Diffuse interface models of nucleation

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Diffuse interface models of nucleation L.
Gránásy, T. Börzsönyi, T. Pusztai Research
Institute for Solid State Physics and
Optics Budapest, Hungary
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?
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Contents - Introduction (nucleation
diffuse intf.) - Examples of superiority of
diff. intf. models 1. homogeneous vapor-liquid
nucleation (perturbative density functional
theory) 2. crystal nucleation in hard-sphere
fluid (density functional theory, GL free
en.) 3. crystal nucleation in unary and binary
liquids (phase field theory) - Multi-domain
freezing (PFT)
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I. Introduction (nucleation)
Heterophase fluctuations
Crystal-liquid intf.
Classical (sharp intf.) picture cluster free
energy (W) volumetric interfacial
contributions
Crit. fluct. typically 10 100 molecules
W Diffuse interface is needed !!!
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Nucleation rate
? net formation rate of supercritical
fluctuations
critical size 1 nm, still J can be measured!
Two-step methods The nucl. growth
rates have maxima at different
distances from equilibrium ? 1. step
nucleation 2. step
development
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Failure of the sharp-interface model
Vapor-liquid nucleation
Classical sharp interface model W(R)
? 4?/3 R3 ?g 4 ? R2 ? W 16?/3 ? 3 / ?g2
R 2 ? / ?g
R
Liquid-crystal nucleation
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More accurate cluster model is needed !
Nucleation rate
J J0 exp ? W / kT
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II. Diffuse interface models
1. LIQUID-VAPOR NUCLEATION (Perturbative
Density Functional Theory)
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Perturbative density functional theory
Oxtoby et al. (1988,1991,1995)
Perturbation theory with respect to HS
reference.
Free energy functional
Prof. Piechór
Random phase approx.
Grand canonical potential
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Nucleus extremum of ?
?
Euler-Lagrange eq.
Yukawa
HS Carnahan-Starling
Remarks - 3 model parameters Yukawa
(?,?) ? , that can be fixed using
Peq, ?l, ?. - The integral Euler-Lagrange
eq. can be solved by self-consistent
iteration.
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Other substances
L. Gránásy, Z. Jurek, D.W. Oxtoby, PRE 2000
 
 
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2. CRYSTAL NUCLEATION IN HS FLUID (Density
Functional Theory)
Colloidal suspensions
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Density functional theory of crystallization
Crystal ? highly inhomogeneous liquid
Fourier expansion ?(r) ?0 1 ?K uK exp( i K
r ) 0 ? K RLVs Shih et al.
- Free energy difference expanded in terms of uK
for DDWs - Dominant DWs (110) RLVs for bcc
(111) for fcc
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Alexander-McTague nth order contribution
nonzero if ?j1n Kj 0 bcc ?? a2
m2 a3 m3 a4 m4 fcc ?? a2 m2 a4 m4
a6 m6 where m u/u0 ? 0,1 1st
order phase transition
?? (1) ? ?g ??? /?m?1 0 ?? a2
(m2 ? 2m3 m4) ? ?g (4m3?3m4) ?? a2 (m2
? 2m4 m6) ? ?g (3m4?2m6) a2 c ?
? d10-90 Calculation without adjustable
parameters! Euler-Lagrange eq. 0
???/?m ? 2c?2m Boundary cond. m ? m0
for ?r? ? ? ?m ? 0 for ?r? ? 0
WDFT ? dr3??m(r) c (?m)2
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Results
SLOW bcc nuclation !
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Computer simulations (MC) for HS system
Direct evaluation of nucleation barrier (Auer
Frenkel, Nature, 2001)
Barrier height shows a minimum as a function of
?? !
a2 ? S ?1(K) c ? ?C(K)
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3. CRYSTAL NUCLEATION GROWTH IN REALISTIC
FLUIDS (Phase Field Theory)
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Nucleation in binary phase-field theory

• Free energy functional (standard theory)
• where
• f(?,c) W T g(?) 1 ? P(?) fS P(?) fL
• W (1 ? c) WA c WB
• g(?) ¼ ?2 (1 ? ?)2,
• P(?) ?3 (10 ? 15? 16?2),
• fL,S (1 ? c) fL,SA c fL,SB R T / v c
  ln(c) (1 ? c) ln(1 ? c),

Model parameters WA, WB ?2 ? intf. free
energy intf. thickness to satisfy all
conditions ?2 ?B2 (?A2 ? ?B2)(T ? TB)/(TA ?
TB)
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Time evolution ? CA CH eqs.
Composition
Phase-field
??
??c
phase field theory mean-field type approach

Modeling of nucleation
- inclusion of noise into governing eqs.
- introduction of critical fluctuations small
amp. noise
defined by
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Nucleation rate in 3D
(a) One-comp. liquids

• MD simulations
• - mod. Lennard-Jones syst.
• - ? ? 0.36
• - d10?90,? ? 3?
• - spec. heat diff. considered
• - J known at 25 ?T
• Exp. ice-water system
• - ? ? 0.297 from GT-eff.
• - d10?90,? from MD
• - spec. heat diff. considered
• - J from emulsion, single
• drop supersonic nozzle

Quantitative agreement without adjustable paramete
rs!
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(b) Binary alloy

• Calculation
• - Cu-Ni properties
• - ? ? 0.6 from dihedral angle
• - d10?90,? ? 3(v/N0)1/3
• - spec. heat diff. considered
• - D (1 - c) DA c DB
• - J 10-4 1 1/drop/s
• Experiment
• EML drops, d 6mm
• (Willnecker et al, 1988)

Homogeneous nucleation!
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III. Multi-domain freezing (Interacting
nucleation growth)
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A. Formal theory of nucleation growth
(Johnson-Mehl-Avrami-Kolmogorov kinetics)
Overlapping transformed fraction Mean field
correction For constant J v Exact if
(i) Infinite medium
(ii) spatially homogeneous J
(iii) common v(t) ? uniformly oriented
convex particles constant J v
p 1 d diffusion controlled growth (?)
p ? 1 d/2
Kolmogorov-scaling
Kolmogorov-exponent
Used in Materials science Chemistry Biology Atmo
spheric sciences Astronomy
The phase field theory is an appropriate tool
to study this
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Multi-domain freezing

Aims - distinguish crystalline particles -
grain boundary energy
Non-conservative orientational field ?
constant ? 0, 1 in solid fluctuates
between 0 and 1 in liquid
f s,tot fs fori where fori M????
Time evolution
where ?? ??,0P(?)
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Limitations
Different length- time scales for ?, c, ?. To
enable calculations ? broader interface
(41.6 nm) reduced ? (1/6?) enhanced Dl
(100?) Thermodynamic data of Cu-Ni is
used (nearly ideal solution) ? only qualitative
results
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Nucleation
(a) Noise induced
? color code
solidus
liquidus
solid
liquid
composition
phase field
orientation
(b) Inserted critical fluctuations
( coupling between c J )
? 0.5
orientation
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Medium-scale simulation
3000 ? 3000 grid
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Medium-scale simulation
3000 ? 3000 grid
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Large-scale simulation
7000 ? 7000 grid
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Transformation kinetics
x (c-cs)/(cl-cs) Solidus x 0 Liquidus x
1
L. Gránásy, T. Börzsönyi, T. Pusztai, PRL in print
(d)
grid 2000 2 ? 7000 2 T 1574 K s0 0.25 M? ?
2T/Dl 0.9 Ds/Dl 1 M?,lM?/Dl 720
M?,sM?/Dl 7.2 ?10-4
Textbook solution diffusion controlled growth
(?) p ? 1 d/2 constant J v
p 1 d
? X/Xmax p dlog?log(1??) / dlnt
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Reduced M? ? multiple orientations nucleate
M?/30
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Se
Spherulitic growth
Ryschenkow Faivre JCG (1988)
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Fractal-like multidomains
Fleury, Nature (1997)
L. Gránásy, T. Börzsönyi, T. Pusztai Research
Institute for Solid State Physics and Optics,
Budapest, Hungary
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Summary The diffuse interface theories
(density functional theory, phase field theory)
proved superior to the classical sharp
interface model for (1) vapor-liquid
nucleation (2) crystal nucleation in HS
liquid (3) crystal nucleation in realistic
liquids Interacting nucleation growth has
been studied in PFT Kolmogorov exponent
depends on composition transformed
fraction Possibility to model multi-domain
growth
Financed by ESA Prodex No. 14613/00/NL/SFe Forms
part of ESA MAP Project AO-99-101