Capillary bridges

Usually, we understand the term capillary bridge as a minimized surface of liquid or membrane, created between two rigid bodies with an arbitrary shape. Capillary bridges also may form between two liquids. Plateau defined a sequence of capillary shapes known as (1) nodoid with 'neck', (2) catenoid, (3) unduloid with 'neck', (4) cylinder, (5) unduloid with 'haunch' (6) sphere and (7) nodoid with 'haunch'. The presence of capillary bridge, depending on their shapes, can lead to attraction or repulsion between the solid bodies. The simplest cases of them are the axisymmetric ones. We distinguished three important classes of bridging, depending on connected bodies surface shapes:Schematic from the tomb of Djehutihotep depicting the transport of a colossal statueAFMSolderingWhite lipped tree frog Δ P = γ ( 1 R 1 + 1 R 2 ) = γ d ( r sin ⁡ ϕ ) r d r = c o n s t a n t , {displaystyle Delta P=gamma left({frac {1}{R_{1}}}+{frac {1}{R_{2}}} ight)=gamma {frac {dleft(rsin phi ight)}{rdr}}={ m {constant}},}     (1) sin ⁡ ϕ = ( r 2 − r 1 r 2 ) r ( r 1 − r 2 ) {displaystyle sin phi ={frac {left(r^{2}-r_{1}r_{2} ight)}{rleft(r_{1}-r_{2} ight)}}}     (2) d z d r = − tan ⁡ ϕ = − sin ⁡ ϕ ( 1 − sin 2 ⁡ ϕ ) {displaystyle {frac {dz}{dr}}=- an phi =-{frac {sin phi }{sqrt {left(1-sin ^{2}phi ight)}}}}     (3) d z d r = r 1 r 2 − r 2 ( r 2 − r 1 2 ) ( r 2 2 − r 2 ) {displaystyle {frac {dz}{dr}}={frac {r_{1}r_{2}-r^{2}}{sqrt {left(r^{2}-r_{1}^{2} ight)left(r_{2}^{2}-r^{2} ight)}}}}     ( 4) z = ± [ r 1 F ( r , ϕ ) − r 2 E ( r , ϕ ) ] + ( 1 − r 0 2 r 2 ) ( r 1 2 r 2 − 1 ) r {displaystyle z=pm left+{frac {sqrt {left(1-{frac {r_{0}^{2}}{r^{2}}} ight)left({frac {r_{1}^{2}}{r^{2}}}-1 ight)}}{r}}}     (5) sin 2 ⁡ ϕ = r 2 − r 2 2 k 2 r 2 {displaystyle sin ^{2}phi ={frac {r^{2}-r_{2}^{2}}{k^{2}r^{2}}}} . d z d r = r 1 r 2 + r 2 ( r 2 − r 1 2 ) ( r 2 2 − r 2 ) {displaystyle {frac {dz}{dr}}={frac {r_{1}r_{2}+r^{2}}{sqrt {left(r^{2}-r_{1}^{2} ight)left(r_{2}^{2}-r^{2} ight)}}}}     (6) z = ± [ r 1 F ( r , ϕ ) + r 2 E ( r , ϕ ) ] + ( 1 − r 0 2 r 2 ) ( r 1 2 r 2 − 1 ) r {displaystyle z=pm left+{frac {sqrt {left(1-{frac {r_{0}^{2}}{r^{2}}} ight)left({frac {r_{1}^{2}}{r^{2}}}-1 ight)}}{r}}}     (7) d ( r sin ⁡ r ) r d r = 0 {displaystyle {frac {dleft(rsin r ight)}{rdr}}=0}     (8) r r 1 = cosh ⁡ ( z r 1 ) {displaystyle {frac {r}{r_{1}}}=cosh left({frac {z}{r_{1}}} ight)}     (9) z = ± [ r 2 F ( r , ϕ ) − ( C − 1 ) r 2 E ( r , ϕ ) ] {displaystyle z=pm left}     (10) P γ = γ d ( r sin ⁡ ϕ ) r d r {displaystyle P_{gamma }=gamma {frac {d(rsin {phi })}{rdr}}}     (11) C = X sin ⁡ θ − 1 X 2 − 1 {displaystyle C={frac {Xsin { heta }-1}{X^{2}-1}}}     (12) d y d x = ± C ( x 2 − 1 ) + 1 x 2 − [ C ( x 2 − 1 ) + 1 ] 2 {displaystyle {frac {dy}{dx}}=pm {frac {Cleft(x^{2}-1 ight)+1}{sqrt {x^{2}-left^{2}}}}}     (13) C ( X = 1 − Δ ) ≈ − 1 − sin ⁡ θ 2 Δ + 1 + sin ⁡ θ 4 {displaystyle Cleft(X=1-Delta ight)approx -{frac {1-sin heta }{2Delta }}+{frac {1+sin heta }{4}}}     (14) d y d x = ± 1 + 2 C ( x − 1 ) 1 − [ 2 C ( x − 1 ) + 1 ] 2 {displaystyle {frac {dy}{dx}}=pm {frac {1+2Cleft(x-1 ight)}{sqrt {1-left^{2}}}}}     (15) y 2 + ( x + 1 ± 1 2 C ) 2 = ( 1 2 C ) 2 {displaystyle y^{2}+left(x+1pm {frac {1}{2C}} ight)^{2}=left({frac {1}{2C}} ight)^{2}}     (16) H ∗ ≡ ( H V 3 ) = 1 X ( R V 3 ) { π 4 C − α ( X , C ) − ∫ 1 X ζ − C ( X 2 − 1 ) + 1 ζ + C ( X 2 − 1 ) + 1 d ζ } {displaystyle H^{*}equiv left({frac {H}{sqrt{V}}} ight)={frac {1}{X}}left({frac {R}{sqrt{V}}} ight)left{{frac {pi }{4C}}-alpha left(X,C ight)-int limits _{1}^{X}{sqrt {frac {zeta -Cleft(X^{2}-1 ight)+1}{zeta +Cleft(X^{2}-1 ight)+1}}}dzeta ight}}     (17) R ∗ ≡ ( R V 3 ) = X 2 π 3 { β ( C ) π 4 C − ( 1 − 2 C ) ( X 2 − 1 ) 2 C 2 − β ( C ) α ( X , C ) − ∫ 1 X ζ 2 ζ − C ( X 2 − 1 ) + 1 ζ + C ( X 2 − 1 ) + 1 d ζ } − 1 3 {displaystyle R^{*}equiv left({frac {R}{sqrt{V}}} ight)={frac {X}{sqrt{2pi }}}left{eta left(C ight){frac {pi }{4C}}-{frac {sqrt {left(1-2C ight)left(X^{2}-1 ight)}}{2C^{2}}}-eta left(C ight)alpha left(X,C ight)-int limits _{1}^{X}zeta ^{2}{sqrt {frac {zeta -Cleft(X^{2}-1 ight)+1}{zeta +Cleft(X^{2}-1 ight)+1}}}dzeta ight}^{-{frac {1}{3}}}}     (18) Usually, we understand the term capillary bridge as a minimized surface of liquid or membrane, created between two rigid bodies with an arbitrary shape. Capillary bridges also may form between two liquids. Plateau defined a sequence of capillary shapes known as (1) nodoid with 'neck', (2) catenoid, (3) unduloid with 'neck', (4) cylinder, (5) unduloid with 'haunch' (6) sphere and (7) nodoid with 'haunch'. The presence of capillary bridge, depending on their shapes, can lead to attraction or repulsion between the solid bodies. The simplest cases of them are the axisymmetric ones. We distinguished three important classes of bridging, depending on connected bodies surface shapes: Capillary bridges and their properties may also be influenced by Earth gravity and by properties of the bridged surfaces. The bridging substance may be a liquid or a gas. The enclosing boundary is called the interface (capillary surface). The interface is characterized by a particular surface tension. Capillary bridges have been studied for over 200 years. The question was raised for the first time by Josef Louis Lagrange in 1760, and interest was further spread by the French astronomer and mathematician C. Delaunay. Delaunay found an entirely new class of axially symmetrical surfaces of constant mean curvature. The formulation and the proof of his theorem had a long story. It began with Euler's proposition of new figure, called catenoid. (Much later, Kenmotsu solved the complex nonlinear equations, describing this class of surfaces. However, his solution is of little practical importance because it has no geometrical interpretation.) J. Plateau showed the existence of such shapes with given boundaries. The problem was named after him Plateau's problem. Many scientists contributed to the solution of the problem. One of them is Thomas Young. Pierre Simon Laplace contributed the notion of capillary tension. Laplace even formulated the widely known nowadays condition for mechanical equilibrium between two fluids, divided by a capillary surface Pγ=ΔP i.e. capillary pressure between two phases is balanced by their adjacent pressure difference.A general survey on capillary bridge behavior in gravity field is completed by Myshkis and Babskii.In the last century a lot of efforts were put of study of surface forces that drive capillary effects of bridging. There was established that these forces result from intermolecular forces and become significant in thin fluid gaps (<10 nm) between two surfaces. The instability of capillary bridges was discussed in first time by Rayleigh. He demonstrated that a liquid jet or capillary cylindrical surface became unstable when the ratio between its length, H to the radius R, becomes bigger than 2π. In these conditions of small sinusoidal perturbations with wavelength bigger than its perimeter, the cylinder surface area becomes larger than the one of unperturbed cylinder with the same volume and thus it becomes unstable. Later, Hove formulated the variational requirements for the stability of axisymmetric capillary surfaces (unbounded) in absence of gravity and with disturbances constrained to constant volume. He first solved Young-Laplace equation for equilibrium shapes and showed that the Legendre condition for the second variation is always satisfied. Therefore, the stability is determined by the absence of negative eigenvalue of the linearized Young-Laplace equation. This approach of determining stability from second variation is used now widely. Perturbation methods became very successful despite that nonlinear nature of capillary interaction can limit their application. Other methods now include direct simulation. To that moment most methods for stability determination required calculation of equilibrium as a basis for perturbations. There appeared a new idea that stability may be deduced from equilibrium states. The proposition was further proven by Pitts for axisymmetric constant volume. In the following years Vogel extended the theory. He examined the case of axisymmetric capillary bridges with constant volumes and the stability changes correspond to turning points. The recent development of bifurcation theory proved that exchange of stability between turning points and branch points is a general phenomenon. Recent studies indicated that ancient Egyptians used the properties of sand to create capillary bridges by using water on it. In this way, they reduced surface friction and were capable to move statues and heavy pyramid stones. Some contemporary arts, like sand art, are also close related to capability of water to bridge particles. In atomic force microscopy, when one works in higher humidity environment, his studies might be affected by the appearance of nano sized capillary bridges. These bridges appear when the working tip approaches the studied sample. Capillary bridges also play important role in soldering process. Capillary bridges also widely spread in living nature. Bugs, flies, grasshoppers and tree frogs are capable to adhere to vertical rough surfaces because of their ability to inject wetting liquid into the pad-substrate contact area. This way is created long range attractive interaction due to the formation of capillary bridges. Many medical problems involving respiratory diseases, and the health of the body joints depend on tiny capillary bridges. Liquid bridges are now commonly used in growth of cell cultures because of the need to mimic work of living tissues in scientific research. General solution for the profile of capillary is known from consideration of unduloid or nodoid curvature. Let's assume the following cylindrical coordinate system: z shows axis of revolution; r represents radial coordinate and φ is the angle between the normal and the positive z axis. The nodoid has vertical tangents at r = r1 and r = r2 and horizontal tangent at r = r3. When φ is the angle between the normal to the interface and positive z axis then φ is equal to 90°, 0°, -90° for nodoid. The Young-Laplace equation may be written in a form convenient for integration for axial symmetry : where R1, R2 are the radii of curvature and γ is interfacial surface tension.The integration of the equation is called the first integral and it yields: