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

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1 Chapter 02: Numercal methods for mcrofludcs Xangyu Hu Techncal Unversty of Munch

2 Possble numercal approaches Macroscopc approaches Fnte volume/element method Thn flm method Mcroscopc approaches Molecular dynamcs (MD) Drect Smulaton Monte Carlo (DSMC) Mesoscopc approaches Lattce Boltzmann method (LBM) Dsspatve partcle dynamcs (DPD)

3 Possble numercal approaches Macroscopc approaches

4 Macroscopc approaches Solvng Naver-Stokes (NS) equaton Fnte volume/element method Contnuty equaton v = 0 v + vv t 1 = p + ρ g η + ρ 2 v + 1 F ρ s Interface/surface force Momentum equaton Pressure gradent Gravty Vscous force Euleran coordnate used Equatons dscretzed on a mesh Macroscopc parameter and states drectly appled Pressure Velocty

Macroscopc approaches Fnte volume/element method Interface treatments Volume of flud (VOF) Most popular Level set method Phase feld 1.0 1.0 0.

5 Macroscopc approaches Fnte volume/element method Interface treatments Volume of flud (VOF) Most popular Level set method Phase feld Complex geometry Structured body ftted mesh Coordnate transformaton Matrx representng Unstructured mesh Lnked lst representng Unstructured mesh VOF descrpton

mcro-channel (MC) n a shear flow Dfferent aspect ratos of crcular or

6 Macroscopc approaches Fnte volume/element method A case on droplet formaton (Kobayash et al 2004, Langmur) Droplet formaton from mcro-channel (MC) n a shear flow Dfferent aspect ratos of crcular or ellptc channel studed Interface treated wth VOF Body ftted mesh for complex geometry

7 Macroscopc approaches Fnte volume/element method Dffcultes n mcro-fludc smulatons Domnant forces Thermal fluctuaton not ncluded Complex fluds Mult-phase Easy: smple nterface (sze comparable to the doman sze) Dffcult: complex nterfcal flow (such as bubbly flow) Polymer or collods soluton Dffcult Complex geometry Easy: statc and not every complcated boundares Dffcult: dynamcally movng or complcated boundares

8 Macroscopc approaches Thn flm method Based on lubrcaton approxmaton of NS equaton Vscosty Surface tenson Flm thckness h η t p = σ h + V ( h) ( m( h) p) = 0 Moblty coeffcent depends of boundary condton Effectve nterface potental h(x) Flm Sold

9 Macroscopc approaches Thn flm method A case on flm rapture (Becker et al. 2004, Nature materals) Nano-meter Polystyrene (PS) flm raptures on an oxdzed S Wafer Studed wth dfferent vscosty and ntal thckness

10 Macroscopc approaches Thn flm method Lmtaton Seems only sutable for flm dynamcs studes.

11 Possble numercal approaches Mcroscopc approaches

12 Mcroscopc approaches Based on nter-molecular forces Molecular dynamcs (MD) Molecule velocty F = Fj = j j dp = F dt p v = m u( r r j j ) e j Total force acted on a molecule Potental of a molecular par u( r j ) Lennard- Jones potental F j F j j r j

13 Mcroscopc approaches Molecular dynamcs (MD) Features of MD Lagrangan coordnates used Trackng all the smulated molecules at the same tme Determnstc n partcle movement & nteracton (collson) Conserve mass, momentum and energy Macroscopc thermodynamc parameters and states Calculatng from MD smulaton results Average Integraton

E) Two fluds and sold walls are smulated Studed the movng

14 Mcroscopc approaches Molecular dynamcs (MD) A case on movng contact lne (Qan et al. 2004, Phys. Rev. E) Two fluds and sold walls are smulated Studed the movng contact lne n Couette flow and Poseulle flow Slp near the contact lne was found

15 Mcroscopc approaches Molecular dynamcs (MD) Advantages Beng extended or appled to many research felds Capable of smulatng almost all complex fluds Capable of very complex geometres Reveal the underlne physcs and useful to verfy physcal models Lmtaton on mcro-fludc smulatons Computatonal neffcent computaton load N 2, where N s the number of molecules Over detaled nformaton than needed Capable maxmum length scale (nm) s near the lower bound of lqud mcro-flows encountered n practcal applcatons

16 Mcroscopc approaches Drect smulaton Monte Carlo (DSMC) Combnaton of MD and Monte Carlo method Translate a molecular Same as MD Number of par tryng for collson n a cell Molecular velocty after a collson r M v v r j tral = + v 1 ( v 2 1 ( v 2 t 2 ρ πd v + v v j j j max 1 ) ) 2 V t, v v c j j e e ρ = N V c c, Collson probablty proportonal to velocty only v j = v v A unformly dstrbuted unt vector j cell

17 Mcroscopc approaches Drect smulaton Monte Carlo (DSMC) Features of DSMC Determnstc n molecular movements Probablstc n molecular collsons (nteracton) Collson pars randomly selected The propertes of collded partcles determned statstcally Conserves momentum and energy Macroscopc thermodynamc states Smlar to MD smulatons Average Integraton

18 Mcroscopc approaches Drect smulaton Monte Carlo (DSMC) A case on dlute gas channel flow (Sun QW. 2003, PhD Thess) Knudsen number comparable to mcro-channel gas flow Modfed DSMC (Informaton Preservng method) used Consderable slp (both velocty and temperature) found on channel walls Velocty profle Temperature profle

19 Mcroscopc approaches Drect smulaton Monte Carlo (DSMC) Advantages More computatonally effcent than MD Complex geometry treatment smlar to fnte volume/element method Hybrd method possble by combnng fnte volume/element method Lmtaton on mcro-fludc smulatons Sutable for gaseous mcro-flows Not effcency and dffcult for lqud or complex flow

20 Possble numercal approaches Mesoscopc approaches

21 Mesoscopc approaches Why mesoscopc approaches? Same physcal scale as mcrofludcs (from nm to µm) Effcency: do not track every molecule but group of molecules Resoluton: resolve mult-phase flud and complex fluds well Thermal fluctuatons ncluded Handle complex geometry wthout dffculty Two man dstngushed methods Lattce Boltzmann method (LBM) Dsspatve partcle dynamcs (DPD) N-S Mesoscopc partcle LBM or DPD Molecule MD or DSMC Macroscopc ρ u r T Mesoscopc Mcroscopc v r Increasng scale

22 Lattce Boltzmann Method (LBM) From lattce gas to LBM Does not track partcle but dstrbuton functon (the probablty of fndng a partcle at a gven locaton at a gven tme) to elmnates nose LBM solvng lattce dscretzed Boltzmann equaton Wth BGK approxmaton Equlbrum dstrbuton determned by macroscopc states Introducton Example of lattce gas collson LBM D2Q9 lattce structure ndcatng velocty drectons

23 Lattce Boltzmann Method (LBM) Contnuous lattce Boltzmann equaton and LBM Introducton Contnuous lattce Boltzmann equaton descrbe the probablty dstrbuton functon n a contnuous phase space LBM s dscretzed n: n tme: tme step δt=1 n space: on lattce node δx=1 n velocty space: dscrete set of b allowed veloctes: f set of f, e.g. b=9 on a D2Q9 Lattce Dscrete veloctes Tme step Equlbrum dstrbuton Df Dt f = t + c f f = t coll. Contnuous Boltzmann equaton f ( x + c δ, t + δ ) = t t f ( x, t) f ( x, t) f τ eq ( x, t) Lattce Boltzmann equaton =0,1,,8 n a D2Q9 lattce Relaxaton tme

24 Lattce Boltzmann Method (LBM) A case on flow nfltraton (Raabe 2004, Modellng Smul. Mater. Sc. Eng.) Flows nfltraton through hghly dealzed porous mcrostructures Suspendng porous partcle used for complex geometry

25 Lattce Boltzmann Method (LBM) Applcaton to mcro-fludc smulaton Smulaton wth complex fluds Two approaches to model mult-phase flud by Introducng speces by colored partcles Free energy approach: a separate dstrbuton for the order parameter Partcle wth dfferent color repel each other more strongly than partcles wth the same color Amphphles and lqud crystals can be modeled Introducng nternal degree of freedom Modelng polymer and collod soluton Suspenson model: sold body descrbed by lattce ponts, only collod can be modeled Hybrd model (combnng wth MD method): sold body modeled by off-lattce partcles, both polymer and collod can be modeled

26 Lattce Boltzmann Method (LBM) Applcaton to mcro-fludc smulaton Smulaton wth complex geometry Smple bounce back algorthm Easy to mplement Valdate for very complex geometres Lmtatons of LBM Lattce artfacts Accuracy ssues Hyper-vscosty Mult-phase flow wth large dfference on vscosty and densty No slp Free slp WALL WALL

27 Dsspatve partcle dynamcs (DPD) From MD to DPD Introducton Orgnal DPD s essentally MD wth a momentum conservng Langevn thermostat Three forces consdered: conservatve force, dsspatve force and random force dr dt dp dt F C = = F = 1 m j C p + F + F C D D αϖ e, F = γϖ e v e, j D j Translaton R Momentum equaton j j j j j Random number wth Gaussan dstrbuton F R = j σξ ϖ e j R j j Conservatve force Dsspatve force Random force

28 Dsspatve partcle dynamcs (DPD) A case on polymer drop (Chen et al 2004, J. Non-Newtonan Flud Mech.) A polymer drop deformng n a perodc shear (Couette) flow FENE chans used to model the polymer molecules Drop deformaton and break are studed

29 Dsspatve partcle dynamcs (DPD) Applcaton to mcro-fludc smulaton Smulaton wth complex fluds Smlar to LBM, partcle wth dfferent color repel each other more strongly than partcles wth the same color Internal degree of freedom can be ncluded for amphphles or lqud crystals modelng polymer and collod soluton Easer than LBM because of off-lattce Lagrangan propertes Smulaton wth complex geometres Boundary partcle or vrtual partcle used

30 Dsspatve partcle dynamcs (DPD) Applcaton to mcro-fludc smulaton Advantages comparng to LBM No lattce artfacts Strctly Gallean nvarant Dffcultes of DPD No drected mplement of macroscopc states Free energy mult-phase approach used n LBM s dffcult to mplement Scale s smaller than LBM and many mcro-fludc applcatons Problems caused by soft sphere nter-partcle force Polymer and collod smulaton, crossng cannot avod Unphyscal densty depleton near the boundary Unphyscal slppage and partcle penetratng nto sold body

31 Dsspatve partcle dynamcs (DPD) New type of DPD method To solvng the dffcultes of the orgnal DPD Allows to mplement macroscopc parameter and states drectly Use equaton of state, vscosty and other transport coeffcents Thermal fluctuaton ncluded n physcal ways by the magntude ncrease as the physcal scale decreases Smulatng flows wth the same scale as LBM or even fnte volume/element Inter-partcle force adjustable to avod unphyscal penetraton or depleton near the boundary Mean deas Deducng the partcle dynamcs drectly from NS equaton Introducng thermal fluctuaton wth GENRIC or Fokker- Planck formulatons

32 Dsspatve partcle dynamcs (DPD) Features Dscretze the contnuum hydrodynamcs equatons (NS equaton) by means of Vorono tessellatons of the computatonal doman and to dentfy each of Vorono element as a mesoscopc partcle Thermal fluctuaton ncluded wth GENRIC or Fokker- Planck formulatons dρ = ρ v dt Vorono tessellatons (1) dv 1 F = g p + F + dt ρ ρ Vorono DPD Isothermal NS equaton n Lagrangan coordnate

33 Dsspatve partcle dynamcs (DPD) Features Dscretze the contnuum hydrodynamcs equatons (NS equaton) wth smoothed partcle hydrodynamcs (SPH) method whch s developed n 1970 s for macroscopc flows Include thermal fluctuatons by GENRIC formulaton Advantages of SDPD Fast and smpler than Vorono DPD Easy for extendng to 3D (Vorono DPD n 3D s very complcate) Smulaton wth complex fluds and complex geometres Requre further nvestgatons Smoothed dsspatve partcle dynamcs (SDPD)

34 Summary The features of mcro-fludcs are dscussed Scale: from nm to mm Complex fluds Complex geometres Dfferent approaches are ntroduced n the stuaton of mcro-fludc smulatons Macroscopc method: fnte volume/element method and thn flm method Mcroscopc method: molecular dynamcs and drect smulaton Monte Carlo Mesoscopc method: lattce Boltzmann method and dsspatve partcle dynamcs The mesoscopc methods are found more powerful than others

(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate

(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1