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Lecture I-4 Outline
Classification of phase transitions
Solutions - Definitions
Types of solutions
Experimental determination of solubilities
Gibbs energy of solutions
Enthalpy of Mixing
Entropy of Mixing
Ideal Solid Solution Model
Substitutional Solutions - Hume-Rothery Rules
Interstitial Solutions - Rules
Phase Transitions (Transformations)
Examples:
Solid - Gas Sublimation
Solid - Liquid Melting
Liquid – Solid Solidification
Liquid - Gas Evaporation
Gas –Liquid Condensation
Definition:
Phase transition is a conversion (transformation) of one phase into another phase (at a constant composition).
Phase Transitions
Classification
Classifications
Structural Thermodynamic
(solid-solid)
Other examples of Phase transitions:
# Conductor – to – insulator
# Normal – to - Superconductor
Phase TransitionsClassification
Solid – Solid phase transitions
Discontinuous Graphit → Diamant # Chemical bond breaking and reconstruction
(reconstructive) NaCl → CsCl-Typ # No Group-Subgroup relations between parent
hcp → fcc, fcc → hcp and product phase
Continuous
Displacive ß-Quarz → α-Quarz # Only small displacements of the atoms
Order-Disorder AuCu # Group – Subroup relations between parent and
product phase (Lecture 9)
7
Change of first coordination number (CN 4 ↔ 3)
No Group – Subgroup relation
P 63/mmc Fd3m
Phase TransitionsStructural Phase Transitions
Reconstructive
8
Space group Space groupP 31 21 P 62 2 2
Phase TransitionsStructural Phase Transitions
Displacive
The first coordinations of Si and Oxygen remains the same,only small rotations of the SiO4 tetrahedra
No change of translational symmetry (both lattices are primitive)Change only of the point-group symmetry ( 622 → 32)
Phase TransitionsThermodynamic Classifications (1)
Entropy Classification:
A) Discontinuous Phase Transitions
# The entropy is discontinuous DS;
# There is a latent heat L = Tc DS;
# Cp = - T(∂2G/∂T2) is finite for T ≠ TC
and undefined at TC.
B) Continuous Phase Transitions
# The entropy is continuous;
# No latent heat;
# Singularities in Cp and kT at TC.
Discontinuities
Divergences
G
G
T
T
TC
TC
TC TC
S
S
P. Ehrenfest
Phase TransitionsThermodynamic Classifications (2)
Ehrenfest Classification:
A) 1st Order Phase Transitions
# The first derivatives of the Gibbs free energy V= (∂G/∂p)T and S = - (∂G/∂T)p are
discontinuous (exhibit ‚jumps‘).
B) 2nd Order Phase Transitions
#The first derivatives of the Gibbs free energy (V and S) are continuous;
# The second derivatives Cp = - T(∂2G/∂T2)p and ßT = - (∂2G/∂p2)T/V
are discontinuous (exhibit ‚jumps‘) at the transition temperature
Phase TransitionsEhrenfest Classification
1. Order 2. Order
Latent heat L = TC(S1 – S2) No latent heat (L = 0)
(Heat is released/absorbed at T = const)
Co-existance of the two phase No coexistance of the two phase
Overheating/Undercooling possible Overheating/Undercooling not possible
Discontinuous phase transition Continuous phase transition
Solid-Solid phase transitons
No Group-Subgroup relations Group – Subgroup relations mandatory
Phase TransitionsEherenfest Classifications
Heat Capacities
cp
T
cp
Tc
1st Order
T
cp
Tc
2nd Order
Implications:
Cp is divergent at Tc.Cp has a ‘jump’ at Tc
Phase TranisitonsExamples: La0.7Ca0.3MnO3
# S is continuous
2nd order magnetic
phase transition
# D Cp discontinuousSouza et al. (2005)
Phase TransitionsEhrenfest Classification
1. Order 2. Order
Coexsistance of the two phase No coexistance of the two phase
● Existence of Interface bewteen the
two phases – Growth of the product
phase through movement of
interfaces.
● Nucleation of the product phase in
the parent phase (Part III)
‚Spontaneous‘ change of structural state
in the whole macroscopic volume
Formation of domains
# change only of the point-group symmetry
Number of domains = PHT/PLT
Phase TransitionsEhrenfest Classification
PbTiO3 pervoskite
SG PG Order
HT P m -3 m m -3 m 48
LT P 4mm 4mm 8
Number of domains 48/8 = 6
Sicron et al. (1995)
Phase TransitionsEhrenfest Classification
Examples
1. Order 2. Order
Crystallization/Melting Feroelectricity
Sublimation Ferromagnetism
Condensation Normal-to-Superconducting state
Solid-Solid phase transformations Solid-solid phase transformation
Ce
Solutions
Definitions:
Solubility: The ability of a given component to dissolve into another component.
Solvent: The component in greater amount, which dissolves the minor component.
Solute: The component in lesser amount, which is dissolved.
Solubility Limit: The maximum amount (concentration) of the solute, which can be dissolved in the solvent.
Mechanical mixture: System consisting of two completely immissible components.
A solution is a homogeneous system ↔ it represents a single phase.
System with C = 2 and P = 1
Solutions
Alcohol Solubility in water at RT
[g Alc/100 mL water]
Methanol ∞
Ethanol ∞
Butanol 8.0
Hexanol 0.7
Solubility of sugar in water
Solubility changes with T and pressure !!!
Solutions
Types of solutions:
Gaseous
Liquid
Solid (Alloys)
Substitutional
Interstitial
Types of solutions:
Binary (C = 2)
Ternary (C = 3)
Multicomponent ( C > 3)B
A
C
SolutionsDetermination of Solubilities
Experimental methods:
# Solubility of metals in metals Diffusion experiments (Part II)
Quenching of liquid specimens
# Composition – Chemical methods, ICP, EDX SEM
# Homogeneity – Optical microscopy, Diffraction
Variation of the Crystallization/Melting temperature
of the solution with composition
DSC, DTA ( at ambient pressure)
SolutionsDetermination of Solubilities
In DSC/DTA the Cp peak broadens due to instrumental factors
(heating rate; mass of the sample, shape of sample, time lag, etc.)
The melting/crystallization temperaturs in DTA/DSC are
determined by the corresponding Onset Temperatures.
Melting is 1st Order Phase transtion → The Entalpy shows a jump.
Heat capacity Cp ~ dH/dT = Ŝ/ß ideally divergies at Tm.
Solutions
Determination of Solubilities
Schematic change of DTA/DSC peak
with alloying
Pure Ag (Tm = 961.8 oC)
DTA curves at three different heating rates (5, 10, 20 K/min)
Shift of onset temperatures, peak maximum with heating rate →
low heating/cooling rates for accurate results
Boettinger et al.
Elsevier, 2007
Boettinger et al.
Elsevier, 2007
Liquidus
SolutionsGibbs Energy
Mechanical mixture (C = 2):
G = nAGA + nBGB; N = nA + nB ( number of moles);
Molar Gibbs energy: Gm = XA GA + XB GB; XA + XB = 1
Implications:
Gm = (1 – XB) GA + XB GB;
The molar Gibbs energy of a mechanical
mixture is a linear sum of the Gibbs energies
XA = nA/(nA + nB); XB = nB/ (nA + nB);
Solution (C = 2):
Gsol = nAGA + nBGB + DGmix; DGmix - Gibbs energy of mixing
Solutions
Gibbs Energy
Implications:
Condition for the formation of a solution (1): DGmix < 0; but G = H – TS
Condition for the formation of a solution (2): DHmix < TDSmix
DHmix - Enthalpy of Mixing (Heat of formation of the solution);
DSmix - Entropy of Mixing (Entropy of formation of the solution).
Solutions
Enthalpy of Mixing
Water - Metanol Water - EthanolWater - Propanol
DH mix m = Q = Cp DT
Peteers et al. (1993)
Solutions
Enthalpy of Mixing
DH mix = Hsol – HA – HB;
H = U + pV ~ U
Solid (Liquid) Solutions
Neglect the kinetic energy of the species; U ~ Potential energy of the species
U = ½ S E2(rij) + E3(r) + ∙∙ ; E2 – pair-wise potential energy between species
E3 – tree-body potential energy
U = ½ S E2(rij) + Emany-body
Van-der-Waals crystals/liquids (noble gases at low T):
non-directional interactions
Lenard-Jones potential E2LJ = 4e[(s/r)12 - (s/r)6];
SolutionsEnthalpy of Mixing
Potentials
Ionic crystals/melts
non-directional, electrostatic interactions
E2ionic ~ C/r12 – e2/(4peor);
eo – dielectric constant of the vacuum
SolutionsEnthalpy of Mixing
Potentials
Metals/metallic melts
non-directional, delocalized electron-electron
+ electron-core interactions
SolutionsEnthalpy of Mixing
Potentials
Metals
Finnis-Sinclair (FS) potential
E2FS = (r - c)2(co + c1r + c2r
2); r ≤ c (cut-off)
Emany-bodyFS = - Si f(ri); r – Electron density of atom i;
f(r) = r ½ ; ri = S j≠i A2 f(rij); f(rij) = (rij-d)2; r ≤ d
A = 0.93 eV/Å
c = 2.96 Å
d = 4.05 Å
Cu a
c
Dai et al. (2006)
Covalent crystals/melts
strongly-directional interactions
Stillinger-Weber (SW) potential
SolutionsEnthalpy of Mixing
Potentials
Covalent crystals
SW potential
E2SW = [B(s/r)4 – 1]exp(s/(r-as))
E3SW = exp(s/(rij – as))exp(s/(rik-as))(cos(Qjik + 1/3)2;
Sisp3 hybridisation
H ~ U ~ ½ S { E2(r) + E3(r) + ∙∙ } ~ ½ S E2(rNN) = ½ Nz eBond; N – Number of atoms
z - Coordination of the atoms
eBond = E2(rNN) – Bond energy between
nearest-neighbours
Substitutional solid-solution model
# Components A and B have the same coordination (z)
# Bond Energies independent of composition
UA = ½ Nz eAA;
UB = ½ Nz eBB;
Hsol ~ Usol = XA2 UA + XB
2 UB + Nz XAXB eAB; XA2 ~ probability to find A-A bond
XB2 ~ probability to find B-B bond
XAXB ~ probability to find A-B bond
SolutionsEnthalpy of Mixing
A A
B A
AB
BA
B B
A B
AA
AB
B A
B B
BB
AB
A A
B A
AA
BA
A B
A A
BA
AB
B A
B B
AA
AB
Random distribution
SolutionsEnthalpy of Mixing
DHmixm = XA
2 UA + XB2 UB + Nz XAXB eAB - XAUA - XBUB=
= ½ Nz eAA XA(1-XB) + ½ Nz eBB XB(1-XA) + Nz XAXB eAB - ½ Nz XAeAA - ½ Nz XBeBB =
= ½ Nz XAXB (2eAB – eAA – eBB) = ½ Nz XAXB De ; De = 2eAB – eAA – eBB .
= WXAXB ; W = ½ NzDe (N = NA)
Implications:
De > 0 DHmix > 0
De < 0 DHmix < 0
Ideal Solution Model: Solution for which the Enthalpy of mixing is zero (DHmix = 0).
DGmix = -TDSmix ;
Implications:
De = 0 → eAB = ½ (eAA + eBB); uniformity of the nearest-neighbhour interactions
eAB = eAA = eBB ; (not necessary condition)!!
DV mix = (∂DG mix/∂p)|T = 0 Vmid = XAVA
o+ XB VBo ;
SolutionsIdeal Solid Solution Model
SolutionsEntropy of Mixing
DS mix = Ssol – nASA – nBSB;
DS mix = kB ln(W)
W = Wvib Welec WConf ; DS mix = DS vib + DS Conf + DS Elec;
Vibrational Entropy:
For a single component system (C = 1) without electronic degrees of freedom and defects, the entropy arrises
from the vibrational degrees of freedom (Lecture 2)
S (vib) = - (∂G/∂T)p ;
Ssol = nASA + nBSB - (∂DGmix/∂p)T(∂p/∂T) = nASA + nBSB - DVmix(∂p/∂T)
DS mix (vib) = -(∂p/∂T) DVmix ; For the ideal solid-solution model DVmix = 0 → DS mixid(vib) ~ 0
# Substitutional solid solutions;
# DS mixid (vib) ~ 0;
# Neglecting electronic contributions to the entropy;
# Only contribution from the different possible arrangements of the A and B atoms in the lattice
considered (configurational entropy)
# ideal solid-solution model: similar forces between different types of atoms → random distribution is
the most probable:
W ~ Wconf = N!/ (XAN)! (XBN)! ; DS mix = kB ln[N!/ (XAN)! (XBN)! ]
SolutionsEntropy of Mixing
SolutionsEntropy of Mixing
DS mix = kB ln[N!/ (XAN)! (XBN)! ] = Sterling formula
= ···
= - kBN { XAlnXA + XBlnXB}; [= - kBN S Xiln(Xi) ] ; i = 1,2, ∙∙∙ C
N = NA; ( 1 mole solution)
Molar Entropy of mixing of the ideal solid solution model:
DS mixid = - R{XAlnXA + XBlnXB}
DS mix = XA SA + XB SB → Si = - R ln(Xi); partial molar entropies of the components
SolutionsEntropy of Mixing
Max
XA = XB = 0.5
DS id mix m = Rln2 = 5.76 J/mol.K
DS mix id > 0
Increase of Entropy by
Mixing!!!
SolutionsGibbs energy of the ideal solution model
Gsol = nAGA + nBGB + DGmix;
Molar Gibbs energy
Gsol m = XAGA + XBGB + (DHmix m – TDSmix m)
Molar Gibbs energy of the ideal solution model:
G mid = XAGA + XBGB + TR {XAlnXA + XBlnXB}
DGmix = TR {XAlnXA + XBlnXB} < 0
The Gibbs energy of the ideal solution model decreases, compared to the
mechanical mixture XAGA + XBGB.
G mid has a minimum at XB = 1/{1 + exp[ (GB-GA)/2RT]}; if GA = GB ; min at XB= 0.5
T2 > T1
SolutionsGibbs energy of ideal solution model
Molar Gibbs energy of the ideal solution model:
1st Derivative: ∂DGidm/∂XB = RTln (XB/1-XB)
XB → 0 ∂DGidm/∂XB → ∞ Logarithmic divergence
# Addition of small amount of solute (vacancies) decreases the Gibbs energy due to the
configurational entropy
2nd Derivative: ∂2DGidm/∂XB
2 = RT(1/XB – 1/(1-XB) =
= RT/XB(1-XB)
Curvature k = ∂2DGidm/∂XB
2 / [(1 + (∂DGidm/∂XB )2]3/2 = RT/ {[XB(1 – XB)]{1 + [RTln (XB/1-XB)]2}3/2 }
maximum curvature kmax = 4RT at XB = ½
SolutionsSubstitutional Solid Solutions
Questions: when are formed ideal substitutional solid solutions,
Hume-Rothery Rules:
1. 100 * | (Rsolute – Rsolvent)/RSolvent | < 15 %; R – atomic radius
2. Crystal structures very similar (identical);
3. Solute and Solvent atoms typically have the same valence;
4. Small difference in the electronegativities : | cA – cB | → 0.
W. Hume-Rothery
Hume – Rothery Rules
Examples:
Proprerty Al Cu
Radius (Å) 1.43 1.28 10%
Structure fcc fcc
Electonegativity 1.6 1.9
Valence state +3 +2(+1)
Limited solubility
SolutionsSubstitutional Solid Solutions
Hume – Rothery Rules
Examples:
Proprerty Ni Cu
Radius (Å) 1.35 1.28 5%
Structure fcc fcc
Electonegativity 1.9 1.9
Valence state +2 +2(+1) +2
Complete Solubility!!
SolutionsSubstitutional Solid Solutions
SolutionsSubstitutional Solid Solutions
Hume – Rothery Rules
Examples:
Proprerty Ag Cu
Radius (Å) 1.44 1.28 11%
Structure fcc fcc
Electonegativity 1.9 1.9
Valence state +2(+1) +2(+1)
Limited solubility
(M) – Solid solution with M as solvent
SolutionsSubstitutional Solid Solutions
Darken – Gurry Maps:
[ (c – cA)/0.2]2 + [ (r – rA)/0.075] 2 = 1
R. Ferro, Intermetallic Chemistry, Elsevier 2008
SolutionsInterstitial Solid Solutions
Size of the Voids
RV ~ 0.7 Ratom;
Criteria for formation of interstitial solid solutions
# Number of Voids, coordination of interstitial atoms
# Degree of distortion of the host (solvent) lattice
SolutionsInterstitial Solid Solutions - bcc
TV
OV
Distorted tetrahedral and octahedral voids
Size of the Voids
TV: RV ~ 0.29 Ratom;
OV: RV ~ 0.155 Ratom
SolutionsInterstitial Solid Solutions - fcc
TV OV
Size of the Voids
TV: RV ~ 0.225 Ratom;
OV: RV ~ 0.414 Ratom
undistorted
SolutionsInterstial Solid Solutions - hcp
Size of the Voids (for ideal c/a ratio)
TV: RV ~ 0.225 Ratom;
OV: RV ~ 0.414 Ratom
SolutionsInterstitial Solid Solutions
Type lattice Type of VoidCubic Tetrahedral Octahedral
SC 1 - -
bcc - 12 6
Fcc - 8 4
hcp - 4 2
SolutionsIntersititial Solid Solutions
RFe ~ 1.26 Å
Size of Voids (Å)
Lattice TV OV
bcc 0.37 0.20
Fcc 0.28 0.52
RC= 0.77 Å; RN = 0.71 Å; RH = 0.46Å
Bond Distortion
eB = (RI – RV) / RA = RI / RA - c;
Total Distortion ~ ½ NV zV eB x 100 (%)
Total DistortionLattice TV OV
bcc 7.7 % 8.1%
Fcc 6.2% 2.4%
Smaller distrotion - Larger Solubility
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