introduction dynamics computer simulations of …condensed matter dynamics d.quigley 3rd year...

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)

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University of York


D.Quigley


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



Langevin Dynamics

Introduction

14/04/2005Slide 3 of 28

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D.Quigley


Introduction

What are we trying to do?

Conclusions

Future Work



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

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D.Quigley


Models

Langevin Dynamics



Conclusions

Future Work


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

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D.Quigley


EnsemblesQuantities we can calculate depend on the ensemble for which the configurations were generated.

Langevin Dynamics



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

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D.Quigley


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



Langevin Dynamics


Smooth trajectory in which time has a clear interpretation.

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University of York


D.Quigley


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



Langevin Dynamics


Discontinuous trajectory –time has no clear meaning.

14/04/2005Slide 8 of 28

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D.Quigley


Energy LandscapesModel defines an energy landscape.

Langevin Dynamics



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

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D.Quigley


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



Conclusions

Future Work


T>Tt

T=Tt

T<Tt

Phase 1 Phase 2

14/04/2005Slide 10 of 28

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D.Quigley


Non-Hamiltonian Systems

Conclusions

Future Work



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!

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D.Quigley


Langevin Dynamics

Conclusions

Future Work



Langevin Dynamics


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.

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Controlling Pressure

Conclusions

Future Work



Langevin Dynamics


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).

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D.Quigley


How well does it work?

Conclusions

Future Work



Langevin Dynamics


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.

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D.Quigley


Graphite

Conclusions

Future Work



Langevin Dynamics


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.

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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



Langevin Dynamics


Introduction

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D.Quigley


Snap!

Conclusions

Future Work



Langevin Dynamics


Introduction

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D.Quigley



Conclusions

Future Work



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.

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Tracing liquid-vapour line

Conclusions

Future Work



Langevin Dynamics

Introduction •Histogram reweighting and multicanonical sampling

•Useful for transitions with a critical point.

•Automated method for tracing a phase boundary.

Gas

Liquid

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Locating Melting Curves

Conclusions

Future Work



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.

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D.Quigley


Crystal Structure

Conclusions

Future Work



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!

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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


–Why?–Solid–Liquid


Langevin Dynamics

Introduction

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D.Quigley


The DFT Slide

Conclusions

Future Work




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.

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D.Quigley


Zero TemperatureStudy structure under pressure using CASTEP.

Conclusions

Future Work




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

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D.Quigley


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




Langevin Dynamics

Introduction

Seems metallic at finite temperature!

•More atoms?

•Anisotropic NPT?

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D.Quigley


HPCSimulations of liquid require large cells – expensive!

Conclusions

Future Work




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

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Liquid

Conclusions

Future Work




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

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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



Langevin Dynamics

Introduction

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University of York


D.Quigley


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



Langevin Dynamics

Introduction

introduction dynamics computer simulations of …condensed matter dynamics d.quigley 3rd year...

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