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

Solving Navier-Stokes (NS) equation –Eulerian coordinate used –Equations discretized on a mesh –Macroscopic parameter and states directly applied Finite volume/element method GravityViscous force Momentum equation Interface/surface force Pressure gradient Continuity equation

Macroscopic approaches Interface treatments –Volume of fluid (VOF) Most popular –Level set method –Phase field Finite volume/element method Complex geometry –Structured body fitted mesh Coordinate transformation Matrix representing –Unstructured mesh Linked list representing VOF description Unstructured mesh

Macroscopic approaches 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 Finite volume/element method

Macroscopic approaches 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 Finite volume/element method

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 Based on lubrication approximation of NS equation Thin film method ViscosityFilm thickness Surface tension Effective interface potential Mobility coefficient depends of boundary condition

Macroscopic approaches 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

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

Possible numerical approaches Microscopic approaches

Based on inter-molecular forces Molecular dynamics (MD) Potential of a molecular pair Total force acted on a molecule Molecule velocity F ij F ji i j Lennard- Jones potential

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  N 2, 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 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

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

Why mesoscopic approaches? –Same physical scale as micro- fluidics (from nm to  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) MD or DSMC N-S Mesoscopic particle Macroscopic Mesoscopic Microscopic Increasing scale LBM or DPD Molecule

Lattice Boltzmann Method (LBM) 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 Introduction Example of lattice gas collision LBM D2Q9 lattice structure indicating velocity directions

Lattice Boltzmann Method (LBM) Introduction 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 f i, 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 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 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 Conservative forceDissipative forceRandom 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

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) 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 Isothermal NS equation in Lagrangian coordinate Voronoi tessellations

Dissipative particle dynamics (DPD) 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 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 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