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Hybrid Particle Simulations: An approach to the spatial multiscale problem

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Hybrid Particle Simulations: An approach to the spatial multiscale problem

Guang Lin, PNNL

Thanks: George Em Karniadakis, Brown University

Igor Pivkin

Vasileios Symeonidis

Dmitry Fedosov

Wenxiao Pan

Flow in heterogeneous porous media.(Darcy flow)

Field Scale

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

G. Lin, et al, PRL, 2007

Mesoscale

Pore Scale

Multiphase flow and transport in complex fractures.(Navier-Stokes flow)

Microscopic Scale

Molecular Scale

2

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

DPD

SPH

venules

Platelet Aggregationactivated platelets

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

Thrombocytopenic haemorrhages in the skin

- Myocardial platelet micro-thrombus

V1

V2

F2dissipative

Outline- Dissipative Particle Dynamics (DPD)
- Biofilaments: DNA, Fibrin
- Platelets
- Red Blood Cells
- Triple-Decker: MD-DPD-NS

Symeonidis, Caswell & Karniadakis, PRL, 2005

Symeonidis, Caswell & Karniadakis

Dissipative Particle Dynamics (DPD)

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

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

MDDPDNavier-Stokes

- MICROscopic level approach
- atomistic approach is often problematic because larger time/length scales are involved

- set of point particles that move off-lattice through prescribed forces
- each particle is a collection of molecules
- MESOscopic scales
- momentum-conserving Brownian dynamics

- continuum fluid mechanics
- MACROscopic modeling

i

j

ri

rj

Pairwise Interactions

Fluctuation-dissipation relation:

σ2 = 2γκΒΤωD=[ ωR]2

forces exerted by particle J on particle I:

Conservative

fluid / system dependent

Dissipative

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

Random

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

MD

Conservative Forceaij

- Linear 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.

The Conservative Force Coefficient: aij

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

From the work of Groot and Warren1, we know for rDPD > 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)

F1dissipative

V1

V2

F2dissipative

F2random

Dissipative and Random Forces- 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

Upscaling Approach DPD: 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

External

Force

Frozen particles

Frozen particles

Periodic

Boundary Conditions in DPD- Lees-Edwards boundary conditions can be used to simulate an infinite but periodic system under shear

- 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

- Density Fluctuations in MD and DPD

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.

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

Adaptive Boundary Conditions

Wallforce

Locally

averaged

density

Target

density

Current

density

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

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

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

Solid-like

Mean-Square Displacement -Large Equilibrium System- The DPD conservative force coefficient a is found by equating the dimensionless compressibility of the systems.

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

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

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

- The interaction forces in M-DPD are:

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

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

- 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.

S.Y. Trofimov, 2003

Pagonabarraga and Frenkel, 2000

Pan and Karniadakis 2007

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

V1

V2

F2dissipative

Outline- Dissipative Particle Dynamics (DPD)
- Biofilaments: DNA, Fibrin
- Platelets
- Red Blood Cells
- Triple-Decker: MD-DPD-NS

Intra-Polymer Forces – Combinations Of the Following:

- Lennard-Jones Repulsion

- Stiff (Fraenkel) / Hookean Spring

- 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

V1

V2

F2dissipative

Outline- Dissipative Particle Dynamics (DPD)
- Biofilaments: DNA, Fibrin
- Platelets
- Red Blood Cells
- Triple-Decker: MD-DPD-NS

activated platelet,

injured vessel wall

Activation delay time,

chosen randomly

between 0.1 and 0.2 s

ACTIVATED

adhesive

PASSIVE

non-adhesive

TRIGGERED

non-adhesive

If not adhered after 5 s

- passive - triggered - activated

Platelet AggregationRBCs are treated as a continuum

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

- passive - triggered - activated

Simulation of Platelet Aggregation in the Presence of Red Blood Cells- Pivkin & Karniadakis, 2007

26

Schematic Model of the RBC Membrane

75 nm

- 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

Motivation 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 108DOFs 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.

0isthe spontaneous curvature anglebetween two adjacent triangles

Li is the lengthof spectrin linkiand A is the area of triangular plaquette.

Coarse-grain by reducing # of points, NThe total Helmholtz free energy of the system:

The bending free energy:

isthe anglebetween two adjacent triangles

The in-plane energy:

- Pivkin & Karniadakis, 2007

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-graining Procedure

Fine model

Coarse-grained model

Nc points, L0c – equilibrium link length

Nf points, L0f – equilibrium link length

- The shape and surface area are preserved, therefore
- The equilibrium length is set as
- The spontaneous angle is set as

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

- The persistence length is set as

- The maximum extension length is set as
- Lcmax=Lfmax (Lc0 / Lf0)

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

Coarse-grained ModelShape at Different Stretch Forces

Optical Tweezers, MIT

23867

points

500

points

100

points

0 pN

90 pN

180 pN

Deformable RBCs

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

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

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

- Pivkin & Karniadakis, 2007

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

Simulation of Red Blood Cell Motion in Shear Flow

High

Low

Tumbling

Tank-Treading

- Pivkin & Karniadakis, 2007

Lonely RBC in Microchannel

Experiment by Stefano Guido,

Università di Napoli Federico II

- Pivkin & Karniadakis, 2007

- Pivkin & Karniadakis, 2007

Experimental Status

- Fabrication of 3-6um wide channels
- Completed room temperature, healthy runs. Still need to be analyzed.

3mm

4mm

5mm

6mm

Higher concentrations of RBCssimulations

- Pivkin & Karniadakis, 2007

NS + DPD + MD

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

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.

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