Crystallization in two dimensions

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

Workshop on Crystallization and Melting in Two-DimensionsMTA-SZFKI,

Budapest, Hungary, May 18, 2010

by 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

1

r)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(g

11

1

lattice triangular for

j

)(O)r(g

eN

)r(

)()r()r(g

)r(i

j

i

*

ii

ij

1

1

0

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

0

1

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

4

1

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(1

2)( 22

3

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 confirmed

binary 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

posi

tion

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 and

2.1/ 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 mixtures

3d, 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)