J. Chem. SOC., Faraday Trans. 1, 1982, 78, 1491-1497 Capillary Phenomena Part 17.-Properties of Fluid Rings Between a Sphere Above a Horizontal Plane in a Gravitational Field BY ERNEST A. BOUCHER* AND TIMOTHY G. J. JONES School of Molecular Sciences, University of Sussex, Brighton BNl 9QJ Received 1st June, 1981 The prediction of liquid ring properties for the sphere/plane system has been approached by obtaining accurate fluid/fluid meridian curves and associated quantities, e.g. capillary pressure differences, fluid ring heights and volumes, and by using approximate expressions for limiting configurations. Reduced quantities, which can be related to terrestrial gravity conditions, are compared with some zero-gravity properties. Provided the horizontal solid is of large extent, fluid rings in a gravitational field (but not at zero g ) , will reach a maximum height, unless they can completely engulf the sphere.It is not possible for the sphere to be completely enveloped if its radius is r 2 0.54[2y(1 -cos8)/Apg]i. For contact angles 8 > Oo, the ring height tends to a limit as its volume increases, having passed through the height maximum. It has become common to call an amount of liquid between two touching solids a pendular ring when the meridian describing the fluid body is axially symmetric,l and to use the term fluid bridge if the solids are held apart.2 In this paper fluid constrained by a sphere resting on a horizontal solid plane is examined, extending the range of capillary phenomena for macroscopic systems of the type recently re~iewed.~ Cross and Picknett4 examined the case of a liquid ring with particular reference to the physicochemical equilibrium between water and air. Mason and Clark5 and Clark et a1.6 have examined properties of zero-gravity bridges between two spheres and a sphere and a plane.Orr et aZ.l examined rings, mostly for zero-gravity, or more strictly zero Bond number ( B = gPAp/ y, where I is a characteristic length for the system), and gave many analytical solutions. They also made some estimates of gravitational effects. Iinoya and Muramoto7 examined liquid rings between two touching spheres and between a sphere touching a cone, using 8 = 0' and arcs of circles as approximate meridians. In this paper the emphasis is on the Laplace equilibrium of systems in a gravitational field, but the differential equations whose numerical solutions give the fluid/fluid meridian curves are given in a form which allows the gravitational field strength to be varied.Approximations to the behaviour of fluid rings are also given. The fluid forming the ring is always more dense than the second fluid. THEORETICAL BACKGROUND Fig. 1 shows the geometrical quantities used to describe the generating circle, horizontal line and fluid ring meridian curve which give the three-dimensional system by rotation about the vertical Z-axis. For identical contact angles 8 at the plane and the sphere, the meridional angles to the horizontal are, respectively, $l = 1800-8, 42 = e+v (1) 1491I492 PLANE-SPHERE FLUID RINGS and the three-phase confluence (Xe, Z*) is related to ry by Computation is conveniently started at ( X e , 20) using ry as the parameter which specifies growth as the ring volume Va is increased.For computation 2 can be used as the independent variable by artificially dealing with the mirror image of fig. 1 for which d Z > 0: the arc length S was used as independent variable for configurations encountering d Z = 0 [see ref. (2) and (3) for details of computational methods]. The pressure difference across the interface at height Z is given in terms of the shape factor H , where < = 1 corresponds to terrestrial gravity and 4' = 0 to zero gravity. Xe=Rsinry, Z e = R(1-cosry). (2) AP = 2(H-[Z) (3) 0 S X' xo x FIG. 1.-Coordinate system representing a pendular ring of fluid between a plane and a sphere.NUMERICAL RESULTS The systems studied are for equal contact angles of 8 = Oo, 30°, 60°, 90' and 180' at the two contacts with reduced sphere radius R = 0.5. The ring phase a is always more dense than phase j?. When 8 = 0' the half-angle ry increases to an upper bound (denoted ymax in table 1) as the ring volume Ya increases as shown in fig. 2. The sphere will not, therefore, be enveloped by phase a. In the limit as the contact radius X' with the plane increases the pressure difference across the interface is 2H --+ 0 as Xo + a, and the meridian becomes that of a holm (e.g. the fluid body formed by dipping a rod into an infinite For all the cases 8 > O', as Ya increases ry increases to a maximum, rymax, and then decreases to a limit, ry, corresponding to the infinitely large ring indicated in fig.3. Values of rymax and woo are given in table 1. The attainment of a maximum pendular ring height, Zgax, is analogous to the existence of sessile drop height maxima (and subsequent limiting heights for large drops) which were recently discussed.8 A critical reduced sphere radius is being anticipated (see below), above which envelopment by phase a cannot occur, i.e. larger spheres will always have a dry patch provided the liquid has sufficient space to spread sideways. In contrast, Orr et a1.l pool).E. A. BOUCHER AND T. G. J. JONES 1493 TABLE ACCURATE AND APPROXIMATE ANGLES v03 AND ymax AND SHAPE FACTORS H CORRESPONDING TO THE LATTER FOR R = 0.5 ~~ ~ ~ ~~ ~~ - Wv,/" WrnaJ" H 01" acc . approx. acc. approx.acc. approx. 0 - - 99.00 98.93 0 0 30 110.0 109.6 1 1 1.28 111.45 0.398 0.395 60 117.0 117.0 120.32 120.92 0.783 0.764 90 122.5 122.3 128.33 128.40 1 .085 1.080 180 113.0 112.6 124.62 125.20 1.517 1.527 0 0.5 1 .o 1.5 2.0 X FIG. 2.-Pendular ring configurations for R = 0.5, 8 = 0' and several values of w : (a) 30°, (b) 60°, (c) 90' and (d) vmax = 99.0'. In the limit the large ring adopts the holm configuration. X 0 1.0 2 .o 3 .O L.0 X FIG. 3.-Pendular rings for R = 0.S and 8 = 30' having values of v / : (a) 60°, (b) 90°, (c) 110'; (d) 1 1 1.28' and (e) tyco = 110'. Curve (d) represents the maximum height.1494 PLANE-SPHERE FLUID RINGS found that for a reduced system under zero-gravity conditions the properties did not depend on reduced sphere radius [fig.3 of ref. (1) for 6 = 40'1. The approximate values of y/, in table I , which are in excellent agreement with accurate values, were obtained as follows. It is assumed that for the infinitely large ring, the lower portion is part of a large sessile drop8 and the upper part is a portion of a holm,g then, approximately (4) KO( 2/ 2 R sin v [ 1 + cos (8 + I,Y,)]~ R(1 -cosy/,) = (1 -cosO)Q K l ( d 2 Rsin w,) with the + sign for 42 < 180° (raised holm) and the - sign for 42 > 180' (submerged holm). In eqn (4), KO and K, are modified Bessel functions of the second kind. A first-integral approximation for holm meridians can be adapted to give the limiting ring height, Zfitax, when 0 = 0'. In the resulting expression Ko(2/2 R sin y / m a x ) / K l ( d 2 R sin u/ma,> = R(1 -COS y/max)/(I +COS vmax)' ( 5 ) vmax is found by trial-and-error solution for a given R value, and Zgax is given by The analogous approximate height can be obtained for 0 > 0' by combining holm and sessile drop first integrals.It is assumed that the sessile drop height maximum occurs in the vicinity where y/ is a maximum, and that there is negligible pressure difference across the interface where the meridian has an inflexion (fig. 3). The com bined approxima tion eqn (2). is used as before to give tymaX for use in eqn (2). The shape factor, H , for the approximation, being one-half of the pressure difference across the interface where it meets the plane solid, is given by the maximum sessile drop height. Accurate and approximate values of vmax are compared in table 1 for R = 0.5.Cross and Picknett4 measured what is in effect vmax as a function of actual sphere radius (their fig. 4). Eqn (5) provides justification for their expectation that y/y,ax -+ 180' as R -+ 0, and it furthermore shows that yma, -+ 0' as R --+ a. Eqn ( 5 ) can also be used to give the contact height Z* (and raised fluid volume) for 6 = 0' when a sphere is just touched against a large liquid surface so that a holm forms spontaneously. Cases of 0 > Oo are dealt with by replacing (1 +cos vmax); by [ 1 + cos(8 + ~max)]'. The upward forcef, exerted by the ring on the horizontal plane is f l = n X o sin 8 - 7 ~ ( X ' ) ~ H f, = nR sin ly sin (8+ u / ) - zR2 sin2 y(H+ R cos I,U - R). y a = f, - f, - p 7 s (7) (8) (9) and the downward force exerted on the sphere is The volume of fluid forming the ring is where Vals is the sphere segment volume (standard formula) surrounded by phase a.The pressure difference A P e across the fluid/fluid interface at the three-phase confluence is 2(H-Z*) = 2(H+ Rcos y/- R). Fig. 4 shows the dependence off, on y/ for R = 0.5 and several values of 6. For 6 = Oo, 30' and 60° the positive force decreases almost linearly as y/ increases to ymax. The limit for 8 = 60' is close to f 2 = 0. For 0 = 90' the force is always negative andE. A. BOUCHER AND T. G. J. JONES 1495 2 f2 0 - 2 - -4- FIG. 4.-Dependence on v/ of the forcef, exerted on the sphere by the presence of the pendular ring, for several values of the contact angle 6: (a) Oo, (b) 30°, (c) 60°, ( d ) 90' and (e) 180'.- 1 0 1 FIG. 5.-Dependence of the pressure difference A@ at the upper contact on ring volume f a for 6 values: (a) O', (b) 30°, (c) 60°, (d) 90" and ( e ) 180".1496 PLANE-SPHERE FLUID RINGS has little dependence on cy, whereas for 0 = 180' the negative force passes through a minimum and then increases, The dependence of AP@ on Va in fig 5 shows quite a different pattern of behaviour. The curve for 0 = 0' begins at A p e = - co and, while remaining negative, terminates at Va = 1.7 when the holm meridian is reached. The curve for 0 = 30'is similar in shape, but it has a very broad maximum before levelling off as Va -+ 00. The curve for 0 = 60' has a distinct maximum at A p e N + 1.5. In contrast, the curves for 0 = 90' and 180' begin at A p e = + 00 when Va = 0 and decrease smoothly to positive A p e limits as Va + GO.By taking the meridian to be an arc of a circle for small Va it can be shown that the initial portions of the curves of the dependence of A P e on Va obey (10) sin 8 cos cy R sin cy cos 0 cos O( 1 + cos v / ) + R - R(1 -cos v / ) A p e = which gives the correct limits as Va -+ 0. Fluid rings of small volume correspond to capillary condensation at relative pressures p/p' of vapour less than unity (saturation) when 0 is small or zero. The difference in some properties between rings in a gravitational field and those where g = 0 ([ = 0) is shown by the comparison made in table 2 with the data of Orr et a1.I TABLE 2.-cOMPARISON OF REDUCED QUANTITIES WITH c = 1 AND c = 0 FOR R = 0.5 AND e = 00 30 0.005" 0.005b 60 0.051 0.073 90 0.561 0.449 99 1 .652" - 140 1 82.75b 0.270 0.270 0.6 16 0.609 1.48 1 1.242 - 33.604 - 12.182 - 12.21 5 - 1.929 - 2.045 -0.187 -0.366 0 - - 0.0002 2.798 2.791 2.462 2.383 2.110 1.875 2.022 - 0.649 a First entry for c = 1, second entry for gravity-free, [ = 0, from Orr et a1.l DISCUSSION The limiting height of the large rings is well described in terms of the approximations involving Bessel functions.The maximum height of a sessile drop grown on a plane surfaces can be used to estimate whether a ring can completely envelop a sphere. It is supposed that the fluid will only completely surround the sphere if the sessile drop of volume Va has a height exceeding 2R, which to a good approximation means that envelopment will only be possible if (1 1) R < 0.54(1 -coSe)B.When R = 0.5, 0 must exceed ca. 82" before the largest ring could become a sessile drop with the sphere completely contained inside it. The possibility of an energyE. A. BOUCHER AND T. G. J. JONES 1497 barrier preventing spontaneous envelopment of this kind has yet to be investigated. The inequality eqn (1 1) implies that for R 2 0.77 a sessile drop can never develop, regardless of 8, from a ring, because the most favourable condition is 8 = 180'. For R < 0.5 the contact angle 0 required to prevent eventual engulfment decreases to the limit 8 = 0" for R = 0, i.e. no sphere of finite radius will be engulfed when 8 = 0'. When Va is small (small ty) it can be shown that the meridian becomes an arc of a circle giving the limit according to eqn (8), fz = 2ncos 8 as Va --+ 0; a conclusion reached without proof by McFarlane and Tabor,1° Cross and Picknett4 and Princen.'l Comparison can be made between the case ([ = 1) 8 = 0" and R = 0.5 and the gravity-free (c = 0) data of Orr et aZ.l.When c = 1 the limiting configuration is that of a holm of finite volume for which ty,,, N 99', whereas for [ = 0 the limiting meridian is a catenary with ly,,, N 140' and Va -+ GO. For rings having volumes such that t,u 5 30" there are no major differences in the meridian curves for the two cases and little difference in quantities such asfz, Va, AP* and X'. As ly increases some differences occur, most notably in A P and Va: for [ = 0, A P e -+ 0 and nlra --+ co, whereas for c = 1, Va tends to a finite limit and A P + - 228. There is an interesting similarity with a pendent drop grown at the end of a tip, which shows a maximum in A P and a limit in Va (detachment), but in the absence of gravity the drop can be grown indefinitely, although A P e still possesses a maximum. F. M. Orr, L. E. Scriven and A. P. Rivas, J. Fluid Mech., 1975, 67, 723. E. A. Boucher and M. J. B. Evans, J. Colloid Interface Sci., 1980, 75, 409. E. A. Boucher, Rep. Prog. Phys., 1980, 43, 497. N. L. Cross and R. G. Picknett, Trans. Faraday SOC., 1963, 59, 846. G. Mason and W. C. Clark, Chem. Eng. Sci., 1965, 20, 859. W. C. Clark, J. M. Haynes and G. Mason, Chem. Eng. Sci., 1968, 23, 810. E. A. Boucher and T. G. J. Jones, J. Chem. Soc., Furaday Trans. I , 1981, 77, 1183. E. A. Boucher and T. G. J. Jones, J. Chem. Soc., Faraday Trans. I , 1980, 76, 1419. ' K. Iionya and H. Muramoto, Zairyo, 1967, 16, 70. lo J. S. McFarlane and D. Tabor, Proc. R . SOC. London, Ser. A , 1950, 202, 224. l 1 H. M. Pnncen, in Surface and Colloid Science, ed. E. Matijevic (Wiley, New York, 1969), vol. 2, p. 1. (PAPER 1 /882) 49 FAR 1