Part 4 - Phase diagrams and phase transitions

Question 1

What is the phase rule proposed by Gibbs?

Answer 1

The phase rule states that for a system in equilibrium:

P + F = C +2

Where P is phases, F is degrees of freedom and C is components.

Question 2

What is a phase?

Answer 2

A state of matter which is uniform throughout in terms of both its chemical composition and its physical state.

In order words, a homogenous part of a chemical system that is bounded by some sort of surface such that it can (in principle) by separated from the other parts of the system.

Question 3

What is a solid solution?

Answer 3

A solid solution (of say component A and B) is a solution in which atoms are homogenously mixed on an atomic length scale.

Question 4

What is a component (as used in the phase rule)?

Answer 4

The components of a system is the minimum number of independently variable chemical constituents that we need to define the overall composition.

Question 5

How is the degrees of freedom defined (as used in the phase rule)?

Answer 5

The number of variables (pressure, temperature, concentration of components) that need to be specified to fully define the condition of the system.

Question 6

How is a single solid phase represented in a one-component phase diagram?

Answer 6

By an area in the phase diagram.

Question 7

How is a single solid phase represented in a two-component phase diagram?

Answer 7

By a vertical line in the phase diagram.

Question 8

How is a single solid phase represented in a three-component phase diagram?

Answer 8

By a point in the phase diagram.

Question 9

In the phase diagram of water, what is the triple point?

Answer 9

The point at which water, vapour and ice coexist in equilibrium. This is an invariant point.

Question 10

What is an invariant point?

Answer 10

An invariant point is a point in the phase diagram where there are no degrees of freedom. Changing any of the parameters, and you change the physical state of the system as well.

Question 11

What is a univariant point?

Answer 11

A univariant point is a point in the phase diagram where there are only one degree of freedom.

Question 12

what is a bivariant point?

Answer 12

A bivariant point is a point in the phase diagram where there are two degrees of freedom.

Question 13

What happens at the critical point in the phase diagram of water?

Answer 13

At the critical point (at the critical temperature and critical pressure) is the point beyond which liquid and vapour have the same density and can’t be dinstinguished. This is called a supercritical fluid, and at this point it is the only phase.

Question 14

What is the phase rule for solid systems, and why does it differ from the general phase rule?

Answer 14

The phase rule for solid systems is:

P + F = C + 1

The reason is that the pressured of the substances involved are negligible compared to atomspheric pressure, and the pressure can thus be considered as constant, and one degree of freedom is removed from the system.

Question 15

What is a eutectic point?

Answer 15

A eutectic point is a point where a solid mixture melts directly to form a liquid. This is the lowest melting point for a system.

Question 16

What is the solidus?

Answer 16

The line in a phase diagram below which there exist only solid phases.

Question 17

What is the liquidus?

Answer 17

The line in a phase diagram above which there is only a liquid phase.

Question 18

What is the lever rule?

Answer 18

The lever rule allows us to calculate the relative fractions of different phases from a point in the phase diagram.

In a two-phase region, we will for a given point have an overall composition corresponding to the point, but the two phases will have different compositions.

The lever rule gives us the relative amount of each of the phases, and we can read out the composition of the phases from the phase boundaries on each side.

Question 19

What’s a pseudoatom formula?

Answer 19

A pseudoatom formula is a normalised formula. That is AB2 would be written A1/3B2/3, so that the total composition is added up to 1.

Question 20

What is a congruent melting point?

Answer 20

A congruent melting point is that a phase melts directly to a liquid without decomposition.

Question 21

What is an incongruent melting point?

Answer 21

An incongruent melting point is where a phase decomposes into a solid and a liquid of different chemical compositions.

Question 22

How can a eutectic point and a peritectic point be dinstinguished in a phase diagram?

Answer 22

A eutectic point is a minimum on the liquidus curve, while the peritectic is simply a kink. Also, the peritectic point lies outside of the compositions range that make up the solid phases the liquid is in equilibrium with.

Question 23

What is a peritectic reaction?

Answer 23

A peritectic reaction can occur on cooling from a region of e.g. B + liquid to a region of AB2 + liquid.

In this case, all of B must react to form AB2.

Question 24

What can happen if you rapidly cool a liquid that in equilibrium would go through a peritectic reaction?

Answer 24

In this case we can get a non-equilibrium reaction, where not all of the solid phase that in a peritectic reaction would react to form a new solid phase actually reacts, so that we are left with more phases than we would in an equilibrium case.

Question 25

What can happen during rapid cooling of a liquid of A and B that shows complete solubility in both liquid and solid phases? (B has a higher melting point than A)

How can this be remedied?

Answer 25

Since B has a higher melting point than A, there is a region where we have a coexistence of a liquid and solid, where we initially get crystals of B.

This means that if we rapidly cool this solution, we can get a core that is very rich in B and a shell that is very rich in A.

If we want to homogenise the sample if this has occurred, we might be able to remedy it through extended annealing.

Question 26

When can we expect a full range of solid solutions?

Answer 26

For components that are chemically very similar.

Question 27

What is the solvus?

Answer 27

The line in a phase diagram that separates a single solid solution from a mixture of two phases.

Question 28

What is an immiscibility dome?

Answer 28

An immiscibility dome is an area in a phase diagram where two different liquid phases coexist as two separate phases.

Question 29

What is the upper consulute temperature?

Answer 29

The temperature at which for no compositions there exist only a single liquid phase.

Question 30

What is a monotectic point?

Answer 30

It is similar to an eutectic point, but instead of a single liquid decomposing into two solids, it is a single liquid decomposing into one liquid and a solid.

Question 31

How are phase diagrams of ternary systems usually represented?

Answer 31

As an equilateral triangle with the pressure held constant, and either with temperature at an axis extending out of the plane (forming a prism) or as slices of constant temperature.

Question 32

What does Viviani’s theorem state?

Answer 32

That from any given point inside an equilateral triangle, the sum of the lengths of normals from all sides to this point adds up to the height of the triangle.

Question 33

Why is it beneficial to draw ternary phase diagrams as equilateral triangles?

Answer 33

Because we can make use of Viviani’s theorem, that the lengths of the normals from each side to a given point inside the triangle must sum to the height. The mole fractions of all components must also sum up to 1.

Question 34

What are three ways of reading compositions of a ternary phase diagram?

Answer 34

1. Drawing lines parallel to the sides, and reading of. Where the line of BC that coincides with a point d would intersect the AB and AC-lines be the composition of A, the line of AB with AC and BC would be the composition of C etc.
2. The triangle rule. This is the equivalent of the level rule in binary systems. We draw lines from each corner of the triangle through our point to the opposite edge. Each component is then given by the relative fraction of the line from its corner on the other side of the point to the full length.
3. Reading the amount from one axis. We draw lines parallel to each side through the point. From any given side we can then read of the composition from the intercepts. If reading the AB line, then the portion from B to the first intercept is the fraction of A, the portion from A to the other intercept is the fraction of B and the portion in between is the fraction of C. Also applies to scalene triangles.

Question 35

In a ternary phase diagram, where can we binary, ternary and quaternary oxides?

Answer 35

The binary oxides are found in each corner (e.g. A(O) ). Ternary oxides are found along the line from one corner to another (e.g. AC(O)) and quaternary oxides are found as points in the middle of the triangle.

Question 36

What is a displacive phase transition?

Answer 36

A transition in which no chemical bonds are broken or formed.

Question 37

What is a reconstructive phase transition?

Answer 37

A phase transition which entail major reorganisation of the structure involving bond breaking and formation?

Question 38

What is an order-disorder phase transition?

Answer 38

A phase transition where there is full disorder in one phase and an ordering in the next.

Question 39

What kind of transitions can order-disorder phase transitions be divided into?

Answer 39

Substitutional disorder, where atoms order at a specfici lattice site.

Orentiational disorder, where atoms of groups of atoms change their positions without breaking the fundamental bonding pattern of the material.

Question 40

How does Ehrenfest classify phase transitions?

Answer 40

In first order transitions and second order transitions.

Question 41

What is a first-order phase transition?

Answer 41

A first order phase transition is a phase transition in which there is a discontinuity in the first derivative of the free energy with respect to temperature and pressure.

dG/dp = V, and dG/dT = -S (and ΔS = ΔH/T)

Phase transitions that show abrupt changes in volume or entropy/enthalpy will thus be first-order transitions.

Question 42

How is constant pressure heat capacity defined, and what will it be during a first-order phase transition?

Answer 42

Cp = dH/dT.

It will be infinite at the phase transition.

Question 43

What is latent heat?

Answer 43

The heat (ΔH) that will be supplied during a phase transition that goes towards the phase transition and will not increase temperature.

Question 44

When will a first-order phase transition show a hysteresis?

Answer 44

When the phase transition involves a change in cooperative forces (e.g. interlocked magnetic, electrostatic or elastic domains).

Question 45

What is a second-order phase transition?

Answer 45

A second-order phase transition is a where there is a discontinuity in the second derivative of the free energy with respect to temperature.

Question 46

What happens to the free energy as a function of temperature during a first and second order phase transition respectively?

Answer 46

First: It has a kink (its derivative is discontinuous)

Second: It is smooth (its derivative is continuous)

Question 47

What happens to the volume as a function of temperature during a first and second order phase transition respectively?

Answer 47

First: Changes abruptly

Second: Does not change at the phase transition, is continuous.

Question 48

What happens to the entropy as a function of temperature during a first and second order phase transition respectively?

Answer 48

First: Changes abruptly

Second: Does not change at the phase transition, is continuous.

Question 49

What happens to the enthalpy as a function of temperature during a first and second order phase transition respectively?

Answer 49

First: Changes abruptly

Second: Does not change at the phase transition, is continuous.

Question 50

What happens to the constant-preesure heat capacity as a function of temperature during a first and second order phase transition respectively?

Answer 50

First: It becomes infinite at the phase transition.

Second: As the change in enthalpy is continuous (but has a kink), the heat capacity is discontinous.

Question 51

What is the Euler strut?

Answer 51

An Euler strut is a simple mechanical system designed to explain the symmetry breaking that occurs during phase transitions. You have a straight rod with a weight on top. As the weight increases (temperature decreases) the strut eventually buckles, and symmetry is broken.

Question 52

What is an order parameter?

Answer 52

An order parameter η is a parameter that describes the ordering in a distorted structure relative to the high-symmetry structure.

An order parameter has the value 0 in the fully disordered case and 1 in the fully ordered case.

Question 53

What is a zone-centre transition?

Answer 53

A zone-centre transition is a transition in which the unit cell merely distorts, and there is no major change in the cell dimensions or reciprocal lattice.

Question 54

What is a zone-boundary transition?

Answer 54

A zone-boundary transition is a transition that change the translational symmetry and give rise to a superlattice.

Question 55

What is the basic assumption used in Landau theory?

Answer 55

That the Gibbs free energy of a phase can be expressed as a simple power series in terms of the order parameter η.

Question 56

In the power series used in Landau theory (see below), what is G_0?

G(η) = G_0 + 1/2 Aη^2 + 1/4 Bη^4 + 1/6 Cη^6 …

Answer 56

G_0 is the part of the free energy that does not change at the phase transition.

Question 57

Why do we only use even values in the power series used in Landau theory?

Answer 57

Because our order parameter is symmetric (that is η = -1 represents the same order as η = 1), and we only model the free energy with even parameters to ensure that our G(η)-cruve is symmetry around the origin.

Question 58

What is the simplest way (using Landau theory) to describe a second order transition?

Answer 58

Keeping the first three terms (G(η) = G_0 + 1/2 Aη^2 + 1/4 Bη^4).

A must be able to be both positive (high T) and negative (low T), so we say that A = a(T - Tc), with positive a.

We need the last term to describe the minima that occurs at η ≠ 0. This means B must be positive as well.

Question 59

How can we get the values of a and B in Landau theory for second order transitions from experiments?

Answer 59

We measure Tc and one of the excess thermodynamic quantities.

Question 60

If we know a and B in Landau theory for second order transitions, what can we then calculate?

Answer 60

If a and B are known for second order transitions, we then can calculate all the thermodynamic quantities associated with the phase transition.

Question 61

How does the Landau expression for second-order and first-order transitions differs?

Answer 61

For first-order we must include a fourth term, and we also change the sign before the B-term to minus (still keeping a, B, and C positive). We also change out Tc with T_0 in the first-order expression.

First: G(η) = G_0 + 1/2 a(T - T_0)η^2 - 1/4 Bη^4 + 1/6 Cη^6

Second: G_0 + 1/2 a(T - T_c)η^2 + 1/4 Bη^4

Question 62

What is the meaning of the temperature T_0 in Landau theory for first-order transitions ?

Answer 62

T_0 is the temperature at which the function ceases to have a local minimum for η = 0.

Question 63

What is the meaning of the temperature T_1 in Landau theory for first-order transitions?

Answer 63

T_1 is the temperature at which we go from having one minimum (at η = 0) to three minima (though the new two minima will be local minima, and be far above the η = 0 minimum).

Alternatively we can say that T_1 is the temperature above which the solutions for η ≠ 0 cease to be real.

Question 64

What are the relative energies of the minima at η = 0 and η ≠ 0 for first-order transitions according to Landay theory at:

a) T > T_1
b) T_1 > T > T_c
c) T = T_c
d) T_c > T > T_0
e) T < T_0

Answer 64

a) When the temeprature is higher than T_1, we only have one minimum at η = 0.
b) At T_1 > T > T_c we have three minima, however the minima at η = 0 is lower than the ones at η ≠ 0.
c) At T = T_c all three minima are the same value.
d) At T_c > T > T_0 the minima at η ≠ 0 are lower than at η = 0.
e) At T > T_0 we only have minima at η ≠ 0.

Question 65

Why does the sign shift of the B-term in the Landau expression for first-order transitions shift relative to the second-order?

Answer 65

Because we want to keep our constants positive, and for first-order transitions would otherwise be negative.

Question 66

Considering the power series used in Landau theory, what is the value of B for first and second order transitions respectively?

G(η) = G_0 + 1/2 Aη^2 + 1/4 Bη^4 + 1/6 Cη^6 …

Answer 66

First order: B is negative

Second order: B is positive

Question 67

What is it called if B is 0 in the below power series Landay expression?

G(η) = G_0 + 1/2 Aη^2 + 1/4 Bη^4 + 1/6 Cη^6 …

Answer 67

It is called a tricritical transition.

Question 68

What is a soft mode?

Answer 68

For displacive transitions, a soft mode is a phonon mode in a high-symmetry structure whose frequency lowers as the temperature cools (it softens). If the frequency of the mode falls to zero, its pattern of dynamic displacements become frozen into the structure as a static distortion.