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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

  • 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

  • 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

  • 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

  • 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

  • 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

  • Background and MotivationHomogenous nucleation

    Heterogenous nucleation

    Local crystal symmetry

    Hard spheres system. = 1.03

    Loading

    Sven Dorosz Nucleation in Hard Spheres Systems

    Lavf52.64.2

    test1800.mp4Media File (video/mp4)

  • 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

  • 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

  • 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

  • Background and MotivationHomogenous nucleation

    Heterogenous nucleation

    two snapshots

    Spontaneous formation of a crystallite

    Sven Dorosz Nucleation in Hard Spheres Systems

  • 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

  • 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

  • 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

  • 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

  • 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

  • Background and MotivationHomogenous nucleation

    Heterogenous nucleation

    Immediate wetting

    a < aeq

    t = 6D t = 150D

    Sven Dorosz Nucleation in Hard Spheres Systems

  • 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

  • Background and MotivationHomogenous nucleation

    Heterogenous nucleation

    3 layer stacking

    Domains of ABA and ABC structure

    Sven Dorosz Nucleation in Hard Spheres Systems

  • Background and MotivationHomogenous nucleation

    Heterogenous nucleation

    5 layer stacking

    Crystal grows in random hexagonal closed packing (RHCP)

    Sven Dorosz Nucleation in Hard Spheres Systems

  • 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

  • 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

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