A simple experimental system to illustrate the nonlinear properties of a linear chain under compression

Abstract

A linear chain of spheres confined by a transverse harmonic potential experiences localized buckling under compression. Here we present simple experiments using gas bubbles in a liquid-filled tube to demonstrate this phenomenon. Our findings are supported semi-quantitatively by numerical simulations. In particular we demonstrate the existence of a critical value of compression for the onset of buckling.

Buckling of a linear chain of hard spheres in a harmonic confining potential: Numerical and analytical results for low and high compression

“…Hard spheres (e.g. ball bearings) placed in a horizontal cylindrical tube, exhibit all of the localisation properties described here (see [22] for full details).…”

Peierls-Nabarro potential for a confined chain of hard spheres under compression

“…-Since the PN potential is readily calculated for this system it provides a convenient prototype for the wider discussion of the concept and its implications. It also provides an elementary platform for experimental investigations such as that suggested by [20,21] which we intend to pursue.…”

“…Many of them can be related to the simpler system under consideration here which is exceptionally tractable and transparent. It concerns the buckling of a line of hard spheres under compression [19][20][21], in the presence of a transverse harmonic potential which opposes their displacement. This system has interesting properties; the linear chain buckles to form a variety of zigzag structures.…”

Description of the buckling of a chain of hard spheres in terms of Jacobi functions

“…Systems in which analogous examples of one-dimensional pattern formation have been observed include cold ions in traps [5][6][7][8][9][10][11][12][13] (with possible relevance to quantum computing [14,15]), dusty plasmas [16], droplets in microfluidic crystals [17], paramagnetic colloidal particles in an external field [18], and linear chains of magnetic spheres [19]. In contrast, the system that we study here, consisting of hard spheres in a harmonic confining potential, is much more tractable in both theory and experiment [3,20], yet displays the rich buckling phenomena observed in the more complex systems listed above. In the experimental system which stimulated the work presented here, a small number of ball-bearings (between five and eight) was placed inside a horizontal cylindrical tube, sealed with stoppers at both ends [4].…”

“…The result is a modulated zig-zag structure of lateral sphere displacements, with a displacement profile which becomes increasingly localized as compression is increased. This interesting nonlinear system is readily accessible to experiment (including some very simple procedures [1][2][3][4]) and may be simulated by straightforward numerical methods [2,4]. Its properties include the appearance of further equilibrium states of sphere displacements at high values of compression and the eventual occurrence of additional contacts between spheres, at which point new structures begin to form [4].…”