3.7 FCC Hard Sphere Model

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来自 Georgia Institute of Technology 的课程

Material Behavior

415 个评分

Georgia Institute of Technology

Material Behavior

415 个评分

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.

从本节课中

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

与讲师见面

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

In this lesson, we're going to continue our discussion of the face centered cubic

model, but what we're going to do, is to add to it the concept of a hard sphere.

When we look at the figure on the left, when we described the lattice, so

we have each one of those points that represent the FCC structure.

And if we move to the figure to the right, what we're seeing are the spheres that

are sitting at each one of those lattice points and

they're developing the hard sphere FCC structure.

Now, it turns out that this is a very useful way of describing

certain materials, for example, copper crystallizes

into a structure that's referred to as face center cubic.

That is, each one of the atoms is represented as a sphere, and

it touches in particular directions, and

it produces a dense structure called the FCC structure.

Now it turns out that regardless of the dimensions of the sphere,

the structures all will look the same and

describe what we consider to be the packing factor and

see that regardless of the radius of the sphere.

When we're describing a face center cubic structure,

the packing is always the same in terms of the packing fracture.

So, if we now look at a section through the FCC

structure where it having the hard sphere,

we see that on the lower left.

And we can see how those corner positions and face positions are cut,

developing the face center cubic structure.

Now what is important for you to remember is that in order for

us to develop this hard sphere structure of face center cubic,

the spheres must touch along a particular direction.

And the direction where they must touch is along one of the diagonals of the face.

We can relate the diagonal of the face to the edge of

the unit cell by recognizing the simple geometry that

the length of the direction on the phase is A0 which is

the edge of the lattice times the square root of two.

So we can relate then the dimension where all of those spheres touch,

that length as a zero onto the square root of two.

Now what we can do is,

we can describe something we refer to as the packing factor.

And the packing factor is going to be the number of spheres and

the volume of the spheres divided by the volume of the unit cell.

And we know already that there is a relationship in the hard sphere

approximation that we can relate a0 and r to one another and

what we and then, enable to do is develop a non-dimensional packing factor.

Here is our FCC structure, we have six faces, each sharing with a half,

eight corners and we turned out with a total of force pierce in the volume.

Now, what we're going to do is to represent the packing factor as

the numbers of spheres times the volume of the spheres

divided by the volume of the cubic unit.

And we know the relationship that exists between the edge of the unit cell and

the radii.

So we know for example that a onto the square root of 2 = 4r.

And therefore, we know the relationship between a and

r for this face centered cubic structure.

So all we then have to do is to substitute so

that we can cancel out the terms of r, and

we wind up with a packing factor of 0.74.

It turns out that this is very important because the packing factor

for a distribution of hard spheres

in the face center cubic geometry is the highest packing factor that we can get.

These are all the same dimension spheres and they pack 74%.

We'll develop a similar relationship in the next lesson

where we describe the hard sphere packing in the body centered cubic arrangement.

Thank you.

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