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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Nucleation in Hard Spheres Systems
Sven Dorosz
University of Luxembourg
August, 2011
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Table of contents
1 Background and Motivation
2 Homogenous nucleation
3 Heterogenous nucleation
Sven Dorosz Nucleation in Hard Spheres Systems
-
Background and MotivationHomogenous nucleation
Heterogenous nucleation
Softmatter
They include liquids, colloids, polymers, foams, gels, granularmaterials, and a number of biological materials.The energy scale comparable with room temperature thermalenergy.Illustration taken from SoftComp website
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Classical Nucleation Theory
G = 4R2 +4
3R3 R =
2
||
Gcrit =163
3(||)2I = exp
[
163
3(kBT||)2
]
source Hamed Maleki, PhD Thesis, University of Mainz, 2011
Sven Dorosz Nucleation in Hard Spheres Systems
-
Background and MotivationHomogenous nucleation
Heterogenous nucleation
Hard Spheres
V (r) =
{
if r < R0 if r R
collisionsare elastic internal energy constant
F = TS
most simple model for a liquid
=1
6R3
The equations of state is given by
P
=
(1 + + 2)
(1 )3
Sven Dorosz Nucleation in Hard Spheres Systems
-
Background and MotivationHomogenous nucleation
Heterogenous nucleation
Phasediagram for Hard Spheres
Rintoul, Md. and Torquato, S., 1996, PRL 77, 20, 4198-4201.
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Rare event Sampling
Stefan Auer and Daan Frenkel, Nature 409, 1020-1023 (2001)
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Motivation Event driven molecular dynamics
time driven MD simulation vs. event driven MD simulation
V is a discontinuous potential free flight in between collisionsNVE ensembleT is trivial parametersetting the time scale, i.e. diffusion constant
Alder, B. J., and Wainwright, T. E., 1957, J. chem. Phys., 27, 1208.
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Algorithm Event driven molecular dynamics
Diagramneighboring boxesFind the first collision for each particleinto the listupdate first collisionupdate the listnext collision
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Local crystal symmetry
Hard spheres system. = 1.03
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Sven Dorosz Nucleation in Hard Spheres Systems
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test1800.mp4Media File (video/mp4)
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Local crystal symmetry
Sven Dorosz Nucleation in Hard Spheres Systems
-
Background and MotivationHomogenous nucleation
Heterogenous nucleation
Definition of the order parameter
Observables for the Local Bond Ordering:In 3d:
q6m(i) :=1
n(i)
n(i)
j=1
Y6m (~rij) , rij < 1.4
where Y6m (~rij) are the spherical harmonics (with l=6).
Crystalline particles (colorcoded green), nb > 10.Low symmetry cluster (LSC)(colorcoded light brown), nb > 5.
P. J. Steinhardt, D. R. Nelson, and M. Ronchetti. Phys. Rev. B, 28(2), 1983
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Local crystal symmetry
~q6(i) ~q6(j) < 0.7 ~q6(i) ~q6(j) > 0.7
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Distribution of the order parameter
source Hamed Maleki, PhD Thesis, University of Mainz, 2011
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
two snapshots
Spontaneous formation of a crystallite
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Radius of gyration
Small crystallites are far from being sphericalSchilling T.; Dorosz S.; Schoepe H. J.; et al. JPCM, 23, 19, 194120, 2011
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Precursor nucleation
Effective two step process. Precursor formation
Schilling T.; Dorosz S.; Schoepe H. J.; et al. JPCM, 23, 19, 194120, 2011
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
nucleation rates
MD and MC simulations produce rates match the experimental results
SPRES as the first full non equilibrium rare event sampling method
Schilling T.; Dorosz S.; Schoepe H. J.; et al. JPCM, 23, 19, 194120, 2011
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
SPRES
time
2
Bin 1 Bin 2 Bin 3
0
J.T. Berryman and T. Schilling. Sampling rare events in nonequilibrium and nonstationary systems.
J Chem Phys, 133(24):244101, 2010.
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Motivation
Will the substrate induce different nucleation pathways?
Where does the nucleation happen?
What are the consequences of the mismatch betweensubstrate and equilibrium crystal lattice constant?
What is the crystal structure of the nucleus?
How does the substrate change the nucleation rate?
Sven Dorosz Nucleation in Hard Spheres Systems
-
Background and MotivationHomogenous nucleation
Heterogenous nucleation
Setup of the system
We studied a super-saturated fluid of hard spheres in contact witha triangular substrate.
1 1,2 1,4
lattice constant a
1,005
1,01
1,015
1,02
1,025
dens
ity
instantaneous growth
nucleation
equi
libri
um c
onst
ant
Sven Dorosz Nucleation in Hard Spheres Systems
-
Background and MotivationHomogenous nucleation
Heterogenous nucleation
Parameters
N = 220200 (216000 bulk + 4200 substrate) particlesN3/V = 1.005 ( = 0.526) 1.02 ( = 0.534)Corresponding chemical potentials 0.50 0.54 kBT
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Immediate wetting
a < aeq
t = 6D t = 150D
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
vertical density profile
0 5 10 15
distance to the substrate z []0
0,002
0,004
0,006
0,008de
nsity
(z
)
t = 600 Dt = 300 Dt = 6 D
Sven Dorosz Nucleation in Hard Spheres Systems
-
Background and MotivationHomogenous nucleation
Heterogenous nucleation
The first layer
at t > 150D
Sven Dorosz Nucleation in Hard Spheres Systems
-
Background and MotivationHomogenous nucleation
Heterogenous nucleation
Defect density
1 2 3 4 5
layer n0,1
0,2
0,3
0,4
0,5de
fect
den
sity
a=1.01a=1.05a=1.10
1 2 3 4 5
layer n3400
3600
3800
4000
4200
# of
par
ticle
s in
laye
r N
(n)
Induced defects are compensated in the first 3 layers
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
3 layer stacking
Domains of ABA and ABC structure
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
5 layer stacking
Crystal grows in random hexagonal closed packing (RHCP)
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Nucleation at the wall
a > aeq
t = 6D t = 80D Droplet formation on the substrate
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Droplet characterization
0 100 200 300 400 500
t D after critical nucleus is formed0
1000
2000
3000
4000
5000
6000
# of
sol
id p
artic
les
in c
lust
er =1.01=1.0175=1.02
100 1000# particles in crystallite
1
10
prin
cipa
l mom
ents
of
gyra
tion
tens
or
Re
z
Re
z
R||e
z
Analysis of the droplets shows non-spherical droplets even up to 4000
particles.
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Nucleation rates
1,005 1,01 1,015 1,02 1,025
density 0
5e-06
1e-05
1,5e-05
2e-05
nucl
eatio
n ra
te [
5/6
Dl]
a=1.15a=1.2a=1.25a=1.3a=1.35a=1.4
a
Decreasing nucleation rate with increasing mismatch to the substrate
Sven Dorosz Nucleation in Hard Spheres Systems
-
Background and MotivationHomogenous nucleation
Heterogenous nucleation
Acknowledgements
Tanja Schilling, University of Luxembourg
Martin Ottel, Frederike Schmid, Hamed Maleki, Hajo Schope,and Kurt Binder, University of Mainz
The present project is supported by the National ResearchFund, Luxembourg and cofunded under the Marie CurieActions of the European Commission (FP7-COFUND)
Sven Dorosz Nucleation in Hard Spheres Systems
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Background and MotivationHomogenous nucleation
Heterogenous nucleation
Group
J. Berryman, M. Anwar, M. Sinha, T. Schilling, S. Dorosz.
M. Radu missing.
Sven Dorosz Nucleation in Hard Spheres Systems
Background and MotivationHomogenous nucleationHeterogenous nucleation
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