3.11 Hard Sphere Packing - Visualization

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Material Behavior

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Georgia Institute of Technology

Material Behavior

326 classificações

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 first 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.

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Crystalline Structure [Level of Difficulty: Medium || Student Effort: 2hrs 30mins]

This module covers how atoms are arranged in crystalline materials. Many of the materials that we deal with on a daily basis are crystalline, meaning that they are made up of a regularly repeating array of atoms. The "building block" of a crystal, which is called the Bravais lattice, dtermines some of the physical properties of a material. An understanding of these crystallographic principles will be vital to discussions of defects and diffusion, which are covered in the next module.

3.7 FCC Hard Sphere Model5:02

3.8 BCC Hard Sphere Model5:40

3.9 Calculating Density6:32

3.10 Hard Sphere Packing3:09

3.11 Hard Sphere Packing - Visualization5:01

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

In this lesson we're going to continue talking about these

two types of structures, the FCC and HCP,

but we're going to visualize them slightly differently.

We'll start out with our A row, and the spheres are all touching one another.

And then what we do is we add the B row.

And those are going to touch, and

will develop a complete row of B that sits in the interstitial positions of the A.

And now, on top of that,

we're going to include the row that we're referring to as C.

So now we have our ABC stacking.

And what I'd like to do then,

is to indicate the directions that tell us where the spheres are touching.

So, I have those two sets of arrows.

And now what I'm going to do, is to go back to the unit cube that we've been

using in the past, and I'm indicating on the cube where those two arrows

that are in the lower visual appear inside of the unit cell.

And now, what I'll do is to place on top of that triangle, in the FCC structure,

I'll place the positions of the spheres as indicated below.

And so, now what we have is the ABCABC stacking.

And it leads to the FCC structure.

And you can see the alternative description that we have of stacking

with the unit cell that ultimately develops.

Now when we look at the HCP structure, we get again, our A row.

And now we're going to put a B row.

And now instead of putting a C row,

this time we're putting another row, which is just above the A.

And we're going to refer to this as the A row.

So now we have ABA, and

our structure then becomes the ABA hexagonal close-packed structure.

When we consider the two sets of arrangements, that is the FCC, what

we see first is, there is a direction that's indicated on this visual and

that direction is telling us the direction of stacking.

And what we have is a vector that we can refer to as a unit.

So this is the unit that tells us the repeat distance in the FCC structure.

Now when we look at the HCP structure, that vector that takes us from

the first A to the second A is two-thirds of that which is in the first visual.

So that distance between the repeat layers for

the ABC are 1, and

the AB sequence for HCP are two-thirds of that.

So now what we can do is we can relate the HCP directly to the FCC.

And by doing that, what we can do is, we can describe

the c-axis of the hexagonal closed pack in terms of

the magnitude of the direction with respect to the FCC.

Remember that the FCC is along a direction of the body diagonal and

so, therefore, it's going to be a0 onto the square root 3, but

now it's two-thirds of that, since we're in the HCP structure.

Now when we look at the a, the A is the in-plane distances between the spheres.

And now when we go back on the right-hand side,

this relates directly to the distances that we see in the FCC,

which is one-half of Aa0 onto the square root of 2.

And if we make that ratio, what we get is the C over A ratio

of the square root of 8 over the square root of 3.

What that means is, it tells us that for ideal spheres

packed in the sequence ABAB, what we're getting is, a C

over A ratio that represents the square root of 8 over the square root of 3.

We refer to this, then, as ideal HCP packing.

Now when we look at traditional metals what we see is,

in this table, we see the ideal packing, which is the 1.633.

And now what we can do, is we can make some comparisons

of the ideal to actual metals that we see in practice.

Cadmium and zinc, for example.

If you look at the C over A ratio, that value is larger.

Which means that there is a change in the spherical symmetry and

it would suggest perhaps that the spheres are elongated in the C.

As opposed to being in the A direction.

On the other hand, when we look at the elements magnesium through barillium

what we see is a reduction.

And as a consequence of that, when we deviate from this ideal positioning,

we're going to see that the structure is changing.

And that's going to have an affect on

ultimately the mechanical behavior of these materials.

Thank you.

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