Modeling from the nanoscale to the macroscale

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Modeling from the nanoscale to the macroscale

• 950 - 1105 T, Th
• Park Shops, Studio 2
• or over the internet by video streaming
• http//engineeringonline.ncsu.edu/onlinecourses/co
  ursehomepages/MSE791K.htm

Instructors
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Overview

• This course will will provide a broad survey of
  modern theory and modeling methods for predicting
  and understanding the properties of materials.
• In particular we will cover
• Commonly used theoretical and simulation methods
  for the modeling of materials properties such as
  structure, electronic behavior, mechanical
  properties, dynamics, etc. from atomistic,
  electronic models up to macroscale continuum
  simulations
• Quantum methods both at the first principles and
  semi-empirical level
• Classical molecular modeling molecular dynamics
  and Montecarlo methods
• Solid defect theory
• Continuum modeling approaches
• Lectures will be complemented by hands-on
  computing using publicly available or personally
  developed scientific software packages

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Goals and objectives

• Provide you with a basic background and the
  skills needed to
• Appreciate and understand the use of theory and
  simulation in research on materials science and
  the physics of nano- to macroscale systems
• Be able to read the simulation literature and
  evaluate it critically
• Identify problems in materials science/condensed
  matter physics amenable to simulation, and decide
  on appropriate theory/simulation strategies to
  study them
• Acquire the basic knowledge to perform a
  materials modeling task using state-of-the-art
  scientific software, analyze critically the
  results and draw scientific conclusions on the
  basis of the computational experiment.

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Course organization

• The course is structured in four different
  sections taught by different instructors, each of
  them expert in a particular area of the
  multi-scale modeling
• Grades for each part of the course will be
  determined by individual instructors, and will be
  based on projects, papers, etc. assigned by that
  instructor. The final course grade will be based
  on an average of these grades.

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Course organization
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Modeling and scientific computing

• Continuous advances in computer technology make
  possible the simulation approach to scientific
  investigation as a third stream together with
  pure theory and pure experiment
• Experiment primary concerned with the
  accumulation of factual information
• Theory mainly directed towards the
  interpretation and ordering of the information in
  coherent patters to provide with predictive laws
  for the behavior of matter through mathematical
  formulations
• Computation push theories and experiments beyond
  the limits of manageable mathematics and feasible
  experiments
• Properties of materials under extreme conditions
  (temperature, pressure, etc.)
• Study of properties of complex systems - a solid
  crystal is already an unmanageable system for a
  microscopic mathematical model!
• Testing of theories vs. experimental observation
• Suggestion of experiments for validation of the
  theory

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Modeling and scientific computing

• Steps to set up a meaningful computational model
• Individuate the physical phenomenon to study
• Develop a theory and a mathematical model to
  describe the phenomenon
• Cast the mathematical model in a discrete form,
  suitable for computer programming
• Develop and/or apply suitable numerical
  algorithms
• Write the simulation program
• Perform the computer experiment
• A good computational scientist has to be a little
  bit of
• A theorist, to to develop new approaches to solve
  new problems
• An applied mathematician, to be able to translate
  the theory in a mathematical form suitable for
  computation
• A computer scientist/programmer, to write new
  scientific codes or modify existing ones to fit
  the needs and deal with the always changing world
  of advanced and high-performance computing
• An experimentalist, to be able to define a
  meaningful path of computer experiments that
  should lead to the description of the physical
  phenomenon

A very demanding task!
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Multi-scale modeling

• Challenge modeling a physical phenomenon from a
  broad range of perspectives, from the atomistic
  to the macroscopic end

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Multi-scale modeling

• Ab initio methods calculate materials properties
  from first principles, solving the
  quantum-mechanical Schrödinger (or Dirac)
  equation numerically
• Pros
• Give information on both the electronic and
  structural/mechanical behavior
• Can handle processes that involve bond
  breaking/formation, or electronic rearrangement
  (e.g. chemical reactions).
• Methods offer ways to systematically improve on
  the results, making it easy to assess their
  quality.
• Can (in principle) obtain essentially exact
  properties without any input but the atoms
  conforming the system.
• Cons
• Can handle only relatively small systems, about
  O(102) atoms.
• Can only study fast processes, usually O(10) ps.
• Numerically expensive!

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Multi-scale modeling

• Semi-empirical methods use simplified versions
  of equations from ab initio methods, e.g. only
  treat valence electrons explicitly include
  parameters fitted to experimental data.
• Pros
• Can also handle processes that involve bond
  breaking/formation, or electronic rearrangement.
• Can handle larger and more complex systems than
  ab initio methods, often of O(103) atoms.
• Can be used to study processes on longer
  timescales than can be studied with ab initio
  methods, of about O(10) ns.
• Cons
• Difficult to assess the quality of the results.
• Need input from experiments or ab initio
  calculations and large parameter sets.

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Multi-scale modeling

• Atomistic methods use empirical or ab initio
  derived force fields, together with
  semi-classical statistical mechanics (SM), to
  determine thermodynamic (MC, MD) and transport
  (MD) properties of systems. SM solved exactly.
• Pros
• Can be used to determine the microscopic
  structure of more complex systems, O(104-6)
  atoms.
• Can study dynamical processes on longer
  timescales, up to O(1) ?s
• Cons
• Results depend on the quality of the force field
  used to represent the system.
• Many physical processes happen on length- and
  time-scales inaccessible by these methods, e.g.
  diffusion in solids, many chemical reactions,
  protein folding, micellization.

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Multi-scale modeling

• Mesoscale methods introduce simplifications to
  atomistic methods to remove the faster degrees of
  freedom, and/or treat groups of atoms (blobs of
  matter) as individual entities interacting
  through effective potentials.
• Pros
• Can be used to study structural features of
  complex systems with O(108-9) atoms.
• Can study dynamical processes on timescales
  inaccessible to classical methods, even up to
  O(1) s.
• Cons
• Can often describe only qualitative tendencies,
  the quality of quantitative results may be
  difficult to ascertain.
• In many cases, the approximations introduced
  limit the ability to physically interpret the
  results.

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Multi-scale modeling

• Continuum methods Assume that matter is
  continuous and treat the properties of the system
  as field quantities. Numerically solve balance
  equations coupled with phenomenological equations
  to predict the properties of the systems.
• Pros
• Can in principle handle systems of any
  (macroscopic) size and dynamic processes on
  longer timescales.
• Cons
• Require input (elastic tensors, diffusion
  coefficients, equations of state, etc.) from
  experiment or from a lower-scale methods that can
  be difficult to obtain.
• Cannot explain results that depend on the
  electronic or molecular level of detail.

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Multi-scale modeling

• Connection between the scales
• Upscaling
• Using results from a lower-scale calculation to
  obtain parameters for a higher-scale method. This
  is relatively easy to do deductive approach.
  Examples
• Calculation of phenomenological coefficients
  (e.g. elastic tensors, viscosities,
  diffusivities) from atomistic simulations for
  later use in a continuum model.
• Fitting of force-fields using ab initio results
  for later use in atomistic simulations.
• Deriving potential energy surface for a chemical
  reaction, to be used in atomistic MD simulations
• Deriving coarse-grained potentials for blobs of
  matter from atomistic simulation, to be used in
  meso-scale simulations

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Multi-scale modeling

• Connection between the scales
• Downscaling
• Using higher-scale information (often
  experimental) to build parameters for lower-scale
  methods. This is more difficult, due to the
  non-uniqueness problem. For example, the results
  from a meso-scale simulation do not contain
  atomistic detail, but it would be desirable to be
  able to use such results to return to the
  atomistic simulation level. Inductive approach.
  Examples
• Fitting of two-electron integrals in
  semiempirical electronic structure methods to
  experimental data (ionization energies, electron
  affinities, etc.)
• Fitting of empirical force fields to reproduce
  experimental thermodynamic properties, e.g.
  second virial coefficients, saturated liquid
  density and vapor pressure

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Multi-scale modeling an example
Behavior of carbon nanotubes under mechanical
deformations

• Carbon nanotubes are an excellent example of a
  physical system whose properties can be described
  at multiple length- and time-scales
• An example mechanical properties of nanotubes
  under deformation or tension

(Ruoff, PRL, 2000)
(Postma et al, Science 2001)
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Multi-scale modeling an example

• Carbon nanotubes under tension the ab initio
  results
• A relatively small nanotube
• A very short simulation time ( 1 ps)

Nanotubes break by first forming a bond rotation
5-7-7-5 defect.
Buongiorno Nardelli, Yakobson, Bernholc PRL 81,
4656 (1998)
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Multi-scale modeling an example

• Carbon nanotubes under tension the atomistic
  results
• Expansion to a larger system and longer
  simulation times allows exploration and discovery
  of new behaviors

For low strain values and high temperatures the
(5775) defect behaves as a dislocation loop made
up of two edge dislocations (57) and (75). The
two dislocations can migrate on the nanotube wall
through a sequence of bond rotations ? PLASTIC
BEHAVIOR
Buongiorno Nardelli, Yakobson, Bernholc PRL 81,
4656 (1998)
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Multi-scale modeling an example

• Carbon nanotubes under tension the atomistic
  results
• Expansion to a larger system and longer
  simulation times allows exploration and discovery
  of new behaviors

Under high tension and low temperature
conditions, additional bond rotations lead to
larger defects and cleavage ? BRITTLE BEHAVIOR
Buongiorno Nardelli, Yakobson, Bernholc PRL 81,
4656 (1998)
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Multi-scale modeling an example

• Carbon nanotubes under compression a mesoscale
  result
• Expansion to a mesoscopic system through finite
  elements methods. Parameters are modeled upon
  atomistic and ab initio calculations

Axial compression simulation of a 9 walls MWNT -
Stress in the axial direction (section view)
Pantano, Parks, Boyce and Buongiorno Nardelli,
submitted (2004)