Chapter 02: Numerical methods for microfluidics

1. Chapter 02:Numerical methods for microfluidics Xiangyu Hu Technical University of Munich

2. 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)

3. Possible numerical approaches • Macroscopic approaches

4. 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 Pressure gradient Gravity Viscous force

5. 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

6. 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

7. 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 fluctuationnot 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

8. 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

9. 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

10. 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

11. Macroscopic approaches Thin film method • Limitation • Seems only suitable for film dynamics studies. • No further details will be considered in current course

12. Possible numerical approaches • Microscopic approaches

13. 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

14. 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

15. 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

16. 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

17. Molecular dynamics in current course • Basic implementation • Multi-phase modeling • SHAKE alogrithm for rigid melocular structures

18. cell 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

19. 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

20. 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

21. 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

22. DSMC in current course • Basic implementation • Introduction on noise decreasing methods • Information preserving (IP) DSMC

23. Possible numerical approaches • Mesoscopic approaches

24. Mesoscopic approaches Macroscopic • 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) N-S Mesoscopic Mesoscopic particle Increasing scale LBM or DPD Microscopic Molecule MD or DSMC

25. 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

26. 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 dt=1 • in space: on lattice node dx=1 • in velocity space: discrete set of b allowed velocities: f  set of fi, e.g. b=9 on a D2Q9 Lattice Equilibrium distribution Time step Discrete velocities Lattce Boltzmann equation Continuous Boltzmann equation i=0,1,…,8 in a D2Q9 lattice Relaxation time

27. 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

28. 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

29. 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

30. LBM in current course • Basic implementation • Multi-phase modeling • Molcular force approach • Phase field model

31. 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

32. 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 2 1 5 6 8 4 3 7

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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