lattice, quasicontinuum & phase transitions...noam bernstein, center for computational materials...
TRANSCRIPT
Lattice, Quasicontinuum & Phase Transitions
Slava Sorkin
Department of Aerospace Engineering and Mechanics, University of Minnesota
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
Part I: Multiscale modelling of materials● The object of multiscale 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 atomicscale
mechanism leads to fracture can be seen. This
picture compliments of the Naval Research Lab
Washington, DC
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 bicrystal. 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
Quasicontinuum method ● The main idea of the QC approach is a
selective representation of atomic degrees of
freedom. A small relevant subset of all atoms
is chosen to represent, by proper waiting, the
total energy of the system as whole.
● The density of 'representative' atoms is
significant in highly deformed regions
(dislocation core, crack tip, free surfaces,
grain boundaries), and significantly reduced
in less deformed regions further away. The
fully atomistic regions can evolve with the
deformation, during the simulation. Selection of representative atoms from
all the atoms near a dislocation core.
Taken from: ”The QC method: overview, application and current directions”,
by R. Miller and E. B. Tadmor, JCAMD, 9, 203
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 interatomic 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.
Complex lattices
Adapted from
http://www.molecularexpression.com
● The extension of the QC method to complex lattices permits the study of many technologically important materials such as semiconductors, ferroelectrics, and shapememory 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 interpenetrating sublattices with the same lattice vectors, but different origin positions.
Complex Lattices
● When a uniform macroscopic deformation is applied, all the sublattices 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 sublattice displacements into account.
● The equilibrium configuration is now obtained by minimizing the total energy with respect to the node and sublattice displacements concurrently.
Complex Lattices ● To run a QC simulation one has to specify the lattice
description by choosing a unit cell. Although this
unit cell can be selected in many possible ways, the
smallest (essential) unit cell is the typical choice.
● This choice can sometimes be too restrictive: some
configurations cannot be described by deforming
this essential unit cell.
● To overcome the restriction we decided to replace
the essential unit cell by a nonessential one 'onfly',
if necessary.
● During the simulation the phonon spectrum is
calculated, and if at least one negative frequency is
detected we select a new unit cell.
essential unit cell
nonessential unit cells
Uniaxial 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 α
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 solidtosolid transformations.
Application ● Using the extended QC method we study
properties of NiTi shapememory 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 shapememory effect is due to temperaturedependent phase transformation from a lowsymmetry martensite to a highsymmetry 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
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
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 leftoriented martensite elements
reversed their orientation. The process started at the bottom
and propagated upward. The simulation was terminated when
all elements are rightoriented.
● Finally, we heated up the sample to T/Tref = 1.1; at this
temperature a reverse martensitetoaustenite transformation
occurred and the original shape of the sample was recovered.
● In conclusion, we demonstrated the capability of the extended
QC method to simulate shapememory 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
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).
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.
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 manybody interaction potential developed
by Finnis and Sinclair (FS) was chosen to
simulate vanadium:
http://www.neyco.fr/images/vanadium.jpg
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.
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.
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.
Part III: Thermodynamic melting
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.
Simulation details
● We modelled the thermodynamic melting
transition of vanadium with a free surface
using MD simulations in canonical
ensemble. The same FS manybody
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 inplane (x and y) directions.
Simulation Details ● Three different samples with various lowindex
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.
Results● We found that the surface region of the
leastpacked 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 closestpacked V(011), this effects was observed only in the close proximity to the melting temperature.
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.
The End
● For help on the Israel InterUniversity 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.