Catenoid Soap Films

The soap film that forms between two parallel circular rings has the shape of a catenoid, which has the equation r = cosh(z) in cylindrical coordinates. For a separation/diameter ratio under 0.66274, there are a thick-necked stable catenoid and a thin-necked unstable catenoid. At the critical ratio, there is one unstable catenoid. Above the critical ratio, there are no catenoids.

The images here were made with my Surface Evolver program. Click on the small images to get larger versions. The corresponding datafiles (extension .fe) may be downloaded by clicking on the links in parentheses.

stable catenoid

Stable catenoid (cat.fe)

If the rings are pulled further apart, the neck will narrow until the critical separation is reached, and then the catenoid will pop into two disks. Note that the surface becomes unstable before the neck reaches zero radius.

unstable catenoid

Unstable catenoid (catbody.fe)

The unstable catenoid on the same rings as above. For separation/diameter ratio above 0.4717945, the unstable catenoid can be stabilized by adding a volume constraint. For a catenoid, the desired volume can easily be calculated, but one could also adjust the volume until the pressure is zero. By adjusting the volume so the pressure is positive or negative, one gets the surfaces known as unduloids and nodoids, respectively.

catenoid with disk

Catenoid with disk (catdisk.fe)

Here the film consists of two sections of catenoid and a flat disk meeting at 120 degrees. Again, for a separation/diameter ratio below a critical value, there are a stable thick-necked film and an unstable thin-necked film. As the rings are pulled apart, the film becomes unstable before the central disk radius reaches zero.

double catenoid

Double catenoid (crosscat.fe)

The wire boundary consists of two pairs of catenoid rings with gaps joined at right angles. The diagonal lines joining the angle vertices are lines of 180 degree rotational symmetry. There is a mathematical theorem that the tangent plane of the film at a corner must be the tangent plane of the corner. Usually the film is on the inside of the corner. Only in unusual cases (like the symmetry here) is the film on the exterior.

double catenoid with 1 disk

Double catenoid with disk (cat1disk.fe)

Having a disk across one of the catenoids spoils the symmetry around the diagonals, the film becomes tangent to the insides of the angles, although that level of detail is far too small to be seen in this image.

corner close-up

Corner close-up (cat1diskpart.fe)

A close-up of a corner of the double catenoid with one disk. This image is magnified several hundred times; the lengths of the edges around the vertex is about 0.0006.

double catenoid with 2 disks

Double catenoid with two disks (cat2disk.fe)

With two disks, the symmetry is restored, and the film is tangent on the exteriors of the angles. Note that the triple lines meet at a tetrahedral point singularity where the diagonals cross.

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