aleksandar donev - nyu courantdonev/flucthydro/nanoblob.handout.pdf · coupling a nano-particle...

Coupling a nano-particle with fluctuating hydrodynamics

Aleksandar Donev

Courant Institute, New York University&

Pep Espanol, UNED, Madrid, Spain

Advances in theory and simulation of non-equilibrium systemsNESC16, Sheffield, England

July 2016

A. Donev (CIMS) Coarse Blob 7/2016 1 / 21

Levels of Coarse-Graining

Figure : From Pep Espanol, “Statistical Mechanics of Coarse-Graining”.


Velocity Autocorrelation Function

Key is the velocity autocorrelation function (VACF) for theimmersed particle

C (t) = 〈V(t0) · V(t0 + t)〉

From equipartition theorem C (0) =⟨V 2⟩

= d kBT/M for acompressible fluid, but for an incompressible fluid the kinetic energyof the particle that is less than equipartition.

Hydrodynamic persistence (conservation) gives a long-timepower-law tail C (t) ∼ (kBT/M)(t/tvisc)−3/2.

Diffusion coefficient is given by the integral of the VACF and is hardto compute in MD even for a single nanocolloidal particle.


Brownian Bead

Classical picture for the following dissipation process: Push a spheresuspended in a liquid with initial velocity Vth ≈

√kBT/M and watch

how the velocity decays:

Sound waves are generated from the sudden compression of the fluidand they take away a fraction of the kinetic energy during a sonic timetsonic ≈ a/c, where c is the (adiabatic) sound speed.Viscous dissipation then takes over and slows the particlenon-exponentially over a viscous time tvisc ≈ ρa2/η, where η is theshear viscosity.Thermal fluctuations get similarly dissipated, but their constantpresence pushes the particle diffusively over a diffusion timetdiff ≈ a2/D, where

D ∼ kBT/(aη) (Stokes-Einstein relation).


Timescale Estimates

The mean collision time is tcoll ≈ λ/Vth ∼ η/(ρc2),

tcoll ∼ 10−15s = 1fs

The sound time

tsonic ∼{

1ns for a ∼ µm1ps for a ∼ nm

, with gaptsonic

tcoll∼ 102 − 105

It is often said that sound waves do not contribute to the long-timediffusive dynamics because their contribution to the VACF integratesto zero.


Estimates contd...

Viscous time estimates

tvisc ∼{

1µs for a ∼ µm1ps for a ∼ nm

, with gaptvisc

tsonic∼ 1− 103

Finally, the diffusion time can be estimated to be

tdiff ∼{

1s for a ∼ µm1ns for a ∼ nm

, with gaptdiff

tvisc∼ 103 − 106

which can now reach macroscopic timescales!

In practice the Schmidt number is very large,

Sc = ν/D = tdiff /tvisc � 1,

which means the diffusive dynamics is overdamped.


Brownian Dynamics

Overdamped equations of Brownian Dynamics (BD) for the particlepositions R (t) are

dR

dt= MF + (2kBTM)

12 W(t) + kBT (∂R ·M) , (1)

where M(R) � 0 is the symmetric positive semidefinite (SPD)hydrodynamic mobility matrix.Hydrodynamic mobility matrix is given by Green-Kubo formula

(kBT )Mij =

∫ τ

0dt 〈Vi (0)·Vj (t)〉eq . (2)

The upper bound τ must satisfy

τ �r2ij

ν∼ L2

ν� tvisc ,

so that the whole VACF power law tail is included in the integral.Therefore computing hydrodynamic interactions is infeasiblewith MD.


Hydrodynamic Diffusion Tensor

Since computing the hydrodynamic mobility is so difficult in MD,usually M is modeled by the Rotne-Prager mobility [1],

Mij ≈ η−1

(I +

a2

6∇2

r

)(I +

a2

6∇2

r′

)G(r − r′)

∣∣r=qj

r′=qi.

where G is the Green’s function for the Stokes problem (Oseentensor for infinite domain).

This is not only an approximate closure neglecting a number ofeffects, but also requires an estimate of the effective hydrodynamicradius a as input.

Our goal will be to split the integral into a short-time piece,computed by feasible MD via Green-Kubo integrals, and a long-timecontribution, computed by fluctuating hydrodynamics coupled to animmersed particle.


“Old” approach: Particle/Continuum Hybrid

Figure : Hybrid method for a polymer chain.


VACF using a hybrid

Split the domain into a particle and acontinuum (hydro) subdomains [2].

Particle solver is a coarse-grained fluidmodel (Isotropic DSMC).

Hydro solver is a simple explicit(fluctuating) compressiblefluctuating hydrodynamics code.

Time scales are limited by the MDpart despite increased efficiency.


Small Bead (˜10 particles)

10.10.01 10

t / tvisc

1

0.1

0.01

0.001

M C

(t)

/ kBT

Stoch. hybrid (L=1)Det. hybrid (L=1)Stoch. hybrid (L=2)Det. hybrid (L=2)Particle (L=1)Theory

10.1 10

t cs / R

1

0.5

0.75

0.25

tL=1


Large Bead (˜1000 particles)

0.01 0.1 1

t / tvisc

1

0.1

0.01

M C

(t)

/ kBT

Stoch. hybrid (L=2)Det. hybrid (L=2)Stoch. hybrid (L=3)Det. hybrid (L=3)Particle (L=2)Theory

0.01 0.1 1t cs / R

1

0.75

0.5

0.25

tL=2


New Approach: Fluctuating Hydrodynamics

Figure : Coarse-Graining a Nanoparticle: Schematic representation of ananoparticle (left) surrounded by molecules of a simple liquid solvent (in blue).The shaded area around node µ located at rµ is the support of the finite elementfunction ψµ(r) and defines the hydrodynamic cell (right).


Notation

Define an orthogonal set of basis functions,

||δµψν || = δµν , (3)

where ||f || ≡∫drf (r).

Continuum fields which are interpolated from discrete “fields”:

ρ(r) = ψµ(r)ρµ (4)

Introduce a regularized Dirac delta function

∆(r, r′) ≡ δµ(r)ψµ(r′) = ∆(r′, r) (5)

Note the exact properties∫drδµ(r) = 1,

∫dr rδµ(r) = rµ (6)∫

dr′∆(r, r′)δµ(r′) = δµ(r)


Slow variables

Key to the Theory of Coarse-Graining is the proper selection of therelevant or slow variables.

We assume that the nanoparticle is smaller than hydrodynamiccells and accordingly choose the coarse-grained variables [3],

R (z = {q,p}) = q0, (7)

We define the mass and momentum densities of the hydrodynamicnode µ according to

ρµ(z) =N∑

i=0

miδµ(qi ), discrete of ρr(z) =N∑

i=0

miδ(qi − r)

gµ(z) =N∑

i=0

piδµ(qi ), discrete of gr(z) =N∑

i=0

piδ(qi − r)

where i = 0 labels the nanoparticle. Note that both mass andmomentum densities include the nanoparticle!


Final Discrete (Closed) Equations

dR

dt= v(R)− D0

kBT

∂F∂R

+D0

kBTFext +

√2kBTD0 W(t)

dρµdt

= ||ρ v ·∇δµ||

dgµdt

= ||g v·∇δµ||+ kBT∇δµ(R)− ||δµ∇P||+ δµ(R)Fext

+ η||δµ∇2v||+(η

3+ ζ)||δµ∇ (∇·v) ||+ d gµ

dt(8)

The pressure equation of state is modeled by

P(r) ' c2

2ρeq

(ρ(r)2 − ρ2

eq

)+ m0

(c20 − c2)

ρeq∆(R, r)ρ(r), (9)

and the gradient of the free energy is modeled by

∂F∂R' m0

(c20 − c2)

ρeq

∫dr∆(R, r)∇ρ(r). (10)


Final Continuum Equations

The same equations can be obtained from a Petrov-Galerkin discretizationof the following system of the fluctuating hydrodynamics SPDEs

d

dtR =

∫dr∆(r,R)v(r) +

D0

kBTFext +

√2kBTD0 W(t)

− D0

kBT

m0(c20 − c2)

ρeq

∫dr∆(R, r)∇ρ(r)

∂tρ(r, t) = −∇·g∂tg(r, t) = −∇·(gv)− kBT∇∆(r,R)

−∇P(r) + Fext∆(r,R)

+ η∇2v +(η

3+ ζ)∇ (∇·v) + ∇·Σαβ

r (11)

where v = g/ρ, and the pressure is given by

P(r) =c2

2ρeq

(ρ(r)2 − ρ2

eq

)+

m0(c20 − c2)

ρeq∆(R, r)ρ(r) (12)


Diffusion Coefficient

The scalar bare diffusion coefficient is grid-dependent,

D0 =1

d

∫ τ

0dt⟨δV(0)·δV(t)

⟩eq

(13)

where the particle excess velocity over the fluid is

δV = V −⟨

V⟩Rρg

≈ V − v(R).

The crucial point is that now the integration time τ � h2/ν, where his the grid spacing, is accessible in MD.

The true or renormalized diffusion coefficient [4] should begrid-independent,

D = D0 + ∆D ≈ D0 +1

d

∫ τ

0dt 〈v(R(0))·v(R(t))〉eq

≈ D0 +1

d

∫ ∞0

dt ψµ(R)⟨vµ(0)·vµ′(t)

⟩eqRψµ′(R)


Numerical VACF

10-4

10-3

10-2

10-1

100

101

t / tν

0.2

0.4

0.6

0.8

1C

(t)

/ (kT

/m)

Rigid spherec=16c=8c=4c=2c=1Incompress.

10-1

100

101

10-2

100

Figure : VACF for a neutrally buoyant particle for D0 = 0 and c = c0, fromcoupling a finite-volume fluctuating hydrodynamic solver [5, 6].


Conclusions

We considered the problem of modeling the Brownian motion of asolvated nanocolloidal particle over a range of time scales.Hydrodynamic scales are not accessible in direct MD socoarse-grained models are necessary.If one eliminates the solvent DOFs one obtains a long-memorynon-Markovian SDE in the inertial case or a long-ranged overdampedSDE in the Brownian limit.If fluctuating hydrodynamic variables are retained in thedescription, one obtains a large system of Markovian S(P)DEs.A concurrent hybrid coupling approach couples MD directly tofluctuating hydrodynamics; time scales are limited by the MD.We derive coarse-grained equations by a combination of Mori-Zwanzigwith physically-informed modeling.It remains to actually try this in practice and see what range ofeffects can be captured correctly and efficiently.It also remains to generalize this to a denser suspension of colloids.


References

S. Delong, F. Balboa Usabiaga, R. Delgado-Buscalioni, B. E. Griffith, and A. Donev.

Brownian Dynamics without Green’s Functions.J. Chem. Phys., 140(13):134110, 2014.Software available at https://github.com/stochasticHydroTools/FIB.

A. Donev, J. B. Bell, A. L. Garcia, and B. J. Alder.

A hybrid particle-continuum method for hydrodynamics of complex fluids.SIAM J. Multiscale Modeling and Simulation, 8(3):871–911, 2010.

P. Espanol and A. Donev.

Coupling a nano-particle with isothermal fluctuating hydrodynamics: Coarse-graining from microscopic to mesoscopicdynamics.J. Chem. Phys., 143(23), 2015.

A. Donev, T. G. Fai, and E. Vanden-Eijnden.

A reversible mesoscopic model of diffusion in liquids: from giant fluctuations to Fick’s law.Journal of Statistical Mechanics: Theory and Experiment, 2014(4):P04004, 2014.

F. Balboa Usabiaga, R. Delgado-Buscalioni, B. E. Griffith, and A. Donev.

Inertial Coupling Method for particles in an incompressible fluctuating fluid.Comput. Methods Appl. Mech. Engrg., 269:139–172, 2014.Code available at https://github.com/fbusabiaga/fluam.

F. Balboa Usabiaga, X. Xie, R. Delgado-Buscalioni, and A. Donev.

The Stokes-Einstein Relation at Moderate Schmidt Number.J. Chem. Phys., 139(21):214113, 2013.


https://github.com/stochasticHydroTools/FIB

https://github.com/fbusabiaga/fluam

aleksandar donev - nyu courantdonev/flucthydro/nanoblob.handout.pdf · coupling a nano-particle...

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