J. Chem. Soc., Faraday Trans. I, 1982, 78, 1499-1506 Capillary Phenomena Part 18.-Conditions for the Flotation of Solid Spheres at Liquid/Liquid and Liquid/Vapour Interfaces in a Gravitational Field BY ERNEST A. BOUCHER* AND TIMOTHY G. J. JONES School of Molecular Sciences, University of Sussex, Brighton BN1 9QJ Received 19th June, 1981 Conditions for unaided sphere flotation at fluid/fluid interfaces are explicitly given in reduced terms using density parameters ka = @"-p")/Ap and kb = CpS-&/Ap, where Ap is the positive density difference between fluids a and b. The method given entails the minimum of numerical computation needed to give the limits for stable flotation: at the limits the sphere detaches and either rises or sinks. Representative accurate data illustrate the range of flotation depending on the contact angle and the sphere radius. Two approximations are also discussed.Previous studies of single solid spheres at fluid/fluid interfaces can be divided into those where the sphere is manipulated by an externally applied f ~ r c e l - ~ and where the sphere can float ~naided.~-lO The unaided flotation of a sphere at a fluid/fluid interface depends on its radius Y, the contact angle 0 (measured through the denser fluid phase), the interfacial tension yap and the densities, pa, pp and ps. The study by Boucher and Kent4? l1 of the equilibrium and stability of spheres mechanically manipu- lated at fluid interfaces showed how these quantities can be used to predict the conditions for stable flotation. Sphere flotation was shown to be a special case of an ideal stress-controlling ~ y s t e m , ~ where equilibrium is stable only between the maximum and minimum values of the applied force.The shapes and related properties of the asymptotic fluid/fluid interfaces meeting the required boundary conditions at the sphere are generated by the accurate numerical integration of the Pascal-Laplace equation.12 A large range of values of the reduced sphere radius R is examined for contact angles in the range 0-180" to determine the limiting values of the sphere density for which stable flotation can occur. Approximate solutions to the problem of sphere flotation are also discussed. THEORETICAL BACKGROUND Fig. 1 shows two different holm-sphere systems which are mirrored in the horizontal plane 2 = 0.The criteria for the stable flotation of the lower system where 0 > 90' may be directly obtained from the criteria for the upper system where 0 < 90". The contact angles of the two systems are 0 and O', where 0' is used to denote the complement of the angle 0. The density ps of the sphere is expressed as a reduced density difference4 ka = @'-p*)/Ap; kp = @'-@)/Ap (1) where Ap is the positive density difference between the two fluid phases and a is always the phase which forms the holm. Reduced quantities of dimension Lv are given by dividing the actual dimension l v by av where a2 = 2yaB/Apg. 49-2 14991500 SPHERE FLOTATION 7 V \ P FIG. 1 .-Coordinate systems for (a) raised and (b) submerged spheres giving mirror-image holm configurations. The reduced external absolute force which must be applied to the sphere to maintain it in an arbitrary position is f:?: = ~Xesind,+n(Xe)~ Z*+ kaP,S+kp5/8$S (2) where ( X e , Z*) are the coordinates of the three-phase confluence, 4 is the meridian angle at the three-phase confluence, and V and F-@3 are the volumes of the sphere in phases a and p, respectively.[The meridian angle is generally @ = arctan (dZ/dX).] The first two terms on the right-hand side of eqn (2) represent the force exerted by the fluid/fluid interface on the sphere. The first term is the force due to the vertically resolved component of the interfacial tension, and the second term is the force due to the pressure difference (Young-Laplace) across the curved interface. The two terms give the holm v01ume,~g~~ including any solid located between 0 and Z e , P = n X 0 sin d, + Z e .(3) Substitution of eqn (3) into eqn (2), noting that v7 = YS - Va, where YS is the sphere volume, gives for a raised-holm configuration f:?; = V"+kBVS- Va*s (4) with an analogous equation for submerged holms. An excess applied forcef:;; is defined as the applied force on the sphere less that, kpVs, on the sphere entirely immersed in the less dense p phase ( 5 ) fext = va- y a , s * Z" =Z0+Rcost# ( 6 ) exc The position 2" of the centre of the sphere from the level 2 = 0 is where w( = 4 - 0) is the angle made between the three-phase confluence and the axis of rotational symmetry; y~ is always measured through the denser phase. The contact radius X* is related to the angle v/ by X e = R sin t# (7)E.A. BOUCHER AND T. G. J. JONES 1501 and the volume Va,s of solid surrounded by the a phase is It can be seen from the above equations that when R and 8 are chosen for a particular system, the holm-sphere configurations depend only on the angle ly. The only unknown is the value of 20, which must generally be obtained from numerical computation. For each pair of 8 and R values and a series of values of tp the corresponding 20 values are found from the holms which satisfy the conditions X = Rsinly and (b = 6+y. FIG. 2.-Dependence of absolute externally applied force on sphere-centre position for R = 0.5,O = 0' and various kfi values: (a) 5 , (b) 0, ( c ) - 3, (d) - 8. Fig. 2 shows the dependence off.$ on 2" for R = 0.5 and 8 = 0' and several values of kb including kb = 0.The shapes of the curves are all identical and independent of the values of kb, which only determine the positions of the curves relative t0f.g; = 0. The shape of the force-displacement curves, and hence the values of Z x wheref:;; is a maximum or minimum, depends only on the values of R and 8. The force- displacement behaviour for any sphere system may, therefore, be characterized by the excess force given by eqn (9, which can be considered as the absolute force when kb = 0. Consequently it is only necessary to carry out numerical computation for this special case. For the raised holm configuration shown in fig. 1 where the contact angle is 8 while for the submerged holmI502 SPHERE FLOTATION The addition of eqn (9) and (10) for the two systems characterized by the angles v/ and t,v' gives (1 1) with Va(0) = - Va(8').Fig. 3 showsf::: as a function of 2" for spheres of reduced radius R = 0.5 and 8 = 40 and 140'. The value of 2" at the force maximum for 8 = 40' is equal to -2" at the force minimum for 8 = 140'. Similarly, 2" at the force minimum for 6 = 40' f:; (8) +f::; (8') = - v s FIG. 3.-Dependence of externally applied force on sphere centre position ( R = 0.5), showing how systems with contact angles 8 and 180'-8 are related: (a) 6 = 40, (b) 8 = 140'. is equal to - 2" at the force maximum for 8 = 140'. The force-displacement curve for 6 = 140' can be obtained from the curve for 8 = 40' by transforming Z x (8) to - 2 (8') and using eqn (1 1) to transform f: (8) to f:;; (6). The essential condition for the unaided flotation of any sphere is that the net force on the sphere is zero,fl$t = 0.It has been shown4 that stable sphere flotation can only occur for holm-sphere configurations where df,e,"dZX > 0, and the onset of instability occurs when df,e,"dZX = 0. Fig. 2 shows that for a given system defined by a pair of R and 8 values, there exists a range of ki values (i = a or /?) for which flgi = 0. The upper value of ki, ka (upper), corresponds tof:;; (min) = 0, where the holm is submerged and ps > fl > pa. From eqn (4) with f;;; (min) = 0 ka (upper) = - f:$ (min)/ VS. (12) When ka > ka(upper) the sphere will detach from the interface and sink into the denser phase. The lower value of ki, kp (lower), occurs for the raised holm where f;$k (max) = 0 and ps < pp < pa.E.A. BOUCHER A N D T . G. J . JONES 1503 From eqn (4) withf:c:(max) = 0 kP (lower) = -f:z; (max)/ VS (13) such that when kp < kp (lower), the sphere will detach from the interface and rise into the less dense p phase. The criterion for the stable flotation of a sphere is ka (upper) 3 ki 3 kp (lower). f::; (max, O)+fz:k (min, 6') = - VS (14) (1 5 ) From eqn (1 1) it can be seen that which combined with eqn (12) and (13) gives Similarly ka (upper, 0) + kP (lower, 6') = 1. kp (lower, 0) + ka (upper, 8') = 1 and consequently the values of ka (upper) and kp (lower) for 8 > 90" can be obtained from the limiting values of ki for 8 < 90". The case of 8 = 8' = 90" is unique in that 2 Vmax) = - Z (fmin) and kP (lower) = 1 - ka (upper).NUMERICAL RESULTS FOR SPHERE FLOTATION ACCURATE PARAMETERS A range of R values from to 5.0 with 8 = 0, 20, 40, 60, 80 and 90" has been studied by accurate numerical computation. For each pair of R and 8 values, the values off:;; (max) andf:;: (min) have been computed to give kP (lower) and ka (upper) using eqn (12) and (1 3), respectively. Eqn (1 5 ) and (1 6) are used to give the limiting values of ki for the range of R values with 8 = 100, 120, 140, 160 and 180". When 6 = 0' it is found that there is no minimum inf:;;, only a lower value off: = - VS, where ly = 180' and the interface is planar. Consequently ka(upper) = I for all values of R. Similarly for 8 = 180' there is no maximum in the excess force, only an upper value off:;; = 0, where ly = 0' and hence kP (lower) = 0 for all R.Table 1 shows the values of ka (upper) for a range of R for all of the values of 0 given above. To our knowledge no data of comparable range are available in the literature. The values of k.8 (lower) can be calculated from eqn (I 6) and (1 7). The table can be considered to be a compilation of the values of -f:$(min)/ Vs, wherefrom f:; (min) and f:; (max) can be obtained. APPROXIMATE PARAMETERS James13 has given an approximate first-integral holm solution for small holm-solid systems. The approximation, which has already been critically discussed,14 gives Z e as a function of 4 and X e . Using this approximation VQ = nR sin cy sin (8+ cy)+nR3 sin2 cy sin (8+ ly) (ln[22/2/(Rsin cy(1 -cos(8+cy)))]-yy) (18) where (here) y = 0.5771 6.. . is Euler's constant. The substitution of eqn (1 8) into (5) with W S given by eqn (8) gives an expression forf:$ which depends only on cy when R and 8 have been specified. For particular values of R and 8 the values of cy which correspond to the stationary values off:;;, and hence the limiting values of ki, are found by trial and error. When R < 0.1 the agreement between the stationary values off:; calculated from1504 SPHERE FLOTATION James' approximation and from accurate computation is excellent. When R = 0.1 and 8 = 0, 20, 40, 60, 80 and 90°, values off:$(max) andf::i(min) given by accurate computation and by the approximation agree to four decimal places. The accuracy of the approximation increases as R decreases. TABLE 1 .-ACCURATE VALUES OF ka(upp~~) FOR SEVERAL CONTACT ANGLES AND SPHERE RADIIa e p R + O .O ~ 0.1 0.3 0.5 1 .o 2.0 5.0 20 40 60 80 90 100 120 140 160 180 224 879 1876 3099 3751 4402 5627 6625 7277 7503 3.26 1.25 9.80 2.00 19.9 3.16 32.3 4.62 38.9 5.41 45.6 6.21 58.1 7.74 68.4 9.0 1 75.2 9.86 77.5 10.2 1.092 1.366 1.804 2.363 2.668 2.978 3.575 4.075 4.410 4.529 I .024 1.097 1.218 1.375 1.461 1.550 1.720 1.864 1.961 1.995 1.006 1.027 1.061 1.105 1.130 1.154 I .202 1.241 1.267 1.275 1.001 1.005 1.01 1 1.019 1.024 1.028 1.036 1.043 1.048 1.049 ~ ~ a Values for R = 0.001 range from 2.4 x lo4 to 7.6 x lo5 over the range of angles. For moderately large spheres, where the holm meridians are tending to be of similar shape but of varying radius at their waist, the volume Va can be obtained from the first-integral holm approximation of Boucher and Jones.l4 The approximate expression for Va is Va = nRsiny/sin(8+y/)+zR2sin2 y/KO(1/2Rsin y/)[l +cos(8+yl)]~/Kl(1/2Rsiny/) where KO and K, are modified Bessel functions of the second kind. By a similar procedure, the stationary values off::; are obtained, and hence the limiting values of ki can be calculated. A comparison of the values off:g:(max) andf::i(min) obtained from the use of eqn (19) and from accurate computation for R = 1 .O shows that the approximation is good to better than 1 % for f::i (max), and that off:;; (min) values agree to four significant figures, except that the discrepancy is -0.002 on -6.122 for 8 = 90'. The accuracy will increase as R increases. The accurate data in table 1 can be represented by approximate empirical expression^,^^ which give explicit relations between the limiting values of ki, R and 8.The expressions for the limiting reduced density differences are (19) ka (upper) = 1 + {cos [( 180' - 8)/2])2.2 R-1.925 kp (lower) = - {cos(8/2)}2.2 R-1.925. (20) (21) and Eqn (20) and (21) rearrange to give the upper values of R for which flotation occurs for given values of ki and 8 R (upper) = (ka - l)-0.519 (cos [( 1 80' - 8)/2])1.143 for p s > pb > pa (22) R (upper) = lkBl-0.51~ [cos ( 0 / 2 ) y ~ ~ for p s < pb < p a . (23) andE. A. BOUCHER AND T. G . J. JONES 1505 When R > R(upper) upper phase when ps range of 8 values for (21) when R, ka and the sphere will detach from the interface and either rise into the < pb < pa, or sink into the denser phase when ps > @ > pa.The which flotation can occur can be determined from eqn (20) and kb are known. The lower value of 0 is given by (24) cos [( 180' - 8 (lower))/2] = (ka - 1)0.455 R0.875 while the upper value is given by cos [8(upper)/2] = lkp1°.455 R0.875. When 8 > 8(upper) or 8 < 8(lower) the sphere will detach from the interface and stable flotation cannot occur. FIG. 4.-Comparison of accurate (lines) and approximate (dots) density parameter limits (see text) as a function of reduced sphere radius R for B values: (a) 180, (b) 90, (c) 40, (d) 140, (e) 0'. DISCUSSION Some of the accurate numerical data given in table 1 are shown in fig. 4, where ka (upper) and kp (lower) are plotted as a function of the reduced sphere radius R for given values of 0.The lines represent the data from accurate computation while the dots for ka (upper) and kp (lower) are given by eqn (20) and (21), respectively. The accuracy of the empirical equations has already been discussed.15 The curves in fig. 4 show that as R + 0, ka (upper) + + 00 and kp (lower) --+ - 00, indicating that spheres which are very much denser than the denser fluid phase and very much less dense than the lighter phase, respectively, can float. When R -+ 00, ka (upper) -+ 1, indicating1506 SPHERE FLOTATION that flotation can only occur by ps -+ pb, where p is the denser phase. Similarly kp (lower) -+ 0 as R -+ co and flotation is only possible as ps + pp, where p is now the less dense phase. The values that ka (upper) and kb (lower) tend to as R -+ 00 show that the surface forces are becoming insignificant in supporting the sphere.In the limit as R -+ 00, flotation can only occur when the density of the sphere is intermediate between the densities of the two fluid phases. The use of the data given in table 1 is exemplified by application to a particular system. A sphere of radius Y = 1.9 mm and density ps = 4 g ~ m - ~ is required to float at a water/air interface for which the capillary constant is a = 3.8 mm, and hence the reduced sphere radius is R = 0.5. The water/air interface makes a contact angle of 100' with the sphere. The reduced sphere density in the less dense phase is kff = 4. For a sphere with R = 0.5 and 8 = 100' table 1 shows that ka(upper) = 2.978, and hence kff > ka (upper) indicating that the sphere cannot float.For the water/air system with R = 0.5 and 8 = loo', the sphere will only float if ps d 2.978 g ~ m - ~ . The range of 8 values for floating the sphere with R = 0.5 and ka = 4 can be estimated by using eqn (24), noting that ps > pp > pa, where the water is the p phase and the air is the a phase. Eqn (24) gives 8 (lower) = 128O, and hence the sphere will only float if 8 > 128'. With 8 = 100' and ka = 4, the upper-sphere radius calculated from eqn (22) is R (upper) = 0.42 and flotation can only occur if R < 0.42 or Y d 1.6 mm. Previous studies of the flotation of spheres have in the main only considered systems of limited range and presented a small amount of numerical data, often obtained by the use of approximations.For example Tovbin et a1.' consideJed holm-sphere systems where the meridian angle <f, was everywhere close to 180°, and hence the meridian curve could be accurately represented by the (conventional) Bessel solution for shallow interfaces. The study closest to that presented here was made by Huh and Mason,s who tabulated values of what we term ka(upper) (in their table 1) from R = 0.177 to R = 1.273 for 8 = 30, 60, 90, 120, 150 and 180'. No direct comparison of the values of ka (upper) from the two studies can be made by using the two sets of published data because Huh and Mason used the capillary constant a = (y/Apg)*, and hence their sphere radii are 2/2 times bigger than the values given here. However, reasonable comparison of the two sets of data can be made using the approximate treatment: eqn (20). For example, in our nomenclature, for R = 0.707 and 8 = 90°, Huh and Mason give ka (upper) = 1.8737 while eqn (20) gives ka (upper) = 1.91. For R = 0.700 and 8 = 90' table 1 gives ka(upper) = 1.890. With R = 0.177 and 8 = 30' Huh and Mason give kff (upper) = 2.6166 while eqn (20) gives a value of 2.44. T. G. J. J. acknowledges an S.R.C. studentship. A. D. Scheludko and A. D. Nikolov, Colloid Polym. Sci., 1975, 253, 396. C. Huh and S . G. Mason, Can. J. Chem., 1976, 54, 969. C. Fieber and H. Sonntag, Colloid Polym. Sci., 1979, 257, 874. E. A. Boucher and H. J. Kent, J. ChLm. SOC., Faraday Trans. 1 , 1978, 74, 846. C . W. Nutt, Chem. Eng. Sci., 1960, 12, 133. S. Hartland and J. D. Robinson, J . Colloid Interface Sci., 1971, 35, 1971. C. Huh and S. G. Mason, J . Colloid Interface Sci., 1974, 47, 271. H. C. Maru, D. T. Wasan and R. C . Kintner, Chem. Eng. Sci., 1971, 26, 1615. ' M. V. Tovbin, I. I. Chesha and S . S . Dukhin, Colloid J. USSR, 1970, 32, 643. lo A. V. Rapachietta and A. W. Neumann, J. Colloid Interface Sci., 1977, 59, 555. l 1 E. A. Boucher and H. J. Kent, Proc. R. SOC. London, Ser. A , 1977, 356, 61. l 2 E. A. Boucher, Rep. Progr. Phys., 1980, 43, 497. l 3 D. F. James, J. Fluid Mech., 1974, 63, 657. l4 E. A. Boucher and T. G. J. Jones, J. Chem. SOC., Faraday Trans. I , 1980, 76, 1419. l 5 E. A. Boucher and T. G. J. Jones, J. Colloid Interface Sci., 1981, 83, 645. (PAPER 1 / 1000)