hybrid particle simulations: an approach to the spatial multiscale problem guang lin, pnnl thanks:...

Hybrid Particle Simulations: An approach to the spatial multiscale problem

Guang Lin, PNNL

Thanks: George Em Karniadakis, Brown UniversityIgor Pivkin

Vasileios SymeonidisDmitry Fedosov

Wenxiao Pan

Upscaling Approach: MD -> DPD -> Navier StokesHybrid Approach: MD + DPD + Navier Stokes

2

Field Scale

Mesoscale

Pore Scale

Microscopic Scale

Molecular Scale

Flow in heterogeneous porous media. (Darcy flow)

Multiphase flow and transport in complex fractures.

(Navier-Stokes flow)

M 1>1

Lif t

Rough S kin

S moot h S kin

G. Lin, et al, PRL, 2007

Numerical Modeling Methods

DPD

SPH

Upscaling Approach: MD -> DPD -> Navier StokesHybrid Approach: MD + DPD + Navier Stokes

Platelet Aggregation

arterioles venules

activated platelets,red blood cells,fibrin fibers

•Platelet diameter is 2-4 µm•Normal platelet concentration in blood is

300,000/mm3

•Functions: activation, adhesion to injured walls, and other platelets

activated platelets

•Myocardial platelet micro-thrombus

•Thrombocytopenic haemorrhages in the skin

Outline

Dissipative Particle Dynamics (DPD)

Biofilaments: DNA, Fibrin

Platelets

Red Blood Cells

Triple-Decker: MD-DPD-NS

V

2

V

1

F1dissipative

F2dissipative

Equilibrium: Static Fluid

Symeonidis, Caswell & Karniadakis, PRL, 2005

Shear Flow

Symeonidis, Caswell & Karniadakis

Dissipative Particle Dynamics (DPD)

• First introduced by Hoogerbrugge & Koelman (1992)• Coarse-grained version of Molecular Dynamics (MD)

MD DPD Navier-Stokes

• MICROscopic level approach

• atomistic approach is often problematic because larger time/length scales are involved

• continuum fluid mechanics

• MACROscopic modeling

• set of point particles that move off-lattice through prescribed forces

• each particle is a collection of molecules

• MESOscopic scales• momentum-conserving Brownian dynamics

Upscaling Approach: MD -> DPD -> Navier StokesHybrid Approach: MD + DPD + Navier Stokes

Conservativefluid / system dependent

Dissipative frictional force, represents viscous resistance within the fluid – accounts for energy loss

forces exerted by particle J on particle I:

Randomstochastic part, makes up for lost degrees of freedom eliminated after the coarse-graining ri

rj

rij i

j

Pairwise Interactions

Fluctuation-dissipation relation:

σ2 = 2 γ κΒΤ ωD= [ ωR]2

MDDPD

Conservative ForceConservative Force

From Forrest and Sutter, 1995

Soft potentials were obtained by averaging the molecular field over the rapidly fluctuating motions of atoms during short time intervals.

This approach leads to an effective potential similar to one used in DPD.

aij

•Linear force

Upscaling ApproachThe Conservative Force Coefficient: aij

The value of aij is determined by matching the dimensionless compressibility1,2 of the DPD system with that of the MD system.

11 11

MDm

TMDDPD

MD

BTDPDBDPD N

p

Tk

p

Tk

Tk

aaTkp

B

DPDijDPDDPDijBDPD

2.01 1.0 12

From the work of Groot and Warren1, we know for DPD > 2:

1. R. D. Groot and P. B. Warren, J. Chem. Phys., 107, 4423 (1997)2. R. D. Groot and K. L. Rabone, Biophys. J., 81, 725 (2001)

Dissipative and Random ForcesDissipative and Random Forces

F1random

F2random

• Dissipative (friction) forces reduce the relative velocity of the pair of particles

• Random forces compensate for eliminated degrees of freedom

• Dissipative and random forces form DPD thermostat

• The magnitude of dissipative and random forces are defined by fluctuation dissipation theorem

V2

V1

F1dissipative

F2dissipative

Upscaling Approach Upscaling Approach DPD: Coarse-graining of MDDPD: Coarse-graining of MD

•The mass of the DPD particle is Nm times the mass of MD particle.

•The cut-off radius can be found by equating mass densities of MD and DPD systems.

•The DPD conservative force coefficient a is found by equating the dimensionless compressibility of the systems.

•The time scale is determined by insisting that the shear viscosities of the DPD and MD fluids are the same.

•The variables marked with the symbol “*” have the same numerical values as in DPD but they have units of MD.

R. D. Groot and P. B. Warren, J. Chem. Phys., 107, 4423 (1997)E.E. Keaveny, I.V. Pivkin, M. Maxey and G.E. Karniadakis, J. Chem. Phys., 123:104107,

2005

Boundary Conditions in DPDBoundary Conditions in DPD

• Lees-Edwards boundary conditions can be used to simulate an infinite but periodic system under shear

ExternalForce

Periodic

Periodic

Fro

zen

pa

rtic

les

Fro

zen

pa

rtic

les

• Revenga et al. (1998) created a solid boundary by freezing the particles on the boundary of solid object; no repulsion between the particles was used.

•Willemsen et al. (2000) used layers of ghost particles to generate no-slip boundary conditions.

•Pivkin & Karniadakis (2005) proposed new wall-fluid interaction forces.

Poiseuille Flow Results: Non-Adaptive BC

MD data taken every time step and averaged over t=100t to t=2000t.DPD data taken every time step and averaged over t=200tDPD to t=4000tDPD.

•Density Fluctuations in MD and DPD

I.V. Pivkin and G.E. Karniadakis, PRL, vol.96, 206001, 2006

Iteratively adjust the wall repulsion force in each bin based on the averaged density values.

Adaptive Boundary ConditionsAdaptive Boundary Conditions

Targetdensity

Currentdensity

Locallyaverageddensity

Wallforce

Adaptive BC:• layers of particles • bounce back reflection• adaptive wall force

I.V. Pivkin and G.E. Karniadakis, PRL, vol.96, 206001, 2006

Mean-Square Displacement -Mean-Square Displacement -Large Equilibrium SystemLarge Equilibrium System

Liquid-like Solid-like

At high levels of coarse-graining the DPD fluid behaves as a solid-like structure

•The DPD conservative force coefficient a is found by equating the dimensionless compressibility of the systems.

I.V. Pivkin and G.E. Karniadakis, 2006

Multi-body DPD (M-DPD) with Attractive Potential

Rij

Dij

Cijij FFFF

•The dissipative force and random forces are exactly the same as in original DPD, but the conservative force is modified:

•The interaction forces in M-DPD are:

•Take a negative-valued parameter A to make the original DPD soft pair potential attractive, and add a repulsive multi-body contribution with a smaller cut-off radius.

ij[ ( ) ( ) ( )]eCi jij C ij C ijF Aw r B w r

3 2( ) 15 /(2 )(1 / )ij C ij Cw r r r r ( ),i ijj i

w r

( ) 1 /C ij ij Cw r r r

S.Y. Trofimov, 2003

Pan and Karniadakis 2007

Pagonabarraga and Frenkel, 2000

A, B are obtained through upscaling process from corresponding MD simulations

M-DPD: Multibody-DPD

M-DPD allows different EOS fluids by prescribing the free energy (DPD fluid has quadratic EOS)

M-DPD has less prominent freezing artifacts allowing higher coarse-graining limit

M-DPD can simulate strongly non-ideal systems & hydrophilic/hydrophobic walls

Outline

Dissipative Particle Dynamics (DPD)

Biofilaments: DNA, Fibrin

Platelets

Red Blood Cells

Triple-Decker: MD-DPD-NS

V

2

V

1

F1dissipative

F2dissipative

Intra-Polymer Forces – Combinations Of the Following:

• Stiff (Fraenkel) / Hookean Spring

• Lennard-Jones Repulsion

• Finitely-Extensible Non-linear Elastic (FENE) Spring

FENE Chains in Poiseuille Flow

30 (20-bead) chains

side view

top view

Symeonidis & Karniadakis, J. Comp. Phys., 2006

Outline

Dissipative Particle Dynamics (DPD)

Biofilaments: DNA, Fibrin

Platelets

Red Blood Cells

Triple-Decker: MD-DPD-NS

V

2

V

1

F1dissipative

F2dissipative

Platelet AggregationPlatelet Aggregation

- passive - triggered - activated

PASSIVEPASSIVEnon-adhesive

TRIGGEREDTRIGGEREDnon-adhesive

ACTIVATEDACTIVATEDadhesive

Interaction withactivated platelet,injured vessel wall

Activation delay time,chosen randomly

between 0.1 and 0.2 s

If not adhered after 5 s

Pivkin, Richardson & Karniadakis, PNAS, 103 (46), 2006

RBCs are treated as a continuum

Simulation of Platelet Aggregation in the Presence of Red Blood Cells

26

- passive - triggered - activated

•Pivkin & Karniadakis, 2007

• The lipid bilayer is the universal basis for cell membrane structure• The walls of mature red blood cells are made tough and flexible because

of skeletal proteins like spectrin and actin, which form a network• Spectrin binds to the cytosolic side of a membrane protein• Spetrin links form 2D mesh• The average length of spectrin link is about 75nm

Source: Hansen et al., Biophys. J., 72, 1997 and http://www.lbl.gov/Science-Articles/Archive/LSD-single-gene.html

75 nm

Schematic Model of the RBC MembraneSchematic Model of the RBC Membrane

Motivation and GoalMotivation and Goal

Spectrin based model has about 3x104 DOFs. It was successfully validated against experiments data, however its application in flow simulations is prohibitively expensive. In arteriole of 50m diameter, 500m length with 35% of volume occupied by RBCs, we would require 108 DOFs for RBCs, and >1011 DOFs to represent the flow.

The goal is to develop a systematic coarse-graining procedure, which will allow us to reduce the number of DOFs in the RBC model. Together with a coarse-grained flow model, such as Dissipative Particle Dynamics (DPD), it will lead to efficient simulations of RBCs in microcirculation.

The total Helmholtz free energy of the system:

The bending free energy:

 0 is the spontaneous curvature angle between two adjacent triangles

Li is the length of spectrin link i and A    is the area of triangular plaquette  .

The in-plane energy:

  is the angle between two adjacent triangles

Worm-like chain:

We need to define:The spontaneous angle between adjacent triangles, 0

Persistence length of the WLC links, p

The maximum extension of the WLC links, Lmax

The equilibrium length of the WLC links, L0

Coarse-grain by reducing # of points, NCoarse-grain by reducing # of points, N

•Pivkin & Karniadakis, 2007

Coarse-graining ProcedureCoarse-graining ProcedureFine model Coarse-grained model

Nf points, L0f – equilibrium link length Nc points, L0

c – equilibrium link length

• The shape and surface area are preserved, therefore– The equilibrium length is set as

– The spontaneous angle is set as

– The persistence length is set as

–The maximum extension length is set as Lc

max=Lfmax (Lc

0 / Lf0)

• The properties of the membrane were derived analytically (Dao et al. 2006)

Dao M. et al, Material Sci and Eng, 26 (2006)

Coarse-grained ModelCoarse-grained ModelShape at Different Stretch ForcesShape at Different Stretch Forces

23867 points

500 points

100 points

0 pN 90 pN 180 pN

Optical Tweezers, MITOptical Tweezers, MIT

Deformable RBCsDeformable RBCs

Membrane model: J. Li et al., Biophys.J, 88 (2005) - WormLike Chain

Coarse RBC model:• 500 DPD particles connected by links• Average length of the link is about 500 nm

•bending and in-plane energies, constraints on surface area and volume

• RBCs are immersed into the DPD fluid• The RBC particles interact with fluid particles through DPD potentials• Temperature is controlled using DPD thermostat

•Pivkin & Karniadakis, 2007

Simulation of Red Blood Cell Simulation of Red Blood Cell Motion in Shear FlowMotion in Shear Flow

HighLow

Tank-TreadingTank-TreadingTumblingTumbling

•Pivkin & Karniadakis, 2007

Lonely RBC in MicrochannelLonely RBC in Microchannel

Experiment by Stefano Guido, Università di Napoli Federico II

•Pivkin & Karniadakis, 2007

RBCs Migrate to the Center

•Pivkin & Karniadakis, 2007

RBC in Microchannel

Experimental Status

Fabrication of 3-6um wide channels

Completed room temperature, healthy runs. Still need to be analyzed.

3m 4m

5m 6m

Higher concentrations of RBCssimulations

•Pivkin & Karniadakis, 2007

NS + DPD + MD

Macro-Meso-Micro Coupling

Geometry of Coupling

• 3 separate overlapping domains: NS, DPD and MD

• integration is done in each domain sequentially

• communication of boundary conditions only, not of the whole overlapping region

•Fedosov & Karniadakis, 2008

MD-DPD-NS Time Progression Coupling

41 •Fedosov & Karniadakis, 2008

Couette and Poiseuille flows

•Fedosov & Karniadakis, 2008

Conclusions

We have introduced the upscaling process for defining DPD parameters through coarse-graining of MD and discussed the limitation of DPD

We have presented the difficulties and some new approaches in constructing no-slip boundary conditions in DPD

DPD has been demonstrated to be able to simulate complex bio-fluid, such as platelets aggregation and the deformation of red blood cells.

A hybrid particle model: MD-DPD-NS has been successfully applied to simple fluids.