Role of symmetry and energy in structural phase...
Transcript of Role of symmetry and energy in structural phase...
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Summer School on Mathematical and Theoretical Crystallography
Palazzo Feltrinelli, Gargnano, Garda Lake (Italy), 27 April - 2 May 2008
Role of symmetry and energy in structural phase transitions
Michele Catti
Dipartimento di Scienza dei Materiali, Universita’ di Milano Bicocca, Milano, Italy
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Thermodynamic order of the phase transition (Ehrenfest’s classification):
smallest order of the derivatives of the Gibbs free energy G (with respect to the relevant variable, T or p) which are discontinuous across the transition
First-order transitions: the first-order derivatives (entropy S and volume V) are discontinuous, leading to enthalpy (latent heat) ∆H and volume ∆V jumps
dG = -SdT + Vdp (1)Second-order transitions: the second-order derivatives (heat capacity Cp and compressibility β) are discontinuous, whereas the first-order ones are continuous (no heat or volume effects):∆S = 0, ∆H = 0, ∆V = 0
d2G = - (Cp/T)dT2 - Vβdp2 -VαdTdp (2)
Phase α stable for T < Tc, phase β stable for T > Tc →
∆G = Gβ - Gα > 0 for T < Tc, ∆G = Gβ - Gα < 0 for T > Tc ?
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But ....... Taylor expansion about Tc of the Gibbs free energy of each phase:
G = G(Tc) - S(T-Tc) – (T-Tc)2 + ....... (3)
∆G = Gβ - Gα = - (T-Tc)2 + ..... same sign for T < Tc and for T > Tc ! (4)
Michele Catti – Gargnano, 1st May 2008
c2TC cp,
c2TC cp,∆
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Buerger's classification of structural phase transitions
reconstructive: primary (first-coordination) chemical bonds are broken and reconstructed ↓
discontinuous enthalpy and volume changes →
first-order thermodynamic character(coexistence of phases at equilibrium, hysteresis and metastability)
displacive: secondary (second-coordination) chemical bonds are broken andreconstructed, primary bonds are not →small or vanishing enthalpy and volume changes →
second-order or weak first-order thermodynamic character
order/disorder: the structural difference is related to different chemical occupation of thesame crystallographic sites, leading to different sets of symmetry operators in the two phases → vanishing enthalpy and volume changes →
second-order thermodynamic character
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Symmetry aspects of Buerger’s phase transitions
Displacive and second-order phase transitions:
- the space group symmetries of the two phases show a group G0 /subgroup G relationship (G ⊂ G0)
- the G low-symmetry phase approaches the transition to G0 higher symmetry continuously;
- the order parameter η measures the 'distance' of the low-symmetry to the high-symmetry (η=0) structure; it may have one or more components (η1, η2, ...)
• T-driven transition:
usually the G symmetry of the l.t. phase is a subgroup of the G0 group of the h.t. phase
• p-driven transition:
it is hard to predict which one of the two phases (l.p. and h.p.) is more symmetric, although in most cases G0 → G with increasing pressure
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Reconstructive phase transitions:
- the space group symmetries of the two phases are unrelated
- the transition is quite abrupt (no order parameter)
Examples of simple reconstructive phase transitions:
• HCP to BCC, FCC to HCP and BCC to FCC in metals and alloys
• rocksalt (Fm-3m) to CsCl-type (Pm-3m) structure in binary AB systems:
the C.N. changes from 6 to 8
• zincblende (F-43m) to rocksalt (Fm-3m) structure in binary AB systems:
the C.N. changes from 4 to 6
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Examples of displacive phase transitions:
NiAs-type (P63/mmc) to MnP-type (Pcmn) of CoAs and VS
Hexagonal (NiAs-type) and orthorhombic (MnP-type) cells outlinedOrder parameter η: length of the vector describing the structural distortion
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Examples of order/disorder phase transitions:
β - β' brass transition in CuZn (Pm-3m to Im-3m)
Pm-3m (η = 1) Pm-3m (1>η>0) → Im-3m (η = 0)
Order parameter: η = 2fCu - 1
fCu: fraction of Cu atoms occupying the site at 0,0,0 →fCu changes from 1 (full ordered Pm-3m structure) to 1/2 (full disordered Im-3m structure)
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Continuous transformation with symmetry group/subgroup relation:
the low-symmetry phase approaches gradually the point where new symmetry elements are acquired
• the low-symmetry phase can not survive, even in a metastable form, above Tc →
the Taylor expansion (3) is not allowed above Tc !
Michele Catti – Gargnano, 1st May 2008
0 1T/Tc
0
1
η
Order parameter
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Elements of Landau's theory of continuous (displacive and order-disorder) phase transitions
- Taylor expansion of G(η) around its stationary point η=0 (simple case of one-component order parameter):
G = G(p,T,η) = G0 + Aη2 + Bη3 + Cη4 + Dη5 + Eη6 + ........
- truncation to the smallest order reproducing the physics of the transition (usually 4 or 6)
Only the powers of η which are invariant with respect to the symmetry operations of the G0 group are allowed to be present in the G(η) Landau polynomial:
↓
even powers are always invariant, whereas odd powers may be invariant (‘Landau condition’)only when G0 belongs to the trigonal, hexagonal or cubic symmetry systems
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Three most important cases of Landau polynomials:
• G – G0 = A(p,T)η2 + C(p,T)η4 (C > 0) 2-4 even powers, second-order transition
• G – G0 = A(p,T)η2 + C(p,T)η4 + E(p,T)η6 (E > 0) 2-4-6 even powers, first-order tr.
• G – G0 = A(p,T)η2 + B(p,T)η3 + C(p,T)η4 (C > 0) 2-3-4 even/odd powers, first-order tr.
∂G/∂η = η(2A + 3Bη + 4Cη2 + 6Eη4)
A > 0 η=0 is a minimum (∂2G/∂η2)η=0 = 2A →
A < 0 η=0 is a maximum
In the surrounding of Tc: B, C and E are approximately constant, only A = A(T)
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1) G – G0 = Aη2 + Cη4 A >0, A=0, A<0
∂G/∂η = 2η(A + 2Cη2) = 0 →
(A<0) η= (-A/2C)1/2
↓order parameter of the low-symmetry
phase as a function of p,T
No equilibrium of two phases at T = Tc, no metastability effects at T ≠ Tc →
second-order thermodynamics Michele Catti – Gargnano, 1st May 2008
0 η
G -
G0
A < 0
A = 0A > 0
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As A(Tc) = 0, and A<0 for T<Tc and A>0 for T>Tc, in the neighbourhood of T=Tc:
A(T) = a(T-Tc), η= ±[a(Tc-T)/2C]1/2 ≈ (Tc-T)1/2 consistent with experimental behaviour for most second-order phase transitions
S – S0 = -∂(G-G0)/∂T = -aη2 = a2(T-Tc)/2C behaves continuously across the phase transition
Cp – Cp,0 = T∂(S-S0)/∂T = a2T/2C
limT→Tc (Cp-Cp,0) = a2Tc/2C = ∆Cp second-order discontinuity
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2) G – G0 = Aη2 + Cη4 + Eη6 A=a(T-Tc) A>0, A=0, A<0 C≤0 E>0
Michele Catti – Gargnano, 1st May 2008
1 T < Tc
2 T = Tc
3 Tc<T<T0
4 T = T0
5 T0<T<T1
6 T = T1
7 T > T1
equilibrium of two phases at T = T0, metastability region for Tc < T < T1
→ first-order thermodynamics
0η
G -
G0
A<0
A>0
A=0
1
2345
67
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• Condition of two-phase equilibrium at T=T0 :
∂G/∂η = 2η(A + 2Cη2 + 3Eη4)= 0 for η ≠ 0→ T0 = Tc + C2/4aE
G – G0 = Aη2 + Cη4 + Eη6 = 0
• Discontinuous jump of the order parameter at T=T0: η(T0) = (-C/2E)1/2
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S – S0 = -∂(G-G0)/∂T = -aη2 = aC/2E behaves discontinuously across the phase transition
latent heat ∆H = T0∆S = (aC/2E)T0 → typical first-order effect
Cp – Cp,0 = T∂(S-S0)/∂T = (a2/2C)T for T ≤ Tc, 0 for T > Tc → second-order
discontinuity, as in the 2-4 case
For E=0 (case 2-4), C=0 (case 2-6, ‘tricritical polynomial’) or C>0 the first-order discontinuity disappears, and the second-order behaviour is recovered: T0 = Tc; η(T0)=0
G – G0 = Aη2 + Eη6 second-order tricritical thermodynamics↓
η = ±[a(Tc-T)/3E]1/4 ≈ (Tc-T)1/4
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3) G – G0 = Aη2 + Bη3 + Cη4 A > 0, B < 0, C > 0
G – G0 = 0 → either one, two or three roots, according to the sign of B2 - 4AC
∂G/∂η = η(2A + 3Bη + 4Cη2)=0 → either one, two or three roots, according to the sign of B2 -(32/9)AC
equilibrium of two phases at T = Tc, metastability at T ≠ Tc
→ first-order thermodynamics
Michele Catti – Gargnano, 1st May 2008
B2>4ACT > Tc
4AC > B2 > (32/9)ACT < Tc
B2=4ACT=Tc
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Thermodynamics: effect of pressure
Static calculation at the athermal limit (T = 0 K)
• Static equation of state: p = -dEst/dV
elastic bulk modulus K = -Vdp/dV; K’= dK/dp
Murnaghan equation of state (K’ = constant):
Est(V) = K0V0 + E0
V0, E0, K0, and K’ parameters: obtained by fitting the numerical E(V) results(constant volume energy minimizations)
Thermodynamics: effect of pressure
Static calculation at the athermal limit (T = 0 K)
• Static equation of state: p = -dEst/dV
elastic bulk modulus K = -Vdp/dV; K’= dK/dp
Murnaghan equation of state (K’ = constant):
Est(V) = K0V0 + E0
V0, E0, K0, and K’ parameters: obtained by fitting the numerical E(V) results(constant volume energy minimizations)
−+
1
111
1
1
K'-0V
V
K'
K'-
V0V
)K'(K'-
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p(V) = ; V(p) =V0
free energy (T =0 K) = enthalpy H = E + pV
H(p) = + E0
Phase equilibrium vs. pressure and pressure-driven transitionbetween phase A and phase B:
∆H(p) = HB(p)-HA(p) = 0 ⇒ p = pt equilibrium pressure
∆H(p) > 0 ⇒ phase A stable; ∆H(p) < 0 ⇒ phase B stable
p(V) = ; V(p) =V0
free energy (T =0 K) = enthalpy H = E + pV
H(p) = + E0
Phase equilibrium vs. pressure and pressure-driven transitionbetween phase A and phase B:
∆H(p) = HB(p)-HA(p) = 0 ⇒ p = pt equilibrium pressure
∆H(p) > 0 ⇒ phase A stable; ∆H(p) < 0 ⇒ phase B stable
−1
K'
V0V
K'0K K'
p0K
K'1
1
−
+
−
−
+−
1
11
11
/K'
p0K
K'
K'0V0K
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Thermodynamics: effect of temperature
Quasi-harmonic lattice-dynamical model
⇓thermal vibrations only are taken into account
(correct for insulating diamagnetic ordered crystals)
Thermodynamics: effect of temperature
Quasi-harmonic lattice-dynamical model
⇓thermal vibrations only are taken into account
(correct for insulating diamagnetic ordered crystals)
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Thermodynamic functions of phase assembly⇓
∆tG(p,T) = ∑hnhGh(p,T) = 0
⇓Thermodynamics of phase transitions / chemical reactions
Athermal limit case:
only pressure is considered, enthalpy replaces G function
∆tH(p) = ∑hnhHh(p) = 0
Thermodynamic functions of phase assembly⇓
∆tG(p,T) = ∑hnhGh(p,T) = 0
⇓Thermodynamics of phase transitions / chemical reactions
Athermal limit case:
only pressure is considered, enthalpy replaces G function
∆tH(p) = ∑hnhHh(p) = 0
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Reconstructive phase transitions: B1/B2 (NaCl and CaO); B3/B1 (ZnS and SiC)
No group-subgroup relation + large enthalpic (heat) effect ⇒strong first-order character
Rocksalt- (B1, Fm-3m, C.N.=6) Zincblende- (B3, F-43m, C.N.=4)↓ ↓
CsCl-type (B2, Pm-3m, C.N.=8) Rocksalt- (B1, Fm-3m, C.N.=6)
pexp pcalc(PBE) pexp pcalc(B3LYP)
NaCl 27 26 GPa ZnS 15 19 GPa
CaO 60 65 SiC ∼100 92
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Enthalpy differences B2-B1 and Cmcm(TlI-like)-B1 vs. pressure for CaO
0 10 20 30 40 50 60 70 80 90p (GPa)
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
H-H
(B1)
(eV)
CaO
B2
Cmcm
∆H(p) = 0Equilibrium condition
⇓p = pt
∆H(p) = 0Equilibrium condition
⇓p = pt
Catti M (2003); Catti M (2004)
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Displacive phase transitions: AgCl and AgI Catti M, Di Piazza L (2006); Catti M (2006)
Group-subgroup relation + small/vanishing enthalpic effect ⇒second-order or weak first-order character
Rocksalt-(Fm-3m) → KOH-(P21/m) → TlI-(Cmcm) → CsCl-type (Pm-3m)
AgCl 7 11 17 GPa
AgI 11 ? ?
Hull and Keen, Phys. Rev. B (1999); Kusaba et al., J. Phys. Chem. Solids (1995)Synchrotron X-ray diffraction with diamond-anvil-cell
pressure
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KOH-type
rocksalt-type
TlI-type
CsCl-type
points towardslower symmetry
points towardshigher pressure
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StructuresStructures of of highhigh--pressurepressure polymorphspolymorphs of of silversilver halideshalides
rocksalt KOH-type TlI-typecubic monoclinic orthorhombic
Ag
X
Z
I1 I2
pressure
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TwoTwo subsequentsubsequent displacivedisplacive ((continuouscontinuous) ) phasephase transitionstransitions
►► symmetrysymmetry groupgroup//subgroupsubgroup relations in the first relations in the first ((FmFm--3m 3m ||P2P211/m/m) and second () and second (CmcmCmcm ||P2P211/m/m) pair of ) pair of phasesphases
►► samesame orderorder parameterparameter ((continuouscontinuous lattice lattice distortiondistortion) ) forforbothboth transformationstransformations
►► a a reconstructivereconstructive ((discontinuousdiscontinuous) ) FmFm--3m3m toto CmcmCmcmtransitiontransition with intermediate with intermediate P2P211/m/m state is state is mimickedmimicked
►► Landau Landau potentialpotential at the at the athermalathermal limitlimit: : enthalpy enthalpy H = E + H = E + pVpV
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TwoTwo orientationorientation statesstates of the of the P2P211/m/m structurestructure, and , and orthorhombicorthorhombic BmmbBmmb ((CmcmCmcm)) lattice in the middlelattice in the middle
a
c
a’
c’
a
c
a’
β β
c’
β β’ ’
Rocksalt: β=90°, η=0 TlI-type: cosβ= -a/2c, η=1
η=-(2c cosβ)/a η = 1 η’= 2- η
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Theoretical method
Crystal quantum-mechanical total energy: E
Enthalpy: H = E + pV = H(a,X,p) = H(a’,X’,η,p)
a’ and X’ : lattice constants and atomic coordinates notrelated to the order parameter η
• enthalpy minimized vs. a’, X’ and η at constant p⇒ H(p) ⇒ equilibrium phase diagram
• enthalpy minimized vs. a’ and X’ at constant η and p⇒ H(η,p) ⇒ Landau theory analysis
Theoretical method
Crystal quantum-mechanical total energy: E
Enthalpy: H = E + pV = H(a,X,p) = H(a’,X’,η,p)
a’ and X’ : lattice constants and atomic coordinates notrelated to the order parameter η
• enthalpy minimized vs. a’, X’ and η at constant p⇒ H(p) ⇒ equilibrium phase diagram
• enthalpy minimized vs. a’ and X’ at constant η and p⇒ H(η,p) ⇒ Landau theory analysis
Michele Catti – Gargnano, 1st May 2008
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Theoretical phasediagram of AgI
15 20 25 30 35 40 45p (GPa)
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10∆
H (e
V)
H(Cmcm)-H(Fm-3m)
H(P21/m)-H(Fm-3m)
H(P21/m)-H(Cmcm)
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pcalc pexp pcalc pcalc pexpB3LYP B3LYP PBE
Calculated and experimental1 phase transition pressures
1 Hull & Keen, Phys.Rev.B (1999)
6
4
23
42
19
----
?
11
Rocksalt-* ↔ TlI-type*(metastable)
KOH- ↔ TlI-typeP21/m ↔ Cmcm
Rocksalt- ↔ KOH-typeFm-3m ↔ P21/m
----12
1115
7 GPa10
AgI AgCl
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-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0η
-0.08
-0.06
-0.04
-0.02
0.00
0.02∆H
(eV)
AgI 18
19.3
24
21
30
35
16 GPa
Enthalpy of the KOH-type phase of AgI vs. order parameter(lower p range) - η = 0 corresponds to the rocksalt-type phase
Michele Catti – Gargnano, 1st May 2008
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Enthalpy of the KOH-type phase of AgI vs. order parameter(upper p range) - η = 1 corresponds to the TlI-type phase
0.7 0.8 0.9 1.0 1.1 1.2 1.3η
-0.006
-0.004
-0.002
0.000
0.002
0.004∆H
(eV)
AgI
30
2435
40
41.7
45 GPa
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BothBoth phasephase transitionstransitions show a show a partlypartly discontinuousdiscontinuous charactercharacter, , withwithactivationactivation barriersbarriers and and possiblepossible metastablemetastable statesstates; ; butbut::
►► in the KOHin the KOH-- to TlIto TlI--type case, the type case, the ∆∆HHaa value (0.8 meV) is negligible with value (0.8 meV) is negligible with respect to thermal respect to thermal energyenergy at RTat RT
►► in the rocksaltin the rocksalt-- to KOHto KOH--type case, the type case, the activationactivation enthalpyenthalpy isis significantsignificant(10.3 (10.3 meVmeV) )
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StructuralStructural parametersparameters optimizedoptimized at the at the rocksaltrocksalt--/KOH/KOH--type (19.3 type (19.3 GPaGPa) and ) and
KOHKOH--//TlITlI--type (41.7 type (41.7 GPaGPa) phase equilibrium pressures) phase equilibrium pressures
ηη ββ ((°°)) aa ((ÅÅ) ) bb ((ÅÅ) ) cc ((ÅÅ) ) xxAgAg zzAgAg xxII zzII ∆∆H H ((meVmeV))
-0.150.72030.31730.18980.14915.7494.0584.06799.570.474
10.220.73320.29290.22070.19145.7464.0264.07995.080.250
00.750.250.250.255.7624.0744.074900
00.71550.35780.17610.08805.6024.0713.716109.371
0.830.71520.34950.17680.09965.5434.0463.734107.040.870
0.000.71520.34440.17760.10655.5064.0163.757105.230.769
p = 41.7 GPa
p =19.3 GPa
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Order parameter of the phases of AgI vs. pressure
KOH-type
Rocksalt-type
TlI-type
Red: equilibrium states(primary enthalpyminima)
Blue: metastable states(secondary enthalpyminima)
15 20 25 30 35 40 45p (GPa)
0.00
0.20
0.40
0.60
0.80
1.00
η
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►► LargeLarge jumpjump ∆η∆η of of orderorder parameterparameter at the at the rocksaltrocksalt--//KOHKOH--typetype transitiontransition,,and and significantsignificant ∆η∆η valuevalue alsoalso at the KOHat the KOH--//TlITlI--typetype transformationtransformation
⇓⇓►► RelativelyRelatively limitedlimited ηη rangerange forfor the the stablestable monoclinicmonoclinic KOHKOH--typetype phasephase
UnexpectedUnexpected resultresult, in , in viewview of the moderate/of the moderate/smallsmallenthalpyenthalpy barriersbarriers forfor eithereither phasephase transitiontransition
⇒⇒ HowHow can the can the largelarge ∆η∆η jumpjump at the at the rocksaltrocksalt--//KOHKOH--typetype phasephase boundaryboundarybebe justifiedjustified on on structuralstructural groundsgrounds ??
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rocksalt KOH-type TlI-typecubic Fm-3m monoclinic P21/m orthorhombic Cmcm
Ag
X
Z
I1 I2
Mechanism of phase transformation: slide of double (001) atomic layers along [100]
C.N. = 6 C.N. = 6+1 C.N. = 7
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►► MonoclinicMonoclinic KOHKOH--typetype structurestructure: : d(Agd(Ag--I2) I2) ≤≤ d(I1d(I1--I2) in I2) in orderorder toto formform the the seventhseventh AgAg--I2 I2 coordinationcoordination bondbond
►► KOH (KOH (experimentalexperimental data data byby JacobsJacobs etet al., Z. al., Z. AnorgAnorg. . AllgAllg. . ChemChem. (1985)):. (1985)):KK--O2 = 3.92, O1O2 = 3.92, O1--O2 = 3.96 O2 = 3.96 ÅÅ
►► AgIAgI (19.3 (19.3 GPaGPa, this work):, this work):AgAg--I2 = 3.96, I1I2 = 3.96, I1--I2 = 4.07 I2 = 4.07 ÅÅ
►► AgIAgI (41.7 (41.7 GPaGPa, this work):, this work):AgAg--I2 = 3.31, I1I2 = 3.31, I1--I2 = 3.76 I2 = 3.76 ÅÅ
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8-th order polynomial fits of the H (η) Landau potential
about the KOH-/TlI-type phase equilibrium (η = 1)
∆H = A (η-1)2+B (η-1)4+C (η-1)6+D (η-1)8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0η
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10∆ H
(eV)
AgI 35 GPa
24
30
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PressurePressure dependencedependence of the Landau of the Landau polynomialpolynomial coefficientscoefficients
20 25 30 35 40 45p (GPa)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
eV
A(2nd)
B(4th)
C(6th)
D(8th)
∆H = A (η-1)2+B (η-1)4+C (η-1)6+D (η-1)8
2° order A:linear behaviour aboutpt for KOH-/TlI-typetransition
4°, 6°, and 8° orderB, C, and D:constant behaviour aboutpt for KOH-/TlI-typetransition
⇓Second-order phasetransition
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♦ First-order transformations:
both discontinuous order parameter and significant activation enthalpy
♦ AgI phase transitions: weak first-order (rocksalt-/KOH-type) and effective second-order (KOH-/TlI-type) characters on the basis of ∆Ha values
♦ H (η) Landau potential expanded by 8-th order even polynomial about the KOH-/TlI-type phase equilibrium
⇓correct pressure dependence of the four coefficientsA, B, C, and D
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Michele Catti – Gargnano, 1st May 2008
Mechanisms of reconstructive phase transitions and symmetry of the intermediate states
G1 (S.G. of phase 1) → G (S.G. of intermediate state) → G2 (S.G. pf phase 2)
G ⊂ G1, G ⊂ G2, G1 ⊄ G2
Let T1, T2 and T be the translation groups of G1, G2 and G, respectively, and T1 ⊆ T2. Then:
T ⊆ T1, T ⊆ T2
In the simplest case T1=T2, so that G1 and G2 have the same translation group (i.e., the
primitive unit-cells of phases 1 and 2 have the same volume, except for a minor difference
due to the ∆V jump of first-order transitions).
The translation group of G may coincide with that of G1 and G2 (T=T1), but it may also
be a subgroup of it (T ⊂ T1, i.e., the volume of the primitive cell of the intermediate state
is an integer multiple of that of the end phases, called the 'index' of the superlattice).
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Michele Catti – Gargnano, 1st May 2008
Let P be the symmetry point group of the superlattice T.
The point group P' of G must be a subgroup of P: P' ⊆ P.
Important constrain on the atomic displacements during the reconstructive phase transition:
Atoms must remain in the same types of Wyckoff positions of G along the entire path G1 → G2.
Example: rocksalt (G1= Fm-3m) to CsCl-type (G2= Pm-3m) structure in NaCl under pressure
Na and Cl in Wyckoff positions (a) and (b), respectively, in both G1 and G2
Fm-3m (B1) Z=4 M 0, 0, 0 (m-3m); X ½, ½, ½ (m-3m) aI
Pm-3m (B2) Z=1 M 0, 0, 0 (m-3m); X ½, ½, ½ (m-3m) aII
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Michele Catti – Gargnano, 1st May 2008
Intermediate state with common subgroup of the Fm-3m and Pm-3m space groups
- Buerger's mechanism: intermediate state with R-3m maximal common subgroup (index k1)
- Watanabe's mechanism: intermediate state with Pmmn maximal common subgroup(index k2)
- new mechanism: intermediate state with P21/m non-maximal common subgroup (index k2), but subgroup of R-3m
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R-3m pathway of the B1/B2 phasetransition
RR--3m3m pathwaypathway of the B1/B2 of the B1/B2 phasephasetransitiontransition
B1 B1 RR--3m3m B2B2Michele Catti – Gargnano, 1st May 2008
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Pmmn pathway of the B1/B2 phasetransition
PmmnPmmn pathwaypathway of the B1/B2 of the B1/B2 phasephasetransitiontransition
B1 B1 PmmnPmmn B2B2
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P21/m pathway of the B1/B2 phase transitionP2P211/m/m pathwaypathway of the B1/B2 of the B1/B2 phasephase transitiontransition
B1
B2
P21/m
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Enthalpy of intermediate states vs. reactioncoordinate for the B1/B2 transition at equilibrium
pressure
EnthalpyEnthalpy of intermediate of intermediate statesstates vs. vs. reactionreactioncoordinate coordinate forfor the B1/B2 the B1/B2 transitiontransition at at equilibriumequilibrium
pressurepressure
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0ξ
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
∆ H (e
V)
R-3m
Pmmn
P21/m
CaO
p=65 GPa
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0ξ
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
∆ H (e
V)
NaCl
p=26 GPa
R-3m
Pmmn
P21/m
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→ The P21/m pathway is most favoured
→ Hierarchy of mechanisms: P21/m > Pmmn > R-3m
→ Activation enthalpies (eV):
P21/m Pmmn R-3m
CaONaCl
⇒ Intermediate metastable phase in the P21/m pathway
→→ The The P2P211/m/m pathwaypathway isis mostmost favouredfavoured
→→ HierarchyHierarchy of of mechanismsmechanisms: : P2P211/m > /m > PmmnPmmn > R> R--3m3m
→→ Activation enthalpies (eV):Activation enthalpies (eV):
P2P211/m Pmmn R/m Pmmn R--3m3m
CaOCaONaClNaCl
⇒⇒ Intermediate metastable phase in the P2Intermediate metastable phase in the P211/m /m pathwaypathway
0.0810.0810.0720.0720.0500.0500.2150.2150.1430.1430.1300.130
Michele Catti – Gargnano, 1st May 2008
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♦ P21/m metastable phase at β =71°:slight monoclinic distortion of the orthorhombic Cmcm TlI (yellow phase) structure
♦ Cmcm structure: stable intermediate phase in the B1→Cmcm→B2 transition of AgCl, NaBr, NaI
♦ C.N. = 7 for both cations and anions, giving a peculiar stability to the structure
Only along the P21/m pathway C.N.=7 isallowed
♦♦ P2P211/m/m metastablemetastable phasephase at at ββ =71=71°°::slight monoclinic distortion of the orthorhombic slight monoclinic distortion of the orthorhombic CmcmCmcm TlI (yellow phase) structure TlI (yellow phase) structure
♦♦ CmcmCmcm structure: stable intermediate phase in the structure: stable intermediate phase in the B1B1→→CmcmCmcm→→B2 transition of AgCl, NaBr, NaIB2 transition of AgCl, NaBr, NaI
♦♦ C.N. = 7 for both cations and C.N. = 7 for both cations and anionsanions, , givinggiving a a peculiarpeculiar stabilitystability toto the the structurestructure
OnlyOnly alongalong the the P2P211/m/m pathwaypathway C.NC.N.=7 .=7 isisallowedallowed
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B3/B1 phase transition
► B3 structure: zinc blende-type (F-43m), C.N.=4low pressure
► B1 structure: rocksalt-type FCC (Fm-3m), C.N.=6high pressure
ZnS: pt = 19 GPa SiC: pt = 92 GPa
Proposed mechanisms:
→ R3m (Buerger, 1948)
→ Imm2 (Catti, 2001)
B3/B1 B3/B1 phasephase transitiontransition
►► B3 B3 structurestructure: : zinczinc blendeblende--typetype ((FF--43m43m), C.N.=4), C.N.=4low pressurelow pressure
►► B1 structure: rocksaltB1 structure: rocksalt--type FCC (type FCC (FmFm--3m3m), C.N.=6), C.N.=6high pressurehigh pressure
ZnS: ZnS: pptt = 19 GPa SiC: = 19 GPa SiC: pptt = 92 = 92 GPaGPa
ProposedProposed mechanismsmechanisms::
→→ R3mR3m ((BuergerBuerger, 1948), 1948)
→→ Imm2Imm2 (Catti, 2001)(Catti, 2001)
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Imm2 and R3m mechanisms of the B3/B1 phasetransition
Imm2Imm2 and and R3mR3m mechanismsmechanisms of the B3/B1 of the B3/B1 phasephasetransitiontransition
Imm2
R3m
Michele Catti – Gargnano, 1st May 2008
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Imm2 pathway of the B3/B1 phase transition of ZnSand SiC
Imm2Imm2 pathwaypathway of the B3/B1 of the B3/B1 phasephase transitiontransition of of ZnSZnSand SiCand SiC
B3 Imm2 B1Michele Catti – Gargnano, 1st May 2008
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Enthalpy of intermediate states vs. reactioncoordinate for the B3/B1 transition of ZnS at
pt=19 GPa
EnthalpyEnthalpy of intermediate of intermediate statesstates vs. vs. reactionreactioncoordinate coordinate forfor the B3/B1 the B3/B1 transitiontransition of of ZnSZnS at at
pptt=19 =19 GPaGPa
Michele Catti – Gargnano, 1st May 2008
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→ The Imm2 mechanism is favoured over R3m
→ Activation enthalpies (eV):
Imm2 R3m
ZnSSiC
→ Imm2: C.N. increases from 4 to 6R3m: C.N. decreases from 4 to 3, then increases to 6
→→ The The Imm2Imm2 mechanismmechanism isis favouredfavoured over over R3mR3m
→→ ActivationActivation enthalpies (eV):enthalpies (eV):
Imm2 R3mImm2 R3m
ZnSZnSSiCSiC
→→ Imm2Imm2: C.N. increases from 4 to 6: C.N. increases from 4 to 6R3mR3m: C.N. decreases from 4 : C.N. decreases from 4 toto 3, 3, thenthen increasesincreases toto 66
Michele Catti – Gargnano, 1st May 2008
2.102.100.750.75
0.490.490.150.15
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Summary
♦ Ab initio simulations are a powerful tool to investigate the thermodynamics and kinetics of high-pressurephase transitions
♦ The first- or second-order character of displacivetransitions can be assessed
♦ Landau theory can be applied quantitatively to displacivetransformations
♦ Comparison of enthalpy barriers can detect the hierarchyof different kinetic mechanisms for reconstructive transitions
SummarySummary
♦♦ AbAb initioinitio simulationssimulations are a are a powerfulpowerful tooltool toto investigate investigate the the thermodynamicsthermodynamics and and kineticskinetics of of highhigh--pressurepressurephasephase transitionstransitions
♦♦ The firstThe first-- or or secondsecond--orderorder charactercharacter of of displacivedisplacivetransitionstransitions can can bebe assessedassessed
♦♦ Landau Landau theorytheory can can bebe appliedapplied quantitativelyquantitatively toto displacivedisplacivetransformationstransformations
♦♦ Comparison of enthalpy barriers can detect the Comparison of enthalpy barriers can detect the hierarchyhierarchyof of differentdifferent kinetickinetic mechanismsmechanisms forfor reconstructivereconstructive transitionstransitions
Michele Catti – Gargnano, 1st May 2008
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References
• Catti M, Phys. Rev. Lett. 87 035504 (2001)• Catti M, Phys. Rev. B 65 224115 (2002)• Catti M, Phys. Rev. B 68 100101(R) (2003) • Catti M, J. Phys.: Condens. Matter 16 3909 (2004)• Catti M., Phys. Rev. B 72, 064105 (2005)• Catti M, Di Piazza L, J. Phys. Chem. B 110, 1576 (2006)• Catti M, Phys. Rev. B 74, 174105 (2006) • Stokes H T and Hatch D M, Phys. Rev. B 65 144114 (2002)• Toledano P and Dmitriev V, 'Reconstructive Phase Transitions' Singapore:
World Scientific (1996)• Toledano J C and Toledano P, 'The Landau Theory of Phase Transitions'
Singapore: World Scientific (1987)
Michele Catti – Gargnano, 1st May 2008