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Telephone: 880-7221 Email: espark@snu.ac.kr
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2020 Spring
Advanced Solidification
Eun Soo Park
04.20.2020
1
Melting and Crystallization are Thermodynamic Transitions
(1st order transition)
Chapter 1 Introduction of Solidification
Glass transition is kinetic Transitions
(pseudo 2nd order transition)
Melting Temp. (Tm) Δ G = 0 1) GL versus GS 2) Interfacial free energy
Glass transition (Tg) “Internal” time scale ≈ “external” time scale
2 Contents for previous class
Liquid Undercooled Liquid Solid
<Thermodynamic>
Solidification:
Liquid Solid• Interfacial energy
Δ T
NMelting:
Liquid Solid• Interfacial energy SV LV
SL
γ γ
γ + <
No superheating required!
No Δ T
NTm
vapor
Melting and Crystallization are Thermodynamic Transitions
Incentive Homework 1:
Example of Superheating (PPT 3 pages)
3
NANO STRUCTURED MATERIALS LAB. 4
The First-Order Transitions
G
T S
T
∆S = L/T
T CP
N P
P T
T S C
,
∂
= ∂ Latent heat
Energy barrier
Discontinuous entropy, heat capacity
Compressibility at constant T or S Heat capacity
at constant P or V Coefficient of
Thermal expansion
The Second Order Transition
G
T S
T
∆S=0 Second-order transition
T CP
No Latent heat Continuous entropy
∞
→
∂
= ∂
N P
P T
T S C
,
Glass Formation is Controlled by Kinetics
• Glass-forming liquids are those that are able to “by-pass” the melting point, Tm
• Liquid may have a “high viscosity”
that makes it difficult for atoms of the liquid to diffuse (rearrange) into the crystalline structure
• Liquid maybe cooled so fast that it does not have enough time to
crystallize
• Two time scales are present
– (1)“Internal” time scale controlled by the viscosity (bonding) of the liquid for atom/molecule
arrangement
– (2) “External” timescale controlled by the cooling rate of the liquid
Temperature M o lar V o lume liquid
glass
Glass transition (1) ≈ (2)
7
Schematic of the glass transition showing the effects of temperature on the entropy, viscosity, specific heat, and free energy. Tx is the crystallization onset temperature.
continuous
(V, S, H) (α TC Pκ T)
discontinuous
8
Q1. Types of Melting (L → S)
9 Contents for today’s class
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- Simple Phase Diagrams
Binary phase diagrams
11
Melting Point
12
Binary phase diagrams
12
Q2. Types of Melting (L → S)
Congruent vs Incongruent
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Congruent vs Incongruent
Congruent phase transformations: no compositional change associated with transformation
Incongruent phase transformation:
at least one phase will experience change in composition
Examples:
• Allotropic phase transformations
• Melting points of pure metals
• Congruent Melting Point
Examples:
• Melting in isomorphous alloys
• Eutectic reactions
• Pertectic Reactions
• Eutectoid reactions
Ni Ti
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15
* Congruent transformations
Congruent transformation:
(a) and (b): a melting point minimum, a melting point maximum, and a critical temperature associated with a order-disorder transformation
(c) and (d): formation of an intermediate phase (next page)
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* Congruent transformations
: More usual type of congruently-melting intermediate phase
→ Partial phase diagram A-X and X-B
Microstructure of a cast Al-22% Si alloy
showing polyhedra of primary Si in eutectic matrix
: Similar with eutectic alloy system/ primary β phase with well-formed crystal facets (does not form dendrite structure)
In many cases, X = normal valency compound such as Mg2Si, Mg2Sn, Mg2Pb
or Laves phase, particularly stable compounds 17
18
: More usual type of congruently-melting intermediate phase
Very little solubility of the components in the β phase
Metatectic reaction: β ↔ L + α
Ex. Co-Os, Co-Re, Co-Ru 1920
Synthetic reaction Liquid
1+Liquid
2↔ α
L
1+L
2α
K-Zn, Na-Zn,
K-Pb, Pb-U, Ca-Cd
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21
a) Eutectic reaction
b) Peritectic reaction
> 0
∆
>
∆Hmixα Hmixl
Considerable difference between the melting points
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22
c) Monotectic reaction:
Liquid1 ↔Liquid2+ Solid
d) Synthetic reaction:
Liquid1+Liquid2 ↔ α
L
1+L
2α
K-Zn, Na-Zn,
K-Pb, Pb-U, Ca-Cd
L2+S L1
e) Metatectic reaction: β ↔ L + α
Ex. Co-Os, Co-Re and Co-Ru(Both α and β are allotropes of A)
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1.4 Gas-Metal Equilibrium
Two main types of gas-metal equilibrium: (a) those in which the liquid and the solid each take the form of (liquid/solid) solutions, and (b) those in which a compound phase is formed.
Q3. Pressure effect of melting
27
The melting point of a pure element or compound is a constant, but it does vary slightly with pressure. This is because the application of a pressure tends to favor the formation of the phase (solid or liquid) which has the smaller specific volume. Most metals expand on melting, the solid being the denser phase. Increase of pressure, therefore, in such cases, raises the melting point. On the other hand, some substances, including water, gallium, germanium, silicon, and bismuth contract on melting. Pressure lowers the melting point of such materials. The change of melting point corresponding to a change in pressure of one atmosphere can be calculated from the Clapeyron equation:
Pressure Effects
29
The equilibrium temperatures discussed so far only apply at a specific pressure (1 atm, say). At other pressures the equilibrium temperatures will differ.
If α & β phase are equilibrium,
dT S dP V dG
dT S dP V dG
β β
β
α α
α
−
=
−
=
At equilibrium,
β
α dG
dG =
V T
H dT
dP
T S H H
V S V
V S S dT
dP
eq eq
eq eq
∆
= ∆
= ∆
∆
∆
= ∆
−
= −
,
ere
α β
α β
For, α→γ ; ∆V (-), ∆H(+) <0
∆
= ∆
V T
H dT
dP
eq
For, γ →liquid; ∆V (+), ∆H(+)
>0
∆
= ∆
V T
H dT
dP
eq
* Clausius-Clapeyron Relation :
(applies to all coexistence curves)
Fcc (0.74)
Bcc (0.68) hcp (0.74)
Fig. 1.5 Effect of pressure on the equilibrium phase diagram for pure iron
P V G
T
=
∂
∂
Pressure Effects
Q4. Ternary Phase Diagram
30
6) 1.5 Ternary and Multicomponent Alloys
www.sjsu.edu/faculty/selvaduray/page/phase/ternary_p_d.pdf
Diagrams that represent the equilibrium
between the various phases that are formed between three components, as a function of temperature.
Normally, pressure is not a viable variable in ternary phase diagram construction, and is therefore held constant at 1 atm.
What are ternary phase diagram?
Page 12
31
Gibbs Phase Rule for 3-component Systems
7) 1.6 The Phase Rule
(constant pressure)
Page 14
32
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Gibbs phase rule : P=(C+2)-F For isobaric systems : P=(C+1)-F For C=3,
① f=3, trivariant equil, p=1 (one phase equilibrium)
② f=2, bivariant equil, p=2 (two phase equilibrium)
③ f=1, monovaiant equil, p=3
(three phase equilibrium)
④ f=0, invariant equil, p=4
(four phase equilibrium)
G=f(comp., temp.)
→ Ternary system : A, B, C
→ G=XAGA+XBGB+XCGC+aXAXB+bXBXC+cXCXA+RT(XAlnXA+XBlnXB+XClnXC)
“Ternary Phase diagram”
Gibbs Triangle
An Equilateral triangle on which the pure components are represented by each corner.
Concentration can be expressed as either “wt. %” or “at.% = molar %”.
XA+XB+XC = 1
Used to determine
the overall composition
34
Overall Composition
35
Overall Composition
36
Ternary Isomorphous System
Isomorphous System: A system (ternary in this case) that has only one solid phase. All components are
totally soluble in the other components. The ternarysystem is therefore made up of three binaries that exhibit total solid solubility.
The Liquidus surface: A plot of the temperatures above which a homogeneous liquid forms for any given overall composition.
The Solidus Surface: A plot of the temperatures below which a (homogeneous) solid phase forms for any given overall composition.
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Ternary Isomorphous System
38
Ternary Isomorphous System
39
Ternary Isomorphous System
40
Ternary Isomorphous System
41
Ternary Isomorphous System
Isothermal section → F = C - P
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Ternary Isomorphous System
Isothermal section
43
Ternary Isomorphous System
Isothermal section → F = C - P
Fig. 1.41 (a) Free energies of a liquid and three solid phases of a ternary system.
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45
Ternary Isomorphous System
Locate overall composition using Gibbs triangle
46
Ternary Eutectic System
(No Solid Solubility)
47
Ternary Eutectic System
Liquidus projection
(No Solid Solubility)
48
Ternary Eutectic System
(No Solid Solubility)
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* Isothermal section
Ternary Eutectic System
(No Solid Solubility)
50
Ternary Eutectic System
(No Solid Solubility)
51
Ternary Eutectic System
(No Solid Solubility)
52
Ternary Eutectic System
(No Solid Solubility)
53
Ternary Eutectic System
(No Solid Solubility)
54
Ternary Eutectic System
(No Solid Solubility)
T= ternary eutectic temp.
A C
B L+A+C
L+A+B L+B+C
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Ternary Eutectic System
(with Solid Solubility)
56
Ternary Eutectic System
(with Solid Solubility)
57
58
Ternary Eutectic System
(with Solid Solubility)
Ternary Eutectic System
(with Solid Solubility)
59
Ternary Eutectic System
(with Solid Solubility)
60
61
Ternary Eutectic System
(with Solid Solubility)
61
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Ternary Eutectic System
(with Solid Solubility)
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Ternary Eutectic System
(with Solid Solubility)
63
Ternary Eutectic System
(with Solid Solubility)
T= ternary eutectic temp.
A C
L+β+γ
L+α+γ L+α+β
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Ternary Eutectic System
정해솔 학생 제공 자료 참조: 실제 isothermal section의 온도에 따른 변화 http://www.youtube.com/watch?v=yzhVomAdetM
(with Solid Solubility)
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Ternary Eutectic System
Solidification Sequence
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Ternary Eutectic System
* The horizontal lines are not tie lines.
(no compositional information)
* Information for equilibrium phases at different tempeatures
* Vertical section
67
Construction of pseudo-binary phase diagram
1MAR2017 / TMS 2017
0 10 20 30 40 50 60 70 80 90 100
600 700 800 900 1000 1100 1200 1300 1400 1500 1600
Temperature ( ℃ )
Cu composition (at. %)
α β
0 10 20 30 40 50 60 70 80 90 100
600 700 800 900 1000 1100 1200 1300 1400 1500 1600
Temperature ( ℃ )
Cu composition (at. %)
α β
0 10 20 30 40 50 60 70 80 90 100
600 700 800 900 1000 1100 1200 1300 1400 1500 1600
Temperature ( ℃ )
Cu composition (at. %)
α β
Phase diagram was expected to optimize composition and
microstructure of phase separating HEAThermodynamic calculation X-ray diffraction
TGA/DSC FE-EPMA
• Expecting approximation of phase diagram
• Determination of phases
• Finding out temperatures of phase transformations
• Confirming invariant reaction points
• Investigation of equilibrium composition at each
temperature
68
Pseudo-binary phase diagram of PS-HEA
1MAR2017 / TMS 2017
Pseudo-binary system between FeCoCrNi and Cu shows
monotectic reaction having liquid separation region.Calculation courtesy of J.H.Kim et al.
Experiment Calculation
L
L 1 + L 2 FCC 1 + L 2
FCC 1 + FCC 2 FCC 2
Dendrite → Droplet → Dendrite
0 10 20 30 40 50 60 70 80 90 100
400 600 800 1000 1200 1400 1600
Temperature ( ℃ )
Cu content (at%) Cu
FeCoCrNi
Posted on L-88 (HEA V: Poster session)
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MoVNbTiZr: Construction of pseudo-ternary phase diagram
Nb,Ti
Mo,V Zr
bcc IC+bc
c
bcc Mo, V, Nb, Ti : Complete S.S.
V-Zr, Mo-Zr Ti-Zr, Nb-Zr
TiNbMoVZr: Construction of pseudo-ternary phase diagram
1500K 1300K
1100K
Calculated pseudo-ternary isothermal sections of the MoNbTiVZr system
MoVNbTiZr: Construction of pseudo-ternary phase diagram
X-ray diffraction analysis of the as-cast samples
Construction of pseudo-ternary phase diagram
Find single phase region without intermetallic compounds BCC1+BCC2+IC
BCC1+IC / BCC2+IC BCC1+BCC2
Single BCC
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.0 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
20
25 26
23 24
22
21
19 11
10
1213
14 15
16 17
18 9 7
8 5
6 4
2 3
MoV Zr
NbTi
Nb Ti Mo
V
Zr
1
Homework 2: Please find ternary phase diagram in the literature.
And explain the detail for the phase diagram in your word.
(within 3 pages PPT)
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< Quaternary phase Diagrams >
Four components: A, B, C, D
A difficulty of four-dimensional geometry
→ further restriction on the system Most common figure:
“ equilateral tetrahedron “ Assuming isobaric conditions, Four variables: XA, XB, XC and T
4 pure components 6 binary systems 4 ternary systems
A quarternary system
a+b+c+d=100
%A=Pt=c,
%B=Pr=a,
%C=Pu=d,
%D=Ps=b
* Draw four small equilateral tetrahedron
→ formed with edge lengths of a, b, c, d
76
Q5. Distribution coefficient
& Van’t Hoff equation
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Page 15
78
Van’t Hoff equation relates the change in the equilibrium constant, Keq, of a chemical reaction to the change in temperature , T, given the standard enthalpy change ΔH, for the process. The equation has been widely utilized to explore the changes in state functions in the
thermodynamic system.
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80 It is probable that the solidus curve is inaccurate.
