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Fall 2012-Lecture 3

Phase Diagrams and Related

Thermodynamics

Prof. Mitra Taheri

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Phase Diagrams:

Useful Tools for Metallurgists!

Phase: A portion of a system which is uniformin structure and/or in composition

Phase Diagram: A plot with axes of

thermodynamic variables (temp, pressure,composition, activity, chemical potential)

Three Main Types

Unary: Example: water (right)

Binary:

Ternarywww.splung.com

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Types of Phase

Diagrams

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forums.dfoggknives.com/uploads

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www.msm.cam.ac.uk

.but well concentrate on binary phase diagrams, because they help us

understand most of the transformations that we study in metallurgylike steel!

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The Phase Rule

The phase rule, first announced by J . Willard Gibbs in 1876, relates thephysical state of a mixture to the number of constituents in the system and toits conditions. It was also Gibbs who first called each homogeneous region ina system by the term phase. When pressure and temperature are the statevariables, the rule can be written as:

.

f= c -p + 2

wherefis the number of independent

variables (called degrees of freedom),

c is the number of components, andp

is the number of stable phases in the

system.

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Binary Phase Diagrams

Number ofcomponents Number ofphases Degree offreedom Equilibrium

2 3 0 Invariant

2 2 1 Univariant

2 1 2 Bivariant

The Gibbs phase rule applies to all states ofmatter (solid, liquid, and gaseous), but whenthe effect of pressure is constant, the rulereduces to:

f=c - p + 1

The stable equilibria for binary systems aresummarized as:

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Understanding Binary Phase Diagrams

For a material of na moles of a, nb moles of b, withmole fractions Xa and Xb:

Xa=(na/na+nb), Xb= (nb/na+nb), so Xa =1-Xb

Tie-line (Lever Rule)always apply to two-phase field to figure out the relative amountsof each phase at a given point (X,T) in thephase diagram.

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Lever RuleA tie line is an imaginary horizontal

line drawn in a two-phase fieldconnecting two points that

represent two coexisting phases in

equilibrium at the temperature

indicated by the line.

Tie lines can be used to determine

the fractional amounts of the

phases in equilibrium by employing

the lever rule.

The lever rule is an expression

derived by the principle of

conservation of matter in which the

phase amounts can be calculated

from the bulk composition of the

alloy and compositions of the

conjugate phases

Portion of a binary phase diagram

containing a two-phase liquid-plus-

solid field illustrating (a)

application of the lever rule to (b)

equilibrium freezing, (c)

nonequilibrium freezing, and (d)heating of a homogenized sample

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Types of Phase Diagrams

Solid-liquid Transformations

Cases where mutual solubility (miscibility) appearsin solid and liquid states

Partial solubility in both liquid and solid

Cases where miscibility appears in liquid state, butpartial miscibility in solid state:

Solid miscibility gap: Alpha alpha+alpha on cooling Eutectic: liquid alpha + beta on cooling

Peritectic: L+alphabeta on cooling

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Continued Solid-solid transformations

Eutectoid: alpha beta + gamma on cooling

Peritectoid: alpha + beta gamma on cooling

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Thermodynamics Review

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Internal Energy. The sum of the kinetic energy (energyof motion) and potential energy (stored energy) of a

system is called its internal energy, U. Internal energy

is characterized solely by the state of the system.

Closed System. A thermodynamic system that

undergoes no interchange of mass (material) with its

surroundings is called a closed system. A closed

system, however, can interchange energy with its

surroundings.

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First Law. The First Law of Thermodynamics, states that energy

can be neither created nor destroyed. Therefore, it is called the

Law of Conservation of Energy.

The total energy of an isolated system remains constant

throughout any operations that are carried out on it; that is,

for any quantity of energy in one form that disappears from

the system, an equal quantity of another form (or other forms)will appear. For example, consider a closed gaseous system to

which a quantity of heat energy, dQ, is added and a quantity of

work, dW, is extracted. The First Law describes that change in

the internal energy, dU, of the system as:

dU= dQ - dW

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Second Law. While the First Law establishes the

relationship between the heat absorbed and the workperformed by a system, it places no restriction on the

source of the heat or its flow direction.

This restriction is set by the Second Law ofThermodynamics.

The Second Law states that the spontaneous flow of

heat always is from the higher temperature body to the

lower temperature body (all naturally occurring

processes tend to take place spontaneously in the

direction that will lead to equilibrium).

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Third Law. A principle advanced by Theodore Richards,

Walter Nernst, Max Planck, and others often called

the Third Law of Thermodynamics, states that the

entropy of all chemically homogeneous materials can betaken as zero at absolute zero temperature (0 K). This

principle allows calculation of the absolute values of

entropy of pure substances solely from heat capacity.

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Enthalpy. Thermal energy changes under constant pressure (again

neglecting any field effects) are most conveniently expressed in

terms of the enthalpy, H, of a system. Enthalpy, also called heatcontent, is defined by:

H= U+pV

Enthalpy, like internal energy, is a function of the state of the

system, as is the productpV.

Heat Capacity The heat capacity, C, of a substance is the amountof heat required to raise its temperature one degree; that is:

C= dQ/dT

However, if the substance is kept at constant volume (dV= 0):

CV= (Q/T)V= (U/T)VIf, instead, the substance is kept at constant pressure (as in many

metallurgical systems),

Cp = (H/T)p

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Entropy. The Second Law is most conveniently stated in

terms of entropy, S. Entropy represents the energy (per

degree of absolute temperature, T) in a system that is notavailable for work.

In terms of entropy, the Second Law states that all natural

processes tend to occur only with an increase in entropy,and the direction of the process is always such as to lead

to an increase in entropy. For processes taking place in a

system in equilibrium with its surrounding, the change in

entropy is defined as:

S =Q/T=U+pV/T

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Gibbs Energy. Because both S and Vare difficult to control

experimentally, an additional term, Gibbs energy, G, is introduced,

whereby:

G = U+pV- TS = H- TSand dG = dU+pdV+ Vdp - TdS - SdT

However, dU= TdS +pdV

Therefore, dG = Vdp - SdT

The change in Gibbs energy of a system undergoing a process is

expressed in terms of two independent variables, pressure and

absolute temperature, which are easily controlled experimentally. If

the process is carried out under conditions of constant pressure and

temperature, the change in Gibbs energy of a system at equilibriumwith its surroundings (a reversible process) is zero. For a spontaneous

(irreversible) process, the change in Gibbs energy is less than zero

(negative); the Gibbs energy decreases during the process, and it

reaches a minimum at equilibrium.

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Clausius-Clapeyron Equation. The theorem of Le

Chatelier was quantified by Benoit Clapeyron andRudolf Clausius to give:

dp/dT=H/TV

where dp/dTis the slope of the univariant line in

ap-Tdiagram, Vis the difference in molar

volume of the two phases in the reaction, and His the difference in molar enthalpy of the two

phases (the heat of reaction).

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Review (again): Gibbs Free Energy

For closed systems (those with fixed mass andcomposition) at constant pressure,temperature: G =H-TS, where H = enthalpy

and S = entropy H = E+PV; E = internal energy and V = volume

Closed systems at equilibrium when dG=0, so

we want low enthalpy and high entropy. Metastable: local minimums in equilibrium(see example on pg.3 of P&E aboutdiamond/graphite

i d

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continued

G,H and S for pure components are

sometimes tabulated as a function of T More commonly by heat capacity, Cp: Cp= (dH/dT)P We can integrate for HH is energy related, and so

it has no absolute value. We have to take it as H=0for a pure element at 298K:

H=

Similarly, Cp/T = (dS/dT)P so S =

T

CpdT298

T

TdTCp0

/

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So, the Gibbs free energy can be found by integrating-S= (dG/dT)P or d(G/T)/dT = -deltaH/T2:

Variation of H, S, G with temperature.

But its easier to do it directly from G=H-TS.

Consider a change of phase, where Hliq>Hsol at all temps,

because the internal energy of liq is always larger than thatof the solid:

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Solutions- Real and Ideal:

Review this in Gaskell for your reading!

Recall dA=RTlnaA Gmix = XAlnA +XBlnB= RT (XAlnaB +XBlnaB)

For a real solution

Recall Gmix = Hmix -TSmix

For an ideal solution, Hmix = 0

For a regular solution, Hmix =XA

XB

where is

the regular solution parameter

= 0 for an ideal solution, positive for clustering, andnegative for ordered solutions.

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Common Tangents

For any mixing quantity, the tangent at anypoint gives intercepts that are relative partial

molar quantities (graphical depictions):

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Solutions. The shape of liquidus, solidus, and solvus curves (or surfaces) in a phase diagram are

determined by the Gibbs energies of the relevant phases. In this instance, the Gibbs energy must

include not only the energy of the constituent components, but also the energy of mixing of these

components in the phase. Consider, for example, the situation of complete miscibility shown in (next

figures). The two phases, liquid and solid, are in stable equilibrium in the two-phase field between

the liquidus and solidus lines. The Gibbs energies at various temperatures are calculated as a function

of composition for ideal liquid solutions and for ideal solid solutions of the two components, A and B.

The result is a series of plots (Next slides).

At temperature T1, the liquid solution has the lower Gibbs energy and, therefore, is the more stable

phase. At T2, the melting temperature for component A, the liquid and solid are equally stable only ata composition of pure A. At temperature T3, between the melting temperatures of components A

and B, the Gibbs energy curves cross. Temperature T4 is the melting temperature of component B,

while T5 is below it.

Construction of the two-phase liquid-plus-solid field of the phase diagram in the next slide is as

follows. According to thermodynamic principles, the compositions of the two phases in equilibrium

with each other at temperature T3 can be determined by constructing a straight line that is tangentialto both curves in figure c. The points of tangency, 1 and 2, are then transferred to the phase diagram

as points on the solidus and liquidus, respectively.

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Solutions

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References

F.N. Rhines, Phase Diagrams in Metallurgy: Their Development and

Application, McGraw-Hill, 1956

Metallography and Microstructures, Vol 9, 9th ed., ASM Handbook, ASM

International, 1985

J.E. Morral, Two-Dimensional Phase Fraction Charts, Scr. Metall., Vol 18

(No. 4), 1984, p 407-410

J.E. Morral and H. Gupta, Phase Boundary, ZPF, and Topological Lines on

Phase Diagrams, Scr. Metall., Vol 25 (No. 6), 1991, p 1393-1396

http://www1.asminternational.org/asmenterprise/apd/help/intro.aspx

www.sv.vt.edu

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Fall 2012 Lecture 4: Diffusion

Prof. Mitra Taheri

Acknowledgements: P&E, Reed-Hill, 4th ed., Udel coursenotes, www.matter.org.uk,

and CMU coursenotes.

http://www.matter.org.uk/http://www.matter.org.uk/

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What Is Diffusion?

Controls the rate at which phase transformationsoccur, to produce a decrease in Gibbs free energy.

Ficks laws are formulated under the assumption that

the driving force for diffusion is a chemical gradient,

and most diffusion is downhill. Specifically, the

driving force is a gradient of chemical potiential.

x

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What IS a concentration gradient (in

an A-B Solution)?

2003Brooks/Cole,adivisiono

fThomsonLearning,Inc.ThomsonLearningisatrademarkusedhereinunder

license.

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Ficks First Law

Equilibrium is reached when the gradient has been

eliminated, thus making the chemical potential uniform

over the whole material (example: A-B solid solution).

JA = -MA[dA/dx]

Where J = flux (atoms/area*time), dA/dx is the chemical potential

gradient (energy/atom*length), and M is a constant

We will use the case for an ideal (dilute) solution,

unless solution is very concentrated, when its only an

approximation.

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Schematic: Ficks 1st Law

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Atomic Diffusion in Metals

2003Brooks/Cole,adivisionof

ThomsonLearning,Inc.ThomsonLearningisa

trademark

usedhereinunderlicense.

The dependence of diffusion

coefficient of Au on

concentration.

Before going on to look at Fick's 2nd law, a

more detailed insight into the term

diffusivity is given. This is explained in

terms of the atomic jump frequency, G,

which is highly temperature-dependent. It

is shown step-by-step how D is related totemperature via the expression:

where D0 is the frequency factor and QID is

equivalent to the enthalpy of interstitial

atom migration, DHm. Both these terms can

be taken as material constants.

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Ficks Second Law: Time Dependence

of the Concentration

Remember, Ficks first law is applicable to steadystate conditions, where concentration, c, isINDEPENDENT of time.

BUT! Even when c is a function of time, you can use the

first law at any instant in time. If c at any given position, x, changes with time, we use

Ficks 2nd law to account for the variation.

2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning is a trademark used herein under license.

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Derivation of Ficks 2nd Law

A derivation for Fick's 2nd law is given. This

applies to non steady-sate conditions, i.e.

those in which interstitial concentration, CB

varies with time. The general form of Fick's2nd law is given by:

For cases in which DB is independent ofcomposition, or where the ranges of

composition are very small, this reduces to:

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What happens

during diffusion

between Cu and

Ni?

We will find

out in Lab

number 2!

Calculation of

Matano

Interface.

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Matano Analysis (your 2nd lab)Since Fick's law cannot be directly integrated for variables (need error function

solutions : ), values must be obtained experimentally. The most

common method is the Matano analysis. A pure diffusion couple is created and

annealed at a constant temperature for a given length of time. After removal from

the furnace, a concentration profile is generated. From this, the Matano interface

is defined as being the plane across which an equal number of atoms have crossed

in both directions. It is shown step-by-step that the interdiffusion coefficient can be

obtained by graphical construction for different compositions, Cusing the

equation:

(13) The integral term, is the area between the profile and Matano

interface, whilst dx/dCis the reciprocal of the curve gradient at C. Some real data is

provided in an additional exercise, from which a plot of versus Ccan be obtained.

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Schematic of Matano Interface

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The Kirkendall Effect: Marker

Movement of a Diffusion Couple (in

sectin 12.2).The discovery was a

great story!

The Discovery and Acceptance of the Kirkendall Effect: The Result of a Short

Research Career -Hideo Nakajima

In the 1940s, it was a common belief that atomic diffusion took place via a direct

exchange or ring mechanism that indicated the equality of diffusion of binary

elements in metals and alloys. However, Ernest Kirkendall first observed inequality

in the diffusion of copper and zinc in interdiffusion between brass and copper. This

article reports how Kirkendall discovered the effect, now known as the Kirkendall

Effect, in his short research career.

http://www.tms.org/pubs/journals/JOM/9706/Nakajima-9706.htmlhttp://www.tms.org/pubs/journals/JOM/9706/Nakajima-9706.html

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High Diffusivity Paths

Dislocations, Grain Boundaries, Free surfaces:

2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning is a trademark used herein under license.