2009 fall Phase Transformation of Materials 09.22.2009 Eun Soo Park Office: 33-316 Telephone: 880-7221 Email: espark@snu.ac.kr Office hours: by an appointment 1
Contents for previous class - inary phase diagrams 합금의평형조성주어진온도에서얻은자유에너지곡선으로얻음평형은온도변화에따라어떻게변화되어가는가? 1) Simple Phase Diagrams 2) Systems with miscibility gap 4) Simple Eutectic Systems 3) Ordered Alloys 5) Phase diagrams containing intermediate phases - Gibbs Phase Rule F = C P + 2 (from T, P) 2
Contents for today s class Effect of Temperature on Solid Solubility Equilibrium Vacancy Concentration Influence of Interfaces on Equilibrium Ternary Equilibrium: Ternary Phase Diagram 3
4 Effect of T on solid solubility o o o o o X RT X G G G G G G X RT X G ln ) (1 ln ) (1 2 0 2 = Ω = = = Δ + + Ω = α α α α β α β α α β α α μ μ μ μ ) exp( ln 1), ( ) (1 ln ln ) (1 2 2 RT G X G X RT X here X G X RT X RT X G e e e + Ω Δ = >> Ω = Δ << Ω = Δ = Ω Δ α β α β α β α β ) )exp( exp( RT H R S X S T H G e + Ω Δ Δ = Δ = Δ Δ α β α β α β α β α β 이므로 Q : heat absorbed (enthalpy) when 1 mole of β dissolves in A rich α as a dilute solution. e X T = RT Q A X e exp
Equilibrium Vacancy Concentration ΔG = ΔH TΔS Vacancies increase the internal energy of crystalline metal due to broken bonds formation. ΔH Vacancies increase entropy because they change the thermal vibration frequency and also the configurational entropy. Total entropy change is thus ΔH X Δ S = ΔSVXV R{XVlnX V + (1 X V)ln(1 X V)} The molar free energy of the crystal containing X v mol of vacancies G= GA +Δ G= GA +ΔHVXV TΔ SVXV + RT{XVlnX V + (1 X V)ln(1 X V)} V V With this information, estimate the equilibrium vacancy concentration. 5
at equilibrium dg dx e V X = X V V = 0 e ΔH TΔ S + RTlnX = 0 V V V e ΔSV ΔHV XV = exp exp R RT putting Δ G = ΔH TΔS e ΔG XV = exp RT V V V V In practice, H V is of the order of 1 ev per atom and X V e reaches a value of about 10-4 ~10-3 at the melting point of the solid 6
Interface (α/β)=γ 2γ ΔP = r ΔG = X X X 2γV r e r= r= X X m 의 effect Gibbs-Thomson effect: 계면에너지로인해자유에너지가증가하는현상 ΔG + Ω = exp( ) RT ΔG + Ω = exp( ) RT r ΔG + Ω 2γV = exp( RT r= 2γVm = X exp( ) RTr r= r r= 2γVm 2γV = exp( ) 1+ RTr RTr m / r ) Ex) γ=200mj/m 2, V m =10-5 m 3,T=500K X r = 1 + X 1 r( nm) r=10nm 이면 10% 증가 m 7
8 β formation in α β Nucleation & growth in α Interface (α/β) : size barrier composition barrier Δ Δ = Δ = *, 2 2 * r T T L T G r m V γ γ Undercooling 이클수록 r* 가작다 Nucleation β 상의수 size barrier (r*) r V G m 2γ = Δ T m T L G Δ Δ
Additional Thermodynamic Relationships for inary Solutions 조성변화로인한화학퍼텐셜의변화계산 : Gibbs-Duhem 식 X dμ + X dμ = A A 0 dμa X = dμ X A = d( μ μa 1 ) dg dx = μ μa 1 2 dg A μa μ A 2 X d = X d = X X dx dx 9
The Gibbs-Duhem Equation be able to calculate the change in chemical potential (dμ) that result from a change in alloy composition (dx). For a regular solution, G= XAGA + XG +Ω XAX + RT(XAlnXA + XlnX ) 2 dg RT 2 2 dx X AX = Ω For a ideal solution, Ω = 0, 2 dg 2 dx μ = G + RTlna = G + RTlnγX = RT X X A dμ RT X dγ RT dln γ = 1+ = 1+ dx X γ dx X d ln X 10
a similar relationship can be derived for dμ A /dx dln γ A dln γ X dμ = X dμ = RT 1+ dx = RT 1+ dx dln X A dln X A A 2 dg A μa μ A 2 X d = X d = X X dx dx 2 dg dln γ A dln γ X AX = RT 1 RT 1 2 + = + dx dln X A dln X 11
Driving force: precipitation * Consider the chemical potential of component in phase alpha compared to in beta. This difference, labeled as G n on the right of the lower diagram is the driving force (expressed as energy per mole, in this case). * To convert to energy/volume, divide by the molar volume for beta: G V = G n /V m. R T e N P T Driving force for the reaction : G 0 Driving force for nucleation : G n S Q U ecause the first nuclei of beta to appear do not ignificantly change the composition of the parent material 12
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What are ternary phase diagram? www.sjsu.edu/faculty/selvaduray/page/phase/ternary_p_d.pdf 14
Gibbs Phase Rule for 3-component Systems 15
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 %. X A +X +X C = 1 Used to determine the overall composition 16
Overall Composition 17
Overall Composition 18
Ternary Isomorphous System 19
Ternary Isomorphous System 20
Ternary Isomorphous System 21
Ternary Isomorphous System 22
Ternary Isomorphous System 23
Ternary Isomorphous System Isothermal section F = C - P 24
Ternary Isomorphous System Isothermal section 25
Ternary Isomorphous System Isothermal section F = C - P 26
Ternary Isomorphous System Locate overall composition using Gibbs triangle 27
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Ternary Eutectic System (No Solid Solubility) 29
Ternary Eutectic System (No Solid Solubility) Liquidus projection 30
Ternary Eutectic System (No Solid Solubility) 31
Ternary Eutectic System (No Solid Solubility) 32
Ternary Eutectic System (No Solid Solubility) 33
Ternary Eutectic System (No Solid Solubility) 34
Ternary Eutectic System (No Solid Solubility) 35
Ternary Eutectic System (No Solid Solubility) 36
Ternary Eutectic System (No Solid Solubility) T= ternary eutectic temp. A C L+A+C L+A+ L++C 37
Ternary Eutectic System (with Solid Solubility) 38
Ternary Eutectic System (with Solid Solubility) 39
Ternary Eutectic System (with Solid Solubility) 40
Ternary Eutectic System (with Solid Solubility) 41
Ternary Eutectic System (with Solid Solubility) 42
Ternary Eutectic System (with Solid Solubility) 43
Ternary Eutectic System (with Solid Solubility) 44
Ternary Eutectic System (with Solid Solubility) 45
Ternary Eutectic System (with Solid Solubility) T= ternary eutectic temp. A L+β+γ C L+α+β L+α+γ 46
Ternary Eutectic System (with Solid Solubility) 정해솔학생제공자료참조 : 실제isothermal section의온도에따른변화 http://www.youtube.com/watch?v=yzhvomadetm 47
Ternary Eutectic System Solidification Sequence 48
Ternary Eutectic System Solidification Sequence 2 상영역에서수직단면이 tie line 과불일치하므로다른온도에서평형상만나타내고조성은표시할수없음. 49
Quarternary 의평형상태를기하학적으로표시한그림 4 성분원소들가운데서임의의 3 성분의농도가독립적으로변할수있는함수이므로여러가지조성의 Quar-ternary alloy 은공간적으로표시된다. 3 원계의조성은정 4 면체의면상에, 그리고 4 원계합금의조성은정 4 면체의내부공간에표시된다. 합금의조성은정 4 면체의기하학적성질에의하여결정된다. 4 원계에서상조성을결정하기위하여 lever rule 을이용한다. 4 원합금의변태과정을고찰할때정 4 면체안의추상적인 4 차원투영을이용한다. 50
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