Maximal liquid bridges between horizontal cylinders
aa r X i v : . [ c ond - m a t . s o f t ] J un Maximal liquid bridges between horizontal cylinders
Himantha Cooray, Herbert E. Huppert,
1, 2, 3 and Jerome A. Neufeld
1, 4, 5 Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK ∗ Faculty of Science, University of Bristol, Bristol BS8 1UH, UK School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia BP Institute, Bullard Laboratories, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK Department of Earth Sciences, Bullard Laboratories,University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
We investigate two–dimensional liquid bridges trapped between pairs of identical horizontal cylin-ders. The cylinders support forces due to surface tension and hydrostatic pressure which balancethe weight of the liquid. The shape of the liquid bridge is determined by analytically solving thenonlinear Laplace–Young equation. Parameters that maximize the trapping capacity (defined asthe cross–sectional area of the liquid bridge) are then determined. The results show that theseparameters can be approximated with simple relationships when the radius of the cylinders is smallcompared to the capillary length. For such small cylinders, liquid bridges with the largest cross sec-tional area occur when the centre–to–centre distance between the cylinders is approximately twicethe capillary length. The maximum trapping capacity for a pair of cylinders at a given separation islinearly related to the separation when it is small compared to the capillary length. The meniscusslope angle of the largest liquid bridge produced in this regime is also a linear function of the sepa-ration. We additionally derive approximate solutions for the profile of a liquid bridge making use ofthe linearized Laplace–Young equation. These solutions analytically verify the above relationshipsobtained for the maximization of the trapping capacity.
I. INTRODUCTION
The trapping of a fluid in contact with a solid is a gen-eral problem with applications in biological, engineering,industrial and geological processes. Generally, a volumeof liquid trapped by two or more solid surfaces and im-mersed in a different fluid is called a “liquid bridge”.The trapping is achieved by balancing the weight of theliquid with the surface tension forces acting along thethree–phase contact lines and the forces of hydrostaticpressure exerted on the solid–liquid contact surfaces. Adetailed review of liquid bridges can be found in Butt andKappl [1]. Liquid bridges are a very common occurrencein granular matter and porous media. Examples includetrapping of water in sand, which acts as an adhesive insand castles [2], and capillary trapping of supercriticalcarbon dioxide in porous rocks [3] during carbon dioxidesequestration.In this paper, we study two-dimensional liquid bridgesproduced between pairs of horizontal cylinders. A studyin this simplified geometry is a first step in the detailedunderstanding of trapping in porous media. It can alsogive insights into the behaviour of a three–dimensionalliquid bridge trapped between cylindrical rods. Liquidabsorption to textiles [4] and retention of water dropletson spider webs are common examples of trapping in thisgeometry. Additionally, it has recently been proposedas a method of handling and mixing small volumes ofliquid in analytical research [5]. Princen [6] and Lukas ∗ [email protected]; http://himantha.freeshell.org/ and Chaloupek [4] solved this problem in two dimen-sions neglecting the effects of gravity. Such solutions losetheir accuracy as the amount of trapped liquid increases.Although three–dimensional profiles of trapped dropletshave been studied experimentally [7, 8] and numerically[9, 10], there is no straightforward method to determinehow much liquid a given system can trap.Capillary trapping in other related geometries has beenstudied using a variety of methods. Urso et al. [11] anal-ysed trapping of a liquid in a two–dimensional porousmedium comprised of horizontal cylinders. They stud-ied trapping in the limit of small liquid volumes, wheregravitational effects can be neglected and the liquid–fluidinterfaces may be approximated by circular arcs. Chenet al. [12] determined the shape of a three–dimensionalliquid bridge trapped between vertical plates using a per-turbation method in which the weight of the liquid wasneglected, and calculated numerically, using a finite el-ement method, cases in which the weight was incorpo-rated. While a two–dimensional liquid bridge is approx-imately symmetric in the vertical if its weight is close tozero, the shape becomes significantly asymmetric whenmore liquid is added. The shape of the lower interfacein this regime can be modelled as a pendant drop. Pro-files of pendant drops have been studied extensively fortwo–dimensional [13, 14] and axially symmetric [15, 16]cases. Although the above solutions take all the phys-ical parameters into account, they are either analyticalsolutions that give complicated expressions or numericalsolutions and, as a result, do not provide direct expres-sions to determine the trapping capacity.The study in this paper starts with an exact solu-tion for the profile of a two–dimensional liquid bridge of FIG. 1. A liquid bridge formed between a pair of horizontal cylinders. θ is the contact angle, ω and ω are the angles fromthe vertical where the liquid meets the cylinder and ψ and ψ are the interfacial slope angles, which are positive if measuredcounter-clockwise. All the lengths are nondimensionalized by dividing by the capillary length. The height y is proportional tothe pressure of the liquid at the liquid–fluid interface relative to the pressure of the fluid. At y = 0, the pressure differencebetween the two phases and the interfacial curvature are 0. R is the radius of the cylinders and d is the half distance betweentheir centres. arbitrary volume. Results obtained using this solutionshow very simple approximate relationships governingthe maximum trapping capacity: the maximum trappingcapacity is linearly related to the separation between thecylinders when the separation is small compared to thecapillary length; and the separation that produces thelargest trapping capacity is twice the capillary length.We then analytically verify these limiting relationshipsusing several approximate solutions for the shape of aliquid bridge. II. THEORETICAL SETTING
We consider a two dimensional, horizontally symmetricliquid bridge produced between a pair of identical hori-zontal cylinders as shown in figure 1. The weight of theliquid is balanced by the forces of surface tension andthe reaction to the hydrostatic pressure exerted by thecylinders. Both liquid–fluid interfaces of the liquid bridgemeet the cylinders at a fixed contact angle θ , which is inpractice locally determined by the fluid and solid surfaceenergies. The interfacial slope angles at the contacts aregiven by ψ i , where the subscript i = 1 denotes the up-per interface and i = 2 denotes the lower interface, and ψ i is positive if the interface slopes upwards leaving thecylinder. The point of contact between a cylinder and aninterface is denoted by the angle ω i to the vertical. Thefollowing relationships between ψ i , θ and ω i are obtainedby consideration of the geometry of the system ψ = θ − ω , (1) ψ = π − θ − ω . (2)The shape of each liquid interface of the liquid bridgeis governed by the nonlinear Laplace–Young equationwhich relates the pressure difference across the interface to its curvature. If the height Y of the interface is givenas a function of the horizontal position X by Y = G ( X ),the Laplace–Young equation is written as Y = ℓ c P ( X ) G XX [ G X + 1] / , (3)where the subscripts denote derivatives and the capillarylength defined as ℓ c ≡ r γ ∆ ρ g , (4)in which γ is the liquid–fluid interfacial tension, ∆ ρ is thedensity difference between the liquid and the fluid and g is the acceleration due to gravity. P ( X ) = ± P ( X ) = sgn { D [ X, G ( X ) − δ ] − D [ X, G ( X ) + δ ] } , (5)where D ( X, Y ) is the density at a location (
X, Y ) coveredby a fluid, which is assumed to be constant within eachphase, and δ is a positive infinitesimal length.Due to the symmetry of the system, we only need tosolve for a half of the bridge to determine its full shape.(In the solution presented here, we only consider the leftside). However, writing the Laplace–Young equation inthe form of (3) has several drawbacks. First, it cannot besolved by direct integration and, secondly, the shape ofthe lower interface can be multivalued relative to X andalso P ( X ) can change sign within a single fluid interface(for example, consider the lower fluid interface of the liq-uid bridge shown in figure 2(b)). These problems can beavoided by instead expressing the interfacial shape as afunction of Y . It is also convenient to nondimensionalizeall the lengths with respect to the capillary length and ✲✶ ✵ ✶✶(cid:0)✶(cid:0)✁ ✲✶ ✵ ✶✲✶✵✶ FIG. 2. Shapes of two liquid bridges obtained using exact and approximate solutions of the Laplace–Young equation. Somevalues of ψ can produce two different liquid bridges because (12) can have two solutions for ψ . Both liquid bridges shownhere are obtained using the same input parameters R = 0 . , d = 0 . , θ = 0 and ψ = − π/ ψ ,i.e. 0 .
44 in ( a ) and − .
17 in ( b ). The black solid curves (–) are obtained from the solution to the nonlinear Laplace–Youngequation (23). The magenta dashed curves ( - - ) show an approximation (33) for the shapes of the upper interfaces of theliquid bridges obtained by solving the linearized Laplace–Young equation. The blue dashed curve ( - - ) in (a) shows a similarapproximation (38) for the lower interfaces which is valid when the interfacial slopes are small. The red dashed curve ( - - ) in(b) is a composite approximation for the shape of the lower interface, valid for distended liquid bridges, given by (59). Theresults show very good agreement between the exact and approximate solutions. define x = X/ℓ c and y = Y /ℓ c . The interfacial shape canthen be written as x = f ( y ) , (6)where x = 0 is the axis of symmetry and y = 0 representsthe vertical coordinate at which d f / d y = 0, which isnot known a priori and has to be determined as a partof the solution. The nondimensionalized Laplace–Youngequation is y = p ( y ) f yy (cid:0) f y + 1 (cid:1) / , (7)with p ( y ) = ± p ( y ) is now deter-mined by whether the liquid phase is located in the right hand side or left hand side of the fluid phase, so that p ( y ) = sgn { D n [ f ( y ) − ǫ, y ] − D n [ f ( y ) + ǫ, y ] } (8)as ǫ →
0, from above, where D n ( x, y ) is the fluid den-sity at a location ( x, y ) which is specified in terms of thenondimensionalized coordinates. Since only a half of aliquid bridge is to be solved, p is constant within eachinterfacial segment we consider and it depends only onthe direction of the meniscus slope at the contact point p = ( sgn( ψ ) for the upper meniscus − sgn( ψ ) for the lower meniscus . (9)The liquid bridge shown in figure 1 is trapped betweencylinders of (nondimensionalized) radius R and a centre-to-centre distance d . If the vertical coordinates of thecontact point and middle point of each interface of theliquid bridge are y = u i and y = v i respectively, theinterfacial slope angle defines a boundary condition ateach contact point f y ( u i ) = − cot( ψ i ) , (10)and the requirement for symmetry provides a boundarycondition at the centre linelim y → v i f y = − sgn( ψ i ) ∞ . (11)Finally, we impose that the free surfaces intersect thecylinder at the points f ( u i ) = d − R sin ω i , (12)and are continuous across the centre line f ( v i ) = 0 . (13)In the following section, we obtain a solution for thefull shape of the liquid bridge given R , θ , d and ω (or ψ ) and predict ω , u i and v i as part of the solution. III. EXACT SOLUTION OF THE NONLINEARLAPLACE–YOUNG EQUATION
The Laplace–Young equation given in (7) may be in-tegrated and rearranged to obtain f y = p ′ y − a i q − (cid:0) y − a i (cid:1) , (14)where a i is a constant of the integration and p ′ = ± p ′ , we differentiate the aboveequation to obtain f yy = p ′ y h − (cid:0) y − a i (cid:1) i . (15)Comparison of this result with (7) shows that p ′ = p. (16) Substitution of f y given by (14) into (11), which de-notes the meniscus slope at the mid–point of each inter-face, yields a i = 12 v i + q, (17)where q = p sgn( ψ i ) . (18)The value of p in (9) is combined with (18) to produce q = ( − . (19)We then combine (10), which gives the meniscus slopeat a contact point, with (14) and (17) to obtain u i = v i + 2 q (1 − cos ψ i ) . (20)The general shape of an interface is determined by inte-gration of (14). This integration is carried out using thesubstitution 12 y − a i = cos α, (21)which transforms (14) to f α = − p cosα √ a i + cos α . (22)The interface may therefore be described completely bythe expression f ( y ) = p n − sgn( y ) g ( y ) + [sgn( y ) − sgn( v i )] g (0)+ sgn( v i ) g ( v i ) o , (23)where g ( y ) = q q ) + v i E (cid:20)
12 cos − (cid:18) y − v i − q (cid:19) , q ) + v i (cid:21) − q + v i p q ) + v i F (cid:20)
12 cos − (cid:18) y − v i − q (cid:19) , q ) + v i (cid:21) , (24)is given in terms of incomplete elliptic integrals E ( σ, k )and F ( σ, k ) [17]. This equation satisfies the boundarycondition f ( v i ) = 0 and remains continuous at y = 0. According to the Laplace–Young equation, the pres-sure in the liquid side of the interface is higher than thepressure in the fluid side when a liquid surface is con-vex. As a result, a convex liquid surface corresponds toa negative y and a concave liquid surface corresponds toa positive y . If the lower interface of the liquid bridgeslopes downwards at the contact point (i.e. ψ < x = 0) to satisfy thesymmetry. This makes v negative. If ψ is positive, theinterface is concave in the middle and v is therefore pos-itive. Using a similar argument for the upper interfaceas well, one can obtain the following general relationshipfor a liquid bridge.sgn( v i ) = − q sgn( ψ i ) , (25) This equation can be used to eliminate sgn( v i ) from (23)to obtain f ( y ) = p n − sgn( y ) g ( y ) + [sgn( y ) + q sgn( ψ i )] g (0) − q sgn( ψ i ) g ( v i ) o , (26)and v i can be eliminated from (24) using (20) to produce g ( y ) = q q cos ψ i ) + u i E (cid:20)
12 cos − (cid:18) y − u i − q cos ψ i (cid:19) , q cos ψ i ) + u i (cid:21) − q cos ψ + u i p q cos ψ i ) + u i F (cid:20)
12 cos − (cid:18) y − u i − q cos ψ i (cid:19) , q cos ψ i ) + u i (cid:21) . (27)We now use the boundary condition that defines thehorizontal position of the contact point of each meniscigiven by (12) to obtain a relationship between ψ i and u i .The geometry of the cylinder gives the relationship be-tween the vertical positions of the upper and lower con-tact points of the menisci u = u − R (cos ω − cos ω ) , (28)from which ω i can be replaced using (1) and (2) to obtain u = u − R [cos ( θ − ψ ) + cos ( θ + ψ )] . (29)Equations (28) and (12) with i = 1 and 2 then representthree equations for ψ , ψ , u and u . If any one ofthese four parameters is known, the other three can bedetermined and the shapes of both the menisci can befound.The following steps show the method used to determinethe shapes of the liquid bridges in this paper.1. Select the upper point of contact with the cylinder ω and determine ψ using (1), or select ψ directly.2. Substitute (25), (23) and (27) into (12) and solvefor u .3. Express u as a function of ψ using (29).4. Determine ψ by solving (12), into which (25),(23)and (27) are substituted.5. Determine ω using (2).6. Obtain the shapes of the menisci using (23).For a given value of ψ , (12) gives only one solution for u . However, for some values of u , the solution is multi-valued, and thus can give two solutions for ψ resulting in two different liquid bridges as shown in figure 2. The firstsolution produces a liquid bridge with approximate ver-tical symmetry and the second solution produces a largerliquid bridge in which the lower interface is significantlydistended, and as a result, contains a larger amount ofliquid compared to the first. Both these solutions areequally valid. IV. APPROXIMATE SOLUTIONS FOR THESHAPES OF THE LIQUID INTERFACESA. Shape of the upper interface as | ψ | → Expressing the shape of the upper meniscus by thefunction y = j ( x ) and assuming the interfacial slopes tobe small ( j x ≪ j = j xx . (30)Solution of this equation with the boundary condition j x (0) = 0 gives j ( x ) = c cosh x, (31)where c is a constant to be determined. Since the ver-tical component of the surface tension force exerted bythe cylinders at the contact points is equal to the weightof a liquid meniscus with vertical edges [18, 19], the forcebalance may be written as Z d − R sin ω j d x = − sin ψ . (32)This gives the correct value for c , and so j ( x ) = − sin ψ cosh x sinh [ d − R sin( θ − ψ )] , (33) ✲✶ ✵ ✶✲✶✵✶ ✲✷ ✲✶ ✵ ✶ ✷✲✶✵✶ ✲✵(cid:0)✁ ✵ ✵(cid:0)✁✲✶(cid:0)✁✲✶✲✵(cid:0)✁✵✵(cid:0)✁✶ ✲✶ ✲✵(cid:0)✁ ✵ ✵(cid:0)✁ ✶✲✶(cid:0)✁✲✶✲✵(cid:0)✁✵✵(cid:0)✁✶ FIG. 3. ( a ) and ( b ) show two liquid bridges carrying the same amount of liquid (cross sectional area, A = 3 .
0) between a pairof horizontal cylinders with R = 0 . θ = 0 located at two different separations. The separation d in ( a ) is 0 .
4, which gives ψ = − π/ ψ = − .
40. Parameters in ( b ) are d = 2 . ψ = − .
01 and ψ = − .
48. ( c ) and ( d ) show shapes of liquidbridges corresponding to the maximum trapping capacities for a pair of cylinders with R = 0 .
063 and θ = π/ d = 0 . A max = 2 . ψ ,A max = − .
10 and ψ = − .
50 in ( c ) and d = 1 . A max = 3 . ψ ,A max = − .
88 and ψ = − .
00 in ( d ). The figures show results obtained using both exact and approximate solutions tothe Laplace–Young equation. The black solid curves (–) are the solutions to the nonlinear Laplace–Young equation given by(26). The magenta dashed curve ( - - ) is the approximation for the shape of the upper interface (33) obtained by solving thelinearized Laplace–Young equation. The cyan dashed curve ( - - ) and the green dashed curve ( - - ) are the approximations forthe shapes of the upper part and the lower part of the lower interface given by (55) and (58) respectively. The red dashed curve( - - ) is the composite approximation for the shape of the lower interface (59) obtained by combining (55) and (58). There isexcellent agreement between the approximate and exact solutions when ψ → | ψ | → π/
2. The composite approximation(59) covers both (55) and (58) very well. which is valid in the region where the meniscus slopesare small. If the absolute value of the meniscus slopeangle | ψ | is small, this solution is valid throughout themeniscus, and if | ψ | is large, the solution is valid far(compared to ℓ c ) away from the contact points. As aresult, the approximation for v obtained using (33) isin general more accurate than the approximation for u obtained using the same equation. The height of themid–point of the meniscus is therefore obtained using(33) as v = − sin ψ cosech [ d − R sin( θ − ψ )] , (34)and u is to be determined using (20), which is a rela-tionship between u and v derived from the nonlinearLaplace–Young equation. The upper interface cannotpass through y = 0 because the interface is convex tothe fluid side when y < y > u ) = sgn( v ) , (35)where sgn( v ) is given by (25). Using (34), (35) and (19)on (20), we obtain u = − sgn( ψ ) × n sin ψ cosech [ d − R sin( θ − ψ )]+ 2 (1 − cos ψ ) o / (36)for the contact height of the meniscus. B. Shape of the lower interface
1. Solution for small liquid volumes, | ψ | → In the limit of small liquid volumes, the upper andlower interfaces are nearly symmetric. If the shape ofthe lower meniscus is given by y = k ( x ), the linearizedLaplace–Young equation is k = − k xx . (37)This is solved in a manner similar to the upper interfaceto obtain k ( x ) = − sin ψ cos x sin [ d − R sin( θ + ψ )] , (38)which gives v = − sin ψ csc [ d − R sin( θ + ψ )] . (39)For small values of | ψ | we havesgn( u ) = sgn( v ) . (40) Substitution of the above two equations into (20) and(25) produces u =sgn( ψ ) × n sin ψ csc [ d − R sin( θ + ψ )] − − cos ψ ) o / . (41)Equation (36) gives the value of u for a given ψ . Thisis substituted into (29) to express u as a function of ψ , u = − sgn( ψ ) × n sin ψ cosech [ d − R sin( θ − ψ )]+ 2 (1 − cos ψ ) o / − R [cos( θ − ψ ) + cos( θ + ψ )] . (42)Equations (42) and (41) together provide an implicitequation for ψ . With this result, (33) and (38) give theshapes of the upper and lower interfaces for any given ψ in the limit of small interfacial slopes. The shape of aliquid bridge determined using this method is shown infigure 2(a) as the magenta and blue dashed curves. Itis a very good approximation for the solution obtainedusing the nonlinear Laplace–Young equation.
2. Approximation of the elliptic integrals
The solution to the nonlinear Laplace–Young equationwas given as a function of elliptic integrals in (23). Herewe introduce an approximation to these integrals for thelower meniscus in order to obtain simpler relationshipsthat can describe the meniscus shapes and the trappingbehaviour. Since g ( v ) = 0 according to (24), the rela-tionship (23) reduces for the lower meniscus to f ( y ) = − sgn( y ) g ( y ) + [sgn( y ) − sgn( v )] g (0) . (43)We now use the values of p and q for the lower meniscus,(9) and (19), on (24) to obtain g ( y ) = | v | E "
12 cos − (cid:18) y − v (cid:19) , (cid:18) v (cid:19) − (cid:18) | v | − | v | (cid:19) F "
12 cos − (cid:18) y − v (cid:19) , (cid:18) v (cid:19) . (44)The elliptic integrals in the above equation can be re-placed using the following transformation formulae [17]: F ( σ, k ) = 1 √ k F (cid:18) β, k (cid:19) , (45) E ( σ, k ) = √ k (cid:20) E (cid:18) β, k (cid:19) − (cid:18) − k (cid:19) F (cid:18) β, k (cid:19)(cid:21) , (46)where β = sin − ( √ k sin σ ) . (47)This produces g ( y ) =2 E " sin − s − y v , (cid:16) v (cid:17) − F " sin − s − y v , (cid:16) v (cid:17) . (48)Substitution of y = 0 gives g (0) =2 E (cid:20)(cid:16) v (cid:17) (cid:21) − K (cid:20)(cid:16) v (cid:17) (cid:21) , (49)where E ( k ) and K ( k ) are complete elliptic integrals.Byrd and Friedman [17] gives series approximations forthese functions. Using the first term of each series, weobtain g ( y ) ≈ s − y v − ln " | v | − p v − y | y | (50)and g (0) ≈ π q − v ) − π p − v . (51)This expression for g (0) is then used in the next sectionto determine an approximate solution for the shape ofthe lower meniscus.
3. Solution for large liquid volumes, | ψ | → π/ The solution given in section (IV B 1) is applicable forsmall | ψ | and therefore represents liquid bridges thatcontain only a small liquid volume. We now introduce a solution for liquid bridges where ψ is close to π/
2, andwhere the trapped volume is large and hence, to counter-balance the weight of the liquid, the vertical componentof the surface tension force is high. In this regime, wefocus on the largest liquid bridges, for which v < u > x = h ( y ) with h y ≪
1. The linearized Laplace–Youngequation for this regime is therefore y = h yy , (52)which we may solve to obtain x = h ( y ) = 16 y + c y + c , (53)where c and c are constants. These constants can nowbe constrained by our solutions to the nonlinear Laplace–Young equation. We first recall the constrains (17) and(14), obtained in the solution of the nonlinear Laplace–Young equation, which gives f y (0) = 1 − v | v | q − v . (54)We use the conditions h y (0) = f y (0) and h (0) = f (0),where f (0) is given by the approximation (51), to deter-mine c and c . Thus, we have h ( y ) = 16 y + 1 − v | v | q − v y + π q − v ) − π p − v , (55)which we may combine with the approximation for theupper meniscus determined for | ψ | → u , (42). Combination of this expression with(20) gives v = n sgn( ψ ) q sin ψ cosech [ d − R sin( θ − ψ )] + 2 (1 − cos ψ ) − R [cos( θ − ψ ) + cos( θ + ψ )] o + 2(1 − cos ψ ) . (56)We then use the boundary condition given in (12), thatthe fluid intersects the cylinder h ( u ) = d − R sin( θ + ψ ) , (57)along with u given by (42) and v given by (56), to getan equation which may be solved to determine ψ . Wenote that h ( y ) is a good approximation for the upper part of the lower meniscus, as demonstrated in figures 3.Once ψ , and hence v , are determined, the shape ofthe lower part of the lower meniscus can be obtainedapproximately. The meniscus slopes in this regime aresmall relative to the x axis, and therefore the linearizedLaplace–Young equation, (37), is applicable. Solutionwith the boundary condition k (0) = v gives y = v cos x , FIG. 4. ( a ) The maximum trapping capacity (the cross sectional area of the largest liquid bridge, A max ) between pairs ofhorizontal cylinders. ( b ) The value of ψ for the largest liquid bridges. Symbols show results obtained for different values of R , θ and d by numerically maximising A (60), determined using the solution of the nonlinear Laplace–Young equation, withrespect to ψ . Each marker represents a cylinder radius. Triangles ( △ ) denote R = 0 .
01, squares ( ) denote R = 0 . ) denote R = 0 .
4. Colours represent different contact angles. Red symbols ( △ , , ) represent θ = 0, green symbols( △ , , ) represent θ = π/ △ , , ) represent θ = π/
2. The black curves are approximations for themaximal trapping parameters. In ( a ), the black solid line (–) denotes (68), which is valid for small d , and the black dashedcurve (- -) denotes (86), which is valid when d is close to 1. The black solid line in ( b ) is the equation (74). The approximatesolutions describe the maximal trapping behaviour very well for small R ( R ≪ A max and ω ,A max are linearly relatedto d when d ≪ x = cos − ( y/v ) . (58)We combine the solutions for the upper part of the lowermeniscus (55) and lower part of the lower meniscus (58)to produce the following empirical expression for themeniscus shape, x = tanh (cid:20)
74 ( y − v ) (cid:21) h ( y )+ { − tanh[2( y − v )] } cos − (cid:18) yv (cid:19) , (59)which is valid for the entirety of the lower meniscus asshown in figures 2 and 3. V. THE MAXIMAL TRAPPING CAPACITY
A quantity of significant interest in a variety of physicalsettings is the volume of fluid that may be trapped as afunction of the imposed geometry and material propertiesthrough the apparent contact angle. Here we calculatethe trapping capacity, which in our two–dimensional ge-ometry is equivalent to the cross–sectional area. We thendetermine the maximum achievable trapping capacity ata given separation between the cylinders and the separa-tion at which the largest liquid bridge can be produced.
A. The maximum trapping capacity at a givenseparation
The cross sectional area A of a liquid bridge can bedetermined using a force balance considering the liquidweight and the forces of surface tension and hydrostaticpressure. A = − ψ + sin ψ )+ 2 R u [sin( θ − ψ ) − sin( θ + ψ )]+ R " ψ − ψ + 2 θ − π + 2 cos( θ − ψ ) sin( θ + ψ ) − sin 2( θ − ψ ) − sin 2( θ + ψ )2 . (60)The quantities ψ and u in this equation can be deter-mined as functions of ψ using the solution of the nonlin-ear Laplace–Young equation described in section III. Bynumerical maximisation of A with respect to ψ , the max-imum trapping capacity ( A max ) and ψ that producesthis trapping capacity ( ψ , A max ) can be determined fora given combination of R, θ and d . Two representativeliquid bridges, corresponding to A max for different valuesof d , are shown in figures 3( c ) and ( d ). This solutionprocess was repeated for a range of R, θ and d , and the behaviour of A max and ψ , A max were analysed. The re-sults are shown by symbols in figure 4( a ) and ( b ).Figure 4( a ) shows that the maximal trapping capacity, A max is linearly proportional to the separation, d , when R ≪ d ≪
1. This relationship can be explainedusing the approximate solution derived in section (IV).We define 2 s i as the distance between the contact pointsof a meniscus, that is, s i = d − R sin ω i , (61)so that when d, R ≪ s i ≪ ω i . In this regime,the shape of the upper meniscus has a nearly constantradius of curvature − s/ sin ψ . As a result we have | u − v | ≤ s . (62)Since s ≪ y = v is negligible and since R ≪ y = u . The cross–sectional area of the part of the liquid bridge below y = u is determined by balancing the non–dimensionalizedweight of the liquid A with the force of surface tensiongiven by − ψ and the force of hydrostatic pressuregiven by 2 u s , A = 2 ( − sin ψ + u s ) . (63)The surface tension force acting on the liquid bridge ismore significant compared to the force of hydrostaticpressure since s ≪ A is therefore maximized when ψ ≈ − π/ , (64)which is the meniscus slope angle that maximizes thevertical component of the force of surface tension. Using(64) and d − R sin( θ + ψ ) = s on (57) and replacing q and v using (19) and (20) respectively, we obtain anequation for u − p − u ! u + π (cid:18) q − u (cid:19) − π p − u = s . (65)As s → u ≈ ω = − cos θ , which in combination with(61) produces s = d + R cos θ. (67)Substitution of (64), (67) and (66) into (63) produces A max ≈ d + R cos θ ) . (68)1 FIG. 5. For small cylinders, the maximum trapping capacity A max maximizes at d ≈
1. Figure shows the value of d at whichthe maximum trapping capacity occurs, obtained using the nonlinear Laplace–Young equation. Red squares ( (cid:4) ) are for θ = 0,green diamonds ( (cid:7) ) are for θ = π/ ⋆ ) are for θ = π/ This is plotted by the black line shown in figure 4( a ).It is a good approximation for small cylinders at closerange.We also observe a linear relationship between and ψ ,A max for small R and d in figure 4( b ). This relation-ship can also be verified using the approximate solutionsto the Laplace–Young equation. If | ψ | is small, (33) isvalid throughout the upper meniscus, which gives u = − sin ψ coth( d − R sin ω ) . (69)Since R ≪ u ≈ u , (70)which gives u ≈ − sin ψ ,A max = tanh( d − R sin ω ,A max ) . (71)Since d − R sin ω ,A max = s ≪
1, the above equationgives − ψ ,A max ≈ d − R sin ω ,A max , (72)where ψ and ω are related by (1), which gives ω ≈ θ (73)for small ψ . Substitution of (73) to (72) gives the rela-tionship − ψ ,A max ≈ d − R sin θ, (74)which is plotted by the black line in figure 4( b ). Thisresult approximates the exact solution very well whenthe cylinder radius and inter–cylinder radius are smallcompared to the capillary length. B. The separation which maximizes the trappingcapacity
Figure 4( a ) shows that the maximum trapping capac-ity A max as a function of d is increasing when d ≪ d . Figure 5 plots the value of d in which A max reaches a maximum ( d ( A max ) max ) as afunction of R for different values of θ . Interestingly, itshows that d ( A max ) max = 1 when R ≪ θ . Inthis section, we analytically explain this result based onthe approximate solutions obtained earlier for the liquidbridge geometry.We assume that (33) gives a sufficiently good approx-imation for u u = − sin ψ coth s, (75)where s = s i ≈ d which is valid when R → R , we also have u ≈ u which gives u ≈ − sin ψ coth s, (76)and (20) then gives v ≈ coth s sin ψ − ψ + 2 . (77)Substitution of the above two expressions obtained for u and v into (57) yields m ( ψ , ψ , s ) = 0 , (78)2 æææææææææææææææææææææææææææææææææ - - - - - Ψ FIG. 6. The expression for ψ given in (83) (solid curve: –) is compared with ψ obtained using a numerical solution of (78)(red symbols: • ). The figure shows that the analytical expression is an accurate solution for (78). - - d A d Ψ H Ψ L H a L - d d s A m a x H b L FIG. 7. (a) shows d A/ d ψ calculated using (85) at ψ = ψ = ψ . The derivative is negative around s = 1 and beyond. Since u ≈ u and the two menisci should not intersect, the minimum possible value of ψ is ψ . The negative derivative means that A maximizes when ψ = ψ = ψ . (b) shows the derivative of d V / d s calculated at ψ = ψ = ψ . A maximizes at s = 1. where m ( ψ , ψ , s ) = −
16 coth s sin ψ − coth s sin ψ (cid:0) ψ − coth s sin ψ (cid:1)q − (cid:0) coth s sin ψ − ψ (cid:1) + 14 π (cid:18) q ψ − coth s sin ψ + 2 (cid:19) − π p ψ − coth s sin ψ + 2 − s. (79)Since the contact points of the upper and lower menisciare very close to each other ( u ≈ u ), we need ψ ≥ ψ to avoid the two menisci intersecting each other. We nowconsider the limit ψ = ψ = ψ , where (78) is written as m ( ψ, s ) = 0 . (80)To solve for ψ , m ( ψ, s ) is expanded in a first orderpower series m ( ψ, s ) = m ( ψ , s ) + ( ψ − ψ ) m ψ ( ψ , s ) . (81) A numerical solution of (80) shows that ψ ≈ − s as s → ψ ≈ − s →
1. We therefore select ψ in(81) as ψ ( s ) = − s (1 − s ) − s (82)= 14 (cid:0) s − s (cid:1) , which gives the solution ψ ( s ) = 14 (cid:0) s − s (cid:1) − m (cid:2) (cid:0) s − s (cid:1) , s (cid:3) m ψ (cid:2) (3 s − s ) , s (cid:3) . (83)To test the accuracy of the solution for ψ given by (83),it is compared with the numerical solution of (80). Asshown in figure 6, the accuracy of the analytical approx-imation is very good for a wide range of s .The force of hydrostatic pressure exerted by smallcylinders on a liquid bridge is negligible compared tothe force of surface tension because the solid–liquid con-tact area is small. The cross–sectional area of the liquidbridge can therefore be calculated by balancing the sur-face tension force with the weight A = − ψ + sin ψ ) . (84)3When ψ = ψ = ψ for a given s we haved A d ψ [ ψ ( s )] = − ψ (cid:18) ∂ψ ∂ψ [ ψ ( s ) , s ] (cid:19) , (85)where ∂ψ /∂ψ is obtained as a function of ψ , ψ and s by differentiating (78) with respect to ψ .Figure 7 (a) shows that ∂ψ /∂ψ [ ψ ( s )] is negativearound s = 1, which means the trapping capacity fora given separation of around 1 is maximized when ψ = ψ = ψ ( s ). The maximum trapping capacity is thereforegiven by A max = − ψ ( s ) . (86)For small cylinders, (86) gives the value of A max at farrange while (68) explains the behaviour at short range asshown in figure 4( a ).Differentiation of (86) givesd A max d s = − ψ d ψ d s , (87)while ψ and d ψ/ d s can be obtained from (83). Accord-ing to figure 7 (b), A max is a maximum when s = 1.According to the results in figure 5, which are obtainedby solving the nonlinear Laplace–Young equation, A max maximizes at d ≈ R ≪
1. Both these results aresimilar since d ≈ s for small R . VI. CONCLUSIONS
We present exact solutions to the nonlinear Laplace–Young equation to determine the equilibrium shape ofa liquid bridge trapped between a pair of infinitely longhorizontal cylinders. We also introduce several simplersolutions that approximate the exact solutions very well. Both the exact and approximate solutions show thatthe maximum amount of liquid that can be trapped ina given system and the conditions of this maximisationcan be approximated by a few simple relationships whenthe cylinder radius is small compared to the capillarylength ( ℓ c ). Regardless of the contact angle, the largestliquid bridges form when the inter–cylinder distance isapproximately 2 ℓ c . If the inter–cylinder distance is smallcompared to ℓ c , the maximum amount of liquid held by apair of cylinders is given by the equation a max ≈ ℓ c (1 + D + r cos θ ), in which a is the cross–sectional area ofthe liquid bridge, 2 D is the inter–cylinder distance, r is the cylinder radius and θ is the contact angle. Atthis maximum trapping, the meniscus slope angle of theupper interface of the liquid bridge can be approximatedby the linear relationship ψ ,a max ≈ ( r sin θ − D ) /ℓ c .The solutions we present here can be extended to deter-mine the equilibrium of fluid ganglia or stringers trappedin a solid matrix, enclosed by a different non–mixingfluid. Although such systems have been studied neglect-ing gravitational effects [20], an analysis considering theweight of the fluid can help determine the residual trap-ping capacity of a porous medium. It can also be usedto characterise deformations of the solid support inducedby the surface tension forces from fluid ganglia and anyfluid movement that result from this. This is a significantfactor in trapping by a flexible solid support, as shownby Duprat et al. [8] for the case of small liquid bridgesbetween cylinders. ACKNOWLEDGMENT
We thank Raphael Blumenfeld for many useful dis-cussions. This work is funded under the EU TRUSTconsortium. HEH was partially funded by a LeverhulmeEmeritus Professorship during this research. JAN is par-tially supported by a Royal Society University ResearchFellowship. [1] Butt, H.-J. and Kappl, M. feb 2009. Normal capillaryforces.
Adv. Colloid Interface Sci. , (1-2):48–60. doi:10.1016/j.cis.2008.10.002.[2] Schiffer, P. oct 2005. Granular physics: A bridgeto sandpile stability. Nat. Phys. , (1):21–22. doi:10.1038/nphys129.[3] Juanes, R., MacMinn, C. W., and Szulczewski, M. L. jun2009. The Footprint of the CO2 Plume during CarbonDioxide Storage in Saline Aquifers: Storage Efficiency forCapillary Trapping at the Basin Scale. Transp. PorousMedia , (1):19–30. doi:10.1007/s11242-009-9420-3.[4] Lukas, D. and Chaloupek, J. jan 2003. Wet-ting between parallel fibres; column-unduloid and col-umn disintegration transitions. Proc. Inst. Mech.Eng. Part H J. Eng. Med. , (4):273–277. doi:10.1243/095441103322060721. [5] Cheong, B. H.-P., Lye, J. K. K., Backhous, S.,Liew, O. W., and Ng, T. W. 2013. Microplates basedon liquid bridges between glass rods. J. Colloid InterfaceSci. , (0):177–184. doi:10.1016/j.jcis.2013.01.043.[6] Princen, H. oct 1970. Capillary phenomena in assem-blies of parallel cylinders: III. Liquid Columns betweenHorizontal Parallel Cylinders. J. Colloid Interface Sci. , (2):171–184. doi:10.1016/0021-9797(70)90167-0.[7] Protiere, S., Duprat, C., and Stone, H. A. nov 2013. Wet-ting on two parallel fibers: drop to column transitions. Soft Matter , (1):271. doi:10.1039/c2sm27075g.[8] Duprat, C., Proti`ere, S., Beebe, A. Y., and Stone, H. A.feb 2012. Wetting of flexible fibre arrays. Nature , (7386):510–3. doi:10.1038/nature10779.[9] Wu, X.-F., Bedarkar, A., and Vaynberg, K. A. jan 2010.Droplets wetting on filament rails: surface energy andmorphology transition. J. Colloid Interface Sci. , (2): Appl. Surf. Sci. , (23):7260–7264. doi:10.1016/j.apsusc.2010.05.061.[11] Urso, M., Lawrence, C., and Adams, M. dec 1999. Pen-dular, Funicular, and Capillary Bridges: Results for TwoDimensions. J. Colloid Interface Sci. , (1):42–56. doi:10.1006/jcis.1999.6512.[12] Chen, T.-Y., Tsamopoulos, J. a., and Good, R. J. jun1992. Capillary bridges between parallel and non-parallelsurfaces and their stability. J. Colloid Interface Sci. , (1):49–69. doi:10.1016/0021-9797(92)90237-G.[13] Pitts, E. 1973. The stability of pendent liquid drops.Part 1. Drops formed in a narrow gap. J. Fluid Mech. [14] Majumdar, S. R. and Michael, D. H. oct 1976. TheEquilibrium and Stability of Two Dimensional PendentDrops.
Proc. R. Soc. A Math. Phys. Eng. Sci. , (1664):89–115. doi:10.1098/rspa.1976.0131. [15] Orr, F. M., Scriven, L. E., and Rivas, A. P. mar1975. Pendular rings between solids: meniscus prop-erties and capillary force. J. Fluid Mech. , (04):723.doi:10.1017/S0022112075000572.[16] Boucher, E. A. and Evans, M. J. B. nov 1975. Pen-dent Drop Profiles and Related Capillary Phenomena. Proc. R. Soc. A Math. Phys. Eng. Sci. , (1646):349–374. doi:10.1098/rspa.1975.0180.[17] Byrd, P. F. and Friedman, M. D. 1971. Handbook ofelliptic integrals for engineers and scientists . Springer,Berlin.[18] Vella, D. and Mahadevan, L. 2005. The “Cheerios effect”.
Am. J. Phys. , (9):817. doi:10.1119/1.1898523.[19] Keller, J. B. nov 1998. Surface tension force on apartly submerged body. Phys. Fluids , (11):3009. doi:10.1063/1.869820.[20] Niven, R. K. jun 2006. Force stability of pore-scalefluid bridges and ganglia in axisymmetric and non-axisymmetric configurations. J. Pet. Sci. Eng. ,52