Critical undercooling for homogeneous nucleation There is a critical undercooling for homogeneous nucleation and homogeneous nucleation rate is very sensitive to interfacial energy Course Name: Phase transformation and heat treatment

Authors M P Gururajan

Learning Objectives

After interacting with this Learning Object, the learner will be able to: • Explain the critical undercooling needed for homogeneous nucleation in terms of critical size of nuclei and statistical fluctuations at any given temperature • Show how small changes in interfacial energy can have an exponential influence on nucleation kinetics Pre-requisites

Before interacting with this Learning Object, the learner is asked to peruse the following animation:

Homogeneous nucleation

Definitions of the components/Keywords:

1 Cluster: a group of atoms that have the structure of the one phase (say solid) in 2 another (say melt); for example, any group of atoms that form a close packed structure in a melt will be called a cluster

3 Embryo: a cluster

Critical nuclei: a cluster or embryo which has 4 reached a size at which even the addition of one atoms makes it stable and even the removal of one atom makes it unstable 5 Definitions of the components/Keywords:

1 Critical radius: radius of the critical nucleus

2 Critical free energy: the change in free energy for a nuclei of critical radius

3 Driving force (∆G): the reduction in free energy due to phase transformation

Undercooling (∆T): the process of cooling a 4 melt below its freezing point without the formation of the solid

5 Definitions of the components/Keywords:

1 Critical undercooling (∆T ): the undercooling N 2 at which the nucleation rate for homogeneous nucleation becomes appreciable (and an “explosion” of nuclei occur)

3 Homogeneous nucleation rate: the rate at which embryos/clusters turn into homogeneous nuclei in a melt 4

5 Expressions

Assuming spherical solid of radius r, the driving force for nucleation is given by −4 ΔG= πr 3 ΔG 4 πr 2 γ 3 v ---(1)

At any temperature T, the number of spherical clusters of radius r (nr) is given by the expression

−ΔGr nr =n0 exp  k B T  ----(2)

Where kB is the Boltzmann constant, T is the absolute temperature of the melt, and n0 is the total number of atoms in the system Expressions

Recall the expressions for critical radius of the nuclei and the critical free energy for homogeneous nucleation are as follows (as we have derived in the homogeneous nucleation animation):

2γT m 1 r=  L  ΔT ---(3)

3 2 16 πγ T m 1 ΔG = ----(4) 3L2   ΔT 2 Derivation

Consider a melt that contains C0 atoms per unit volume. Using the expression (2) above, the number of clusters that have reached the critical size can be calcualted as −ΔG C =C exp r clustersm−3 0 ----(5)  k B T 

Since an addition of one or more atoms makes these clusters nuclei, if the frequency of such addition is given by f0, then, the homogeneous nucleation rate is where

−ΔG − A −3 −1 N hom =f 0 C 0 exp =f 0 C 0 exp 2 nucleim s ---(6)  k B T    ΔT   3 2 16 πγ T m A is given by A= 2 ---(7) 3L k B T Master Layout: Part 1 1 2

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5 1 Action

In this part of the animation we will show two curves and their 2 intersection; these curves correspond to the critical radii for the nuclui and the maximum sized 3 particle that can be found at the given undercooling with an appreciable probability; the 4 intersection point defines the critical undercooling in the undercooling versus radius plot

5 T1: Plot of the undercooling versus critical radius 1 Step 1:

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5 Decription of the animation/action

Draw the x and y-axes. The x-axes spans both positive and negative values. Mark the y-axis as r and the x-axis as ∆T. Mark zero and the – and + on the x-axis. Draw a curve that goes as 1/x and mark it as r*

Audio narration

Let us consider the variation of critical radius as a function of undercooling. It has an inverse relation to ∆T and is as shown. Note that the critical radius is not defined on the negative side of the ∆T axis.

Text to be displayed

Equation 3

T1: Maximum cluster size versus undercooling 1 Step 2:

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5 Decription of the animation/action

Draw the x and y-axes. The x-axes spans both positive and negative values. Mark the y-axis as r and the x-axis as ∆T. Mark zero and the – and + on the x-axis. Draw a curve that goes as 1/x and mark it as r*.

Draw another curve which starts from negative values of ∆T. Mark this curve as rmax. After the point at which it intersects the r* curve, draw the curve with dotted lines. ∆ Mark the intersection point as TN on the x-axis.

Audio narration

As described in the introduction section of this animation, the number of clusters of a given radius can be calculated. At any given temperature, let us calculate the driving force and hence the average number of particles of a given radius. Let us consider the biggest radius particle that can exist at the given temperature and call it r-max. For any particle of radius bigger than r-max, nr is less than one. This r-max curve intersects the r* curve as shown at a particular undercooling – which is the critical undercooling, which, when reached, there will be an explostion of homogeneous nuclei. This is because, beyond this undercooling, the critical radii of nuclei becomes smaller than the clusters that would form because of the thermal fluctuations. Hence, the probability of these clusters turning into nuclei are large.

Text to be displayed

Equation 2 Master Layout: Part 2 1 2

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5 Decription of the animation/action

Draw the x and y-axes. The x-axes spans only the positive side.

Mark the y-axis as N_{hom} and the x-axis as ∆T. Mark zero at the x- and y-axes intersection. Draw a curve that grows exponentially as shown at a point ∆ away from zero on the xaxis and mark the point as TN on the x-axis.

Audio narration

It is possible to show the critical undercooling in another way. If you plot the homogeneous nucleation rate as a function of undercooling, the critical undercooling at which the explosion of nuclei happen is clearly seen. Text to be displayed

Equation 6 Credits

What will you learn Play/pause Restart

Lets Learn!

Definitions

Concepts

Assumptions (if any) Show the plots here

Formula with derivation (if any) Graphs/Diagram (for reference) Test your understanding (questionnaire)

Lets Sum up (summary)

Want to know more… (Further Reading)

Show equations here

1 Interactivity and Boundary limits •In this section, I want to add the ‘Interactivity’ options to the animation. The interactivity involves a drop-down menu to 2 choose any percentage change to the interfacial energy. • The interfacial energy should change between -10% to 10%. • Results: For the given material parameters (that correspond 3 to that of copper), at a given undercooling the homogeneous nucleation rate changes by several orders of magnitude on changing the interfacial energy by about plus or minus 10%.

4 Homogeneous nucleation rate Drop down menu In per cubic metres per second

∆ γ List the values used in the calculation (f0,C0, T,Tm, ,L,kB) 5 Interactivity option 1: 1 Step No: 1

2 Drop down menu of a set of numbers

Interactivity Instruction Boundary type to the Instructions for the animator Results and Output limits 3 (IO1/IO2..) learner

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5 Interactivity type: Drop down menu

Instruction to the learner: Change the interfacial energy by a small amount; the change is in percentage; you can reduce or increase the interfacial energy up to 10%. Boundary limits: -10,-9,-8,....-1,0,1,2,...,9.10 should show up in the drop down menu Instructions to the animator: Change the interfacial energy by the given amount and calculate the homogeneous nucleation rate using the formula and given values in the next slide and report the homogeneous nucleation rate per cubic metre per second. Also, show on the screen all the other values used in the calculation and the formula Results and output: We have used the values for copper soldification from its melt and have shown the effect of small changes in interfacial energy on homogeneous nucleation rate. You can see that the homogeneous nucleation rate changes by about 18 orders of magnitude for a small change of plus or minus 10%. This is because the interfacial energy enters the exponential term as a cube. So, during nucleation, the system is very sensitive to interfacial energy considerations.

∆T = 200 K Tm = 1356 K γ = 0.177 J/m2 L = 1.8368x109 J/m3 11 -1 f0 = 1x10 s 29 -3 C0 = 1x10 m -23 kB = 1.38x10

− A −3 −1 N hom =f 0 C0 exp nucleim s   ΔT 2  3 2 16 πγ T m A= 2 3L k B ΔT + T m  Questionnaire for users to test their understanding [1] The critical radius of the nucleus increases with undercooling. True or false? [2] The probability of finding a cluster of relatively bigger radius increases with increasing undercooling. True or false? [3] Clusters always grow and become nuclei. True or false? [4] The homogeneous nucleation rate is very sensitive to changes in interfacial energy. True or false? [5] Critical undercooling is the undercooling at which the critical radius is much larger than the maximum cluster size formed due to thermal fluctuations. True or false? Questionnaire 1 1. False

2. True 2 3. False Only clusters which reach the critical size grow. 3 4. True 5. False The two values are comparable at the critical 4 underrcooling.

5 Links for further reading

Books:

Phase transformations in metals and alloys, D A Porter, K E Easterling and M Y Sherif, Third edition, CRC Press, 2009 (First Indian edition).

Materials science and engineering: A first course, V. Raghavan, Fifth edition, Prentice-Hall of India Private Ltd., 2008.

Summary

• There is a critical undercooling for homogeneous nucleation. • At and beyond critical undercooling, the thermal fluctuations produce clusters of a size which are larger than the critical radius of the nuclei at the given temperature. • The nucleation rate is very sensitive to the interfacial energy values.