v'vinogradkristall'unifrankfurt'de - PowerPoint PPT Presentation

Title: v'vinogradkristall'unifrankfurt'de


1
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nkfurt.de
2
?????? ??????? ?? ??????????? ?? ?????? ?????
N 6x6
ltngt 36/2
???????? ???????????...
Number of the greens, n
ltngt 900/2
N 30x30
Number of flips (time)
1. ??????????? ?????????? ???? ????? ???????
2. ???????????? ????? ??? ????????????
???????????? ???????
3. ?????????? ?????????? ??? ?????????? ???????
???????
3
?????? ??????? ?? ??????????? ?? ?????? ?????
N 6x6
ltngt 36/2
???????? ???????????...
Number of the greens, n
ltngt 900/2
N 30x30
Number of flips (time)

• ??????????? ?????? ????????? ?? ?????? ????.

• ??o???????????? ?????? ?????????
• ?????????????? ?????? ????????? ????? ??????,
• ?????? ?????????

??????? ?????????? ? ?????? ????????? ?
???????????? ??o?????????????.
4
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????? ??????????????
??o???????????? ??????? ??? ???????
???????????????? ??????????????
...? ???? ?? ??? ?????????????? ??????????????
???????? ??o???????????? ???????????
????? ??????? ????? ????????? ??o????????????? ??
??? ??????????.
??????????? ???????? ??o???????????? ???????
(???????????????? ?????????????? ??
??????????????)
?????????? ???????? ????????? S
5
???????? ?????????? ????????
??? ?????????????? ?????????????, ???????
??????? ?????????
6
???????? ???????? ?????????? ????????
6
5.7 J/mol/K
5
4
3
Entropy (J/mol K)
2
1
R 8.314 J/mol K
0
0
0,25
0,5
0,75
1
Composition, x
7
????????? ??????? ?????????? ????????
-TSideal
?????? ?? ???? ???????? ????? ??????????
?????, ?? ????????? ??????? ????????????? ?????
?????? ?? ????????? ? ????????? ??? ????????
???????????? ??? ???????? Tconst, Pconst
8
????? ????? ????????? ???????? ? ???????????
mB
0
mB
Free energy
mA
G
1-xA
xA
1.0
0.0
0.5
A
B
Mole fraction of A
????????? ????????? ??????? ???????? ??????????
????????, -TSid, ???????
9
????????? ???????
a x
a x
??????????? ???????
a f(x, T, P)
??????????? ??????????-?????? ???
??????????? ???????????????? ????????? ????????
???? ???????????
10
??? ????????/?????????? ????????? ????????
???????????
??????
????? ?? ??????????
????????????
?? ?????????? ??????? ?????? ????????? ????????
????????
11
????????????? ?????????
????????????? ??????? ??? ?????????? ???????????
T const
const

max
? ???????? ????????? ???????? ??? ???????
?????????? ??????? ??????? ???????
?????????? ??????? (T,V) ????????????
???????????? ?????????? S
12
????? ????????????? ????????? ?? ?????????
????????
????? ???????? ????????? ????????????? ?????????,
????????? ???????? ????????? ???????????
????????.
????? ????????? ????????? ??????? ? ?????????
????????, ????? ????????????? ?????????????
?????????. ??? ????? ????? ????? ?????????
??????? ????????? ?????????-????????????.
13
????????????? ??????? ????????????
??????????? ?????? ?????????
-
-

-
-
14
?????????? ??????????
????????????? ??????????? ?????????? ??????? ?
???????????? ??????????? ???????? ?????? ??
??????? ???????????
n
THB-?????? ????? ?????????? ??????????? ????? ?
?????????????
r
-
r0

?????? ? ??????????? ???????, ??????
????????????? ?? ??????? ???????????? ????
??????????? ??????????
15
????????????? ????????
?????????????? ???????????? ?????????
16
??? ??????? ???????????
?????????? ????????? ???????? ? ?????????
?????????
??????????
??????
500
?????????
400
300
Prediction, GPa
200
100
0
0
100
200
300
400
500
Experiment, GPa
17
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18
??? ??????? ??????????? ??????? ???????????
???????????? ?????????? ????????? GULP.
????????? ????????? ????? ?????????? ab initio
Structure
Randomization
??? ??????? ????????????? ????????? ????? ?????
??????? ???? ???????????? ? ?????????? ???????
???????????.
?? ??????? ?????...
19
???????????????? ?????
20
?????????? ??????????
Connolly Williams, 1983
I
II
III
21
?????????? ??????????
????????? ??????? ??????????????
22
? ??? ???????????? ??????????? ???????????
Structure
Randomization
Cluster expansion
?????????? ????? ??????? ???? ?????????? ?????
????????????.
??????? ????????? ???????????? ????? ??????????
?? ?????????? ??????? ????? ?????? ????????? ?
??????? ??????????!
23
Ca3Al2Si3O12 Mg3Al2Si3O12
?????????? ?????????? ?????? ?????????????
?????????? ??????? ???????
GULP
CE
? ????????? ?????? ??????? ??????? ?????
????????????.
??? ??????, ??? ????? ?????
24
? ????? ?????!
E1
E2
???????? ???????????
then
if
then
if
???????? ??????????? ???????????? ? ?????????????
?????????
25
???????? ????????? ???????? ?? ????? ?????
Newton et al. (1977)
Pyr
Gr
26
???????? ???????
????????? ??????? ? ?????? ????????????? ??????
???????????? ???????? ???? ???????????? ????? ?
???????????? ????????? ?????????
???????????? ???????? ???????????? ??
?????? ???????????? ??????????
27
Ca3Al2Si3O12 Mg3Al2Si3O12
?????????? ????????? ??????? ?????????? ?
????????????
?????????? ?????????? ???????????? ????????
28
Ca3Al2Si3O12 Mg3Al2Si3O12
???????????? ???????? ?? ?????????...
HW
29
Ca3Al2Si3O12 Mg3Al2Si3O12
? ?????????? ????? ?? ?????????...
30
????????? ????? ??????? ? ?????????
Sexcess
Vexcess
????????? w
????? ?? ?? ???????, ??? ? ???????????
?????????? ?????????? ?????? ?????????
?????? ???? ???????? ????????? ?????????
???????
31
????????????? ?????????? ????????
DV
??????????? ???????? ???????? ?
??????/?????????? ??? ???????.
x
decompression
?????????? ??????????? ?????? ??? ????????????
?????? ???????? ? ????? ??? ???????????
????????. ??????????????? ?????? ??? ??????????
????? ????.
compression
300 K
32
????????????? ?????????? ?????????
DV
compression
Newton et al. (1977)
decompression
???????????? ?????? ? ?????? ?????????? ???????
?????? ???????.
33
????????????? ?????????? ?????????
Global strain
Local strain relaxation
Enthalpy
Residual strain Hmix
????????? ???????? ??? ????????? ????????????
?????????? ? ??? ???????? ??????????
34
????????????? ????????? ??????????
?? ????????????? ??????????? Ca-Mg ????
?????????? ??????? ?????????? ?? ????????? ?
?????? Ca-Ca ? Mg-Mg
Volume
0.5
Pyr
Gr
J lt 0 ???????? ???????????? ??????????? 2 ??? ??
?????? ?????????? ?????
Distance
???? ???? ??? AB ??? ???????? ???????? ?
?????????? ????????? ??????????
35
?????? ??????????? ????????
I.
?????????? ??????? ???????
Enthalpy
??????? ?????????? ???????
?????? ???????????? ????????? ???????? ???
??????????
?????? ??????
A
A
A
A
B
B
B
B
B
B
B
B
A
A
A
A
??? ????? ????? ???? ?????
36
?????? ??????????? ????????
I.
?????????? ??????? ???????
Enthalpy
IIa.
????????? ?????
????????? ????? ?????????? ½ ?????? ????? ???
A
A
A
A
A
A
B
B
A
B
B
B
B
B
B
B
B
A
A
A
A
37
?????? ??????????? ????????
I.
?????????? ??????? ???????
Enthalpy
IIa.
????????? ?????
????????????
IIb.
0.5
???????????? ?????????? ????? ???? ???
B
B
B
A
B
B
B
A
A
A
A
A
B
B
B
B
A
B
A
B
A
A
A
A
38
?????????? ?????????? ?????????
??????? ?????????? ??????? ?????????????
?????????? H0
??????? ????????? ?????????? ?????????????
??????????? Jn
39
?????? ?????? ????????
????????????? ?????????? ????? ????????????
????? ??? J3, J4b ? J5.
40
????????????? ?????????? ? ???????
?????-?????????!..
I 41 2 2
SLECMC
LDMC
41
????????????????? ?????????????? ?????????
????????? ????????? ??????? ????????
? ??????????? ???????????? ??????????????
??????????
42
????????????????? ?????????????? ?????????
????????? ???????? ???????????????? ????????
Ideal mixing
43
?-x ????????? ?????-?????????
0 GPa
3 GPa
????????????? ?????????? ????????? ???? 650 K
44
??????????? ??????????-?????? ? ?-??
?????????-????? ?????????? ? ???????
??????????????? ??????? ? ????????? ????????
????????
Gr
Py
45
??????? ?? ?????? ????????? ????????????
??????? ?? ?????? ???????????? ???????????
46
Ca-Mg ?????????
???????????? ???????? ???????? ???????? ????????
Mg
Ca
47
??????? 36 ????????? ????????????? ??????????
(Ca,Mg)CO3 ????????????? ?? ????????????
??????????? ? ????????? ? ???????? ?? ??????
?????????
????? ?????????? - ???????!
48
?????????? ??????? ? ?????????? ?? ??????
???????????? ???????????
MgCO3
CaCO3
49
?????????? ?????????? ? ??? ????????
50
????????? ??????????????? ????????
51
????????? ???????? ? ??????? ????????
CaCO3
MgCO3
52
???????? ???????? ????????
CaCO3
MgCO3
53
????????? ??????? ???????? ? ?????????? ???????
54
??????? ????????? ???????-???????? ?? ??????
????????????? ? ? ????????? ? ?????????????
Goldsmith, 1983
CaCO3
MgCO3
????? ?? ??????? ??????????? ?????????? ???????
?????????
55
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56
?????-???????
Mg3Al2Si3O12 Mg3MgSiSi3O12
?????? ?????????
Al3Al3 Mg2Si4
???????????? ???????? ???????? ???????
57
?????????? ??????? ? ??????????? ???????? ??
?????? ???????????? ???????????
I41/a
58
???????? ????????? ???????? ?? ??
Ia3d
I41/a
??????? ?????????????????? ??? ????? ???????
?-??. ??? ??????? ? ??????? ?????????? ?
???????. ?????????? ???? Al ??????? ?-?? ????????.
59
???????? ?????? ???????? ???????? ??????? ???????
-1
-1
-2
Ca-Mg
-0
Å
??? ?????? ???????? ???????, ??? ???????
????????????
60
????????? ??????? ? ???????? ???????? ??
?????? ?????????????????? ??????????????
61
???????????? ? ?-?? ?????-???????
Heinemann et al. (1997)
????????? ?????? ????? ???? ??????????????...
?????????????? ???? ????? ????? ????-??????
?????? ? ???????? ?????? ?????????.
62
??????????? ??????????-?????? ? ?-??
?????-??????? ?????????? ? ???????
??????????????? ??????? ? ????????? ????????
????????
Py
Maj
63
??????? ?????????? ? ??????? ?????-???????
??????? ?????????? ? ?????? ????? ???? ?????????
P6/mmc / Cccm ?????????, ???? ??????????? ?
?????????? ???? ????????? 2000 K ? ???? ?????? ??
?????? ? Mg-Al-Si-O.
64
??????????????? ?????????? ???????????? ? ????????
Ca3Fe2Ge3O12 Ca3CaGeGe3O12
Iezzi et al. 2005
65
Ca/Na ? Mg/Al ???????????? ? ???????????????
M1
M2
M1
M2
CaCa NaNa 2CaNa
MgMg AlAl 2MgAl
MgNa CaAl NaAlCaMg
66
?????????? ???????
CaMgSi2O6 NaAlSi2O6
GULP
?????????? ??????????
67
?????? ?????? ????????
CaAl NaMg
CaNa
MgAl
68
????????? ???????? ?? ????? ????? ?????????????
Wood et al., 1980
69
???????? ???????? ?? ?? ??????????????
70
????????? ??????? ???????? ?? ?? ??????????????
71
T-X ????????? ?? ????????? ??????? ????????
Jd
Di
72
?????????????? ??????? ??????? ? ????????
Q
73
????????????? ????????????? ???? Q, ??????? ????
?? ??????????...
P
om
om
om
om
P2/n
Q
om
di
a-om
jd
Q-phase
C
74
???????? ???????
CaMgSi2O6 NaAlSi2O6
CaMgSi2O6 KAlSi2O6
K-Jd
Di
Jd
Di
??? ????????????????? ??????????? ??? ????????? K
? ???????
75
??????? ??????????? ??????????? ??????? ?????????
SLEC (GULP)
Monte Carlo
Cluster expansion
Transferable interatomic potentials
Thermodynamic integration
Free energy phase diagram
Structure elasticity data
VSSL
76
? ?? ????? ????????????????????????
77
?????????
???????? ??, ??????? ?, ?????? ?, ???? ??, ??????
?? (2006) ????????????? ???????????????
?????????? ? ???????? ??????????? ???????????
???? ? ????????????? ??????????. ?????????? (to
be submitted)
Vinograd VL, Winkler B, Wilson, D, Putnis A, Gale
JD (2006) Monte Carlo simulation of mixing in
Ca3Fe2Ge3O12 Ca4Ge4O12 garnets and implications
for the thermodynamic stability of
pyrope-majorite solid solution. PCM (in press)
Vinograd VL, Winkler B, Putnis A, Kroll H, Milman
V, Gale JD, Fabrichnaya OB (2006)
Thermodynamics of pyrope - majorite, Mg3Al2Si3O12
- Mg4Si4O12, solid solution from atomistic model
calculations. Molecular Simulations, 32(2)85-99
Vinograd VL, Sluiter MHF (2006) Thermodynamics of
mixing in pyrope - grossular, Mg3Al2Si3O12
Ca3Al2Si3O12, solid solution from lattice
dynamics calculations and Monte Carlo
simulations. Am Min (in press)
Vinograd VL, Burton BP, Gale JD, Allan NL,
Winkler B (2006) Activity-compositionrelations
in the system calcite - magnesite predicted from
static structure energycalculations and Monte
Carlo simulations. GCA (submitted)
78
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79
?????????
Mg3Al4Si5O18
??????-??? ????????????? ????????
???????? P6/mmc Cccm
80
??????? ??????? P6/mmc Cccm ?? ?????????
????? ????? ?????????????
??????? ???????????? ???????? Al ?? T1 ? T2
????????? ???????? ???????????? ????????????????
81
????????????? ????????? ???????????? ????????
82
????????????? Cccm ? P6/mmc ???
??????????? ????????
Cccm
P6/mmc
Cccm
P6/mmc
83
??????????? ????????????? ??????? ????????
???????
Grt Qz Opx Cord
????????? ??????? ??????????? ? ??????????????
????????
84
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85
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86
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87
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88
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89
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90
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91
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92
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93
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94
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95
Static Lattice Energy Calculations with GULP
Interatomic potentials
for all structural parameters xi
96
The mixing and ordering effects in
pyrope-majorite solution are related to the
difference in charges
Sexcess
Vexcess
DV 0
1)
Hexcess (high T)
HVD
2)
DV 0
Hexcess (low T)
97
The pair expansion fails
Higher order terms are needed
J-Q expansion
98
Phase diagram of Na-Eu-substituted powellites
Ca2Mo2O8- NaEuMo2O8
I 41/a
I -4
Temperature, oC
SRO
Mole fraction of Eu-powellite
99
CaMgSi2O6 NaAlSi2O6
C2/c
P2/n
C2/c P2/n
C2/c P2/n
100
CaMgSi2O6 KAlSi2O6
C2/c
P2/n
C2/c P2/n
C2/c P2/n
Mole fraction of K-jadeite
101
Order/disorder transition in cordierite,
Mg2Al4Si5O18
102
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103
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104
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105
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106
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107
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108
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109
Ca3Al2Si3O12 Mg3Al2Si3O12
The asymmetry of the excess volume
Geiger, 1999
110
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111
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112
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113
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114
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115
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116
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117
Order/disorder in, SrYbSi2Al2O2N5
118
VSSL works!
SLEC (GULP)
Monte Carlo
Cluster expansion
Transferable interatomic potentials
Thermodynamic integration
Free energy phase diagram
Structure elasticity data
119
Why garnets order at 50/50?
Ca-Mg pairs experience lesser strain
at intermediate compositions
Volume
I.
0.5
Pyr
Gr
Only at the 50/50 ratio the number of Ca-Mg
pairs can increase to 100
II.
Distance
120
Why the J3 is the strongest?
J4a
J6
J7
J2
J1
J4b
J5
J3
121
The 6 most important pair interactions in garnets
122
Mg
J4b
Al
J3
Ca
Si
J4a
J3
Si
J4b
Ca
Al
Mg
Si and Al happen to lie on J3 and J4b bonds
these bonds are stiffer then the others. The
tendency to alternation is stronger.
The magnitude of the Js reflects the local
stiffness of the structure
123
The ordering temperature of dolomites scales
with the enthalpy of disorder
The ordering force arises from the global srain
124
Mixing enthalpy in carbonates according to SLEC
and Impurity calculations
CaCO3-MgCO3
CaCO3-MnCO3
CaCO3-FeCO3
The magnitude of the ordering enthalpy correlates
with size mismatch of the end-members
125
The Js formalism applied to Ca-Mg carbonates
126
The ordering energies and the elastic term
coefficients in carbonates
The ordering energies
Aij (Ca-Mn)
Aij (Ca-Fe)
Aij (Ca-Mg)
ij
The configuration independent term
15.7
23.1
47.3
12
10.5
14.8
31.9
21
127
Monte Carlo simulation of temperature dependent
disorder in dolomite-like compounds
The ordering temperatures scale with the enthalpy
of disorder and both scale with the size mismatch
128
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129
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130
Powellite, Ca2Mo2O8 NaEu-powellite, NaEuMo2O8
Mo
Ca
131
The expansion of the excess enthalpy
The configuration independent term
absorbs nonlinear virtual crystal effects
132
The expansion of the excess Gibbs free energy
The configuration independent term
absorbs nonlinear virtual crystal effects
133
Analogy between CaCaNaEu and AlAlMgSi
substitution effects
Excess energy, kJ/mol
Excess energy, kJ/mol
134
NaEu-powellite, s.g. I -4
Mo
Na
Eu
135
The origin of the excess volume
Ca
O
Ca
DDexcess
O
DDexcess
Pyr
Gr
136
Pair cluster expansion for Ca-NaEu powellite
137
The Js spectrum
138
MC simulation results for 2Ca-NaEu substitution
Enthalpy
Entropy
139
MC simulation results
Free energy of mixing
140
Powellite, CaMoO4
Mo
Ca
S.g. I41/a
141
The results of the curvature analysis
Temperature, oC
Mole fraction of Eu-powellite
142
Observed and predicted structural parameters of
powellite and Eu-powellite
Note Experimental data by D.Bosbach. The lattice
parameters are in Å units
143
Observed and predicted structural parameters of
powellite and Eu-powellite
Eu2Si2O7
Eu2O3
Exp. data Felsche, J. (1973)
Saiki et al. (1985)
144
Observed and predicted elastic stiffness
constants of powellite
Exp.data Bass (1995)
145
The potentials used in the study
Type atom atom A (eV) B (Å) C
(eVÅ6) Buckingham Ca O-shell 3285.2403
0.271592 0 Buckingham Mg O-shell 1432.8544
0.277265 0 Buckingham Mo O-shell 951.78780
0.363893 0 Buckingham Na O-shell 46421.687
0.197214 0 Buckingham Eu O-shell 4206.0812
0.282787 0 Buckingham Al O-shell 1263.1396
0.286376 0 Buckingham Si O-shell 1073.4668
0.298398 0 Buckingham O-shell O-shell 598.89960
0.314947 26.89746
Type atom atom K (eVÅ-2) String O-core O-shell
56.559808
Type atom atom atom k (eVdegree-2) g
(degree) Three-body Si(4) O-shell O-shell
0.77664 109.47 Three-body Si(6) O-shell O-shell
2.29550 90 Three-body Al(4) O-shell O-shell
1.73260 109.47 Three-body Al(6) O-shell O-shell
1.94750 90
Note O core charge 0.746527 shell charge
2.446527, the charges on cations are
0.85Z.
146
THB potentials set for silicates
Sanders et al., 1984
qi qj
Electrostatic potential
r
Buckingham potential
A exp(- r/B ) - Cr-6
qs - 2.848
-
Core shell interaction
1/2 K r2

qc 0.848
Bond-bending interaction
1/2 k (g - g0)2

g
-
-
147
Monte Carlo
The excess free energies
Cluster expansion
Monte Carlo
Js, Aij
Thermo- dynamic integration
Phase diagram
148
The ordering energies and the elastic term
coefficients in garnets
149
Energy of disorder of (Ca,Ge) vs (Mg,Si)
The effect of the size mismatch
The effect of the charge difference
150
Why the strongest interaction is at the
3rd-neighbor distance?
J4a
J6
J7
J2
J1
J4b
J5
J3
151
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152
Thermodynamic integration analysis The free
energy of mixing
Based on SLEC
Based on LD
153
The positive vibrational entropy
stabilizes the intermediate phase
The Js are fitted to the excess free energies
The Js are fitted to the excess static energies
154
The activity-composition relations at zero
pressure from lattice dynamics calculations and
MC simulations
155
What is the physical meaning of the Js?
8-J- expansion
156
LRO from Monte Carlo simulations vs. experiment
157
Determining LRO parameter from Monte Carlo
simulations
158
Activity-composition relations from
Redlich-Kister fit to Monte Carlo free energies
159
Si4
Si6
Mg
Si4
The translation destroys the tetragonal symmetry
160
New expansion
Mg6
Mg6
Va
Si4
Mg6
Si6
Mg6
Si6
Si6
Si6
Si4 Va Va Si4
161
The pair expansion fails for majorite
162
Pair energies (kJ/mol)
The elastic term (kJ/mol)
163
Quaternary energies (kJ/mol)
164
Why I41/a ?
I41/a C2/c
P 41 3 2
P 41 3 2 has too few 3rd-neighbor MgSi pairs
165
The effect of the quaternary interactions is to
increase number of tetrahedra consistent with
4-fold symmetry
166
Volume can be also cluster expanded
167
The comparison of the SLEC and ab initio results
Empirical potentials
Ab initio DFT GGA
eV/atom
eV/atom
D
Energy differences
Empirical potentials
Ab initio DFT GGA
D (AB)/2
B C
168
Conclusion 2
A The cubic/tetragonal transformation in
majorite-rich garnets is driven by a very
strong ordering of Mg2 and Si4

• B Phase relations in the transition zone can
• be affected by the cubic-tetragonal
• transformation

169
GULP excess energies and the enthalpy of mixing
from MC simulations
170
Correlation between the excess energies
calculated with GULP and fitted with the cluster
expansion
171
The configurational entropy from the
thermodynamic integration analysis
172
The free energy of mixing from the thermodynamic
integration analysis
173
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174
Thermodynamics of mixing in Ca3Fe2Ge3O12 -
Ca4Ge4O12 garnets from atomistic simulations
(to be submitted to Phys. Chem. Minerals)
Victor L. Vinograd1, Bjoern Winkler1 Julian D.
Gale2
1University of Frankfurt, Institute of
Mineralogy, 2Nanochemistry Research Institute,
Curtin University of Technology
175
The phase diagram of grossular-pyrope from MC
simulations
Based on SLEC
Based on LD
I41 2 2
I41 2 2
Gr
Pyr
Pyr
Gr
176
Determining LRO parameter from Monte Carlo
simulations
SLECMC
LDMC
177
Cluster expansion
AA BB 2AB
178
Scaling the charges improves the transferability
0.85
Scaled-charge set
Formal-charge set
179
Ca3Al2Si3O12 Mg3Al2Si3O12
Free energy minimization The effect of pressure
180
Miscibility gap in Ca-Fe-Ge garnets
0 GPa
10 GPa
I41/a
I a 3 d
Iezzi et al. 2005 1373 K
Mole fraction of Ca-Ge-majorite
181
The energetics of order/disorder is very
similar in Ca-Fe-Ge and Mg-Al-Si garnets
182
Conclusion 3

• A Miscibility gaps in couple-substitution s.s.
• are driven by the strong ordering of the
• cations with dissimilar charges

B Mixing phenomena in Ca-Fe-Ge- and
Mg-Al-Si systems are fundamentally the same

183
Virtual Solid Solution Laboratory
SLEC (GULP)
Monte Carlo
Cluster expansion
Empirical interatomic potentials
Thermodynamic integration
Free energy phase diagram
Structure elasticity data
184
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185
New expansion
Mg6
Mg6
Va
Si4
Mg6
Si6
Mg6
Si6
Si6
Si6
Si4 Va Va Si4
186
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187
Miscibility gap in pyrope-majorite s.s.
0GPa
20GPa
188
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189
NaEu-powellite, s.g. I -4
Mo
Na
Eu
190
Analogy between CaCaNaEu and AlAlMgSi
substitution effects
CaCa
AlAl
NaEu
MgSi
191
Pair cluster expansion for Ca-NaEu powellite
192
All mixing and ordering effects in
grossular-pyrope solution are related to the
difference in volumes
Vexcess
Sexcess
DV
Vexcess
1)
HVD
2)
Hexcess (high T)
DV
Hexcess (low T)
193
Enthalpy and entropy of mixing from Monte Carlo
simulations
CaCa
NaEu
CaCa
NaEu
194
The free energy of mixing and the phase diagram
from Monte Carlo simulations
I -4
I 41/a
SRO
195
Conclusions
The developed technique together with the
available potentials now permits quantitative
prediction of mixing effects in many
petrologically important silicate minerals
The extension of the empirical potentials
database will permit predicting mixing properties
in many industrially important ceramic phases
196
Miscibility gap in Ca-Fe-Ge garnets
I41/a
I a 3 d
Iezzi et al. 2005
Mole fraction of Ca-Ge-majorite
197
Cluster expansion for diopside-jadeite s.s.
CaMgSi2O6 NaAlSi2O6
198
Monte Carlo simulation results for
diopside-jadeite
199
Excess energy and configurational entropy in
diopside-jadeite solid solution from Monte Carlo
simulations
200
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New expansion
202
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203
D
D (AB)/2
B C
204
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205
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206
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207
Free energy and entropy of mixing from
thermodynamic integration
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209
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210
Si
Mg
Al
J1
J3
J2
J4a
Ca
J2
J4a
J4b
J3
J4a
J4b
J4b
J1
J3
J1
J1
211
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212
Monte Carlo simulation confirms that the
structure (II) has the lowest energy
But is this structure a stable compound?
We need to know its free energy
213
Thermodynamic integration
Initial distribution
MC

Interaction constants
Monte Carlo simulation leads to the equilibrium
(Boltzmanns) distribution
J (J1,J2,J3Jn)
We can scale the Js with a factor of 0 lt l lt 1
and get a new distribution which is closer to
that of a random solid solution
Jl (lJ1, lJ2, lJ3lJn)
Now we will think of F as a continuous function
of l
Full derivation can be found in Dove, 2001
214
Pair cluster expansion
J3
J2
J1
The configuration independent term
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216
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217
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218
THB model for Mg-Si-O
Price et al., 1987
O
-
2
, Mg
2
Si
4
Si
4
219
THB transferable set for aluminosilicates
Winkler et al., 1991
O
-
2
, Mg
2
Si
4
Si
4


Ca
2
, Al
3
, Na
, K
220
New scaled-charge potentials for oxides
Type atom atom A(eV) B(Å) C(eVÅ6) Buckingham Ca
O-shell 2895.68 0.2811 0 Buckingham Mg O-shell 107
7.55 0.2899 0 Buckingham Na O-shell 30267.4 0.1997
0 Buckingham K O-shell 65164.8 0.2122 0 Buckingha
m Al O-shell 1226.70 0.2893 0 Buckingham Si O-shel
l 1096.42 0.2999 0 Buckingham O-shell O-shell 614.
71 0.3016 27.07
Type atom atom K(eVÅ-2) String O-core O-shell
54.70
Type atom atom atom k(eVgrad-2)
g(grad) Three-body Si(4) O-shell O-shell
3.79 109.47 Three-body Si(6) O-shell O-shell
3.77 90 Three-body Al(4) O-shell O-shell
0.67 109.47 Three-body Al(6) O-shell O-shell
1.89 90
Note O core charge 0.7465 shell charge
2.4465, the charges on cations are
0.85Z.
Available also for Fe2, Fe3, Mn2, Ge, Zr, Ti,
Y, P, F-, C, OH-
221
Configurational entropy of a coupled substitution
s.s.
From standard model assumptions
From atomistic simulation
222
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223
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224
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225
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226
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227
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228
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229
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230
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231
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232
Virtual Solid Solution Laboratory
SLEC (GULP)
Monte Carlo
Cluster expansion
Transferable interatomic potentials
Thermodynamic integration
Free energy phase diagram
Structure elasticity data
233
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234
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235
Static Lattice Energy Calculations with GULP
Interatomic potentials
for all structural parameters xi
236
Thermodynamics of solid solutions
237
Enumeration of microstates
There are 3 possibilities to locate the blue
band. For each of these 3 possibilities there are
2 possibilities to locate the red band. For each
of the 6 variants there is just one possibility
to locate the white band.
N!N(N-1)(N-2)...(N-(N-1))
238
Number of arrangements within distinguishable
objects
N!N(N-1)(N-2)...(N-(N-1))
Now suppose we cannot distinguish blue from red
and white from yellow
239
Thus, we have over-counted the number of the
distinguishable ones. We can correct the result
by counting the possible arrangements of red
bands and white bands using the N! formula
2! 2! 4
arrangements of red bands
arrangements of white bands
240
The distinguishable configurations
4!
6
2!
2!
Number of distinguishable configurations within i
groups of undistinguishable objects
241
Ideal solution
Stirlings formula
242
Entropy of mixing in an ideal solution
6
5.7 J/mol/K
5
4
3
Entropy (J/mol K)
2
1
R 8.314 J/mol K
0
0
0,25
0,5
0,75
1
Composition, x
243
How does the configurational entropy depend on
temperature?
The most general equation for the entropy
In a temperature-equilibrated system the
microstates are distributed according to
Boltzmanns law
when Ei decreases, pi increases
high T tends, to equalize pi
In the high-temperature limit