Crystallization in two dimensions
Workshop on Crystallization and Melting in Two-Dimensions
MTA-SZFKI,
Budapest, Hungary, May 18, 2010
by Hartmut Löwen
(Heinrich-Heine-Universität Düsseldorf)
Outline:
1) Introduction: crystallization and melting in two dimensions
2) Dynamical density functional theory
3) Glass transitions in magnetic colloids
3) Crystallization in 2d binary mixtures
4) Conclusions
Classical many body system in strict two dimensions
Temperature T=0: Ground state of a repulsive potential
- hard disks,
- repulsive dipoles V ( r ) 
1
r3
- plasma V ( r )  1 , V ( r )  ln( r /  )
r
is a triangular lattice
Long-range translational order, periodic density field
potential energy minimization
Temperature T>0:
long-range translational order does not exist in 2d under certain
general conditions
Mermin, Wagner
PRL 17, 1133 (1968) spin systems
Mermin
PRE 176, 250 (1968)
more general mathematical proof by
Fröhlich, Pfister,
Communications in Mathematical Physics
81, 277 (1981)
(not for plasma and hard disks)
but long-ranged bond orientational order exists!
Communications in Mathematical Physics 81, 277 (1981)
More quantitative: correlation functions
translational order, pair correlation fucnction
1
g( r ) 

N

N

g( r )

  ( r  ( ri  rj )) 
i , j 1
i 1
band-orientational order
r
 *
g 6 ( r )   i ( r ) i ( 0 ) 
1

 i( r ) 
Nj
e

i 6 ij ( r )
j
ij
i
j
fixed reference
axis
g 6 ( r )  O( 1 ) for triangularlattice
fluid:
g( r )  1  e  r /  r   w ith the bulk correlatio n length 
g 6 ( r )  e r / 
solid:
6
( r   ) w ith a different correlatio n length 6
g ( r )  1 decays algebraica lly  r ( r   ).
g 6 ( r )  0 ( r   )  long - range orientatio nal order.
s
The debate about two-dimensional melting/crystallization
coexistence
1) first order
fluid
solid

1. order
2) Kosterlitz-Thouless-Nelson-Halperin-Young (KTNHY) scenario
thermal unbinding of defects
disclinations
fluid
dislocations
hexatic
2. order
2. order
intermediate hexatic phase
solid
g( r )  1  e r / 
g 6 ( r )  r 
6 
1
4
6
h
(r  )
(r  )
Experimental realization of classical two-dimensional systems
a) Colloids at an air-water interface
(or between two parallel glass plates)
b) Granulates on a vibrating horizontal table
c) Dusty (complex) plasma sheets
a) 2d colloidal dispersions
(Keim, Maret, Zahn et al.)
tilt angle
  
• spherical colloids
B
confined to water/air surface normal n
interface
• superparamagnetic due to
Fe2O3 doping

• external magnetic field B
 0
  54.70
 induced dipole 

repulsive
no interaction
moments m  B
( for   0)
 tunable interparticle
potential 2

m 1
2
2
u (r )  u HS 
(
1

3
cos

cos
)
3
2 r

 
(r , B )
  900
attractive
Particle configurations for different fields
(Maret, Keim, Eisermann 2004)
f B perp. to surface, liquid
in-plane B
i
k
B perp. to surface, crystal
KTNHY scenario confirmed
binary mixtures also realizable
b) granulates on a vibrating table
one-component hard disks: consistent with KTNHY (Shattuck et al, 2006)
binary mixtures
M.B. Hay, R.K. Workman, S. Manne, Phys. Rev. E 67, 012401 (2003)
G.K. Kaufman, S.W. Thomas III, M. Reches, B.F. Shaw, J. Feng, G.M. Whitesides
Soft Matter 5, 1188 (2009)
c) dusty complex plasmas
R.A. Quinn, J. Goree, Phys. Rev. E 64, 051404 (2001)
Donko, Hartmann: theoretical work on 2d Yukawa
consistent with KTNHY
2) Dynamical density functional theory
Equilibrium Density Functional Theory (DFT)
Basic variational principle:
There exists a unique grand-canonical free energy-density-functional
which gets minimal for the equilibrium density
and then coincides with the real grandcanoncial free energy.
→ is also valid for systems which are inhomogeneous on a microscopic scale.
In principle, all fluctuations are included in an external potential which breaks all
symmetries.
For interacting systems, in 3d (2d),
is not known.
,
exceptions:
i) soft potentials in the high density limit, ideal gas (how density 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
→ bulk phase diagram
with respect to all variational parameters
EPL 22, 245 (1993)
b) approximations for the density functional
+
defines the excess free energy functional
A) Ramakrishnan-Youssuf (RY) 1979
results in a first order solid-fluid transition
(for hard spheres)
dynamical density functional theory (DDFT)
Starting point: Smoluchowski equation (exact)
integrate out
adiabatic approximation:
for Brownian dynamics (colloids)
(Archer and Evans, JCP 2004)
 
( 2)  
 ( r , r , t)   ( r , r )
( 2)
equi
such that time-dependent one particle density field

 ( r , tis) the same

1  ( r , t )      F 
     ( r , t )
 

t
 ( r , t ) 

F : equilibrium free energydensityfunctional
(in excellent agreement with BD computer simulations)
Dynamics of 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
fluctuations, here we prescribe them
How does nucleation depend on cluster size and shape?
(S. van Teeffelen, C.N. Likos, H. Löwen, PRL, 100,108302 (2008))
equilibrium functional by Ramakrishnan-Yussouff (2d)
hexatic phase??
(S. van Teeffelen et al, EPL 75, 583 (2006); J. Phys.: Condensed Matter, 20, 404217 (2008))
u0
2d V(r)  3
r
(magnetic colloids with dipole moments)
coupling parameter
equilibrium freezing for
  u03/ 2 / k BT
 f  36
connection to phase field crystal models (L. Granasy et al) by gradient expansion
(van Teeffelen, Backofen, Voigt, HL, Phys. Rev. E 79, 051404 (2009)
procedure
a)
particles in an external trapping potential
Vext (r) at high temperatures(   10  for
)f t < 0
b)
release Vext (r)
and decrease T instantaneously
for t > 0(enhance  towards 
=63)
imposed nucleation seed
cut-out of a rhombic crystal with N=19 particles
nucleation + growth
  600
A  0.7
no nucleation
  600
A  0.6
„island“ for heterogeneous nucleation in
(cos , A) space.
Brownian dynamics
computer simulation
strongly asymmetric in A

symmetric in
3) Crystallization in 2d binary mixtures
• binary spherical colloids
confined to water/air
interface
• superparamagnetic due to
Fe2O3 doping

• external magnetic field B
 induced dipole
moments mi  i B
 tunable interparticle
potential
mi m j 1
Vij (r ) 
2
r3
composition
phase diagram at zero temperature
m2 / m1
A. Lahcen, R. Messina and HL, EPL 80 48001 (2007)
Some important phases
X=0
X=1
X=2/3
X=1/3
X=1/2
X=1/2
Experimental
snapshots at
m2 / m1  0.1 and
2 / 1  1.2
11  20  100
(F. Ebert, P. Keim, G. Maret, EPJE 26, 161 (2008)
composition
m2 / m1
Found in experiment
Ultra-fast temperature quench can be
realized by increasing the magnetic field
11  5  11  78 (m  0.1)
Brownian Dynamics computer simulation in agreement with experiments
“patches“ of
crystallites
is this a
glass??
dynamical
heterogeneities
L. Assoud, F. Ebert, P. Keim, R. Messina, G. Maret, H. Löwen, Phys. Rev. Lett. 102, 238301 (2009)
non-monotonic behaviour (in time) for 2-2- structure
4) Ground state of 2d oppositely charged mixtures
3d, textbook knowledge: NaCl, CsCl, ZnS structures are stable
Lattice sum minimization
(penalty method for hard spheres)
L. Assoud, R. Messina, H.L., EPL 89, 36001 (2010)
5) Conclusions
2d melting/crystallization is still interesting
- mixtures
- tetratic phase?
- (2+  ) confinement (e.g. between plates
with finite spacing)