Document Sample

1 CE 530 Molecular Simulation Lecture 24 Non-Equilibrium Molecular Dynamics David A. Kofke Department of Chemical Engineering SUNY Buffalo kofke@eng.buffalo.edu 2 Summary from Lecture 12 Dynamical properties describe the way collective behaviors cause macroscopic observables to redistribute or decay Evaluation of transport coefficients requires non-equilibrium condition • NEMD imposes macroscopic non-equilibrium steady state • EMD approach uses natural fluctuations from equilibrium Two formulations to connect macroscopic to microscopic • Einstein relation describes long-time asymptotic behavior • Green-Kubo relation connects to time correlation function Several approaches to evaluation of correlation functions • direct: simple but inefficient • Fourier transform: less simple, more efficient • coarse graining: least simple, most efficient, approximate 3 Limitations of Equilibrium Methods Response to naturally occurring (small) fluctuations Signal-to-noise particularly bad at long times • but may have significant contributions to transport coefficient here Finite system size limits time that correlations can be calculated reliably correlations between …lose meaning once these two… they’ve traveled the length of the system 4 Non-Equilibrium Molecular Dynamics Introduce much larger fluctuation artificially • dramatically improve signal-to-noise of response Measure steady-state response Corresponds more closely to experimental procedure • create flow of momentum, energy, mass, etc. to measure… • …shear viscosity, thermal conductivity, diffusivity, etc. Advantages • better quality of measurement • can also examine nonlinear response Disadvantages • limited to one transport process at a time • may need to extrapolate to linear response 5 One (Disfavored) Approach Introduce boundaries in which molecules interact with inhomogeneous momentum/mass/energy reservoirs Disadvantages • incompatible with PBC • introduces surface effects • inhomogeneous • difficult to analyze to obtain transport coefficients correctly Have a look with a thermal conductivity applet Better methods rely on linear response theory 6 Linear Response Theory: Static Linear Response Theory forms the theoretical basis for evaluation of transport properties by molecular simulation Consider first a static linear response Examine how average of a mechanical property A changes in the presence of an external perturbation f • Unperturbed value A 0 • Apply perturbation to Hamiltonian H H0 B( p N , q N ) d Ae 0 H B • New value of A A 0 A H0 B d e Susceptibility • Linearize (A) describes first-order AB A B 0 0 0 0 static response to perturbation 7 Example of Static Linear Response Dielectric response to an external electric field • coupling to dipole moment of system, Mx H E y M y (q N ) • interest in net polarization induced by field M y • thus A = B = My No field Field on E My 0 My 0 2 2 • susceptibility M y 0 My 0 8 Linear Response Theory: Dynamic 1. Time-dependent perturbation Fe(t) Consider situation in which Fe is non-zero for t < 0, then is switched off at t = 0 Response A decays to zero H 0 B A(t ) d A(t )e Ensemble average over (perturbation- H0 B d e weighted) initial conditions B (0) A(t ) 9 Linear Response Theory: Dynamic 2. Now consider a more general time-dependent perturbation Fe(t) Simplest general form of linear response t Value at time t is a sum of the A(t ) dt AB (t t ) Fe (t ) responses to the perturbation over the entire history of the system For the protocol previously discussed (shut off field at t = 0) 0 A(t ) dt AB (t t) d AB ( ) t • thus d AB ( ) B(0) A(t ) AB (t ) B(0) A(t ) t 10 t Perturbation-Response Protocols A(t ) dt B(0) A(t ) Fe (t ) Turn on perturbation at t = 0, and keep constant thereafter • measured response is proportional to integral of time-integrated correlation function 0 t A(t ) dt B(0) A(t ) Apply as -function pulse at t = 0, 0 subsequent evolution proceeding normally • measured response proportional to time correlation function itself 0 A(t ) B(0) A(t ) Use a sinusoidally oscillating perturbation • measured response proportional to Fourier- Laplace transformed correlation functions at the applied frequency 0 t • extrapolate results from several frequencies A(t ) dt eit B(0) A(t ) to zero-frequency limit 0 11 Synthetic NEMD Perturb usual equations of motion in some way • Artificial “synthetic” perturbation need not exist in nature For transport coefficient of interest Lij, Ji = LijXj • Identify the Green-Kubo relation for the transport coefficient Lij J i ( ) J j (0) d e.g., D vx ( ) vx (0) d 0 0 • Invent a fictitious field Fe, and its coupling to the system such that the dissipative flux is Jj H 0 J j Fe ad • ensure that equations of motion correspond to an incompressible phase space equations of motion are consistent with periodic boundaries equations of motion do not introduce inhomogeneities • apply a thermostat • couple Fe to the system and compute the steady-state average J i (t ) • then J i (t ) Lij lim lim Fe 0 t Fe 12 Phase Space Underlying development assumes that equations of motion correspond to an incompressible phase space q q p p 0 This can be ensured by having the perturbation derivable from a Hamiltonian H ne H A (p, q) f (t ) H ne q p / m Ap f (t ) p H ne p F(q) Aq f (t ) q Most often the equations of motion are not derivable from a Hamiltonian • but are still formulated to be compatible with an incompressible phase space 13 Diffusion: An Inhomogeneous Approach Artificially distinguish particles by “color” Introduce a species-changing plane Molecules moving this way across wall get colored red Those crossing this way get blue 14 Diffusion: An Inhomogeneous Approach Artificially distinguish particles by “color” Introduce a species-changing plane Considering periodic boundaries, this creates a color gradient Problems • Difficult to know form of inhomogeneity in color profile • Cannot be extended to multicomponent diffusion 15 Self-Diffusion: Perturbation Green-Kubo relation D vx ( ) vx (0) d rx ( ) vx (0) d 0 0 Label each molecule with one of two “colors” • each color given to half the molecules f Apply Hamiltonian perturbation f N H H 0 ci rix f (t ) i 1 New equations of motion q p/m pix Fix ci f (t ) p F(q) Aq f (t ) pi y , z Fi y , z System remains homogeneous 16 Self-Diffusion: Response Appropriate response variable is the “color current” 1 N J x (t ) ci vix (t ) V i 1 According to linear response theory t J x (t ) V ds J x (t s) J x (0) 0 f (s) f 0 f In the canonical ensemble 1 2 i j xi J x (t ) J x (0) c c v (t )vxj (0) V i, j 1 ci2 vxi (t )vxi (0) V2 i N vx (t )vx (0) V2 1 J x (t ) D lim lim Back to Green-Kubo relation t f 0 F 17 Thermostatting External field does work on the system • this must be dissipated to reach steady state Thermostat based on velocity relative to total current density • “peculiar velocity” pix pix ci ˆ 1 Nm c j p jx pix ci J x / m • constrain kinetic energy p2 / m 3NkT ˆ • modified equations of motion qi pi / m pi Fi e xci f pi ˆ • thermostatting multiplier mFi pi ˆ pi pi ˆ Shear Viscosity: 18 Boundary-Driven Algorithm Homogeneous algorithm for boundary-driven shear is possible • unique to shear viscosity Lees-Edwards shearing periodic boundaries (sliding brick) • Image cells in plane above and below central cell move dv • Image velocity given by shear rate dy x • Peculiar velocity of all images equal Ly L pix pix Ly ˆ Ly L Shear Viscosity: 19 Boundary-Driven Algorithm Homogeneous algorithm for boundary-driven shear is possible • unique to shear viscosity Lees-Edwards shearing periodic boundaries (sliding brick) • Image cells in plane above and below central cell move dv • Image velocity given by shear rate dy x • Peculiar velocity of all images equal pix pix Ly ˆ Shear Viscosity: 20 Boundary-Driven Algorithm Homogeneous algorithm for boundary-driven shear is possible • unique to shear viscosity Lees-Edwards shearing periodic boundaries (sliding brick) • Image cells in plane above and below central cell move dv • Image velocity given by shear rate dy x • Peculiar velocity of all images equal pix pix Ly ˆ Shear Viscosity: 21 Boundary-Driven Algorithm Homogeneous algorithm for boundary-driven shear is possible • unique to shear viscosity Lees-Edwards shearing periodic boundaries (sliding brick) • Image cells in plane above and below central cell move dv • Image velocity given by shear rate dy x • Peculiar velocity of all images equal pix pix Ly ˆ Try the applet 22 Lees-Edwards Boundary Conditions dvx L dy Molecule exiting here, in middle of central cell L 23 Lees-Edwards Boundary Conditions dvx L dy Is replaced by one here, shifted over toward the edge of L the cell Shift distance = Lt 24 Lees-Edwards Boundary Conditions dvx L dy And with a velocity that is modified according to the L shear rate 25 Lees-Edwards Boundary: API User's Perspective on the Molecular Simulation API Simulation Space Controller Phase Species Potential Display Device Integrator MeterAbstract Boundary Configuration 26 Lees-Edwards Boundary: Java Code public class Space2D.BoundarySlidingBrick extends Space2D.BoundaryPeriodicSquare public void nearestImage(Vector dr) { double delrx = delvx*timer.currentValue(); double cory; cory = (dr.y > 0.0) ? Math.floor(dr.y/dimensions.y+0.5):Math.ceil(dr.y/dimensions.y-0.5); dr.x -= cory*delrx; dr.x -= dimensions.x * ((dr.x > 0.0) ? Math.floor(dr.x/dimensions.x+0.5) : Math.ceil(dr.x/dimensions.x-0.5)); dr.y -= dimensions.y * cory; } public void centralImage(Coordinate c) { Vector r = c.r; double cory = (r.y > 0.0) ? Math.floor(r.y/dimensions.y) : Math.ceil(r.y/dimensions.y-1.0); double corx = (r.x > 0.0) ? Math.floor(r.x/dimensions.x) : Math.ceil(r.x/dimensions.x-1.0); if(corx==0.0 && cory==0.0) return; double delrx = delvx*timer.currentValue(); Vector p = c.p; r.x -= cory*delrx; r.x -= dimensions.x * corx; r.y -= dimensions.y * cory; p.x -= cory*delvx; } 27 Limitations of Boundary-Driven Shear No external field in equations of motion • cannot employ response theory to link to viscosity Lag time in response of system to initiation of shear • cannot be used to examine time-dependent flows A fictitious-force method is preferable 28 DOLLS-Tensor Hamiltonian: Perturbation An arbitrary fictitious shear field can be imposed via the DOLLS-tensor Hamiltonian N H H 0 qipi u(t ) T i 1 Equations of motion qi pi / m qi u pi Fi u pi • must be implemented with compatible PBC Example: Simple Couette shear 0 0 0 qi pi / m qiy e x u 0 0 pi Fi pixe y 0 0 0 29 DOLLS-Tensor Hamiltonian: Response Appropriate response variable is the pressure tensor 1 N 1 N P (t ) m pipi 2 rij Fij 1 V i 1 i, j According to linear response theory t P(t ) V ds P(t s)P(0) 0 u( s) 0 Shear viscosity, via Green-Kubo Pxy (t ) lim lim t 0 30 SLLOD Formulation DOLLS-tensor formulation fails in more complex situations • non-linear regime • evaluation of normal-stress differences • a simple change fixes things up SLLOD Equations of motion DOLLS qi pi / m qi u qi pi / m qi u pi Fi pi u pi Fi u pi Only change Example: Simple Couette shear 0 0 0 qi pi / m qiy e x qi pi / m qiy e x u 0 0 pi Fi piy e x pi Fi pixe y 0 0 0 Methods equivalent for irrotational flows u u T 31 Application NEMD usually introduces exceptionally large strain rates • 108 sec-1 or greater 1/ 2 ms 2 • dimensionless strain rate * • thus, e.g., m = 30g/mol; s = 3A; /k = 100K; * = 1.0 = 51011 sec-1 Shear-thinning observed even in simple fluids at these rates Very important to extrapolate to zero shear Newtonian *1/ 2

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 6 |

posted: | 3/23/2011 |

language: | French |

pages: | 31 |

OTHER DOCS BY sanmelody

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.