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VILA
VÍLA - VItrum LAugaricioJoint Glass Center
of Institute of Inorganic Chemistry of SAS, A. Dubček University of Trenčín, and RONA, j.s.c.
Študentská 2, Trenčín, SK-911 50, Slovak Republic
VILA & FU AVČR & UPCE
KLASICKÁ NUKLEAČNE RASTOVÁ TEÓRIA A JEJ APLIKÁCIE
Marek Liška, Mária Chromčíková, Miroslava Rodová, Karel Nitsch, Zdeněk Černošek
OUTLINE
• Gibbs energy controls the [P, T] world
• Driving force for crystallization
• Crystal nucleation - embryo
• Crystal growth
• Nucleation and growth – TTT
• Mathematical model
• Application to ZnCl2
PERSISTING WARNING
NOT
TOO
MUCH
THEORY!!!!!!!!!!!!
GIBBS ENERGY
S
S
S
L
L
L
As2Se3 - G vs T(K)FACT53 - FACT53 - FactSage 5.3 compound database (Jan. 2007)
T(K)
G(K
J/m
ol)
300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500-650
-600
-550
-500
-450
-400
-350
-300
-250
-200
-150
-100
-50
Tliq
Undercooling
VOLUME VERSUS SURFACE23 4
34
rSrV
fpmelt,crystal, 0
)()(TT
V
TGTGG
m
mmV
0 SG
23 434
)( rGrGSGVrG VSV
2484d
)(d 2 VV GrrrGr
rrG
VV G
rGrrG 2
20max)( ***
Critical nucleus – critical radius r*:
VOLUME
VERSUS
SURFACE
Thermodynamic activation barrier:MISSION IMPOSSIBLE
Entropy of mixing decreases G:
N
N
N
NkTrGN
NN
N
NN
N
NN
N
NN
NkTrGNG
rrr
rrr
r
r
rr
ln)(
lnln)(
kTGN
kT
GNN r
2V
3*
*
3
16expexp
2mfpmfp
2fp
3
2V
3*
3
16
3
16
VTTH
T
GG
Vmfpfpmfpmfpmfpmfp )( GVTTTHSTHTG
Proportional to # of neighbors, Ns, frequency, kT/h, Boltzman factor of kinetic activation energy, and Nr*:
HOMOGENEOUS NUCLEATION RATE
kTHT
TVN
kT
G
h
kTNI
2mfp
2fp
2m
3
as
3
16expexp
2fp
as
11)/ln()ln(ln
TTTB
Tk
GhkTNNI
kHTVB 2mfp
2fp
2m
3 316
HOMOGENEOUS NUCLEATION RATE
TTT fp
kTHT
TV2
mfp
2fp
2m
3
3
16exp
kTG
hkT aexp
Decreases with T
Increases with T
Increases with T
= 0 for T=Tfp
RATE OF CRYSTAL GROWTH
kTNGGfNdt
dnAmfpas exp)lc(
kTGfNdtdn
as exp)cl(
hkT
f
uNRTGkTGfNdt
dn
dt
dns mfpas exp1exp)lc()cl(
fp
fpmfpa exp1exp
TTR
TTHkTG
h
kTu
NUCLEATION vs CRYSTAL GROWTH
T Tfp fpGm 0 u 0T 0 u 0
Tfp
Tg
NUCLEATION AND GROWTHConstant p, T, single component => u, I = const.
31 34
utV Volume after growth from to t
dutIVtVt
0
3liqcr 3
4)( Vcr<<Vliq=> Vliq= const
43
0
33
liq
cr
3
1d
3
4
)0(
)()( tuItuI
tV
tVt
t
TTT = Temperature – Time – Transformation
.)0(
)()(
liq
cr consttV
tVt
TTT DIAGRAMS
TTT diagram of salol
(A) = 10-6 (B) = 10-8
THE MODEL
)3/ln(ln4)10ln()ln( exp6
exp3 tIu
TC
TTTBAWIu
11)ln( 2
fp
exp3
fp
2fpmfpexp1ln3)/ln(4TTR
TTHhkTW
3ss lnln3ln3ln3 NNNNA
k
GC a4
kHTVB 2
mfp2
fp2
m3 316
64expexp
3 1031
)( tuIt
REGRESSION TREATMENT
3ss lnln3ln3ln3 NNNNA
k
GC a4
CzBxAWIuy )ln( 3
kHTVB 2mfp
2fp
2m
3 316
2fp1 TTTx
Tz 1
RESULTS
A = - 49.11
B = 46916278
C = 46795
Ga = 97 kJ/mol
sapr = 3.4 K
RESULTS
2 3 4 5 6 7 8 9150
200
250
300
Calculated Experiment
T /
oC
log(t / s)
CONCLUSION
The plausible semi-empirical method (e.g. the general form of the regression
function)was proposed for the quantitative description of the TTT diagrams.
GENERAL CONCLUSIONSSome people do not like theory...
AT LEASTONE
EXPERIMENTPLEASE!!!
(Prayerof
MiaFarrow)
Photo
© Z. Černošek
THANK YOU FOR YOUR PATIENT ATTENTION