Instabilities in two-dimensional flower and chain clusters of bubbles

We assess the stability of simple two-dimensional clusters of bubbles relative to small displacements of the vertices, at fixed bubble areas. The clusters analysed are: 1) flower clusters consisting of a central bubble of area l surrounded by N shells each containing n bubbles of unit area, 2) periodic chain clusters consisting of N 'parallel' rows of n bubbles of unit area and width w. The energy and bubble pressures of the symmetrical, unbuckled clusters are found analytically as a function of l and w for given N and n. Both types of clusters studied show a single energy minimum at a critical lm or wm. At the energy minimum for flower clusters, the pressure in the central bubble vanishes. The clusters show a symmetry-breaking buckling instability under compression at a critical lb or wb. The corresponding critical energy Eb was determined with the Surface Evolver software. While for N=1 the conditions lb = lm, wb = wm and Eb = Em hold, for N>1 buckling requires further compression beyond the minimum, for which the energy increases with increasing compression (decreasing l or w), and the excess pressure in the central bubble of the flower clusters becomes negative.