Modeling of Micro-Fluidics by a Dissipative Particle Dynamics Method. Justyna Czerwinska

Transcription

1 Modeling of Micro-Fluidics by a Dissipative Particle Dynamics Method Justyna Czerwinska

2 Scales and Physical Models years Time hours Engineering Design Limit Process Design minutes Continious Mechanics Process Simulation seconds Meso-scale Modeling (Clusters) Material Science, Bio-Science microseconds (10-6 s) nanoseconds (10-9 s) picoseconds (10-12 s) femtoseconds (10-15 s) Molecular Mechanics (Atoms) Material Science, Nano-Technology Quantum Mechanics (Electrons) Electronics 1Å (10-10s) Distance 1nm 1µm 1mm 1m

3 Micro- and Nanoscale effects length surface volume 1m 1m2 1m3 1µm 10-6 m 1µm m 1µm m S1 S2 S3 increasing importance of surface to volume effects (surface tension important, gravity unimportant) slip, no-slip boundary conditions gas, liquid differences (1μm3-25 millions of air molecules, 34 billion water molecules) non-linear effects, thermodynamical nonequilibrium 1mm - MACRO - MICRO PHYSICS BARRIER

4 Nano- and Micro- Effects

5 Models of fluids Continuum Global parameters: density, velocity, energy, temperature Meso-scale Microscopic averaged properties, local description, Brownian mechanics, molecules kinetic stochastic equations of energy and motion intermolecular interaction potential ction a r e t n i

6 Meso-scale physics Brownian motion of coiled DNA Micro-biological motor (1.50 µm/s)

7 Meso-scale physics Langevin description of Brownian motion (1908) dv M = ξ M V + F (R ) + Θ( t ) dt V velocity R position; dr/dt = V ξ M V systematic contribution of forces x ξ M friction coefficient F(R) external forces Θ(t)- random forces τm v m 1/ξ relaxation time, without random forces Assumption: Typical time scale on which collisions take place is very small compared to the evolution of the average velocity Θα (t ) = 0 Θα (t ) Θ β (t ) = 2Θ 0δ αβ δ (t t ' ) uncorrelated character of collisions

8 Meso-scale physics Langevin description of Brownian motion (1908) Langevin equation as an evolution equation for PDF of velocity 1/ξ relaxation time scale <V(t) V(0)> <(R(t)-R(0))2> 3kbT/M 1/ξ t 1/ξ t Evolution from ballistic to the diffusive motion Diffusion D = kbt/ M ξ Fokker-Plank equation; Kramers Smoluchowski: SDE -> PDE for PDF P( V, t ) Θ0 ξ V P (V, t ) + = 2 P (V, t ) t V V M

9 Fluctuation-Dissipation Theorem General solution of the Langevin equation 1 V (t ) = V (0) exp( ξ t ) + M lim t V (t ) 2 = t ds exp( ξ (t s)) Θ( s) 0 3Θ 0 M 2ξ from equipartition theorem lim t V (t ) 2 = 3k bt M Fluctuation-Dissipation theorem ensures for equilibrium that PDF is equal to Maxwell distribution M ξ kb T = Θ 0

10 Numerical methods Continuum fluid mechanics Global parameters: density, velocity, energy, temperature Meso-scale mechanics lattice methods sacrifice detail in potential model, simpler interactions and motion rules; LBM particle methods particles as mesoscopic object; replace small particle with random forces; DPD, SPH Microscopic mechanics Molecular Dynamics computation of forces, based on the interaction potential; particle move according to Newton s equation of motion Monte Carlo methods Set a configuration, make a trial move; acceptance/rejection procedure, and accumulation of averages

11 Dissipative Particle Dynamics Spherical formulation two particle interaction Three type of forces conservative (purely repulsive, represents pressure ) dissipative (reducing velocity difference between particles, friction forces ) stochastic ( degree of freedom removed by coarsegraining procedure) ( ) FD = γ M ω ( rij ) ( eij vij ) eij ; δ θ ij FB = ω ( rij ) eij ; FC = π ω rij eij ; t

12 Dissipative Particle Dynamics Aij Rj+ 1 τij Ri Rij ωjj Rj+2 Vi Ri Rj+5 Rj+3 Rj+4 Rij distance between centers of Voronois i and j; ϖij unit vector normal to the face ij, originated from the Voronoi center i; Aij area of the contact surface: length in 2D; rij vector indicating the mass center of the contact surface ij originated from the center of the surface ij: edge in 2D ; Vi is Voronoi volume: surface in 2D.

13 Topological Properties

14 Dissipative Particle Dynamics Classical DPD [ ( ) ( ( )] ) dp i = f i ( t ) Fij C rij + Fij D rij, v ij + Fij R rij dt j i dri = vi (t ) dt dqij dqij D dqij R = + + dt dt dt dt j i DPDE energy conservation deij Voronoi DPD reverssible part irreverssible part Ri = vi Mi = j Aij ρ i + ρ j Rij ( τ ij v i v j 2 (. P i = Aij ϖ ij pi p j ) j dpi ) ( Aij ρ i + ρ j v i + v j 2+ τ ij v i v j + ˆ 2 2 j Rij ( ) ( ( ) = σ ij Π j + Π i 1 dt + dpi ( ) j ) ρi + ρ j si + s j τ ij pi p j µi µ j + Ti T j 2 2 j Rij. Aij si + s j Si = τ ij v i v j 2 j Rij + Aij irr ) ( ) kb 2η ξ Ti ds i irr = 1 i Ωi Ω i + i Ψi2 dtˆ + σ ij Φ j dt Cˆ Vˆ Vˆi j i i 2ηi ξ i κj 2 σ ij kb 2 σ ij T j dt Ti + dt + Ti dsˆi Vˆ ˆ Ti Ci j Vj M j i Vi j 1 1 T ds i = σ ij dφ j dθ σ ji v j Ti j Ti j

15 Channel Flow

16 Channel Flow

17 DNA in the Flow

18 DNA in the Flow

19 Numerical Issues

20 Fluctuation-Relaxation Model ideal reflecting wall diffusively reflecting wall Slip velocity Maxwell v gas v wall= 2 σ v σv v λ wall y Slip flow and temperature jump Smoluchowski v gas v wall= 2 σ v σv Tgas T wall = 2 σ σ λ T T v 3 μ T y wall 4 ρtgas y wall [ ] 2γ λ T γ 1 Pr y wall

21 Fluctuation-Relaxation Model Relaxation time determines if velocity slip or thermal jump Relaxation occurs; it defines scale for which meso-scopic (nonequilibrium) and continuum (equilibrium)boundary condition occurs Fluctuation Theorem (FDT) compensates molecular level interactions, which has been lost in coarse-grained procedure Gas-solid, liquid-solid interface need to be treated similarly; fluid-solid boundary interaction is represented as a nonequilibrium reological process, the difference between liquid and gas would be mainly in the relaxation time of this phenomena. t = 0 t d t = et /

22 Slip : No-Slip Phenomena

23 Self-organization

24 Conclusion Mesoscopic description of fluid; Time scale and relaxation phenomena slip, self-organization; Mesoscopic solid-fluid interaction the most important scale for nano- and micro fluidics