PHASE TRANSFORMATIONS

Transcription

1 PHASE TRANSFORMATIONS Nucleation & Growth TTT and CCT Diagrams APPLICATIONS Transformations in Steel Precipitation Solidification & crystallization Glass transition Recovery, Recrystallization & Grain growth Phase Transformations in Metals and Alloys David Porter & Kenneth Esterling Van Nostrand Reinhold Co. Ltd., New York (1981)

2 Phase Transformations: an overview When one phase transforms to another phase it is called phase transformation. Often the word phase transition is used to describe transformations where there is no change in composition. In a phase transformation we could be concerned about phases defined based on: Structure e.g. cubic to tetragonal phase Property e.g. ferromagnetic to paramagnetic phase Phase transformations could be classified based on (pictorial view in next page): Kinetic: Mass transport Diffusional or Diffusionless Thermodynamic: Order (of the transformation) 1 st order, 2 nd order, higher order. Often subtler aspects are considered under the preview of transformations. E.g. (i) roughening transition of surfaces, (ii) coherent to semi-coherent transition of interfaces.

3 Based on Mass transport PHASE TRANSFORMATIONS Diffusional Diffusionless Involves long range mass transport E.g. Martensitic Based on order PHASE TRANSFORMATIONS 1 nd order nucleation & growth 2 nd (& higher) order Entire volume transforms

4 Transformations in Materials Phase transformations are associated with change in one or more properties. Hence for microstructure dependent properties we would like to additionally worry about subtler transformations, which involve defect structure and stress state (apart from phases). Therefore the broader subject of interest is Microstructural Transformations. Phases Defects Residual stress Phases can transform Defect structures can change Stress state can be altered Phase Transformation Defect Structure Transformation Stress-State Transformation Geometrical Structural Physical Property Phases Phases Transformations Microstructure Microstructural Transformations

5 Some of the questions we would like to have an answer for What is a Phase? What kind of phases exist? What constitutes a transformation? How can we cause a phase transformation to occur? The stimuli: P, T, Magnetic field, Electric field etc. What kind of phase transformations are there? Why does a phase transformation occur? Energy considerations of the system? Thermodynamic potentials (G, A ) Answers for some these questions may be found in other chapters

6 Revise concepts of surface and interface energy before starting on these topics When a volume of material (V) transforms three energies have to be considered : (i) reduction in G (assume we are working at constant T & P), (ii) increase in (interface free-energy), (iii) increase in strain energy. In a liquid to solid phase transformation the strain energy term can be neglected (as the liquid can flow and accommodate the volume/shape change involved in the transformationassume we are working at constant T & P). New interface created Bulk Gibbs free energy Volume of transformed material Energies involved Interfacial energy Strain energy Important in solid to solid transformations

7 The origin of the strain energy can be understood using the schematics as below. Eshelby construction is used for this purpose. In general a solid state phase transformation can involve a change in both volume and shape. I.e. both dilatational and shear strains may be involved. For simplicity we consider only change in volume of the material, leading to an increase in the strain energy of the system (in future considerations). Bulk Gibbs free energy Energies involved Interfacial energy Strain energy (a) (b) Only volume change (c) Considering only volume change (d) Schematic of the Eshelby construction to understand the origin of the stresses due to phase transformation of a volume (V): (a) region V before transformation, (b) the region V is cut out of the matrix and allowed to transform (the transformation could involve both shape and volume changes), (c) the transformed volume (V - shown to be larger in the figure) is inserted into the hole (here only volume change is shown for simplicity), (c) the system is allowed to equilibrate. The continuity of the system is maintained during the transformation. The system is strained as a larger volume V is inserted into the hole of volume V.

8 Let us start understanding phase transformations using the example of the solidification of a pure metal. (This process is a first order transformation*. First order transformations involve nucleation and growth**). There is no change in composition involved as we are considering a pure metal. If we solidify an alloy this will involve long range diffusion. Strain energy term can be neglected as the liquid melt can flow to accommodate the volume change (assume we are working at constant T & P). The process can start only below the melting point of the liquid (as only below the melting point the G Liquid < G Solid ). I.e. we need to Undercool the system. As we shall note, under suitable conditions (e.g. container-less solidification in zero gravity conditions), melts can be undercooled to a large extent without solidification taking place. Bulk Gibbs free energy Energies involved Interfacial energy Strain energy Solid-solid transformation ** 1 nd order nucleation & growth Trasformation = Nucleation of phase + Growth till is exhausted * Click here to know more about order of a phase transformation

9 1 2 Video snap shots of solidification of stearic acid Liquid Solid phase transformation: Solidification Liquid Two crystal going to join to form grain boundary 5 6 Growth of nucleated crystal Solid 3 4 Growth of Crystal Grain boundary Caution: here we are seeing an increase time experiment and soon we will be talking of increasing undercooling experiments t Crude schematic! See video here See video here Solidification complete For sufficient Undercooling

10 Liquid Solid phase transformation On cooling just below T m solid becomes stable, i.e. G Liquid < G Solid. But even when we are just below T m solidification does not start. E.g. liquid Ni can be undercooled 250 K below T m. We will try to understand Why? The figure below shows G vs T curves for melt and a crystal. The undercooling is marked as T and the G difference between the liquid and the solid (which will be released on solidification) is marked as G v (the subscript indicates that the quantity G is per unit volume). Hence, G v is a function of undercooling (T) Solid stable Liquid stable G v Solid (G S) G G ve T T - Undercooling T m Liquid (G L ) G +ve T

11 As pointed out before solidification is a first order phase transformation involving nucleation (of crystal from melt) and growth (of crystals such that the entire liquid is exhausted). Nucleation is a technical term and we will try to understand that soon. In solid solid phase transformation, which involve strain energy, heterogeneous nucleation (defined below) is highly preferred. Even in liquid solid transformations heterogeneous nucleation plays an very important role. Solidification = Nucleation + Growth of crystals from melt of nucleated crystals till liquid is exhausted Nucleation Homogenous Heterogeneous Heterogenous nucleation sites Liquid solid walls of container, inclusions Solid solid inclusions, grain boundaries, dislocations, stacking faults In Homogenous nucleation the probability of nucleation occurring at point in the parent phase is same throughout the parent phase. In heterogeneous nucleation there are some preferred sites in the parent phase where nucleation can occur

12 Homogenous nucleation Let us start with a text-book description of nucleation before taking up an alternate perspective Let us consider LS transformation taking place by homogenous nucleation. Let the system be undercooled to a fixed temperature T. Let us consider the formation of a spherical crystal of radius r from the melt. We can neglect the strain energy contribution. Let the change in G during the process be G. This is equal to the decrease in bulk free energy + the increase in surface free energy. This can be computed for a spherical nucleus as below. Free energy change on nucleation f() r ΔG (Volume).( G ) (Surface).( ) Reduction in bulk free energy increase in surface energy increase in strain energy G v f ( T ) r 3 r 1 r 2 V ΔG 4 3 r.( Gv ) 3 Note that G V is negative Neglected in L S transformations 2 4r.( ) Note that below a value of 1 the lower power of r dominates; while above 1 the higher power of r dominates. In the above equation these powers are weighed with other factors/parameters, but the essential logic remains.

13 Funda Check A note on minimization versus criticality conditions. ΔG 4 3 r.( Gv ) r.( ) In the above equation, the r 3 term is +ve and the r 2 term is ve. Such kinds of equations are often encountered in materials science, where one term is opposing the process and the other is supporting it. Example of such processes are crack growth (where surface energy opposes the process and the strain energy stored in the material supports crack growth). In the current case it is the higher power is supporting the phase transformation. Since the higher power dominates above 1, the function will go through a maximum as in fig. below. This implies the G function will go through a maximum. I.e. if the process just even starts it will lead to an increase in G! (more about this soon). On the other hand the function with ve contribution from the lower power (to G) will go through a minimum (fig. below) and such a process will take place down-hill in G and stop x^2 x^ (x - x^2) (x^2 - x) Goes through a maximum x^n f(x) x Goes through a minimum x

14 ΔG 4 3 r.( Gv ) 3 2 4r.( ) As we have noted previously G vs r plot will go through a maximum (implying that as a small crystal forms G will increase and hence it will tend to dissolve). The maximum of G vs r plot is obtained by by setting dg/dr = 0. The maximum value of G corresponds to a value of r called the critical radius (denoted by superscript *). If by some accident (technically a statistical random fluctuation ) a crystal (of preferred crystal structure) size > r* (called supercritical nuclei) forms then it can grow down-hill in G. Crystals smaller than r* (called embryos) will tend to shrink to reduce G. The critical value of G at r* is called G*. Reduction in G (below the liquid state) is obtained only after r 0 is obtained (which can be obtained by setting G = 0). dg dr r * G 0 2 * G G v G v r * 1 r * 2 r Trivial solution G v 3 G v Note that G is a function of T, r & As G v is ve, r * is +ve Note that we are at a constant T G Shrink Embryos dg dr * r 0 Grow r 0 Supercritical nuclei r G 0

15 What is the effect of undercooling (T) on r* and G*? We have noted that G V is a fucntion of undercooling (T). At larger undercoolings G V increases and hence r* and G* decrease. This is evident from the equations for r* and G* as below (derived before). At T m G V is zero and r* is infinity! That the melting point is not the same as the freezing point!! This energy (G) barrier to nucleation is called the nucleation barrier. Decreasing G * G T m Decreasing r * Increasing T G v f ( T ) The bulk free energy reduction is a function of undercooling r Using the Turnbull approximation (linearizing the G-T curve close to T m ), we can get the value of G interms of the enthalpy of solidification. r G * G 2 * G v G v T 2 T m 2 2 H

16 Turnbull s approximation T T T T m G H f H f Tm m G G T Solid (G S) Liquid (G L ) G ΔH 16 T 3 H f * 3 m f heat of fusion 2 T T m T

17 Quantum Jump How are atoms assembled to form a nucleus of r* Statistical Random Fluctuation To cause nucleation (or even to form an embryo) atoms of the liquid (which are randomly moving about) have to come together in a order, which resembles the crystalline order, at a given instant of time. Typically, this crystalline order is very different from the order (local order), which exists in the liquid. This coming together is a random process, which is statistical in nature i.e. the liquid is exploring locally many different possible configurations and randomly (by chance), in some location in the liquid, this order may resemble the preferred crystalline order. Since this process is random (& statistical) in nature, the probability that a larger sized crystalline order is assembled is lower than that to assemble a smaller sized crystal. Hence, at smaller undercoolings (where the value of r* is large) the chance of the formation of a supercritical nucleus is smaller and so is the probability of solidification (as at least one nucleus is needed which can grow to cause solidification). At larger undercoolings, where r* value is relatively smaller, the chance of solidification is higher. T m T r* Chances of nucleation increases

18 What is meant by the Nucleation Barrier an alternate perspective Funda Check Here we try to understand: What exactly is meant by the nucleation barrier?. It is sometime difficult to fathom out as to the surface energy can make freezing of a small embryo energetically infeasible (as we have already noted that unless the crystallite size is > r 0 the energy of the system is higher). Agreed that for the surface the energy lowering is not as much as that for the bulk*, but even the surface (with some unsaturated bonds ) is expected to have a lower energy than the liquid state (where the crystal is energetically favoured). I.e. the specific concern being: can state-1 in figure below be above the zero level (now considered for the liquid state)? Is the surface so bad that it even negates the effect of the bulk lowering? We will approach this mystery from a different angle by first asking the question: what is meant by melting point? & what is meant by undercooling?. * refer to surface energy and surface tension slides.

19 Melting point, undercooling, freezing point (in the realm of homogenous nucleation) The plot below shows melting point of Au nanoparticles, plotted as a function of the particle radius. It is to be noted that the melting point of nanoparticles decreases below the bulk melting point (a 5nm particle melts more than 100C below T bulk m ). This is due to surface effects (surface is expected to have a lower melting point than bulk!?*) actually, the current understanding is that the whole nanoparticle melts simultaneously (not surface layer by layer). Let us continue to use the example of Au. Suppose we are below T bulk m (1337K=1064C, i.e. system is undercooled w.r.t the bulk melting point) at T 1 (=1300K T = 37K) and suppose a small crystal of r 2 = 5nm forms in the liquid. Now the melting point of this crystal is ~1200K this crystal will melt-away. Now we have to assemble a crystal of size of about 15nm (= r 1 ) for it not to melt. This needless to say is much less probable (and it is better to undercool even further so that the value of r* decreases). Thus the mystery of nucleation barrier vanishes and we can think of melting point freezing point (for a given size of particle)! T m is in heating for the bulk material and in cooling if we take into account the size dependence of melting point everything sort-of falls into place. T 1 r 1 Other materials like Pb, Cu, Bi, Si show similar trend lines * Surface atoms are loosely bound as compared to the bulk atoms.

20 Atomic perspective of nucleation: Nucleation Rate The process of nucleation (of a crystal from a liquid melt, below T m bulk ) we have described so far is a dynamic one. Various atomic configurations are being explored in the liquid state some of which resemble the stable crystalline order. Some of these crystallites are of a critical size r* T for a given undercooling (T). These crystallites can grow to transform the melt to a solid by becoming supercritical. Crystallites smaller than r* (embryos) tend to dissolve. As the whole process is dynamic, we need to describe the process in terms of rate the nucleation rate [dn/dt number of nucleation events/time]. Also, true nucleation is the rate at which crystallites become supercritical. To find the nucleation rate we have to find the number of critical sized crystallites (N*) and multiply it by the frequency/rate at which they become supercritical. If the total number of particles (which can act like potential nucleation sites in homogenous nucleation for now) is N t, then the number of critical sized particles given by an Arrhenius type function with a activation barrier of G*. G kt * N * N e t

21 The number of potential atoms, which can jump to make the critical nucleus supercritical are the atoms which are adjacent to the liquid let this number be s*. If the lattice vibration frequency is and the activation barrier for an atom facing the nucleus (i.e. atom belonging to s*) to jump into the nucleus (to make in supercritical) is H d, the frequency with which nuclei become supercritical due atomic jumps into the nucleus is given by: H d ' s * e kt Rate of nucleation = No. of critical sized particles Frequency with which they become supercritical I dn dt G kt * N * N e t ' s * e H d kt No. of particles/volume in L lattice vibration frequency (~10 13 /s) s * atoms of the liquid facing the nucleus Critical sized nucleus (r*) Jump taking particle to supercriticality nucleated (enthalpy of activation = H d ) Outline of critical sized nucleus

22 The nucleation rate (I = dn/dt) can be written as a product of the two terms as in the equation below. How does the plot of this function look with temperature? At T m, G* is I = 0 (as expected if there is no undercooling there is no nucleation). At T = 0K again I = 0 This implies that the function should reach a maximum between T = T m and T = 0. A schematic plot of I(T) (or I(T)) is given in the figure below. An important point to note is that the nucleation rate is not a monotonic function of undercooling. T m Increasing T T (K) 0 T = T m G * = I = 0 T = 0 I = 0 I I N t s * e G * H kt d G * I Note: G * is a function of T T I

23 Heterogenous nucleation We have already talked about the nucleation barrier and the difficulty in the nucleation process. This is all the more so for fully solid state phase transformations, where the strain energy term is also involved (which opposes the transformation). The nucleation process is often made easier by the presence of defects in the system. In the solidification of a liquid this could be the mold walls. For solid state transformation suitable nucleation sites are: non-equilibrium defects such as excess vacancies, dislocations, grain boundaries, stacking faults, inclusions and surfaces. One way to visualize the ease of heterogeneous nucleation heterogeneous nucleation at a defect will lead to destruction/modification of the defect (make it less defective ). This will lead to some free energy G d being released thus reducing the activation barrier (equation below). hetro,defect ΔG (V) G G A ( G ) v s d Increasing G d (i.e. decreasing G * ) Homogenous sites Vacancies Dislocations Stacking Faults Grain boundaries (triple junction ), Interphase boundaries Free Surface

24 Heterogenous nucleation Consider the nucleation of from on a planar surface of inclusion. The nucleus will have the shape of a lens (as in the figure below). Surface tension force balance equation can be written as in equation (1) below. The contact angle can be calculated from this equation (as in equation (3)). Keeping in view the interface areas created and lost we can write the G equation as below (2). Interfacial Energies is the contact angle ΔG (V lens ) G Cos v (A Surface tension force balance lens ) Created Created (3) Lost ( A circle Cos ) ( A circle ) A lens A circle A circle V lens = h 2 (3r-h)/3 A lens = 2rh h = (1-Cos)r r circle = r Sin (1) (2)

25 Using the procedure as before (for the case of the homogenous nucleation) we can find r * for heterogeneous nucleation. Using the surface tension balance equation we can write the formulae for r * and G * using a single interfacial energy (and contact angle ). * * Further we can write down in terms of and contact angle. dg dr G * hetero G * homo r hetero 2 3 * * 4 G 2 3Cos Cos 3 G G hetero v 3 2 3Cos Cos hetero G G 3 * hetero * homo G homo G 2 v Cos Cos Just a function of the contact angle Increasing contact angle Decreasing tendency to wet the substrate G G * hetero * homo ( ) Cos Cos f 4 = 0f() = 0 = 90f() = ½ = 180f() = 1 Complete wetting Partial wetting * * The plot of / is shown in the next page. G hetero G homo No wetting

26 Plot of G * hetero/g * homo is shown below. This brings out the benefit of heterogeneous nucleation vs homogenous nucleation. If the phase nucleus (lens shaped) completely wets the substrate/inclusion (-phase) (i.e. = 0) then G * hetero = 0 there is no barrier to nucleation. On the other extreme if -phase does not we the substrate (i.e. = 180) then G * hetero = G* homo there is no benefit of the substrate. In reality the wetting angle is somewhere between Hence, we have to chose a heterogeneous nucleating agent with a minimum value. G * hetero / G* homo Complete wetting 0 G * hetero (0o ) = 0 no barrier to nucleation G * hetero (90o ) = G * homo /2 Partial wetting No wetting (degrees) G * hetero (180o ) = G * homo no benefit Cos

27 Choice of heterogeneous nucleating agent Heterogeneous nucleation has many practical applications. During the solidification of a melt if only a few nuclei form and these nuclei grow, we will have a coarse grained material (which will have a lower strength as compared to a fine grained material- due to Hall-Petch effect). Hence, nucleating agents are added to the melt (e.g. Ti for Al alloys, Zr for Mg alloys) for grain refinement. Cos How to get a small value of? (so that easy heterogeneous nucleation). Choosing a nucleating agent with a low value of (low energy interface) (Actually the value of ( ) will determine the effectiveness of the heterogeneous nucleating agent high or low ) How to get a low value of? We can get a low value of if: (i) crystal structure of and are similar and (ii) lattice parameters are as close as possible Examples of such choices: In seeding rain-bearing clouds AgI or NaCl are used for nucleation of ice crystals Ni (FCC, a = 3.52 Å) is used a heterogeneous nucleating agent in the production of artificial diamonds (FCC, a = 3.57 Å) from graphite.

28 Why does heterogeneous nucleation dominate? (aren t there more number of homogenous nucleation sites?) To understand the above questions, let us write the nucleation rate for both cases as a preexponential term and an exponential term. The pre-exponential term is a function of the number of nucleation sites. However, the term that dominates is the exponential term and due to a lower G * the heterogeneous nucleation rate is typically higher. I homo I 0 homo e * G kt homo I hetero I 0 hetero e * G kt hetero = f(number of nucleation sites) = f(number of nucleation sites) ~ ~ BUT the exponential term dominates I hetero > I homo

29 Heterogeneous nucleation in AlMgZn alloy

30 Growth Diffusional transformations involve nucleation and growth. Nucleation involves the formation of a different phase from a parent phase (e.g. crystal from melt). Growth involves attachment of atoms belonging to the matrix to the new phase (e.g. atoms belonging to the liquid phase attach to the crystal phase). Nucleation we have noted is uphill in G process, while growth is downhill in G. Growth can proceed till all the prescribed product phase forms (by consuming the parent phase). Transformation = Nucleation of phase + Growth of phase till is exhausted*

31 Growth At transformation temperature the probability of jump of atom from (across the interface) is same as the reverse jump Growth proceeds below the transformation temperature, wherein the activation barrier for the reverse jump is higher than that for the forward jump. H d H d v atom G v phase Transformation rate phase As expected transformation rate (T r ) is a function of nucleation rate (I) and growth rate (U). In a transformation, if X is the fraction of -phase formed, then dx /dt is the transformation rate. The derivation of T r as a function of I & U is carried using some assumptions (e.g. Johnson-Mehl and Avarami models).

32 We have already seen the curve for the nucleation rate (I) as a function of the undercooling. The growth rate (U) curve as a function of undercooling looks similar. The key difference being that the maximum of U-T* curve is typically above the I-T curve*. This fact that T(U max ) > T(I max ) give us an important handle on the scale of the transformed phases forming. We will see examples of the utility of this information later. Transformation rate T m f(nucleation rate, Growth rate) U dx Tr f( I, U) dt Maximum of growth rate usually at higher temperature than maximum of nucleation rate Increasing T T r T (K) I 0 I, U, T r [rate sec 1 ] * The U-T curve is an alternate way of stating the U-T curve

33 Fraction of the product () phase forming with time the sigmoidal growth curve Many processes in nature (etc.), e.g. growth of bacteria in a culture (number of bacteria with time), marks obtained versus study time(!), etc. tend to follow a universal curve the sigmoidal growth curve. In the context of phase transformation, the fraction of the product phase (X ) forming with time follows a sigmoidal curve (function and curve as below). 1.0 X β 1 e π I U 3 3 t 4 Saturation phase decreasing growth rate with time 0.5 Linear growth regime ~constant high growth rate X 0 Incubation period slow growth (but with increasing growth rate with time) t

34 From Rate to time : the origin of Time Temperature Transformation (TTT) diagrams A type of phase diagram The transformation rate curve (T r -T plot) has hidden in it the I-T and U-T curves. An alternate way of plotting the Transformation rate (T r ) curve is to plot Transformation time (T t ) [i.e. go from frequency domain to time domain]. Such a plot is called the Time- Temperature-Transformation diagram (TTT diagram). High rates correspond to short times and vice-versa. Zero rate implies time (no transformation). This T t -T plot looks like the C alphabet and is often called the C-curve. The minimum time part is called the nose of the curve. T m Rate f ( T, t) T m Small driving force for nucleation Nose of the C-curve T (K) T r Replot T (K) T t Time for transformation Sluggish growth 0 T r (rate sec 1 ) 0 T t (time sec)

35 Understanding the TTT diagram Though we are labeling the transformation temperature T m, it represents other transformations, in addition to melting. Clearly the T t function is not monotonic in undercooling. At T m it takes infinite time for transformation. Till T 3 the time for transformation decreases (with undercooling) [i.e. T 3 < T 2 < T 1 ] due to small driving force for nucleation. After T 3 (the minimum) the time for transformation increases [i.e. T 3 < T 4 < T 5 ] due to sluggish growth. This is a phase diagram where the blue region is the Liquid (parent) phase field and purplish region is the transformed product (crystalline solid). The diagram is called the TTT diagram because it plots the time required for transformation if we hold the sample at fixed temperature (say T 1 ) or fixed undercooling (T 1 ). The time taken at T 1 is t 1. To plot these diagrams we have to isothermally hold at various undercoolings and note the transformation time. I.e. instantaneous quench followed by isothermal hold. Hence, these diagrams are also called Isothermal Transformation Diagrams. Similar curves can be drawn for (solid state) transformation.

36 Clearly the picture of TTT diagram presented before is incomplete transformations may start at a particular time, but will take time to be completed (i.e. between the L-phase field and solid phase field there must be a two phase region L+S!). This implies that we need two C curves one for start of transformation and one for completion. A practical problem in this regard is related to the issue of how to define start and finish (is start the first nucleus which forms? Does finish correspond to 100%?). Since practically it is difficult to find % and 100%, we use practical measures of start and finish, which can be measured experimentally. Typically this is done using optical metallography and a reliable resolution of the technique is about 1% for start and 99% for finish. Another obvious point: as x-axis is time any transformation paths have to be drawn such that it is from left to right (i.e. in increasing time). TTT diagram phase transformation Increasing % transformation How do we define the fractions transformed? T (K) 1% = start t (sec) 99% = finish Fraction transformed f f volume fractionof at t final volumeof volume fractionof

37 How can we compute T t (T) (transformation time for each T) The C curve depends on various factors as listed in diagram below. Some common assumptions used in the derivation are: (i) constant number of nuclei, (ii) constant nucleation rate, (iii) constant growth rate. Nucleation rate Growth rate f(t,t) determined by Density and distribution of nucleation sites Overlap of diffusion fields from adjacent transformed volumes Impingement of transformed volumes

38 Constant number of nuclei (these form at the beginning of the transformation) One assumption to simplify the derivation is to assume that the number of nucleation sites remain constant and these form at the beginning of the transformation. This situation may be approximately valid for example if a nucleating agent (inoculant) is added to a melt (the number of inoculant particles remain constant). In this case the transformation rate is a function of the number of nucleation sites (fixed) and the growth rate (U). Growth rate is expected to decrease with time. In Avrami model the growth rate is assumed to be constant (till impingement). f F( number of nucleation sites, growthrate) growth rate withtime

39 Derivation of f(t,t): Avrami Model Parent phase has a fixed number of nucleation sites N n per unit volume (and these sites are exhausted in a very short period of time Growth rate (U = dr/dt) constant and isotropic (as spherical particles) till particles impinge on one another At time t the particle that nucleated at t = 0 will have a radius r = Ut Between time t = t and t = t + dt the radius increases by dr = Udt The corresponding volume increase dv = 4r 2 dr Without impingement, the transformed volume fraction (f) (the extended transformed volume fraction) of particles that nucleated between t = t and t = t + dt is: 2 dr Ut Udt f N 4r N 4 N 4U t dt n n n This fraction (f) has to be corrected for impingement. The corrected transformed volume fraction (X) is lower than f by a factor (1X) as contribution to transformed volume fraction comes from untransformed regions only: f dx 1 X dx 1 X N n U t dt

40 X dx 1 X tt 0 t0 N n U t dt X 1 e β 4π NnU t Based on the assumptions note that the growth rate is not part of the equation it is only the number of nuclei.

41 Where do we see constant growth rate? In cellular transformations constant growth rate is observed. Termination of transformation does not occur by a gradual reduction in the growth rate but by the impingement of the adjacent cells growing with a constant velocity. E.g.: Pearlitic transformation, Cellular precipitation, Massive transformation, recrystallization. Cellular Transformations Constant growth rate All of the parent phase is consumed by the product phase Pearlitic transformation Cellular Precipitation Massive Transformation Recrystallization

42 Constant nucleation rate Another common assumption is that the nucleation rate (I) is constant. In this case the transformation rate is a function of both the nucleation rate (fixed) and the growth rate (U). Growth rate decreases with time. If we further assume that the growth rate is constant (till impingement), then we get the Johnson-Mehl model. f F( nucleation rate, growth rate) growth rate withtime

43 Derivation of f(t,t): Johnson-Mehl Model Parent phase completely transforms to product phase ( ) Homogenous Nucleation rate of in untransformed volume is constant (I) Growth rate (U = dr/dt) constant and isotropic (as spherical particles) till particles impinge on one another At time t the particle that nucleated at t = 0 will have a radius r = Ut The particle which nucleated at t = will have a radius r = U(t ) Number of nuclei formed between time t = and t = + d Id Without impingement, the transformed volume fraction (f) (called the extended transformed volume fraction) of particles that nucleated between t = and t = + d is: f r Id U t Id ( ) This fraction (f) has to be corrected for impingement. The corrected transformed volume fraction (X) is lower than f by a factor (1X) as contribution to transformed volume fraction comes from untransformed regions only: f dx 1 X dx X r Id U( t ) Id

44 X t dx 4 1 X Ut ( ) Id π I U 3 X β 1 e 3 t 4 Note that X is both a function of I and U. I & U are assumed constant 1.0 For a isothermal transformation 3 π I U 3 is a constant during isothermal transformation 0.5 X 0 t

45 APPLICATIONS of the concepts of nucleation & growth TTT/CCT diagrams Phase Transformations in Steel Precipitation Solidification, Crystallization and Glass Transition Recovery recrystallization & grain growth As hyperlinks

46 Phase Transformations in Steel Now we have the necessary wherewithal to understand phase transformations in steel Phase diagram (Fe-Fe 3 C) and Concept of TTT diagrams We shall specifically consider TTT and CCT diagrams for eutectoid, hypo- and hypereutectoid steels. Further we will consider the use of these diagrams to design heat treatments to get a specific microstructure (each microstructure will give us a different set of properties).

47 We have already seen the Fe-Fe 3 C phase diagram (please have a second look!) Fe-Cementite diagram Peritectic L ºC Eutectic L + Fe 3 C L L %C ºC Eutectoid + Fe 3 C + Fe 3 C 723ºC %C T + Fe 3 C Fe %C 4.3 Fe 3 C 6.7

48 For every composition of steel we should draw a different TTT diagram. To the left of the start C curve is the Austenite () phase field. To the right of finish C curve is the ( + Fe 3 C) phase field. TTT diagram for Eutectoid steel (0.8%C) C Eutectoid temperature Eutectoid steel (0.8%C) Above Eutectoid temperature there is no transformation Important points to be noted: The x-axis is log scale. Nose of the C curve is in ~sec and just below T E transformation times may be ~day. The starting phase has to. The ( + Fe 3 C) phase field has more labels included. There are horizontal lines labeled M s and M f. T 600 Nose of C curve Pearlite + Fe 3 C Pearlite + Bainite Austenite Bainite M s M f Martensite t (s)

49 As pointed out before one of the important utilities of the TTT diagrams comes from the overlay of microconstituents (microstructures) on the diagram. Depending on the T, the ( + Fe 3 C) phase field is labeled with microconstituents like Pearlite, Bainite. We had seen that TTT diagrams are drawn by instantaneous quench to a temperature followed by isothermal hold. Suppose we quench below (~225C, below the temperature marked M s ), then Austenite transforms via a diffusionless transformation (involving shear) to a (hard) phase known as Martensite. Below a temperature marked M f this transformation to Martensite is complete. Once is exhausted it cannot transform to ( + Fe 3 C). Hence, we have a new phase field for Martensite. The fraction of Martensite formed is not a function of the time of hold, but the temperature to which we quench (between M s and M f ) C Eutectoid temperature Eutectoid steel (0.8%C) Fe 3 C Pearlite Pearlite + Bainite T Austenite M s Bainite 100 M f Martensite t (s)

50 Strictly speaking cooling curves (including finite quenching rates) should not be overlaid on TTT diagrams (remember that TTT diagrams are drawn for isothermal holds!). Isothermal hold at: (i) T 1 gives us Pearlite, (ii) T 2 gives Pearlite+Bainite, (iii) T 3 gives Bainite. Note that Pearlite and Bainite are both +Fe 3 C (but their morphologies are different). To produce Martensite we should quench at a rate such as to avoid the nose of the start C curve. Called the critical cooling rate. If we quench between M s and M f we will get a mixture of Martensite and (called retained Austenite) C Eutectoid temperature Austenite T 1 Eutectoid steel (0.8%C) Coarse Pearlite Fine 500 T 2 Pearlite + Bainite 400 T 3 Bainite Not an isothermal transformation T M s M f Austenite Martensite t (s)

51 Funda Check For the transformations to both Pearlite and Bainite, why do we have only one C curve? In principle two curves exist for Pearlitic and Bainitic transformations they are usually not resolved in plain C steel (In alloy steels they can be distinct). Eutectoid steel (0.8%C)

52 TTT Diagram: hypoeutectoid steel In hypo- (and hyper-) eutectoid steels (say composition C 1 ) there is one more branch to the C curve-np (marked in red). The part of the curve lying between T 1 and T E (marked in figs. below) is clear, because in this range of temperatures we expect only pro-eutectoid to form and the final microstructure will consist of and.(e.g. if we cool to T x and hold- left figure). The part of the curve below T E is a bit of a mystery (since we are instantaneously cooling to below T Hypo-Eutectoid steel E, we should get a mix of + Fe 3 C what is the meaning of a pro -eutectoid phase in a TTT diagram? (remember pro- implies pre- ).(Considered next) C 1 Atlas of Isothermal Transformation and Cooling Transformation Diagrams, ASM International, Metals Park, OH, 1977.

53 Funda Check Why do we get pro-eutectoid phase below T E? Suppose we quench instantaneously an hypo-eutectoid composition (C 1 ) to T x we should expect the formation of +Fe 3 C (and not pro-eutectoid first). The reason we see the formation of pro-eutectoid first is that the undercooling w.r.t to A cm is more than the undercooling w.r.t to A 1. Hence, there is a higher propensity for the formation of pro-eutectoid. undercooling wrt A 1 line (formation of + Fe 3 C) Undercooling wrt A cm (formation of pro-eutectoid ) C 1

54 Similar to the hypo-eutectoid case, hyper-eutectoid compositions (e.g. C 2 in fig. below) have a +Fe 3 C branch. For a temperature between T 2 and T E (say T m (not melting point- just a label)) we land up with +Fe 3 C. For a temperature below T E (but above the nose of the C curve) (say T n ), first we have the formation of pro-eutectoid Fe 3 C followed by the formation of eutectoid +Fe 3 C. T 2 T E Hyper-Eutectoid steel C 2

55 Continuous Cooling Transformation (CCT) Curves The TTT diagrams are also called Isothermal Transformation Diagrams, because the transformation times are representative of isothermal hold treatment (following a instantaneous quench). In practical situations we follow heat treatments (T-t procedures/cycles) in which (typically) there are steps involving cooling of the sample. The cooling rate may or may not be constant. The rate of cooling may be slow (as in a furnace which has been switch off) or rapid (like quenching in water). Hence, in terms of practical utility TTT curves have a limitation and we need to draw separate diagrams called Continuous Cooling Transformation diagrams (CCT), wherein transformation times (also: products & microstructure) are noted using constant rate cooling treatments. A diagram drawn for a given cooling rate (dt/dt) is typically used for a range of cooling rates (thus avoiding the need for a separate diagram for every cooling rate). However, often TTT diagrams are also used for constant cooling rate experiments keeping in view the assumptions & approximations involved. The CCT diagram for eutectoid steel is considered next. Blue curve is the CCT curve and TTT curve is overlaid for comparison. Important difference between the CCT & TTT transformations is that in the CCT case Bainite cannot form.

56 Continuous Cooling Transformation (CCT) Curves Important points to be noted: As before the x-axis is log scale. Bainite cannot form by continuous cooling. Constant rate cooling curves look like curves due to log scale in x- axis. The higher cooling rate curve has a higher (negative) slope. As time is one of the axes, no treatment curve can be drawn where time decreases or remains constant. Cooling curves Constant rate T Constant Cooling rate Eutectoid temperature M s M f Martensite T 1 > 2 Start T dt T dt Eutectoid steel (0.8%C) Pearlite Finish Original TTT lines T T t (s)

57 Funda Check The CCT curves are to the right of the corresponding TTT curves. Why? As the cooled sample has spent more time at higher temperature, before it intersects the TTT curve (virtually superimposed) and the transformation time is longer at higher T (above the nose) CCT curves should be to the right of TTT curves Eutectoid temperature Eutectoid steel (0.8%C) 600 Pearlite T Original TTT lines Cooling curves Constant rate M s M f Martensite T T t (s)

58 Common heat treatments involving cooling Common cooling heat treatment labels (with increasing cooling rate) are: Full anneal < Normalizing < Oil quench < Water quench The microstructures produced for these treatments are: Full Anneal (furnace cooling) Coarse Pearlite Normalizing (Air cooling) Fine Pearlite Oil Quench Matensite (M) + Pearlite (P) Water Quench Matensite To produce full martensite we have to avoid the nose of the TTT diagram (i.e. the quenching rate should be fast enough). Within water or oil quench further parameters determine the actual quench rate (e.g. was the sample shaken?).

59 Different cooling treatments It is important to note that for a single composition, different cooling treatments give different microstructures these give rise to a varied set of properties. After even water quench to produce Martensite, further heat treatment (tempering) can be given to optimize properties like strength and ductility Eutectoid steel (0.8%C) Water quench Normalizing Full anneal T Oil quench 200 Coarse P P = Pearlite M = Martensite 100 M M + P Fine P t (s)

60 What are the typical cooling rates of various processes? Process Cooling rate (K/s) Furnace cooling (Annealing) Air Cooling 1 10 Oil Quenching* ~100 Water Quenching* ~500 Splat Quenching 10 5 Melt-Spinning Evaporation, sputtering 10 9 (expected) * Depends on conditions discussed later

61 Pearlite + Fe 3 C Lamellae of Pearlite in ~0.8% carbon steel Nucleation and growth Heterogeneous nucleation at grain boundaries Interlamellar spacing is a function of the temperature of transformation Lower temperature finer spacing higher hardness

62 Mechanism of Pearlitic transformation: arising of the lamellar microstructure 1 Let us consider the heterogeneous nucleation of one of the phases of the pearlitic microconstituent (say Fe 3 C), at a grain boundary of Austenite (). Further let this precipitate be bound by a coherent interface on one side and a incoherent interface on the other side. The incoherent interface will be glissile (mobile) and will grow into the corresponding grain ( 2 ). The orientation relation (OR) between and Fe 3 C is refered to as the Kurdyumov- Sachs OR (as in fig. below). 2,3 The region surrounding this Fe 3 C precipitate will be depleted in Carbon and the conditions will be right for the nucleation of adjacent to it. 4 The process is repeated to give rise to a pearlitic colony. Branching of an advancing plate may also be observed (100) C (111) Orientation Relation: Kurdyumov-Sachs (010) C (110) (001) C (112) Branching mechanism

63 Bainite + Fe 3 C ** Micrograph courtesy: Prof. Sandeep Sangal Bainite formed at high temperature (~ 350C) has a feathery appearance and is called Feathery Bainite. Bainite formed at lower temperature (~ 275C) has a needle-like appearance and is called acicular Bainite. The process of formation of bainite involves nucleation and growth Acicular, accompanied by surface distortions ** Lower temperature carbide could be ε carbide (hexagonal structure, 8.4% C) Bainite plates have irrational habit planes Ferrite in Bainite plates possess different orientation relationship relative to the parent Austenite than does the Ferrite in Pearlite

64 More images of Bainite Micrograph courtesy: Prof. Sandeep Sangal, Swati Sharma AFM image 0.8% C steel, the sample was quenched in a salt bath having 400 C temperature and then it was held for 2 hours. Micrograph courtesy: Prof. Sandeep Sangal, Swati Sharma

65 Characteristic of Martensitic transformations Shape of the Martensite formed Lenticular (or thin parallel plates) Associated with shape change (shear) But: Invariant plane strain (observed experimentally) Interface plane between Martensite and Parent remains undistorted and unrotated This condition requires: 1) Bain distortion Expansion or contraction of the lattice along certain crystallographic directions leading to homogenous pure dilation 2) Secondary Shear Distortion Slip or twinning 3) Rigid Body rotation Surface deformations caused by the Martensitic plate

66 Martensite Change in Crystal Structure ( FCC) Quench ' ( BCT ) 0.8 % C 0.8 % C Martensitic transformation can be understood by first considering an alternate unit cell for the Austenite phase as shown in the figure below. If there is no carbon in the Austenite (as in the schematic below), then the Martensitic transformation can be understood as a ~20% contraction along the c-axis and a ~12% expansion of the a-axis accompanied by no volume change and the resultant structure has a BCC lattice (the usual BCC-Fe) c/a ratio of 1.0. FCC Austenite alternate choice of Cell ~20% contraction of c-axis ~12% expansion of a-axis In Pure Fe after the Matensitic transformation c = a FCC BCC

67 Martensite In the presence of Carbon in the octahedral voids of CCP (FCC) -Fe (as in the schematic below) the contraction along the c-axis is impeded by the carbon atoms. (Note that only a fraction of the octahedral voids are filled with carbon as the percentage of C in Fe is small). However the a 1 and a 2 axis can expand freely. This leads to a product with c/a ratio (c /a ) > In this case there is an overall increase in volume of ~4.3% (depends on the carbon content) the Bain distortion*. C along the c-axis obstructs the contraction Austenite to Martensite ~4.3 % volume increase Tetragonal Martensite * Homogenous dilation of the lattice (expansion/contraction along crystallographic axis) leading to the formation of a new lattice is called Bain distortion. This involves minimum atomic movements.

68 Martensite in 0.6%C steel

69 But shear will distort the lattice! Slip Twinning Average shape remains undistorted

70 Summary of characteristics of Martensitic transformation The martensitic transformation occurs without composition change The transformation occurs by shear without need for diffusion The atomic movements required are only a fraction of the interatomic spacing The shear changes the shape of the transforming region results in considerable amount of shear energy plate-like shape of Martensite The amount of martensite formed is a function of the temperature to which the sample is quenched and not of time Hardness of martensite is a function of the carbon content but high hardness steel is very brittle as martensite is brittle Steel is reheated to increase its ductility this process is called TEMPERING

71 Hardness (R c ) Harness of Martensite as a function of Carbon content % Carbon Properties of 0.8% C steel Constituent Hardness (R c ) Tensile strength (MN / m 2 ) Coarse pearlite Fine pearlite Bainite Martensite 65 - Martensite tempered at 250 o C

72 Element Added Plain Carbon Steel ROLE OF ALLOYING ELEMENTS Segregation / phase separation Compound (new crystal structure) Solid solution + Simplicity of heat treatment and lower cost Low hardenability Loss of hardness on tempering Low corrosion and oxidation resistance Low strength at high temperatures Interstitial Substitutional Alloying elements hardenability Provide a fine distribution of alloy carbides during tempering resistance to softening on tempering corrosion and oxidation resistance strength at high temperatures Strengthen steels that cannot be quenched Make easier to obtain the properties throughout a larger section Elastic limit (no increase in toughness) Alter temperature at which the transformation occurs Alter solubility of C in or Iron Alter the rate of various reactions

73 Sample elements and their role P Dissolves in ferrite, larger quantities form iron phosphide brittle (cold-shortness) S Forms iron sulphide, locates at grain boundaries of ferrite and pearlite poor ductility at forging temperatures (hot-shortness) Si ( %) increases elastic modulus and UTS Cu 0.8 % soluble in ferrite, can be used for precipitation hardening Pb Insoluble in steel Cr Corrosion resistance, Ferrite stabilizer, hardness/strength, > 11% forms passive films, carbide former Ni Austenite stabilizer, strength ductility and toughness, Mo Dissolves in &, forms carbide, high temperature strength, temper embrittlement, strength, hardenability

74 v Brinell Hardness Mn +0.1% C Addition of Carbon Mn Addition of Carbon Cr + 0.1%C Cr Alloying Element (%) Alloying elements increase hardenability but the major contribution to hardness comes from Carbon

75 Mn, Ni are Austenite stabilizers Temperature 0.35% Mn 6.5% Mn C (%) Temperature 15% Cr 12% Cr 5% Cr 0% Cr Outline of the phase field Cr is Ferrite stabilizer Shrinking phase field with Cr C (%)

76 TTT diagram for Ni-Cr-Mo low alloy steel Austenite Pearlite T Bainite M s M f Martensite ~1 min t

77 TTT diagram of low alloy steel (0.42% C, 0.78% Mn, 1.79% Ni, 0.80% Cr, 0.33% Mo) U.S.S Carilloy Steels, United States Steel Corporation, Pittsburgh, 1948)

78 Effect of carbon content and heat treatment on properties of steel 1000 Engineering Stress (s) [MPa] % C - Slow cooled 0.8% C - Slow cooled 0.8% C - quenched Tensile Test Engineering Strain (e) Hardness Vikers Hardness Slowly cooled- 0.6%C Quenched- 0.8% C Slowly cooled- 0.8% C Slowly cooled- 1.0% C C %

79 Precipitation

80 Precipitation Hardening The presence of dislocation weakens the crystal easy plastic deformation. Putting hindrance to dislocation motion increases the strength of the crystal. Fine precipitates dispersed in the matrix provide such an impediment. Strength of Al 100 MPa Strength of Duralumin with proper heat treatment (Al + 4% Cu + other alloying elements) 500 MPa

81 Al-Cu phase diagram: the sloping solvus line and the design of heat treatments L 600 T (ºC) Sloping Solvus line high T high solubility low T low solubility of Cu in Al Al % Cu

82 + + Slow equilibrium cooling gives rise to coarse precipitates which is not good in impeding dislocation motion. * 4 % Cu ( FCC) ( FCC) CuAl2( Tetragonal) slow cool 4 % Cu 0.5 % Cu 52 % Cu o 550 C RT RT *Also refer section on Double Ended Frank-Read Source in the chapter on plasticity: max = Gb/L

83 To obtain a fine distribution of precipitates the cycle A B C is used B Note: Treatments A, B, C are for the same composition A C + 4 % Cu A Heat (to 550 o C) solid solution B Quench (to RT) supersaturated solution Increased vacancy concentration C Age (reheat to 200 o C) fine precipitates

84 100 o C Schematic curves Real experimental curves are in later slides Hardness 180 o C 20 o C Log(t) Higher temperature less time of aging to obtain peak hardness Lower temperature increased peak hardness optimization between time and hardness required Note: Schematic curves shown- real curves considered later

85 180 o C Peak-aged Hardness Dispersion of fine precipitates (closely spaced) Coarsening of precipitates with increased interparticle spacing Underaged Overaged Region of solid solution strengthening (no precipitation hardening) Log(t) Region of precipitation hardening (but little/some solid solution strengthening)

86 180 o C Peak-aged In-coherent (precipitates) Coherent (GP zones) Hardness Section of GP zone parallel to (200) plane Particle shearing r r Particle radius (r) Particle By-pass Log(t) r f (t) CRSS Increase

87 A complex set of events are happening in parallel/sequentially during the aging process These are shown schematically in the figure below Hardness Log(t) Increasing size of precipitates with increasing interparticle (inter-precipitate) spacing Interface goes from coherent to semi-coherent to incoherent Precipitate goes from GP zone

88 GP Zones Cu rich zones fully coherent with the matrix low interfacial energy (Equilibrium phase has a complex tetragonal crystal structure which has incoherent interfaces) Zones minimize their strain energy by choosing disc-shape to the elastically soft <100> directions in the FCC matrix The driving force (G v G s ) is less but the barrier to nucleation is much less (G*) 2 atomic layers thick, 10nm in diameter with a spacing of ~10nm The zones seem to be homogenously nucleated (excess vacancies seem to play an important role in their nucleation)

89 Selected area diffraction (SAD) pattern, showing streaks arising from the zones. 5nm Bright field TEM micrograph of an Al-4% Cu alloy (solutionized and aged) GP zones. 5nm Atomic image of Cu layers in Al matrix

90 Due to large surface to volume ratio the fine precipitates have a tendency to coarsen small precipitates dissolve and large precipitates grow Coarsening in number of precipitate in interparticle (inter-precipitate) spacing reduced hindrance to dislocation motion ( max = Gb/L)

91 '' Distorted FCC 10 nmthick,100 nm diameter UC composition Al 6 Cu 2 = Al 3 Cu (001) (001) '' [100] [100] '' (001) (001) ' [100] [100] ' Becomes incoherent as ppt. grows Tetragonal UC composition Al 4 Cu 2 = Al 2 Cu ' BCT, I4/mcm (140), a = 6.06Å, c = 4.87Å, ti12 UC composition Al 8 Cu 4 = Al 2 Cu Phase Transformations in Metals and Alloys, D.A. Porter and K.E. Easterling, Chapman & Hall, London, 1992.

92 Successive lowering if free energy of the system Schematic diagram showing the lowering of the Gibbs free energy of the system on sequential transformation: GP zones Phase Transformations in Metals and Alloys, D.A. Porter and K.E. Easterling,Chapman & Hall, London, 1992.

93 The activation barrier for precipitation of equilibrium () phase is large But, the free energy benefit in each step is small compared to the overall single step process If the transformation is broken down into a series of steps with smaller activation barrier the processes can take place even with low thermal activation Single step ( equilibrium ) process Schematic plot

94 In this diagram additionally information has been superposed onto the phase diagram (which strictly do not belong there- hence this diagram should be interpreted with care) The diagram shows that on aging at various temperatures in the + region of the phase diagram various precipitates are obtained first At higher temperatures the stable phase is produced directly At slightly lower temperatures is produced first At even lower temperatures is produced first The normal artificial aging is usually done in this temperature range to give rise to GP zones first Precipitation processes in solids, K.C. Russell, H.I. Aaronson (Eds.), The Metallurgical Society of AMIE, 1978, p.87

95

96 Precipitation Sequence in some precipitation hardening systems (Morphology and compound stoichiometry are given in brackets) Base Alloy Precipitation Sequence Al Al-Ag GPZ (Spheres) ' (plates) (Ag 2 Al) Al-Cu GPZ (Discs) '' (Discs) ' (Plates) (CuAl 2 ) Al-Cu-Mg GPZ (Rods) S' (Laths) S (Laths, CuMgAl 2 ) Al-Zn-Mg GPZ (Spheres) ' (Plates) (Plates/Rods, Zn 2 Mg) Cu Cu-Be GPZ (Discs) ' (CuBe) Cu-Co GPZ (Spheres) (Plates, Co) Fe Fe-C -carbide (Discs) Fe 3 C (Plates) Fe-N '' (Discs) Fe 4 N (Plates) Ni Ni-Cr-Ti-Al ' (Cubes/Spheres)

97 Details in practical aging curves at start Points to be noted: In low T aging (130C) The aging curves have more detail than the single peak as discussed schematically before. In low T aging (130C) the full sequence of precipitation is observed (GPZ '' '). At high T aging (190C)'' directly forms (i.e. the full precipitation sequence is not observed). Peak hardness increases with increasing Cu%. For the same Cu%, the peak hardness is lower for the 190C aging treatment as compared to the 130C aging treatment. Peak hardness is achieved when the microstructure consists of a ' or combination of (' + ''). [1] J.M. Silcock, T.J. Heal and H.K. Hardy, J. Inst. Metal. 82 ( ) 239.

98 Particle/precipitate Coarsening There will be a range of particle sizes due to time of nucleation and rate of growth As the curvature increases the solute concentration (X B ) in the matrix adjacent to the particle increases Concentration gradients are setup in the matrix solute diffuses from near the small particles towards the large particles small particles shrink and large particles grow with increasing time * Total number of particles decrease * Mean radius (r avg ) increases with time Gibbs-Thomson effect

99 Gibbs-Thomson effect