, 151. (7) Isenberg, C. The Science of Soap Films and Soap Bubbles; Tieto, Ltd.: Clevedon, England, 1978.
-1
5
-1 0
-0 5
00
05
10
15
Figure 2. Three different loci in catenoid families: ordinate,z; abscissa,r. (A) Lindel6f envelope. (B)MacNeish locus. Transition between catenoidal area being absolute and relative minimum. (C)Sinclair locus. All catenoidsfrom the generating circle have equal area and are perpendicular to the locus.
the catenoid which touches the envelope between the fixed point and the variable point, does not represent a minimum surface. Only the catenoid which touches the envelope beyond the two points is a true minimum. Inside the envelope none of the catenoids of the family can penetrate. For points on the envelope itself, the two catenoids coincide. The location of the envelope is determined bySb
The curve obtained by solving eq 4 is shown by A in Figure 2. The Lindel6f envelope is cusp-shaped, and the three-dimensionalbody it forms will be termed a cuspoid. There exists an interesting relationship between surface areas of catenoids to which the envelope is tangent and the area of the envelope. This Lindelof envelope theorem, as it was called by Bliss,%statesthat the difference between areas of two arbitrary catenoids up to the point where they touch the envelope, just equals the area of the part of the envelope they surround. Figure 3 gives a three-dimensional illustration of the envelope and of the meaning of the envelope theorem. Part of two catenoids touching the envelope of their
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r:
+ r2- b(z - zl) + 2(z, - a ) E[ sinh 7- sinh 2(2-a)] = 0 (6) 2 b
The MacNeish locus is indicated by curve B in Figure 2. It is a cuspoid, similar to the Lindel6f envelope, but less pointed.
Figure 3. Three-dimensional impression of Lindel6f cuspoid. Lindeliif s theorem states that the area difference between the catenoids, partially drawn at A and B, equals the shaded area on the envelope.
common family is shown on the left-hand (A) and on the right-hand (B) side of the drawing. The darkened area on the envelope indicates the width of the collar which, according to the theorem, has an area which equals the area differencebetween the two catenoids. We will return below to the significance of this theorem for the behavior of fluid surfaces on solids.
Catenoidal Instability If a ring with radius r2 coaxial to the catenoids of the family passing through a ring with radius rl (Figure 1)is moved upward, no minimal and stable surface can exist between the two rings as soon as the moving ring reaches the envelope of the family determined by the fixed ring. The fact that muthemutics dictates that no connecting minimal surface can exist beyond this point means that any physical surface should become unstable. This is illustrated by experiments with soap films stretched between two coaxial rings. Such films always undergo an instability at a critical separation. This well-known phenomenon, therefore, has as a background the mathematical properties of catenoids and is not related to the instabilities in soap filmswhich precede rupture. Rupture can, and sometimes does, occur as a side effect due to the abrupt nature of the catenoidal instability. Usually though, the film does not break but adopts the shape of the nearest minimal surface. This involves its separation into two planar films inside the two supporting rings. The total area of the rings in that case corresponds to the Goldschmidt8discontinuous solution of the surface minimization problem. In fact, well before the ring separation reachesthe critical point, this Goldschmidt solution already represents an absolute minimum in area (and thus in energy for a soap film), while the catenoid only gives a relative minimum. The locus of those points on the catenoids in the family at which the transition takes place from catenoidal to planar films, giving the absolute minimum, has been calculated by Ma~Neish.~ The transition should obey the condition
Minimal Surfaces of Revolution Terminating on a Fixed Surface From Figure 1it can be seen immediatelywhich catenoid is supported by a moving ring. Such a ring intersects the projection of the catenoid family in mathematical points. When the catenoid is instead supported not by such an infinitesimally thin ring, but by a solid surface with macroscopic dimensions, the intersection is a line instead of a point. Selection of the catenoid with minimum area now requires the condition of orthogonality to be fulfilled; for the liquid surface to be minimal, it has to approach the solid surface at a 90° angle. This condition, dictated by mathematics, is obeyed by soap films for physical reasons. Because the tensions of the two opposite surfaces of the film will be equal, the condition of force equilibrium requires a 90° approach angle (or contact angle) between the film and any fixed surface of which the tension can be considered uniform. In that case the energyminimization depends on the area minimization of the film alone. The calculation of the equilibrium position and of the point of instability for catenoidal films between a ring and a cone is more complicated than it is for the case of two coaxial rings. This is because the orthogonality condition causes the film to slide over the conical surface as the distance between ring and cone is varied. The problem was solved by Mary Emily Sinc1air.G She performed experiments with soap films,which was an unusual thing to do for a mathematician, to confirm her theoretical predictions. The more general problem of soap films on cones will be discussed in a separate paper, in which specialattention will be paid to the special case of a cone with a 90° top angle, i.e., the horizontal surface of a soap solution from which a catenoidal film can be drawn with a ring. The Special Case of Indifferent Equilibrium In the case of conical solid surfaces there is usually only one member of a catenoid family which can adopt an equilibrium position on a conical surface. We want to show that there are solid surfaces of a particular shape, on which an infinite number of catenoids of one family can be at equilibrium. Such surfaces should obey the condition that all catenoids up to the point of intersection have the same area. In the case of soap films, and single interfaces with a 90° contact angle outside the gravity field, this means that all positions are energeticallyequally favorable and that the equilibrium should therefore be indifferent. A special case of this indifferent equilibrium is obtained when all areas are also equal to the area of the generating circle (with radius rl) of the family. This leads to a condition which is very similar to the condition used before to obtain the MacNeish locus or
or (8)Goldschmidt, B.Determinatw superfkei minimue rotatione curuae data duo puncta jungentis circa datum axem ortae; Gettingen, 1831. (9) MacNeish, H. F. Ann. Math. 1905, 7,72.
The locus, which can be calculated by solving the transcendental eq 8, is illustrated by curve C in Figure 2.
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Figure 4. Three-dimensional impression of Sinclair cuspoid.
All catenoidsare perpendicularto this cuspoid.and can therefore be modeled by soap f i i .
It is cusp-shaped, very much like the Lindeldf envelope and the MacNeish locus. The main difference is that it curves back toward the z = 0 plane, while the catenoid supported by it degenerates into a flat, ring-shaped film outside the generating circle with an area equal to that of the circle. We propose to call this locus the Shclair locus, in recognition of her pioneering contribution in using soap films to study minimal surfaces terminating on solids. A three-dimensional impressionof the corresponding Sinclair cuspoid is shown in Figure 4. When a soap filmheld in a ring of radius rl, coaxial with the cuspoid, just touches the cuspoid's point, then this film can be expected to adopt an arbitrary equilibrium position on the cuspoid surface. In all cases, the total area is the same,the film intersects the cuspoid at the required 90° angle, and therefore there is no free energy difference between the different positions. However, an infinitesimal displacement of the top of the cuspoid toward the film should be enough to set the film in motion and make it move upward over the surface of the cuspoid away from ita tip. When it is subsequently withdrawn to a position just beyond ita starting point, the film can be expected to move all the way back to ita original position. If we consider the soap film as a model of a fluid surface which has a 90° contact angle on a solid, the phenomenon described could be considered as critical wetting. Other solid surfaces of indifferent equilibrium for soap filmscan be designed by still requiring that all catenoids have the same surface area but dropping the condition that this area should be equal to that of the generating ring. Examples of such surfaces are shown in Figure 5. For catenoidal areas smaller than that of the generating ring, the solid approaches the shape of a torus with the generating ring as axis. When the area is larger than that of the ring, the solid surface intersects the Lindeldf envelope. At the point of intersection, indicated by A and B in Figure 5, there should be a transition from indifferent equilibrium to instability. LindeliiPs Envelope as a Surface of Critical Wetting The shape of the cuspoid surface needed to give an indifferent wetting equilibrium for a 90° contact angle, as provided by a soap film,is easy to calculate because the energy minimization only involves one surface area. The solid surfaces on both sides of the film have equal surface tensions and, therefore, do not feature in the minimization. This would be the same,of course, if the filmwere replaced by a fluid-fluid interface which makes a 90° contact angle with the solid. The situation becomes significantly different, however, if the contact angle is not 90°, as would usually be the case. In general, the calculation of a surface of indifferent equilibrium, leading to critical wetting conditions, in the case of an arbitrary contact angle is not quite straightforward. The minimization involves three
Figure 6. Loci in catenoid familiesfor which areae are different fractions of the area of the generating ring: (A) catenoid area = 1.15 X ring area; (B)catenoid area * 1.075 X ring area (in these two cases, the dotted region in the center is inside the Lindel6f envelopeand can, therefore,not support equilibriumcatenoids); (C)catenoid area = ring area (the Sinclair locus); (D)catenoid area = 0.925 X ring area. interfacial areas, each of them multiplied by a constant factor, the surface tension. Only the fluid-fluid interface is a catenoid. For the solid surface no general equation is known. We now want to show that there is an exception for the case of complete wetting (0' contact angle). By combination of Young's law and Lindelaf s envelope theorem, it can easily be shown that the Lindeldf enveloperepresents a surface of indifferent wetting equilibrium for that case. We imagine the envelope as depicted in Figure 2 to be a solid body wetted by a liquid. Young's law for complete wetting gives (9) = QSA The change in free energy caused by moving the catenoid to a higher position on the envelopesurface,i.e., by wetting the collar on the envelope indicated by the black area in Figure 3, is given by QSL + QLA
AF = QLAAQ - QUA1+ USLA, - USA& (10) where the areas A1 and A2 correspond to the higher and lower catenoid (see Figure 3) and where A3 is the increase in wetted area on the envelope. Lindeldfs theorem gives AZ-A, = A , (11) Combination of eqs 9-11 shows that for any change in position of the catenoid on the envelope
AF=O (12) Thus, Lindelaf s envelope (Figure 3) is for contact angle zero what the locus shown in Figure 4 is for the case of a 90° angle. The following thought experimentshows what this means in practical terms. If, in gravity-free surroundings, the tip of a solid, shaped as a Lindelaf cuspoid, is brought in contact with a liquid which web the solid, and if the surface of this liquid is contained in the generating circle of the catenoid family determining this envelope, then the same phenomena can be expected as described above for the soap film.A small movement of the tip through this surface should bring about complete wetting while a reversal of the movement should again give complete dewetting. Experimentalverification of the wetting behavior of a solid body shaped like a Lindelaf envelope would require surroundings in which gravity can be neglected: a small system, agpacecraft, or a fluid interface between equally dense liquids.
Generalization for Arbitrary Contact Angles It is obvious that the solid surface shapes which lead to indifferent wetting equilibria, as calculated above for contact angles of 90 and Oo, are special cases and that similar surfaces do exist-and can be calculated in principle-for any arbitrary contact angle 8. They should intersect all catenoids of a family at an angle 8, and the surface areas
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Figure 6. Sinclair cuspoid.
should obey a generalized,Lindelijf theorem FILM MOVES TOWRDS LEFT
FILM MOVES T O W R D S RIGHT
A,-A,=A,cos8 (13) These solid surfaces should once again be cuspoidswith their tip at the origin (0,O).Work is under way to calculate the shape of the surfaces for this general case.
Experimental Verification of Indifferent Equilibrium for Soap Films The central part of a Sinclair cuspoid was machined out of PVC (Figure 6). It was based on a generating circle with a radius of 4.3 cm. The outer edge of the cuspoid was taken at the maximum of curve C in Figure 2. To this purpose, eq 8 was solved with a simple iteration program to give 600 coordinates of the surface contour at about equal intervals. These data were subsequently fed into a computer-steered lathe. A ring with the required radius of 4.3 cm, which is the radius of the circle from which the cuspoid shape had been calculated, was also constructed from PVC. The ring had a sharp inner edge in order to controlthe position of films formed in it. When the center of such films coincides with the cuspoid's tip, any film formed between the ring and the cuspoid surface should give the expected indifferent equilibrium. The distance between ring and cuspoid could be varied by means of a micromanipulator. The following experiment was performed. A circular soap film was formed inside the ring by withdrawing it from a surfactant solution. As surfactant we used Teepol with some added glycerol to increase the film's long-term stability. The ring and the Sinclaircuspoid were adjusted into a coaxialposition, and the tip of the prewetted cuspoid was made to touch the center of the film. Subsequently the cuspoid tip is moved over a distance of about 2 mm toward the film. The film can be seen to move gradually over the cuspoid surface away from its tip. The top left picture of Figure 7 shows the film during this process. When the cuspoid is now moved in the opposite direction, to a point just beyond its starting position, the f i b beginsto retreat. The top rightchandpicture of Figure 7 shows the film just before it detaches from the cuspoid. If, however, at any stage during the film's movement, the cuspoid tip is returned to its starting position in the z = 0 plane, the film movement stops altogether. The bottom picture of Figure 7 shows the film in a stationary position with the cuspoid tip in the plane of the ring. This last observation proves that the equilibrium is of a genuinely indifferent nature. Discussion The experiment with the soap film on the Sinclair cuspoid confirmsthat solid surfacescan be shaped in such
F I L M STAT ION A RY (IN DIF F E RE NT EQ UIL I BR I UM)
Figure 7. Behavior of soap film on Sinclair cuspid.
a way that the contact angle equilibriumis of an indifferent character. Although,strictly speaking,this has so far only been proven for a 90' contact angle, cuspoids with the same behavior for arbitrary contact angles must exist. It is well-known that the wetting characteristics of solid surfacesstronglydepend on surfacetexture.1° Roughening of surfaces causes a polarization of the apparent, macroscopic,contact angle. When the microscopiccontact angle lies between 0 and 90°, the macroscopicvaluetends toward ' 0 . For values between 90 and 180°,the polarizationcauses the apparent angle to shift toward 180'. This behavior can be clarified on the basis of a model surface which has a 90' microscopic contact angle. If such a surface is made rough by coveringit with small cuspoidsgiving indifferent equilibriumfor this angle, such as the one shown in Figure 4, and if the distance between adjoining cuspoids is of the order of the diameter of their generating circle, then the surface as a whole can be expected to display indifferent or critical wetting properties. Making all cuspoids a bit steeper, which is similar to increasing the surface roughness, leads to complete repellence of the fluid; making them a bit less steep leads to complete wetting. For surfaces on which microscopic contact angles are different from 90°, wetting reversal willrequire the control of the distance between cuspoid point and liquid surface where the notional generating ring of the catenoids is supposed to be. In other words, we have to ensure that the distance between cuspoid and surface is alwaya an (10) Adameon, A. W.Physical Chemistry of Surfaces, 4th ed.;John Wiey and Sone: New York, 1982; Chapter X.
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Figure 8. Three-dimensional impremion of a surface covered with cuepoids for critical wetting. The case for a zero contact angle is shown.
independent variable and can be kept under control. This is because the cuspoid relevant for the contact angle under consideration will only display critical wetting behavior when catenoids approachingit originate from a generating ring at one particular distance. A surface coated with Sinclair cuspoids will be wetted when the microscopic contact angle is smaller than 90° and when the distance between solid and fluid surfaces is allowed to vary; abrupt wetting reversal w i l l only take place when the cuspoid points are at the level of the liquid surface and then moved toward it or away from it. Plant leaves, insect skins, or bird feathers” can be expected to be fully water repellent if the microscopic contact angle is larger than 90O and if they are coated with cuspoids-or asperities with a similar shape--‘sharper” than the one depicted in Figure 3. In order to get sudden wetting reversal for microscopic angles smaller than 90°, water should first be drained off the surface so as to force the tips against the liquid-air surface. The formation of holes in supported liquid films and the subsequent dewetting caused by hydrophobic solid particles present on the supporting solid are familiar everyday phenomena. The theory developed in the present paper shows that by coating solid surfaces with particles-or asperities-with a predetermined shape, orientation and separation, as illustrated by Figure 8, dewetting (or wetting) could be made into a more controllableprocess. The surface roughness created by these asperities should be such that after &wetting has started at the asperity tips, the fluid surfaces should remain close to indifferent wetting equilibrium. This permits the speed of the (11) Baxter, S.;Casaie, A. B. D.Tram. Faraday SOC.1944,40, 646.
Lucagsen et al. dewetting process which follows the initial hole formation to be kept under control. Dewetting at the high speeds which accompany instability growth far from equilibrium is usually an incomplete process, as pockets of liquid are left behind. There is another application for which solid particles with a %apillary-engineered’’ shapecould possibly be used. It is well-known that hydrophobicparticles suspended in liquid can trigger the rupture of free liquid f i i (soap films) formed out of this liquid. Their action has been ascribed to a capillary instability.12 The theory presented in this paper should be able to indicate the shape of those particles which would give them optimal effectiveness. For example, if spiky solid particles (like rolled-up hedgehogs) with cuspoid points would be present inside foam lamellae, they can be expected to initiate dewetting and capillary instability when forced against the two surfaces of the film.
Biographical Notes Joseph Antoine Ferdinand Plateau (1801-1883),Physics Professor at Gand, Belgium, from 1835. Leonard Lorenz Lindelgf (1827-19081,Mathematics Professor at Helsingfors from 1857 to 1874. Mary Emily Sinclair (1878-19551,Mathematics Professor at Oberlin College, Ohio (1907-1944). Gilbert Ames Bliss (1876-19611,MathematicsProfessor at Chicago. Although Jean Baptiste Perrin (1870-1942) was a contemporary of Bliss, lived in the same country for a couple of years, and studied another aspect of the properties of soap films, it is doubtful if they knew about each other’s work. It is perhaps interesting to note, as a curiosity, that Bliss applied his knowledge of calculus of variations to construct improved f i i tables during World War I, while Perrin worked at the same time on acoustic detection devices for submarines. Acknowledgment. We thank Mr. H. de Rooy of WWeningen APicdtural University for the accurate construction of the Sinclair cuspoid with the aid of a ComPute-teered lathe, and Ir. c. G. J. BisPerink for the skillful performance of the soap film experiment. (12)Garrett, P.R. J. Colloid Interface Sci. 1979,69, 107.
