lecture 5 - university of michigan
TRANSCRIPT
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 1
Event-based or collision-basedmethods and
Brownian dynamics methods
Lecture 5
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 2
Tim
e Sc
ale
(sec
)
Length Scale
pico
Angstroms nanometers microns mm
AbInitio
metersfemto
nano
micro
millise
c MacroscaleSimulation
Electronic Structure - MO & DFTAb initio MDQuantum MC
MesoscaleSimulation Finite
element
ClassicalMolecularSimulation
Brownian dynamicsLattice BoltzmannCellular automataDPDDDFTMolecular dynamics
Monte Carlo
CFDMech
Simulating Across the Scales
The methods of computational materials science.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 3
Simulations of colloidal nanoparticles
❏ Colloidal particles suspended in solution (“colloids”)❖ Diameters can be nanometers to microns❖ E.g. PMMA, PS, silica, gold, ….❖ Effective interaction potential
• Can be charged, so “bare potential” is Coulomb 1/r– Screened by adding salt to solution, grafting short polymers
to surface, etc. (resulting interaction: Yukawa, etc.)• Also van der Waals: -1/r when summed
– Can be screened by matching indices of refraction betweenparticles and solvent
• If both interactions screened, only repulsive hard coreinteractions left.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 4
“MD” Simulations of Hard Spheres
❏ Discontinuity in potential❖ Potential not differentiable to get forces❖ Discontinuous change in velocities
❏ Cannot use traditional molecular dynamics
U(r)U(r) = 0 for r > σU(r) = ∞ for r ≤ σ
rσ
σ
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 5
“MD” Simulations of Hard Spheres
❏ Since no force acts on particles between collisions, thenbetween collisions a particle’s position changes by viδt, whereδt is the time between collisions.
❏ Event-based collision dynamics:
(a) Identify location and time of next collision(b) Calculate new positions of all particles at collision time(c) Implement collision dynamics for colliding pair
• Calculate new positions and velocities for colliding pairbased on conservation of linear momentum for elasticcollisions
(d) Return to (a)
cf. Allen and Tildesley, pp 105-106.EtomicaDemo
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 6
“MD” Simulations of Hard Spheres
❏ First “MD” simulation: Alder & Wainright 1957❖ Predicted the hard-sphere phase diagram prior to expts.
❏ Used for studying❖ Dense fluids and glasses❖ Nucleation and crystallization❖ Colloidal suspensions❖ Granular materials
0 0.494 0.545
0.58 0.64
0.74φ1 φ2
φg φrcp
φhcp
Fluid Fluid+Crystal
Crystal
Glass
Why do crystal phases form?
Dynamicsdifferent!
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 7
“MD” Simulations of Attractive Hard Spheres
❏ Square well potential (sticky hard spheres)
❏ Used to model colloidal nanoparticles with short-rangedelectrostatic interactions, which can be caused by depletionforces or surface treatment.
❏ “Stickiness” causes gelation of particles.
U(r)U(r) = 0 for r > σ2U(r) = ε for σ1 < r < σ2U(r) = ∞ for r ≤ σ1
r
σ1
σ2σ1ε
Also not differentiable.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 8
“MD” Simulations of Attractive Hard Spheres
❏ Algorithm similar to hard spheres, but now there are two“collision times” to calculate, one at σ1 and one at σ2.❖ Collisions at the inner sphere obey normal hard-sphere
dynamics.❖ Collisions at outer sphere follow conservation of energy as
well as momentum:• For particles approaching each other, U drops when r
< σ2, so KE increases.• For particles moving away from each other, there are
two possibilities:– If KE is sufficient, the particles cross the boundary with a
loss in KE to compensate the increase in U.– If KE is not sufficient, reflection occurs at σ2 and the
particles remain “bound”.
EtomicaDemo 1
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 9
Hard Sphere Model
❏ The hard sphere and attractive hard spheremodel may also be solved using Monte Carlomethods.
❏ Both discontinuous MD and MC methodsgive the same phase diagram (e.g. samethermodynamics), but kinetics could bedifferent.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 10
Brownian dynamics simulationsof nanoparticles
❏ In many instances of solute-solvent systems, the behavior ofthe solute is desired, and the behavior of the solvent isuninteresting (e.g. proteins, DNA, dendrimers, nanoparticles insolution.)
❏ Instead of modeling solvent explicitly (by individualmolecules), include its effects implicitly, and use appropriate,simplified force fields to describe interactions between solute.
❏ For nano-objects suspended in solvent, diffusive motion(Brownian dynamics) typically observed in quiescent state:stochastic dynamics.
Reference: Allen and Tildesley is good place to start.Also: Grest, et al JCP 105, 10583 (1996)
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 11
Brownian Dynamics Simulations of Nanoparticles
❏ To model nanoparticle-nanoparticle interaction, can usepotential of mean force appropriate to system.
❏ The solvent influences the dynamics of the nanoparticles viarandom collisions, and by imposing a frictional drag force onthe motion of the nanoparticle in the solvent.
❏ Stochastic dynamics models incorporate these two effects viathe Langevin equation of motion.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 12
Brownian dynamics simulations
❏ In Brownian dynamics, the force on a particle is assumed tocome from three sources:❖ Conservative potential of mean force Fc between particles❖ Random collisions❖ Drag force
❏ Conservative force: Fc depends only on distance betweenparticles (e.g. soft sphere 1/r12, vdW 1/r, etc.)
❏ Random force: R(t) due to random fluctuations caused byinteractions with solvent molecules, which bombard theparticle constantly on all sides, giving rise to Brownian motionof the particle.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 13
❏ Drag force: Frictional drag force on a particlearises from motion of particle in solvent.❖ Ffrictional = - ξ v❖ Friction coefficient ξ is related to collision frequency γ of
molecules in the solvent by γ = ξ / m.❖ 1/ γ is the velocity autocorrelation time associated with the
solvent.❖ For a spherical particle of radius a, ξ is related to viscosity
of solvent by Stokes’ law:
❖ The frictional force is then
!
" = 6#$a
!
F frictional = 6"a#v
Brownian dynamics simulations
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 14
Brownian dynamics method
❏ “Particle”-based MD-like simulation method❖ One type of Langevin dynamics method❖ Reproduces canonical [NVT] ensemble.❖ Equation of motion for particle positions
contains Newtonian and stochastic terms.
Usual NVE MD Addt’l stochastic termsLangevin equation
Total Force Frictional Force
Conservative Force Random Force
!
mi
d2xi(t)
dt2
= Fi{x
i(t)}" #
i
dxi(t)
dtm
i+ R
i(t)
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 15
Brownian dynamics method
❏ Stochastic terms❖ If solvent present, included implicitly through stochastic
terms:• Drag force• Random force
NB: Momentum not conserved, so hydrodynamics not fullyincluded.
❖ For a neat system (no solvent), stochastic terms represent a(non-momentum-conserving) thermostat with heat sourceand sink. Kremer and Grest 1996
Total Force Frictional Force
Conservative Force Random Force
!
mi
d2xi(t)
dt2
= Fi{x
i(t)}" #
i
dxi(t)
dtm
i+ R
i(t)
Usual NVE MD Addt’l stochastic terms
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 16
Brownian dynamics method
❏ Stochastic terms❖ Assumes no spatial or temporal correlations (i.e. Gaussian
white noise process).❖ Allows access to longer time scales than MD.
• No explicit solvent• Friction term dampens equation of motion, allows larger time
step
❏ Implementation❖ Particle positions and velocities updated each
time step by calculating forces acting on eachparticle due to other particles, and due to stochastic terms.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 17
❏ One method of implementation is called the DirectForce method.❖ Calculate conservative force on i due to all particles j.❖ Calculate frictional force on i.❖ Calculate random force on i such that it satisfies the
fluctuation dissipation theorem:
❖ Use any MD integration scheme (e.g. leapfrog or velocityVerlet) to calculate new position and velocity of particle i.
!
Fi
Rt( )F j
R " t ( ) = 6kBTmi# i$ij$ t % " t ( )
Fi
Rt( ) = 0 spatially
uncorrelatedtemporally uncorrelated
Brownian dynamics method
Random number (zeromean, unit variance)
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 18
!
vi( t+"t) = c
ovi(t)+ c
1ai( t)"t+"vR
!
"xR = A "xR( )2
!
"vR = "vR( )2
cxvA+ B 1# c
xv
2( )( )
!
xi(t + "t) = x(t) + c
1vi(t)"t + c
2ai(t)"t 2 + "xR
!
cxv"x"v
= #xR#vR =kT
$m1% e%$#t( )
2
Brownian dynamics method
❏ Or use direct integration method via leapfrog (or velocity verlet):
❏ Random position and random velocity are calculated via thefollowing equations with “A” and “B” random numbers of zeromean and unit variance generated from a bivariate Gaussiandistribution.
Use Box-Mueller toget two Gaussiannumbers at once
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 19
MD vs. BD
❏ For the right problems, BD wins in three ways:❖ Eliminating many atoms and including them implicitly
means far fewer computations per time step.• If solvent not important, then BD may be a good choice.
❖ Can choose timestep δt roughly 2-3 times larger than inMD due to dissipative term, because damping termstabilizes the equations of motion.
❖ Because in BD the fastest frequency motions in the realsystem are replaced by stochastic terms, δt is now chosento resolve the slower degrees of freedom, and thus δt isseveral hundred times larger than in MD.• Thus BD starts at picoseconds and can access into
microseconds, but for effectively larger systems than couldbe done with MD.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 20
BD Issues
❏ Simple BD contains no hydrodynamic interactions.❏ Excluded volume effects of solvent not included.❏ Not trivial to implement drag force for non-spherical
particles.❏ Solvent molecules must be small compared to
smallest molecules explicitly considered.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 21
Examples
❏ BD for a monoatomic mixture❖Motion diffusive, except at high density
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 22
Example: Binary Mixture Phase Diagram
T
φφ1 φs1 φs2 φ2
one phase(miscible)
For each T < Tc, plot φ1, φ2,
φs1, and φs2.
This defines boundaries between (i) a miscible and immiscible regionand (ii) a metastable and unstable immiscible region.
two-phase(immiscible)
binodal or coexistence curve
spinodal
unstable metastable
UCST
Demo Glotzilla
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 23
BD simulations for nanoscience
❏ Example of a Brownian dynamics simulation❖Tethered nanospheres❖Tethered nanorods❖Tethered nanotriangles
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 24
Model tethered nano building blocks
❏ Use a coarse-grainedrepresentation whereby agroup of atoms in thenanoparticle or tether isreplaced by a single “atom”❖ Reduces force calculations❖ Retains nanoscale roughness
❏ Empirical pair potentialsbetween “atoms”❖ van der Waals interactions❖ Excluded volume interactions
❏ Minimal model❖ Thermodynamic immiscibility❖ Geometrical constraints
Our goal: discover trends and construct general design strategies.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 25
Minimal model of a tethered nanoparticle
❏ Spherical nanoparticles modeled as spheres.Non-spherical nanoparticles modeled as smallerspheres rigidly bound to one another.
❏ Polymer tethers modeled as bead-spring chains of Nmonomers connected by FENE springs.
❏ Tethers connected to specific “atoms” on thenanoparticles via a FENE spring.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 26
Minimal model of a tethered nanoparticle
❏ In neutral solvent, “atoms” or monomers of the sametype interact via 12-6 LJ potential.
❏ “Atoms” or monomers of different type: WCA.❏ Solvent selectivity modeled by describing interactions
between favored species via WCA and those betweenunfavored species via LJ.
❏ Degree of immiscibility and solvent quality determinedby reciprocal temperature ε/kBT.
Computational Nanoscience of Soft [email protected]
ChE/MSE 557 Lecture 5 October 24, 2006 27
Increasing level of detail
❏ Ab initio computations of portions of a nanocrystal ornanostructured molecule near solvent to getinteraction energies.
❏ Classical, explicit-atom MD simulations of severalnanostructured molecules or portions ofnanocrystals with explicit solvent to obtain preferreddistances, orientations, etc.
❏ Use this info to parameterize mesoscopic interactionpotentials.