Lattice, Quasi-continuum

1
Lattice, Quasi-continuum Phase
Transitions

• Slava Sorkin

Department of Aerospace Engineering and
Mechanics, University of Minnesota

2
Acknowledgements

• Ellad B. Tadmor, University of Minnesota,
  Aerospace Engineering and Mechanics
• Ryan Elliot, University of Minnesota, Aerospace
  Engineering and Mechanics
• Emil Polturak, Physics Department, Technion
• Joan Adler, Physics Department, Technion
• Noam Bernstein, Center for Computational
  Materials Science and Technology
  Division Naval Research Laboratory Washington
• Gábor Csányi, University of Cambridge,
  Engineering Laboratory
• Mitchell Luskin, University of Minnesota, School
  of Mathematics
• David Ceperley, National Center for
  Supercomputing Applications at Illinois
• Matteo Cococcioni, University of Minnesota,
  Chemical Engineering and Materials Science
• David Landau, UGA Physics and Astronomy School

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Part I Multi-scale modelling of materials

• The object of multi-scale modelling is to predict
  behaviour of materials using theoretical and
  computational techniques that link across spatial
  and temporal scales.
• This approach can be considered as an alternative
  to the empirical methods of today. It offers
  great opportunities for tomorrow technological
  advances.

The series of images shows details of the crack
tip at the different scales. The atomic-scale
mechanism leads to fracture can be seen. This
picture compliments of the Naval Research Lab
Washington, DC

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Quasicontinuum (QC) method

• The QC method is one of the best possible
  strategies devised to couple concurrently micro
  and macro scales.
• This techniques is a mixed continuum and
  atomistic approach for simulating the mechanical
  response of crystalline materials. With QC one
  can reproduce the results of full atomistic
  calculation at a fraction of computational cost.

A crack tip approaching a grain boundary in a
nickel bi-crystal. For frames are shown at
increasing level of external load. The snapshots
show dislocation emission from the grain
boundary, followed by crack extension, and,
finally, grain boundary migration toward the
crack tip. Taken from www.qcmethod.com

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Quasicontinuum method

• The energies of the 'representative' atoms are
  calculated based on their environment either by
  using atomistic methodology, or as befitting to
  a continuum model. The total energy is
  calculated without any assumptions beyond the
  form of inter-atomic potentials.
• With a knowledge of the total energy one can
  study mechanical response of crystalline material
  to external load. This can be done by minimizing
  the total energy with respect to the
  displacements of the 'representative' atoms.
• Currently, one of the most important direction
  for our research is to extend the QC method from
  simple to complex lattices.

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Complex lattices

• The extension of the QC method to complex
  lattices permits the study of many
  technologically important materials such as
  semiconductors, ferroelectrics, and shape-memory
  materials.
• Unit cell of complex lattices contains more than
  one basis atom per Bravais lattice site. In
  general, complex lattice can be described as a
  set of inter-penetrating sub-lattices with the
  same lattice vectors, but different origin
  positions.

Adapted from http//www.molecularexpression.com

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Complex Lattices

• When a uniform macroscopic deformation is
  applied, all the sub-lattices undergo the same
  uniform deformation, but in addition they can
  slide relative each other. Therefore, to describe
  complex lattices we increase the number of
  degrees of freedom to include sub-lattice
  displacements into account.
• The equilibrium configuration is now obtained by
  minimizing the total energy with respect to the
  node and sub-lattice displacements concurrently.

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Uni-axial Stress

• To test the new approach we applied uniaxial
  stress to the NiTi crystal with a cubic lattice
  structure. An essential unit cell containing two
  basis atoms was initially chosen.
• The external load was gradually incremented until
  the first phonon instability was detected. At
  this point the two basis atom unit cell was
  replaced by a four basis atom unit cell. The
  subsequent minimization led to a drastic increase
  in the nominal strain new stable a-IrV phase
  was identified. The old approach failed to
  reproduce the proper transition.
• This simple test illustrates necessity of
  flexible description of the underlying lattice
  for correct modelling of solid-to-solid
  transformations.

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Application

• Using the extended QC method we study properties
  of NiTi shape-memory alloy. The shape memory
  alloy 'remembers' its shape it can be returned
  to that shape after being deformed, by applying
  heat to the alloy.
• The shape-memory effect is due to
  temperature-dependent phase transformation from a
  low-symmetry martensite to a high-symmetry
  austenite.
• In order to validate the capability of the QC
  method to reproduce the shape memory effect we
  decided to simulate the shape memory cycle
  cooling deformation - heating.

http//everythang.wordpress.com
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Shape memory cycle

• Elliot's temperature dependent potentials were
  used to model the prototypical NiTi alloy. A two
  basis atom unit cell was used to describe the
  initial austenite structure of the alloy.
• At the first stage temperature of the sample was
  gradually reduced. At each step the energy
  minimization was accompanied by phonon stability
  analysis.
• When the temperature reached the critical value
  T/Tref 0.667, the phonon instability was
  detected, and the unit cell was extended to
  include four basis atoms. The subsequent
  minimization led to the transformation from the
  austenite phase to the martensite phase. This
  martensite phase contained both left (blue color)
  and right (red color) oriented variants.

(a) Austenite structure of the sample at T/Tref
0.8 (b) Sample is cooled down to T/Tref 0.667
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Shape memory cycle

• Next, we sheared the sample by displacing the
  top row nodes to the right. The sample
  temperature was kept constant T/Tref 0.65 at
  this stage.
• As the shear progressed all left-oriented
  martensite elements reversed their orientation.
  The process started at the bottom and propagated
  upward. The simulation was terminated when all
  elements are right-oriented.
• Finally, we heated up the sample to T/Tref 1.1
  at this temperature a reverse martensite-to-auste
  nite transformation occurred and the original
  shape of the sample was recovered.
• In conclusion, we demonstrated the capability of
  the extended QC method to simulate shape-memory
  cycle.

(c) The sample is deformed at T/Tref 0.65 (d)
The sample is heated up to T/Tref 1.1 and then
cooled down to T/Tref 0.8

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Part II Mechanical Melting

• Melting is a fundamental process, but despite
  its common occurrence, understanding this process
  is still a challenge.
• Over the years, several theories explaining the
  mechanism of melting have been proposed. This
  research has evolved to a state where two
  possible scenarios exist the first scenario of
  mechanical melting resulting from lattice
  instability, and the second scenario of
  thermodynamic melting which begins at a free
  surface or at an internal interface (grain
  boundary, void).

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Mechanical Melting

• Mechanical melting occurs when the crystal loses
  its ability to resist shear. This rigidity
  catastrophe is caused by vanishing one of the
  elastic shear moduli. At this point the crystal
  expands up to a critical specific volume, which
  is close to that of the melt. This condition
  determines the mechanical melting temperature Ts
  of a bulk crystal as it was confirmed in
  extensive studies of FCC metals.
• The critical volume at which FCC metals melt is
  independent of the path through phase space by
  which it is reached whether one heats the
  perfect crystal or adds point defects to expand
  the solid at a constant temperature.
• Our aim was to verify whether this scenario of
  mechanical melting developed for FCC crystals is
  also applicable to crystals with BCC lattice
  structure.

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Method and Model

• Mechanical melting transition of vanadium was
  modeled using molecular dynamics (MD)
  simulations.
• Since we are interested in the generic features
  of metallic solids with a BCC structure, the
  choice of vanadium has no special significance.
• The many-body interaction potential developed by
  Finnis and Sinclair (FS) was chosen to simulate
  vanadium

http//www.neyco.fr/images/vanadium.jpg
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Geometry and Boundary Conditions

• The samples used for the simulations contained
  2000 atoms, initially arranged as a perfect BCC
  crystal.
• We introduced point defects to the samples either
  by insertion of extra atoms (self
  interstitials) or by removal of atoms from the
  lattice (vacancies).
• Since solids can undergo mechanical melting only
  if they have no free surfaces, periodic boundary
  conditions were applied in all three directions.

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Melting Transition

• We carried out our simulations using samples with
  various concentrations of point defects.
• The initial temperature was chosen far below the
  expected melting point. The samples were
  gradually heated, and at some point we observed
  an abrupt decrease of the order parameter,
  together with a simultaneous increase of the
  specific volume and the total energy.
• This event determined the mechanical melting
  temperature, Ts 2500 K.

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Results

• We found that once point defects are introduced,
  the melting temperature becomes a function of
  their concentration, which has been confirmed
  experimentally.
• Using the dependency of shear modulus C' (C11
  C12)/2 on specific volume we extracted the value
  of the critical volume at which the system
  melts.
• Our results show that the Born model of melting
  applies equally to BCC and FCC metals in both
  the nominally perfect state and in the case where
  point defects are present.

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Part III Thermodynamic melting

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Thermodynamic melting

• The mechanical melting transition cannot be
  observed in the laboratory since it is preempted
  by the thermodynamic melting transition. Long
  before the melting temperature is reached a thin
  quasiliquid layer appears at the free surface.
  Numerous experiments and computer simulations
  confirm that FCC metals start to melt from the
  surface.
• Our primary motivation was to answer the question
  whether premelting phenomena, extensively studied
  for FCC metals, are also present in BCC metals.
  In addition, our goal was to calculate the
  thermodynamic melting temperature, since
  the temperature Ts 2500 K at which mechanical
  melting occurs is far above the experimental
  value Tm 2183K.

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Simulation details

• We modelled the thermodynamic melting transition
  of vanadium with a free surface using MD
  simulations in canonical ensemble. The same FS
  many-body potential was applied for vanadium.
• A crystal with a surface was modelled as a thick
  slab with the fixed bottom layers to mimic the
  presence of the infinite bulk. On top of those
  layers there were 24 layers in which atoms are
  free to move. Periodic boundary conditions were
  imposed along the in-plane (x and y) directions.

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Simulation Details

• Three different samples with various low-index
  surfaces were constructed V(001), V(011) and
  V(111). All the samples contained about 3000
  atoms initially arranged a perfect BCC crystal.
• Each simulations started from a low-temperature
  solid, and then the temperature was gradually
  raised up to a specific value. At this
  temperature the samples were equilibrated.
• The structural, transport, and energetic
  properties were measured in the thermal
  equilibrium at various temperatures up to the
  melting point.

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Results

• We found that the surface region of the
  least-packed V(111) surface began to disorder
  first via generation of defect pairs and the
  formation of an additional layer at temperature
  above T 1000 K. At higher temperatures, the
  surface region became quasiliquid.
• This process began above T 1600 K for the
  V(001) surface.
• For the closest-packed V(011), this effects was
  observed only in the close proximity to the
  melting temperature.

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Results

• We determined the thermodynamic melting
  temperature of vanadium as Tm 2220 K, in a
  good agreement with experimental value Tm 2183
  K.
• The results of our simulations of surface
  premelting of the BCC metal, vanadium, are
  similar to the results obtained for various FCC
  metals, in the sense that the onset of disorder
  is seen first at the surface with the lowest
  density.

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The End

• For help on the Israel Inter-University
  Computation Center supercomputers thanks to Dr.
  Moshe Goldberg, Dr. Anne Weill, Gabi Koren and
  Jonathan Tal
• For help on the Minnesota Supercomputing
  Institute machines thanks to Dr. Haoyu Yu, Dr.
  Shuxia Zhang and Dr. Benjamin Lynch.