Nucleation versus spinodal

Transcription

1 Nucleation versus spinodal In this animation, we show two different mechanisms of phase separation Course Name: Phase transformations and heat treatment Authors M P Gururajan

2 Learning Objectives After interacting with this Learning Object, the learner will be able to: Show the microstructural evolution during nucelation process Show the microstructural evolution during spinodal decomposition Show the free energy versus composition diagrams and identify the regions where nucleation occurs and those where spinodal decomposition occurs Pre-requisites It is recommended that the learner peruses the following animations before perusing this one: Driving force for diffusion Three types of equilibrium Phase diagram and free energy versus composition curves

3 Definitions of the components/keywords: 1 Miscibility gap: region of a phase diagram in which the material separates into a mechanical mixture of two different phases Temperature, T Miscibility gap 5 Composition, x B

4 Definitions of the components/keywords: 1 Chemical spinodal: region inside a miscibility gap in which phase separation takes place through the spinodal mechanism Temperature, T Miscibility gap Chemical spinodal 5 Composition, x B

5 Definitions of the components/keywords: 1 Spinodal decomposition: Process in which alloys that are unstable against small fluctuations phase separate into two phases without overcoming any barriers, and in which, the diffusion is up hill (or, against the 2 concentration gradient) Nucleation: process in which an alloy that is metastable phase separates into two phases by overcoming a barrier, an in which, the diffusion is down hill (or, classical, that is, along the concentration gradient) Spinodal point: a point on the free energy versus composition curve at which the second derivative of free energy with respect to composition is zero: that is 2 G x B 2 =0

6 1 Master Layout: Part

7 Analogy / Scenario / Action Draw the phase diagram Choose a temperature in the miscibility gap Draw the free energy versus composition diagram Mark the chemical spinodal and the points of inflection Mark the minima and maxima 5

8 1 Step 1: T1: Phase diagram

9 Instructions to the animator Draw a box. Mark the x axis on the bottom as composition. Mark the y axis at the left as temperature. Draw a big parabola. Inscribe a smaller parabola inside the bigger one. Make sure that the two noses of the parabola coincide. Write alpha (the Greek symbol) outside the parabolas in the box. Mark alpha1+alpha2 inside the parabolas. Mark the outer parabola as Miscibility gap. Mark the inner parabola as chemical spinodal. Draw a dotted horizontal line that cuts the two parabolas at four different points. Mark the line as T. Audio narration Consider a binary system which undergoes phase separation from alpha to alpha1 and alpha2 whose phase diagram is as shown. The two phase region is known as miscibility gap. Inside the miscibility gap, there is a region known as chemical spinodal, in which, the phase separation occurs without overcoling any barrier. To understand why there is barrierless transformation inside the chemical spinodal, let us consider a temperature T and draw the free energy versus composition diagram at this temperature T.

10 1 Step 1: T1: Chemical spinodal

11 Instructions to the animator Draw a box below the box drawn in Step 1. Mark the x axis on the bottom as composition. Mark the y axis at the left as G. Draw a curve with two minima and one maxima. The two minima should correpsond to the points where the dotted T line cuts the outer parabola called Miscibility gap. The maxima should be corresponding to the nose of the parabola. The minima points at the left and right respectively on the bottom box should be marked x1 and x2. Draw five vertical lines. The line 1 passes through the minima on the left hand side. The line 3 passes through the maxima. The line 5 passes through the minima on the right hand side. The line 2 passes through the point of intersection of the dotted T line with the inner parabola (called chemical spinodal) on the left hand side and the line 4 passes though the point of intersection of the dotted T line with the inner parabola on the right hand side. Text to be displayed: At the point where the line 1 intersects the curve (which is the minima on the left hand side), and where the line 5 intersects the curve (which is the minima on the right hand side) write 2 G x 2 0 At the point where line 3 intersects the curve (which is the maxima at the centre) write 2 G x 2 0 At the point where the lines 2 and 4 intersects the curve (on the left and right hand sides respectively, write 2 G x 2 =0

12 Audio narration The free energy versus composition curve has two minima because at this temperature there is coexistence of a mechanical mixture of two phases. The minima correspond to these two phases. In between the minima is the maxima. At the minima, the second derivative of the free energy with respect to composition is positive and at the maxima this quantity is negative. This implies that there are two points of inflection on this surve at which the second derivative of free energy with respect to composition becomes zero. These points are known as spinodal points. The locus of spinodal points at different temepratures is marked as chemical spinodal in the phase diagram. Inside the spinodal, the system is unstable and outside of it is metastable. As we have seen in the driving force for diffusion animation, inside the spinodal the diffusion is up hill and outside of it, it is down hill. Further, since any small change in composition (with alpha 1 phase becoming richer in A and alpha 2 phase becoming richer in B) reduces the free energy, the process is barrierless and spontaneous. However, in the metastable region, there is a barrier for the formation of a small region of alpha1 or alpha2 to form inside alpha before the process can proceed. In the next section, we schematically show the change in composition as a function of position in two systems marked in this diagram, namely, one with an overall composition of x N and another with an overall composition of x S. The subscripts N and S stand for nucleation and spinodal respectively.

13 1 Master Layout: Part

14 Instructions to the animator Clear the animation area. Draw a rectangle on the left hand side of the animation area. Mark the x axis at the bottom of the box as composition and mark the y axis to the left of the box as Temperature. Draw to parabolas one bigger and another smaller; let the noses of the two parabolas be touching. Mark the outside of the parabolas as alpha and the inside as alpha1 + alpha2. Draw two dotted horizontal lines; the top one should be above the parabolas and should be marked as T0. The bottom one should cut the two parabolas at four points and should be marked as T. The outermost points of intersection of the T line with the parabolas should be marked as x1 and x2 on the left and right hand sides, repsectively. Draw two vertical lines; both should start at T0 and reach the x axis. One should lie inside the inner parabola. The other should lie between the inner and outer parabolas. The line through the inner parabola should be marked as xs on the x axis and the other should be marked as xn. Draw two boxes, one below the other on the right hand side of the animation area. Mark them as xs and xn. Draw an arrow towards the bottom next to these two boxes and write Cool next to the arrow. Mark two horizontal lines through the boxes.

15 Audio narration Consider the cooling of two blocks of material with compositions xs and xn from a high temperature of T0 to a low temperature, say T. At this temperature, the system will consist of a mechanical mixture of two phases with composition x1 and x2 respectively (for the phases alpha1 and alpha2). We want to know how the composition evolves from their respective values to x1 and x2 in these two cases, along, say the dotted lines. In the next part of the animation, we show this composition evolution, which explains the nucleation and growth and spinodal modes of phase separation.

16 1 Master Layout: Part

17 Instructions to the animator Clear the animation area. Write on the left side top xn and on the right side top xs. Draw a downward arrow at the left extreme and mark it as increasing time. Draw the top two plots. Then, the next two plots below that and so on till the fourth row of plots as shown in the figure.

18 Audio narration Consider what happens as a function of time in the two systems, namely one with composition xn and another with composition xs. In the case of the system with composition xn, the initial changes in composition are localised, but are very large. This is nucleation. A very small region of the system undergoes a very large composition fluctuation. On the other hand, in the case of the system with composition xs, the changes are small but occur throughout the system. The further evolution of the system with overall composition xn is by the classical diffusion by ferrying material from the matrix to the interface which makes the interfaces move and hence the particles to grow. On the other hand, in the case of the system with composition xs, the composition modulations become more and more pronounced leading to the final composition at the end of the process.this is known as spinodal decomposition. As seen in the second row, in the case of spinodal decomposition, the diffusion of B atoms is up hill.

19 Credits What will you learn Lets Learn! Definitions Concepts Assumptions (if any) Animation Area Formula with derivation (if any) Graphs/Diagram (for reference) Test your understanding (questionnaire) Lets Sum up (summary) Want to know more (Further Reading)

20 INSTRUCTIONS SLIDE Self- Assessment Questionnaire for Learners At the spinodal point, the second derivative of the free energy with respect to composition becomes zero. True or false? The locus of the points of inflection on the free energy curve at different temperatures is called the miscibility gap. True or false? For a system to show spinodal decomposition, there should exist a miscibility gap. True or false? During spinodal decomposition, the diffusion is down hill. True or false? The system that undergoes nucleation and growth is in unstable state. True or false?

21 1 2 3 Questionnaire 1. True; the spinodal point is the point of inflection on the free energy versus composition curve. 2. False; it is called chemical spinodal. Miscibility gap is the locus of the minima points as a function of temperature. 3. True; if there exist no miscibility gap, (in general) the free energy versus composition curve will not have the double well shape and hence, one does not expect the point of inflection (and hence, no spinodal is expected. 4. False 4 5. False. It is in metastable state. The system that undergoes spinodal is the unstable one. 5

22 Links for further reading Books: Phase transformations in metals and alloys, D A Porter, K E Easterling, and MY Sherif, Third edition, CRC Press, 2009 (First Indian edition).

23 INSTRUCTIONS SLIDE Summary Phase separation can take place through two different mechanisms. System which are in a metastable state undergo phase separation by nucleation and growth mechanism. Systems which are unstable undergo phase separation by the spinodal mechanism. The diffusion during spinodal is up-hill. During the early stages of spinodal, small fluctuations take place over large spatial dimensions; on the other hand, at the beginning of nucleation, large fluctuations take place over small spatial dimensions.