chapter 02: numerical methods for microfluidics

Chapter 02:Numerical methods for microfluidics

Xiangyu Hu

Technical University of Munich

Possible numerical approaches

• Macroscopic approaches– Finite volume/element method– Thin film method

• Microscopic approaches– Molecular dynamics (MD)– Direct Simulation Monte Carlo (DSMC)

• Mesoscopic approaches– Lattice Boltzmann method (LBM)– Dissipative particle dynamics (DPD)


• Macroscopic approaches

Macroscopic approaches

• Solving Navier-Stokes (NS) equation

– Eulerian coordinate used– Equations discretized on a mes

h– Macroscopic parameter and st

ates directly applied

Finite volume/element method

sgpt

Fvvvv

v

11

0

2

Gravity Viscous force

Momentum equation

Interface/surface force

Pressure gradient

Continuity equation

P re s s u re

V e lo c i t y


• Interface treatments– Volume of fluid (VOF)

• Most popular

– Level set method– Phase field

0 .9 5

0 .6 4

0 .3 2

1 .01 .0

1 .0

0 .0 7 0 0

0

0

0 .11


• Complex geometry– Structured body fitted mesh

• Coordinate transformation• Matrix representing

– Unstructured mesh• Linked list representing

VOF description

Unstructured mesh


• A case on droplet formation (Kobayashi et al 2004, Langmuir)

– Droplet formation from micro-channel (MC) in a shear flow– Different aspect ratios of circular or elliptic channel studied– Interface treated with VOF– Body fitted mesh for complex geometry



• Application in micro-fluidic simulations– Simple or multi-phase flows in micro-meter scale channels

• Difficulties in micro-fluidic simulations– Dominant forces

• Thermal fluctuation not included

– Complex fluids• Multi-phase

– Easy: simple interface (size comparable to the domain size)

– Difficult: complex interficial flow (such as bubbly flow)

• Polymer or colloids solution– Difficult

– Complex geometry• Easy: static and not every complicated boundaries• Difficult: dynamically moving or complicated boundaries


Macroscopic approaches in current course

• Numerical modeling for multi-phase flows– VOF method– Level set method– Phase field method– Immersed interface method – Vortex sheet method


• Based on lubrication approximation of NS equation

Thin film method

)(

0)(

hVhp

phmt

h

Viscosity

Film thickness

Surface tension

Effective interface potential

Mobility coefficient depends of boundary condition

S o l id

h (x ) F i lm


• A case on film rapture (Becker et al. 2004, Nature materials)

– Nano-meter Polystyrene (PS) film raptures on an oxidized Si Wafer

– Studied with different viscosity and initial thickness

Thin film method


• Limitation

– Seems only suitable for film dynamics studies.

• No further details will be considered in current course

Thin film method


• Microscopic approaches

Microscopic approaches

• Based on inter-molecular forces

Molecular dynamics (MD)

i

ii

ii

ijij

ij

ij

ijiji

m

dt

d

r

ru

pv

Fp

eFF

)(Potential of a molecular pair

Total force acted on a molecule

Molecule velocity

Fij

Fji

i

j

)( ijru

ijr

Lennard-Jones

potential

Microscopic approachesMolecular dynamics (MD)

• Features of MD– Lagrangain coordinates used – Tracking all the “simulated” molecules at the same time– Deterministic in particle movement & interaction (collision)– Conserve mass, momentum and energy

• Macroscopic thermodynamic parameters and states– Calculating from MD simulation results

• Average

• Integration


• A case on moving contact line (Qian et al. 2004, Phys. Rev. E)

– Two fluids and solid walls are simulated– Studied the moving contact line in Couette flow and Poiseuille flow– Slip near the contact line was found


• Advantages – Being extended or applied to many research fields– Capable of simulating almost all complex fluids– Capable of very complex geometries– Reveal the underline physics and useful to verify physical models

• Limitation on micro-fluidic simulations– Computational inefficient computation load N2, where N is the

number of molecules– Over detailed information than needed– Capable maximum length scale (nm) is near the lower bound of li

quid micro-flows encountered in practical applications

Molecular dynamics in current course

• Basic implementation • Multi-phase modeling• SHAKE alogrithm for rigid melocular structures

Microscopic approachesDirect simulation Monte Carlo (DSMC)

• Combination of MD and Monte Carlo method

evvvv

evvvv

vvvv

vrr

ijjij

ijjii

jiijc

cc

ij

trial

iii

V

NtV

dM

t

2

1)(

2

12

1)(

2

1

,,2

max

22

Number of pair trying for collision in a cell

Translate a molecular

Same as MD

Collision probability proportional to velocity only

Molecular velocity after a collision

A uniformly distributed unit vector

cell


• Features of DSMC– Deterministic in molecular movements– Probabilistic in molecular collisions (interaction)

• Collision pairs randomly selected

• The properties of collided particles determined statistically

– Conserves momentum and energy

• Macroscopic thermodynamic states– Similar to MD simulations

• Average

• Integration


• A case on dilute gas channel flow (Sun QW. 2003, PhD Thesis)

– Knudsen number comparable to micro-channel gas flow– Modified DSMC (Information Preserving method) used– Considerable slip (both velocity and temperature) found on cha

nnel walls

Velocity profile Temperature profile


• Advantages – More computationally efficient than MD

– Complex geometry treatment similar to finite volume/element method

– Hybrid method possible by combining finite volume/element method

• Limitation on micro-fluidic simulations– Suitable for gaseous micro-flows

– Not efficiency and difficult for liquid or complex flow

DSMC in current course

• Basic implementation • Introduction on noise decreasing methods

– Information preserving (IP) DSMC


• Mesoscopic approaches

Mesoscopic approaches

• Why mesoscopic approaches?– Same physical scale as micro-flui

dics (from nm to m) – Efficiency: do not track every mole

cule but group of molecules– Resolution: resolve multi-phase fl

uid and complex fluids well– Thermal fluctuations included– Handle complex geometry without

difficulty

• Two main distinguished methods– Lattice Boltzmann method (LBM)– Dissipative particle dynamics (DP

D)

MD or DSMC

N-S

Mesoscopic particle

u

T

v

Macroscopic

Mesoscopic

Microscopic

Increasing scaleLBM or DPD

Molecule

Lattice Boltzmann Method (LBM)

• From lattice gas to LBM – Does not track particle but distribution function (the probabilit

y of finding a particle at a given location at a given time) to eliminates noise

• LBM solving lattice discretized Boltzmann equation– With BGK approximation– Equilibrium distribution determined by macroscopic states

Introduction

Example of lattice gas collisionLBM D2Q9 lattice structure indicating velocity directions

Lattice Boltzmann Method (LBM)Introduction

.collt

ffc

t

f

Dt

Df

),(),(),(),(

tftftftf

eqii

ittii

xxxcx

Continuous Boltzmann equation Lattce Boltzmann equation

• Continuous lattice Boltzmann equation and LBM– Continuous lattice Boltzmann equation describe the

probability distribution function in a continuous phase space– LBM is discretized in:

• in time: time step t=1

• in space: on lattice node x=1

• in velocity space: discrete set of b allowed velocities: f set of fi, e.g. b=9 on a D2Q9 Lattice

i=0,1,…,8 in a D2Q9 lattice

Discrete velocities

Time step

Relaxation time

Equilibrium distribution

Lattice Boltzmann Method (LBM)

• A case on flow infiltration (Raabe 2004, Modelling Simul. Mater. Sci. Eng.)

– Flows infiltration through highly idealized porous microstructures– Suspending porous particle used for complex geometry

Lattice Boltzmann Method (LBM)Application to micro-fluidic simulation

• Simulation with complex fluids– Two approaches to model multi-phase fluid by Introducing species

by colored particles• Free energy approach: a separate distribution for the order parameter• Particle with different color repel each other more strongly than particl

es with the same color

– Amphiphiles and liquid crystals can be modeled• Introducing internal degree of freedom

– Modeling polymer and colloid solution• Suspension model: solid body described by lattice points, only colloid

can be modeled• Hybrid model (combining with MD method): solid body modeled by off-

lattice particles, both polymer and colloid can be modeled

Lattice Boltzmann Method (LBM)Application to micro-fluidic simulation

• Simulation with complex geometry– Simple bounce back algorithm

• Easy to implement• Validate for very complex

geometries

• Limitations of LBM– Lattice artifacts

– Accuracy issues• Hyper-viscosity• Multi-phase flow with large

difference on viscosity and density

No slip

WALL

Free slip

WALL

LBM in current course

• Basic implementation • Multi-phase modeling

– Molcular force approach– Phase field model

Dissipative particle dynamics (DPD)Introduction

• From MD to DPD– Original DPD is essentially MD with a momentum conserving

Langevin thermostat– Three forces considered: conservative force, dissipative force

and random force

ijij

ij

Rijij

Riijijij

ij

Dij

Diij

Cij

Ci

Ri

Di

Ci

i

ii

i

FF

dt

d

mdt

d

eeveeF

FFFp

pr

,,

1

Conservative force

Dissipative force

Random force

Random number with Gaussian distribution

Translation

Momentum equation

Dissipative particle dynamics (DPD)

• A case on polymer drop (Chen et al 2004, J. Non-Newtonian Fluid Mech.)

– A polymer drop deforming in a periodic shear (Couette) flow– FENE chains used to model the polymer molecules– Drop deformation and break are studied

1 2

3 4

5 6

7 8

Dissipative particle dynamics (DPD)Application to micro-fluidic simulation

• Simulation with complex fluids– Similar to LBM, particle with different color repel each other

more strongly than particles with the same color– Internal degree of freedom can be included for amphiphiles o

r liquid crystals– modeling polymer and colloid solution

• Easier than LBM because of off-lattice Lagrangian properties

• Simulation with complex geometries– Boundary particle or virtual particle used

Dissipative particle dynamics (DPD)Application to micro-fluidic simulation

• Advantages comparing to LBM– No lattice artifacts

– Strictly Galilean invariant

• Difficulties of DPD– No directed implement of macroscopic states

• Free energy multi-phase approach used in LBM is difficult to implement

• Scale is smaller than LBM and many micro-fluidic applications

– Problems caused by soft sphere inter-particle force• Polymer and colloid simulation, crossing cannot avoid• Unphysical density depletion near the boundary • Unphysical slippage and particle penetrating into solid body

Dissipative particle dynamics (DPD)New type of DPD method

• To solving the difficulties of the original DPD– Allows to implement macroscopic parameter and states

directly• Use equation of state, viscosity and other transport coefficients

• Thermal fluctuation included in physical ways by the magnitude increase as the physical scale decreases

• Simulating flows with the same scale as LBM or even finite volume/element

– Inter-particle force adjustable to avoid unphysical penetration or depletion near the boundary

• Mean ideas– Deducing the particle dynamics directly from NS equation– Introducing thermal fluctuation with GENRIC or Fokker-

Planck formulations


• Features– Discretize the continuum hydrodynamics equations (NS equation)

by means of Voronoi tessellations of the computational domain and to identify each of Voronoi element as a mesoscopic particle

– Thermal fluctuation included with GENRIC or Fokker-Planck formulations

Voronoi DPD

)1(1 FFg

v

v

pdt

d

dt

d

Isothermal NS equation in Lagrangian coordinate

Voronoi tessellations


• Features– Discretize the continuum hydrodynamics equations (NS e

quation) with smoothed particle hydrodynamics (SPH) method which is developed in 1970’s for macroscopic flows

– Include thermal fluctuations by GENRIC formulation

• Advantages of SDPD– Fast and simpler than Voronoi DPD– Easy for extending to 3D (Voronoi DPD in 3D is very com

plicate)

• Simulation with complex fluids and complex geometries– Require further investigations

Smoothed dissipative particle dynamics (SDPD)

DPD in current course

• DPD is the main focus in current course– Implementation of traditional DPD– Implementation of SDPD

• Multi-phase modeling

• Multi-scale simulations with DPD and MD– Micro-flows with immersed nano-strcutres

Summary

• The features of micro-fluidics are discussed– Scale: from nm to mm– Complex fluids– Complex geometries

• Different approaches are introduced in the situation of micro-fluidic simulations– Macroscopic method: finite volume/element method and t

hin film method– Microscopic method: molecular dynamics and direct simul

ation Monte Carlo– Mesoscopic method: lattice Boltzmann method and dissip

ative particle dynamics

• The mesoscopic methods are found more powerful than others

chapter 02: numerical methods for microfluidics

Documents

film dynamics studies

film rapture becker

numerical methods

microchannel mc

complex interficial

meshmacroscopic parameter

langmuirdroplet formation

simulated molecules