nucleation kinetics in closed systems - fzu
TRANSCRIPT
Nucleation kinetics in closed systemsCOST Action CM1402: From molecules to crystals – how do organic
molecules form crystals?
Zdenek Kožíšek
Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic
[email protected]://www.fzu.cz/˜kozisek/
2-6 November 2015, Lyon, France
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 1 / 37
PragueCharles Bridge, Prague Castle
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 2 / 37
Institute of Physics CAS, Prague
Phase transition theory group (Dept. of Optical Materials):Zdenek Kožíšek, Pavel Demo, Alexei Sveshnikov, Jan Kulveit (PhD Student)
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 3 / 37
Contents
1 Introduction
2 Standard nucleation modelKinetic equationsWork of formation of clusters
3 Selected ApplicationsHomogeneous nucleationHeterogeneous nucleation on active centers
4 Summary
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 4 / 37
IntroductionNucleation
process leading to the formation of a new phase (solid, liquid) within metastableoriginal phase (undercooled melt, supersaturated vapor or solution)
first step in crystallization process; plays a decisive role in determining thecrystal structure and the size distribution of nuclei
u--
uuu ut srr
rr
nuclei of a new phase (droplet, crystal)
parent phase (vapor, solution or liquid)
homogeneous nucleation (HON)(at random sites in the bulk of a parent phase)
heterogeneous nucleation (HEN)(on foreign substrate, impurities, defects, active centres)
nucleus:�� ��? the smallest observable “particle”
�� ��? (often ≈ 1µm)
Clusters of a new phase are formed on nucleation sites due to fluctuations and afterovercoming some critical size (< 1nm) become nuclei (overcritical clusters).Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 5 / 37
IntroductionCritical supersaturation
Experimental data: H.R. Pruppacher: A new look at homogeneous ice nucleationin supercooled water drops, J. Atmospheric Sciences 52(11) (1995) 1924.
Supersaturation (supercooling) increases with volume decrease.⇒ nucleation kinetics in confined volumesZ. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 6 / 37
Standard nucleation modelUNARY NUCLEATION
(single component system of a new phase)
-� -� -� -� q q qu uu uuu uu uuk+1
k−2
k+2
k−3
k+3
k−4
k+4
k−5
BINARY NUCLEATION(A, B components)
-� -�u uu uuu
6
?
6
?
-�
-�
6
?
6
?
-�
6
?
u uu uuu
uu uuu
uuu
k+A(1, 0)
k−A(2, 0)
k+A(2, 0)
k−A(3, 0)
k+A(0, 1)
k−A(1, 1)
k+A(1, 1)
k−A(2, 1)
k−A(1, 2)
k+A(0, 2)
k+B(0, 2) k−B(0, 3)
k+B(0, 1) k−B(0, 2) k+B(1, 1) k−B(1, 2)
k+B(1, 0) k−B(1, 1) k+B(2, 0) k−B(2, 1)
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 7 / 37
Standard nucleation modelKinetic equations
k+i (k−i ) – attachment (detachment) frequencies of molecules
dFi
dt= k+
i−1Fi−1 − [k+i + k−i ]Fi + k−i+1Fi+1 = Ji−1(t)− Ji (t) (1)
Ji (t) = k+i Fi (t)− k−i+1Fi+1(t) (2)
Fi – number density of clusters of size iJi - cluster flux density (nucleation rate for i∗)k+
i (k−i ) – attachment (detachment) frequencies
Total number of nuclei greater than m : Zm(t) =∑
i>m Fi (t) =∫ t
0 Jm(t ′)dt ′
Local equilibrium:
Ji = 0 ⇒ k+i F 0
i = k−i+1F 0i+1 ⇒�
�k−i+1 = k+
iF 0
iF 0
i+1= k+
i exp(
W i+1−W ikBT
)(3)
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 8 / 37
Kinetic equationsEquilibrium
Ji = 0 ⇒ k+i F 0
i = k−i+1F 0i+1 (4)
⇒ F 0i = F 0
1k+
1 k+2 k+
3 . . . k+i−1
k−2 k−3 k−4 . . . k−iF 0
i – equilibrium number of cluster formed by i molecules It can be shown that
F 0i = B2 exp
(−Wi
kT
)Homogeneous nucleation, self-consistent model: B2 = N1 exp
(W1kT
)N1 – number of molecules within parent phaseKnowing F 0
i and k+i ⇒ k−i from Eq. (4).
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 9 / 37
Standard nucleation modelClassical nucleation theory (CNT)
Initial and boundary conditions
N1 - number of molecules (number of nucleation sites)F1(t = 0) = N1, Fi>1(t = 0) = 0
F1(t) = N1 CNT
t � induction time⇒Stationary nucleation (steady-state): Ji (t) = JS = const .
JS =
( ∞∑i=1
1k+
i F 0i
)−1
exact analytical formula
JS = k+i∗ zF 0
i∗ , where Zeldovich factor: z =
√1
2πkT
(−d2Wi
di2
)i=i∗
F 0i = B exp
(−Wi
kT
); B = N1 exp
(W1
kT
) �� ��k+i ,W i =?; small clusters?
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 10 / 37
Standard nucleation modelStationary nucleation (steady-state)
Ji = JS = const . for any cluster size i (N1 = const .!)⇒Boundary conditions:
F Si → F 0
i for i → 1; F Si → 0 for i →∞
JS = k+i F S
i − k−i+1F Si+1
= k+i F 0
iF S
i
F 0i− k−i+1F 0
i+1︸ ︷︷ ︸F Si+1
F 0i+1
= k+i F 0
i
(F S
i
F 0i− F S
i+1
F 0i+1
)k+
i F 0i
M−1∑i=1
JS
k+i F 0
i=
(F S
1
F 01− F S
2
F 02
)+
(F S
2
F 02− F S
3
F 03
)+
(F S
3
F 03− F S
4
F 04
)+ . . . = 1+
F SM
F 0M→ 1
for M →∞���
���
���
���
���
JS =
( ∞∑i=1
1k+
i F 0i
)−1
exact analytical formula
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 11 / 37
Standard nucleation modelStationary nucleation (steady-state)
Ji = JS = const . for any cluster size i (N1 = const .!)⇒Boundary conditions:
F Si → F 0
i for i → 1; F Si → 0 for i →∞
JS = k+i F S
i − k−i+1F Si+1 = k+
i F 0i
F Si
F 0i− k−i+1F 0
i+1︸ ︷︷ ︸F Si+1
F 0i+1
= k+i F 0
i
(F S
i
F 0i− F S
i+1
F 0i+1
)k+
i F 0i
M−1∑i=1
JS
k+i F 0
i
=
(F S
1
F 01− F S
2
F 02
)+
(F S
2
F 02− F S
3
F 03
)+
(F S
3
F 03− F S
4
F 04
)+ . . . = 1+
F SM
F 0M→ 1
for M →∞���
���
���
���
���
JS =
( ∞∑i=1
1k+
i F 0i
)−1
exact analytical formula
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 11 / 37
Standard nucleation modelStationary nucleation (steady-state)
Ji = JS = const . for any cluster size i (N1 = const .!)⇒Boundary conditions:
F Si → F 0
i for i → 1; F Si → 0 for i →∞
JS = k+i F S
i − k−i+1F Si+1 = k+
i F 0i
F Si
F 0i− k−i+1F 0
i+1︸ ︷︷ ︸F Si+1
F 0i+1
= k+i F 0
i
(F S
i
F 0i− F S
i+1
F 0i+1
)k+
i F 0i
M−1∑i=1
JS
k+i F 0
i=
(F S
1
F 01− F S
2
F 02
)+
(F S
2
F 02− F S
3
F 03
)+
(F S
3
F 03− F S
4
F 04
)+ . . .
= 1+F S
M
F 0M→ 1
for M →∞���
���
���
���
���
JS =
( ∞∑i=1
1k+
i F 0i
)−1
exact analytical formula
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 11 / 37
Standard nucleation modelStationary nucleation (steady-state)
Ji = JS = const . for any cluster size i (N1 = const .!)⇒Boundary conditions:
F Si → F 0
i for i → 1; F Si → 0 for i →∞
JS = k+i F S
i − k−i+1F Si+1 = k+
i F 0i
F Si
F 0i− k−i+1F 0
i+1︸ ︷︷ ︸F Si+1
F 0i+1
= k+i F 0
i
(F S
i
F 0i− F S
i+1
F 0i+1
)k+
i F 0i
M−1∑i=1
JS
k+i F 0
i=
(F S
1
F 01− F S
2
F 02
)+
(F S
2
F 02− F S
3
F 03
)+
(F S
3
F 03− F S
4
F 04
)+ . . . = 1+
F SM
F 0M→ 1
for M →∞���
���
���
���
���
JS =
( ∞∑i=1
1k+
i F 0i
)−1
exact analytical formula
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 11 / 37
Standard nucleation modelStationary nucleation (steady-state)
Ji = JS = const . for any cluster size i (N1 = const .!)⇒Boundary conditions:
F Si → F 0
i for i → 1; F Si → 0 for i →∞
JS = k+i F S
i − k−i+1F Si+1 = k+
i F 0i
F Si
F 0i− k−i+1F 0
i+1︸ ︷︷ ︸F Si+1
F 0i+1
= k+i F 0
i
(F S
i
F 0i− F S
i+1
F 0i+1
)k+
i F 0i
M−1∑i=1
JS
k+i F 0
i=
(F S
1
F 01− F S
2
F 02
)+
(F S
2
F 02− F S
3
F 03
)+
(F S
3
F 03− F S
4
F 04
)+ . . . = 1+
F SM
F 0M→ 1
for M →∞���
���
���
���
���
JS =
( ∞∑i=1
1k+
i F 0i
)−1
exact analytical formula
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 11 / 37
Standard nucleation modelk+
i - attachment frequency
k−i follows from the principle of local equilibirum (3)
Crystal nucleation
Crystal phase corresponds a stable phase, liquid a metastable phase, and in between is thediffusion activation energy.
From: Yukio Saito,Statistical Physics of Crystal Growth, Word Scientific (1996).
k+i = RDAi exp
(−
ED
kT
)exp
(−
q(Wi+1 −Wi )
kT
); q = 0.5[1 + sign(Wi+1 −Wi )]
Ai = γi2/3 = 4πr2 – surface areaRD – mean number of molecules striking on unit nucleus surface
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 12 / 37
Standard nucleation modelk+
i - attachment frequency
Vapor→ crystal
RD =P√
2πm1kT(deposition rate); P – vapor pressure; m1 – molecular mass
Melt→ crystal (unary parent phase)D. Turnbull, J. Fisher, Rate of nucleation in condensed systems, J. Chem. Phys. 17 (1949) 145.
RD = NS
(kTh
); NS = %SAi
NS – number of nucleation sites on the nucelus surface; %S – surface density of molecules
Solution→ crystalHON: (A) Direct-impingement control / (B) Volume-diffusion control
RD = CNS
(kTh
); C - concentration; (A) nucleation kinetics is restrictive
RD − incoming diffusion flux of monomers; (B) HON is controlled by volume diffusion
Details: D. Kashiev, Cryst. Res. Technol. 38 (2003) 555.Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 13 / 37
Standard nucleation modelWork of formation of clusters
Formation of phase interface is energetically disadvantageous
Homogeneous nucleation:
Capillarity approximation
Wi = −i∆µ+ γi2/3σ︸ ︷︷ ︸ = −43πr3
v1∆µ+ 4πr2σ
W S =∑
k Akσk – surface energy
i – cluster size (number of molecules within cluster)r – cluster radius; σ – interfacial energy; v1 – molecular volume∆µ – difference of chemical potentialsAk – surface areas; σk – corresponding interfacial energies
Vn% = im1 ⇒ r(i)
Vn - nucleus volume; % – density of crystal phase; m1 – molecular mass
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 14 / 37
Standard nucleation modelWork of formation of clusters
Homogeneous nucleation
0
5
10
15
20
25
30
0 10 20 30 40 50 60
W (
in k
BT
unit
s)
i
∂Wi
∂i= 0 ⇒ i∗ =
(2γσ3∆µ
)3
; i∗ – critical size; W∗ = Wi∗ – nucleation barrier
melt→ crystal: ∆µ =∆hE
NATE(TE − T ) solution→ crystal: ∆µ = kBT ln S
∆hE – heat of fusion; NA – Avogadro constant; TE – equilibrium temperature; T – temperaturekB – Boltzmann constant; S – supersaturation;
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 15 / 37
Standard nucleation modelWork of formation of clusters
Nucleation in polymer systems: thermodynamic aspects
M. Nishi et al., Polymer Journal 31 (1999) 749.Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 16 / 37
Standard nucleation modelClassical nucleation theory
Classical nucleation theory (CNT): fails to explain exp. data (Wi = ??)
Argon Lennard-Jones nucleation – MC simulations
δ∆Wn = Wn −Wn−1
⇒ Down to very small cluster sizes,classical nucleation theory built on theliquid drop model can be used veryaccurately to describe the workrequired to add a monomer to thecluster!
However, erroneous absolute value forthe cluster work of formation, Wi .
B. Hale, G. Wilemski, 18th ICNAA conference (2009) 593.J. Merikanto et al., PRL 98 (2007) 145702.Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 17 / 37
Standard nucleation modelConfined systems
Initial and boundary conditionsFi (t = 0) = F 0
i pro i ≤ i0 (usually i0 ≈ i∗/2)
F 0i - equilibrium distribution of nuclei
Fi (t = 0) = 0 pro i > i0
F1(t) = F 01 = Nn = const . CNT model (HON+HEN)
Nn – number of nucleation sites (Nn = N1 for HON) Nn �∑
i>1 iFi
F1(t) = N1(t = 0)−∑
i>1 iFi (t) confined system (HON)
F1(t) = Nn(t = 0)−∑i>1
Nsi (t)
︸ ︷︷ ︸confined system (HEN)free substrate surface ↓
number of nucleation sites occupied by nuclei
N1(t) = NT −∑
i>1 iFi (t) confined system (HEN)volume of parent phase ↓
N1 - number of molecules within parent phase
NT - total number of molecules within system (liquid + solid phase)
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 18 / 37
Standard nucleation modelConfined systems
Initial and boundary conditionsFi (t = 0) = F 0
i pro i ≤ i0 (usually i0 ≈ i∗/2)
F 0i - equilibrium distribution of nuclei
Fi (t = 0) = 0 pro i > i0
F1(t) = F 01 = Nn = const . CNT model (HON+HEN)
Nn – number of nucleation sites (Nn = N1 for HON) Nn �∑
i>1 iFi
F1(t) = N1(t = 0)−∑
i>1 iFi (t) confined system (HON)
F1(t) = Nn(t = 0)−∑i>1
Nsi (t)
︸ ︷︷ ︸confined system (HEN)free substrate surface ↓
number of nucleation sites occupied by nuclei
N1(t) = NT −∑
i>1 iFi (t) confined system (HEN)volume of parent phase ↓
N1 - number of molecules within parent phase
NT - total number of molecules within system (liquid + solid phase)
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 18 / 37
Selected ApplicationsHomogeneous nucleation
HON: ethanol, V→L transition (T = 260 K)[Z. Kožíšek et al., J. Chem. Phys. 125 (2006) 114504]
0
0.5
1
1.5
2
2.5
3
50 100 150 200 250 300 350 400
k+, k
- x 1
0-9 (
s-1)
i
ki+
ki+1-
S=1
S=2
S=4
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 19 / 37
Selected ApplicationsHomogeneous nucleation
Size distribution of nuclei
0
5
10
15
20
0 50 100 150 200 250 300 350 400 450 500
Log 1
0 F
(m
-3)
i
S=3, i* = 75
υ = 11121314151F0
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 20 / 37
Selected ApplicationsHomogeneous nucleation
Distribution function - time dependence
1
10
100
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
0 20 40 60 80 100
F (
m-3
)
Dimensionless time
Sini = 3
75
150
closed system
open system
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80 100 120 140F
x 1
0-17 (
m-3
)Dimensionless time
Sini = 5
24
40
100
closed systemopen system
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 21 / 37
Selected ApplicationsHomogeneous nucleation
Distribution function - size dependence (Sini = 5)
0
1
2
3
0 10000 20000 30000
F x
10-1
5 (m
-3)
i
40 80 120 140160
0
2
4
6
8
10
12
14
0 20000 40000 60000F
x 10
-14 (
m-3
)i
160
200
240
320
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 22 / 37
Selected ApplicationsHomogeneous nucleation
Nucleation rate - time dependence
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120 140
J/JS (
m-3
s-1)
Dimensionless time
Sini=3
75 150
closed systemopen system
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120 140 160
J/JS (
m-3
s-1)
Dimensionless time
Sini = 5
24
100
300
closed systemopen system
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 23 / 37
Selected ApplicationsHomogeneous nucleation
Nucleation rate (Sini = 5, JS0 = JS at Sini )
Open system (Sini = const .)
0
1
2
3530
2520
30
20
10
0
1
2
J/JS
i
υ
J/JS
Closed system
0
0.5
1
60000 40000
20000
400
300
200
100
0
0.5
1
J/JS0
i
υ
J/JS0
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 24 / 37
Selected ApplicationsHomogeneous nucleation
Nucleation rate (Sini = 5)
0
0.1
0.2
0.3
0 20000 40000 60000
J/JS 0
i
180
200
240
300-0.01
-0.005
0
0.005
0 40000 80000 120000
J/JS 0
i
400
1000
2000
3000
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 25 / 37
Selected ApplicationsHomogeneous nucleation
Supersaturation
1.0
2.0
3.0
4.0
5.0
0 200 400 600 800
Supe
rsat
urat
ion
Dimensionless time
1.1
1.2
0 1000 2000 3000
Supe
rsat
urat
ion
Dimensionless time
0
10000
20000
30000
40000
50000
0 500 1000 1500 2000 2500 3000
Cri
tical
siz
e
Dimensionless timeZ. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 26 / 37
Selected ApplicationsHomogeneous nucleation
HON: Liquid/Solution→Solid transitionZ. Kožíšek, CrystEngComm 15 (2013) 2269
Size distribution of nuclei
CNT
0
2
4
6
8
10
12
2 4 6 8 10 12 14 16
F x
10
-19 (
m-3
)
r (nm)
200600
10001100120013001400
N1(t) = const .
Encapsulated system
0
2
4
6
8
10
12
2 4 6 8 10 12 14 16
F x
10
-19 (
m-3
)
r (nm)
200600
10001100120013001400
N1 decreases with time
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 27 / 37
Selected ApplicationsHomogeneous nucleation
Model system: nucleation of Ni melt
DSC experiments and MC simulations [J. Bokeloh et al., PRL 107 (2011) 145701J. Bokeloh et al., Eur. Phys. J. Special Topics 223 (2014) 511]J was obtained from a statistical evaluation of crystallization behavior during continuous cooling.A single Ni sample was repeatedly heated up to 1773 K an subsequently cooled down to 1373 K.
sample masses: 23 µg – 63 mg
survivorship function:Fsur (∆T ) = 1− exp(
∫J(∆T )dt)
∆T ⇒ J = Γ exp(−∆W∗
kBT )
MC simulations show a deviation of theenergy of formation ∆Wi from CNT.
However, the actual height of the energybarrier is in good agreement with CNT.
All system parameters are known⇒ we can determine the size distribution of nuclei Fi
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 28 / 37
Selected ApplicationsHomogeneous nucleation
Ni melt: ∆hE = 17.29 kJ mol−1, TE = 1748 K , %S = 8357 kg m−3, σ = σmT/TE ,where σm = 0.275 J m−2, E = 29.085 kJ mol−1
T = 1449 K ⇒ i∗ = 456
0
5
10
15
20
25
30
0 0.1 0.2
F x 1
0-4
(m
-3
)
Time (s)
456
500
600
1000
�� ��Fi V = 1 – one i-sized nucleus is formed in V
F S456 = 250000 m−3 ⇒ V1 = 1/FS
i∗ = 4 cm3
sample masses: 23 µg – 63 mg⇒ V = 7.15× 10−9 m3 − 2.9× 10−12 m3
In difference of experimental data, no critical nuclei are formed in Ni melt.Solution: reduce the interfacial energy or take into account σ(i) dependency⇒
lower nucleation barrierMaybe heterogeneous nucleation occurs ?
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 29 / 37
Selected ApplicationsNucleation on active centers
CNT model of nucleation on active centers – one additional equation is needed[S. Toshev, I. Gutzow, Krist. Tech. 7 (1972) 43; P. Ascarelli, S. Fontana: Diamond Rel. Mater. 2 (1993) 990-996; D. Kashchiev:
Nucleation, Butterworth-Heinemann Boston, 2000]
0
0.2
0.4
0.6
0.8
1
1.2
0 2000 4000 6000
Z/N
0
Dimensionless time
Our approachAvrami modelOpen system
Additional equation:
dZ (t)dt
= [N0 − Z (t)]JS(t)
Z (t) - the total number of nuclei at time tN0 - the number of active sites;JS - time dependent nucleation frequency
(usually taken as fit<F8> parameter)
Our new model does not need additional equation!!→ only modification of boundary conditions
Z. Kozisek et al., Transient nucleation on inhomogeneous foreign substrate, J. Chem. Phys. 108 (1998) 9835; Nucleation on
active sites: evolution of size distribution, J. Cryst. Growth 209 (2000) 198-202.
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 30 / 37
Selected Applications
Nucleation on active centersHEN: Vapor→Solid transition
0
0.001
1 2 3 4 5 6 7 8
F/N
A
r (nm)
20406080
100120140
160200240280320360400
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 31 / 37
Selected Applications
Nucleation on active centersHEN: Vapor→Solid transitionH. Kumomi, F.G. Shi: Phys. Rev. Lett. 82 (1999) 2717.
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 32 / 37
Selected ApplicationsNucleation on active centers
HEN: Vapor→Solid transition
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2 3 4 5 6 7 8 9 10 11
F/N
A x
10
4
r (nm)
240280320360
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 33 / 37
Selected ApplicationsNucleation on active centers
HEN: polymer nucleation from melt (polyethylene) ∆T = 10.4 K
0
1
2
3
4
5
6
7
8
0 50 100 150 200 250 300
Z x
10
-13(m
-3)
Time (s)
567data
Z. Kožíšek et al. Nucleation kinetics of folded chain crystals of polyethylene on active centers, J. Chem. Phys. 121 (2004) 1587.
Z. Kožíšek et al. Nucleation on active centers in confined volumes, J. Chem. Phys. 134 (2011) 114904
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 34 / 37
Selected ApplicationsNucleation on active centers
HEN: polymer nucleation from melt (polyethylene)2007: K. Okada et al.: Polymer 48 (2007) 1116-1126.
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 10 20 30 40 50 60 70 80 90 100
Log 1
0 F
(i, t)
t (min)
Experimental data
i=20940
6900
-6
-4
-2
0
2
4
0 1 2 3 4 5 6
Log 1
0 F
Log10 i
data: Okada et al.: Polymer 48 (2007) 382-392
7 min14 min21 min35 min64 min98 min
σnano (instead of σ) introduced to fit F Si ⇒ k∗ = 9, l∗ = 24.9, m∗ = 4.5⇒ 3D model
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 35 / 37
Selected ApplicationsPolymer nucleation on active centers (polyethylene)
-6
-5
-4
-3
-2
-1
0
1
2
3
0 10 20 30 40 50 60 70 80 90 100
Log 1
0 F
t (min)
2Dmodel: macro parameters, i*=666, cN = 4.4 10-15
20*cN940*cN
6900*cN -6
-5
-4
-3
-2
-1
0
1
2
0 10 20 30 40 50 60 70 80 90 100Lo
g 10
F (
F*c
n in
m-3
)
t (min)
3D: nano parameters, i*=487, iexp=500,630
50063020
9406900
Z. Kožíšek, M. Hikosaka, K. Okada, P. Demo: J. Chem. Phys. 134 (2011) 114904 & 136 (2012) 164506
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 36 / 37
SummaryAdvantages
Relatively simple model enables to determine basiccharacteristics of nucleation in real time.Model takes into account depletion of the parent phase duringphase transformation.Model includes exhaustion of active centres(new approach to heterogeneous nucleation).
This work was supported by the Project No. LD15004 (VES15COST CZ) of the Ministry of Education of the Czech Republic.
Thank you for your attention.
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 37 / 37
SummaryAdvantages
Relatively simple model enables to determine basiccharacteristics of nucleation in real time.Model takes into account depletion of the parent phase duringphase transformation.Model includes exhaustion of active centres(new approach to heterogeneous nucleation).
This work was supported by the Project No. LD15004 (VES15COST CZ) of the Ministry of Education of the Czech Republic.
Thank you for your attention.
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 37 / 37
SummaryAdvantages
Relatively simple model enables to determine basiccharacteristics of nucleation in real time.Model takes into account depletion of the parent phase duringphase transformation.Model includes exhaustion of active centres(new approach to heterogeneous nucleation).
This work was supported by the Project No. LD15004 (VES15COST CZ) of the Ministry of Education of the Czech Republic.
Thank you for your attention.
Z. Kozisek (Prague, Czech Republic) Nucleation kinetics 2-6 Nov 2015 Lyon 37 / 37