Hartmut-Loewen.pdf

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1.5) Density Functional Theory (DFT) SSee also: l Lectures L t of fB Bob b Evans E a) Basics Basic variational principle: There exists a unique grand-canonical grand canonical free energy energy-density-functional density functional , which gets minimal for the equilibrium density and then coincides with the real grandcanoncial g free energy. gy → is also valid for systems which are inhomogeneous on a microscopic scale. In principle, principle all fluctuations are included in an external potential which breaks all symmetries. For interacting gsystems, y , in 3d, , is not known.
exceptions: i) ) soft potentials p in the high g density y limit, , ideal gas g (how ( density y limit) ) ii) 1d: hard rod fluid, exact Percus functional strategy: 1) chose an approximation 2) parametrize the density field with variational parameters gas, liquid: solid: with lattice vectors of bcc or fcc or ... crystals, spacing sets , vacancies? variational parameter Gaussian approximation for the solid density orbital is an excellent approximation 3) ) minimize with respect p to all variational parameters p � bulk phase diagram EPL 22, 245 (1993) ρρ
Page 1 and 2: 3rd Warsaw School of Statictical Ph
Page 3 and 4: 1) Density functional theory of fre
Page 5 and 6: ii) Hansen-Verlet rule of freezing
Page 7 and 8: Maxwell-double tangent construction
Page 9 and 10: 1.4) Phase diagrams of simple poten
Page 11 and 12: ) plasma (with neutralizing backgro
Page 13 and 14: e) Lennard-Jones-system phase p dia
Page 15: g) ultrasoft interactions repulsive
Page 19 and 20: contour plot of
Page 21 and 22: approximations on different levels
Page 23 and 24: 3) Ramakrishnan-Youssuf (RY) 1979 i
Page 25 and 26: 5) Rosenfeld functional (for hard s
Page 27 and 28: advantages - excludes overlapping h
Page 29 and 30: 2) Brownian Dynamics and dynamical
Page 31 and 32: Thi This is i valid lid for f ffree
Page 33 and 34: Smoluchowski equation Diffusion in
Page 35 and 36: linear relation between and One can
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 86 and 87:
3.3) Brownian dynamics of rod-like
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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