Crystallization in two dimensions
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Transcript of Crystallization in two dimensions
Crystallization in two dimensions
1) Introduction: crystallization and melting in two dimensions2) Dynamical density functional theory3) Glass transitions in magnetic colloids3) Crystallization in 2d binary mixtures4) Conclusions
Workshop on Crystallization and Melting in Two-DimensionsMTA-SZFKI,
Budapest, Hungary, May 18, 2010by Hartmut Löwen
(Heinrich-Heine-Universität Düsseldorf)
Outline:
Classical many body system in strict two dimensions
Temperature T=0: Ground state of a repulsive potential
- hard disks,
- repulsive dipoles
- plasma ,
is a triangular lattice
3
1r
)r(V
r)r(V 1
)/rln()r(V
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 Physics81, 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
band-orientational order
N
ij,i
ji ))rr(r(N
)r(g1
1
1
lattice triangular for j
)(O)r(g
eN
)r(
)()r()r(g
)r(i
ji
*ii
ij
1
10
6
6
6
j
iji
fixed reference axis
fluid:
solid:
666
1
length ncorrelatio different a with
length ncorrelatio bulk the with
)r(e)r(g
re)r(g/r
/r
order. nalorientatio range-long lly algebraica decays
)r()r(g).r(r)r(g s
01
6
)r(g
r
The debate about two-dimensional melting/crystallization
1) first order
2) Kosterlitz-Thouless-Nelson-Halperin-Young (KTNHY) scenario thermal unbinding of defects
intermediate hexatic phase
fluid solid
1. order
coexistence
41
1
6
66
)r(r)r(g
)r(e)r(g h/r
fluid solidhexatic
2. order 2. order
disclinations dislocations
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
)coscos31(12
)( 223
2
r
muru HS ),( Br
• spherical colloids confined to water/air interface
• superparamagnetic due to Fe2O3 doping
• external magnetic field induced dipole
moments tunable interparticle potential
Bm
surface normal Btilt angle
n
B
repulsive no interaction attractive
0 07.54 090
)0( for
a) 2d colloidal dispersions (Keim, Maret, Zahn et al.)
Particle configurations for different fields
B perp. to surface, liquid
B perp. to surface, crystal
in-plane B
(Maret, Keim, Eisermann 2004)
KTNHY scenario confirmedbinary mixtures also realizable
M.B. Hay, R.K. Workman, S. Manne, Phys. Rev. E 67, 012401 (2003)
b) granulates on a vibrating table
one-component hard disks: consistent with KTNHY (Shattuck et al, 2006)
binary mixtures
G.K. Kaufman, S.W. Thomas III, M. Reches, B.F. Shaw, J. Feng, G.M. WhitesidesSoft Matter 5, 1188 (2009)
R.A. Quinn, J. Goree, Phys. Rev. E 64, 051404 (2001)
c) dusty complex plasmas
Donko, Hartmann: theoretical work on 2d Yukawa
consistent with KTNHY
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.
2) Dynamical density functional theory
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 with respect to all variational parameters
→ bulk phase diagram
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)
Starting point: Smoluchowski equation (exact)
integrate out
adiabatic approximation:
such that time-dependent one particle density field is the same
)r,r()t,r,r( )2()2(equi
)t,r(
(Archer and Evans, JCP 2004)
)t,r(F)t,r(
t)t,r(1
dynamical density functional theory (DDFT)
for Brownian dynamics (colloids)
(in excellent agreement with BD computer simulations)
functional density energy free mequilibriu:F
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)
coupling parameter
equilibrium freezing for
(magnetic colloids with dipole moments)03
u2d V(r)r
3/ 20 Bu / k T
f 36
(S. van Teeffelen et al, EPL 75, 583 (2006); J. Phys.: Condensed Matter, 20, 404217 (2008))
hexatic phase??
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
ext fV (r) 10 a)particles inanexternal trappingpotential
at hightemperatures( )for t<0
extV (r)
b)release and decrease T instantaneously for t>0( enhance =63)towards
cut-out of a rhombic crystal with N=19 particles
imposed nucleation seed
nucleation + growth
060 A 0.7
060 A 0.6
no nucleation
„island“ for heterogeneous nucleation in
,A ) . (cos space
strongly asymmetric in A symmetric in
Brownian dynamics computer simulation
3
1( )2i j
ij
m mV r
r
• binary spherical colloids confined to water/air interface
• superparamagnetic due to Fe2O3 doping
• external magnetic field induced dipole
moments tunable interparticle potential
i im B
B
3) Crystallization in 2d binary mixtures
phase diagram at zero temperature
A. Lahcen, R. Messina and HL, EPL 80 48001 (2007)
com
positi
on
12 / mm
Some important phasesX=0
X=1
X=2/3
X=1/3 X=1/2
X=1/2
Experimental snapshots at
1.0/ 12 mm and2.1/ 12
1002011
(F. Ebert, P. Keim, G. Maret, EPJE 26, 161 (2008)
Found in experiment
com
posi
tion
12 / mm
Ultra-fast temperature quench can be realized by increasing the magnetic field
11 115 78 ( 0.1)m
“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)
Brownian Dynamics computer simulation in agreement with experiments
non-monotonic behaviour (in time) for 2-2- structure
4) Ground state of 2d oppositely charged mixtures3d, textbook knowledge: NaCl, CsCl, ZnS structures are stable
L. Assoud, R. Messina, H.L., EPL 89, 36001 (2010)
Lattice sum minimization
(penalty method for hard spheres)
5) Conclusions
2d melting/crystallization is still interesting- mixtures- tetratic phase?- (2+ ) confinement (e.g. between plates with finite spacing)