Did You Shatter a Plate? Physics Already Predicts the Fragment Sizes.

Every time something falls and breaks, it leaves behind an apparent mess. However, this chaos obeys a rigorous mathematical logic. Whether a piece of sugar is crushed, a vase shatters or a bubble explodes, the distribution of fragments follows a recurring pattern, regardless of the material or the cause of the rupture. This regularity, long suspected but never formally explained, has just been described by a team led by Emmanuel Villermaux, physicist at Aix-Marseille University and member of the Institut Universitaire de France.

Published in the journal Physical Review Letters, the study establishes a universal law of fragmentation, based on a principle of maximum disorder. This model unifies phenomena as varied as the breakage of a solid object, the bursting of a liquid bubble or the disintegration of plastic waste, revealing an underlying structure in what seemed purely random.

An equation for all fractures

The fragmentation of an object, when it breaks, does not produce a random distribution of debris. By studying this question, physicist Emmanuel Villermaux (Aix-Marseille University, Institut Universitaire de France) proposed a universal equation capable of predicting the distribution of fragment sizes in many cases of rupture. His work is based on a probabilistic and statistical approach. An approach that goes against the usual studies that attempt to model the precise breakage mechanisms (such as crack propagation or local mechanical stresses).

This equation describes a power law, in which the probability of obtaining a fragment of a given size decreases according to a precise mathematical function. The model is based on two constraints. Namely: conservation of the total mass of the initial object, and an assumption of maximum random distribution of fragment sizes — the system explores all cutting possibilities, but retains those which maximize entropy. The result? A predictable pattern: lots of small fragments and a few larger ones, in a constant proportion.

This law remains valid whatever the material, whether it is brittle like glass, crumbly like sugar, or liquid like a bubble. It makes it possible to predict not how the cracks will appear, but how the final fragments will be distributed. This transforms a seemingly chaotic phenomenon into a situation governed by a simple and general law. A law applicable to solids, liquids and hybrid structures such as thin shells. The elegance of the model lies in its abstraction. It ignores the details of the breaking process to focus on the end result, which is universally observable.

The principle of maximum disorder

Rather than modeling the shocks or cracks causing the fragmentation, Emmanuel Villermaux chooses to consider the object already broken. This change in perspective is based on a central principle. The system naturally tends towards the most probable state, that is to say the one which maximizes its entropy, or disorder. This concept, from statistical thermodynamics, implies that, during a rupture, the object generates the most disorganized configuration of fragments possible.

This principle of “maximum randomness” has a direct implication. How an object breaks is not only dictated by physical forces, but also by a universal statistical law. This is not pure chaos, but disorder governed by probability. In other words, it is not the initial shock which dictates the number or size of the pieces. But the tendency of the system to explore as many cutting combinations as possible.

We grant you that this model may seem abstract. However, it applies concretely to very different objects: soap bubbles, burst drops of water, thin exploding shells, solid spheres, blocks of sugar or even rock fragments. It explains why, regardless of the material or the context, we observe a majority of small pieces and a minority of larger fragments.

According to Villermaux, this regularity only depends on the size of the initial object. A 3D object generates a different distribution than a 2D object (like a dinner plate). The equation relates this dimension to the power law exponent. This mathematical link underpins the predictability of the model. By maximizing disorder, nature tends toward the most statistically probable fragmentation, thus revealing a deep logic in the apparent chaos.

A law valid for solids, liquids and beyond

What also makes Villermaux’s model particularly remarkable is its universality. It is not limited to solid materials. The equation also applies to liquids, gases in the form of bubbles, and even to hybrid structures such as liquid shells or explosive droplets Clarifications: a soap bubble that bursts is a liquid shell, a drop of water that explodes on hot metal, an explosive droplet. Regardless of the nature of the object, its fragmentation follows this same law of maximum disorder.

For example, when an air bubble bursts, it generates microdroplets of varying sizes. This phenomenon, although apparently distinct from the breakage of a glass or a piece of stone, respects a similar distribution. Even dry spaghetti that breaks when twisted shows behavior consistent with this pattern. From macroscopy to plastic micro-debris, this regularity crosses scales.

The model also allows us to better understand the practical consequences of certain phenomena. In the oceans, for example, the degradation of plastic waste into microplastics follows progressive fragmentation. Villermaux's law can help predict the size of fragments and their distribution, which is crucial for studying their dispersion and ecological impact.

Physicist Ferenc Kun, from the University of Debrecen (Hungary), emphasizes in a comment to New Scientistthat this approach could extend to the shape of the fragments, not just their size. If this link is confirmed, it would further strengthen the scope of the model.

Concrete applications and open perspectives

One of the essential contributions of the Villermaux model therefore lies in its concrete applications, well beyond the theoretical framework. By better understanding how objects fragment, we can optimize numerous technical processes, predict dangerous natural phenomena or even refine our understanding of environmental processes.

In industry, grinding materials like ore is energy-intensive. If we know in advance how a material will fragment, it becomes possible to adjust crushing methods to minimize energy losses and maximize yield. The distribution of fragment sizes also makes it possible to calibrate downstream machines and filters. This gain in precision is particularly useful in sectors where the particle size influences the quality of the final product.

In a completely different field, this law can be used to model geological collapses, such as landslides or rock falls in mountains. Knowing the probability of the appearance of large blocks or fine debris allows the risks to be assessed more accurately.

Scientifically, the model still raises many open questions. What is the minimum size that a fragment can reach before the law no longer applies? This limit could depend on the properties of the material at the molecular scale. Furthermore, if we succeed in linking the shape of the fragments to this same principle of maximum disorder, this would open a new field of research in matter physics.

Source: Emmanuel Villermaux, “Fragmentation: Principles versus Mechanisms”. Physical Review Letters (2025).