An algorithm for determining the shapes of floating fluid lenses

An Algorithm for Determining the Shapes of Floating Fluid Lenses DAVID S. ROSS Applied Mathematics and Statistics Group, Computational Science Laboratory, Eastman Kodak Company, Rochester, New York 14650-2205 Received September 19, 1991; accepted March 30, 1992

The problem of determining the shape of a fluid lens floating at the interface of two other fluids is considered. The problem is formulated mathematically, then reformulated in a m a n n e r that facilitates its numerical solution. An iterative scheme for the solution of the problem is described. The results of calculations done with a computer program based on this algorithm are presented and discussed. © 1992AcademicPress,Inc.

wet the interface, fluid C may wet the surface of the drop, or fluid B may wet the surface of the drop, depending on which of the surface tensions is the largest ( 1 ). If each surface tension is less than the sum of the other two, then a triple interface intersection is possible. If the density orB is neither too much greater than that of C, nor too much less than that of A, the drop will form a lens at the interface. An example of a lens is shown in Fig. 1. A lens may form even if the density of B is somewhat greater than that of C or somewhat less than that of A; surface tension may restrain a drop that would otherwise sink or float away. An analysis of this problem for sessile lenses, the case in which the density of B lies between the densities of A and C, is presented in the paper of Pujado and Scriven (2). They discuss the history of the problem and analytic solutions of it in certain limiting cases, and they outline a numerical scheme for solving it. In the general case, the problem requires the solution of a system of coupled nonlinear differential and algebraic equations that are intractable by analytic methods; a numerical solution is needed. Their scheme is a shooting method, a method in which boundary value problems are solved by repeated solution of

INTRODUCTION

Consider a drop of a fluid, let us call this fluid B, placed at the interface of two other fluids in a gravitational field. The interface of the other two fluids will be planar, except in the vicinity of the drop of B, and it will be orthogonal to the gravity vector. The denser of the other two fluids, let us call it fluid C, will lie below the interface, where "below" means in the direction of gravitational acceleration. The less dense fluid, we shall call it fluid A, will lie above the interface. One of A, B, and C may be a gas, but at least two of these must be liquids; gases would mix. What happens to the drop of B? The answer depends on the surface tensions of the three interfaces and on the densities of the fluids ( 1 ). If any one of the surface tensions is greater than the sum of the other two, there can be no point at which the three interfaces intersect; the balance ofinterfacial forces that is required at such a point cannot be achieved. In this case, the drop must lie completely above or below the A - C interface. If the density of B is greater than that of C, or less than that of A, the drop will sink or float away. If the density of B is between that of A and C, the drop may 66 0021-9797/92 $5.00 Copyright© i992 by AcademicPress,Inc. All rightsof reproductionin any formreserved.

Journal ofColloidand InterfaceScience, Vol. 154,No. 1, November1992

SHAPES OF FLOATING

67

FLUID LENSES

initial value problems, with the initial data ad- of the fluids are 0a, Pb, and Pc. The surface justed iteratively until boundary conditions are tensions are O'ab, ffac,and O'bc.The acceleration satisfied. of gravity is g. The lens will be radially symIn this paper we present a general method metric. We shall use an x-y coordinate system, for computing the shapes of lenses, sessile or in which the y axis is the axis of symmetry of otherwise. Our numerical method is a relax- the lens, and the x axis is the radial axis, which ation method (shooting and relaxation are the is parallel to the undisturbed A-C interface, two broad classes of methods for the numerical as in Fig. 1. solution of boundary value problems for orWe shall regard each of the interfaces as a dinary differential equations (3)). We for- parameterized curve, parameterized by arc mulate the problem as a system of two-point length. For this purpose, we shall measure the boundary value problems for second-order length of the B-C interface from its intersecordinary differential equations (with some tion with the y axis, and we shall measure the unusual features). We then use a finite differ- lengths of the A-B and A-C interfaces from ence discretization to approximate these sec- the triple intersection point. We shall denote ond-order equations. We solve the resulting these parameterized curves by pairs of coordifference equations by relaxation. Finally, we dinate functions, (Xab(S), Yab(S)), (Xae(S), use a binary search to locate the lens with the Yac(S)), and (Xbc(S), Ybc(S)). Here, we have correct volume. used the same parameter, s, to indicate the arc Our method can also be applied to situa- lengths for all three curves, although the dotions in which one of the surface tensions is main of s will generally be different for the greater than the sum of the other two, and the three curves. In order to express the physical density of fluid B is neither greater than that principles that determine the shapes of the inof fluid A nor less than that of fluid C. In such terfaces, we shall use the tangent angles, 0ab(S), situations, if fluid B does not wet the A-C in- 0ao(S), and 0b~(S), the angles between the tanterface, the drop assumes a shape that is a lim- gents to the interfaces and the positive x axis. iting case of a lens. In this context, our method There are four basic physical principles from provides a more general solution of the prob- which we shall derive our equations, four lem treated in (4), in which fluids A and B principles that determine the shape of a lens: are identical. Examples of such configurations 1. The lens is radially symmetric. are presented in the section on computational 2. The hydrostatic pressure gradient in each results. Since such cases can be treated as limfluid is the product of the fluid's density and iting cases of lenses, our method yields a comthe gravity vector. This is the standard asplete solution to the problem of determining sumption for the hydrostatic pressure of an the equilibrium shape of a drop placed at an incompressible fluid. It implies that the presinterface. A copy of the source code of a F O R T R A N program that implements the algorithm dey scribed in this paper is available from the auA thor. FORMULATION

OF THE PROBLEM

We shall use subscripts a, b, and c to indicate quantities in fluids A, B, and C, respectively. The double subscripts ab, ac, and bc will be used to indicate quantities at the A-B, A-C, and B-C interfaces, respectively. The densities

j

x

FIG. 1. Typical floating lens. Journal of Colloid and Interface Science, Vol. 154, No. 1, November I992

68

DAVID S. ROSS

sure has the form P~ - pagY, with P~ a constant, in fluid A, and analogous forms in the other fluids.

3. Force balance at the triple intersection is determined by Neumann's triangle, i.e., at this point, each interface exerts a force, with magnitude equal to its surface tension, that lies in its plane. In making this assumption, we

In order to make ~,c finite, we shall introduce a right circular cylindrical container of radius x~ that is coaxial with the lens. We shall take x~ to be so large that the presence of the cylinder has no effect on the lens shape. Geometrical considerations yield several auxiliary conditions on the functions. Principle 1 implies that

are ignoring the line tension, which is generally negligible in practical problems.

0ab(~b) = ~r

[4]

0bd0) = 0.

[5]

4. The jump in hydrostatic pressure across an interface is balanced by the force due to surface tension, which is directed normal to the interface, and is equal in magnitude to the product of the surface tension and twice the mean curvature of the interface. This is Laplace's law, the usual statement of force balance at an interface. The mean curvature of, for example, the A-B interface is (5)

and

Our assumption on the size ofxo~ implies that the value of 0ac(~ac) does not affect the lens shape, so we shall take 0ac(~]ac)

+

-

-

Xab

0.

[6]

We also have

1 [ dOab sin(0ab) /

g

=

Xab(TIab) = 0

[7]

Xbd0) = 0

[8]

]

and It follows from this, from the definition of 0ab, and from the second and fourth principles that along this interface we have

[ dOab sin(0ab)] "~\as X~b ]

O'ab|~

= APab q-

dXab ds =

Apabgyab [ 1]

COS(0ab)

[2]

ds = sin(0ab)

[3]

dyab

0 ~ S ~ ~ab.

Here, we have introduced the length, ~ab, of the A-B interface. We have also introduced, for notational economy, the constants z~kPab = P b - - P. and A p a b = Pa -- Pb- We shall use analogous constants for the other interfaces, AP, c = P, - Pc, AOac = Pc - Pa, ~kPbc = Pb -Pc, and Apb~ = Pc -- Oh. The analogous equations along the other interfaces are obtained by substituting ac or bc for the subscripts in [1], [2], and [3]. lournal of Colloid and Interface Science,

Vol. 154, No. 1, November1992

Xac(~]ac) = Xoo.

[9]

If the triple intersection point is to actually be an intersection point, the y coordinates of the interfaces must be the same at that point. We have not yet specified the location of the x axis, the line y = 0. We shall choose it so that the triple intersection point lies on the x axis, Y a h ( 0 ) = Ybc(~/bc) = Y a c ( 0 ) = 0.

[10]

We now have nine initial or final values, one for each function. Along with the system of nine differential equations these might appear to constitute a determined system. However, note that the equations contain two independent constants, APab and APac (APbc is just the sum of these two), which remain undetermined. Moreover, the arc lengths, n,b, nao, and nbc, are not known. Thus, we need five more conditions to specify these five constants. Two of the needed conditions follow directly from principle 3:

SHAPES O F F L O A T I N G ffabSin(0ab(0))

"ff

aacSin(0ac(0))

-- 0"bcSin(0bc(r/bc)) = 0

[ 11]

O'abCOS(0ab(0)) q- O'acCOS(0ac(0)) -- ffbcCOS(0bc(T]bc)) = 0.

[12]

These equations express the balance of surface tension forces at the triple intersection point. If, and only if, each surface tension is less than the sum of the other two, these equations can be solved to express the angles between the interfaces at the triple intersection point in terms of the surface tensions; 0 a b ( 0 ) -- 0bc(Tlbc) = 17A 0"2c ~ 0 " 2 b ~ ffac = cos -i

2ffbc O'ab

0ab(0 ) -- 0 a c ( 0 ) = I~B

~c + ~u - ~c = COS -1

"~ ffac-"~ab

•

The x coordinates of the interfaces at the triple intersection point must be equal; this yields two more conditions, X a b ( 0 ) = Xbc(nbc ) = Xac(0 ).

[13]

Finally, we know the volume of the drop, let us call it V,

69

F L U I D LENSES

grounds, that a unique solution of this system of equations exists. In fact, for fixed values of the other six parameters, we expect such a solution to exist also for values of Pb somewhat less than Pa and somewhat greater than pc, as we discussed above. However, to our knowledge, there is no rigorous mathematical proof of the existence of such solutions. Neither is there a theory that predicts the range of values of Pb. This system is nonstandard for many reasons. The differential equations contain parameters that must be determined as part of the problem's solution. The intervals on which the differential equations are to be solved are themselves to be determined as part of the problem's solution. The constraints whose enforcement is to determine the values of these parameters include nonlinear and nonlocal functions of the solutions of the differential equations and of the parameters themselves. Two of the differential equations (Eq. [ 1] and its analog for the B-C interface) are singular (at s = ~ab and at s = 0, respectively). In the next section, we present a reformulation of the problem that eliminates many of these difficulties, and that lends itself to solution by finite difference methods. In the following section, we present a numerical scheme that treats the remainder of the difficulties, and which converges for parameters in the range discussed above.

V = --27r[fo'r~XbcYbcCOS(Obc)dS REFORMULATION

q- fonabXabYabCOS(Oab)dS].

[14]

The system comprising Eqs. [ 1] - [ 14 ] (including the duplications of [ 1], [ 2 ], and [ 3 ] for the other interfaces) constitutes a complete formulation of the problem of determining the nine unknown functions Xab, Yah, Xac, Yac, Xbc, Ybc, 0ab, 0ac, and 0bc, and the five unknown constants ~ P a b and A P a c , ~ab, T]ac,and nbc, in terms of the seven parameters Pa, Pb, Pc, aab, aac, trbc, and V. If each of the surface tensions is less than the sum of the other two, and ifpa < Pc < Pc, it is reasonable to expect, on physical

We reformulate the problem by introducing coordinate transformations that map the three interfaces onto a unit interval and by differentiating [ 1] and its analogs for the other interfaces in order to eliminate the constants APab and APac. The coordinate transformation for the A-B interface is s = ~abq, d/ds = ( 1/ ~lab)(d/dq), Nab = ~abZab, Jab = Tlab~ab. The transformations for the other interfaces are obtained by substituting the appropriate subscripts. In the new coordinates, and after differentiation, Eq. [ 1] becomes Journal of Colloid and Interface Science,

Vol. 154, No. 1, November 1992

70

DAVID S. ROSS { d20ab

d

sin(0ab)]

= 7~bAOaugSin(O~b),

0 ~< q ~ 1.

[151

Again, the equations for the other interfaces have the same form after the change of subscript. Under the coordinate transformation, Eq. [13 ] becomes 7abZab(0) = 7acZac(0) = 7bcZbc(I) = 7.

[161

Here, we have introduced 7, the x coordinate of the triple intersection point, a single parameter that will replace the three arc lengths 7

7

Tab = Zab(0) ,

7ac -- Zac(0) ,

7bc

Zbc ( 1 )

[17]

By substituting these expressions in [15] and its analogs for the other interfaces, we obtain [ d20ab

d sin(0___ab)] Zab ]

Zab(0) 2

,

dOab

O'abZab(O)W

(0) +

dOac

0"acZac(0 ) W

0 ~< q ~< 1,

[18]

(0)

dObc

--ab~Zbc(1)--~q ( 1 ) = 0 .

[191

This condition is a compatibility condition; it ensures that solutions of the second-order equations will, in fact, be solutions of the original, first-order system. Were we to use some condition other than Eq. [ 19 ], there would be no guarantee that the constant contributions to the pressures would stand in the proper relation. Physically, this would mean that the forces might not balance. The other conditions that can be naturally regarded as boundary conditions for the second-order differential equations are the transformed versions o f [ 4 ] , [5], and [6], 0ab(1) = 71", 0ac(1) = 0,

trab~--~-qe+dq

= 72 Apabg sin(0ab)

need one more. We obtain it by adding Eq. [ 1] to its analog for the A - C interface and subtracting from that sum the B-C analog of [ 1]. In the transformed coordinates, this yields

0bc(0 ) = 0

[20]

and the transformed expression of the Neumann triangle conditions, ffabSin(0ab(0)) + O'acSin(0ac(0))

and analogous equations for the other interfaces. The net result of these manipulations is that we have reduced the number of constants to be determined from five to one. That one constant is 7. However, we have paid a price; we now have three second-order ordinary differential equations, each of which requires two boundary conditions, where before we had three first-order equations, which required only one initial or final condition each. We used the two conditions expressed in Eq. [ 16 ] to replace the three arc lengths with 7; two conditions used, two parameters eliminated. The other two parameters that we have eliminated, APab and &Pa~, we eliminated by differentiating; we did not use any of the conditions. So this leaves us with two conditions that can be considered boundary conditions for the second-order differential equations. We Journalof Colloidand InterfaceScience, Vol. 154,No. 1, November1992

--aucsin(Obc(1))=O

[21]

0"abCOS(0ab(0) ) -1- 0"acCOS(0ac(0))

- ~b~COS(0bc(1)) = 0.

[22]

The remaining differential equations retain their forms under the coordinate transformation, dZab

dq

--

d~ab

COS(0ab),

dq

-- sin(0ab), 0 < q ~ < 1,

[23]

for the coordinates of the A-B interface, and their analogs for the other interfaces. The initial/final conditions for these equations become Zab(1 ) = 0

7zac(1) =

x~zac(O)

[24a]

[24b]

SHAPES OF F L O A T I N G

Zbc(0) = 0

[24C]

71

F L U I D LENSES

We replace Eq. [ 18 ] with 0~g 1 - 20]b + 0Jb 1

~ a b ( 0 ) = ~ a c ( 0 ) : ~ b c ( 1 ) = 0.

[251

Note that the equations for ~ab(q), ~ac(q), and ~bc(q) are uncoupled from the rest of the equations. Once the other functions are known, these functions can be determined simply by integration• For fixed 7, given the densities and the surface tensions, the system comprising Eqs. [ 18 ] - [ 25 ] (including the analogs of [ 18 ] and [23] for the other interfaces) constitutes a complete problem. It is the not-particularlyuseful problem of computing the shape of a lens whose triple intersection point is a specified distance from its axis of symmetry. What is needed for a solution of the original problem is an equation that allows us to determine the volume of the lens as a function of 7. This equation is the transformed version of Eq. [14], V = --27rn 3 +

fo I [ Zbc(q)~bc( q)coS( Obc(q) ) Zbc( 1 )3

Zab(q)~ab( q)cos( Oab(q) )J] d q. [26] ~

The form of this equation may be deceptive. Vis not a simple function of 7; the integrand in [26] depends on ~7in a complex manner, via the solution of the system [ 18 ]- [ 25 ].

Aq 2 sin(0~b) + sin(0~g I) • "4- ~_ j ab +l ZJab

sin(O Jab) + sin(O~g 1) Z~b + ZJa~1 Aq

+

=

rl 2 A p a b g

-O'ab

SCHEME

We shall first present a scheme to solve Eqs. [ 18 ] - [ 25 ] for given values of the densities, surface tensions, and n. Then we shall present a simple binary search that inverts Eq. [ 26]. We begin by dividing the interval 0 ~ q 1 into n subintervals of length Aq = 1/n. We shall refer to the point q = jAq as the jth grid point, for 0 < j ~< n. We shall use superscripts to denote the finite difference approximations to functions at the grid points; e.g., Z~b is the finite difference approximation to zab(q) at the jth grid point.

~ (Zab)

l~
1

[27]

'

Again, the equations for the other interfaces are obtained by substituting the appropriate subscripts. In this equation, the index i in the denominator on the right-hand side is equal to 0 for the A-B and A-C interfaces and is equal to n for the B-C interface• Note that the averaged form of the discretization [ sin (0 ~c) • j+l j j+l + S l n ( 0 b c ) ] / ( Z b c -1- Zbc ) e l i m i n a t e s t h e s i n g u l a r i t y in the equations for the A-B and B-

C interfaces, on the assumption that Z~c and Zanb -1 are nonzero. Discretization of Eq. [ 19 ] yields 0 1 0 1 O'abZabOab "~- O'acZac(Oac + ~ A ) n n-1 - - O ' b c Z b c ( O b c + I~B) 0a0b = 0 O'abZab "4- O'acZ0c -- O-bcZg c

[28]

Equations [ 20 ] - [ 22] become 00ac = 00ab -- I~A, 0nb = 71",

NUMERICAL

sin(0~b)

0~c = 00b --

0nc = 0,

00c = O.

FB

[291 [30]

The first-order equations, Eq. [23] and its analogs for the other interfaces, are integrated using the trapezoid rule, and each has an initial or final condition z j +abI

" -- Z Jab

Aq

j + l ) -t- C O S ( 0 J b ) COS(0ab

2

~ jab + l -- ~Jb

Aq = sin(0~ -l) + sin(0~b) 2

O<~j<~n - 1 [311 Journal of Colloid and Interface Science,

Vol. 154, No. 1, November 1992

72

DAVID S. ROSS

Z2b=0, .zj+l ac

Zjc

- -

(Oh=0

[32]

j+l ) + COS(0L) COS(0ac

Aq

2

'

O<~j
1.

[331

The initial condition for Eq. [33 ] will be the finite difference analog of Eq. [ 24b ]. However, Eq. [24b] is a relation between the initial and final values of Z a b ( q ) . To obtain a condition in a more standard form, we must do some work. If we integrate both sides of the A-C interface analog of Eq. [23] with respect to q, and consider the result along with Eq. [24b1, we have a system of two equations in the two unknowns Zac(0 ) and Zac(1),

in this scheme with the superscript k; e.g., Z(a~'k) is the k th iterative approximation to ZJab. We need an initial guess to begin the iteration. In our calculations, we have used two types of initial guesses: a lens shape composed of two cones with a planar A-C interface, and a spherical lens with a planar A-C interface. However, in practice, we have found that the convergence of the scheme has not depended on the initial guess, within the range of reasonable initial guesses.

z(.g'~) = 0

z~,~ = Za~÷',~ -

Aq

COS(0a~+~'k-') + CoS(Oa(~k-~)) 2

*/Za~( 1 ) -- XooZ~¢(O) = 0 Zac(1) -- Zac(0)

=

f0

O<~j~ncos(Obe(q))dq.

= z(O,k) ac

By solving these two equations, we obtain Zac(O) - - -~

Xoo

n

Z0c Xoo

--

n-I COS(0(b~J,k-1)) Av COS(0(bJ+l,k-l))

xE

~01 COS(Obe(q))dq,

%1 COS(0£c) _[_ COS(0~ +1) ~] j=O

~(j,k) z ac( j + l =, k~ac ) COS(0(J+l,k-1)) -]- COS(0(J,k-1))

+ Aq Aq.

2

O~j
[341 The rest of the equations are simply the analogs of [31] and [32] with the appropriate subscript substitutions. Equations [ 27 ] - [ 34 ] constitute a complete set of finite difference equations. Once they are solved, we can determine the approximation to the lens volume from the finite difference analog of [ 261, V=-2r~

n [ J J J 3 ~ [ Zbc~bcCOS(0bc)

j=o

Zbe

j Aq.

[351

Z(bJ+l,k ) = ~be~(J'k)

+ Aq

cos(0~{+',k-')) + cos(0 ~b{,k-l)) 2 O<<.j~n-

Journal of Colloid and Interface Science, Vol. 154, No. 1, November1992

1

o(O,k)

ab

ra. ~.(O,k)/~(l,k-1) + tr .r(O,k)[~9(1,k-l) .-}- pA ) t~ao yap vac~ac xvac _

z(n,k)to(n-lk-l)

0 ac (O,k) ~--- Oab' a (0 k) -- FA t~(n,k)

We use an iterative scheme to solve the finite difference system. We shall denote the iterates

1

= 0

Obc bc t be ' 7]ff-~at,ZaD(O'k) + uac~'T v'(0,k)ac __ O'bcz-(n'k)be

z~ucos(0~u)] (zOb)3

(O,k)

--

(z~) 3 +

Aq

2

j=0

which, in finite difference form, is the initial condition for Eq. [331,

1

I/bc

a (O,k)

Uab

0 ( ~ 'k) -~ 7r

-- P~

FB)

SHAPES OF FLOATING

0 ~ 'k) = 0

o(O,k) bc =

0

0a~,~ = 0a~+~'k-~ + 0a<~-"k~ 2

Aq ( s i n ( 0 ~ ,k-~)) + sin(0a<{+1,~-]))

+T /

+

_ sin(Oa(~'k-l))_+_ sin(0a(~-l'~))]

+

)

Aq 2 ~/2 (Pa - Pb)g sin(0~ 'k-l)) 2 (Tab (z(~k)) 2

'

O<~j<~n - 1. Again, the equations for the other interfaces are obtained by making the correct index substitutions. The superscript i in the denominator of the last term in this equation has the value 0 for the A-B and A - C interfaces and is equal to n for the B-C interface. This iterative scheme is a straightforward extension to this nonlinear problem of the standard Gauss-Seidel method (3). It has the advantage that no inversions of nonlinear functions are necessary; at each step, the value of the variable being computed is given explicitly as a function of known quantities. In our experience, this iteration converges for physically realistic values of the parameters. However, there is no convergence proof for the scheme. Finally, we regard the entire iterative scheme as a means of computing the volume of the lens in terms o f n using Eq. [35]. Then, given the volume V of the lens, we compute n (and the shape of the lens of that volume) by a binary search. We guess at a value, nl, that is small enough that the volume VI of the lens associated with that value of ~ is less than V. We guess another value, nu, that is large enough that the volume V~ of the lens associated with that value of n is greater than V. We then compute the volume of the lens with ~Tm= (~7~ + nu)/2. If this volume is less than V, it becomes VI and nm becomes ~7]. If this volume is greater than V, it becomes V~ and

FLUID

73

LENSES

7/m becomes 7/u. We repeat this process until we have a lens whose volume is sufficiently close to V. What we have presented is something of a "bare essentials" algorithm. We have used a uniform grid, the same grid for all functions; one obvious enhancement would be to refine the grid locally where interface curvatures are high. We have used the simplest possible iterative scheme; another, simple, enhancement would be to introduce an overrelaxation parameter (3). An even more sophisticated option would be to use Newton's method or some other more sophisticated method for the solution of the nonlinear difference equations. Certainly, there are abundant faster alternatives to the binary search (3). However, given the speed of modern computers, we think that for most applications such improvements would be pointless. The scheme we have described converges rapidly. COMPUTATIONAL

EXAMPLES

The sequence of graphs in Fig. 2 shows configurations with O'ab = 45 d y n / c m , ~c = 72 d y n / c m , and ~bc = 50 d y n / c m . These values are in the ranges of typical air-oil, air-water, and oil-water surface tensions, respectively. The densities Pa = 0.00 lg/cc (approximately the density of air), and pc = 1.0g/cc (the density of water) are fixed, as is the volume, V = 2 cc, and the acceleration of gravity, g = 980.0 cm/s 2. The lens density Pb is different in each of the five graphs. In Fig. 2a, Pb = 0.0g/cc, in Fig. 2b, Pb = 0.3g/cc, in Fig. 2c, Pb = 0.7g/ CC, in Fig. 2d, Pb = 1.0g/cc, in Fig. 2e, Pb = 1.1 g/CC. For values of Ob greater than 1.1 g~ cc, the algorithm diverges. Thus, 1.1 is a lower bound for the density of the most dense lens that can be supported in such a configuration. The sequence of graphs in Fig. 3 shows configurations with the same parameter values, except that Pb is fixed at 0.7g/cc, and the volume of the drop is different in each of the graphs. In Fig. 3a, V = 0.1 cc, in Fig. 3b, V--1.0 ee, in Fig. 3c, V = 10.0 cc. This sequence depicts the transition from the small-lens limit, Journal of Colloid and Interface Science, Vol. 154, No. 1, November 1992

74

DAVID S. ROSS

i

.......................................................... i...............................

O.8

~..............................

~.............................

~..............................

.................................................. i............................................................... i......................................................... O.4

O.6

..........................

~.............................

i ............................

~............................................................................................

0.4

.............................

, ............................

i ...............................

: ...............................

~ o

o

-o.~ ........................i

......................

-0.4

...........................

...........................

.......................................................................

i ...........................

~ .............................

iiiiiii

0,2

i

iiiiiiiii

....

'- ............................

-0.4

i......................................................................................... F

-O,6

-O,8

-~.8

..............................................

-1_3

-2

~............................ -i

} i

i i

o

1

............................... I ~I-3

-2

-i

i

i

x

O.B

o.~

0.6

i

i

........................... i............................. i........................... i .............................. i ............................... ~..............................

0.4

0.2!

~,

o

0.2 i

-

I .....................

i ........................................................................................

~ ............................

-

-o.2

-o.4

-0.6

-0.8

........................................................

-1_3

-2

i ..........................

i

'~.............................

-i

-3

i

-2

0 x

-1

0.8

..................... ,........................................................ ,............................. "......................... ~.........................

0.6

..................... ,..................................................... ,............................. ............................ ~..........................

.............................................................................................................

0.4

1

!........................................................

0.2

-0.2

-0.4

-0.6

-0.8

-i

-2

-3

-i

0 x

FIG. 2. aab = 45 dyn/cm, a~c = 72 dyn/cm, ~r~ = 50 dyn/cm, P~ = 0 . 0 0 1 g / c o , g = 980.0 cm/s=. (a) Pb = 0.001g/cc. ( b ) p b = 0.3glcc. (e) oh = 0.7g/co. (d) pb =

in w h i c h the lens is c o m p o s e d o f t w o spherical caps, to the large-lens l i m i t , in w h i c h the lens is a p a n c a k e . T h e final e x a m p l e s are l i m i t i n g cases that w e d i s c u s s e d in the I n t r o d u c t i o n , cases in w h i c h o n e o f the surface t e n s i o n s e x c e e d s t h e Journal

of

Colloid and I n t e r f a c e

Science,

Vol.

154,

No.

1, November

1992

Pc = 1 . 0 g / c c ,

1.Og/cc.(e) Pb =

2 cc, 1.1g/cc.

V =

s u m o f the other t w o , a n d Pa ~< Pb ~ Pc. I f aao > O'ab "~ O'bc, fluid B w i l l w e t the A - C interface;

w e shall a s s u m e that aac < aab + (rbc. Strictly speaking, w e d o n o t h a v e a lens in s u c h cases, w e h a v e a drop ( 1 ). H o w e v e r , w e can m o d e l s u c h c o n f i g u r a t i o n s as l e n s e s w i t h the three

SHAPES

O.6

.......................................................

0.4

........................... , .............................. i ............................ ~...........................

0.2

.........................

-0.2

,. . . . . . . . . . . . . . . . . . . .

.............................................

OF FLOATING

i"

FLUID

~ ...........................

,. . . . . . . . . . . . . . . . . . . . .

,..........................

, ...............................

75

LENSES

o.8

...................

o. 6

........................ i .........................

~...........................

o.4

.........................

i ..........................

i.........................

........................

i ......................

i ...........................

-o.2

.... iZiiiiiiiilliZiiiiiiiiiiiiiiiiiiililZilZiiZiilLi

......................

- ....................

i ............................

~

.....

. . . . . . . . . . . . . . . . . . . . . . . . . .

~ .......................

~. . . . . . . . . . . . . . . . . . . . . .

-0.6

i-

............................. i ........................... ~-.......................... ~

~. . . . . . . . . . . . . . . . . . . . . .

...... ~ . . . . . . . . . . . . . . . . . . . . . . .

-o.4

................

;..................

-o.6

............

i ........................

.....................

i

.............

. . . . . . . . . . . . . . . . . . . . . . . . . . i. . . . . . . . . . . . . . . . . . . . . . . .

i ........................

~ ............................

- ....................

.......................

- ......................

-o.s

...................................................... f..............................!...........................i .................................................... _~

-3

-2

i

i

-1

0 x

-

i

o.8

o.6

.........................

1I

0.4

-2

-3. 13

-1

i ...........................

T.............................

~.........................

~ ..........................

i

'~

i

i

i

.........................

( ..........................

i ........................

) ........................

i ...........................

~ ..........................

.......................

) .............................

i ..........................

i ..........................

i ..........................

i ..........................

.

i

..............................................

O.2

-0.4

* ...........................

i

...........................................................................

....................

i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.....................

i

....

i...................... i...............................................

....................................................................

-13

-2

-i

i

i

i

o

1

×

F I G . 3. a.b = 45 d y n / c m , aac = 72 d y n / c m , aM = 50 d y n / c m , Pa = 0 . 0 0 1 g / c c , Pc = 1 . 0 g / c c , Pb = cc, g = 9 8 0 . 0 c m / s 2. ( a ) V = 0.1 cc. ( b ) V = 1.0 cc. ( c ) V = 10.0 cc.

0 . 7 g /

1

O.8

....................... ~.......................... ~......................... ~..........................................................

o.6

.....................

i .............................

i ........................

i ...............................

i

~".............................

0.6

i

i .......................................................

0.4 ........................i...............................i............................i.............................i..............................~..............................

o.4

o,2

o

-o.2 ................. i......................... . . . . . . . . . .

i......................... ~...........................

-0.4

i ...............................

~

1

2

.............

-O.6

i ........................

i. . . . . . . . . . . . . . . .

-o.2

.............................

.....................................................................................

1

i

-3

-2

-1

o

i

-t

i

-3

~2

-Z

x

F I G . 4 ( a ) . ~,c = 72 d y n / c m , aM = 50 d y n / c m , a,b > 122 d y n / c m , p~ = 0 . 0 0 1 g / c c , Ob = 0 . 7 g / c c , Pc = 1 . 0 g / c c , V = 1.0 cc, g = 9 8 0 . 0 c m / s 2. B e c a u s e ~,b is tOO large, n o A - B i n t e r f a c e c a n exist, a n d fluid C w e t s t h e d r o p . T h i s c a s e is m o d e l e d b y s e t t i n g O'ab = 122 d y n / c m . ( b ) S a m e c o n f i g u r a t i o n , e x c e p t aab = 50 d y n / c m , Crb~> 122 d y n / c m ,

a n d fluid A w e t s t h e d r o p .

Journal

of

Colloid

and

Interface

Science,

Vol.

154,

No.

1, November

1992

76

DAVID S. ROSS

interfaces confluent at the triple intersection point. If, say, aab > O'ac"]- O'bc, fluid C will wet the drop. T h e A - B interface will disappear. In its place, we will have a very t h i n layer o f fluid C separating the d r o p f r o m fluid A; the A - B interface has b e e n r e p l a c e d b y a n A - C interface a n d a B - C interface s e p a r a t e d b y a t h i n layer o f fluid C. I f we a s s u m e t h a t this layer is infinitesimally thin, we can treat it as the l i m iting case o f a lens, with O'ab = O'ac q- ffbc. T h e g r a p h in Fig. 4a shows a case with aac = 72 d y n / c m , Crbc = 50 d y n / c m , a n d ~ab > 122 d y n / c m . W e m o d e l it, as discussed above, b y p u t t i n g gab = 122 d y n / c m . T h e densities are Pa = 0 . 0 0 1 g / c c , Pb = 0 . 7 g / c c , a n d Pc = 1 . 0 g / c c . T h e v o l u m e , V = 1.0 cc, a n d the acceleration o f gravity, g = 980.0 c m / s 2. T h e graph in Fig. 4b shows a configuration with the s a m e data, except aab = 50 d y n / c m , a n d

Journal of Colloid and Interface Science. Vol. 154, No. 1, November 1992

abc > 122 d y n / c m , which we m o d e l b y putting abe = 122 d y n / c m . ACKNOWLEDGMENTS The author thanks Rich Dempsey, Dave Miller, K. C. Ng, and Tom Whitesides for helpful discussions of this problem, and he thanks the referees for several useful suggestions. This research was supported by Eastman Kodak Company. REFERENCES 1. Princen, H. M., in "Surface and Colloid Science" (E. Matijevir, Ed.), Vol 2, p. 57. Wiley-Interscience, New York, 1969. 2. Pujado, P. R., and Scriven, L. E., J. Colloidlnterface Sci. 40, 82 (1972). 3. Stoer, J., and Bulirsch, R., "Introduction to Numerical Analysis." Springer-Verlag, New York, 1980. 4. Princen, H. M., J. ColloidSci. 18, 178 (1963). 5. DoCarmo, M. P., "Differential Geometry of Curves and Surfaces." Prentice-Hall, New York, 1976.