On the attraction of floating particles

Chemical Engineering Science, 1971, Vol. 26, pp. 287-297. Pergamon Press. Printed in Great Britain.

On the attraction of floating particles w . A. GIFFORD and L. E. SCRIVEN Department of Chemical Engineeringand Materials Science, University of Minnesota, Minneapolis, Minn. 55455, U.S.A. (First received 8 August 1970; accepted 17 August 1970)

Abstract-Capillary attraction between floating particles, a phenomenon of everyday experience as well as technological importance, is caused by interfacial tension and buoyancy forces that have defied calculation so far owing to the complicated shape of the intervening meniscus. The one exception is treated here: parallel, stationary cylinders of infinite length. Computations show that the attractive force falls off nearly exponentially with separation. The parallel configuration is unstable until cylinders make contact, as experiments with finite rods confirm. INTRODUCTION SMALL particles floating in a gas-liquid or liquid-liquid interface tend to clump together. T h e attraction they feel for one another is responsible for the scums that fine powders sometimes form on fluid interfaces. Emulsions can be stabilized against coalescence by such scums; to be an effective emulsifying agent a powder must consist of particles which are much smaller than the droplets and are not well wet by either fluid [la, 2]. Cohesion of particle layers on the surfaces of droplets on collision courses apparently inhibits drainage of the liquid film between them. Observant outdoorsmen see wavelets and tipples damped by pollen scums on natural bodies of water, and talc films produce similar effects in the laboratory[cf. 3]; one explanation regards the surface film as an inextensible yet flexible sheet[4]. When subjected to sufficient tension to rend it a scum under some conditions may heal the fracture with astonishing rapidity [5, 6]. Coherent powder layers have been seen encasing fast-moving laminar liquid jets [7]. T h e economically most significant examples of surface coagulation may well arise in froth flotation. This important ore concentration process depends on the selective attachment of small, pretreated mineral particles to rising gas bubbles [8]. Small air-filled soap bubbles floating on a

water surface draw together into rafts in which they are close-packed on hexagonal centers, a p h e n o m e n o n which led to a laboratory model of dislocation dynamics which was very influential in the development of that field[9]. It was in this connection that Nicolson[10] over twenty years ago attempted to analyze the interactions between floating particles. H e is, so far as the writers know, the only persion to have done so previously. Others, however, have analyzed the "force of adhesion" developed by pendular tings and liquid bridges between pairs of particles having simple shapes: slab and parallel cylinder[l 1], parallel cylinders[l 1-12], slab and s p h e r e [ 1 3 15], and spheres[13, 16-20]. In general there are two contributions to the "force of adhesion": a component of the interfacial tension force that acts everywhere along the line of contact at which the meniscus meets the solid and, secondly, a component of the differential pressure force acting on the wetted solid, the differential pressure being the "capillary pressure" developed by the curvature of the meniscus. Thus the entire force of adhesion depends on the shape of the meniscus of the pendular liquid ring between the pair of solid particles. But apart from a few exceptions the meniscus shapes for which explicit analytical or numerical descriptions are available are either two287

W. A. G I F F O R D and L. E. S C R I V E N

dimensional, i.e. cylindrical (in the general sense) or axisymmetric (circularly symmetric, strictly). And only when gravity and other body forces have negligible influence are there convenient analytical expressions, for only then are the shapes surfaces of uniform mean curvature: planes, cylinders, spheres, nodoids, catenoids, and unduloids [l l]. So it is that all the work cited above treats cylindrical and axisymmetric pendular rings and liquid bridges under conditions in which the effect of gravity on meniscus shape is negligible. Without gravity or other body forces, capillary effects cause no attraction between particles that happen to be caught in a perfectly fiat meniscus. But in the presence of gravity, virtually any floating particle depresses or elevates the meniscus that surrounds it, depending on its density in relation to that of the fluid aboveand that of the liquid below the interface. A second particle of the same sort then tends to sink down into the depression or to rise up into the elevation created by the first. The configuration of the meniscus is fully three-dimensional except for one special circumstance, which is the subject of this paper. Before we turn to that, we should describe Nicolson's [ 10] work. When two floating objects with vertical symmetry axes are widely separated, the threedimensional meniscus they create can be described to a good approximation by superposition of solutions of Bessel's equation, this equation being the linearized version which

is appropriate to the nearly flat interface [cf. 21]. Nicolson employed this approximation to estimate the effective force of attraction (actually the potential of the attractive interaction) between two floating bubbles that are far apart. He also found it necessary to assume that the bubbles do not tilt at all, i.e., that the contact circle on each bubble remains horizontal, and that the contact angle all around the contact circle remains zero. Consequently the force of attraction he obtained is due exclusively to a perturbed pressure field, or, in other words, to a buoyancy effect. Nicolson devised an entirely different approximation for estimating the effective force of repulsion between floating soap bubbles pressed so closely together as to be deformed with a flat septum between them. Only when the menisci surrounding the particles are cylindrical can they be described exactly in terms of standard functions. This special circumstance arises when the particles are themselves cylindrical, lie horizontal and parallel to each other in the meniscus, and are indefinitely long. In the simplest case two infinitely long, circular cylinders of the same radius and density lie alongside each other and are wet over the same arc, as shown in crosssection in Fig. 1. ANALYSIS

Two cylinders of radius R and uniform density Ps are separated by center-to-center distance d as they float in an interface between liquid B and

Fig. 1. Parallel, stationary floating cylinders of equal radius.

288

On the attraction of floating particles fluid A. L e t the effective force of attraction, F, be defined as the force per unit length of cylinder which would have to be applied horizontally at the axis of both cylinders to k e e p them from moving t o w a r d one another. This force depends on the extent of the wetted arc 01 + 02 (or, equivalently, the "equilibrium" contact angle e t 0 - s e e below), on the interfacial tension o- and the densities PA and pB, as well as on Ps, R , and d. Its calculation is n o w described. In the first place there is an obvious plane of reflection s y m m e t r y midway between the two cylinders, so that only one need be considered in detail. Secondly the contact angles? O~1 and as at which the interface meets the cylinder surface must be equal, although together they depend on other parameters. F o r at mechanical equilibrium with respect to rotation, the torque about its axis experienced by the cylinder must vanish in the a b s e n c e of externally imposed torque: o R ( c o s a 2 - - cos ¢q) = 0. Because both contact angles are defined to fall in the interval 0 ° <~ a ~< 180 ° it follows that oq = 0~2 = a. T h e procedure is now to analyze the forces acting on the cylinder and on the interfaces, all per unit length of cylinder. Vertical force balance on cylinder. T h e r e are three contributions. T h e b u o y a n c y force component is

Fb= f

T h e contribution from interfacial tension is F~ = - - o - [ s i n ( a + O l - - l r ) + s i n ( o ~ + 0 2 - - I r ) ] and the force of gravity is

F u = 7rR2psgL/ge .

i=A,B

(4)

Altogether these give 2h* (sin 01 + sin 02) + 01 + sin 01 cos 01 + 02 + sin 02 cos 02 = 2 ( B o ) - l [ s i n ( a + 0 1 ) + s i n ( a + 0 2 ) ] + 2 r r D (5) where h* =-h2/R, the Bond n u m b e r is Bo =-gL(PB--Pa)R2/gc o', and the density p a r a m e t e r is D =- ( P s - - P A ) ] ( P B - - P a ) . Horizontal force balance on cylinder. Similarly,

F * - F/o-Bo = h * ( c o s 01-- cos 02) + (sin s 02 - - s i n s 01)/2+ [cos ( a + O 2 ) -- cos ( a + 01) ]/Bo.

(6)

Horizontal force balance on exterior liquid. F o r a certain well-known choice o f control volume, the shape of the unbounded meniscus is immaterial in the balance of the contributions f r o m interfacial tension and pressure [22]t:

21r

(Po+pizgL/gc)RCosOdO,

(3)

(h* + cos 02) 2 = 2 [ 1 + cos ( a + 02) ] / B o (7)

(1)

(2)

Condition o f hydrostatic equilibrium. T h e differential equation of local hydrostatic equilibrium can be integrated around any closed path leading to P 3 - - Po = ( P 3 - - P2) + (P2 - - P 1 ) "q- (P1 -Po) in order to arrive at a relation b e t w e e n the radius of curvature b at the apex of the interior meniscus, where P2--P1 = o/b, and the depth there, which is D z - - D 1 in Fig. l; thus b = R2/ Bo(D2--D1). N o w since D2 = h2 + R and D1 = H i -q-R(1 -- cos 01) it follows that

tThese contact angles correspond to contact angles measured through Phase B, in the convention of surface chemistry [ lb].

tAs is sometimes the case, a shrewd choice of control volume leads one to an integral of the underlying differential equation, the latter being Laplace's equation of capillarity in each of these cases.

where P0 is the pressure at the distant interface, z = 0, and the depth z is related to 0 by z = h2 + R cos 0. Dividing the range of 0 into parts appropriate to the two phases and integrating yields •~2 gL {

__ lab -- Pa [

Fb = - - a - - / T r p a ~ gc \

~

A

01 + 02 + sin 01 COS 01

+ sin 02 cos 02 + - 2h2 ~ - - (sin 01 + sin 02)])

289

W. A. G I F F O R D and L. E. SCRIVEN

1/b*Bo = h* + cos 01-- H *

(8)

where b* - b/R and H * =- H1/R. Horizontal force balance on interior fluid. A fairly obvious choice of control volume, part of its boundary lying on the plane of symmetry, yieldsT H * 2 + 2 H * / b * B o = 2 ( 1 - - c o s d~l)/Bo

b* = ± [ B o Z ( h * + c o s 01) 2 - 2 B o ( 1 - - c o s ~bl) ]-1/2

(10) where b* is taken as having the same sign as ~bl. Condition o f interfacial equilibrium. Laplace's equation of capillarity is the differential equation of local static equilibrium of the interface. Integrating it once relates the horizontal semichord of the interior meniscus with the local inclination of the meniscus [23; cf. 1 1]: cot 9 cos [1 +½b*2Bo(1 - - c o s ~)]--1/2.

(11)

It is convenient to introduce q2 =_ 4b.2[(4b.2+ B o -1) and 2~ - 7r--~p. T h e n integrating (1 1) between t~ = 7r/2 and ~ = ~ =- (Tr--14~11)/2 yields for the reduced horizontal distance between contact lines, S* =- S / R ,

s.

q)] (12)

where F and E are elliptic integrals of the first and second kind, respectively. T h e reduced distance between cylinder centers is then d* -- d/R = 2 sin 0 1 + S * . TSee preceding footnote.

01+ 02 = 200,

(9)

where thl -= a + 01 - 7r. Combining this result with (8) and eliminating H * gives

= ~

T h e interior meniscus has the shape of a symmetrical segment of nodal elastica [24]. M e t h o d o f calculation. Equations (5-8), (10), (12) and (13) are seven independent relations among the nine parameters 01, 02, t~, h ' H * , b*, S*, F * and d*. If the wetted arc is specified, that is, if

(13)

(14)

the force of attraction can be calculated as a function o f separation. Because the equations involve implicit functions of 01 and 02 it is expedient to parametrize calculations with one of them in order to avoid trial-and-error procedures. Calculation is begun by setting 00, Bo and D and then choosing 01; 02 follows from (14); h* is eliminated from (5) and (7) and the result is squared twice to produce a fourth-order polynomial in sin or. O f the four roots any that do not lie on the real axis between 0 and 1 are automatically discarded. Because sin -1 a corresponds to two angles in the range considered the number of remaining possible solutions is doubled. H o w ever, all but one of the extraneous roots are eliminated by the requirements that (i) h* must have the same value whether calculated from (5) or (7) and must have same sign as th2; (ii) b*, which is calculated from (10), is real; and (iii) H *, which is calculated from (8), must be of the same sign as ~bl. Finally, S*, F * , and d* are found from (12), (6), and (13) respectively. This method of calculation can be derived systematically, as is explained in the Appendix. T h e method has a'variant, noted below, in which F * is found from a combination of (5), (6) and (14) from which both Bo and a are absent. All computations were made on the C D C 6600 computer with the aid of library subroutines for roots of polynomials and for elliptic integrals. T h e average time required per computation, i.e. for each choice of 01, was less than 0.05 sec. When the cylinders are so far apart that they do not 'affect each other the contact angle a0 and wetted arc 20o must satisfy the balance of vertical forces on a lone cylinder [22], which follows from (5) when 02 = 01 = 00, and the balance of

290

On the attraction of floatingparticles horizontal forces on the liquid, which is (7) with 02 = 00. T h e s e two equations can be solved for Cto in terms of 00; therefore specifying ~0, the contact angle at infinite separation, or "equilibrium Contact angle", is equivalent to setting the wetted arc 200. Both parameters appear in the figures and tables. T h e set of eight equations can be shown to be invariant under the transformationt D--> D' = I--D Og0, 00, 01, 02, O1~-"> respective supplemen-

tary angles h*, b*, H *, ~bl, ~bz ---> respective negatives S *, d*, F * --> unchanged.

(15)

T h e underlying symmetry operation is reflection in the plane of the distant interface, but cylinder density must be transformed at the same time. T h e invariance means that a single computation describes both the attraction between a pair of cylinders of density Os > pB and the attraction between a pair of density p~ < Pa, all in the same interface but under different conditions of wetting. Redundant calculations can accordingly be avoided. (It should be noted when Ps > Pa + pB the second interpretation is empty because then

0.) RESULTS Besides wetted semi-arc O0 or "equilibrium contact angle" %, the Bond number Bo gL(pB--pa)R2[gco " and density parameter D = ( P S - - P A ) / ( P n - - P A ) enter the relation between attractive force and separation. Rather than many working charts covering the full ranges of parameters, sample results are reported here. Computation for other combinations of densities, cylinder radius, and interfacial tension can be programmed and run as needed [23]. Figure 2 shows the reduced force F * vs. d * for B o = 0.137 and D = 2, parameter values that tThis symmetry of the system was inferred intuitively and then demonstrated mathematically; no systematic procedure for uncovering all such symmetries seems to have been published. The analogous symmetry of a single floating cylinder was independently noted by Princen [ 11].

correspond to a cylinder of radius R = 0.1 cm and density Ps = 2g/cms floating on the airwater interface (o- = 71.5 dyn/cm, pa = 1 g/cm ~, PA ----0). T h e uppermost curve pertains to cylinders only slightly wet by water (ao = 170"9°, 00 = 30°); along it the attractive force varies from approximately 0.3 dyn/cm at a center-tocenter separation of 1-2 cm (12 radii) to almost 12 dyn/cm at contact (2 radii). Data for a representative curve in Fig. 2 are given in Table 1. It can be seen that 01 decreases and 02 increases as separation diminishes: the cylinders roll slightly toward one another as they get closer together, provided the contact lines do not migrate over their surfaces. T h e submergence h2 = h * R of course increases and the contact angle o~, inclination ~bl and radius of curvature b = b * R all decrease as the particles get closer together, regardless of the extent of wetting. Figure 3 displays curves of B o = 0.137 and D----½, values that correspond to cylinders of radius R = 0.1 cm and density Ps = 0.5 g/cm 3 floating on the air-water interface. Owing to (15) each curve represents two supplementary wetted arcs or "equilibrium contact angles". In one case, O0 < 90 °, or0 > 90 °, water wets the solid poorly and the meniscus is depressed whereas in the other, 0 0 > 9 0 ° , a 0 < 9 0 °, water wets better and the meniscus is elevated; nevertheless in both cases the horizontal force between the particles is attractive. Data for a representative curve appear in the lower portion of Table 1 (values in parentheses pertain to center-tocenter separation less than cylinder diameter, which is unlikely in reality). When wetting is better (O0 > 90 °) the cylinders roll slightly away from one another and their emergence, - h 2 = - - h ' R , increases as they get closer together. T h e way attractive force depends on Bond number B o and separation is shown by Fig. 4, which is for cylinders all with D = 2, and wetted semi-arc 00 = 75 °. Figure 5 is a similar plot showing the effect of the density parameter D when the Bond number B o = 0.01. In both figures the ordinate is the logarithm of reduced force divided by the Bond number, i.e., F * / B o = trFgc2/

291

W. A. G I F F O R D [

I

and L. E. S C R 1 V E N

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l

I

I

I

I

I

eo,=o i 1.2- 30 ° , 170.9°

l ~° , 6137.1 0 °.

-

I.O

-

90= ' 1 0 3 " 6 ° ~

0.8

~

F*

]\\\ 120" , 71.~

o,

\\\

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0.2 i

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0)

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2

I

3

I

4

I

5

I

6

7

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8

9

I0

d4. Fig. 2. Reduced force of attraction vs. dimensionless center-to-center separation of cylinders floating metastably. Typical curves for Bo = 0' 137, D = 2. If R = 1 ram, P B Pa = I g/cm 3 (P8 = 2 g/cm ) and o - = 71-5 dyn/cm (air-water interface), then F = 9.8 F* dyn/cm and d = 0.1 d* cm. Oo

ao

'

J

'

'

i

i

i

,

3o-=, ,549 `= I

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0.07 ~ 150-= 25.1,= / !

0.06

45°'

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i

0.04

~,22.~-= °.°3 '2°°' 5~2"=ib',,,"

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1

12

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3

~ I

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4

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5

6

----t-'~:

7

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de Fig. 3. Reduced force of attraction vs. dimensionless center-to-center separation of cylinders floating stably. Typical curves for Bo = 0-137, D = ½. If R = 1 ram, On--Pa = 1 g/cm 3 ( p s = ½ g / c m a) and t r = 7 1 . 5 d y n / c m (air-water interface), then F = 9 . 8 F * dyn/cm and d = 0.1 d* cm.

292

I0

On the attraction of floating particles Table 1. Sample results. Bo = 0.137, D = ½, 00 = 45 °, c~0 = 139"001 °

d*

ct

oo 8"269 5.290 3.899 2"978 2"287 (1.996)

139"001 ° 138"935 138"805 138"674 138.544 138.414 (138"350)

01

O=

~bl

h*

1/b*

F*

45"00 ° 44.75 44.25 43"75 43"25 42"75 (42-50)

45"00 ° 45"25 45"75 46.25 46"75 47.25 (47"50)

4'001 ° 3'685 3'055 2.424 1"794 1.164 (0'850)

--0"5185 --0-5067 --0"4831 --0.4594 --0.4356 --0"4118 (--0.3999)

0 0"01451 0"02513 0.03244 0-03839 0"04354 (0"04589)

0 0.00561 0-01683 0-02805 0.03927 0.05050 (0'05612)

1/b*

F*

0 0"04309 0"06094 0"1056 0" 1364 0" 1616 0" 1836 0.2010

0 0"04947 0"09895 0"2971 0"4960 0"6961 0"8979 1"0763

Bo = 0-137, D = 2, 0o = 60"0 °, c=0 = 137"052 ° Oz thl h*

d*

tx

oo 10"45 8"584 5"608 4"206 3"269 2.560 2"053

137"05 ° 136"89 136"73 136"09 135"48 134"89 134'33 133"86

60"00 ° 59"50 59"00 57"00 55"00 53"00 51.00 49"25

60"00 ° 60'50 61"00 63"00 65"00 67"00 69-00 70'75

17'05 ° 16"39 15'73 13'09 10"48 7'890 5"327 3.105

gL2(/3B--pA)2R 4, the abscissa is dimensionless separation times square root of Bond number, i.e., d * V ~ - o = d ~ / g L ( p B - - p a ) l g c o ' . Remarkably the curves are n e a r l y - b u t not exactly-straight 20 Bo<

0"3012 0"3244 0"3477 0"4421 0"5379 0"6353 0"7341 0"8217

and parallel and fall almost on top of one another. Thus F

~--

AR4(pB--pA)20 "-1 e -xdvg~O~"-p")/g~

(16)

where K is nearly independent of D and Bo and A is nearly independent of Bo. The results in Table 1 indicate that K depends slightly and A, strongly, on the wetting parameters. The decrease of attractive force with increasing separation is exponential to within 1 per cent in the results shown. There apparently is no obvious way from the original equations to a justification of this striking result. However, by combining (5), (6) and (14) one can show that

I0 "?-

I0 ~\/,=--Bo • 0.1

o

*=. F, = 0.5

0.2

0"10

I

I

i I

i

i 2

i

, 3

,

. 4

.

.

.

.

5

.

6

d*.v/'Bo Fig. 4. Dependence of attractive force on cylinder Bond number, Bo. Typical curves for D = 2 and 00 = 75 °.

00 +

sin 20o tan

Therefore for given densities (in D) and wetted arc 200, the force of attraction depends directly only on the tangent of the angle (02--01)/2 through which each cylinder has rolled. Decreasing cylinder radius merely shifts the curves in Figs. 2 and 3 to the left and expands the scale of the abscissa. The smaller the radius the smaller the rotation at contact, all else being the same. 293

W. A. G I F F O R D I00

~NN I ~

~

X

X

I

'

I

'

I

'

I

and L. E. S C R I V E N

v

2(1

IO

--

5

u.

21 _

-I0

0.5

0.2

0.10

=

i

I

I

i

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2

I

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3 d"~,V~.O

z

I

4

i

I

5

z

6

Fig. 5. D e p e n d e n c e o f attractive force on density parameter, D. Typical curves f o r B o = 0.01 and 00 = 75 °. DISCUSSION

The capillary force that causes small floating particles to clump together can be remarkably high. Even comparatively large, 2-mm-dia. cylinders of moderate density floating on water may adhere with a force as high as 10 dyn per centimeter of length when they are shoulder-toshoulder. As many as 12 dia. apart they feel an attractive force as high as 0-3 dyn/cm. Depending on the extent to which they are wet, microscopic cylinders of 20/z (2 x 10-~ cm) diameter and the same density adhere with a force of about 2 x 10-rdyn/cm by capillary attraction and feel almost as strong attraction when separated by 12 dia. The mass of these microscopic cylinders is only 6.3 x 10-6 g/cm, and so if their inertia were controlling they would require

roughly ½second to come together from rest at 12 dia. separation. The comparable time interval for 2 mm-dia, cylinders of 2 glcm s density initially separated by 12dia. would be about ½sec if their inertial resistance were controlling.~ Casual observations-they are a common experience-show that floating needles and many other sorts of particles do indeed come together with astonishing acceleration. The unsteady flow fields that are generated challenge analysis by both experiment and theory. They will have to be understood before the commonplace, "capillary attraction", can be more than a mere label, so far as dynamic processes are concerned. One dynamical aspect underlies the choice of 00 as the parameter for Fig. 2 and 3. This has to do with the contact lines at which the fluid interface intersects the solid cylinders. If O0 remains constant as the cylinders approach one another the contact lines are fixed on the solid surfaces. However, the cylinders rotate a little as they approach and at the contact line the contact angle changes from its initial value at infinite separation, which is the "equilibrium contact angle" a0. According to the present calculations the contact angle et actually changes very little: less than 4 ° in cases of 1-mm-radius cylinders and still less in cases of smaller cylinders. The total change between infinite separation and shoulder-to-shoulder contact for the 10-/z-radius cylinders is less than 0.001°! But change the contact angle does. All of the values calculated from Eqs. (5) and (7) are contact angles at mechanical equilibrium. If a unique equilibrium contact angle exists for a given solid-liquid-fluid system, as is often asserted in surface chemistry[cf, lb], it is attained only after the long times required for equilibrium with respect to diffusional processes. Were the contact angle to change too much from its initial value the contact line would break

294

t l f the attractive force follows (16) the time is given by d M Bol12( d~* -- d p ) tan -1 N/exp [ KBoll2( do* -- dfl ) [2 ] -- 1 X/ K B o 1t2(do* -- d~ )/2

On the attraction of floating particles

loose and shift along the cylinder surface, i.e., the wetted arc would change. This is known from experiments with a liquid index in a capillary tube across which increasing pressure differential is applied[25.26] and from more numerous experiments with a liquid drop on a flat solid the tilt of which is gradually increased[27-30]. Typically the critical change in contact angle is 8 ° or 10°, which is rather more than what is seen here, especially in cases of small cylinders. No shifting of contact lines could be detected when pairs of glass cylinders 0.18 cm in dia. and of steel ones 0.06 cm in diameter were seen at low magnification as they drew together from initial separations of as much as 2 cm on an airwater interface. More refined experimental studies are needed, however. Even the most casual observations of a pair of floating needles brings home what should be obvious from Figs. 2 and 3: when cylinders of finite length float parallel to one another they are in an unstable configuration. For if one pair of ends should happen to approach slightly closer together than the other, the force of attraction

@

per unit length will be greater at that pair of ends, which will draw together still closer. This is the explanation for the first stages of the behavior diagrammed in Fig. 6. By the third stage the meniscus is decidedly three-dimensional and no theoretical model is yet available for the subsequent events, of which the most striking is the final slide into near-perfect alignment. This movement, from 5 to 6, does not always occur, but when it does it is brisk. It must be driven by comparatively strong forces developed in the highly curved, three-dimensional menisci around the ends of the cylinders. Another type of instability is important when the solid is more dense than the liquid or less dense than the upper fluid, for in either circumstance floating is at best metastable with respect to sinking-downwards or upwards, as the case may be. Princen and Mason (see Princen [ l l]) investigated one mode of instability of an isolated floating cylinder on the assumption that the contact angle remains constant. Gifford[23] has examined the same mode when the contact lines remain fixed. How motion of cylinders along an

® LENTO

l

® ANDANTE

ALLEGRETTO

®

®

®

Fig. 6. Stages of mutual attraction and contact between floaEng cylinders of finite and equal length. The parallel configuration is inherently unstable. Successive stages are reached with increasing tempo. The final brisk alignment is not always seen.

295 CES-Vol. 26 No. 3-C

W. A. GIFFORD

interface affects their stability sinking is not known.

with respect

and L. E. SCRIVEN

to PI

Acknowledgment-This research was supported by National Science Foundation Grant GK-41. For helpful discussions the authors are indebted to C. Huh, R. Aris and P. R. Pujado.

4

R NOTATION

b Bo d D

s

radius of curvature of apex: b* = b/R Bond number, Bo = gL(pe - pa)R2/g,a center-to-center distance; d* = d/R density parameter, D = ( pS - p,J/ (PB--PA)

F

attractive force per unit length; F* = F/Boa Fo attractive force per unit length at initial separation g, proportionality factor in Newton’s second law of mechanics gL local gravitational acceleration h, center-line submergence below distant interface; h* = h,/R

&Z

apex elevation above contact line; H* = HI/R pressures at reference planes indicated in Fig. 1 V46*2/(4b*2 + Bo-1) cylinder radius contact line-to-contact line separation of cylinders; S * = SIR Cartesian coordinates

Greek symbols a7 % contact angles % wetted semi-arcs 01992 contact line positions density of fluid, liquid, and solid, pA~pB,fb respectively interfacial tension ; interface inclination from horizontal &, 42 contact angle with respect to the horizontal

REFERENCES HI ADAM N. K., The Physics and Chemistry ofSurfaces, 3rd Edn. University Press, Oxford 1941; Dover reprint 1968. (a) pp. 206-208; (b) pp. 178-188. P. R., The theory of stability of emulsions, in Emulsion Science (Edited by PI KITCHENER J. A. and MUSSELLWHITE SHERMAN P.), pp. 77-130. Academic Press, New York 1968. R. J., Surface films.. . , in Sur$uce Phenomena in Chemistry and Biology (Edited by DANIELLI J. F. [31 GOLDACRE et al.), pp. 278-298. Pergamon Press, Oxford 1958. 6th Edn., pp. 631-632. University Press, Cambridge 1932; Dover reprint 1945. 141 LAMB H., Hydrodynamics, [51 GRGURICH D. A., Unpublished observations, Dept. of Chem. Engng, University of Minnesota 1963. [6j DAVIES J. T. and KHAN W., Chem. Engng Sci. 1965 20 713. [7] CULLEN E. J. and DAVIDSON J. F., Trans. Faraday Sot. 1957 53 113. [8] GAUDIN A. M., Flotation, 2nd Edn., Chap. 11. McGraw-Hill, New York 1957. [9] BRAGG L., LOMER W. M. and NYE J. F., Experiments with the Bubble Model ofa Metal Structure, motion picture film oroduced bv N. S. Macaueen, Cambridge Universitv 1954. NldOLSON G. M., Proc. damb. Phil. Soc_l949 45 28i. PRINCEN H. M., The equilibrium shape of interfaces, drops, and bubbles. Rigid and deformable particles at interfaces, in Surj&e and ColloidScience (Edited by MATIJEVIC E.), Vol. 2, pp. l-85. Wiley-Interscience, New York 1969. WI CRISP D. J., Trans. Faraday Sot. 1950 46 228. [I31 CROSS N. L. and PICKNETT R. G., Trans. Faraday Sot. 1963 59 846. u41 CLARK W. C., HAYNES J. M. and MASON G., Chem. Engng Sci. 1968 23 810. of Solids, Vol. 1, pp. 299-302. University Press, Oxford H51 BOWDEN F. B. and TABOR D., The Friction andlubrication 1954. [I61 FISHER R. A.,J.Aan’c. Sci. 1926 16492; 1928 18406. W., Proc. 4th World Petrol. Congr. (Rome), Sec. l/C, p, 399, 1956. [I71 VON ENGELHACDT [I81 WOODROW J.. CHELTON H. and HAWES R. 1.. Reactor Tech. 1961 1229. 1191 MASON G. and CLARK C. W., Chem. Engnn - - Sci.‘1965 20 859. MELROSEJ.C.,A.I.Ch.EJll96612986. w 1211 HUH C. and SCRIVEN L. E.. J. Colloid Interface Sci. 1969 30 323. Part 1. Hydrostatics. Deighton, Bell and Co., Cambridge 1891. WI BESANT W. H., Treatise on hydromechanici. B31 GIFFORD W. A., M.S. Thesis, Dept. of Chem. Engng, University of Minnesota 1970. A. G., The Applications of Elliptic Functions, pp. 87-89. Macmillan, London 1892; Dover reprint 1959. t241 GREENHILL 3rd Edn., Vol. 1, pp. 230-232. Gauthier, Paris 1868. See also P51 JAMIN J., Cours de Physique de I’Ecole Polytechnique, Compt.

rendu 186050

172,311,385. .

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On the attraction of floating particles I261 SCHWARTZ A. M., RADER C. A. and HUEY E., Resistance to flow in capillary systems of positive contact angle, in Conracr Angle, Weftability and Adhesion, Advances in Chemistry Series No. 43, pp. 250-267. American Chemical Society, Washington, D.C. 1964. 1271 ABLETT R. and SUMMER C. G., General discussion, in Wetting and Detergency, Sympos. Brit. Sect. Internatl. Sot. Leather Trades’ Chemists, pp. 45-52. Chemical Publishing Co., New York 1937. [28] MACDOUGALL G. and OCKRENT C., Proc. R. Sm. 1942 A180 15 1. [29] BIKERMAN J. J.,J. CoNoidSci. 1950 5 349. [30] FURMIDGE C. G. L.,J. CoZloidSci. 1962 17 309.

APPENDIX The foregoing analysis turns on seven independent equations in eleven variables and parameters (before H * is eliminated from (8) and (9) to give (10) there are eight equations in twelve quantities). This system can be represented by a rectangular matrix in which a J-entry denotes that a given quantity (column heading) is present in a given equation (row heading), and a O-entry denotes that it is not. By means of permutations of rows and of columns, the matrix can be put in a form which minimizes the number and size of sets of equations that must be solved simultaneously:

The sequence of columns from left to right and of rows from top to bottom are the order in which quantities should be considered and equations solved, respectively. Inspection of the boxed diagonal blocks shows that the method of calculation should be: (i) choose two of &, ~9,and 0, and solve (14) for the other; (ii) choose two of the Bo, h* and OL and solve (7) for the other; (iii) solve (10) for b*, (12) for S*, (5) for D, (6) for F* and (13) for d*, in any order, except that b* must be obtained before S* and S* before d*, as indicated by the blocks enclosed by broken lines below the diagonal.

However, for the present paper the authors decided that f&, Bo, and D should be treated as independent parameters and set before calculation begins (these were chosen on physical grounds; mathematical reasons may be even more compelling, as when a variable enters an equation transcendentally). If the corresponding columns are deleted the conditioned form of the resulting rectangular matrix is

(14) (7) (5) (10) (12) (6) (13) Inspection shows that the strategy of computation should be: (i) choose B1or 0, and solve (14). which happens to be trivial, for the other; solve together (5) and (7). which happens to be nonlinear, for h* and a (it turns out to be possible to eliminate h* at the expense of extraneous solutions for a); solve (10) for b*, (12) for S*, (6) for F* and (13) ford*, in any order, except that b* must still be determined before S* can be and S * before d * can be. (According to ( 17) F * can be computed from or, e, and Q alone, if desired.) Our scheme for systematically devising efficient strategies of solving simultaneous, independent, nonlinear equations is more powerful than any other available, so far as we know. The algorithm and its relation to others will be described elsewhere.

RCsumC-L’attraction capillaire entre des particules flottantes, un phenomene quotidien mais aussi d’importance technologique, est cause par la tension a l’interface et les forces de poussee qui ont, jusqu’ici, deft& toute tentative de calcul a cause de la forme compliquee du m&risque intervenant dam le phenomtne. La seule exception est trait&e ici: le cylindres stationnaires paralleles de longueur indelinie. Le calcul montre que la force d’attraction decroit d’une man&e presque exponentielle avec la separation. La configuration parallele est instable jusqu’a ce que les cylindres soient en contact, comme le conflrment les experiences avec des tiges de longueurs dtterminees. Zusammenfassung - Die kapillare Anziehung zwischen schwebenden Teilchen, eine alltiiglich beobachtete Erscheinung, die jedoch such technologische Bedeutung besitzt, wird durch GrenztHchenspannungs- und Auftriebskrafte verursacht, die sich bisher infolge der komplizierten Form des dazwischenliegenden Meniskus der genauen Berechnung entzogen haben. In diesem Artikel wird die einzige Ausnahme, namlich der Fall paralleler, stationlrer Zylinder unendlicher Lange, behandelt. Berechnungen zeigen, dass die Anziehungskraft beinahe exponential mit der Trennung abfallt. Die parallele Konfiguration ist unbestanding bis die Zylinder in Beriihrung kommen, was durch Versuche mit endlichen Staben bestatigt wird.

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