Classical simulations with particles

Additional Info

  • ECTS credits: 6
  • Semester: 1
  • University: Gdansk University of Technology
  • Prerequisites:

     

    Basic concepts of classical physics -- force, acceleration, potential. Basic knowledge of calculus (Riemann's
    integral, minimisation of a function, partial and total derivatives).

  • Objectives:

     

    We introduce the basics of physics of materials, with particular attention to the relationships between atomic
    structure and macroscopic physical properties. Classical computational particle methods are covered, mainly
    the molecular dynamics (MD) approach -- its basic theory (integration of e.o.m.) and practicalities (potentials,
    boundary conditions, initialisation, neighbourhood, cut-off radius) followed by a brief tour of more advanced
    concepts of MD (rigid molecules, shell model, constrained dynamics, thermostats, barostats, Ewald method).

  • Topics:

     

    Section devoted to physics of materials:

    Crystalline and glassy materials (short-range, medium-range and long-range order, radial and angular
    distribution functions); thermodynamics of phase transitions; glass transition; gels (classification and
    applications); quasicrystals; liquid crystals; auxetics.
    Basic concepts of crystallography (Bravais lattice, primitive and elementary cell, simple and complex
    lattice, Miller indices, etc.); symmetry operations; crystallographic point groups and space groups; models of
    amorphous systems (CRN, RCP, random-coil); reciprocal lattice and its properties; conditions
    for Bragg’s diffraction and Laue diffraction.
    Bonding in crystals (ionic, covalent, metallic, molecular and hydrogen); binding energies (lattice sums,
    Madelung energy, the Evjen method and Ewald method); fluctuation-dissipation effects.
    Structural defects: point defects (Schottky, Frenkel, substitutions, vacancies, intercalations); line defects
    (screw and edge dislocations, Frank network, mechanisms of dislocation generation, relationship with the
    strength of materials), planar defects (low-angle boundaries, stacking faults, twinning).
    Defects in the electronic structure (plasmons, excitons, polarons, magnons, F-centers).
    Lattice vibrations (mono- and diatomic chain, optical and acoustic branches, dispersion relations); normal
    vibrations; models of lattice heat capacity (classical, Einstein, Debye); the most significant
    anharmonic effects.
    Principles of the Drude model, electrical conductivity of metals, magnetoresistive effect and the Hall
    effect.
    The Fermi gas of free electrons, the Fermi-Dirac distribution, Fermi level and chemical potential, degenerate
    and non-degenerate gas, density of states, Wiedemann-Franz law.
    Thermoemission and cold emission from metal to vacuum; contact voltage.
    The model principles of the band theory; Bloch’s theorem; classification of solids on the basis of the band
    theory; effective mass and quasi-momentum.
    Dependence of electrical conductivity on temperature in semiconductors and metals (due to changes
    in the carrier densities and in the relaxation time). Deviations from Ohm’s law (collisional ionisation, Zener
    effect, Poole-Frenkel effect, field dependence of relaxation time).
    Section devoted to the molecular dynamics method:
    Motivation behind computational approaches to nanotechnology, continuum and particle methods, classical
    and quantum-based methods, scaling of computational effort.
    The molecular dynamics method, its advantages and limitations. Conservation of energy in Newtonian
    mechanics. Phase space and trajectories.
    Periodic, open and mixed boundary conditions, minimum image convention, quasiinifinity, limitations of
    PBCs. Cut-off radius and its consequences. Hockney’s linked cells and Verlet neighbour list.
    Initializing an MD simulation (positions, velocities), skew start, equilibration.
    Integration of the equations of motion. Verlet, leapfrog and predictor-corrector methods. Sources of error in
    integrating the equations of motion.
    Visualization in MD, calculating macroscopic quantities (energy, temperature, virial, pressure, specific heat,
    RDF, ADF, S(k), MSD, D(T)).
    Potential and its relationship with force. General and particular forms of potentials. Selected potentials: LJ,
    Born-Mayer, harmonic, Morse, Stillinger-Weber, Sutton-Chen, GAFF, AMOEBA).

    Polarizability and shell models (Cochran, Fincham).
    Constrained dynamics, formal approach, SHAKE, RATTLE, QSHAKE.
    (Optionally): Rigid molecules in MD simulations, Euler angles, rotation matrix, vector transformations,
    quaternions.
    Coulombic interactions in MD, Ewald method.
    NVT and NpT ensembles, primitive thermostats, ESM and CSM thermo- and barostats: Andersen,
    Berendsen, Hoover-Evans, Nose-Hoover, Nose-Andersen, Parrinello-Rahman.
    (Optionally): Hybrid (QM/MM) methods.

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