A linear chain of spheres confined by a transverse harmonic potential experiences localized buckling under compression. 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.

## I. INTRODUCTION

Linear chains of particles have long been popular in providing simple examples for analysis using classical mechanics. Consequentially, numerous classroom demonstrations entail the study of such chains; examples include the problem of determining the force exerted by a falling chain^{1–3} (a long-standing problem, which continues to provoke debate^{4}) the vibrations (and normal modes^{5}) of a chain of particles,^{6} as a means of demonstrating the properties of the catenary^{7} (and related curves), the physics of collisions and shock waves,^{8} as well as numerous other interesting problems suitable for the undergraduate physics curriculum.^{9}

Much of this work has been largely confined to linear elastic theory and dynamics.^{5,10} The pedagogical value of such models lies in their essentially one-dimensional nature, which is helpful for observation, analysis, and theory. In many respects, they share the properties of two- and three-dimensional systems and therefore provide an easy introduction to these.

Here, we extend the suite of classroom demonstrations to linear chains of mutually repelling particles. The particles are compressed along the length of the chain (corresponding to being trapped by an axial potential in the related physical systems mentioned below), while also being confined in the radial (or transverse) direction by a cylindrically symmetric potential. We will focus on the case of static equilibrium, for compressions large enough to induce complex nonlinear properties.

Our demonstration experiments and the accompanying theory and simulations are connected with ongoing research in a number of areas, where they may serve as illustration of the underlying physics, but can also offer inspiration for further measurements. Relevant research includes that on laser-cooled ions in Penning traps^{11} and dusty plasmas.^{12} Related structures have also been observed in experiments with colloids,^{13} microfluidic crystals,^{14} and magnetic particles.^{15} A more accessible system was introduced in Ref. 16, using buoyant plastic spheres in a water-filled tube, rotated by a lathe; structures for a wide range of compression were reported and were further analyzed theoretically in Ref. 17.

The type of arrangement formed by the particles depends on the competition between radial and axial confinement. When radial confinement dominates, the particles form a straight linear chain; however, on reducing the radial force, the preferred (minimum energy) state of the system transitions from a linear chain to a modulated zigzag structure.^{18} Such systems have many interesting properties, including buckling, localization (sometimes described in terms of “kinks” or “solitons”), a variety of alternative (meta)stable structures, topological changes, bifurcation diagrams, and a Peierls-Nabarro potential for transitions between them.^{18,19} The buckling of a linear chain has also been found to be relevant to mechanical properties of engineered materials^{20} and to active colloidal chains in biology.^{21} As mentioned, buckled structures commonly occur in formations of cooled ions in traps; these in turn find a range of advanced applications in spectroscopy, quantum computing, and reaction kinematics (see Ref. 22 for a recent review).

In the present paper, we describe for the first time a very simple experimental set-up that may be used to demonstrate and measure many of the generic nonlinear properties of such a system. It is easily realized with the simplest equipment available in the class-room (test tube with stoppers, aquarium pump, and dish-washing solution) (Fig. 1). The experimental arrangement consists of gas bubbles trapped in a horizontal liquid-filled tube. The bubbles are confined axially by opposing walls (stoppers) at either end of the tube. Compressing the linear chain of bubbles leads to buckling. A further increase in compression generates a sequence of different modulated zigzag structures. These are also related to previous studies of the packings of hard spheres in cylinders.^{23}

This new type of experiment will enable many fine details to be explored, which have not so far been analysed for any of the more sophisticated systems mentioned above, especially when combined with the numerical simulations of the kind presented here.

## II. EXPERIMENTAL METHOD AND RESULTS

Bubbles of equal size are produced by blowing air through a nozzle into a solution of commercial detergent (*“Fairy Liquid”*) using an air pump with a flow-control valve. The bubbles are introduced into a perspex tube (inner diameter 6.7 mm, outer diameter 8.0 mm), which is placed horizontally at the bottom of the container filled with the surfactant solution (Fig. 1), and stoppers are inserted. For a certain separation *L*_{0} of these stoppers, *N* bubbles are only just in contact with one another and the two stoppers. The uncompressed axial extension *D* of the bubbles is then $ D = L 0 / N$. In the experiments reported below, we have used *N *=* *19 bubbles with $ D = 2.3 \u2009 mm$.

Decreasing the length of confinement *L* by manually pushing the stoppers, we may observe and record (as photos or videos) the structures that are formed; for an example, see Fig. 2. For small values of compression Δ, defined as $ \Delta = N \u2212 L / D$, the chain of bubbles remains straight, with all bubbles suffering equal deformation. However, at some critical value of compression Δ, buckling occurs (see Fig. 2). The critical value of Δ is *zero* for hard spheres and finite for soft (elastic) spheres, as in the case of bubbles.

In this regime, the buckled structures are found to be *planar* for rotating cylinders.^{16,17} They are *approximately* so for the technique here introduced. Further examples are shown in Fig. 3 and numbered for later reference.

To characterize these structures under compression in a simple way, we have determined the transverse width *W* of the minimal rectangular box, which contains all the bubbles of a particular chain; for an example, see Fig. 4 (top). This is a convenient parameter for measurements by hand from photographs. However, the data reported below were obtained using the image processing software *imagej.*^{24}

Figure 4 shows the rescaled width *W*/*D* for ten different values of compression Δ, for all the structures shown in Fig. 3. The width increases strongly once the compression exceeds its critical value.

Before describing the data in detail, we will comment on a particular feature of the experimental set-up. In the case of an uncompressed chain (Δ = 0) of *hard* spheres the width *W* is simply *D* (which in this case coincides with the sphere diameter). However, two effects play a role when interpreting *W* in our experiments with bubbles. First, optical distortion arising from using liquid-filled tubes leads to a small increase in the ratio *W*/*D*, also in the case of hard spheres for these experiments. (We found $ W / D \u2243 1.04$ for a chain of hard plastic spheres of diameter 3 mm, placed in the water-filled tube within the container used for the bubble experiments.) Second, our gas bubbles are not spherical even under zero compression Δ, due to the effect of buoyancy, pressing them against the tube surface.

The combination of these two effects can account for the value of $ W / D \u2243 1.14$ found for small compression, $ \Delta \u2243 0.13$, see Fig. 4. Upon further compression the width increases slightly to about $ W / D \u2243 1.23$ for $ \Delta \u2243 1.56$. At $ \Delta = 2.13$, the chain has clearly buckled, causing a large increase in the width to about $ W / D \u2243 1.41$. A further increase in compression results in a roughly linear increase in *W*, as the profile of lateral displacement becomes increasingly *localized* (see the photograph in Fig. 4).

At values of compression exceeding $ \Delta \u2243 2.7$, the localized zig-zag structure gives way to a chain containing a “doublet,” a transverse pair of bubbles (e.g., photograph 10 in Fig. 3).

A linear extrapolation of the width variation of the buckled structures would identify the onset of buckling at around $ \Delta c \u2243 1.75$. However, buckling is generally associated with a square-root scaling in compression, visible in the simulations described in Sec. III. Taking this into account, we estimate the critical value of compression to lie somewhere in the range $ 1.8 < \Delta c < 2.0$ (Fig. 4).

## III. THEORY AND SIMULATIONS

We have made a preliminary comparison of the above data with the results from an elementary numerical simulation. The basis for this is explained below. We should emphasize that the simple model for bubble-bubble interactions, which we will employ, is *not* intended to be accurate, and so comparison will not be fully quantitative.

We will be concerned with structures of length *L*, made up of *N* idealized spherical particles of diameter *D*; see Fig. 5. We will restrict our analysis to structures formed under low compression, $ \Delta = N \u2212 L / D$. We have already shown one simulation result in Fig. 2.

To obtain such numerical results, we have used the Durian Model.^{25,26} This represents bubbles as spheres whose overlap is associated with a repulsive force between the bubble centres. (A similar approach was suggested earlier.^{27}) For a pair of bubbles of equal size, the interaction energy *E _{i}* is $ E i = k 1 / 2 ( | R \u2192 i \u2212 R \u2192 i + 1 | \u2212 D ) 2$, where $ R i \u2192$ are sphere centres and

*k*

_{1}is the spring constant for bubble-bubble interaction. The crude model has proved to be useful in foam physics

^{28,29}in providing qualitative and semi-quantitative insights.

*contacts*, including the contribution of the two bubbles in contact with the confining walls (

*i*=

*1) and (*

*i*=

*N*), in the approximate form,

Only the coordinates *X _{i}* and

*Y*enter, an approximation that makes the system planar and is valid for small values of compression.

_{i}*buoyancy*of a particle held in place by the cylindrical surface is

*k*

_{2}is given by

*g*is the acceleration due to gravity, and

*r*is the radius of the cylinder, see Fig. 5(b).

The system has been reduced to two dimensions. The situation is rather different in the other physical systems to which we referred in Sec. I, where planar structures are found to arise for only low compressions, but are not imposed by geometry at the outset (as we have done here). That is, planar structures are found in practice and become twisted at higher compression.

Starting from a small value of compression Δ, and a straight linear chain, we progressively increase Δ. For each step, the previous equilibrium structure is used as the starting structure for minimization (in accord with the experimental procedure). Energy *E*, Eq. (4), is minimized numerically with respect to the coordinates *x _{i}* and

*y*.

_{i}Below a critical value of compression (which depends on the value for the ratio *k* of the force constant, Eq. (5)), the minimum energy arrangement corresponds to that of a straight linear chain, but this buckles to form a zig-zag chain at a critical value of compression, as in the experiment. (A small perturbation is necessary to promote the instability.)

We performed computations for increments of $ \delta \Delta = 0.01$ up to compression $ \Delta = 3.0$ for various values of *k*. The results for *k *=* *2.5 and *N *=* *19 are collated in Fig. 6, in terms of the dimensionless maximum transverse displacement, $ y max = max ( | y i | )$. We have found this to be a more straightforward quantity for comparison with experiment, rather than width *W*, since *W* is affected by both optical distortion and bubble deformation, as discussed above.

For values of compression slightly exceeding $ \Delta c = 1.83$, we find *y _{max}* to vary as $ ( \Delta \u2212 \Delta c ) 1 / 2$, as is generally the case in buckling transitions. In this range, the envelope of the displacement profile is broad (roughly of cosine form).

For higher values of Δ, there is increased localization of buckling, as in the example shown in Fig. 2. Finally, there is a sudden jump in the maximum transverse displacement with increasing compression; at this point, the doublet structure (with a transverse pair of spheres) becomes favourable. The maximum transverse displacement associated with this increases very slightly with compression, before encountering a further transition. Full details of this rich scenario, as well as a comprehensive overview of theory and simulation, are reserved for a future paper.

## IV. COMPARISON WITH EXPERIMENT

Figure 6 also presents experimental data for comparison. Here, the maximum lateral bubble displacement $ y max = Y max / D$ was obtained by first determining the lateral midpoint of each bubble and then measuring its distance to the tube axis, using the photographs in Fig. 3. (The representation of the buckling of a bubble chain using its width *W*, as in Fig. 4, might be more suited in the context of a class-room since it requires fewer measurements.)

There is broad agreement between experiment and theory for *k *=* *2.5. Increasing *k* moves the critical value of compression Δ_{c} towards zero, the value found for the case of hard spheres ( $ k 1 \u2192 \u221e$).^{17} The theory also correctly predicts the occurrence of a doublet structure (number 10 in Fig. 3).

We may also seek to estimate *k* from the relevant experimental parameters. Setting the dimensional spring constant $ k 1 = \gamma / 2$,^{26} where $ \gamma \u2243 0.03 \u2009 N / m$ is the surface tension of our surfactant solution, we can evaluate $ k = k 1 / k 2$ using Eq. (3). Substituting $ \rho = 1000 \u2009 kg / m 3 , \u2009 D = 2.3 \u2009 mm$, and $ r = 3.35 \u2009 mm$, we obtain $ k \u2243 0.5$, i.e., a value of the same order of magnitude as the one found from comparison with numerical data (Fig. 6).

## V. FURTHER DEMONSTRATION EXPERIMENTS

The effects of buckling in a chain of particles can also be illustrated using even simpler experimental set-ups.

Figure 7 shows an example of a buckled chain of 30 steel spheres (ball bearings) in a tube, closed with two stoppers. In order to reduce friction, we immersed the spheres in vegetable oil. Related structures can also be investigated using golf or tennis balls in a perspex tube, or even in a section of roof gutter, and doubtless other ingredients await discovery and exploitation.

## VI. SUMMARY AND OUTLOOK

We have described a simple experimental set-up, suitable for the class- or lecture room, for the exploration of the nonlinear properties of a chain of spheres under compression. The experiment demonstrates these properties, which have recently led to a number of publications on nano-scale systems.^{11,12,30} The simulation method described is straightforward and reproduces key features of the experiment. It might also lend itself to exploration in the context of a computational physics laboratory.

The use of bubbles offers an additional dimension to the experiment, which could be explored: the effective softness of the bubbles is a function of their size. In the present preliminary work, we have used only a single bubble size and treat the scaled softness parameter (constant *k* in Eq. (4)) as adjustable. Note that *k* can also be varied by varying the cylinder radius.

In previous work, we analysed the desk-top toy called “Newton's Cradle,” i.e., a linear chain of contacting metal balls, suspended from a railing by attached strings, and thus subject to a harmonic confining potential, albeit in the direction of the chain.^{31} As is the case in the present work, this system proved to be remarkably rich when analyzed in detail. In particular, the break-up of the line of balls following the initial impact is generally overlooked in physics textbook descriptions. It is hoped that the bubble chain experiment presented here, which shares with the cradle an economy of effort and expense, provides similar stimulation for students to look for non-trivial phenomena in chains of confined spheres.

## ACKNOWLEDGMENTS

This work was supported by EPSRC Grant Nos. EP/K032208/1 and EP/R014604/1 and Science Foundation Ireland (SFI) Grant No. 13/IA/1926. A.I. acknowledges funding from the Trinity College Dublin Provost's Ph.D. Project Awards.

## REFERENCES

*200 Puzzling Physics Problems: With Hints and Solutions*

*Foams: Structure and Dynamics*