Phase Transformations Flashcards
What is a phase transformation?
The conversion of a material from one phase (parent) to one or more new phases (products)
List the most important phase transformations
Solidification Eutectic Eutectoid Precipitation Polymorphic/allotropic
What is a glissile interface?
Parent and product phases have the same composition.
Rate limited by interface mobility
Temperature effects negligible
Military transformation
Slides easily (often by dislocation motion)
Give an example of a transformation where glissile interfaces form.
Martensite and twinning
List features of a non-glissile transformation where the parent and product phases have the same composition.
Rate is limited by interface mobility
Transport is thermally activated
Civilian
Example of a phase transformation where non-glissile interfaces formand the parent and products have the same compositions..
Grain growth
Solidification of pure metal
List features of a non-glissile transformation where the parent and product phases do not have the same composition.
Long range diffusion required
Interface controlled, diffusion controlled or mixed control depending on the balance of thermodynamics and kinetics
Transport is thermally activated thus strongly temperature dependent.
Example of a phase transformation where a non-glissile interface forms and the parent and products have different compositions.
Eutectic/eutectoid transformations
Where does the driving force for solidification come from?
Difference in free energies of the parent and product phases.
What are the 4 stages of a precipitation reaction?
Nucleation
Growth
Impingement
Coarsening
How are thermodynamics and kinetics balanced in solidification when considering microstructure size?
Kinetics is often determined by diffusion, we would expect the reaction to be faster for finer microstructure as shorter diffusion distances. However finer microstructure reduces the interface energy so the reduces the driving force of the reaction and the reaction is slowed.
Drive the term for the Gibbs-Thompson effect for the growth of a dendrite.
Check
Why do dendrites grow faster than a flat interface if there is a lower driving force?
Thermal diffusion of latent heat away from the curved end of a dendrite is faster than from a flat surface.
Derive an expression for growth velocity of a dendrite in terms of latent heat and undercooling.
Check
Does diffusion only occur in alloys?
No, you can have self-diffusion.
In an ideal solution, diffusion leads to what?
Mixing
In an alloy with a miscibility gap, diffusion leads to what?
Separation.
Diffusion is governed by differences in what?
Chemical potential!!!
Not composition.
Give the equation for Fick 1
Check
Give an expression relating distance travelled and diffusion coefficient for Fick 1
Check
Give an expression for jump frequency in Fick 1 using activation energy (interstitial).
Check
Give an expression for the diffusion coefficient in Fick 1 in terms of jump frequency.
Check
Give an expression for jump frequency in Fick 1 using activation energy (substitutional).
Check
Derive Fick 1 in terms of mobility and chemical potential.
Check
Derive an expression relating diffusion coefficient to mobility for an ideal solution.
Check
Derive an expression relating diffusion coefficient to mobility for an non-ideal solution.
Check
Write the Gibbs-Duhem equation for a non-ideal solution
Check
How can we relate diffusion coefficient to self diffusion coefficient?
With the thermodynamic factor F.
Give an expression for the thermodynamic factor F in diffusion.
Check
Relate the total diffusion of a species in a mixture to the two components’ self diffusions.
Check
Explain the phenomena of carbon diffusion in steel Si alloys.
Carbon has a higher chemical potential in steel with Si than without so it will tend to diffuse out of Si rich regions.
How does diffusion occur in ceramics?
Difference in charge and size of cation and anions mean that species stay on their own sublattices.
Diffusion by vacancy migration within each sub-lattice
Smaller cationalso diffuse as interstitials.
How can we increase diffusion in ceramics?
Increase temperature (more intrinsic defects) Add dopants for increased diffusion at lower T.
Why is diffusion in ceramics important?
Sintering of engineering components.
Sensors (ionic diffusion = current flow, conc difference
What is diffusion in intermetallic compounds like?
Between dilute alloys and ceramics, atoms on ‘wrong sites’ have high energies.
Diffusion typically occurs through correlated atomic motions and complex defects.
How does diffusion occur in polymers.
Lateral motion prevented due to tangling of chains.
Diffusion occurs through an imaginary tube in a snake like motion.
Give the equation got the time taken to escape a tube length L.
t = L^2/D
Rearranges to same as Fick 1
Why is diffusion in liquid polymers important?
There is a critical entanglement length where viscosity is linear to long chains where viscosity goes with length^3.4. Viscosity is important for control in many manufacturing processes.
What is diffusion in semiconductors like?
Diffusion of dopants by substitution are slow as high Ea and need high T or long times. Some interstitial impurities diffuse very fast.
Is diffusion quicker down grain boundaries?
Yes, a lower Ea and dominates dissusion at low T.
Derive an expression for the apparent diffusion coefficient in a polycrystalline material (include diffusion down gbs).
Check
What is electromigration?
The movement of atoms in response to an electric current.
What process does the diffusion in electromigration follow?
Grain boundary diffusion.
Derive an expression for the diffusive flux in electromigration.
Check
Derive Fick 2.
Check
What does Fick 2 descirbe?
A reduction of curvature, so concentration profiles tend to flatten.
What is the integral of the Gaussian function?
erf
erf(1/2) =
1/2
What solution is required to solve Fick 2 for a thin layer of material between semi-infinite blocks?
Gaussian distribution
What solution is required to solve Fick 2 for diffusion into a thick specimen from a surface of constant composition?
erf
What solution is required to solve Fick 2 for diffusion at the interface between two thick slabs?
erf
What solution is required to solve Fick 2 for diffusion into thin specimens from two surfaces at constant composition?
Fourier series with exponential decay
What solution is required to solve Fick 2 for microsegregation in cast material?
Exponentially decaying sine wave.
What solution is required to solve Fick 2 for multilayer film?
Fourier series with exponential decay.
In an alloy when for substitutional diffusion Da ≠ Db, what happens to the vacancies?
There is a net flow of vacancies in one direction.
What is lattice migration/Kirkendall effect?
When vacancies are created more on one side of the lattice than the other and there is a net flow in one direction there is lattice movement.
The Kirkendall effect leads to what unwanted phenomena?
Vacancies clump together creating voids in the lattice.
Derive the Draken equations.
Check
Give the equation for Matano analysis.
Check
When using Matano analysis and the axis (Matano interface) you are integrating around has been set, what do the areas either side of the axis sum to?
0
Derive an expression for the velocity of the markers in the Kirkendall effect.
Check
How can we find the intrinsic diffusion coefficients for components in an alloy?
Having found the interdiffusion coefficient from Matano analysis and the velocity of the markers can solve the Draken equation and the velocity equation simultaneously to find the intrinsic diffusion coefficients.
In ternary systems, what does the liquid composition follow?
The slope downwards.
How to find volume fraction in ternary systems.
You know fa fb and fc sum to 1.
Then you also know that the overall composition of each component is the sum of the volume fractions of each phase multiplied by the amount of component in that phases. Can repeat for different components then solve simultaneously.
How does the first solid tend to form in a ternary system?
Dedritically.
Sketch the microstructure of a single eutectic in a ternary system.
Should have dendrites and lamella between.
Sketch the microstructure of a peritectic in a binary system.
Check pg 138
Sketch the microstructure from peritectic solidification in a ternary system.
Cored dendrites then non-equilibrium solidification around
What are the three invariant reactions in a ternary system?
Ternary Eutectic
Quasi Peritectic
Ternary Peritectic
What is a TE made from?
Three eutectic into the invarient point.
What is a QP made from?
2 peritectic in and a eutectic out of the invariant point.
What is a TP made from?
One eutectic in and 2 peritectics out.
What is a dishonest reaction?
One that changes character as it progresses across the ternary triangle.
Give the Young-Dupré equation for wetting.
Check
In wetting what happens when theta equals 180°?
The liquid drop does not wet the solid at all and balls us. Happens when sigmaSL is much greater than sigmaSV and sigmaLV.
In wetting what happens when theta equals 0°?
The liquid drop spreads out completely and forms a thin liquid layer. Occurs when sigma SV ≥ sigmaLV + sigmaSL
Sketch a plot of grain boundary energy against misorientation angle.
Check
In a two phases system, define the Gibbs free energy of each homogenous phase.
Check pg179
Define chemical potential.
The change in Gibbs free energy when one more atom of the component is added at constant P and T.
Give an equation for the chemical potential of component 2 in phase alpha.
Check pg180
What are the conditions for equilibrium between two phases?
The temperature is the same throughout both phases.
The pressure is the same throughout both phases.
Chemical potential of each component is the same throughout both phases.
Derive the Gibbs-Duhem equation.
Check pg184
Give an expression for the internal energy of a system that includes an interface.
Check pg187
Derive an expression for the excess free energy of the interface in terms of the Gibbs free energy of homogenous alpha and beta phases.
Check pg187
Derive the Gibbs-Duhem equation for a system containing an interface.
Check pg188
Describe the zero creep method.
Stack of transverse grain boundaries and mass hanging on wire of bamboo structure stack. Try to find the mass where the the wire is in equilibrium and the reduction in grain boundary are and increase in surface area of the wire are balanced with the opposition of elongation from the applied force.
Derive an expression for the mechanical equilibrium of the zero creep method.
Check pg 192
Give the equilibrium expression at a triple junction when a grain boundary reaches a free surface.
Check pg193
How can we measure relative surface energy?
With a sessile drop and the Young-Durpé equation.
In nickel superalloys, what is the interface energy between the gamma and gamma’ phases like?
Very low to avoid coarsening,
Derive an expression for the interface energy in terms of entropy and excess volume.
Check pg202
Give expressions for change in entropy and volume in terms of interface energy per unit area.
Check pg203
Interfaces arrange themselves to do what with interface energy?
Minimise interface energy.
How can we find the shape of interfaces?
Using a Wulff plot.
Why is controlling the size and shape of faceted nanoparticles important?
Modifying the shape is important for catalytic properties.
Derive the equilibrium condition for grains meeting at a triple junction.
Check pg 217
Describe the Jackson model for liquid/solid interfaces.
A balance exists between enthalpy and entropy for free energy.
If the enthalpy term is large then “broken bonds” is unfavourable thus free energy is high for partially filled layers and the system prefers full layers resulting in faceted interfaces.
If the enthalpy term is small, entropy dominates and a less well ordered system is preferred. The free energy is high for fully occupied layers and low for partially occupied resulting in non-facetted interfaces.
Give the expression for the Jackson factor.
Check pg226
If the Jackson factor is > 2 what kind of interfaces are there?
Facetted
If the Jackson factor < 2 what kind of interfaces are there?
Non-facetted
What is wrong with the Jackson model for liquid/solid interfaces?
Only a single layer model, still an effective predictor of solid-liquid morphology.
Describe a coherent interface.
Have a reference lattice that is continuous over the interface.
Usually low energy.
Describe an incoherent interface.
Misfit sufficiently high that it is favourable to form a large number of dislocations at the interface to localise the misfit strain.
Higher energy
Describe a semicoherent interface
Low density of interface dislocations.
Can you get a coherent but non-facetted interface?
Yes, if the interface energy is low in all directions and coherent in every direction then wherever you put an interface there will be no facets.
Use the Gibbs-Duhem equation to derive the Gibbs adsorption isotherm.
Check pg236
Derive the Gibbs adsorption isotherm for a solute obeying Henry’s Law.
Check pg 237
Where is segregation higher?
It is higher at average free surfaces as opposed to average grain boundaries. This is because the interface at the grain boundaries is closer to bulk environments (in terms of coordination).
What is the relationship between degree of segregation and bulk solubility?
An inverse relationship as for example large solute atoms with small solubilities in the bulk will find attractive sites at internal interfaces and free surfaces with loose open environements.
What is co-segregation?
When chemical potentials of solutes are coupled, a solute may segregate due to the presence of another where it would not normally segregate on its own. It can also cause site competition where one solute is expelled in preference for another.
How does facetted growth occur?
By lateral growth.
How does non-facetted growth occur?
Continuous growth.
Describe the Cahn model of interface motion.
When there is a driving force for growth, energy decreases as the interface moves forward. This can be represented as a slope in the interface energy curve. For a rough surface, there is only a small barrier to motion thus continuous growth. For a smooth interface, there is a large energy variation wrt to the position. Even though there is a driving force, there is still a significant energy barrier to continuous growth.
Derive an expression for net continuous growth rate.
Check pg253
Derive an expression for rate of interface migration in terms of undercooling for continuous growth.
Check
Give an expression for the rate of interface migration for lateral growth.
Check
Give an expression for rate of interface migration for lateral growth dependent on nucleation.
Check
Give an expression for lateral growth rate of interface migration for sprical growth.
Check
Sketch a plot of growth rate against undercooling comparing different growth mechanisms.
Check pg258
Make a sketch of facetted growth in a material that twins easily.
Check pg259
What is solute drag of the grain boundaries?
Where solute atoms segregated at the interface have to be dragged when the boundary moves. This tends to reduce grain boundary mobility. Only under high driving forces can the gbs break away from the solute.
What is military transformation?
A transformation where motion results from shearing one phase to form another.
What is civilian transformation?
Where uncoordinated diffusion results in motion.
How does interface migration occur at glissile interfaces?
By coorperative dislocation motion. No diffusion and is only weakly T dependent thus athermal.
Is the martensitic transformation a result of glissile interface motion?
Yes as shear along plane coorperatively.
Derive an expression for nucleation in the solid state including Henry’s law.
Check pg280
2 things required to find the rate of homogeneous nucleation.
How many critical clusters
How fast do atoms attach to the critical clusters
Derive an expression for homogeneous nucleation rate including a diffusion based term and a term for number of critical nuclei.
Check pg 284
What is incubation time?
The time needed for clusters to grow to their critical size for nucleation to occur.
Incubation times are inversely proportional to what?
Undercooling ^4
Sketch a plot of lnI against undercooling at two different temperatures.
Check pg 290
What determines shape of growing precipitates (ignore kinetics)?
Interfacial energy
Elastic strain
Determined by the minimisation of the total energy of the system which means minimising the above.
Give expressions for the excess strain caused by a coherent interface in a lattice
Check pg293
Plot strain energy and surface energy against ppt radius for coherent and incoherent interfaces.
Check pg295
Give expressions for the excess strain caused by a incoherent interface in a lattice
Check pg297
Give the expression for work of heterogeneous nucleation based on the Young-Dupré equation.
Check pg301
Give the geometric factor that turns the work of homogeneous nucleation into the work of heterogeneous nucleation.
Check pg303
Which starts first, homogeneous or heterogeneous nucleation?
Heterogeneous nucleation as there is a lower activation barrier.
What is inoculation in casting?
Where high nucleation rates and small grain sizes are encouraged by adding a fine refractory powder.
When nucleating on a grain boundary, what happens to the geometric factor for work of nucleation?
It is doubled.
In stainless steel, the unwanted sigma phase tends to nucleate where?
At the grain boundaries.
What are the 3 conditions during solidification?
Equilibrium conditions.
Liquid mixing (but no solid diffusion)
No liquid mixing (and no solid diffusion)
What do we use for liquid mixing but no solid diffusion during solidification?
The Scheil eqn
Under the Scheil equation, the liquid becomes enriched with solute when the partition coefficient is what?
k<1
Derive the Scheil equation
Check pg319
What do we call it when solidifcation happens with limited liquid diffusion and no solid diffusion?
Bridgeman growth.
Give an expression for Bridgeman growth
Check pg 321
Hoe can undercooling take place in a pure metal?
Temperature variation in the melt.
How can undercooling take place in an alloy?
Compotitional variation can cause compositional supercooling
How can the growth interface break down in an alloy?
If the compositional supercooling causes the real T to be below the eqm freezing T the interface will break down.
When does constitutional supercooling occur?
When the slope of the composition profile ahead pf the interface is greater than the real slope imposed by the temperature gradient.
3 zones in an as-cast structure.
Chill zone
Columnar zone
Equiaxed zone
What is microsegregation?
Compositional differences on the scale of dendrites.
Why does microsegregation occur?
Limited diffusion in solid leads to non-equilibrium solidification
Composition variations are generated between the centre of the dendrite and outside (coring)
Does the Scheil equation do a good job in predicting the compositions in microsegregation?
Yes, but it overestimates the variation as it assumes absolutely no solid diffusion.
Give an expression for homogenisation time.
Check pg339
What are typical limits of values for crystallinity in polymers?
40% to 95%.
How can XDR be used to determine % crystallinity in polymers?
Measuring the intensity of the Bragg peaks indicates and can be used to determine crystallinity as lower crystallinity polymers have less intense Bragg peaks.
Polymer crystals tend to form with what structure?
Spherulites formed from lamella from folding chains.
Fast and slow cooling form what types of lamella in polymers?
Fast - thin crystals
Slow - thick crystals
During the solidification of polymers, are chain folds favourable?
No as the large surfaces of lamella cause excess energy.
What is a typical observation of polymer spherulites?
They polarise optical like and characteristic maltese crosses are observed.
How does the flexibility of a polymer molecule control Tm?
Greater flexibility means lower Tm, higher solubility and lower melt and solution viscosities.
Flexibility is determined by single bonds vs multiple bonds, the presence of bulky side groups and hybridisation in the backbone.
Factors that affect Tm of a polymer.
Backbone flexibility
Hydrogen bonds
Polar groups (esters)
Relationship between Tm and Tg.
Tg ≈ (0.5-0.8)Tm
Give an equation for the mobility of an interface.
Check pg356
How does the Gibbs-Thomson effect affect growth?
It reduces the amount of driving force for interface mobility as it creates an interface free energy.
Give the general growth rate equation including the critically sized nucleus.
Check pg360
What steps must we take to calculate the growth rate?
Work out driving forces
Estimate the slope of the diffusion profile
Calculate interface velocity.
Derive an expression for interface velocity in terms of compositions.
Check pg363
Derive an expression for the interface velocity of spherical particulate growing using the Zener approximation.
Check pg365
Derive an expression for the interface velocity of needle shaped particulate growing using the Zener approximation.
Check pg367
Derive an expression for the length of needle-shaped particulate growing using the Zener approximation.
Check pg368
Typical features of the growth of plate-shaped particles with coherent interfaces.
Interface assumed to be mobile
No composition gradient
Velocity constant until the matrix solute is depleted giving a linear growth law.
Typical features of the growth of plate-shaped particles with incoherent interfaces.
The length of the diffusion profile increases with the thickness of the plate.
Similar to spherical precipitate in that it follows a parabolic growth law.
What is precipitate coarsening?
When the matric solute has become depleted further growth of precipitates becomes competitive and larger precipitates grow at the expense of smaller ones.
Derive an expression for the radius variation with time for precipitation coarsening.
Check pg376
What are the three phases of a precipitation reaction?
Interface controlled growth
Diffusion controlled growth
Diffusion controlled coarsening
States the Laplace-Young equation.
Check pg383
Give an expression for the free energy of a growing grain.
Check pg383
During grain growth, is there driving force for small grains to grow and consume larger grains?
No it is the opposite, large grains are driven to consume smaller grains.
Give an expression for the velocity of a grain boundary when a grain grows.
Check pg385
What temperature should you stay below to keep grain size small?
Below 0.4Tm
For precipitation in the Al-Cu system, if there is a lower driving force, what do we have to do for a specific precipitation reaction to occur?
Undercool it more.
Describe the Widmanstätten morphology.
Solid precipitate caused by heterogeneous nucleation, facetting and selection of mobile interfaces. Caused by fast cooling and preferential movement of incoherent, mobile interface at the tip of the needle.
Derive the simple Avrami model
Check pg415
Derive an expression for full Avrami analysis.
Check pg418
What modifications can be made to Avrami analysis to fit a wider variety of precipitation reactions?
Change the exponent of t.
What can the n=3 exponent of t be used for in Avrami analysis?
Heterogeneous nucleation from limited sites.
What can the n=2 exponent of t be used for in Avrami analysis?
1D growth
What can the n=4 exponent of t be used for in Avrami analysis?
Incubation period
What can the n=2.5 exponent of t be used for in Avrami analysis?
Diffusion controlled growth
How does solidification in polymers differ from metals?
All processes are slower as big molecules
Homogeneous nucleation is improbable
Heterogeneous nucleation often associated with impurities.
Molecular orientation changes the kinetics of nucleation/growth
What is recrystallisation?
The transformation of a cold-worked materials to one with no significant dislocation density.
What happens during recrystallisation?
High angle boundaries move.
Ther driving force is constant from the stored energy from deformation so no undercooling effects.
The driving force is low meaning interface mobility must be high.
Sketch a diagram of recrystallisation nucleation.
Check,
Should show bulge nucleation
The kinetics of recrystallisation follow what model well?
Avrami kinetics
Describe the kinetics of recyrstallisation.
Diffusion is thermally activated
No undercooling required so increased T means increased nucleation and growth rate.
More cold work increases driving force meaning lower recrystallization T and reduced grain size as higher nucleation rate.
What do impurities do to the recrystallisation temperature?
Increase it
Give an expression for estimating the final grain size after recrystallisation.
Check pg439
How can nucleation of ordered phases in alloys take place?
Either
A gradual increase in order based on existing short-range order.
Homogeneous nucleation and growth of fully ordered regions.
If lattice parameters match what happens to interfaces ac coarsening?
The interfaces are coherent and low energy so there is no driving force for coarsening.
What is coupled growth?
Where new phases are formed cooperatively with lateral diffusion ahead of the growth front.
During coupled growth, is there a long range diffusion profile ahead of the interface?
No meaning the growth rate is constant and only depends on the width of the 2 new solid phases.
By what process does eutectic nucleation occur?
One solid phase forms (with the lowest surface energy)
The second phase nucleated heterogeneously on the first.
Renucleation and branching allows for plates of the two phases to grow out as a spherical nodule.
By what process does eutectoid nucleation occur?
First phase nucleates heterogeneously on the grain boundary.
Second phase nucleates heterogeneously on the first.
Fixed orientation relationship exists to reduce surface energy.
Give an expression for the minimum lamella spacing in eutectic solidification.
Check pg455
Derive an expression for the growth rate of the eutectic.
Check pg459
What is modification?
Adding other phases (Na in Al-Si) to reduce the size of the microstructure.
How does Na work in the Al-Si system?
It suppresses Si nucleation by reacting with P to give poor heterogeneous nucleation sites.
This means large undercooling needed for homogeneous nucleation resulting in high nucleation and growth rate which leads to a finer microstructure as opposed to large facets of Si.
What is cellular precipitation?
Where precipitation occurs unexpectedly inside a matrix (often unwanted).
Is cellular precipitation a continuous or discontinuous reaction?
Discontinuous as the composition of phases change abruptly at the original interface.
Conventional precipitation is a continuous reaction.
What is massive transformation?
Where there is a change in structure but no change in composition, often unwanted.
How does massive transformation work?
Nucleation at gbs.
Growth by the migration of interfaces that are usually incoherent.
Results in the formation of large grains.
What is the difference between a massive transformation and a martensitic transformation?
Massive is thermally activated (short diffusion steps at the interface) but martensitic is thermal (glissile interface: the motion of dislocations).
How do massive and martensitic transformations appear on the TTT?
Massive are C curves
Martensitic are horizontal lines separated by temperature not time as athermal.
