Download - 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
+=+= ∑=
1. Thermodynamical model
2. Gibbs free enthalpy
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 =
1. Thermodynamical model
2. Gibbs free enthalpy
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*
1. Thermodynamical model
2. Gibbs free enthalpy
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−=
∂∂
1. Thermodynamical model
2. Gibbs free enthalpy
3. Interpretation
Gibbs free enthalpy of the nucleating system (2/3)
G
MM *
Characteristics of plot G(M)v G decreasing function of M
M < M* xi
iM i*
xi
iMi*
M > M*
1. Thermodynamical model
2. Gibbs free enthalpy
3. Interpretation
-dG/dM
M
v Inflexion point for M=M*
M*
-dG/dM : nucleation "driving force"
Gibbs free enthalpy of the nucleating system (3/3)
Influence of supersaturation
G
MM *
M*
M*
Supersaturation
M*< M * < M*
1. Thermodynamical model
2. Gibbs free enthalpy
3. Interpretation
Interpretation (1/2)
1. Thermodynamical model
2. Gibbs free enthalpy
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
1. Thermodynamical model
2. Gibbs free enthalpy
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