Shattering transitions in collision-induced fragmentation

Krapivsky, P. L.; Ben‐Naim, E.

doi:10.1103/physreve.68.021102

Cited by 39 publications

(49 citation statements)

References 33 publications

(67 reference statements)

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“…Collisions with erosion. In collisions with erosion only a small fraction of a particle mass is chipped off (31,37,38). Here we consider a simplified model of such collisions: It takes place when the relative kinetic energy exceeds the threshold energy E eros , which is smaller than the fragmentation energy E frag .…”

Section: [4]mentioning

confidence: 99%

Size distribution of particles in Saturn’s rings from aggregation and fragmentation

Brilliantov

Krapivsky

Bodrova

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

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129

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Saturn's rings consist of a huge number of water ice particles, with a tiny addition of rocky material. They form a flat disk, as the result of an interplay of angular momentum conservation and the steady loss of energy in dissipative interparticle collisions. For particles in the size range from a few centimeters to a few meters, a power-law distribution of radii, ∼ r −q with q ≈ 3, has been inferred; for larger sizes, the distribution has a steep cutoff. It has been suggested that this size distribution may arise from a balance between aggregation and fragmentation of ring particles, yet neither the power-law dependence nor the upper size cutoff have been established on theoretical grounds. Here we propose a model for the particle size distribution that quantitatively explains the observations. In accordance with data, our model predicts the exponent q to be constrained to the interval 2.75 ≤ q ≤ 3.5. Also an exponential cutoff for larger particle sizes establishes naturally with the cutoff radius being set by the relative frequency of aggregating and disruptive collisions. This cutoff is much smaller than the typical scale of microstructures seen in Saturn's rings. (1-3) and the observation of rapid processes in the ring system (4) indicate that the shape of the particle size distribution is likely not primordial or a direct result of the ring-creating event. Rather, ring particles are believed to be involved in active accretion-destruction dynamics (5-13) and their sizes vary over a few orders of magnitude as a power law (14-17), with a sharp cutoff for large sizes (18-21). Moreover, tidal forces fail to explain the abrupt decay of the size distribution for house-sized particles (22). One wishes to understand the following: (i) Can the interplay between aggregation and fragmentation lead to the observed size distribution? And (ii) is this distribution peculiar for Saturn's rings, or is it universal for planetary rings in general? To answer these questions quantitatively, one needs to elaborate a detailed model of the kinetic processes in which the ring particles are involved. Here we develop a theory that quantitatively explains the observed properties of the particle size distribution and show that these properties are generic for a steady state, when a balance between aggregation and fragmentation holds. Our model is based on the hypothesis that collisions are binary and that they may be classified as aggregative, restitutive, or disruptive (including collisions with erosion); which type of collision is realized depends on the relative speed of colliding particles and their masses. We apply the kinetic theory of granular gases (23, 24) for the multicomponent system of ring particles to quantify the collision rate and the type of collision. Results and DiscussionModel. Ring particles may be treated as aggregates built up of primary grains (9) of a certain size r 1 and mass m 1 . [Observations indicate that particles below a certain radius are absent in dense rings (16).] Denote by m k = km 1 the mass of rin...

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Section: [4]mentioning

confidence: 99%

Size distribution of particles in Saturn’s rings from aggregation and fragmentation

Brilliantov

Krapivsky

Bodrova

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

129

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“…Above 0.5 mm, the curves had an exponential shape (decay), with an exponential distribution parameter of about 0.75 mm. For shorter lengths, the distributions followed a power-law with exponents close to −1.2 (Krapivsky and Ben-Naim 2003), down to very small fragments of 20 µm and below, thus of the fiber diameter scale. This exponent of -1.2 corresponds to the value 0.6 for the fragmentation parameter p. To get an idea of the significance of p, if one assumes that each fiber meets 3000 grits before fiber liberation, and this means that each single contact between a pulpstone grit and spruce wood cuts fiber in two parts with a probability of 2 × 10 −4 .…”

Section: Resultsmentioning

confidence: 93%

“…This study's result with an exponent of -1.2 for the small fragments differed from the expected exponent -3, which would be characteristic for steady source processes (Ben-Naim and Krapivsky 2000). The exponent -1.2 is peculiar to shattering fragmentation when the smaller of two colliding particles splits (Krapivsky and Ben-Naim 2003). According to the theoretical computations, the only asymptotically relevant parameter would be the feed rate , and these grinding trials revealed a faint correlation of feed rate with fiber length distribution.…”

Section: Resultsmentioning

confidence: 94%

Ground Wood Fiber Length Distributions

2014

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This study considers ground wood fiber length distributions arising from pilot grindings. The empirical fiber length distributions appear to be independent of wood fiber length as well as feeding velocity. In terms of mathematics the fiber fragment distributions of ground wood pulp combine an exponential distribution for high-length fragments and a power-law distribution for smaller lengths. This implies that the fiber length distribution is influenced by the stone surface. A fragmentation-based model is presented that allows reproduction of the empirical results.

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“…We can assume this event as the form of two successive steps: formation of an extremely unstable particle in a coagulation step that swiftly breaks into two or more smaller ones. Therefore, the reaction rate of this is taken to be proportional to the numbers of particle per unit volume of the colliding particles . Nonlinear breakage has significant impact in many engineering processes such as fluidized beds, the bulk distribution of raindrops, comminution systems, and several types of milling process .…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the reaction rate of this is taken to be proportional to the numbers of particle per unit volume of the colliding particles. 1 Nonlinear breakage has significant impact in many engineering processes such as fluidized beds, [2][3][4][5] the bulk distribution of raindrops, [6][7][8][9] comminution systems, [10][11][12][13] and several types of milling process. [14][15][16][17] Binary collisional breakage is the specific case of the nonlinear breakage, one where the breaking behavior is influenced through the sole of collisions between two particles.…”

Section: Introductionmentioning

confidence: 99%

An existence‐uniqueness result for the pure binary collisional breakage equation

Paul

Kumar

2018

Math Methods in App Sciences

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The study of collision‐induced breakage phenomenon in the particulate process has much current interest. This is an important process arising in many engineering disciplines. In this work, the existence of continuous solution of the pure collisional breakage model is developed beneath some restrictions on the breakage kernels. Furthermore, the mass conservation and uniqueness of solution are investigated in the absence of “shattering transition.” The underlying theory is based on the compactness result of Arzelà‐Ascoli and Banach contraction mapping principle.

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Shattering transitions in collision-induced fragmentation

Cited by 39 publications

References 33 publications

Size distribution of particles in Saturn’s rings from aggregation and fragmentation

Size distribution of particles in Saturn’s rings from aggregation and fragmentation

Ground Wood Fiber Length Distributions

An existence‐uniqueness result for the pure binary collisional breakage equation

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