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

Upscaling Approach: MD -> DPD -> Navier Stokes Hybrid 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) G. Lin, et al, PRL, 2007

Numerical Modeling Methods DPD SPH Upscaling Approach: MD -> DPD -> Navier Stokes Hybrid Approach: MD + DPD + Navier Stokes

Platelet Aggregation arteriolesvenules activated platelets, red blood cells, fibrin fibers Platelet diameter is 2-4 µm Normal platelet concentration in blood is 300,000/mm 3 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 V2V2 V1V1 F 1 dissipative F 2 dissipative

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) MDDPDNavier-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 Stokes Hybrid Approach: MD + DPD + Navier Stokes

Conservative fluid / system dependent Dissipative frictional force, represents viscous resistance within the fluid – accounts for energy loss forces exerted by particle J on particle I: Random stochastic part, makes up for lost degrees of freedom eliminated after the coarse-graining riri rjrj r ij i j Pairwise Interactions Fluctuation-dissipation relation: σ 2 = 2 γ κ Β Τ ω D = [ ω R ] 2

MD DPD Conservative 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. a ij Linear force

Upscaling Approach The Conservative Force Coefficient: a ij The value of a ij is determined by matching the dimensionless compressibility 1,2 of the DPD system with that of the MD system. From the work of Groot and Warren 1, 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 Forces F 1 random F 2 random 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 V2V2 V1V1 F 1 dissipative F 2 dissipative

Upscaling Approach DPD: Coarse-graining of MD The mass of the DPD particle is N m 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 DPD Lees-Edwards boundary conditions can be used to simulate an infinite but periodic system under shear External Force Periodic Frozen particles 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=200t DPD to t=4000t DPD. Density Fluctuations in MD and DPD I.V. Pivkin and G.E. Karniadakis, PRL, vol.96, , 2006

Iteratively adjust the wall repulsion force in each bin based on the averaged density values. Adaptive Boundary Conditions Target density Current density Locally averaged density Wall force Adaptive BC: layers of particles bounce back reflection adaptive wall force I.V. Pivkin and G.E. Karniadakis, PRL, vol.96, , 2006

Mean-Square Displacement - Large 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 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. 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 V2V2 V1V1 F 1 dissipative F 2 dissipative

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 V2V2 V1V1 F 1 dissipative F 2 dissipative

Platelet Aggregation - passive - triggered - activated PASSIVE non-adhesiveTRIGGERED ACTIVATED adhesive Interaction with activated 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 75 nm Schematic Model of the RBC Membrane

Motivation and Goal Spectrin based model has about 3x10 4 DOFs. It was successfully validated against experiments data, however its application in flow simulations is prohibitively expensive. In arteriole of 50  m diameter, 500  m length with 35% of volume occupied by RBCs, we would require 10 8 DOFs for RBCs, and >10 11 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 L i 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, L max The equilibrium length of the WLC links, L 0 Coarse-grain by reducing # of points, N Pivkin & Karniadakis, 2007

Coarse-graining Procedure Fine model Coarse-grained model N f points, L 0 f – equilibrium link length N c points, L 0 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 L c max =L f max (L c 0 / L f 0 ) The properties of the membrane were derived analytically (Dao et al. 2006) Dao M. et al, Material Sci and Eng, 26 (2006)

Coarse-grained Model Shape at Different Stretch Forces points 500 points 100 points 0 pN 90 pN 180 pN Optical Tweezers, MIT

Deformable 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 Motion in Shear Flow High Low Tank-Treading Tumbling Pivkin & Karniadakis, 2007

Lonely 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. 3m3m4m4m 5m5m6m6m

Higher concentrations of RBCs simulations 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.