Hartmut-Loewen.pdf

3rd Warsaw School of Statictical Physics
Kazimierz Dolny, Poland
Hartmut Löwen
Heinrich-Heine Heinrich Heine University Düsseldorf, Düsseldorf Germany
27 June – 3 July 2009

Outline:
1) DDensity it ffunctional ti l theory th of f freezing: f i spheres h
1.1 Phenomenological results
11.2 2 Independent treatment of the different phases
1.3 Unifying Microscopic theories
1.4 Phase diagrams of simple potentials
1.5 Density Functional Theory (DFT)
2) Brownian Dynamics and dynamical density functional theory
2.1 Brownian dynamics (BD)
2.2 Dynamical density functional theory (DDFT)
2.3 Hydrodynamic interactions
3) ) Density y functional theory yfor rod-like particles p
3.1 Statistical mechanics of rod-like particles
3.2 Simple models
3.3 Brownian dynamics of rod-like particles

1) Density functional theory of freezing: spheres
1.1) Phenomenological results:
experiments:
- liquids crystallize into periodic structures at low temperatures or/and
high densities
- translational symmetry is broken
- one of fth the most ti important t tphase h ttransitions iti iin nature t
- when does it happen?

empirical facts:
i) Lindemann-criterion of melting (1910)
root mean square displacement
ltti lattice vectors t Li Lindemann d parameter: t
computer simulation
hard spheres
OCP
Yk Yukawa it interaction ti
criterion: phenomenological rule

ii) Hansen-Verlet rule of freezing (1963)
at freezing, all are similar
criterion:
li liquid id freezes, f if the h first fi maximum i
of exceeds the value 2.85
→ confirmed for Yukawa, OCP,
Lennard-Jones, etc...

1.2) Independent treatment of the different phases
(a) for the solid
impose a prescribed solid with lattice constant .
harmonic lattice theory (phonons)
→ free energy of solid state:
(b) ffor the th li liquid/fluid id/fl id
lattice sum of potential energy per particle.
liquid theory (e.g. HNC closure).
→ free energy of the liquid state.

Maxwell-double tangent construction
( (ensures equality li of fpressure and d chemical h i lpotentials i l in i the h different diff phases) h )
→ phase diagram (typically 1st order freezing)

1.3) Unifying Microscopic theories
i) ) density yfunctional theory y( (in 3d) )
based on liquid
solid lid ≡ condensation d ti of f liquid li id density d it modes d
ii) crystal-based theory (in 2d) (Kosterlitz-Thouless)
defects in solid
liquid ≡ solid with an accumulation of defects

1.4) Phase diagrams of simple potentials
(known from computer simulations + theory)
a) hard spheres
iinternal l energy
averaged potential energy
there is only yentropy, py,ppacking geffects

phase diagram
above freezing: state with higher entropy (solid), has higher order

) plasma (with neutralizing background)
only the combination determines correlations, phase diagram, etc.
coupling parameter
c) soft spheres
phase diagram
freezing into fcc lattice
freezing g
into bcc lattice

d) Yukawa-system
only 2 reduced parameters:

e) Lennard-Jones-system
phase p diagram: g
2 reduced parameters:

f) sticky hard spheres
iis needed d dto get a li liquid id
is needed to get an isostructural
solid-solid
transition
phase diagram:

g) ultrasoft interactions
repulsive, realized for star polymers
exhibit hibit “ “exotic“ ti “ stable t bl solid lid lattices l tti lik like bco, b sc, A15, A15 diamond, di d etc. t
PRL 80, 4450 (1998)

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

example:
solid/liquid coexistence
coexistence it implies:
i li

contour plot of

) approximations for the density functional
1) ideal gas, gas
since
„generalized barometric law“
Thi This is i indeed i d dthe h equilibrium ilib i density d i of fan inhomogeneous i h
gas.
2) in the interacting case,
defines the excess free energy functional

approximations on different levels
1) LDA LDA, llocal l density d i approximation: i i
where is the excess free energy density in a homogeneous (bulk)
system, input, valid only for small inhomogeneites
2) LDA + mean field:
with
� homogeneous limit is respected valid for ”moderate“ inhomogeneites, but not
for density variations on the microscopic scale

(DH approx.)

3) Ramakrishnan-Youssuf (RY) 1979
is reproducing the direct correlation function
exactly at
results l in i a solid-fluid lid fl id transition i i (f (for hard h d spheres)
h )

4) Weighted density approximation (WDA) Curtin & Ashcroft, 1985
free energy per particle; weighted density
determine such that
for all
WDA yields excellent data for hard sphere freezing, etc. problem with
WDA WDA: overlapping l i hard h d sphere h configurations fi ti are not t excluded
l d d

5) Rosenfeld functional (for hard spheres)
(fundamental measure theory (FMT))
with
geometrical measures

6 weight functions:
hard sphere diameter
and
ffunctional ti lsurvives i dimensional di i lcrossover 3D 3D→2D→1D→0D
2D 1D 0D
PRE 55, 4245 (1997)

advantages
- excludes overlapping hard sphere configurations
- yields as output
- excellent data for hard sphere freezing
results results, hard spheres

6) Hard sphere pertubation theory
then
approximate
good phase diagram for Lennard-Jones (LJ)
by a hard sphere
potential t ti l with ith an effective ff ti diameter di t
(Barker, Henderson)

2) Brownian Dynamics and dynamical density
ffunctional ti lth theory
2.1) Brownian dynamics (BD)
literature: M.Doi, , S.F. Edwards, , “Theory yof Polymer y Dynamics“, y , Oxford, , 1986
colloidal particles will be randomly kicked by the solvent
Smoluchowski picture
Brownian motion is diffusion. time-dependent density field of the particle(s)

Fick´s law
current density
example:
phenomenological diffusion coefficient
mass conservation → continuity equation
→ combined with Fick´s law:
diffusion equation

Thi This is i valid lid for f ffree particles ti l with ith a given i intial i ti l density d it
With an external potential , there is the force
acting on the particles
→ drift velocity of the particles resp. an additional current density
assumption: totally overdamped motion
friction coefficient
remark: for a sphere (radius R) in a viscous solvent, solvent Stokes equation yields:
shear viscosity

total current density
in equilibrium:
i)
ii) the total current has to vanish, i.e.:
→
hence
→ continuity equation
St Stokes-Einstein-relation
k Ei t i lti
(special case for fluctuation-dissipation-theorem)

Smoluchowski equation
Diffusion in phase space
→ non-interacting particles
→ normalized:
→
→
is probality to find a particle at position and time
is identical to
except normalization:
Smoluchowski-equation (Focker-Planck-equation)

now: N interacting particles
more compact notation: (analogue for all other vectors)
velocity of particle induces
flow of solvent
→ (force) ( ) / movement of other pparticle
→ hhydrodynamic d d i iinteraction t ti

linear relation between and
One can show:
positive definite
mobility matrix,
calculated from
Navier-Stokes-eq. q
probability density for interacting particles with equation:
with
→ generalized Smoluchowski equation for interacting particles

One can write:
often:
Smoluchowski operator (compare Liouville operator)
(no hydrodynamic interactions,
good for low packing fractions )

Langevin-picture
Smoluchowski-picture: with diffusion dynamics
Langevin-picture: stochastic trajectories in real-space
assume: system overdamped
probability distribution for
First, we consider only one particle
in an external potential
with random force
→ stochastic differential equation:
comes from random kicks of the
solvent

observable strategy
1) solve (�) for a given 2) average with
Now: is Gaussian distributed
(i (in principal i i l functional f ti l integral) i t l)
Equivalence of Langevin and Smoluchowski picture for
Equivalence of Langevin and Smoluchowski picture for
interacting particles (no hydrodynamic interactions)

with hydrodynamic interactions(HI.):
depends on
The Smoluchowski equation (��) is obtained from the
following Langevin equations:

2.2) Dynamical density functional theory (DDFT)
1) Derivation from the Smoluchowski equation
no HI.
(Archer, Evans, J.Chem.Phys. 121, 4246 (2004))
recall Smoluchowski equation for the N-particle density
idea: integrate out degrees of freedom

integration yields
2-particle density:
integrating the Smol Smoluchowski cho ski equation eq ation with
ith

1)
now:

3)
in equilibrium, necessarily

i.e.
(Yvon (Yvon, Born Born, Green hierarchy YBG)

in equilibrium, DFT says:
apply
combined with YBG:
We postulate that this argument holds also in nonequilibrium. In doing so, nonequilibrium
correlations are approximated by equilibrium ones at the same
(via a suitable in equilibrium)

hence:
applications:
DDFT
– time-dependent external potentials
DDFT makes very good approximations for the dynamical density fields.
eeven en for free freezing, ing glass transitions transitions, crystal cr stal growth gro th when hen tested against
BD computer simulations
example: dynamics of freezing, crystal growth

colloidal particles
colloidal dispersions
(from A. Imhof and D. Pine)

Equilibrium
Non-equilibrium
& in colloidal dispersions p
excellent model
systems
phase transformation Kinetics
(crystallization, glass transition,
hhomogeneous and d heterogeneous
h
nucleation)
plus p an external field ( (shear, ,ggravity, y, electric, , laser-optical, p , walls etc.) )
SPP 1296 Heterogene Keim- und Mikrostrukturbildung:
Schritte zu einem system- y
und skalenübergreifenden Verständnis
(coordinator: H. Emmerich)
German German-Dutch Dutch network SFB TR6

The road map of complexity
H. Löwen, Journal of Physics: Condensed Matter 13, R415 (2001)

Colloids - controlled by an external magnetic field
• spherical colloids confined
to water/air interface
• superparamagnetic ti due d to t
Fe2O3 doping
• external magnetic field B v
• external magnetic field B
� induced dipole
moments
v v
� tunable interparticle m χB
potential
tilt angle φφ
surface normal B v
n v
φ =
0
φ =
0
54.
7
φ =
0
90
= repulsive no interaction attractive
potential ( for θθ
= 0 )
r
u( r ) = uHS
m
+
2
2
1
r
3
( 1
2 2
− 3cos
φ cos θ )
θ = ( r, B )
v v

particle p configurations g for different fields
�
B perp. perp to surface, surface crystal
(P. Keim, G. Maret et al)
� B perp. to surface, liquid
in-plane B
��

Crystal growth at externally imposed nucleation clusters
Idea: impose a cluster of fixed colloidal particles
(e.g. by optical tweezer)
Does this cluster act as a nucleation seed for further crystal
growth?
cf: homogeneous nucleation: the cluster occurs by thermal
fl fluctuations, t ti here h we prescribe ib them th
How does nucleation depend p on cluster size and shape? p
(S (S. van Teeffelen Teeffelen, C.N. C N Likos, Likos H. H Löwen, Löwen PRL, PRL 100,108302 100 108302 (2008))

DDFT, , equilibrium q functional byy Ramakrishnan-Yussouff
(S (S. van Teeffelen et al, al EPL 75 75, 583 (2006); J. J Phys.: Phys : Condensed Matter Matter, 20 20, 404217 (2008))
2d V(r)
u 0
3
coupling parameter
= (magnetic colloids with dipole moments)
equilibrium q freezing gfor
r
Γ = u ρ /k T
Γ =Γ =
3/2
0 B
f
36

imposed nucleation seed
cut-out of a rhombic crystal with N=19 particles
A
φ
t0: undercooled liquid

imposed nucleation seed
cut-out of a rhombic crystal with N=19 particles
A
φ
t0: undercooled liquid

nucleation + growth
0
φ= 60
Aρ =
0.7

no nucleation
0
φ= 60
Aρ =
0.6

„island“ for heterogeneous nucleation in (cosφ,A ρ)
space.
strongly asymmetric in A
symmetric ti in i φ
Brownian dynamics
computer simulation

Two stage process:
- sub-Brownian time: relaxation to “ideal“ crystal positions
- Brownian time: crystal growth
x () t
Time evolution of the position of the linear array‘s three rows of crystalline particles i and the position of the
crystal front x () t as a function of time.
f
Dynamical density functional theory results are compared against Brownian dynamics simulation data. data
The arrows indicate the typical time scales on which the relaxation is occurring and on which the crystal growth sets
in, respectively.

Two linear arrays separated by an empty core
• Snapshots of the central region of the dimensionless density field
ρ(,)/ rt ρ of a linear hollow nucleus of two times three infinite rows of hexagonally
crystalline y p particles at times t / r B
= 0, 0.01, 0.1, 0.63, 1.0 (from ( top p to bottom). )
Note that the images display twice the system‘s central region of dimensions
L x L
for better visiblity.
/4 2
x y

2.3) Hydrodynamic interactions
How does look like explicitely?
Solve Stokes/Navier-Stokes Stokes/Navier Stokes equations, difficult problems:
1) is long-ranged
H.I are important for volume fractions
2) H.I. have many-body character, pair expansion only possible at low concentrations
3) H.I. have quite different near-field behaviour. They are divergent, lubrication

standard approximations:
each quantity is a 3 × 3 matrix
0) no H.I.
1) Oseen-Tensor
with ihO Oseen tensor
dyadic product or tensor product
far field term, , long granged g of H.I.
hydrodynamic radius
shear viscosity of solvent

2) Rotne-Prager-tensor
with
3) higher higher-order order expansions
4) tilt triplet contributions tibti (Beenaker, (B k Mazur)
M )

DDFT for hydrodynamic y y interactions
(M. Rex, H. Löwen, Phys. Rev. Letters 101, 148302 (2008))
starting point: Smoluchowski equation total potential energy
two particle
approximation:
configuration configuration-dependent dependent mobility tensor which
describes hydrodynamic interactions

Rotne-Prager expression
( r ) = 0
v
ω v 3 σ H vˆ
vˆ
1 σ H 3 v
( ) ( 1 ) ( ) ( 1 3ˆ
vˆ
σ H 7
ω = + ⊗ + − ⊗ ) + 0((
) )
11
12
r
8
r
r
r
Integrating Smoluchowski equation (Archer, Evans, 2004)
16
r
r
r
r

Possible closure
easier:
with
− β
suitably averaged density

Application:
colloids in an oscillating trap
R 1 = 4σ , R 2 = σ
V = 10k
T , V = k
1
B
2
τ = 0.
5τ
, τ = σ /
(H) (N)
hard spheres with interaction diameter σσ
=
σσ
H
4
3
B
B
2
B
T
D
0

the breathing mode
green/red: / d (N) blue/purple: (H)

stroboscopic view
t i /t B = 2.5, 2.6, 2.7, 2.75, 2.85, 2.9, 3.0
i = 0 1 2 3 4 5 6
with
HI
(H)
without
HI
(N)
good agreement between simulation (red) and DDFT (black)

central
ddensity it
second
moment
(D) curve D (φ)
relaxation to the steady state
o
C. P. Royall et al, PRL 98, 188304 (2007))

3) Density functional theory for rod-like particles
3.1) Statistical mechanics of rod-like particles
(1) molecular dipolar fluids
(2) rod-like colloids
(3) molecular fluids without dipole moment (apolar), (apolar) e.g. e g molecule
(4) plate-like objects (clays)
now: additional orientational degree of freedom

partition function:
with:
while is the inertia tensor and the unit-sphere in 3d.
central quantity:
central quantity:
one-particle density

density of center-of-masses:
orientational order:

pair correlation function:
Different phases are conceivable:
(1) fluid (disordered) phase isotropic phase
(1) fluid (disordered) phase, isotropic phase
center-of-mass-positions and orientations are disordered

(2) nematic phase
positions iti are disordered di d d and d orientations i t ti are ordered d d
nematic director

nematic order parameter
3x3 tensor:
with:

three eigenvalues with
largest eigenvalue:
nematic director: corresponding eigenvector
perfect orientation: for all nematic director
→ if the h two lower l eigenvalues i l are identical, id i l uniaxial i i lnematics i
→ if biaxial nematics
→ in isotropic phase:
experimental p effect: birefringence g

(3) smectic A phase
position ordered along orientation ordered
(4) smectic B phase
as smectic A phase but in plane triangular lattice

(5) columnar phase:
(6) plastic crystal:
positions ordered,
positions ordered,
orientations disordered

(7) full crystalline phases
further more „exotic“ phases:
positions and orientations ordered

3.2) Simple models
hard objects
A) Analytical results by Onsager, 1948
consider limit
virial expansion up to 2 order gets exact result:
there is an isotrop-nematic transition, first order (with density jump)
at coexistence
correlations to finite
irrelevant only for

B) Computer simulations
phase diagram of hard
spherocylinders
hard ellipsoids p

C) Density functional theory
There exists a unique grandcanonical free energy functional
(functional of the one-particle density) which becomes minimal for the equilibrium
density and equals then the real grand canonical free energy

for spherocylinders:
1) SMA (smoothed density approximations) R. Holyst et al, 1988
→ yields several stable liquid crystalline phases
(isotropic, ( p nematic, smectic A, crystalline) y )
2) MWDA (H. Graf, 1999)
→ iimproved d results l with i h plastic, l i AAA phase h
3) extension of Rosenfeld theory (K (K.Mecke Mecke and HH.Hansen-Goos) Hansen Goos)
other interaction (beyond hard body)
perturbation theory within mean-field approach

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))

“adiabatic“ approximation: assume the pair correlations in
nonequilibrium are the same as those for an equilibrium system with
v
the same one-body density profile (established by a suitable V ˆ ext (r, ω,t))
v
∂ρ(
r,
ˆ ω,
t)
v t
v
⎡ v v δF
= ∇ v • D(
ˆ)
• ⎢ ( r,
ˆ
r ω ρ ω,
t)
∇ v
r
∂ t
⎢
⎣
δδρ
( r,
ˆ ω,
t )
v
⎡
[ ] ⎤
+ ˆ v ˆ δF
ρ(
r,
ˆ ω,
t)
D ⎢ ( , ˆ
r rR⎢
ρρ ( r,
ω , t ) R v ⎥⎦
⎣
δ ( ˆ )
⎥
⎣
δρ(
r,
ˆ ω,
t)
⎦
[ ρ(
r,
ˆ ω,
t)
] ⎤
v ⎥⎦
⎥
⎦
DDFT
with the equilibrium Helmholtz free energy density functional
v
v
3 v v
[ ( ) ] [ ] d r d ˆ ωρ(
r,
ˆ ω)
V ( r,
ˆ ω,
t)
3
3
[ ρ]
= k T d r d ˆ ( r,
ˆ ) ln ( r,
ˆ
B ∫ ∫ ωρ
ω Λ ρ ω)
− 1 + Fext
ρ + ∫
F ext
(M. Rex, H.H. Wensink, H.L., PRE 76, 021403 (2007))
∫

approximation for the density functional
mean mean-field field
F exc
1
v v v
= ω )
2
3 3
[ ρ] ∫ d r ∫ d r′
∫ d ˆ ω ∫ dω′
ρ(
r,
ˆ ω′
) v ( , ′ , ˆ,
ˆ′
2 r r ω
(caveat: ( brings g ideal rotational dynamics) y )
time t e independent depe de t

Model
Gaussian segment-segment segment segment interaction
K K
v v
v ( , , ˆ , ˆ
2 r i i, r i i,
ωω
i i,
ωω
i ) = εε
∑ ∑ exp( −
V ext
αβ
( 2
α= −K
β= −K
σ
Δ
L
v
( r,
ˆ ω,
t)
=
∑ ∑
V
0
Z
( )
Z(
t)
Z
V0
( )
Z(
t)
10
10
N s
r
2
)
= 3L
= σ
confining slit
Δ
=
L
N s − 1
V B
0 = 10k T = Φ
2
0 cos − Φ ϑ aligning field
0

Z
ϑ
z
ω
V nem

full density
orientationally
averaged g
density
orientational
ordering
second moment
Results
ρ(
z, ϑ,
t)
π / 2
∫
ρ( z,
t)
= dϑ
sin ϑρ(
z,
ϑ,
t)
0
π / 2
1
⎡3
2 1⎤
S( z,
t)
= dϑ
sin ϑ cos ϑ ρ(
z,
ϑ,
t)
ρ(
z,
t)
∫ ⎢
−
ρ( , ) 0 ⎣2
2⎥
0 ⎣
⎦
∑ ∞
∑
−∞
2
m ( t)
= dzz ( ρ(
z,
t)
− ρ(
z,
t = ∞))
2

slow compression (set-up A) full density
ρ(z, ϑ,t)

σ 3 ρ(z)
10
8
set-up A DDFT (solid curves) and BD (dashed curves)
12 (a)
6
4
2
0
t
0
t 1
t 2
t
3
t 4
t
5
t
6
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
z/σ
σ 3 S(z)
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
t
0
t 1
t 2
t 3
t
4
t
5
t
6
(b)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
t0 = 0.0 t1 = 0.2 τ B, t 2 = 0.4 τ B, t3 = 0.6 τ B, t4 = 0.8 τ B, t5 = 0.9 τ B, t6 = 15.0τB
z/σ

σ z)
3 ρ(z)
set-up B DDFT (solid curves) and BD (dashed curves)
12
(a)
10
8
6
4
2
0
t
6
5
t t 4
t
3
t 2
t 1
t
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
z/σ
σ )
3 S(z)
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
t
6
t 1
t
0
t
5
t 4
t 3
t 2
(b)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
t0 = 0.0 t1 = 0.2 τ B, t 2 = 0.4 τ B, t3 = 0.6 τ B, t4 = 0.8 τ B, t5 = 0.9 τ B, t6 = 15.0τB
z/σ

expansion compression
transient parallel order transient homeotropic order

σ )
3 ρ(z)
set-up C DDFT (solid curves) and BD (dashed curves)
12
t
0
(a)
0.5
0.4
10
0.3
0.2
8
t
1
0.1
6
4
t
6
t
5
t
4
t
3
t
2
0
-0.1
-0.2
2
-0.3
-0.4
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-0.5
z/σ
σ )
3 S(z)
t
6
t
5
t 4
t
0
t
1
t
2
t
3
(b)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
t0 = 0.0 t1 = 0.02 τ B, t 2 = 0.04 τ B, t3 = 0.06 τ B, t4 = 0.08 τ B, t5 = 0.25 τ B, t6 = 15.0τB
z/σ

mm (t) (
2
7.5
5
2.5
0
-2.5
-5
m 2 (t)
0.2
0.1
0
-0.1
-0.2
0 5 t/τ
B 10
-7.5
0 0.5 1
t/τ
B
1.5 2
short-dashed: slow compression (set-up A)
llong-dashed: d h d slow l expansion i (set-up ( t B)
full curve: instantaneous expansion (set-up C)

Conclusions
- generalization of DDFT towards anisotropic colloidal
particles
- good agreement with BD simulation for nontrivial
relaxation problems
future:
• more realistic density functionals (FMT) as proposed by
Mecke and Hansen-Goos.
work by A. Härtel, HL.

From „passive“ to „active“ particles
inert particle
in an external field
Self-propelled particles with an
external motor
- bacteria (E. ( coli) ) - sperm p
- colloidal microswimmers („micromotors“)
GENUINE NONEQUILIBRIUM

COLLOIDAL MICROSWIMMERS
anisotropic (capped)
colloidal ll id l particles ti l
suspended in water
+
H
2O2 O
trajectories
Brownian motion is
relevant
arXiv: 0807.1619, L. Baraban, P. Leiderer, A. Erbe et al
catalytic chemical
reaction at the
surface
⇒ drive
plus external
magnetic field

L. Baraban, private communication
mixture of active“ and passive“ particles in confining geometry
mixture of „active and „passive particles in confining geometry
wall aggregation?

) Collective behaviour (no torque)
Self-propelled Self propelled Brownian rods in a confining channel
F = 1 KBT / L
F = 5 KBT / L
F KT L
0
0
-aggregation gg g near system y walls
- transient hedgehog clusters
- iin line li with i ha microscopic i i
dynamical density functional theory
= 10 B / 0
H. H. Wensink, HL, Phys. Rev. E. 78,
031409 (2008)