Thermodynamics of a nucleating system

Thermodynamics of a nucleating system

Considerations on the concept of critical nucleus

Michel Cournil

Ecole des Mines de Saint-Etienne

Centre SPIN LPMG- URA CNRS 2021

Objectives

v to obtain a thermodynamic description of the differentstages of a nucleating system (typically initiallysupersaturated liquid solution)

v to clarify different points concerning nucleationthermodynamics (and kinetics):

üspontaneous evolution (or not)...

üstatus and role of the critical nucleus

M. Cournil and P. Gohar, Journal of Colloid and Interface Science, 132, 1989, 188-199

Contents

1. Thermodynamical model of the nucleating system assumptions, model of solution, composition, activity coefficient

2. Gibbs free enthalpy of the nucleating system calculation, characteristics,...

3. Interpretation, comments and tracks of reflectionspontaneous evolution, stability, critical nucleus, consequences on kinetics,...

1. Thermodynamical model

2. Gibbs free enthalpy

3. Interpretation

Thermodynamical model of thenucleating system (1/4)

System definition:

üClosed system (supersaturated solution) in equilibrium

üHomogeneous solution N0 moles of solvent (water), NA moles of solute A

üA is liable to associate into clusters: A1, A2,...Ai, AM (M maximum size)

“Associated solution“ model: (Prigogine and Defay, 1950)

ü Two thermodynamical description for the solution:

1. Nonideal solution of A in solvent: µA=µA* + RTln(γAxA)

2. Ideal solution of solvent, A1, A2, Ai…µAi=µAi* + RTln(xAi)

ü Two expressions of Gibbs function:'00

,100 µµµµ NNNNG

MiAAAA ii

+=+= ∑=



3. Interpretation Thermodynamical model of thenucleating system (2/4)

Mole balance: ∑=

=Mi

AA iiNN

,1

Equilibrium condition:

Ai = iA1

1AA ii

µµ =

⇓⇓

'0 01µµ NNG AA +=

00µµ NNG AA +=⇒⇒

AA µµ =1

'00 µµ =



3. Interpretation Thermodynamical model of the nucleating system (3/4)

Size distribution of the clusters:

Equilibrium condition: ⇒ ⇒1AA ii

µµ = ( )11

ln*ln* AAAA xRTixRTii

+=+ µµ

( ) **

exp : with 1

1

−==⇒

RT

iKKxx i

i

AAii

iAA

µµ

'exp 32

is

ix

i

K

−

=⇒

σ σ’ = σ/(RT)

xS : saturation mole ratio

32

0 - **1

iGiÄgi *iAA i

σµµ +∆−==−

σ ∝ surface tension

xi

i

xA1 < xS

M

xi

i

xA1 > xS

M

Critical nucleus

i*



3. Interpretation

Thermodynamical model of the nucleating system (4/4)

Determination of xA1

∑=

=Mi

AA iiNN

,1

( ) ii

A

MiA

AA Kx

NN

Nx

i

ii 1

,10

=+

=∑

=

⇒

( )0,1 0

1 NN

KxNN

i Ai

iA

Mi

A =

+∑

=

( )01 0

d 1 N

NiKx

NN

i AM

ii

AA =

+∫

or

Activity coefficient of A

Associated solutions model ⇒01

1

xx

x

A

AA =γ

x10 = lim xA1 when xA 1→1

Gibbs free enthalpy of the nucleating system (1/3)

Calculation of G at the different stages of the association (nucleation) process

Each association stage is characterized by : NA, N0, M

At the equilibrium, all xAi are known (see above)

Thus, G can be calculated:

−+++≈+= ∫

M

AAAAAAA ixNxNRTNNNNMGi

10

'0

*'0 d1lnln*)(

10101µµµµ

'00 x

xRTN

MG M−=

∂∂



3. Interpretation


G

MM *

Characteristics of plot G(M)v G decreasing function of M

M < M* xi

iM i*

xi

iMi*

M > M*



3. Interpretation

-dG/dM

M

v Inflexion point for M=M*

M*

-dG/dM : nucleation "driving force"


Influence of supersaturation

G

MM *

M*

M*

Supersaturation

M*< M * < M*



3. Interpretation

Interpretation (1/2)



3. Interpretation

G(M) decreasing function of M ⇒ clustering is spontaneous and no activation is required

Association (clustering, nucleation) process:

i. initially at decreasing driving force dG/dM and cluster concentration

ii. minimum driving force and cluster concentration is reached when crossing the critical nucleus zone

iii. then driving force and cluster concentration increase again

Interpretation (2/2)

The critical nucleus defines a sort of bottleneck both for cluster concentration and nucleation driving force



3. Interpretation

At low supersaturation, cluster concentration and nucleation driving force reach very low values:

i. Nucleation becomes very slow : very long induction period

ii. Or can be kinetically blocked before the critical nucleus zone (metastability zone)

Validity of our approach ?

-Assumption of succession of equilibrium states questionnable if theforward association rate is high

- Good agreement at low association rates: low supersaturation levelsor neighborhood of the critical nucleus

Conclusion

The model of associated solutions provides a quite convenient framework for the modelling of nucleating solutions

When considered as a succession of equilibrium steps, nucleation isproved to be spontaneous from a thermodynamical point of view

Critical nucleus appears as a bottleneck (or rate-determinig step) in thekinetics of nucleation

Nucleation driving force can be calculated throughout the whole process

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Thermodynamics of a nucleating system