Capillary bridges
A capillary bridge is a minimized surface of liquid or membrane created between two rigid bodies of arbitrary shape.
The presence of capillary bridge, depending on their shapes, can lead to attraction or repulsion between the solid bodies.
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.
[8]In the last century a lot of efforts were put of study of surface forces that drive capillary effects of bridging.
[11] 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.
Later, Hove [12] 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.
[8] Perturbation methods became very successful despite that nonlinear nature of capillary interaction can limit their application.
[13][14] To that moment most methods for stability determination required calculation of equilibrium as a basis for perturbations.
He examined the case of axisymmetric capillary bridges with constant volumes and the stability changes correspond to turning points.
[20][21] Recent studies indicated that ancient Egyptians used the properties of sand to create capillary bridges by using water on it.
In atomic force microscopy, when one works in higher humidity environment, their studies might be affected by the appearance of nano sized capillary bridges.
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.
[25] Many medical problems involving respiratory diseases, and the health of the body joints depend on tiny capillary bridges.
[26] Liquid bridges are now commonly used in growth of cell cultures because of the need to mimic work of living tissues in scientific research.
In a completely analogous way: The second integral for unduloid is obtained: where the relation between parameters k and φ are defined the same way as above.
When typical capillary bridge comes to catenoidal state of C = 0, despite that it surface properties are the same as classical catenoid, it is more appropriate to be presented as scaled by cube root of its volume rather than the radius, R. The solution of the second integral is different in cases of oblate capillary bridges (nodoid and unduloid): where: F and E are again elliptic integrals of first and second kind,
It is important to note that all described curves are found by rolling a conic section without slip along z axis.
The unduloid is described by the focus of rolling ellipse, which can degenerate into a line, a sphere or a parabola, leading to the corresponding limiting cases.
In contrast to cases with increasing height of capillary bridges, that poses variety of profile shapes, the flattening (thinning) toward zero thickness has much more universal character.
The positive sign '+' represents generatrix profile of concave bridge and negative '-', oblate.
For the convex capillary bridges, the circular generatrix is retained until the boundary of definition domain is reached while stretching.
Near the beginning of self-initiated breakage kinetics, the bridge profile evolves consequently to an ellipse, parabola and possibly to hyperbola.
The estimation of definition domain requires manipulation of integrated equations for capillary bridge height and its volume.
6 are shown number of stable static states of liquid capillary bridge, represented by two characteristic parameters: (i) dimensionless height that is obtained by scaling of capillary bridge height by cubic root of its volume Eq.
The solutions somehow differs from widely accepted Plateau's approach [by elliptical functions, Eq.
These solutions became further a basis for prediction of capillary bridges quasi-equilibrium stretching and breakage for contact angles below 45°
The practical implementation allows to be identified not only the end of definition domain but also the exact behavior during the capillary bridge stretching,[32] because in coordinates
Equilibrium shapes and stability limits for capillary liquid bridges are subject to many theoretical and experimental studies.
(where: g is Earth gravitational acceleration, γ is the surface tension and R is radius of the contact) the stability diagram can be represented by a single closed piecewise curve on the slenderness/dimensionless volume plane.
