Chapter 02: Numerical methods for microfluidics

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Chapter 02Numerical methods for microfluidics

• Xiangyu Hu
• Technical University of Munich

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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)

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Possible numerical approaches

• Macroscopic approaches

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Macroscopic approaches
Finite volume/element method

• Solving Navier-Stokes (NS) equation
• Eulerian coordinate used
• Equations discretized on a mesh
• Macroscopic parameter and states directly applied

Continuity equation
Interface/surface force
Momentum equation
Gravity
Viscous force
Pressure gradient
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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

Unstructured mesh
VOF description
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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

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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

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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

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Macroscopic approaches
Thin film method

• Based on lubrication approximation of NS equation

Viscosity
Film thickness
Mobility coefficient depends of boundary condition
Effective interface potential
Surface tension
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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

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Macroscopic approaches
Thin film method

• Limitation
• Seems only suitable for film dynamics studies.
• No further details will be considered in current
  course

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Possible numerical approaches

• Microscopic approaches

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Microscopic approaches
Molecular dynamics (MD)

• Based on inter-molecular forces

Potential of a molecular pair
Total force acted on a molecule
Molecule velocity
Lennard-Jones potential
Fji
j
Fij
i
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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

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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

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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 ? N2,
  where N is the number of molecules
• Over detailed information than needed
• Capable maximum length scale (nm) is near the
  lower bound of liquid micro-flows encountered in
  practical applications

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Molecular dynamics in current course

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

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Microscopic approaches
Direct simulation Monte Carlo (DSMC)

• Combination of MD and Monte Carlo method

Translate a molecular Same as MD
Collision probability proportional to velocity
only
Number of pair trying for collision in a cell
Molecular velocity after a collision
A uniformly distributed unit vector
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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

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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
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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

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DSMC in current course

• Basic implementation
• Introduction on noise decreasing methods
• Information preserving (IP) DSMC

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Possible numerical approaches

• Mesoscopic approaches

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Mesoscopic approaches

• Why mesoscopic approaches?
• Same physical scale as micro-fluidics (from nm to
  mm)
• 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)

Macroscopic
N-S
Mesoscopic
Mesoscopic particle
Increasing scale
LBM or DPD
Microscopic
Molecule
MD or DSMC
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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

LBM D2Q9 lattice structure indicating velocity
directions
Example of lattice gas collision
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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 dt1
• in space on lattice node dx1
• in velocity space discrete set of b allowed
  velocities f ? set of fi, e.g. b9 on a D2Q9
  Lattice

Equilibrium distribution
Time step
Discrete velocities
Lattce Boltzmann equation
Continuous Boltzmann equation
i0,1,,8 in a D2Q9 lattice
Relaxation time
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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

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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

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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
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LBM in current course

• Basic implementation
• Multi-phase modeling
• Molcular force approach
• Phase field model

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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
Random number with Gaussian distribution
Momentum equation
Conservative force
Dissipative force
Random force
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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
8
4
3
7
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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

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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

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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

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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

Voronoi tessellations
Isothermal NS equation in Lagrangian coordinate
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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
  1970s 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

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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

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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