## Abstract

A cohesive object will eventually break into fragment when experiencing a strong deformation, during an impact for instance. Using necklaces of cohesive magnetized spheres suddenly expanded, we have shown that the fragmentation of this one-dimensional material results from an inverse aggregation cascade (Vledouts *et al.* 2015 *Proc. R. Soc. A* 471, 20150678. (doi:10.1098/rspa.2015.0678)). Here, we explore a variant of this process by changing the force law between the attracting spheres, using hydrogel beads linked by capillary bridges. We also investigate the role of (weak) defects in the cohesion strength and the consequences of a distribution of forces between the beads. It is found that fragment do form by a cascade of aggregations, which is interrupted earlier when the force disorder is stronger.

## 1. Introduction

The solid objects in our environment, may they be natural or artefacts, often result from a fragmentation process. This operation is routinely used in industry to extract minerals by the explosive shattering of rocks and to reduce their size by various crushings and grindings [1]. For instance, the wealthy industry of abrasives (most manufactured objects are polished at some stage of their construction) is keen to control the fragment size of alumina or zirconia particles designed for different abrasion processes.

Fragmentation is also a major partner in the evolution of mass distribution in the Universe through collisions of celestial bodies of very different natures and scales—from dust grains involved in erosion [2] to large bodies undergoing catastrophic disruption [3]—or in geophysics [4,5]. Fragments may, in these different contexts, form by the repeated action of some crusher, or may result from a single break-up event, such as an impact.

The question of the sizes produced by a single breaking event (like dropping a glass on the floor, by opposition to the case of solid comminution by repeated stress application) has been addressed thoroughly since the 1930s. The paradigm for most of the studies is the presence of flaws in the material, an idea going back to Griffith's theory of fracture [6]. The flaws, or defects, are pre-existing in the sample. They are randomly distributed [7] and, when a stress is applied, some of these flaws are expected to be ‘activated’. These ideas have profoundly influenced the literature on fracture since the pioneering contributions of Prandtl [8], Lienau [9], Weibull [10], Mott [11] and Gilvarry [12], up to more recent reformulations and extensions to the concept of ‘damage’ and crack interaction [13], a phenomenon also encountered in atomizing liquids [14].

The Mott ring experiment [11,15,16] is a remarkable framework to elaborate our understanding of fragmentation [17]. A ring made of a cohesive material is initially communicated a radial speed, leading to a uniform tensile hoop stress along the ring. The one-dimensional geometry further simplifies the analysis, because the complicated problem of the extension of cracks, their branching and interaction with the stress field does not have to be addressed in detail. The analysis of the fragmentation dynamics proposed by Mott relies on the following sequence of events: the stress increases in the sample until it reaches the critical stress corresponding to the weakest flaw in the sample. At this location, the ring breaks and two release waves propagate on each side of the break. In the domain not yet attained by the wave, the stress still increases, and provided the strain rate *S*_{Y} is the stress at failure, *σ* is a measure of the standard deviation of the strain at failure, *ρ* is the density of the material and *c* is the speed of sound. These results are obtained for an *ad hoc* exponential hazard function, meaning that the probability of failure of a segment of unit length when the strain is increased from *ϵ* to *ϵ*+*dϵ* is

An alternative description of fragmentation has been proposed by Grady and Kipp [20,21] based on an energy argument. The amount of kinetic energy initially communicated to the ring is available to create a new interface. Grady's approach extends from the classical von Rittinger's law [22], which simply matches the available kinetic energy and the surface energy, by recognizing that the amount of kinetic energy available for fragmentation is the material dilatational energy (see also [23]). This balance leads to a fragment size
*Γ* is a measure of the surface cohesion energy. The theory has been extended to include the energy stored before failure [24], allowing for a better description of the fragmentation at low strain rates. Surface energy is also a core element of numerical simulations based on a cohesive-law fracture model that have been used to address the fragmentation of rings [25–27] or brittle objects of more complex geometries [28]. In these approaches, cohesive links which open following for example linear or exponential laws are introduced. Such cohesive laws have also been used in one-dimensional models of fragmentation [29].

The detailed construction of the fragment size distribution from an expanded ring of discrete particles linked by a cohesive force decreasing as a power law with distance has been developed in an accompanying paper [17]. By watching the process in real time, it was found that the distribution evolves in a self-similar manner through an ‘inverse cascade’ of aggregations from small to large fragment sizes by the progressive closing of cohesive links until the exhaustion of the dilatational kinetic energy. At that point, the distribution has converged onto a universal shape well represented by a Gamma distribution

This paper aims at extending this previous work in addressing the consequences of changing the force law between the attracting spheres, and studies the influence of inhomogeneities in the cohesion forces, or defects in the overall process.

## 2. Fragmentation of a ring of spheres in capillary interactions

### (a) Experimental set-up

Building on our previous work on the fragmentation of a ring of interacting spherical magnets [17], we study here a ring made of water absorbent polymer spheres linked by capillary bridges. The spheres or ‘water beads’ are made of a polyacrylamide hydrogel (http://autourdelafleur.fr (2015)). Their diameter can be controlled by adjusting the water concentration in salt (here KCl) in which they are inflated. We use diameters between *a*=5 mm and 6 mm. For a given experiment, spheres with identical diameters (within less than 0.1 mm) were used. The high content of water in the spheres ensures that they are perfectly wetting. A ring of such spheres is roughly assembled on a polydimethylsiloxane (PDMS) membrane of thickness 1 mm lying on a rigid plate. A bubble is then inflated at the centre of the ring. When the bubble reaches the ring, the spheres self-position to form a perfectly circular assembly (figure 1). The bubble is then burst, leaving a ring of spheres linked together by capillary bridges. An impactor is launched at the centre of the membrane. A transverse wave propagates on the impacted PDMS membrane [30]. When the ring is reached by the wave, the spheres are ejected from the membrane. Each sphere is thus communicated an initial velocity with vertical and radial (with respect to the centre of the ring) components. The typical radial speeds used in the experiments are of the order of 0.1 m s^{−1}. The spheres separate from each other but they still interact through capillary bridges (figure 2). This attracting effect competes with the diverging effect of the initial momentum, thus resulting in an orthoradial dynamics, leading to the formation of several separated fragment. The dynamics is similar to the dynamics of an expanding ring of interacting spherical magnets, but the nature of the force is different here.

### (b) Capillary interactions

The interaction between two solid spheres linked together by a capillary bridge is a classical problem [31–33] with practical applications ranging from spore dispersal from plants [34,35] to the formation of micro-particle clusters [36]. Assuming a perfectly wetting sphere, the force between the spheres linked together by a bridge of volume *Ω* can be written, with the notations of figure 3, as
*γ* is the surface tension of the liquid (*γ*=32×10^{−3} *J* *m*^{−2} for the soapy water used in the experiments). In the case of a small liquid bridge (compared with the volume of the sphere), writing the wetting length *d*=(*h*/2){[1+2*Ω*/(*πah*^{2}/2)]^{1/2}−1]}, the force can be written *F*=−*πaγ*[1−*h*/(*h*^{2}+*l*^{2})^{1/2}] with the range *l* given by

Using *z*=*h*+*a*, the force can be approximated by
*F*(*a*)=*πaγ* the contact force. This formula is an excellent approximation of the true *z*-dependence of the force (figure 3*b*). It emphasizes the influence of the volume of the liquid bridge through the range *l* (equation (2.2)). The force decreases rapidly for *z*−*a*>*l*, thus mimicking the irreversible break-up of the bridge.

The cohesion energy associated with this force is *lF*(*a*) and the corresponding Weber number, measuring the kinetic divergence energy of the imposed motion in units of the cohesion, is
*U* is the divergence speed of the ring linked to the radial speed *V* and the radial strain rate *N* is the number of spheres (*N*=30 in the experiments) and *R*=*Na*/(2*π*).

### (c) Dynamics of interacting fragment

Two fragment made of *p* and *q* spheres of mass *m* interacting through a capillary bridge are separated by a distance *z* whose dynamics is given by
*ζ*=(*z*−*a*)/*l* with initial condition *ζ*(0)=0 and d*z*/d*t*(0)=*U*(*p*,*q*)/*l*, the equation can be integrated once
*U*_{e}(*p*,*q*) is the divergence velocity above which two fragment initially in contact will not reconnect. From equation (2.6) with d*ζ*/d*t*=0 for

Of particular interest is the time needed for the trajectory to close, that is, for two fragment to reconnect after they have diverged initially. This can only occur for *β*>1. The trajectory is symmetrical in time and the aggregation time *t*_{a}(*p*,*q*) is given by
*β*, and of the force persistence length *l*. For *β*→1, we have *β* we have *U*(*p*,*q*)*t*_{a}/*l*≈4/*β*. This aggregation time, and in particular its dependence on *p* and *q*, is a key parameter to understand the fragmentation overall dynamics, and statistics, as will be seen in the next section. It also yields a relation between the final mean fragment size

## 3. Fragment size distribution and the role of defects

### (a) Fragment size distribution

After the initial impulse, the ring extends radially. The length of the capillary bridges increases and the attracting force between the spheres leads to an aggregation dynamics that results in the formation of fragment. During the dynamics, some of the bridges break. After some time (typically of the order of 50 ms), the fragment are formed and the distribution is frozen. Three different Weber numbers (*We*=0.0225, 0.0505 and 0.0898) were used corresponding to three different radial speeds (*V* =0.08, 0.12, 0.16 m s^{−1}). The divergence speed is measured on the experiments at the end of the dynamics (it may be slightly different only at the beginning of the experiment because the Weber number is high and we do not expect much deceleration [17]). The dynamics is recorded and repeated (about 20 times for each Weber number) for the fragment size distribution to converge. The obtained mean fragment sizes are consistent with Grady's scaling law 〈*n*〉∼*We*^{−1/3} in equations (1.3) and (2.11) while the normalized fragment size distributions plotted against the normalized fragment size *x*=*n*/〈*n*〉 are identical.

All these observations are in quantitative agreement with the aggregation dynamics characteristic of a ring of spherical magnets except for the width of the distribution. In this case, the distribution is broader. It is well approximated by a Gamma distribution of order 4 (instead of 8 in the case of spherical magnets). However, as shown below, according to the theory developed for a ring of interacting magnets, the distribution should be unchanged with an order *ν*=8. Indeed, the order of the distribution reflects the dynamics of aggregation. In particular, the width of the distribution, i.e. the order *ν* of the Gamma distribution (equation (1.4)), is given by
*n*^{⋆} is a critical size above which fragment contribute to the overall aggregation dynamics only by attracting other fragment, while fragment smaller than *n*^{⋆} aggregate with all sizes [17]. The standard deviation *σ* of the distribution is such that *n*^{⋆} aggregate faster than the typical aggregation time of the distribution. Using *n*^{⋆}=*x*^{⋆}〈*n*〉, i.e. considering *n*^{⋆} as a constant fraction of the mean fragment size, the aggregation kinetics based on Smoluchowski's equations yields a relation between *x*^{⋆} and the order of the distribution, namely *ν*≈2/*x*^{⋆}.

The aggregation time *t*_{a} given in equation (2.10) becomes *t*_{a}(*p*,*q*)≈(4*pq*/(*p*+*q*)^{2})(*mU*/*F*(*a*)) in the limit *β*≫1. Thus, from equation (3.1)
*x*^{⋆}=*n*^{⋆}/〈*n*〉
*x*^{⋆} and *ν* to obtain
*σ*, equivalently the order *ν*, or the threshold aggregation size *x*^{⋆}) is obtained by the criterion in equation (3.1) involving a *ratio* of aggregation times whose singular behaviour in *β* is identical in the two cases (magnetic or capillary forces, see §5a in [17]).

The broader distribution obtained in the present case must thus be understood from another source, namely defects and inhomogeneities of the force between the spheres, for which the analysis above will be adapted.

### (b) Fragmentation in the presence of defects

The force between neighbouring spheres depends on the volume of the liquid bridge through the range *l*. Despite the rigorous protocol used in the experiment, the volumes of the bridges vary. They are measured directly from the experiments, and a typical distribution is shown in figure 5.

To further explore the role of defects and to check our hypothesis that defects are responsible for the broadening of the distribution, we have fragmented rings of spherical magnets with a non-uniform magnetization. The experimental set-up is similar to the set-up in [17]. Before the experiments, the spherical magnets are heated on a heating plate up to temperatures of about 400°C, approaching their Curie point. The result of this process is a variable demagnetization of the spheres introducing a variability in the interaction forces between the spheres. The magnetization of each sphere is obtained by measuring the contact force with a reference magnet. For two spheres of magnetization *M*_{i} and *M*_{j}, the contact force is *F*_{ij}=*a*^{2}*πμ*_{0}*M*_{i}*M*_{j}/6. This formula is used to compute the distribution of contact forces (shown in the inset of figure 6). The distribution of forces has a mean 〈*F*_{ij}〉=1.46 *N* and a standard deviation *σ*_{F}=0.47 *N*. The distribution of the contact forces induces a distribution of the (local) Weber number, of the aggregation time, and thus the effect of this statistical variability is similar to the effect of a distribution of force ranges in the case of capillary bridges. The comparison between the two systems is made further relevant by noticing that *σ*_{l}/〈*l*〉≈*σ*_{F}/〈*F*_{ij}〉≈0.33. Figure 6 shows that, indeed, the fragment size distribution obtained with capillary bridges is similar to the distribution obtained with magnets with a variable magnetization. The latter experiment has been performed with *N*=50 spheres at a Weber number *We*=0.0063 leading to a mean fragment size 〈*n*〉=5.56. The distributions are well approximated by a Gamma distribution of order *ν*=4.

## 4. Discussion

### (a) A model for the role of defects

We extend here the aggregation model above to account for a statistical distribution of the intensity of the contact forces (the reasoning being equally valid for the force range *l*). The aggregation time computed by considering the dynamics of two fragment of size *p* and *q* is a function of the intensity of the force, and/or of its persistence length. The fragment which are more likely to aggregate with their neighbours are the small fragment, because they are light, but also the fragment that are linked by a ‘strong’ link, that is, a link with a strong attractive force.

Let us first discuss the case of distributed contact forces, relevant to the magnetic spheres, and let *P*(*F*) d*F* be the probability that a contact force lies in the range [*F*,*F*+d*F*]. The force *F* is typically distributed around a mean 〈*F*〉, with standard deviation *σ*_{F}. If forces and fragment sizes are uncorrelated during the aggregation process, an adaptation of the aggregation criterion in equation (3.1) above consists in restricting the population of fragment liable to reconnect to those having links stronger than the mean strength 〈*F*〉
*P*(*F*)=*δ*(*F*−〈*F*〉). Since the aggregation time *t*_{a} is a slowly decreasing function of the force *F*, a further caricature of this idealization is to represent the successful fraction of strong enough links in the distribution by *ν*=8 as before while the consequence of a distribution in the force is to interrupt the aggregation cascade earlier, leading to larger value of *x*^{⋆}, and a smaller value of *ν* reflecting a broader distribution of fragment sizes. With *σ*_{F}/〈*F*〉=0.33, we have *x*^{⋆}≈0.37 and *ν*≈5.

The same reasoning and results hold with liquid bridges with variable volumes, and persistence force range *l*: since the aggregation time in (2.10) is decreasing with *l*, only those spheres linked by a sufficiently big bridge will pass the aggregation criterion in (4.1) replacing, *mutatis mutandis*, *F* by *l*. Consistently, and because the relative width of the persistence length distribution *P*(*l*) is identical to that of the forces (*σ*_{l}/〈*l*〉=0.33, figure 5*b*), the fragment size distribution of the gel beads linked by capillary bridges is superimposable with that of the magnetic spheres, as seen in figure 6.

As anticipated from equation (4.3), a superimposed noise on a mean cohesion force broadens the fragment size distribution, while leaving the fundamental aggregation dynamics of the distribution construction unchanged. The distribution is broader because only the fragment linked by sufficiently strong bonds have a chance to aggregate, and they represent a subsample of those which would have aggregated at constant force. The modified criterion in equation (4.1) is in line with the one without noise as it emphasizes the importance of the hierarchy of the aggregation times, not only with size but with attractive strength as well.

The argument above neglects the evolution of the distribution of the forces during the aggregation process. This is obviously unrealistic, precisely because strong links are more likely to reconnect, depleting the force distribution with its most intense components for further aggregations. However, it provides the initial trend, quantifying properly this aggregation cascade which, anyway, is interrupted earlier.

### (b) Comparison with simulations

As a matter of further illustration, it is instructive to consider the case of a uniform distribution of forces in the interval [〈*F*〉−δ*F*,〈*F*〉+δ*F*]. The Gamma distribution of order *ν* expected from equation (4.3) is *et al*. [17]. In the present case, the force between the end spheres of each fragment is exponential as given by equation (2.3), with a uniform range *l*=0.1 (in units of sphere diameter) and the simulations are carried out with *N*=50 spheres. The intensity of the contact force of each link is randomly chosen with a uniform distribution at the beginning of the simulation. For different Weber numbers, we observe a systematic broadening of the distribution as the contact forces become more widely distributed (figure 8), consistent with the model above. It should be noted though that the distribution of forces evolves significantly during the simulation. This is apparent in figure 8*c*. The distribution of forces evolves in time during the aggregation process since stronger links close faster, thus leading to a distribution of surviving links biased towards weak links, an observation reminiscent of Mott's viewpoint on fragmentation.

## 5. Conclusion

As an extension and complement to our study on the fragmentation of a ring of magnetic spheres [17], we have explored here the case of a ring of spheres linked together by capillary bridges. This system presents several differences from the previous example with magnets: in particular, the force linking the spheres now has a finite range because a capillary bridge breaks up as it is elongated. Also, the functional dependence of the force with the distance between the spheres is different from the dipole magnetic force case, namely it is exponential rather than a power law. Nevertheless, we have shown that the phenomenology and final fragment size distribution are consistent with our previous observations, and analysis. Fragmentation in this system results from an *inverse cascade* of aggregations rather than from a direct cascade of sequential break-ups (figure 9), the conventional view inherited from Kolmogorov, whose success has gone well over the context of ore processing, where it had been originally conceived [37,38], and extended up to the analysis of turbulent flows [39], where it has besides also been criticized [40].

This study has also been the opportunity to consider the influence of inhomogeneities in the linking forces. We have shown that, again, the same aggregation scenario holds in the presence of distributed forces among the links, but that the cascade is interrupted earlier, leading to a broader distribution of sizes, but still pertaining to the class of the Gamma distributions arising from aggregation mechanisms. We finally note that this description obviously holds in the limit of a ‘weak’ disorder, namely for *σ*_{F}/〈*F*〉<1, a condition fulfilled here. Severe faults (i.e. *F*≈0) in the material will preferentially concentrate irreversibly broken links, and the problem in that case amounts to understanding the spatial distribution of these major faults in the material (this is the Lienau limit [9]) rather than, as is the case here, wondering how weakened linking bounds alter an aggregation process.

## Authors' contributions

All authors contributed to the experimental, numerical and theoretical aspects of this study, originally suggested by E.V.

## Competing interests

We have no competing interests.

## Funding

We acknowledge support from the Agence Nationale de la Recherche through grant no. ANR-11-JS09-0005

- Received September 29, 2015.
- Accepted December 9, 2015.

- © 2016 The Author(s)