2.9 Heterogeneous Nucleation - Spherical Cap Approximation

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From the course by Georgia Institute of Technology

Material Processing

158 ratings

Georgia Institute of Technology

Material Processing

158 ratings

Have you ever wondered why ceramics are hard and brittle while metals tend to be ductile? Why some materials conduct heat or electricity while others are insulators? Why adding just a small amount of carbon to iron results in an alloy that is so much stronger than the base metal? In this course, you will learn how a material’s properties are determined by the microstructure of the material, which is in turn determined by composition and the processing that the material has undergone. This is the second of three Coursera courses that mirror the Introduction to Materials Science class that is taken by most engineering undergrads at Georgia Tech. The aim of the course is to help students better understand the engineering materials that are used in the world around them. This first section covers the fundamentals of materials science including atomic structure and bonding, crystal structure, atomic and microscopic defects, and noncrystalline materials such as glasses, rubbers, and polymers.

From the lesson

Kinetics of Structural Transformations

If thermodynamics, which we covered in the previous module, tells us how a material wants to change, then kinetics tells us how and how quickly that transformation occurs. This module starts by explaining the driving force for phase transformations. We will cover the nucleation and growth of precipitates, solidification, and sintering. Finally, there are a number of lessons which apply all that has been covered in the course to understanding carbon steels.

2.7 Wetting5:54

2.8 Heterogeneous Nucleation5:52

2.9 Heterogeneous Nucleation - Spherical Cap Approximation7:40

2.10 Heterogeneous Nucleation - Sodium Acetate Demonstration2:53

2.11 Heterogeneous Nucleation - Applications4:28

2.12 Homogeneous and Heterogeneous Nucleation6:41

2.13 Types of Interfaces12:03

Meet the Instructors

Thomas H. Sanders, Jr.
Regents' Professor
School of Materials Science and Engineering

Welcome back.

In this lesson,

we're going to develop the equations to describe heterogeneous nucleation,

much like we did in the development of the equations for homogenous nucleation.

Remember in the case of homogeneous nucleation,

we said that there were in fact two competing terms.

One term was a volume term, and the other was a surface area term.

When we describe the process of heterogeneous nucleation, we're going to

see that those type of terms reappear, but with a bit more complexity.

I've written the equation for

the heterogeneous nucleation process in the screen above.

And what it tells me is I have a volume term, which is related to how much

undercooling I have, and the volume of the particular structure that has nucleated.

So, in this case, we're talking about the volume of the spherical cap.

Now when we look at the next term, which is the term in red,

that term has to do with the surface area of the spherical cap,

where we have the interface between the solid phase and the liquid phase.

Then we have a term that's associated with the solid and the mold

and its area times the solid mold interfacial energy.

And the last term we have, again, is the solid mold and

the interfacial energy between the mold and the liquid.

And notice, what we are able to do is to relate the solid and

mold to both the appearance and

the disappearance of the interface where the spherical cap is present.

So now what we've done is to go down and rewrite those expressions.

And we've done that as a result of the diagram that lies to the right.

We have three surface energy terms that are important.

One of the surface energy term is between the mold and the liquid.

The other is between the solid and the liquid.

And the third is between the solid and the mold.

So what we're going to do is to sum all of the horizontal forces

associated with those interfacial energy terms.

When we do that, what we'll find is that we can express the interfacial

energy between the mold and the liquid and the interfacial

energy of the solid in the mold, and come up with and expression that tells us that

the interfacial energy associated with the solid and the liquid times the cosine

of theta is equal to that difference in the two interfacial energies.

And when we include this into our equations above,

what we have is the surface area of the spherical cap,

and the volume of the spherical cap.

And remember that the geometry of the spherical cap depends on the angle theta.

So it's going to tell us something about the surface area between the solid and

the mold, and the surface area associated with the solid and the liquid.

So using that geometry, now we have these two equations and

those are the equations that we need to pay particular attention to,

to help us develop an understanding of

the spherical cap approximation to determine heterogeneous nucleation.

So I've written the equations for

heterogeneous nucleation in terms of two terms,

the volume term and the associated geometric term.

So those are the result of inserting those equations that we had on the previous,

visual.

And now what I've done is to multiply and divide the equation by 4.

The reason that I've done that is, if you look at the first term,

that first term is four-thirds pi r cubed delta G sub v +

4 pi r squared times the interfacial energy between the solid and the liquid.

That term should look familiar to you.

That term is, in effect, the homogeneous nucleation

equation that we described in the earlier lessons.

So that's our delta G of homogeneous nucleation.

The next term is what we're calling our wedding term, which is f of theta.

And so it's going to be,

as we change the angle theta we're going to change the values of f of theta.

And now we need to examine how those variables change, that is with f of theta.

So now what we've done is we're going to write the barrier to the heterogeneous

nucleation process in terms of first, homogeneous nucleation, and secondly,

times the associated wedding of that particular interface.

And what we're going to see is that that function,

f of theta, lies between two values.

It goes between 0 and 1 as you change the wedding angle from 0 to 180 degrees.

So what this tells us is, if we look at the bottom of the equation,

what that's telling [COUGH] us is that in one case,

when the value of f of theta is equal to 1,

it means we have no wedding associated with that particular interface.

And consequently, the only type of nucleation that we can have in

the absence of something that is wedding is homogenous nucleation.

Alternatively, if we look at the lower limit, when that value is equal to 0,

what that tells me is I have effectively complete wedding, and

the barrier to the nucleation process, is essentially equal to 0.

And one of the things that becomes important is the fact that

the radius is not going to be changed or altered that we calculated for

homogeneous nucleation, it's going to be the same r star.

And the reason, of course, for that is,

even though we have introduced the substrate, the behavior associated with

the interface separating the liquid and the solid has not changed.

But what has changed is the barrier to the nucleation process.

And the barrier looks exactly the same as

the homogeneous barrier that we described previously, but

this time it's modified by the presence of that wedding angle f of theta.

So by beginning the process of heterogeneous nucleation,

we're able to use homogeneous nucleation as a starting point and

develop how we can modify that particular behavior with respect

to the effectiveness of the wedding of the material as the substrate f of theta.

Thank you.

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