Hybrid Particle Simulations: An approach to the spatial multiscale problem Guang Lin, PNNL Thanks:...
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Transcript of 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.