Heterogenous Nucleation in Hard Spheres Systems
Sven Dorosz and Tanja Schilling
University of Luxembourg
Theory of Soft Condensed Matter,
Universit´e du Luxembourg, Luxembourg, L-1511 Luxembourg sven.dorosz@uni.lu
1 Setup
We studied a super- saturated fluid of hard spheres in contact with a triangular substrate.
1 1,1 1,2 1,3 1,4
lattice constant a [σ]
1,005 1,01 1,015 1,02
number density ρ instantaneous growth
nucleation limit of stability
• Will the substrate induce different nucleation pathways?
• Where does the nucleation happen?
• What are the consequences of the mismatch between substrate and equilibrium crystal lattice constant?
• What is the crystal structure of the nucleus?
• How does the substrate change the nucleation rate?
• Related work [1, 5, 3, 2]
2 How does one detect crystallites?
Observables for the Local Bond Ordering [4]:
In 3d: For all particles i with n(i) neighbours, define
¯
q6m(i) := 1 n(i)
n(i)
X
j=1
Y6m ~rij
,
where Y6m ~rij
are the spherical harmonics (with l=6).
In 2d: For all particle i with n(i) neighbours, define
ψ¯6(i) := 1 n(i)
n(i)
X
j=1
ei6θ(rij) ,
with θ(rij) angle between vector ~rij and an arbitrary fixed axis in the plane.
Allow rij < 1.4 between particle i and its neighbour j. Crystalline particles
colorcoded ”green”) if n > 10.
colorcoded ”light brown” if n > 5.
(left) q~6(i) · q~6∗(j) < 0.7 and (right) q~6(i) · q~6∗(j) > 0.7
3 Simulation details
N = 220200 (216000 bulk + 4200 substrate) particles N σ3/V = 1.005 (η = 0.526) − 1.02 (η = 0.534)
Corresponding chemical potentials −∆µ ≃ 0.50 − 0.54 kBT D denotes the diffusion constant
4 Snapshots of the system a < a
spSystem made of 216000 + 4200 particles. a=1.1 and ρ = 1.01
(left) t = 5 · D and (right) t = 100 · D
⇒ Complete layer growth onto the substrate
Analysis of the fully wetted substrate
(left) perpendicular density profile (right) defect density η for the nth layer
0 5 10 15
distance to the substrate z [σ]
0 0,002 0,004 0,006 0,008
density ρ(z)
t = 600 D t = 300 D t = 6 D
1 2 3 4 5
layer n
0,1 0,2 0,3 0,4 0,5
defect density η
a=1.01σ a=1.05σ a=1.10σ
1 2 3 4 5
layer n
3400 3600 3800 4000 4200
# of particles in layer N(n)
⇒ Induced defects are compensated in the first 3 layers
(left) top view (right) side view
⇒ Crystal grows in random hexagonal closed packing (RHCP)
5 Snapshots of the system a > a
spSystem made of 216000 + 4200 particles. a=1.4 and ρ = 1.01
(left) t = 20 · D and (right) t = 150 · D
⇒ Droplet formation on the substrate
Time evolution of the largest cluster/droplet
0 100 200 300 400 500
t D after critical nucleus is formed
0 1000 2000 3000 4000 5000 6000
# of solid particles in cluster
ρ=1.01 ρ=1.0175 ρ=1.02
100 1000
# particles in crystallite
1 10
principal moments of gyration tensor
R¬e
z
R¬e
z
R||e
z
⇒ Analysis of the droplets shows non-spherical droplets even up to 4000 particles.
Effective nucelation rates due to the substrate
1,005 1,01 1,015 1,02 1,025
denstiy ρ
0 5e-06 1e-05 1,5e-05 2e-05
nucleation rate [σ5 /6D l]
a=1.15σ a=1.2σ a=1.25σ a=1.3σ a=1.35σ a=1.4σ
a
1,1 1,15 1,2 1,25 1,3 1,35 1,4
lattice constant a [σ]
5e-06 1e-05 1,5e-05
nucleation rate [σ5 /6D l] ρ=1.01ρ=1.015
ρ=1.02
⇒ Decreasing nucleation rate with increasing mismatch to the substrate
References
[1] M. Heni and H. L¨owen. Phys. Rev. Lett., 85:3668–3671, 2000.
[2] S. Pronk and D. Frenkel. Phys. Rev. Lett., 90(25):255501, 2003.
[3] T. Schilling, S. Dorosz, H. J. Sch¨ope, and G. Opletal. J. of Phys: Cond. Matter, 23(19), 2011.
[4] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti. Phys. Rev. B, 28(2), 1983.
[5] W.-S. Xu, Z.-Y. Sun, and L.-J. An. Journal of Chemical Physics, 132(14):144506, 2010.
