BASIC RESULTS AND APPLICATIONS BASIC RESULTS AND APPLICATIONS OF NUCLEATION THEORY OF NUCLEATION THEORY Lectures by Dimo Kashchiev Lectures by Dimo Kashchiev (EMSE Nucleation Workshop, Saint-Etienne, June 2003) 1. Thermodynamics of nucleation 2. Kinetics of nucleation 3. Applications of nucleation theory Further reading: Further reading: D. Kashchiev, D. Kashchiev, “ Nucleation: Basic Theory with Nucleation: Basic Theory with Applications Applications” , Butterworth , Butterworth- Heinemann, Oxford, 2000 Heinemann, Oxford, 2000 http://www.ipc.bas.bg/PPages/Kash/Monograph.htm
28
Embed
BASIC RESULTS AND APPLICATIONS OF NUCLEATION …dependence of the nucleus size and the nucleation work. The kinetic considerations show how the general formula for the nucleation rate
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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.
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
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
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
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
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.