Fragmentation

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Fragmentation

• Heli Hietala
• Teoreettisen fysiikan laudatur-seminaari
• 18.9.2007

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1 Introduction

• imagine you drop a plate or a vase onto the
  kitchen floor
• it stays intact
• it gets damaged
• it fragments into several pieces
• it shatters completely

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1.1 Motivation

• any system can be fragmented when the imparted
  energy is larger than the cohesive energy of the
  system
• a ubiquitous phenomenon in nature
• rocks, asteroids, planetesimals
• industrial importance
• building, mining,oil shale
• physics
• out-of-equilibrium process
• connections to turbulence, earthquakes

• liquid droplets, heavy atomic nuclei

• pharmaceuticals, chemicals, food

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1.2 Basic concepts

• fragment size distribution ns
• complementary cumulative distribution
• size of the largest fragment, second largest,
  etc.
• moments of the size distribution

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1.3 Scaling laws

• fragment size distribution is usually observed to
  follow a power law
• if so, the cumulative size distribution
• the observed t vary from 1 to around 2

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1.4 Open questions

• fragmentation may depend on
• form, size and material of the object
• energy used in the breaking
• the way the object is broken
• etc.
• answers simulations
• easy determination of fragment sizes
• toying with parameters

Kun and Herrmann (1996)
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2 Modelling the problem

• analytical models
• geometric fragmentation
• binary fission
• renormalization group techniques
• applying theoretical ideas

Grady and Kipp (1985)
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2.1 Few words on percolation
(Stauffer and Aharony, Introduction to
Percolation Theory, (1994).)

• bond percolation
• two lattice cites connected with a bond that is
  active with probability p
• for p gt pc, a large cluster percolates through
  the system
• pc depends on the lattice type
• e.g. pc 0.5 for square lattice
• scaling near the critical point
• fragmentation of microscopic systems
• heavy nuclei and liquid droplets (Campi et al.
  2000)

p 0.3
p 0.525
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2.2 Phase transition approach
(following the ideas presented by F. Kun and H.J.
Herrmann (1999).)

• size of the largest fragment smax
• decreases as the impact velocity increases
• size of the second largest fragment smax2
• has a maximum
• same for m2
• implies divergence when N -gt 8
• by analogy to percolation theory critical
  velocity Vc (?)
• 2nd order phase transition?

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2.3 Conceptual problems

• power law behaviour for all V gt Vc
• and not just near Vc
• alternatives to 2nd order phase transition
• some (known) out-of-equilibrium phase transition?
• self-organised criticality (SOC)?
• ????

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2.4 Energy dependence

• Does the scaling exponent depend on the impact
  energy?
• simple analytical models
• one exponent, depends on the dimension of the
  space
• most simulations, few experiments
• exponent stays virtually constant
• most experiments and few simulations (Ching et
  al. (1999), Moukarzel et al. (2007))
• exponent increases with increasing energy input
• But what is the exact energy dependence?

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3 A simple numeric model
(N. Sator, S. Mechkov, F. Sausset and H. Hietala)

• meso-scale model
• a large disc of 1000 pieces
• launched into a wall with velocity V control
  parameter
• motion simulated with the Molecular Dynamics
  method (MD)
• interaction between the pieces
• fragments self-bound clusters
• irreversible fractures

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3.1 Adding disorder

• sample the eij from a probability distribution
• vary the amount of disorder by varying the
  standard deviation
• for this work, we chose the gamma distribution
• possible to vary standard deviation independently
  of the mean

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3.2 Simulations

• Basic steps of the simulations (1)-(3)
• Statistics
• create 4-5 discs with the same parameter of
  disorder
• each disc is launched into the wall 500 times at
  different angles

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4 Results
Ordered system
Disordered system
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4.1 Qualitative differences

• ordered system
• smooth fractures
• smooth fragments
• disordered system
• rugged fractures
• fragments with complex surface structure

Order t 1
t 20
Disorder t 1
t 20
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4.2 Composite power law for all V gt Vc
and not just for
Standard deviation 0.7, impact velocity 2.0.
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4.3 Scaling exponent

• scaling exponent in the large fragment region t
  has a logarithmic dependence on the impact
  velocity
• the slope is independent of disorder
• the curve deviates from the line for smaller
  velocity the more ordered the system is
• experimentally observed by Matsui et al. (1982)
  while studying planetesimal collisions (rock
  impacts)

dev 0.7 dev 0.2 order
Matsui et al. (1982)
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5 Summary and prospects

• key ideas
• fragmentation occurs everywhere in nature
• it is not known which parameters it depends on
• fragment size distribution is observed to follow
  a power law -gt scaling
• there might be a phase transition
• in this work
• observed a critical velocity Vc (?)
• fragment size distribution follows a composite
  power law for V gt Vc
• found a logarithmic dependence on the impact
  velocity of the large fragment scaling exponent,
  with a slope independent of disorder
• future work
• consider other probability distributions -gt
  universality?
• analyze the accumulated data with the method of
  maximum likelihood