Transcript Chapter 2 - Xiangyu Hu
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)
Possible numerical approaches
• Macroscopic approaches
Macroscopic approaches
Finite volume/element method
• Solving Navier-Stokes (NS) equation Continuity equation
v
0
v
t
vv
1
p
g
2
v
1
F
s
Interface/surface force Momentum equation Pressure gradient Gravity Viscous force Pressure Velocity – Eulerian coordinate used – Equations discretized on a mesh – Macroscopic parameter and states directly applied
Macroscopic approaches
Finite volume/element method
• Interface treatments – Volume of fluid (VOF) • Most popular – Level set method – Phase field • Complex geometry – Structured body fitted mesh • Coordinate transformation • Matrix representing – Unstructured mesh • Linked list representing 1.0
1.0
0.64
0 1.0
0.95
0.32
0 0.11
0.07
0 0 VOF description Unstructured mesh
Macroscopic approaches
Finite volume/element method
• 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
Macroscopic approaches
Finite volume/element method
• 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
Macroscopic approaches
Thin film method
• Based on lubrication approximation of NS equation Viscosity
p
h
t
h
Film thickness
m
(
h
)
p
V
(
h
) 0 Mobility coefficient depends of boundary condition Effective interface potential Surface tension h(x) Film Solid
Macroscopic approaches
Thin film method
• 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
Macroscopic approaches
Thin film method
• Limitation – Seems only suitable for film dynamics studies.
• No further details will be considered in current course
Possible numerical approaches
• Microscopic approaches
Microscopic approaches
Molecular dynamics (MD)
• Based on inter-molecular forces Molecule velocity Potential of a molecular pair
F
i
j
i
F
ij
j
i
u
(
r ij
)
r ij
e
ij
Total force acted on a molecule
d
p
i dt
v
i
p
m i i
F
i u
(
r ij
)
Lennard Jones potential
F ji j F ij i
r ij
Microscopic approaches
Molecular 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
Microscopic approaches
Molecular dynamics (MD)
• 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
Microscopic approaches
Molecular dynamics (MD)
• 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 the number of molecules N 2 , where N is – Over detailed information than needed – Capable maximum length scale (nm) is near the lower bound of liquid micro-flows encountered in practical applications
Molecular dynamics in current course
• Basic implementation • Multi-phase modeling • SHAKE alogrithm for rigid melocular structures
Microscopic approaches
Direct simulation Monte Carlo (DSMC)
• Combination of MD and Monte Carlo method Translate a molecular Same as MD Number of pair trying for collision in a cell Molecular velocity after a collision
r
i
r
i
v
i
t M trial
2
d
2
v
ij
max 2
V c
t
,
v
i
1 2 (
v
i
v
j
) 1 2
v
j
1 2 (
v
i
v
j
) 1 2
v
ij
e v
ij
e
N c
,
V c
Collision probability proportional to velocity only
v
ij
v
i
v
j
A uniformly distributed unit vector
cell
Microscopic approaches
Direct simulation Monte Carlo (DSMC)
• 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
Microscopic approaches
Direct simulation Monte Carlo (DSMC)
• 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 channel walls Velocity profile Temperature profile
Microscopic approaches
Direct simulation Monte Carlo (DSMC)
• 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
Possible numerical approaches
• Mesoscopic approaches
Mesoscopic approaches
• Why mesoscopic approaches?
– Same physical scale as micro fluidics (from nm to m m) – Efficiency: do not track every molecule but group of molecules – Resolution: resolve multi-phase fluid and complex fluids well – Thermal fluctuations included – Handle complex geometry without difficulty • Two main distinguished methods – Lattice Boltzmann method (LBM) – Dissipative particle dynamics (DPD)
N-S Mesoscopic particle LBM or DPD Molecule Macroscopic
T u
Mesoscopic Microscopic
Increasing scale
MD or DSMC
v
Lattice Boltzmann Method (LBM)
Introduction
• From lattice gas to LBM – Does not track particle but distribution function (the probability 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 Example of lattice gas collision LBM D2Q9 lattice structure indicating velocity directions
Lattice Boltzmann Method (LBM)
Introduction
• 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
f i , e.g. b=9 on a D2Q9 Lattice
set of Equilibrium distribution Time step Discrete velocities
Df Dt
f
t
c
f
f
t
coll
.
Continuous Boltzmann equation
f i
(
x
c
i
t
,
t
t
)
f i
(
x
,
t
)
f i
(
x
,
t
)
f i eq
(
x
,
t
)
Lattce Boltzmann equation
i=0,1, … ,8 in a D2Q9 lattice Relaxation time
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 particles 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 Free slip WALL 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 Translation
d
r
i dt
1
m i
p
i
Momentum equation Random number with Gaussian distribution
d
p
i
F
i C
F
i D
F
i R dt
F
i C
j
i
ij C
e
ij
,
F i D
j
i
ij D
e
ij
v
ij
e
ij
,
F i R
j
i
ij
ij R
e
ij
Conservative force Dissipative force Random force
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 5 6 3 4 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 or 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
Dissipative particle dynamics (DPD)
Voronoi DPD
• 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
d
dt
v
Voronoi tessellations
d
v
g
dt
1
p
F
F
( 1 ) Isothermal NS equation in Lagrangian coordinate
Dissipative particle dynamics (DPD)
Smoothed dissipative particle dynamics (SDPD)
• Features – Discretize the continuum hydrodynamics equations (NS equation) 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 complicate) • Simulation with complex fluids and complex geometries – Require further investigations
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 thin film method – Microscopic method: molecular dynamics and direct simulation Monte Carlo – Mesoscopic method: lattice Boltzmann method and dissipative particle dynamics • The mesoscopic methods are found more powerful than others