Post on 01-Apr-2018
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”.
A. Donev (CIMS) Coarse Blob 7/2016 2 / 21
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.
A. Donev (CIMS) Coarse Blob 7/2016 3 / 21
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).
A. Donev (CIMS) Coarse Blob 7/2016 4 / 21
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.
A. Donev (CIMS) Coarse Blob 7/2016 5 / 21
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.
A. Donev (CIMS) Coarse Blob 7/2016 6 / 21
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.
A. Donev (CIMS) Coarse Blob 7/2016 7 / 21
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.
A. Donev (CIMS) Coarse Blob 7/2016 8 / 21
“Old” approach: Particle/Continuum Hybrid
Figure : Hybrid method for a polymer chain.
A. Donev (CIMS) Coarse Blob 7/2016 9 / 21
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.
A. Donev (CIMS) Coarse Blob 7/2016 10 / 21
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
A. Donev (CIMS) Coarse Blob 7/2016 11 / 21
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
A. Donev (CIMS) Coarse Blob 7/2016 12 / 21
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).
A. Donev (CIMS) Coarse Blob 7/2016 13 / 21
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)
A. Donev (CIMS) Coarse Blob 7/2016 14 / 21
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!
A. Donev (CIMS) Coarse Blob 7/2016 15 / 21
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)
A. Donev (CIMS) Coarse Blob 7/2016 16 / 21
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)
A. Donev (CIMS) Coarse Blob 7/2016 17 / 21
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)
A. Donev (CIMS) Coarse Blob 7/2016 18 / 21
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].
A. Donev (CIMS) Coarse Blob 7/2016 19 / 21
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.
A. Donev (CIMS) Coarse Blob 7/2016 20 / 21
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.
A. Donev (CIMS) Coarse Blob 7/2016 21 / 21