BASIC RESULTS AND APPLICATIONSBASIC RESULTS AND APPLICATIONSOF NUCLEATION THEORYOF NUCLEATION THEORY

Lectures by Dimo KashchievLectures by Dimo Kashchiev(EMSE Nucleation Workshop, Saint-Etienne, June 2003)

1. Thermodynamics of nucleation2. Kinetics of nucleation

3. Applications of nucleation theory

Further reading:Further reading: D. Kashchiev, D. Kashchiev, ““Nucleation: Basic Theory with Nucleation: Basic Theory with ApplicationsApplications””, Butterworth, Butterworth--Heinemann, Oxford, 2000Heinemann, Oxford, 2000

http://www.ipc.bas.bg/PPages/Kash/Monograph.htm

AbstractAbstract

The lectures provide an introduction to basic results in the thermodynamics and kinetics of nucleation and to some applications of the theory. The thermodynamic considerations are focused on the supersaturation dependence of the nucleus size and the nucleation work. The kinetic considerations show how the general formula for the nucleation rate is obtained in the scope of the generally accepted molecular model of nucleation. Finally, the application of the nucleation theory to the process of overall crystallization and to the problem of the induction time in this process is considered.

2

Lecture 1Lecture 1

THERMODYNAMICSTHERMODYNAMICSOF NUCLEATIONOF NUCLEATION

3

- Gas to liquid or crystalµgas ≈ µe + kT ln(p/pe)µliq or crys ≈ µe

∆µ ≡≡ µgas – µliq or crys = kT ln(p/pe)

- Solute to crystalµsolute ≈ µe + kT ln(C/Ce)µcrys ≈ µe

∆µ ≡≡ µsolute – µcrys = kT ln(C/Ce)

1. Supersaturation ∆µ∆µ

µold – chemical potential of bulk old phaseµnew – chemical potential of bulk new phaseµe – equilibrium chemical potentialµgas – chemical potential of gasµliq or crys – chem. potential of liquid or crystalµsolute – chemical potential of soluteµcrys – chemical potential of crystalk – Boltzmann constantT – absolute temperaturep – actual pressurepe – equilibrium (saturation) pressureC – actual concentration of soluteCe – equilibrium concentration (solubility)

Approximations:- ideal gas- dilute solution- liquid or crystal incompressibility

pe or Ce p or C

µe

µliq or crys

µgas or solute∆µ

gas or solute

liq or crys

chem

. pot

entia

l

pressure, concentration

4

∆µ ≡≡ µold − µnewDefinition:

Examples:

- Melt to crystalµmelt ≈ µe + smelt(Te − T)µcrys ≈ µe + scrys(Te − T)

∆µ ≡≡ µmelt – µcrys = ∆se(Te – T)

Undercooling ∆∆T(also used to express the supersaturation)

Definition: ∆T ≡≡ Te – T

Approximations:- smelt and scrys are T-independent

µmelt – chemical potential of meltsmelt – entropy per molecule of meltscrys – entropy per molecule of crystal∆se – melting entropyTe – equilibrium or melting temperature

Hence: ∆µ = ∆se ∆T (∆se = smelt – scrys)

T Te

µe

µcrys

µmelt∆µ

crys

melt

chem

. pot

entia

l

temperature

5

Supersaturation ratio S(also used to express the supersaturation)

S ≡≡ p/pe or C/CeDefinition:

Hence: ∆µ∆µ = kT lnS

2. Work W for cluster formation

Nucleation is the process of randomgeneration of such nanoscopicallysmall formations of the new phase thathave the ability for irreversible growthto macroscopically large sizes.

What is nucleation?

Definition: W(n) ≡≡ Gfin(n) – Gini

x – distance in spaceρ – molecular densityρold – molecular density of old phaseρrnew – molecular density of new phaseW – work to form a cluster of n moleculesn – number of molecules in clusterGfin – final Gibbs free energy of the system

(after the cluster formation)Gini – initial Gibbs free energy of the system

(before the cluster formation)

x

(b)

x

ρρold

ρρnewρρ

(a)phase

boundary

phaseboundary

(a) cluster, (b) its density profile

6

- Homogeneous nucleation (HON)

Gini = MµoldGfin(n) = (M − n)µold + G(n)G(n) = nµnew + Gex(n)

W(n) = − n∆µ + Gex(n) (n=1,2,3,…)

The cluster has (1) pn>p, (2) µn>µnew, (3) surface.

For that reason: Gex(n) = − (pn−p)Vn + (µn−µnew)n + Φ(n)

Gex(n) ≈ Φ(n)

Approximationfor incompressible phases:

Hence:

W(n) = −− n∆µ∆µ + ΦΦ(n) ΦΦ(n) = ?

HON: system in (a) initialand (b) final state

n

M−−nM

(a) (b)

M – total number of moleculesG – Gibbs free energy of clusterGex – excess Gibbs free energy of clusterpn – pressure inside clusterµn – chem. potential of molecule in clusterVn – volume of clusterΦ – total surface energy of cluster

7

According to Gibbs,Φ(n) = σnAn

Approximation of the Classical Nucleation Theory:

σn = σ (σ is n-independent)

Hence:

Φ(n) = σAn = aσn2/3

W(n) = −− n∆µ∆µ + aσσn2/3 (a=(36πv02)1/3 for spheres)

An – area of cluster surfaceσn – specific surface energy of the interface

between n-sized cluster and the old phaseσ – specific surface energy of the interface

between macroscopically large clusterand the old phase

a – cluster shape factorv0 – volume of molecule in cluster

- Gas or solute condensationW(n) = −− nkT ln S + aσσn2/3

- Melt crystallizationW(n) = −− n∆∆se∆∆T + aσσn2/3

8

(Stefan-Skapski-Turnbull)

Empirical estimation of σ:σ:

σ = βλ/v02/3

(β=0.2 to 0.6) λ– molecular heat of evaporation,sublimation, dissolution or melting

9

W vs. n for HON of water dropletsat T=293 K

1 50 100 150 200 2500

10

20

30

40

50

W /

kT

n

S = 4

S = 6

- Heterogeneous nucleation (HEN)

3D HEN: system in (a) initial and(b) final state

n

M−−n

(b)

substrate

M

(a)

substrateσσi σs

θ

substrate

n

Cap-shaped cluster on a substrate

a) 3D HEN

W(n) = −− n∆µ∆µ + aσσefn2/3

σef ≡ ψ1/3(θ)σψ(θ) = ¼(2 + cosθ)(1 – cosθ)2

cosθ = (σs – σi)/σ (Young)

(Volmer)

σef – effective specific surface energyθ – wetting angleψ – activity factor (0≤ψ≤1)σs – specific surface energy of the substrate

when contacting the bulk old phaseσi – specific surface energy of the interface

between macroscopically large clusterand the substrate

θ=180o, ψ=1 → HON (complete non-wetting)θ<180o, ψ<1 → HEN (incomplete wetting)θ=0, ψ=0 → HEN (complete wetting)

Hence: σef = σ → HONσef < σ → HEN

10

Approximation of theClassical Nucleation Theory:

θθ is n-independent.

11

W vs. n for HON and 3D HEN of water dropletsat T=293 K

1 20 40 60 80 1000

10

20

30

40

50

W /

kT

n

spheres (θ=180o, HON)

caps (θ=60o, 3D HEN)

S = 4

b) 2D HEN of monolayers M

(a)

substrate

M−−n

(b)

substraten

2D HEN: system in (a) initial and(b) final stateDisk-shaped cluster on a substrate

σσi σs

substraten

κ

- own substrate (∆σ∆σ=0)

W(n) = −− n∆µ∆µ + bκκn1/2

a0 – area of molecule in clusterd0 – diameter of molecule in cluster∆σ – wetting parameterκ – specific edge energy of clusterb – cluster shape factor

W(n) = −− n(∆µ∆µ−−a0∆σ∆σ) + bκκn1/2

- foreign substrate (∆σ∆σ≠≠0)

∆σ ≡ σ + σi – σs

(b=2(πa0)1/2 for disks)

κ ≈ σd0

12

Approximation of theClassical Nucleation Theory:∆σ∆σ and κκ are n-independent.

13

W vs. n for HON, 3D HEN and 2D HEN of water dropletsat T=293 K. As in HEN the substrate is the same,3D HEN of caps is more likely than 2D HEN of

disks (W for caps is lower than W for disks).

1 20 40 60 80 1000

10

20

30

40

50

W /

kT

n

S = 4spheres (θ=180o, HON)

caps (θ=60o, 3D HEN)

disks (2D HEN)

3. Cluster size distribution C(n)

C(n) = C0 e−W(n)/kT (Boltzmann-type)

C0 = ?C(n) – equilibrium concentration of

n-sized clustersC0 – concentration of nucleation sites

14

Approximation:no cluster-cluster interactions.

Examples:

- HON or 3D HEN

C(n) = C0 en∆µ/kT exp(−a σef n2/3/kT)

- 2D HEN on own substrate

C(n) = C0 en∆µ/kT exp(−bκn1/2/kT)

HEN:

C0 = 1/a0 ≈ 1019 m−2

(surface nucleation, no active centres)

C0 = Na/V < 1028−29 m−3

(volume nucleation on Na active centres)

C0 = Na/As < 1019 m−2

(surface nucleation on active centres)

HON:

C0 = 1/v0 ≈ 1028−29 m−3

(volume nucleation, no active centres)

(volume nucleation on Mseed seeds each with Na active centres)C0 = NaMseed/V < 1028−29 m−3

Na – number of active centresV – volume of old phaseMseed – number of seeds in the old phaseAs – area of substrate surface

15

C vs. n for HON of water droplets at T=293 Kin undersaturated (S=0.2), saturated (S=1)

and supersaturated (S=4.5) vapours

16

1 20 40 60 80 100

1025

1020

1015

1010

105

10.2

1

4.5C

(n)

(m-3)

n

17

4. Nucleus and nucleation work W*

Nucleus: the cluster that requires maximum work for itsformation.

Nucleation work: the work done to form the nucleus.

To remember:

The nucleus is in labile thermodynamic equilibrium – the freeenergy of the system diminishes when the nucleus either losesor gains molecules.

The nucleation work is the energy barrier to nucleation.

n* = ? W* = ? n* – number of molecules in nucleus(or nucleus size)

W* – nucleation work

Definitions:

18

n* and W* can be determined from the condition for maximum:

(dW/dn)n=n* = 0

W* = W(n*)

- HON or 3D HEN of condensed phasesFrom W(n) = − n∆µ + a σef n2/3

n* = 8a3σef3/27∆µ3 (Gibbs-Thomson equation)

n

HONσσi σs

θ

substrate

n

3D HENW* = 4a3σef

3/27∆µ2 Also: W* = (1/2)n*∆µ

In particular, for HON (σef=σ) of spheres (then a3=36πv02)

n* = 32πv02σ3/3∆µ3 W* = 16πv0

2σ3/3∆µ2

For the nucleus radius R*=(3v0/4π)1/3n*1/3 it follows that

R* = 2v0σ/∆µ (Gibbs-Thomson equation)R* – nucleus radius

19

Examples (spherical or cap-shaped nuclei):

- gas or solute condensation

W* = 16πv02σef

3/3(kT)2 ln2 S

- melt crystallization

W* = 16πv02σef

3/3∆se2∆T2

R* = 2v0σ/kT ln S n* = 32πv02σef

3/3(kT)3 ln3 S(Gibbs-Thomson equations)

R* = 2v0σ/∆se∆T n* = 32πv02σef

3/3∆se3∆T3

(Gibbs-Thomson equations)

20

R*, n* and W* vs. S for nucleation ofwater droplets in vapours at T=293 K

1 5 10 150

20

40

60(c)

lnS

W*

/ kT

120406080

100 (b)

n*

0

0.5

1.0

1.5

2.0(a)

R*

(nm

)

3D HEN (cap, θ=90o)

HON (sphere)

HON (sphere)

3D HEN (cap, θ=90o)

S

21

5 10 15 20 251

10

20

30

40

50

60 (b)

S

n*

1

10

20

30

40

50

60

(a)n*

(a) n* vs. S in HON ofwater droplets in vapours at

T=217 to 259 K.Y.Viisanen et al., J.Chem.Phys.

99(1993)4680

(b) n* vs. S in HON ofn-butanol droplets in vapours at

T=225 to 265 K.R.Strey et al., J.Phys.Chem.

98(1994)7748

Circles – experimental data;dashes – Gibbs-Thomson equation

without free parameters

22

- 2D HEN of monolayers of condensed phases

From W(n) = − n(∆µ−a0∆σ) + bκn1/2

W* = b2κ2/4(∆µ−a0∆σ) W* = n*(∆µ−a0∆σ)Also:

n* = b2κ2/4(∆µ−a0∆σ)2 (Gibbs-Thomson equation)

σs

σσi

substraten

κ

2D HEN

In particular, for 2D HEN of monolayer disks (b2=4πa0)on own substrate (∆σ=0)

W* = πa0κ2/∆µ

own substraten

κ

Monolayer disk onown substrateAlso: W* = n*∆µ

For the nucleus radius R*=(a0/π)1/2n*1/2 it follows that

R* = a0κ/∆µ (Gibbs-Thomson equation)

n* = πa0κ2/∆µ2 (Gibbs-Thomson equation)

23

- crystal face in vapours or solution

W* = πa0κ2/kT ln S

- crystal face in melt

W* = πa0κ2/∆se∆T

Examples (monolayer disk-shaped nuclei on own crystal face):

R* = a0κ/∆se∆T n* = πa0κ2/∆se2∆T2

(Gibbs-Thomson equations)

R* = a0κ/kT ln S n* = πa0κ2/(kT)2 ln2 S(Gibbs-Thomson equations)

1.6 2.0 2.4 2.8 3.21

2

3

4

5

6

7

lnS

n*

n* vs. S in 2D HEN of crystalline monolayers on a perfect (100)face of Kossel crystal at constant T: circles – data obtained byD.Kashchiev, J.Chem.Phys. 76(1982)5098 from Monte Carlo

simulation of J.D.Weeks, G.H.Gilmer, Adv.Chem.Phys.40(1979)157; line – Gibbs-Thomson equation

without free parameters

24

25

5. Concentration C* of nuclei

Hence, at n=n*

C* = C0 e−W*/kT

- HON or 3D HEN (spherical or cap-shaped nuclei)

- condensation of gas or solute

C* = C0 exp[−16πv02σef

3/3(kT)3 ln2 S]

- melt crystallization

C* = C0 exp[−16πv02σef

3/3∆se2 kT∆T2]

C(n) = C0 e−W(n)/kT (equilibrium cluster size distribution)

26

C* vs. S for nucleation of water droplets at T=293 K(the numbers indicate the nucleus size

at the corresponding S value)

0 1 2 3 4

1025

1020

1015

1010

105

1

50

100

20

40

5

10

35

70

lnS

C*

(m-3, m

-2)

HON (sphere)

3D HEN (cap, θ=90o)

27

- 2D HEN (disk-shaped nuclei on own crystal face)

- crystal face in melt

C* = C0 exp(−πa0κ2/∆se kT ∆T)

C* = C0 exp[−πa0κ2/(kT)2 ln S]

- crystal face in vapours or solution

To remember:HON dominates at high supersaturations,HEN dominates at low supersaturations.

28

6. Conclusion

- Thermodynamic considerations allow the determinationof the nucleus size n* and the nucleation work W* whichis the energy barrier to nucleation.

- The use of the general formulae for n* and W* requiresknowledge of the supersaturation ∆µ and the specificsurface or edge energy σ or κ in 3D or 2D nucleation,respectively.