3. Chem. SOC., Faraday Trans. I , 1982,78, 2643-2648 Capillary Phenomena Part 19.-Systems Subjected to Various Gravitational Field Strengths BY ERNEST A. BOUCHER* AND TIMOTHY G. J. JONES School of Molecular Sciences, University of Sussex, Brighton BNl 9QJ Received 2nd September, 198 1 A scheme is described which allows quantities relating to axially symmetric capillary systems to be computed for various values of the gravitational field strength, or of the fluid-phase density difference, or of the fluid/fluid interfacial tension. Examples are given using pendent and sessile drops. The treatment deals with zero-gravity conditions (or Ap = 0) under which these drops become spherical, but fluid bridges, for instance, possess a variety of shapes. Analytical formulae exist for many gravity-free axially symmetric cases, but they are not always convenient to use and may not be available for sufficient quantities.The approach used here involving an explicit capillary parameter is quite general. In dealing with the behaviour of axially symmetric fluid bodies, such as pendent and sessile drops and fluid bridges,l the interface shape is given by combining the equations which relate fluid-phase (a, 8) hydrostatic pressures pa and fl at height z to the fluid densities pa and and the mean surface curvature j, and involve the acceleration g due to gravity and the interfacial tension y@. It is customary to use reduced or dimensionless quantities, e.g. 2 = z / a where a = (2yap/Apg)i: whether or not the factor 2 is used is immaterial, and Ap is the fluid-phase density difference.Having obtained numerical results, e.g. the maximum stable volume of a pendent drop in reduced terms,2* the actual maximum volume for a drop at the end of a tip of radius r is simply obtained by evaluating a, or, if the volume is already known from experiment, a can be used to give (say) yaB. However, this common approach is not suitable for ' gravity-free ' cases, including the neutral buoyancy condition Ap = 0, because as a 4 m the reduced quantities degenerate to zero. What is required is a scheme whereby: (i) a pendent drop, for example, becomes a spherical segment of finite volume as g + 0; (ii) the solution by numerical integration of the appropriate Laplace differential form of eqn (1) is compatible with known analytical solutions, e.g.involving elliptic integrals, for gravity-free systems of simple symmetry; and (iii) systems can be described in terms of varying gravitational field strengths, including the zero-gravity limit (or near-zero for space experiments*), terrestrial gravity and greater or lesser field strengths. Analogous variations in Ap or in yap would be similarly accommodated, e.g. the slow decrease in yap leading eventually to pendent drop detachment as described experimentally by Tornberg and L ~ n g h . ~ The purpose of this paper is to show briefly, by example, in terms of varying gravitational field strength, how these objectives can be met. The method of calculation by numerical integration was first used to obtain the generating curves for the axially symmetric zero-gravity bodies the unduloid, the nodoid and the catenoid, given on a small scale in fig. 7 of ref.(1): it was considerably more convenient than 26432644 CAPILLARITY I N VARYING GRAVITY checking existing analytical meridian equations for misprints before evaluating the elliptic integrals. Interfacial areas, fluid volumes and centres of mass where appropriate can also be evaluated by the direct computational technique to be described [see ref. (1) for relevant formulae]. Axially symmetric systems are convenient for study experimentally and theoretically, but the principle of allowing for variation in the quantities in the conventional capillary constant a is quite general. It can be argued that computed meridian curves can be unreduced by using a sequence of a values corresponding to a range of gravitational field strengths.While this is true in principle, except for a = 0, one cannot expect tabulated values, e.g. of Hartland and Hartley,e to cover all the meridian shapes and fluid-body quantities that are required. Furthermore, an ad hoc approach hinders the interpretation of capillary phenomena. The variations using c are explicit and lead to the occurrence of [ in thermodynamic formulae for varying conditions including the zero-gravity case. COMPUTATIONAL TECHNIQUE The essence of the technique is to scale an actual linear dimension 1 by a*, giving L = l/a* as before,192 using the usual values for the density difference Ap*, the interfacial tension y* and the acceleration g* due to terrestrial gravity to define a*.More generally, however, there is a capillary variable a = (2y/Apg)a which can cover a range of conditions. Starting from a chosen vertical position (in reduced terms 2 = z/a*) 2 = 0, where the pressure difference APo is 2H, the shape factor H specifies the mean surface curvature of the fluid/fluid interface. The variation of the pressure difference A P with 2 depends on the value of the scaling or capillary parameter c, introduced to take account of the values of Ap, y and g defining a for the system A P = 2(H--Z); C2 = a*/a. (2) One now obtains by numerical computation' an interface shape or meridian curve for each [ and a given H instead of a single meridian for [ = 1. Finite ( X , 2) meridians are obtained where [ = 0, corresponding to zero-gravity conditions, or to Ap = 0 or y + CQ. Eqn (2) is reminiscent of the scheme introduced earlier1y2 to give pendent drops, emergent bubbles, sessile drops, captive bubbles and fluid bridges from a single set of equations: a quantity d (= _+ 1) was used.It is now seen how the general parameter c can give, e.g., pendent drops for [ > 0 and sessile drop for [ c 0, using these as examples in the following account. The equation to be solved numerically when the meridian arc length S is the independent variable is d@/dS = 2(H-[Z)-sin@/X (3) dX/dS = COSQ,, dZ/dS = sin@,. (4) with the usual relationships The meridian angle is defined as arctan(dZ/dX). Computation is started at X = 2 = S = 0 and Q, = Oo for chosen H and [. The computational procedure is similar to that already de~cribed.l-~ EXAMPLES AND DISCUSSION Fig.1 shows meridian curves for H = 2 and five values of c, including the gravity-free case of [ = 0. Without the procedure introduced here the meridians would have culminated in a point at the origin when [ = 0. A set of meridians meeting variousE. A. BOUCHER AND T. G. J. JONES 2645 2 z 1 0 0.5 1 1.5 X FIG. 1.-Set of pendent-drop meridian curves for shape factor H = 2 and the C values indicated: (a) 0, (6) 0.5, (4 1, (4 3, (4 5. 1 1 I I FIG. 2.-Set of pendent-drop meridian curves for R = 1 and Va = 2.09 beneath a solid tip when [ takes the values: (a) 0, (b) 1, (c) 1.4, ( d ) 1.473, (e) 1.3.2646 CAPILLARITY I N VARYING GRAVITY z FIG. 3.-Dependence of the meridian angle 4 at the tip on C for R = 1 and (a) Va = 2.09, (b) Va = 0.60 and (c) Va = 0.38.prescribed physical conditions can be obtained with varying c as shown in fig. 2 for fixed pendent drop volume [Va([ = 1) = 2.091 and solid tip radius R = 1 (c = 1). c reaches a definite maximum of 1.473 for the example, as is shown in fig. 3 in terms of the three-phase-confluence (tip) meridian angle #. The free energy F of the pendent drop can be written as an extension of earlier treatmentsly where A afi is the reduced area of the drop a/#l interface, 2. is the drop centre of mass and Z* is the drop height. An increase in c is regarded as an increase in the gravitational field strength or in Ap, or a decrease in y : the computations leading to the diagrams herein represent equilibrium configurations, but [ can be used in variational tests for thermodynamic stability.Fig. 4 shows that for the case Va = 2.09 there is a distinct cusp in F as a function of c at the limit of [ corresponding to the strain-controlled limit of stability, i.e. the drop will detach when 5 reaches 1.473. Taking the normal a/mm x 4 for the air/water interface, the tip radius r is 4 mm and the volume of the water drop is ua/mm3 = 130.6. The field strength can be steadily increased to give nearly 14 times terrestrial g before this drop must (partially) detach. Fig. 3 and 4 also show, respectively, plots of # and Fagainst c for the cases Va = 0.6 and 0.38. In neither of these cases is there a maximum in c or a cusp in F: c reaches an extreme value corresponding to # = Oo at the tip when detachment will occur.The inset in fig. 4 shows the case where the cusp occurs at [ = 1 for Va = 2.907, i.e. this is Vgax for terrestrial conditions. The case of Va = 0.6 and 4 x 0' at the limit in c corresponds very closely to the 'Lohnstein point', where the curves of maximum volume at constant contact angle and at constant tip radius coincide at 8 = # = Oo, for the limiting value, [ = 5. Finally, fig. 5 shows how a sessile drop of fixed volume (Va = 4.1) meeting a plane solid at contact angle 4 (= 1 50°) changes shape as [ is systematically varied. F = Aap+2cVa(Z*-Ze) ( 5 )E. A. BOUCHER AND T. G. J. JONES 2647 3 2 F 1 0 - 1 5 FIG. 4.-Dependence of pendent-drop free energy F on [ for (c) V" = 2.09, (b) V" = 0.60, (c) V" = 0.28 and (inset) ( d ) YOr = 2.91. 0 1 X 2 FIG. 5.-Sessile-drop meridians for Va = 4.1 and 6 = 150° and the lrl values: (a) 0, (b) 0.3, (c) 1, ( d ) 4.4, (e) 10.6.2648 CAPILLARITY I N VARYING GRAVITY T. G. J. Jones acknowledges a S.R.C. studentship, and discussions with Dr M. J. B. Evans are gratefully acknowledged. E. A. Boucher, Rep. Prog. Phys., 1980, 43, 497. E. A. Boucher and M. J. B. Evans, Proc. R . Soc. London, Ser. A , 1975,346,49. E. A. Boucher, M. J. B. Evans and H. J. Kent, Proc. R. Soc. London, Ser. A , 1976, 349, 81. European Space Agency, Special Publication no. 114 (1976). E. Tornberg and G. Lungh, J. Colloid. Interface Sci., 1981, 79, 76. S. Hartland and R. W. Hartley, Axisymmetric Fluid-Fluid Interfaces (Elsevier, Amsterdam, 1976). (PAPER 1 / 138 1)