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14/04/2005Slide 1 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Computer simulations of novel systems using constant pressure
Langevin dynamics and Monte-Carlo methods.Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction
Or
“101 Uses for a Binary Supercomputer”
(sorry)
14/04/2005Slide 2 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Contents• Introduction
– Modelling atomic systems, Monte-Carlo and Molecular Dynamics– Phase transitions
• Langevin Dynamics at Constant Pressure– Theory of diffusion-drift in non-Hamiltonian systems– Simulations using model systems ( stretching nanotubes )
• Core-Softened Fluids – model with unusual properties– Locating the melting line– Locating the liquid-vapour line– Finding crystal structures
• High Pressure Iodine – Ab-initio simulation of a ‘real’ system– Constant Pressure Langevin Dynamics simulations of solid– Liquid-Liquid phase transition?
• Future Work and Conclusions
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction
14/04/2005Slide 3 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Introduction
What are we trying to do?
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction–Aims–Models–Tools–Phases
Obtain information on novel properties of materials using atom-scale computer simulation.
Statistical Mechanics
Construct a model for atomic interactions
Generate a number of sample configurations with the appropriate probability
Averages over configurations are equal to
averages in bulk
Specifically interested in liquid-liquid phase transitions in single component systems
14/04/2005Slide 4 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Models
Langevin Dynamics
Core-Softened Fluids
High Pressure Iodine
Conclusions
Future Work
Introduction–Aims–Models–Tools–Phases
Statistically Useful?
AccuracyCPU TimeModel
Pair-Potential
Bond-Order Potential
Tight-Binding
?Density Functional Theory
XMany Body PT / GW
XQMC
•Trade-off between accuracy and speed
Incr
easi
ng
Dec
reas
ing
14/04/2005Slide 5 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
EnsemblesQuantities we can calculate depend on the ensemble for which the configurations were generated.
Langevin Dynamics
Core-Softened Fluids
High Pressure Iodine
Conclusions
Future Work
Introduction–Aims–Models–Tools–Phases Micro-canonical (NVE) Ensemble – Fixed
volume, energy and particle number
Canonical (NVT) Ensemble. Coupled to heat-bath at temperature T. Energy fluctuates
Isobaric-Isothermal (NPT) Ensemble. Pressure now regulated. Volume fluctuates
Grand-Canonical (µVT) Ensemble. Particle number fluctuates.
14/04/2005Slide 6 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Molecular Dynamics•Integrate Newton’s equations numerically, using forces calculated from the model in question.
•Samples configuration and velocity space. 6N dimensional phasespace.
•BUT – Newton’s equations sample NVE ensemble only. Must be modified for NVT or NPT.
•Resulting NVT/NPT dynamics are fictitious, but (hopefully) sample phase space with correct probability.
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction–Aims–Models–Tools–Phases
Smooth trajectory in which time has a clear interpretation.
14/04/2005Slide 7 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Monte-Carlo• Make random trial displacements in the degrees of freedom.
• Accept/reject with probability for the appropriate ensemble.
• Samples configuration space only.
• Generally more efficient than molecular dynamics for sampling configurations due to shorter correlation “times” and hence moreindependent sampling.
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction–Aims–Models–Tools–Phases
Discontinuous trajectory –time has no clear meaning.
14/04/2005Slide 8 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Energy LandscapesModel defines an energy landscape.
Langevin Dynamics
Core-Softened Fluids
High Pressure Iodine
Conclusions
Future Work
Introduction–Aims–Models–Tools–Phases 2D in this case.
System will have one axis per degree of freedom, i.e. many dimensions.
free energy is ensemble dependent
NVE
NVTNPT
µVT
Hamiltonian i.e. KE + PEHelmholtz free energy
Gibbs free energy
Free energy
14/04/2005Slide 9 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Phase Transitions•Our simulations explore the phase/configuration space around a minimum of free energy with correct Boltzmann probability. •Temperature and pressure change the free energy landscape.
•BUT – our simulations only generate a small number of configurations, hence high energy ( low probability ) barrier isnever traversed. System remains in meta-stable state.•Locating phase transitions can hence be difficult.
Langevin Dynamics
Core-Softened Fluids
High Pressure Iodine
Conclusions
Future Work
Introduction–Aims–Models–Tools–Phases
T>Tt
T=Tt
T<Tt
Phase 1 Phase 2
14/04/2005Slide 10 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Non-Hamiltonian Systems
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
–Theory–Simulation
IntroductionConsider Hamilton’s equations (NVE)
To explore NVT or NPT ensembles we must modify these e.g. Nosé-Hoover thermostat for NVT
•In some cases, these equations cannot be derived from anyHamiltonian and do not always obey Boltzmann statistics.
•Equal energy configurations may no longer be equally probable –must generalise statistical mechanics in order to cope!
14/04/2005Slide 11 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Langevin Dynamics
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
–Theory–Simulation
Introduction The simplest form of Langevin dynamics
p
r
Ri
Diffusing away from
constant energy
trajectory
p
r
-γpiDrifting
toward p=0 axis
Random force
Damping constant, relaxation time is 1/γ
By balancing diffusion and drift (Stokes-Einstein relation) we sample the NVT ensemble. Hence Langevin dynamics simulates coupling to a heat bath.
14/04/2005Slide 12 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Controlling Pressure
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
–Theory–Simulation
Introduction The following equations of motion regulate pressure with isotropic fluctuations in the simulation cell.
+ similar system for anisotropic fluctuations( matrix equations )
Both are non-Hamiltonian
Questions
•Can we perform Langevin dynamics in this system?•Will the resulting dynamics correctly sample the NPT ensemble?
Answers
•Yes and Yes – after much consideration of Langevin dynamics in non-Hamiltonian systems. See DQ&MIJP J.Chem.Phys (2004).
14/04/2005Slide 13 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
How well does it work?
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
–Theory–Simulation
Introduction In some cases Nosé-Hoover based NPT schemes sample phase space at a much slower rate than our Langevin dynamics scheme.
Example: Both runs seeking 600 K, 5 MPa with equal relaxation times – No prior knowledge of dynamics.
NPT Langevin dynamics Conventional Nosé-Hoover based scheme.
14/04/2005Slide 14 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Graphite
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
–Theory–Simulation
Introduction
T
•Using bond-order potential for carbon
•First model large system at NVT•Calculate memory function ξ.
•Estimate parameters for NPT run.
•NPT Langevin dynamics run samples phase space correctly and efficiently.
14/04/2005Slide 15 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Carbon Nanotubes
40
41
42
43
44
45
46
47
48
49
50
-3500 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500Pressure (MPa)
Leng
th (B
ohr)
41.655
41.66
41.665
41.67
41.675
41.68
41.685
-15 -10 -5 0 5 10 15
Pressure (MPa)
Leng
th (B
ohr)
Scalar observable – tube length vs Pressure
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
–Theory–Simulation
Introduction
14/04/2005Slide 16 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Snap!
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
–Theory–Simulation
Introduction
14/04/2005Slide 17 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Core-Softened Fluids
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction Perturbation on well studied Lennard-Jones pair-potential.
Add a 2nd Gaussian well at r0 with depth Aand width w.
•Phase behaviour of these fluids is not well understood.
•Significant debate in the Literature.
•Possible liquid-liquid phase transition.
•Evidence for water-like density anomaly.
•Different groups use different strength perturbations. We hope to map phase behaviour as function of perturbation.
14/04/2005Slide 18 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Tracing liquid-vapour line
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction •Histogram reweighting and multicanonical sampling
•Useful for transitions with a critical point.
•Automated method for tracing a phase boundary.
Gas
Liquid
14/04/2005Slide 19 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Locating Melting Curves
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction
•Enforce a high energy interface between solid and liquid.
•Requires prior knowledge of crystal structure.
Liquid
Solid
System is able to access both phases – will melt or freeze depending on T&P.
14/04/2005Slide 20 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Crystal Structure
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction
FCC
Simple Cubic
Diamond
FCCSC
Diamond
α-Hg
?
Need crystal structure on melting line. Energy-volume curves at zero T are a starting point.
Unperturbed Lennard-Jones
A Core-softened potential. Many competing structures!
14/04/2005Slide 21 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Iodine
Experimental phase diagram indicates three liquid phases!
•What are these phases?
•How does the M-I transition fit in with that in the solid?
Solid at zero pressure consists of dimers arranged in layered sheets. Force between the sheets is weak.
Conclusions
Future Work
High Pressure Iodine
–Why?–Solid–Liquid
Core-Softened Fluids
Langevin Dynamics
Introduction
14/04/2005Slide 22 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
The DFT Slide
Conclusions
Future Work
High Pressure Iodine
–Why?–Solid–Liquid
Core-Softened Fluids
Langevin Dynamics
Introduction
BUT – we don’t know what all the bits of E[n] are! Details of DFT are in how we cheat to get round this. Write
To model our system quantum-mechanically we must minimise
w.r.t Ψ. Can then calculate forces e.t.c. for MD.
This is a HUGE problem even for a single atom!
Instead work with density n – function of 3 variables only
Hohenberg-Kohn Theorem
Where ψ are the solutions of a single particle Schrödingerequation yielding the same density as the real system.
14/04/2005Slide 23 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Zero TemperatureStudy structure under pressure using CASTEP.
Conclusions
Future Work
High Pressure Iodine
–Why?–Solid–Liquid
Core-Softened Fluids
Langevin Dynamics
Introduction
Metallisation occurs at 10 GPa, c.f. 18 GPa in experiment. Classic DFT band-gap under-estimation.
2 GPa
72 k-points
Ecut-off = 320 eV
12 nodes of erik
8 GPa
10 GPa
12 GPa
14 GPa
16 GPa
14/04/2005Slide 24 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Finite Temperature•Constant pressure Langevin dynamics simulations.•6 GPa and 900 K – experimentally an insulator.•Ensure temperature and volume distributions converged as graphite case earlier.
•Plot density of states as distribution of eigenvalues over simulation snapshots – compare to smeared zero T case.
Conclusions
Future Work
High Pressure Iodine
–Why?–Solid–Liquid
Core-Softened Fluids
Langevin Dynamics
Introduction
Seems metallic at finite temperature!
•More atoms?
•Anisotropic NPT?
14/04/2005Slide 25 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
HPCSimulations of liquid require large cells – expensive!
Conclusions
Future Work
High Pressure Iodine
–Why?–Solid–Liquid
Core-Softened Fluids
Langevin Dynamics
Introduction
GOLDILOCS ( DQ code for MD/MC )
6–8 nodes of erik~ 30 s / step~ 1000 atoms Bond-order pot
Desktop PC ~ 2-3 s / step~ 100 atoms Bond-order pot
3-4 nodes of erik~ 10 s / step~ 10,000 atomsPair potential
Desktop PC< 1 s / step~ 1000 atoms Pair-potential
CASTEP ( DFT code by MIJP + others )
128+ nodes of HPCx
~ 10 min / step~ 100 atoms
8-12 nodes of erik
~ 1 min / step~ 10 atoms
14/04/2005Slide 26 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Liquid
Conclusions
Future Work
High Pressure Iodine
–Why?–Solid–Liquid
Core-Softened Fluids
Langevin Dynamics
Introduction•72 atom unit cell.
•Superheated solid at constant volume to beyond point of thermal instability.
•10 minutes/time-step on 128 CPUs.
Found that L´´ is atomic liquid. Need to simulate liquid L also – expected to consist of diatomic molecules.
•Constant pressure MD now useful on this scale – MIJP.
•Need some way of beating SOLID-L or L´´-L energy barrier.
Free energy augmentation?
Brute force using mixed ab-initio / semi-empirical model?
Run 1
Run 2
14/04/2005Slide 27 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Future Work
• Further testing of constant pressure Langevin Dynamics scheme.– More ab-initio simulations– Band-gaps + other properties at high temperature
• Continued mapping of core-softened fluid phase diagram.– Meta dynamics – Thermodynamic integration– Analysis of MD data for 2nd order effects– Moving position of 2nd minimum
• Simulations of Iodine– Constrained dynamics of the solid M-I transition– Simulation of the low pressure liquid L.
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction
14/04/2005Slide 28 of 28
University of York
Condensed Matter Dynamics
D.Quigley
3rd Year Graduate Seminar
Conclusions
• A method for performing Langevin dynamics at constant pressure has been developed and tested.
• Phase diagram mapping using both MD and MC methods is underway for a family of core-softened model systems with interesting properties.
• Investigation of unusual phase behaviour in iodine using ab-initio molecular dynamics is progressing.
Conclusions
Future Work
High Pressure Iodine
Core-Softened Fluids
Langevin Dynamics
Introduction
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