1 Attraction and repulsion of floating particles M. A. FORTES Deptrrtrr~~lc.~lto tle Metcrllrrgitr, 111stitlrto Srrpc>rior T6oiic.o; Cetltro tic MecBtrictr c,mntc~ritris tlrr Ut~icc~rsitltrtle Tt;ctlictr de LO11ocr. Ao. Ro~.isco Prris, 1096 Lishorr Cotle.r, Port~rgt~l Received March 17, 1982 M. A. FORTES. Can. J. Chem. 60, 2889 (1982). The horizontal force that has to be applied to floating cylinders to equilibrate the capillary force is calculated for a general geometry with cylindrical symmetry and for any wetting characteristics of the cylinders. The force on a cylinder can be related to a single parameter of each of the two menisci that contact the cylinder. It is found that the force between parallel vertical plates or parallelipipeds can be repulsive if one of the contact angles is acute and the other obtuse. The force on cylinders between fixed walls is also calculated and the results suggest that it may be possible to separate floating particles with different wetting properties by placing them between two walls with different contact angles. M. A. FORTES. Can. J. Chem. 60, 2889 (1982). On a calculc la force horizontale que doit &tre appliquee a des cylindres flottants pour Cquilibrer les forces capillaires, pour une geometric gcncrale avec une symetrie cylindrique et pour n'importe quelle caracteristique de rnouillage des cylindres. On peut relier la force sur un cylindre a un parametre unique pour chacun des deux menisques qui sont en contact avec le cylindre. On a trouve que les forces entre les plaques verticales paralleles ou parallelcpipedes peuvent &tre repulsives si un des angles de contact est aigu et I'autre obtu. On a egalement calcule la force qui s'exerce sur le cylindre entre deux parois fixes; les resultats suggtrent qu'il est possible de separer des particules flottantes ayant des proprietes diffkrentes de mouillage en les pla~ant entre deux parois avec des angles de contact differents. [Traduit par le journal] 1. Introduction It is frequently observed that particles floating at the surface of a liquid at rest are submitted to forces which in most cases tend to produce clusters of particles. These may be either attracted to a wall or find an equilibrium position near the centre of the surface. These effects are due to the horizontal component of the capillary forces associated with the curved liquid surface. The calculation of such forces requires the determination of the equilibrium shape of the liquid surface (more generally of a fluid interface) by solving the Laplace differential equation submitted to the appropriate boundary conditions, for example, a constant contact angle of the liquid with the solid particles. When gravity is included, the only geometry for which the shape of the fluid interface can be determined in closed form is one with cylindrical symmetry. Gifford and Scriven (1) have analysed this situation for two "parallel" identical circular cylinders in free flotation and were able to determine the horizontal force as a function of separation by taking into account the vertical displacement experienced by the cylinders as their separation changes. They have assumed, however, that the arc wetted on each cylinder remained constant, instead of considering a constant contact angle; therefore they were not able to draw conclusions on the effect of wetting characteristics on the horizontal force. Under those conditions the (attractive) force was found to fall off nearly exponentially with separation. More recently, Chan et a/. (2) found a similar variation for the case of two free floating spheres. They used approximate solutions for the Laplace equation, applicable to small Bond numbers (small particles). Again no special attention was given to the effect of the wetting properties of the floating bodies on the horizontal force. The purpose of this paper is to analyse, for a cylindrical geometry, the effect of the wetting characteristics on the horizontal force between two or more floating bodies. The vertical component of the capillary forces will not be included in the analysis, that is, the positions of the floating bodies relative to some reference level are imposed by externally applied vertical forces. To simplify the treatment we shall assume frequently that the bodies are plates or prisms such that the wetted surfaces are planar and vertical. However, application to other geometries is straightforward. The analysis is a static one, as in previous treatments (1, 2), in that equilibrium of the bodies is always assumed; what is calculated is in fact the horizontal force that must be applied to the bodies to maintain equilibrium. In section 2 we obtain the relevant equations for the fluid interface profile and in section 3 we derive general expressions for the horizontal force. It turns out that this force can be simply related to a single parameter of each of the two menisci contact $0 1.OO/O National Research Council of CanadalConseil national de recherches du Canada
2 2890 CAN. J. CHEM. VOL. 60, 1982 ing a floating cylinder. Examples of calculation of such forces between two cylinders or between a cylinder and the walls of a container are given in section 4, which show the effect of wetting properties. These calculations further suggest that in favourable cases it may be possible to separate floating particles of two different materials by placing them between two parallel walls with properly chosen contact angles. 2. Shape of the fluid interface We use cartesian coordinates x, z, as in Fig. 1, with the origin at the point of contact of the fluid interface with the solid; x is horizontal and z vertical directed in such a way that the profile is in the region x > 0, z > 0 near the origin; therefore i = dzldx > 0 in this region. There are four possibilities, indicated in Fig. 1, as regards the sign of z near the origin and the direction of z relative to that of g (the gravity acceleration). Let pa be the radius of curvature at the origin, with the sign of 2 at or near the origin; y is the fluid interface tension and Ap(> 0) is the difference between the densities of the two fluids. By noting the way the pressure difference across the interface varies with z, it is easy to conclude that the Laplace equation can be written as where the + sign applies to the profile region between the origin and the point (if it occurs) with i = dxldz = 0, and the - sign "after" this point. To integrate eq. [l] we first note that where the - sign applies ifi > 0. Introducing the capillary constant, a, defined by FIG. 1. Orientation of the coordinate systemr, z and definition of the angle 0; g is the gravity acceleration. we may put [l] in the form applicable in the region with x > 0. In other regions the - sign may have to be replaced by +. But x [61 = cos 8 (1 + k2)llz where 8 (-n I 8 I n) is the angle defined in Fig. 1 between the profile and the X-axis. He0 (0 I 8,s ni2) is the value of this angle at the origin we obtain upon integration of : It is easy to see that, with the definition of 8 given above, this equation applies to the entire profile. Equation  can be integrated to obtain X(Z). Writing Z Z2  cos8=coseo----= f(z) Ro 2 we obtain The integral may have to be decomposed if x changes sign. Introducing a new variable 4 defined by and the adimensional quantities X, Z (reduced cartesian coordinates) and R (reduced radius of curvature of meniscus) the integral  can be put in the form
3 FORTES with the meniscus constant k defined as I = ~ + c o s ~, (i=o,e=o) 2Ro 1 + cos 8, + 112~~' The i-solutions have an inflection point (i-point),  cc. = arc sin (k sin 4) For I = 0 (k = 1) the solution for 8 > 0 is  fx = G(4,) - G(4) where (  G(4) = 2 sin 4 - In tg i (f  k = z = 0. At this point Z = -1/R, (eq. [(I) and The + sign applies if RO > O (4 < 40 in the necessarily 0 < cos e I 1; eq.  then gives the region) and the - sign if RO < O (4 > 40). Equation following condition for an i-solution: [l 11 is applicable in all cases. Introducing elliptic integrals of the first (F) and second (E) kind we 1 obtain (3)  I = ~ + c o s ~, - ~ I o (i=o) 2Ro Both the a-point or the i-point may not be in the region with Z > 0. This will occur only for R,< 0. The two types of solutions are schematically indicated in Fig. 2 up to the points with z = 0, 8 = n. Each a-solution can be characterized by the value, R,, of the reduced radius of curvature at the where apex; and each i-solution by the value of 8, (0 I 8, I n/2), the slope at the i-point. Using the property  we have y) For 8 < 0, X is obtained as the sum of X(8 = 0) with X(181). Equation  incidentally shows that the quantity cos 8 + 1/(2R2), where 8, R are the values at any point of a profile, is constant for each profile. This property is a direct consequence of eq. . We may then introduce the quantity I (cf. eq. ), the invariant of a meniscus, defined as 1 [I81 I= cos8 + Y= constant 2R and state that the various solutions form a oneparameter, I, family of curves. Since at the origin cos 8,2 0, the parameter I may vary in the interval [- 1, + XI. (As discussed below, for every I there are in fact two solutions which are the mirror image of each other on a vertical or horizontal plane, according to the cases.) We will now show that the solutions can be grouped into two types, which we term a-solutions (or a-menisci) and i-solutions (or i-menisci). The a-solutions have an apex (or a-point) z = 0,8 = 0. (Note that all solutions have a point with 2 = 0, 8 = n at the end of the region with z < 0.) The condition for an a-solution can be obtained from eq. , by making 8 = 0 and requiring that there is a solution for Z, with the result (a-solution) 2R, 1  I= cos8, + - 2RO2 = cos 8, - 1 (i-solution) For the purposes of computation it is convenient to place the origin at the a-point or i-point, as in Fig. 2, and obtain the curve X(Z) from eqs. , FIG. 2. Examples of a-solutions ((I, b) and i-solutions (c, (1). The coordinate system indicated is appropriate to the r.h.s. of the curves; the origin is at the a-point or i-point. The angle 0 and the vertical contact angle 0," at various points are also indicated; o-solutions are even and i-solutions are odd.
4 2892 CAN. J. CHEM. VOL. 60, 1982 , and , with 0 in the interval 0, n. We note that 4, = n/2 at an apex and a. = n/2 at an i-point. For I = 0 the profile has a horizontal inflection point and is determined from eq. . The two a- solutions in Fig. 2 are identical if they have the same R, (>0); and the two i-solutions in Fig. 2 are identical if they have the same Oi (2 0). 3. Horizontal force Consider a cylindrical body placed at a fluid interface with menisci on both sides 1,2 of the body characterized by the values eo1, Rol and OO2, Ro2 at the lines of contact 1 and 2, respectively (Fig. 3). The horizontal force FI2 per unit length of the body, directed from side 1 to side 2, is the resultant of the horizontal components of the surface tension forces, y, acting on ihe lines 1 and 2, and of the pressure difference forces, due to the curvature of the menisci, acting on the plane defined by the lines 1 and 2 (4). To keep equilibrium a force, -FI2, must be applied to the body. Let h be the difference in the levels of the two contact lines (h > 0 if line 2 is above line I). The pressure difference at the line of contact 1 is ylrol and at line 2 it is ylro2. Equilibrium of pressure requires that The horizontal resultant of the pressure difference forces on plane 12 is easily found to be The horizontal force due to surface tension is Both these forces are expressed in y units and refer to unit length of the cylinder. The total force is F,, + F, which can be written, from eq.  or, using eq. , This equation has a simple interpretation. Each meniscus exerts a horizontal force, directed to the side of the meniscus, with a value, in y units and per unit length, equal to the invariant I of the meniscus profile. The net force is the resultant of FIG. 3. Floating cylinder with menisci on both sides 1, 2. The horizontal force F,, is the resultant of surface tension forces y and pressure difference forces acting on the inclined plane 12 and varying from ylr,, to ylr,,. Ra=m R,= oo la) FIG. 4. Examples of menisci between two isolated floating cylinders. If the connecting meniscus is of the o-type (o, b, (I) the force is attractive; if it is of the i-type (c) the force is repulsive. the forces exerted by the two menisci. With this decomposition of F12 in forces associated with each meniscus we may assign to an a-meniscus a force  Fly = 1/2Ra2 and to an i-meniscus a force  Fly = cos ei - 1 The first is positive and vanishes for R, = x. The other is negative and vanishes for ei = 0. The special menisci for which F = 0 are of course free menisci connecting two distant cylinders or walls. For two floating cylinders isolated at a liquid surface, the force between them is attractive if they are connected by an a-meniscus (Fig. 4a, b, d) and repulsive for an i-meniscus (Fig. 4c). This is true even if the a-point or i-point are not in the meniscus region. In the first case, if the free meniscus (side 1)
5 FORTES is itself of the u-type (as in Fig. 4a, b), the radius of curvature at the apex of the connecting meniscus (side 2) must be equal to the height difference A (in n units) between the two apexes, so that Yg 80:oo The force is directed to side Force versus distance In this section examples are given of the variation of the force F with the width X of a meniscus for fixed values of 0 at the two ends of the meniscus. The results apply to any geometry of the solids, but in general the relation between the actual contact angle and the angle 0 involves the slope angle of the solid surface at the line of contact. It is convenient to introduce what we term the vertical contact angle (v.c.a.) at the solid surface. This is the angle 0," defined in Fin. 2, between the fluid interface aid a vertical half-plane below the line of FIG. 5. Force versus separation for two floating cylinders with 'Ontact, measured through the liquid phase (denser the same vertical contact angle I,. Each curve also applies to fluid). 0cv can vary between and 3x12. The n - Io.. Different curves do not intersect. All c.urves are asymprelation between 0," and 0 (as defined in Fig. 2) is totic to the coordinate axes. where the + sign applies when the z-axis is parallel to g. Each meniscus can therefore be characterized F,~ 60,120 by the values 0,,, 0,,' of the v.c.a. at its ends and by the value of X, the horizontal reduced distance between the end points. The force due to the meniscus is, for the same X, equal to that due to a meniscus with v.c.a. (n.- Ocv), (n.- O,,') because of the identity of the menisci with the same I, as discussed above in relation to Fig. 2. When two cylindrical bodies are connected by a 10 - meniscus such that the v.c.a.'s at each cylinder, respectively 0,, and O,,', are both acute (0 I 0,,, 30,~. 0,,' < n.12) or both obtuse, the meniscus is necessarily of the n-type and the force due to it is attractive. In all other cases, and in particular if one 5. of the v.c.a. is acute and the other obtuse, the force may be repulsive or attractive depending on the type of meniscus, and therefore of distance. In the following examples which illustrate these conclusions we shall consider only the interval 0, \ x for the v.c.a. Note that the v.c.a. coincides with the actual contact angle if the solid surface is vertical at the region of contact. The examples then give directly the force on vertical plates or parallelipipeds placed at the fluid interface, with contact angles FIG. 6. Force versus separation for two floating cylinders with equal to the v.c.a. different vertical contact angles0,, and I,' indicated in degrees. Each curve also applies to (n - I,), (n - ~i~~~~~ 5 and 6 show the horizontal capillary e,,'). The insert shows curves in a smaller scale; the continuation of the two force, F, between two floating cylinders far from top curves as x + r, F + 0, intersect each other. All curves are any other bodies, as a function of the horizontal asymptotic to the coordinate axes. 10
6 2894 CAN. J. CHEM. VOL. 60, 1982 distance X between the lines of contact, for various combinations of 8,,, O,,'. Each curve also applies to the complementary v.c.a. In Fig. 5, the v.c.a.'s are the same at the two cylinders. The force is attractive at all separations (F > 0). The function F(X) cannot be represented approximately by an exponential law, even at large distances. However, a power law, with F proportional to X-2, is a good approximation at small separations (see below). Figure 6 applies to different angles. The curves for pairs acute-obtuse show the expected equilibrium position (F = 0) and a minimum (repulsive) force. At the equilibrium separation I = 0, and the corresponding value of X can be obtained from eq. . Figure 7 is a plot of the function G(4) defined by eq. , from which the equilibrium position can be determined from the absolute value of the difference G(8,,) - G(OCv1). For example, for 8,, = 0", 8,,' = 120", the equilibrium The equilibrium is unstable since the force is repulsive for separations larger than equilibrium separa- zio zio z i ~ b o so o i e;v FIG. 7. The function G(4) defined by eq. . The equilibrium separation between two floating cylinders with vertical contact angles O,, and 8,"' is the absolute value of G(O,,) - G(OCv1). FIG. 8. The horizontal capillary force (in the direction of the left wall) on a floating prismatic particle placed between two walls as a function of the position. Wetting characteristics are indicated in the insert; ((I) wall separation X, + X, = 30; (b) wall separation 50. tion and vice-versa. The value of the maximum repulsive force is 1 - Isin8,,1, where8,, is the v.c.a. with smaller value of lsin 8,,1. The repulsive force is at most equal to y, per unit length. The limiting behavior of F(X) as X + 0, F + = is easily found from eq. [Ill by expanding the integrands in a power series of k. The result is As X + x, F + 0, and the leading term in X is
7 FORTES 2895 F(4, k) for 4 = n/2, k = 1, which cannot be approximated by any simple function. The meniscus in this case contains an apex (with R, +x) or an inflection point (with Oi + 0) between the two cylinders. A different situation is that of a single cylindrical body between two walls at a given distance (Fig. 8, insert). The force on the cylinder as a function of its position between the walls can be obtained by combination of the curves, such as those in Figs. 5 and 6, for the appropriate v.c.a.'s at the walls (0,,,, 0,,,) and at the cylinder (0,,, assuming equal v.c.a. on both sides). Figures 80, 86 apply to vertical parallelipipedic particles with contact angles 30" and 170", respectively, for the left and right wall. In Fig. 8a the distance between the walls is 3a and in Fig. 86 it is 5a. F > 0 indicates that the particle is attracted to the left wall. Note that the equilibrium position is stable. 5. Discussion All the calculations were made on the assumption that the necessary forces are applied to keep the floating bodies at equilibrium. In addition to the horizontal force -Flz that was calculated, a vertical force must also be applied if the bodies are to stay at equilibrium. This force must be varied as the relative position of the bodies changes, if their vertical positions relative to some reference level are to stay unchanged. Even in this case, the vertical contact angles may change as the bodies change their places, except for vertical solid surfaces. The situation is more complicated in free flotation (zero vertical applied force). When two free floating cylinders approach each other they may either tend to sink or to emerge depending on the geometry of their cross-sections. An analysis of this situation was undertaken by Gifford and Scriven (1) for free floating circular cylinders. The variation of the vertical contact angles implies that the function F,,(X) will depend also on the geometry and densities, so that no general limiting laws are expected to hold. The analysis of the horizontal force on a parallelipipedic particle between fixed vertical walls shows that there is a stable equilibrium position which depends on the wetting characteristics of the particle and walls. As the examples of Fig. 8 show it is in principle possible to separate particles with different wetting characteristics by placing the mixture between parallel walls of different contact angles. In the case of Fig. 80, if the particles are placed at a distance 2.50 from the left wall (8,,, = 30") the wettable particles (0,, = 30") will experience a small attractive force to the left wall, while the liquid repellent particles (0,, = 170") will be submitted to a larger force (approximately 12 times larger) in the opposite direction. The equilibrium positions of the two types of particles are approximately 2.8a apart. Although the results of this paper apply exclusively to a cylindrical geometry, simple experiments show that the same type of effect of contact angles appears with particles of arbitrary noncylindrical shapes; in particular, the horizontal force between two such particles may be repulsive if their wetting characteristics are considerably different. I. W. A. GIFFORD and L. E. SCRIVEN. Chem. Eng. Sci. 26, 287 (1971). 2. D. Y. C. CHAN, J. D. HENRY, JR., and L. R. WHITE. J. Colloid Interface Sci. 79,410 (1981). 3. I. S. GRADSHTEYN and I. M. RYZHIK. Table of integrals, series and products. Academic Press p M. A. FORTES. Rev. Port. Quim. In press.