Interactions of magnetic particles in a rotational magnetic field

Transcription

1 Bielefeld Universiy Deparmen of Physics Presened a he COMSOL Conference 2008 Hannover A. Weddemann, A. Auge, F. Wibrach, S. Herh, A. Hüen Ineracions of magneic paricles in a roaional magneic field D2 PHYSICS Bielefeld Universiy

2 Ouline 2 A. Weddemann, A. Auge, F. Wibrach, S. Herh, A. Hüen 1. Moivaion 2. Governing equaion 1. Paricle moion 2. Technical realizaion ALE-approach 1. Second domain riangulaion 3. Ineracions of beads in fluids 1. Simple sysem 2. Comparison beween magneic and hydrodynamic forces 4. Conclusions and Oulook

3 Moivaion 3 Magneic micro- or nanoparicles can inerac very srongly: Experimenal observaions Under he influence of an exernal homogenous magneic field paricle creae chains Quesion: Can magneic ineracions be negleced when modeling paricles in microfluidic sysems?

4 Moivaion 4 Expec ferro- or superparamagneic paricles in a soluion, being manipulaed by an exernal magneic field. Model sysem f cri = f cri (r par,m,η) B criical frequency low frequency field Breaking of paricle chains, repulsive forces high frequency field Paricle agglomeraions roaing wih exernal field Paricles oscillaing close o heir iniial posiion

5 Paricle moion 5 Governing equaions: B ex Governing equaions Paricle Magneic moion: paricles in srong exernal field: d M B ex ()= mag M =M + visc s + pen d U F F F Sray field calculaions: Fmag = f dx = grad M, B dx paricle paricle grad ψ,grad A dx F visc Ω F pen viscous Aforce erm z z ψa ψ z Az μ0 My Mx dx =0 par par par T x x y Ω y x force erm prevening paricles from overlapping U( ) ( v, v, v,...) velociy vecor = M B ex M =Ms Ω ψa ψ z Az grad ψa,grad A x x z z d μ0 My Mx d = 0 x y Ω d M U() = F + F + F d mag visc pen Paricle movemen requires mesh displacemen ALE-formalism

6 ALE-formulaion 6 Governing equaions The basic idea of ALE-mehods is o use differen coordinae sysems, a reference and a spaial sysem. ξ 2 Ω x 2 0 A( ξ, ) Calculaion ransformed o reference sysem Example: u (, )+ [ u](, )=0 x L x A -1 Ω Ω ( x, ) ξ 1 x 1 Ω 0 Ω u ψ( x,) ( x,) dx+ ψ( x,) L[ u]( x,) dx=0 u ψ( A( ξ, )) ( A( ξ, ), ) de JA ( ξ, ) dξ + ψ( A( ξ,)) L[ u]( A( ξ,),)de J ( ξ,) dξ =0 Ω 0 weak formulaion domain ransformaion A

7 ALE-formulaion 7 Limiaions of ALE-mehods: - Topological changes: Governing equaions paricles moving owards each oher - Very srong displacemens: paricle moving oo far in one direcion

8 ALE-formulaion 8 Calculaion of ALE-mesh-displacemen: v mesh = v paricle v v mesh = v par,1 Λ 1 Governing equaions v mesh = v par,2 Λ 2 v = v mesh par, iλi nodes second FEM-riangulaion wih linear basis se Λ

9 Second domain riangulaion s The parameer funcions Λ can be calculaed by sandard FEMmehods: uni simplex { si 0, si 1} Φ( s1, s 2 ) y p 3 arbirarily shaped riangle Governing equaions 1 s 1 p 1 p 2 x wih affine linear mapping x1 x2 x1 x x ( s1, s 2) = +s 1 +s2 y1 y2 y1 y3 y1 Φ 3 1 xx x Λ θ1 f (Λ, θ1, θ2) = Θ(Λ θ1) Θ(1 Λ θ2)+ Θ(Λ (1 θ2)) 1 ( θ + θ ) ( x3 x)( y3 y2) ( x3 x2)( y3 y) Λ( x ) = Λ ( Φxx 1 2 x 3 ( x )) = ( x x )( y y ) ( x x )( y y )

10 Paricle moion 10 Governing equaions: Paricles induce fluid flow: ρ u + ρ( u ) u= grad p+ ηδ u+ ρ f div u= 0 Governing equaions Toal mesh displacemen: B ex M B ex M =Ms Ω ψa ψ z Az grad ψa,grad A x x z z d μ0 My Mx d = 0 x y Ω d M U() = F + F + F d mag visc pen r= ( ri ξi) f(λ i(), r θ, θ ) Δ 1 2 i Addiional remeshing condiion: min qual T < σ T T

11 Ineracions of beads in fluids f = f 0 f = 1.5f 0 Differen phenomena: - chain creaion - paricles oscillaing agains each oher

12 Ineracions of beads in fluids 12 Ineracions of beads in fluids Frequency dependence for differen iniial condiions: M disance in µm 2µm 2µm M s = 1000kAm -1 r = 1µm in µs

13 Ineracions of beads in fluids 13 Ineracions of beads in fluids Frequency dependence for differen paricle diameers: M 10µm 20µm disance in µm M s = 1000kAm -1 r = 10µm 20µm in µs

14 Ineracions of beads in fluids 14 Ineracions of beads in fluids Frequency dependence for differen paricle magneizaions: M 20µm disance in µm f = 20kHz r = 20µm in µs

15 Ineracions of beads in fluids 15 Ineracions of beads in fluids Observaion: M s = 1000kAm -1 r = 10µm v in ms -1 in µs high velociies migh lead o non-laminar fluid behaviour on microscale!!

16 Comparison beween forces 16 Model discussion r 0 = 1µm M s = 1000kAm -1 r = r 0 r = 0.75r 0 r = 0.5r

17 Conclusion & Oulook D2 PHYSICS 17 Bielefeld Universiy Conclusion We have developed a model o describe he dynamic behaviour of magneic beads We have simulaed experimenally known effecs (chain creaion) We have shown ha he magneic ineracion of paricles can induce srong fluidic paricle ineracions ha gain imporance when dealing wih differen paricle sizes Oulook Finding proper clearcus for differen force regimes Implemening ferromagneic paricles