Hartmut-Loewen.pdf

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3.3) Brownian dynamics of rod-like particles start from Smoluchowski picture ωˆ full probability density distribution Smoluchowski equation: (Textbook J.K.G. Dhont) ∂ ∂t P = L Lˆ ûu ≡ ω v v r ,..., ; ˆ ,..., ˆ 1 rN ω ω , t) 123 142 243 P( 1 N S P 2 v N N r ˆ ω total potential energy Smoluchowski operator ∑ = ⎩ ⎨⎧ N v t ⎡ v 1 v N N ⎤ Lˆ v = ∇ v s r • D( ˆ ωi ) • ⎢∇ v r + ∇ v rU ( r , ˆ ω , t) i i i ⎥ i 1 ⎣ kBT ⎦ r ˆ ˆ ∇ Ri = ωi × ˆ ω ⎡ ⎤ + ˆ • ˆ 1 ⎢ + ˆ v N N D ( , ˆ rRi Ri RiU r ω , t) ⎥ ⎣ kBT ⎦ i D( ˆ ω ) ˆ ˆ ( 1 ˆ ˆ i = D ωi ⊗ ωi + D − ωi ⊗ ωi ) t ll ⊥ t
idea of Archer and Evans JCP 121, 4246 (2004) integrate Smoluchowski equation N dr dr d ˆ ω ... d ˆ ω ... 2 ∫ 2 ∫ N ∫ ∫ N v v v ∂ρρ ( r , ˆ ω , t ) v t ⎡ v v 1 v v v F ( r , ˆ ω , t ) ⎤ = ∇ v • D( ˆ) • ⎢ ∇ ( ˆ t) + ( ˆ t) ∇ V ( ˆ r D( ω) v r ρ( r, ω, t) ρ( r, ω, t) v rVext ( r, ω, t ) − ⎥ ∂t ⎣ kBT kBT ⎦ ⎡ ⎤ + ˆ • ˆ v 1 v v 1 v v D ⎢ ( , , ) + ( , ˆ, ) ∇ ( , ˆ, ) − T ( r, ˆ rR Rρ r ω t ρ r ω t v ⎢ rVext r ω t ω, t) ⎥ ⎣ k kB T k kB T ⎦ with average force and torque v r F( r, ˆ ω, t) v v T ( r, ˆ ω, t) = 3 = −∫dr′ ∫ − 2 d ω′ ρ ( 2) v v v v v v v v v ( r, r′ , ˆ ω, ˆ ω′ , t) ∇ vV ( r, r′ , ω, ω′ , t) = ρ ( r, ˆ ω) ∇ v v ( r, r′ , ˆ ω, ˆ ω′ , t) Rˆ [ ρ] δFexc δρ ( r, ˆ ω) r 2 0 v r v 0 v v v v ( r, r′ , ω, ω′ , t) [ ρ] v ˆ δFexc ρ ( , ˆ 0 r ω) R δρ ( r, ˆ ω) 3 2 ( 2) ∫dr′ ∫ d ω′ ρ v rV2 = v 0 in general unknown in equilibrium (Gubbins CPL 76, 329 (1980))
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3rd Warsaw School of Statictical Ph
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1) Density functional theory of fre
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ii) Hansen-Verlet rule of freezing
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Maxwell-double tangent construction
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1.4) Phase diagrams of simple poten
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) plasma (with neutralizing backgro
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e) Lennard-Jones-system phase p dia
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g) ultrasoft interactions repulsive
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exceptions: i) ) soft potentials p
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contour plot of
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approximations on different levels
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3) Ramakrishnan-Youssuf (RY) 1979 i
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5) Rosenfeld functional (for hard s
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advantages - excludes overlapping h
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2) Brownian Dynamics and dynamical
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Thi This is i valid lid for f ffree
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Smoluchowski equation Diffusion in
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Page 37 and 38: Langevin-picture Smoluchowski-pictu
Page 39 and 40: with hydrodynamic interactions(HI.)
Page 41 and 42: integration yields 2-particle densi
Page 44 and 45: 3) in equilibrium, necessarily
Page 46 and 47: in equilibrium, DFT says: apply com
Page 48 and 49: colloidal particles colloidal dispe
Page 50 and 51: The road map of complexity H. Löwe
Page 52 and 53: particle p configurations g for dif
Page 54 and 55: DDFT, , equilibrium q functional by
Page 56 and 57: imposed nucleation seed cut-out of
Page 58 and 59: no nucleation 0 φ= 60 Aρ = 0.6
Page 60 and 61: Two stage process: - sub-Brownian t
Page 62 and 63: 2.3) Hydrodynamic interactions How
Page 64 and 65: 2) Rotne-Prager-tensor with 3) high
Page 66 and 67: Rotne-Prager expression ( r ) = 0 v
Page 68 and 69: Application: colloids in an oscilla
Page 70 and 71: stroboscopic view t i /t B = 2.5, 2
Page 72 and 73: 3) Density functional theory for ro
Page 74 and 75: density of center-of-masses: orient
Page 76 and 77: (2) nematic phase positions iti are
Page 78 and 79: three eigenvalues with largest eige
Page 80 and 81: (5) columnar phase: (6) plastic cry
Page 82 and 83: 3.2) Simple models hard objects A)
Page 84 and 85: C) Density functional theory There
Page 88 and 89: “adiabatic“ approximation: assu
Page 90 and 91: Model Gaussian segment-segment segm
Page 93 and 94: full density orientationally averag
Page 95 and 96: σ 3 ρ(z) 10 8 set-up A DDFT (soli
Page 97 and 98: expansion compression transient par
Page 99 and 100: mm (t) ( 2 7.5 5 2.5 0 -2.5 -5 m 2
Page 101 and 102: From „passive“ to „active“
Page 103 and 104: L. Baraban, private communication m
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