Abstract
We formulate the statistics of the discrete multicomponent fragmentation event using a methodology borrowed from statistical mechanics. We generate the ensemble of all feasible distributions that can be formed when a single integer multicomponent mass is broken into fixed number of fragments and calculate the combinatorial multiplicity of all distributions in the set. We define random fragmentation by the condition that the probability of distribution be proportional to its multiplicity, and obtain the partition function and the mean distribution in closed form. We then introduce a functional that biases the probability of distribution to produce in a systematic manner fragment distributions that deviate to any arbitrary degree from the random case. We corroborate the results of the theory by Monte Carlo simulation, and demonstrate examples in which components in sieve cuts of the fragment distribution undergo preferential mixing or segregation relative to the parent particle.
Highlights
Objects disintegrate into fragments via impact, detonation, degradation, or cleavage of the bonds that hold the structure together
We start with a multicomponent particle that is made of discrete units of any number of components, subject it to one fragmentation event with fixed number of fragments, and construct the set of all fragment distributions that can be obtained in this manner
We have presented a treatment of multicomponent fragmentation on the basis of random fragmentation in combination with a functional that biases the ensemble of feasible distributions
Summary
Objects disintegrate into fragments via impact, detonation, degradation, or cleavage of the bonds that hold the structure together. The primary input to this formulation is a breakup model that specifies the distribution of fragments produced by a given parent size and the relative rate at which different sizes break up This population balance approach forms the basis for the mathematical treatment and numerical modeling of fragmentation in granular, colloid, and polymeric systems [11,12,13,14,15,16,17,18]. We start with a multicomponent particle that is made of discrete units of any number of components, subject it to one fragmentation event with fixed number of fragments, and construct the set of all fragment distributions that can be obtained in this manner. We present results from Monte Carlo simulations to corroborate the theory and show that components may preferentially mix or unmix in the fragments depending on the choice of the bias functional
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