## Abstract

When severely impacted, a cohesive object deforms and eventually breaks into fragments. Cohesion forces keeping the material together and momentum driving the fragmentation couple through a complicated process involving crack propagation on a deforming substrate, so that a comprehensive scenario for the build-up of the full fragment size distribution of broken objects is still lacking. We use necklaces of cohesive particles (magnetized spheres) as an experimental model of a one-dimensional material, which we expand radially in an impulsive way. Exploring in real time the intermediate state where the particles are no longer in contact, but still in interaction as they separate, we demonstrate that the final fragments result from the self-assembly of individual particles and that their size distribution converges to a stable self-similar distribution whose parameters, interpreted from first principles, depend on the expansion and cohesion strengths.

## 1. Introduction

The shapes and sizes of condensed matter fragments in our environment result from two competing, complementary phenomena, namely fragmentation and self-assembly [1,2]. Relevant to natural and man-made processes, their combination may produce intriguing objects at all scales, from aggregated atoms in molecules to the formation of planets [3,4].

Although partners in matter cohesion and in its disgregation (following R. Clausius neologism [5]), fragmentation and aggregation are widely recognized as different paradigms. Fragmentation often results from a sequential cascade of break-ups as in crushing and grinding [6], each sequence involving an impact with an object [7], or between the fragments' themselves [8]. Stresses overcome cohesive forces and cracks propagate in an inhomogeneous, time-varying stress field [9,10], making the prediction of the resulting fragments' size distribution extremely difficult, other than resorting to lumped probabilistic arguments [11] where the details of cohesion forces are elusive. These forces are, on the other hand, responsible for the ultimate aggregation of particles like molecules [3], biological cells [12], colloids [13], haze, dust [14] and planets [15]. The rate at which particles aggregate [16–18] depends on how they move, how the medium is stirred [19], and is often independent of the cohesion itself.

The expanding ring geometry proposed by Mott [20] is a remarkably simple configuration to study the interplay between crack nucleation and elastic or plastic wave propagation in building fragments of an initially cohesive material: waves travel along one dimension, and the complicated problem of crack branching does not need to be solved. It has been used to explore scaling laws and fragment size distributions in various limits [21], emphasizing the role of preexisting defects [22]. We study here the discrete version of it, using necklaces of spheres linked by magnetic forces.

## 2. Observations and results

A set of spherical magnets of diameter *a*=5 mm, mass *m*=0.50 g is assembled in a necklace of *N*=99 beads (necklaces of *N*=50 and *N*=200 beads are also used). Two different sets of spheres are used with a magnetic force between the spheres *F*(*a*)=4.6 N or *F*(*a*)=3.9 N (within less than 5%) when the spheres are in contact. The necklace is positioned on a cone with an incline at 45°. The cone, guided by an axle, is released from rest at a height ranging from 15 to 300 cm. When it hits the ground, the cone is suddenly stopped and an initial impulse in the radial direction is communicated to each sphere. The whole dynamics occurring after impact, as detailed below, lasting about 10^{−2} s is recorded by a high-speed camera at 3000 frames per second placed above the experiment. Custom-made image analysis software is used to record the position of each sphere.

After having been communicated, an initial radial speed *V* (in the experiment *V* is typically of the order of a few m s^{−1}), the spheres separate from each other, and start concomitantly to aggregate because of the attractive force between them. Aggregation proceeds at an ever slowed pace to finally stop, and fragments of various sizes are observed. The experiment is repeated and recorded to construct the fragment size distribution as a function of time. Figure 1 shows the fragmentation of a ring made of spherical magnets.

The force between two aligned spherical particles of diameter *a*, magnetization density *M* and separated by a distance (centre to centre) *z* is [23]
*μ*_{0} is the vacuum permeability. The only parameter is the Weber number (familiar in the liquid atomization context [24]), namely the ratio between kinetic and cohesion energies,
*m* is the mass and *ρ* is the density of the particles. *U*=2*πV*/*N* is the inter-particle divergence velocity, which depends on the radial velocity *V* and the radius of curvature of the necklace (i.e. the number of particles *N*).

Shortly after impact, the necklace is atomized (figure 1*a*) and the fragment size distribution is narrow, centred around a mean fragment size of a few particles (figure 2*a*). As fragments aggregate, their mean size increases and the distribution becomes broader, making the fragments both heavier and more distant from each other. Aggregation concomitantly slows down, until it stops, thus completing the fragmentation process.

The time evolution of the fragment size distribution (figure 2) reflects the aggregation dynamics. The distributions, rescaled by the (time dependent) mean size, collapses in a self-similar way [14,25–27] onto a Gamma distribution
*ν*=8.

## 3. Aggregation dynamics

The law above is understood in terms of a model of *incomplete convolution* where fragments of different sizes interact at random, with the small sizes, being lighter, aggregating faster. In other words, the small fragments are more likely to aggregate to any other fragment than bigger fragments are.

### (a) Rate equations

To study the aggregation dynamics of the fragments, we use Smoluchowski's description of coagulation phenomena [17]. In the limit of a continuous spectrum for the fragment size *n*, let *n* and *n*+*dn* at time *t*. The rate equation is
*n*≤0. The first term on the right-hand side of equation (3.1) is the fraction of the distribution that recombines into larger size, the second term represents the aggregated part and *K*(*n*,*n*′,*t*) is the collision frequency of fragments of size *n* and *n*′. A solution to this equation is known for a limited number of kernels only and in general one relies on asymptotic techniques to study the asymptotic behaviour of

Equation (3.1) can be written in terms of the probability density function *N*_{f} can be deduced from (3.1),
*P*(*n*,*t*):

Such a model can only be valid if, at any time, there is no correlation between the sizes of the aggregating fragments (i.e. no spatial correlation between neighbours). This condition was probed on the experiments. In figure 3, we plot the correlation between the size of a fragment and the size of the *p*th neighbour at time *t*

### (b) Partial aggregation, incomplete convolution

In the present problem, the rate of aggregation is expected to decrease with the fragment sizes: small fragments are lighter and reconnect at a much faster pace than massive fragments. We thus make the following caricature: large fragments are massive and thus only contribute to the global dynamics by attracting small fragments while small fragments are likely to aggregate to any other fragment. Large/small depends on the fragment's size *n* compared with a threshold *n*^{⋆}. The aggregation up to *n*^{⋆} complies with Smoluchowski's dynamics at a constant rate *r* while it is zero among sizes larger than *n*^{⋆}. Thus equation (3.4) reduces to
*r*), the model is the classical aggregation equation [17] whose solution is an exponential distribution [25].

### (c) Self-similar distribution

Introducing the Laplace transform *P*(*n*,*t*)
*n*^{⋆} is finite, the rate equation for *ν* a function of *n*^{⋆}. Expanding *e*^{−sn}≈1−*ns*+*n*^{2}*s*^{2}/2+*O*(*s*^{3}), equation (3.8) writes
*P*(*n*,*t*), that is, *s*
*s*^{2}, using the previous result,
*x*=*n*/〈*n*〉[30]. We also define the distribution function *f*(*x*) of the scaled sizes *x*=*n*/〈*n*〉 such that *P*(*n*,*t*)=(1/〈*n*〉)*f*(*x*=*n*/〈*n*〉). Equation (3.13) writes
*ν* (see equation (2.3)) whose second moment is 〈*x*^{2}〉=1+1/*ν*, we have
*ν* to the cut-off size normalized by the mean size *x*^{⋆}. The approximation has been obtained in the limit of large *ν*, for which the Gamma distribution in equation (2.3) approaches a Dirac distribution.

We have thus shown that the partial aggregation model described by equation (3.6) admits the approximate solution in the form of a Gamma function, provided that the sizes that are allowed to aggregate are a fraction of the current mean size. This model emphasizes the mechanism by which the distribution is built: through aggregation of fragments of small sizes with the rest of the distribution. We now proceed to analyse more thoroughly the dynamics of interacting fragments, with the objective to relate the order *ν* of the distribution and the cohesive forces.

## 4. Ring dynamics

We first address the dynamics at short time to describe the early stage of the dynamics of the ring, which is characterized by an instability of the uniform distribution of particles and a decrease of the radial speed. We then address the dynamics at a later stage when the fragments interact by pairs. The aim of this analysis is to obtain the typical aggregation time of two fragments as a function of their sizes.

### (a) Short-time dynamics

#### (i) Stability of the radial expansion

At early times, we consider that the spheres are separated but close to each other. If we consider the interaction of a sphere with its neighbours, the dynamics of the *i*th sphere, in the orthoradial direction, writes (at a fixed radius)
*θ*_{i}=*iθ*_{0}+*α*_{i}

This equation, which in its continuous limit reduces to an ‘anti-diffusion’ equation, reveals that all wavelengths are unstable, the larger wavenumbers (and thus shorter wavelengths) being the most unstable. The typical time scale associated with the instability is [*ma*/4*F*(*a*)]^{1/2}, which is of the order of 0.35×10^{−3} s in experiments. This equation explains the rapidly developing initial instability but it is not useful to further explore the dynamics, when fragments interact by pairs.

#### (ii) Radial speed

Another consequence of the triplet interactions between spheres at short times is the decrease in the radial speed. Attractive forces on a curved substrate induce a Laplace surface tension force decelerating the cohesive necklace in its radial expansion. In the radial direction, assuming that the angle between the spheres is *θ*_{0}≈*a*/*R*_{0}, the equation of motion of a sphere is
*z* is the distance between the centres of the spheres. In the mean-field approximation, the size of the fragments is 〈*n*〉, and *z*(*t*)=2*πR*(*t*)〈*n*〉/*N*. In the absence of an evolution law for 〈*n*〉, to gain insight into the radial dynamics, we consider that the fragment size remains equal to 1 until a typical time *t*^{⋆} at which the radial dynamics stops. With this crude approximation, equation (4.3) can be integrated
*R*(*t*)≈*R*_{0}+*V* _{0}*t* with *V* _{0} the initial radial velocity and *R*_{0} the initial radius. This dynamics takes place until a time *t*^{⋆} at which the evolution stops. The speed has then reached its final value *t*^{⋆} is a fraction of the characteristic time *a*/*V* _{0}. Experiments suggest *t*^{⋆}=0.6*a*/*V* _{0} yielding in equation (4.3) a final radial speed *V*^{2}_{c}=2*a*^{2}*F*(*a*)/(*mR*_{0})=2*πaF*(*a*)/(*mN*) the critical speed below which the ring is not fragmented. In terms of the Weber number, the final radial speed writes
*a*/*V* and thus the speed that is relevant to discuss the aggregation dynamics is the final speed

### (b) Interaction of two fragments

Let two fragments of sizes *p* and *q* separated by a distance *z* between the centre of their facing end spheres interact through magnetic forces. Taking into account only the force between the end spheres, the equation of evolution for *z* reads
*z*(0)=*z*_{0} and d*z*/d*t*(0)=*U*(*p*,*q*)>0

Let *U*_{e} be the escape velocity. It is defined as the divergence initial velocity above which two fragments initially in contact (i.e. *z*_{0}=*a*) will not reconnect. From equation (4.7), we get
*ζ*=*z*/*a* using the non-dimensional time *τ*=*U*(*p*,*q*)*t*/*a*
*ζ*(0)=1 (corresponding to *z*(0)=*a*) and d*ζ*/d*τ*(0)=1. The solution of equation (4.6) is compared with the dynamics of two fragments extracted from an experiment in figure 5. This solution cannot be written in terms of simple functions. We can however compute the aggregation time, i.e. the time at which the fragment reconnects (*ζ*=1). The maximal distance *ζ*_{m} between the fragments is obtained from equation (4.7)
*τ*_{a} is then obtained by integrating the first integral of equation (4.9)
*β*(*p*,*q*), one obtains
*β*(*p*,*q*) approaching 1,

### (c) Aggregation of fragments: shape of the distribution

The knowledge of the relation between the fragment sizes *p*,*q* and the aggregation time *t*_{a} in the above equation can then be used in the aggregation model of §3 since this dependence also rules the order *ν* of the Gamma distribution.

#### (iii) Order of the distribution

A fragment of size *p* will aggregate with another of size 2*n*−*p* to form a fragment of size 2*n* if the typical aggregation time for these two fragments is less than a fraction of the aggregation time between two typical fragments of the distribution, say of sizes 〈*n*〉±*σ*. Assuming that the aggregation time should be less than half the typical aggregation time of the distribution, the aggregation criterion can be written
*p* and *q*, the typical (geometric) divergence velocity is *U*(*p*,*q*)=(*p*+*q*)*U*/2 leading to *β*(*p*,*q*)=4/[*We*(*p*+*q*)*pq*]. These two fragments reconnect at a time *t*_{a}(*p*,*q*) written, in the limit *β*(*p*,*q*)≥1 where most of the rearrangement dynamics occurs, as
*p*=*n*^{⋆} for which the criterion in equation (4.15) is fulfilled relates to 〈*n*〉 and *σ* through
*x*^{⋆}=*n*^{⋆}/〈*n*〉
*x*^{⋆} and *ν* to obtain
*ν*=8 agrees with observations (figure 2) confirming the consistency of the proposed scenario. Fragments smaller than 1/4 the mean are indeed virtually absent from the experimental distributions, and the ratio *σ*/〈*n*〉 converges quickly to its final value *σ*/〈*n*〉 (0.32 in [31], 0.35 here).

#### (iv) Evolution of the mean

The evolution of the mean fragment size, in the large time (*β*(*p*,*q*)→1) limit readily follows. The aggregation time between two fragments of sizes *p* and *q* now writes

The total number of individual particles *n*〉=*N*/*N*_{f}(*t*) obeys, from equation (3.1), to

Using the self-similar character of the distribution of fragment sizes, we have *x*=*x*′=1 and with *K*(〈*n*〉,〈*n*〉)=*N*_{f}/[2*t*_{a}(〈*n*〉,〈*n*〉)], one obtains the equation for the mean

Aggregation slows down as *β*(〈*n*〉,〈*n*〉) approaches unity (i.e. when the divergence velocity approaches the escape velocity) and is frozen for *β* and equation (4.8)

Equation (4.23) can be written in a non-dimensional form. With *U*(〈*n*〉,〈*n*〉)=〈*n*〉*U*, one obtains the following equation:
*α* a number equal to 1/1.5, obtained in the limit of *β*→1. The solution to this equation is shown in figure 6, together with the data obtained for different values of *N* and *We*. Note that the coefficient *α* has been adjusted to obtain superposition of the theoretical curve with the experimental data. The data rescale approximately, but we note that the model was obtained with the approximation of a constant radial speed, while the experiments exhibit a diminution of the radial speed with time.

## 5. Discussion

### (a) Robustness

Through the analysis of §4, we have related the order of the distribution with the aggregation dynamics. It shows good agreement with experimental data and we show here that the analysis is robust. The criterion expressed in equation (4.15) can be used with different forms of the aggregation time. For example, the aggregation time proposed in equation (4.20) can be replaced by the aggregation time at shorter times (for large values of *β*) *t*_{a}=4*a*/[3*U*(*p*,*q*)*β*(*p*,*q*)] without modifying the subsequent analysis. Therefore, the same order of the distribution is obtained.

Furthermore, this argument based on aggregation times can also be extended to the more general case of a force of the form
*τ*_{a}≈4/[(*μ*−1)*β*] for large *β* and *τ*∼(*β*−1)^{−(μ+1)/[2(μ−1)]} as *β* approaches 1. The most remarkable consequence of this result is that if the criterion of equation (4.15), involving a *ratio* of aggregation times, is used, then the order of the distribution does not depend on the exponent *μ*.

We have performed numerical simulations of the dynamics of an assembly of interacting fragments. The fragments interact through attractive forces between their end particles (see appendix A for details on the numerical method). The results are shown in figure 7 for different values of the Weber number. The scaling law (4.24) holds for the different Weber number, number of spheres and force law exponents. Moreover, the distributions obtained for the different parameters (omitting the situations with small fragment sizes for which finite size effects alter the dynamics) are consistent with the analysis of §4c. In particular, the shape of the distributions does not depend on the nature of the force. This result shows the robustness of the proposed scenario, whose main ingredient is the decreasing (with distance) character of the force.

Finally, it is worth mentioning that Smoluchovski's model (equation 3.1) and the ring dynamics present similarities. In discrete form, Smoluchovski's equation writes
*n* and *K*(*p*,*q*)=2/[*N*_{f}*t*_{a}(*p*,*q*)]. In this discrete form, aggregation times *t*_{a} that present a sharp transition to zero will lead to a non-smooth distribution. This is, for example, the case for the form of equation (4.20). For this reason, the discrete Somulchovski equation is not a useful tool to explore the dynamics near the saturation, i.e. when *β*→1. However for large values of *β*, corresponding to the early stages of the dynamics, one can use the aggregation time *τ*≈4/(3*β*). Such a model yields a self-similar evolution of the fragment size. Figure 8 shows the evolution of the distribution, starting from an exponential distribution. The convergence to a self-similar probability density function is fast, as shown by the fast convergence of the order *ν*=(*σ*/〈*n*〉)^{−2}. The distribution of fragment size is shown superimposed with a Gamma distribution of order 4.

### (b) Energetics of fragmentation

The scaling law for the mean fragment size agrees with the energy-based approach of fragmentation developed by Grady [21]. We note however that there is no quasi-static limit of the ring fragmentation: if the impact speed is reduced, a threshold is reached: below this threshold, the ring does not break. This is different from the quasi-static extension of the energy-based fragmentation model [34] which considers that the elastic energy is available to drive the fragmentation. In the present example, at low strain rate, the energy is not sufficient to trigger the fragmentation. Indeed when plotted as a function of the Weber number based on initial radial speed *We*_{0}, the mean fragment size shows a dependence on *N* which is accounted for when the final Weber number is written as a function of the initial Weber number
*η* is the ratio of the surface energy created by the fragmentation process to the input kinetic energy (e.g. [35]). In the present case, the efficiency writes
*F*(*z*)=*F*(*a*)(*a*/*z*)^{4}. At low impact speed, there is no fragmentation and thus the efficiency is zero. At large impact speed, *η* in between these two extremes. Indeed, using for the mean final fragment size *k*=1.8 measures the threshold for fragmentation. The efficiency, shown in figure 9, is independent of *N* once plotted against the variable *N*^{3}We_{0}/(2*π*)^{3}, which measures the ratio of the total kinetic energy *F*(*a*)/*a*; it is way below 1 in absolute value, and exhibits a maximum (at *η*≈0.3), thus contradicting von Rittinger's principle stating that all input energy is used for breaking bonds in a cohesive material [6], thereby suggesting that the efficiency should be constant, and of order unity.

### (c) Maximum entropy

A system of particles interacting at random with no correlation in a kind of ‘molecular chaos’ as in the present case is liable for a statistical description in the sense of Gibbs and Boltzmann. When there is a clear separation of time scales between the one characteristic of the evolution of the global statistics *P*(*n*,*t*) (which converges towards a *time-independent* distribution *f*(*n*/〈*n*〉) if fragment sizes are rescaled by the current mean) and the characteristic times of the microscopic aggregation dynamics *t*_{a}(*p*,*q*), the distribution describing the fragment sizes may be sought as a maximum entropy distribution, subject to some constraints. In the present configuration involving a ring, the simplest way to understand how the periodic boundary condition imposes a constraint on the spatial rearrangements is to introduce a mass density *ψ*(*s*,*t*) aligned with the orthoradial coordinate *s* along the support of the aggregating clusters (figure 10)
*s*. At a given instant of time in its radial expansion, the mass support length is *L*=2*πR*, and we have obviously
*s*, the density *ψ*(*s*,*t*) is subjected to a transport equation. Denoting *u*(*s*,*t*) the local velocity of the density distribution *ψ*(*s*,*t*), we have by continuity
*u*(*s*,*t*) remains continuous on this circular *periodic* support,
*conserved* as the mass rearranges spatially by aggregation.

At a given instant of time during the expansion when the ring has average density 〈*ψ*〉=*mN*/*L*, if the fast local rearrangements have driven the mass distribution *p*(*ψ*) around its maximal likelihood state, its entropy [36]
*δp* in the distribution shape will leave the entropy stationary, that is, *δS*=0, and enforcing the conservation of the average density *α* and *β* must be computed from the exact values of 〈*ψ*〉 and

## 6. Conclusion

Conventionally viewed as a process where the arrow of time points towards ever smaller sizes by the repeated action of various stresses, the fragmentation of a cohesive material has been shown to result, on the contrary, from an *inverse cascade* of aggregations, starting with the smaller atoms, up to stable bigger fragments. The cascade is interrupted when separation forces overcome cohesive forces, leaving the broken material in a dispersed state whose statistics is interpretable from first principles. This scenario emphasizes the role of cohesive forces in the building of the fragment size distributions, rather than pre-existing material flaws or an initial disorder, a subject examined in a companion paper [38]. It is expected to describe the formation of small fragments, also called *fines* in solid comminution, of sizes about *F*(*a*)/*a*=1 J m^{−2} as the cohesion energy with *a*=10^{−10} m as the molecule spacing and *U*/*a*∼10^{2} s^{−1} as the strain rate at impact). These observations also suggest that, contrary to a common belief [39], fragments as small as the elementary molecule can be formed by comminution (provided they escape, by chance, the aggregation dynamics), precisely because they are formed from the start, and not at the end of an interrupted direct cascade.

## 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.

## Appendix A. Simulations of interacting spheres

To accompany the experiments and to further explore the dynamics of the ring of interacting particles, we have carried out numerical simulations of the dynamics of a set of interacting fragments. We consider an assembly of *n* fragments interacting between each other through a nearest neighbour interaction, i.e. fragment *i* interacts with fragments *i*+1 and *i*−1 (figure 11). Starting from an initial condition, we solve the set of differential equations for the fragment distances *x*_{i}
*n* fragments until a reconnection event occurs. A reconnection occurs when one of the distances between the centres of the end spheres *δ*_{i}=*x*_{i}−(*n*_{i}−1)/2−(*n*_{i+1}−1)/2 is equal to 1. There are initially *N* isolated particles of size 1, of mass 1 and the initial divergence velocity is 1/*N*. An initial noise with a uniform distribution in the range [−5×10^{−3}/*N*,5×10^{−3}/*N*] is added to the initial divergence speed. The force between fragments is the force between its end spheres and it writes
*F*(*x*)∼(1/*x*)^{μ}. The dynamics is taking place on a line. To mimic the ring dynamics, periodic boundary conditions are imposed, the spatial period being time dependent and equal to *N*+*t*. The difference between this numerical model and the experiment is the absence of radial geometry and radial speed dynamics and the simplified fragment interactions. As shown in figure 7, the simulations exhibit the same dynamics as the experiments in terms of the scaling law for the final fragment size

- Received September 29, 2015.
- Accepted November 18, 2015.

- © 2015 The Author(s)