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
1

1
   vv   p  g   2 v  Fs
t



Interface/surface force
Momentum equation
Pressure gradient
Gravity
Viscous force
Pressure
– Eulerian coordinate used
– Equations discretized on a
mesh
– Macroscopic parameter and
states directly applied
Velocity
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
1.0
1.0
0.64
0
1.0
0.95
0.32
0
0.07
0
0
0.11
• Linked list representing
Unstructured mesh
VOF description
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
Film thickness
h
   m(h)p   0
t
Mobility coefficient depends of
boundary condition
p  h  V (h)

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
Fi   Fij   
j i
Molecule velocity
u (rij )
dp i
 Fi
dt
p
vi  i
mi
u (rij )
rij
j i
Potential of a molecular pair
eij
Total force acted on a molecule
LennardJones
potential
Fji
Fij
rij
j
i
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  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
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
Collision probability
proportional to velocity only
ri  ri  v i t
M trial 
 2d 2 v ij max
2
Vc t ,  
1
1
v i  ( v i  v j )  v ij e
2
2
1
1
v j  ( v i  v j )  v ij e
2
2
Nc
, v ij  v i  v j
Vc
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 microfluidics (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
Macroscopic
N-S

u
T
Mesoscopic particle
Mesoscopic
Increasing
scale
LBM or DPD
– Lattice Boltzmann method (LBM)
– Dissipative particle dynamics
(DPD)
Molecule
MD or DSMC
Microscopic
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  set of
fi, e.g. b=9 on a D2Q9 Lattice
Discrete velocities
Df f
 f 
  c  f   
Dt t
 t coll.
Continuous Boltzmann equation
Time step
f i (x  ci t , t   t )  f i (x, t ) 
Equilibrium
distribution
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
No slip
– Simple bounce back algorithm
WALL
• 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
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
Translation
dri
1
 pi
dt mi
Momentum equation
dp i
 FiC  FiD  FiR
dt
FiC    ijC eij ,
Fi D     ijDeij  v ijeij ,
j i
Conservative force
j i
Dissipative force
Random number with
Gaussian distribution
Fi R    ij ijR eij
j i
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
3
4
7
6
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 FokkerPlanck 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 FokkerPlanck formulations
d
    v
Voronoi tessellations
dt
dv
1
F (1)
 g  p  F 
dt


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