nucleation kinetics in closed systems - fzu

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


Institute of Physics CAS, Prague

Phase transition theory group (Dept. of Optical Materials):Zdenek Kožíšek, Pavel Demo, Alexei Sveshnikov, Jan Kulveit (PhD Student)


Contents

1 Introduction

2 Standard nucleation modelKinetic equationsWork of formation of clusters

3 Selected ApplicationsHomogeneous nucleationHeterogeneous nucleation on active centers

4 Summary


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)


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)


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).


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?


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





F Si → F 0


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





F Si → F 0


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





F Si → F 0


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



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


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



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;



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)


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



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



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



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



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


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




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



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



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


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



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



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 ?


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.


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


Selected Applications

Nucleation on active centersHEN: Vapor→Solid transitionH. Kumomi, F.G. Shi: Phys. Rev. Lett. 82 (1999) 2717.



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



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



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


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


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.


nucleation kinetics in closed systems - fzu

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