366-2012-lecture3-4
TRANSCRIPT
-
7/29/2019 366-2012-Lecture3-4
1/41
Fall 2012-Lecture 3
Phase Diagrams and Related
Thermodynamics
Prof. Mitra Taheri
-
7/29/2019 366-2012-Lecture3-4
2/41
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
-
7/29/2019 366-2012-Lecture3-4
3/41
Types of Phase
Diagrams
-
7/29/2019 366-2012-Lecture3-4
4/41
forums.dfoggknives.com/uploads
-
7/29/2019 366-2012-Lecture3-4
5/41
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!
-
7/29/2019 366-2012-Lecture3-4
6/41
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.
-
7/29/2019 366-2012-Lecture3-4
7/41
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:
-
7/29/2019 366-2012-Lecture3-4
8/41
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.
-
7/29/2019 366-2012-Lecture3-4
9/41
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
-
7/29/2019 366-2012-Lecture3-4
10/41
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
-
7/29/2019 366-2012-Lecture3-4
11/41
Continued Solid-solid transformations
Eutectoid: alpha beta + gamma on cooling
Peritectoid: alpha + beta gamma on cooling
-
7/29/2019 366-2012-Lecture3-4
12/41
Thermodynamics Review
-
7/29/2019 366-2012-Lecture3-4
13/41
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.
-
7/29/2019 366-2012-Lecture3-4
14/41
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
-
7/29/2019 366-2012-Lecture3-4
15/41
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).
-
7/29/2019 366-2012-Lecture3-4
16/41
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.
-
7/29/2019 366-2012-Lecture3-4
17/41
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
-
7/29/2019 366-2012-Lecture3-4
18/41
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
-
7/29/2019 366-2012-Lecture3-4
19/41
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.
-
7/29/2019 366-2012-Lecture3-4
20/41
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).
-
7/29/2019 366-2012-Lecture3-4
21/41
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
-
7/29/2019 366-2012-Lecture3-4
22/41
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
/
-
7/29/2019 366-2012-Lecture3-4
23/41
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:
-
7/29/2019 366-2012-Lecture3-4
24/41
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.
-
7/29/2019 366-2012-Lecture3-4
25/41
Common Tangents
For any mixing quantity, the tangent at anypoint gives intercepts that are relative partial
molar quantities (graphical depictions):
-
7/29/2019 366-2012-Lecture3-4
26/41
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.
-
7/29/2019 366-2012-Lecture3-4
27/41
Solutions
-
7/29/2019 366-2012-Lecture3-4
28/41
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
-
7/29/2019 366-2012-Lecture3-4
29/41
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/ -
7/29/2019 366-2012-Lecture3-4
30/41
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
-
7/29/2019 366-2012-Lecture3-4
31/41
What IS a concentration gradient (in
an A-B Solution)?
2003Brooks/Cole,adivisiono
fThomsonLearning,Inc.ThomsonLearningisatrademarkusedhereinunder
license.
-
7/29/2019 366-2012-Lecture3-4
32/41
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.
-
7/29/2019 366-2012-Lecture3-4
33/41
Schematic: Ficks 1st Law
-
7/29/2019 366-2012-Lecture3-4
34/41
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.
-
7/29/2019 366-2012-Lecture3-4
35/41
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.
-
7/29/2019 366-2012-Lecture3-4
36/41
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:
-
7/29/2019 366-2012-Lecture3-4
37/41
What happens
during diffusion
between Cu and
Ni?
We will find
out in Lab
number 2!
Calculation of
Matano
Interface.
-
7/29/2019 366-2012-Lecture3-4
38/41
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.
-
7/29/2019 366-2012-Lecture3-4
39/41
Schematic of Matano Interface
-
7/29/2019 366-2012-Lecture3-4
40/41
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 -
7/29/2019 366-2012-Lecture3-4
41/41
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.