Liquid Bridges

Liquid Bridges

 

Notation and 2nd order differential equation

A liquid bridge formed by capillary condensation, between two spheres is a radial symmetric "object of revolution". Its shape may be described by its radial profile r_(Z), specifying the radius r_(Z) in terms of the axial co-ordinate z. According to the equation of Young and Laplace, the liquid bridge is a "surface of constant mean curvature". According to Kelvin's equation, the mean curvature is negative. Such a radially symmetric, surface of constant, negative mean curvature is referred to as a nodoid.

By expressing the mean curvature, H, at a point of a surface in terms of the local behaviour of r_(Z), it is possible to obtain a second order differential equation for r_(Z).

The two principal curvatures are k_z in the axial direction and k_r in the radial direction. The mean curvature is then

41

It is convenient to introduce the slope angle, Φ. This is related to the gradient or the first derivative, dr/dz, by dr/dz=tan Φ. Then k_z and k_r may be obtained as

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These results may be combined to obtain the expression for H as a second order differential operator,

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Given that sec²Φ = 1 + tan² Φ where tanΦ=dr/dz and that sec Φ = 1/cos Φ, then cosΦ may be directly expressed in terms of dr/dz as cos Φ = 1 / √(1 + (dr/dz)²). Thus the expression for H expresses H as a function of r, dr/dz and d²r/dz. Equating this to the required constant, h, of the mean curvature provides the second order differential equation of the form

i.e.

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First order differential equation.

A first integral of this equation may be obtained to provide the first order equation as

46

where r₀ = r(0) is the radius at z=0. (This can be seen to be a first order differential equation for r, given that cos Φ= .

This equation can be written explicitly as an expression for r as function of Φ. before doing so, it is convenient to introduce the notation of elliptic integrals. Firstly we introduce the "Jacobian elliptic functions", cnu = cos Φ and snu = sin Φ. Defining a = -1/2H and the modulus k by

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then equation (46) may be solved as a quadratic in r to as the radius may be expressed in parametric form, as a function of Φ, as

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where dnu is the delta amplitude, defined by dnu = √1 - k²sn²u. The corresponding parametric expression for z as a function of Φ, may be obtained by integration. Using the form k_Z = -d snu/dz , we have H = ½ ((d snu/dz) + (cnu /r)) and so the second order equation becomes

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When z=0, Φ=0 and so snu=0. Hence, z( Φ) can be expressed as the integral z( Φ)

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and so

The integral can be expressed in terms of elliptic integrals to obtain

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where k' is the complementary amplitude, defined by k'²=1-k ², where

is the "elliptic integral of the first kind" and

is the "elliptic integral of the second kind".