One announcement. There will be Quiz 9,

the weekly quiz tomorrow based on the content of Homework 9 which is,

I don't remember what it is, but you know what it is.

I made it up, but I am already thinking about the next glorious

event. Now I remember. There is chemical kinetics and

glasses. That's right, chemical kinetics and amorphous

solids. That's right. Last day we started

talking about diffusion which is solid state mass transport by random

atomic motion. And we were drawn to the paper that

is up on the display which was published in 1855 by Adolf Fick

which gave us the law that bears his name describing how matter diffuses

through matter. Fick was a very talented individual,

and I want to draw attention to something else.

Remember, he was a physiologist. He actually did some work in

medicine as well. In 1870, he described what survives

to this day as the Fick Principle for determining cardiac output and

basically equates the amount of uptake of oxygen by the lungs with

the amount that should be distributed in the blood.

But you have to take arterial oxygen pressure minus venal oxygen pressure

to get the efficiency of the heart. And so I found this little cartoon

off the Web that just briefly encapsulates the notion of the Fick

Principle for measuring cardiac output. His nephew,

by the way, who was orphaned as a young child also has

the name Adolf Fick. The young Adolf Fick,

the nephew was strongly influenced by the uncle. He went on to invent

the contact lens. This family has been quite prolific

in ways that influence many of us. How many in this room are wearing

contact lenses? I think we are all thankful that

there are some cardiac efficiencies. I think we are all beneficiaries.

Let's take a look, in more detail, at what Fick taught

us. He said that if we have ingress of

species i into some solid, the rate of ingress is given by the

equation shown here which expresses the flux which is mass per unit time

per unit cross-sectional area as proportional to the concentration

gradient. And the proportionality constant is the diffusion

coefficient. And what we see here is a sketch of what that

might look like. Some initial surface concentration

held constant with ingress of material. And since the flux is the

derivative, you can see that the flux, but for some multiplication

factor, tracks with the concentration.

We have the steepest concentration gradient at the surface,

and deep inside the specimen there is essentially no concentration

gradient so the flux attenuates. Flux shown here in fuchsia and

concentration shown here. And when I say profile it means

something as a function of distance. Profile, in this case, is the

concentration profile or a flux profile. You can see those two.

And then we started looking at some of the atomistics and reasoned that

there is some jumping involved. And with the jumping comes the

notion of activation. We looked at this.

And we concluded that there is an activation energy associated with

jumping through that saddle point, an Arrhenius type behavior, natural

log of d, linear in reciprocal of the absolute temperature.

Remember, this is not the Rydberg constant, this is the gas constant

which is the product of the Boltzmann constant and the Avogadro

number. And units will give you a clue as to which to use.

If the activation energy is in units per mole then obviously use

the gas constant. Otherwise, you use the Boltzmann

constant. And when we looked at the atomistics further,

we reasoned that when we have diffusion by vacancy jump mechanism,

this activation energy up here in the equation is the sum of two

components, one being the enthalpy of vacancy formation,

which is really the negative of the bonding energy,

and then the energy associated with the atom migration which is the

energy of squeezing through that saddle point shown in the figure.

In the case of interstitial diffusion, we do not have to form

vacancies. There are enough vacant sites, by virtue of the fact that we

have so much free volume, even in a close-packed solid.

We are really just paying for the enthalpy of migration.

Lastly, we looked at how these two are related to the degree of

confinement. In other words, the atom is restricted by the ease

with which it can run down its appropriate raceway.

And we reasoned that going through the bulk lattice is the most

tortuous path and has the highest activation energy,

lowest diffusion coefficient. Grain boundaries have atoms with

fewer nearest neighbors, lower coordination number,

so the diffusion coefficient is higher in the grain boundary and is

highest along the free surface. And we saw last day that the

diffusion coefficient along the free surface of a solid approaches values

equivalent to that of diffusion in a liquid state.

What I want to do today is step back to the continuum and use Fick's law

to describe what is going on mathematically in the macro world.

I want to first start with diffusion across a membrane.

This is a membrane of some thickness L. And I am going to put

a gas to the left, P1, and I am going to hold it

constant. And to the right, I am going to have a gas at P2 also

being held constant. And, just so that we are all on the

same page, I am going to make P1 greater than P2.

That means matter will diffuse from high pressure to low pressure,

high concentration to low concentration.

That means the flux will be left to right. And we can use the Universal

Gas Law, pressure volume equals the product of mole number times gas

constant times temperature in order to convert pressure units into

concentration units. And we know that the concentration

of a species i is simply equal to the mole number i divided by the

total volume of the system. This is moles per unit volume,

so that is a concentration. And so that means then that if we go to the

Universal Gas Law that will give us P over RT. If you take a pressure,

divide it by the product of the gas constant and temperature,

it will move from pressure units to concentration units.

And so what I would like to do is show what this system looks like

after some period of time. I am going to put the convention

with zero on the left surface. This is distance moving from left

to right. X equals zero is the left surface of the membrane.

X equals L is the right surface of the membrane. And now let's put

concentration c1 on the left surface and c2 on the right surface.

And what we know is we expect that after some period of time,

if c1 is held constant and c2 is held constant then,

after some period of time, we expect to have a profile that is

linear. And we should be able to infer that from Fick's First Law.

We know that Fick's First Law tells us that J is equal to

minus D, dc by dx. If I am holding the concentrations

on both sides constant then it tells me then that the gradient is going

to be constant. Otherwise, I have some sources or

sinks which we do not allow for. After this short time we have this

profile which we term steady state. And it is termed steady state

because it is invariant with time. The flux is not a function of time.

The flux entering the free surface on the left equals the flux exiting

on the free surface on the right. And, therefore, the flux must be

constant throughout the membrane. Because if J is constant, and I am

assuming that the diffusion coefficient doesn't change with time,

then it means that this gradient must be constant.

The slope must be the same all the way along, reinforcing the notion

that we have a straight line profile.

Now, there is one assumption here, of course. And that is that the

diffusion coefficient is independent of concentration.

It could be. And we will return to that later and see what happens if

we have the concentration depend on diffusivity. We can describe at

steady state everything we need to know about the behavior of this

system with Fick's First Law alone. Fick's First Law alone is

sufficient. What are the kinds of things that we might want to know?

We might want to ask how much material diffuses over time,

mass per unit time passing through the membrane. All we need is Fick's

First Law because we know that we can say that the mass will simply

equal the flux, which is mass per unit area per unit

time. If I know the total area of the membrane and I know the time

then that is all I need. Fick's First Law is very thorough in

describing a system at steady state. Last thing about this concentration

profile, the operative definition of steady state is that J does not

change with time. In other words, if I look at this

at some later time, as long as I maintain constant end

compositions, I will have this same profile. But suppose I had a

situation like this where the diffusion coefficient is not

independent of concentration. I will give you an example.

The diffusion coefficient of carbon in iron, this is steel,

as a function of the concentration of carbon in iron.

It turns out that as more and more carbon goes into iron,

the diffusion coefficient changes. You would expect that because now

some of the sites are occupied. Which is it? Do you think that the

diffusion coefficient, as you get more and more carbon into

iron, starts to fall like this, or do you think the diffusion

coefficient rises like this so that at higher concentrations it becomes

more difficult to diffuse? Well, this might sound appealing

because the sites are getting blocked. It make sense.

There are fewer empty sites. But, in point of fact, the

diffusion coefficient actually rises.

Why? Well, I have told you that the carbon atom is larger than the

interstitial site. When the carbons go into the

interstitial sites they act as wedges, and they actually wedge open

the system and make it easier for the succeeding atoms to diffuse

through. In point of fact, this is what happens. But it could

be either. Let's imagine, suppose we have case number one,

where the diffusion coefficient falls as concentration rises.

And now I have a steady state concentration profile.

I have c1 constant on the left, c2 constant on the right, and here

is the equation we have to look at. Minus D, dc by dx. But this is now

a function of concentration. If I have case number one, when I

have the highest concentration up here, if this is going to remain

constant, if the concentration dependence of the diffusion

coefficient says D is low when c is high then that suggests to keep a

constant flux, dc/dx must be high when c is high.

That means instead of a straight line, we are going to see something

that looks like this. This is case number one.

How about case number two where the concentration dependence of the

diffusivity is indicated with a rise?

Well, that means at high concentration my diffusion

coefficient is going to be higher than it is at low concentration.

If this number is higher at high concentration then this number has

to be lower so that the product will remain constant.

This is case number two. These are all variants of steady

state. J is not a function of time. J is independent of time.

Now I want to ask what happens in that initial moment when we first

pressurize the system. Let's say we start off and there is

nothing inside the membrane. Just to keep things simple, I am

going to make P equal zero on the right-hand side.

Left side is P1. And at time zero,

we initially pressurize so we have a concentration c1 instantly at the

surface. What is the set of events that occurs leading up to the

establishment of the steady state profile?

I would expect, after a short time,

we would see concentration profile shown like this.

And then after a longer period of time like this.

And then after a longer period of time like this.

And then finally, of course, we establish.

This is time is increasing from low to high.

This could be t1, t2, t3. And so we can then say

let's park ourselves at some value x1 on zero to L.

And look at what is happening. Initially, if I take the slope it

is a low number. If I take the slope at a later time

it is a higher number. If I take the slope at a later time

it is an even higher number. And, ultimately, it is going to be

as high as the steady state value. I could take that data set and plot

the flux at x equals x1. I am sitting here watching the

profile evolve with time. Initially, it is zero because there

is nothing in the membrane. And then it slowly rises and

reaches the steady state value. This is J at steady state here,

and this could be t1, t2 and t3. You can see that coming off the

slope. Clearly, when we are moving through this time

period, J is very much a function of time. Because,

if you close your eyes and wait, you will see that the profile has

changed. And, if the profile has changed,

it means that the instant value of flux has changed.

Flux is a function of time, so this is not steady state.

Instead it is transient. We are in the transient regime.

And so, as is always the case, we can draw these curves.

They start at a high value, they attenuate to zero. What is the

magical question? What is the shape?

That is the only reason I need math, to tell me the shape of those curves.

Is this an exponential decay? Is this half of a sinusoid?

What is the shape of the curves? I want to know c versus x at all

times because it is changing as a function of time.

That means I need to know c is a function of x and t.

I have got to solve this. And Fick gave us the solution to

this as well. The solution to this is the solution to an equation known

as Fick's Second Law. Let's look at Fick's Second Law.

I am going to write it in the special case when D is independent

of composition. D has to be independent of

composition for me to do what I am going to do next.

And it is a pretty good assumption in many industrial settings.

And so I am just going to write it. It is a partial differential

equation. What it is telling you is that the

partial, with respect to time, is equal to the product of the

diffusion coefficient times the partial of the gradient in

concentration. And so, clearly,

this is a partial differential equation. And that is bad.

You cannot solve this one, not yet. But the good news is it is a linear

partial differential equation. That is good because I can show you

a solution that applies for a given setting, and then you can play with

the boundary conditions and use the same solution.

That is the good news, it is linear, but the solution of

this is both in x and t. And the solution looks like this.

Let's say we have an initial concentration c naught.

C is a function of x. I have an initial nonzero

concentration c naught and I have a surface concentration fixed at cs,

and this is the system that we are trying to model.

And the general solution to Fick's Second Law is c of x and t is some

constant A plus Aaconstant B times a function, which I am going to define

for you in a second, the Gaussian error function of x

divided by square roots of Dt These are both constants to be

determined. This is the Gaussian error function.

X is position, t is time, and d is the diffusion coefficient.

And this is nothing but a transcendental function that varies

from zero to one. And I could have said sine.

If I had said it is sine x over two roots of Dt --

You know the definition of sine. It has some weird integral and blah,

blah, blah. Nobody pays any attention to it.

You know it goes from zero to one. It is linear at low values of theta.

So what. We are going to learn what the properties of this are,

but I can tell you something right off the bat. What do you think the

shape of the error function is? What is the shape of the error

function? That is the shape. That is what math is. Math is

finding the shape. Otherwise, it doesn't make any

sense. And what are these?

These are scale factors. You have done this on a computer

where you take some image and you pull it, stretch it and so on,

and you can lock the aspect ratio. Well, that is all you are doing

here mathematically. We are going to pull this.

Why are we going to pull it? Because at a later time it looks

like this. But the relationship between all of these points is the

same. That is what math does. If it doesn't fire it. Math works

for you. You don't work for math. Let's put this thing to work,

lazy. What we are going to do is solve for A and B.

You know that when you differentiate,

you throw away information. Every time you throw away

information we need to add some. We need to have one time-based

boundary condition, and we need to have two spatially

based boundary conditions because we threw away three bits of information.

That is what we are going to do. And, if we go ahead and do that,

here is what A and B will solve to. And this works for all cases,

c minus the surface concentration divided by the initial concentration

minus the surface concentration equals error function of x over two

roots of Dt. C is what you are looking for. This is c as a

function of x and t, cs is surface concentration,

c0 is the initial concentration. And let's take a look at what that

function plots out to be. I am going to plot erf of z as a

function of z. It varies from zero to one.

And it starts off, like so many functions, it is linear.

For low values of z, erf of z equals z. And,

in fact, up to about 0. you are good to within 1%.

Erf of z equals z to within 1%. And then once it gets up to 0.

then it starts to veer off and then asymptotically approach erf z equals

one. In fact, we can write here erf of 1 equals 0.

4. You can see now it is not linear. And erf of 2 equals 0.

95. Once you get out here to two, you are practically at the asymptote.

Here is the integral. And it is defined erf of z is the

integral from zero to z of e to the minus u squared du.

That is this, e to minus u squared as a function of u is the Gaussian

error function. That is the bell curve.

This is the area under the bell curve. If we start at u equals zero,

we are going to integrate as far as we need to out here.

And the integral from zero to infinity of this thing is root pi

over two. And we want this to go from zero to one,

so we will put the factor two over root pi out in front.

That way, when we integrate this from zero to infinity,

erf of infinity becomes one. Erf runs from zero to one. And

that is the area under this curve. That is why some people call this

the Gaussian error function because the Gaussian curve is associated

with random statistics in terms of errors.

This now is the template. If I have any reaction this is all

I have to fit using these multiplication factors.

Let's play "Mr. Dress Up" here. We are going to use this function

to describe all of the typical industrial processes.

There are only two types of processes, those that are diffusing

substance in and those that are pulling substance out.

Let's look at the two cases. Let's look first of all at the one

that is shown right up there. This would be for something like

out-gassing or drying. Many processes involve drying which

is to get the water out of the content of a solid piece.

And so, as you can imagine, you have some initial concentration

c naught. And you take the surface

concentration, whatever this is,

and obviously if you want to draw something out and move matter from

right to left then the surface concentration,

by necessity, must be less than the initial concentration.

Otherwise, why is something going to move? This I will call effusion,

something is effusing. This is effusion. The flux is moving from

right to left, and the determinant here is c naught

is greater than cs. The surface concentration is less

than the initial concentration. I am going to pin the concentration

at cs. I have c0 deep inside. And this is telling me what the

shape of the curve is. I just write with impunity c minus

cs over c naught minus cs equals error function x over two roots of

Dt. **C-Cs / (C0-Cs) = erf (x / (2 sqrt(Dt))** And that is the answer

for all such problems. And because it is linear it doesn't

matter. If I change the initial concentration to a new value,

I can just add. We can add and multiply because it is a linear

equation, linear functionality. I think here we looked at those.

This is the tabulation. This is in the supplemental text.

There is a table of error function values so that if you happen to get

into this regime beyond z equals 0. you will have the values that you

need. Let's look at another case.

Let's look at driving material in. That, for example, is something

like doping or nitriding, carburizing. If we want to case

harden a material, we will raise the carbon content at

the free surface, have something that is soft and

ductile inside. For that here is the scheme.

We have a surface concentration cs. We have an initial concentration c

naught. And in this case c naught is less than cs.

Surface concentration is higher so material now wants to move

from left to right. And so this is infusion.

And what is the curve going to look like? It looks like this,

which is simply that one flipped upside down by subtracting it from

one, which I will show you in a second. Again,

this same general solution applies. The surface concentration is higher

or lower than the initial bulk concentration, it comes

out in the math here. Again, we write c minus cs over c

naught minus cs equals erf x over two roots of Dt.

And this one comes up very often. This is the way it works in doping

of semiconductors and comes up so often that it is convenient to

define another function where, in many instances, c naught equals

zero. When we are doping pure silicon, the initial concentration

of boron or phosphorus in pure silicon is going to be zero.

For many of those cases, you will end up with something that

if you go through the math this is a zero, transpose,

you will end up with this as the solution, c equals cs times the

complimentary error function, erf compliment x over two roots of

Dt where the complimentary error function, erf compliment of z is

simply one minus erf of z. If you see that in the literature,

that is all they are doing. Now, the question is,

I have told you that there are special conditions that validate the

error function solutions. What are the conditions? When to

use erf? Because that is not the only solution to Fick's second law.

When these are the conditions, c of x at time zero is a constant.

This is boundary condition number one. We threw away some time data,

so here is data at time equals zero. This is called the initial

condition. The initial condition must be a constant.

And that is OK. There are many systems for which the initial value

of the material that is being diffused is constant throughout the

material, as opposed to bopping all over the place or maybe already

having some kind of a gradient. As long as you can make this

assumption, you are on the road. Now, we need two more boundary

conditions. We need boundary conditions two and three over here

because we have a double derivative. You have some function plus

constant. When you do d by dx, constant goes into the trash bin.

We need to recover that information somehow. Here is what I have.

I have c for x equals zero at all time is a constant.

The surface concentration is fixed as a constant.

And you say, well, what else could it be?

Well, I could have a slowly rising surface concentration or I could

have a surface concentration that varies sinusoidally or something.

Those won't work. It must be fixed at a constant value.

That is one. This is boundary condition number one.

This is boundary condition number two. And then I need a third one.

The third one I am going to get is this one.

It looks funny when I write it mathematically,

but you will understand what I mean in a second. At infinity the

concentration does not change. You say, well, duh. This really

has some meaning. We know that the size of our piece

is not infinite, but here is what we are assuming.

We are assuming that over the time scale of the process that,

for all intents and purposes, the dimension of the silicon wafer,

for example, is so large in comparison to the depth of the

diffusion profile that assuming the wafer is semi-infinite,

because we have it diffusing in from both sides, but let's just assume

from the one side, that we can assume that it is

infinite. Ultimately, what happens? If we wait long

enough, we start to have a tail over here because the system still thinks

that the wafer exists. But we are saying no, it stops here.

But this value is so small during the time scale of the process that

we can neglect it. And by allowing ourselves to assume

that the material is semi-infinite it simplifies the math.

If we have a finite-sized specimen, we cannot use this simplified

solution and things get very, very messy. But there is another

way of expressing it. You see this relationship?

It is always erf x over two times the square root of Dt.

Something would be infinite would mean it is very large.

Large dimension. Let's say large length scale. That is a large

physical dimension. That means the argument of this is

very, very large. But there is another way to get the

same result: short time scale. That means every diffusion

experiment initially is operating in a material that,

for all intents and purposes, is infinite. Because when those

first phosphorus atoms start diffusing into the silicon,

they don't know the silicon is only a millimeter thick.

As far as they are concerned it is infinitely thick.

This solution is valid everywhere. As long as you have concentration

initial fixed, concentration surface fixed at short

times, this solution is valid. Valid over a short time.

That means it has great utility. Now I want to show you something

that takes everything we have learned and crashes it into a simple

rule of thumb. And these are the kinds of rules I

hope you will retain long after. What was it Franklin said?

Education is what remains when you have forgotten all of your schooling.

This is what you will remember. You have c minus cs over c naught

minus cs, and that is equal to over here erf x over two roots of Dt.

That is the accurate solution. If I wanted to capture the whole

diffusion process in one tiny little sentence, what would be an average

value? This is a normalized ratio, isn't it? It goes from zero to one.

What is an average value between zero and one? I don't know.

How about a half? Suppose I just choose this as an

average value for the whole diffusion process,

everything that can ever happen in diffusion, well,

a half is less than 0. . Since a half is less than 0.

, I can approximate erf x over two roots of Dt as simply x over two

roots of Dt. Now I have this equals this. I will cancel out the twos

and cross-multiply. That gives me x goes as two roots of

Dt as an overarching equation for everything that happens in

short-term diffusion. **x = 2 sqrt (Dt)** I cannot tell

you how many times I have been in situations where I am in some

conference room and a bunch of people are sitting around trying to

figure out what to do and somebody is getting ready to do some finite

element calculation to go mathematically model this whole

process, and I sit back and go, gee, well, I know the length scales.

Well, why don't we do one? Let's do one. Have you ever been

told this thing about cathedral windows, why the glass at the bottom

of a cathedral window is thicker than it is at the top?

It has to do with the fact that glass is really a viscous liquid and

over 500 years it trickles and it gets wider. That is nuts.

And I can show you like this. It's very simple. You already know

that the average -- What is the average temperature in a

cathedral in Northern Europe? It is zero in the winter and 20

degrees in the summer. Let's pick a number, 10 degrees C.

What do you think the diffusion coefficient of silicate glass at 10

degrees C would have to be for this thing to go anywhere?

Here is what I did. I said let's use this. Here is my simple model

of the cathedral window. This is the cross-section.

And I figure I need a length scale. I chose a length scale,

if something is going to diffuse, on the order of ten centimeters.

You can quibble. If you want to make it one

centimeter, I don't care. I am going to blow this thing so

far out of the water it won't matter. This is about ten centimeters,

and it is going to get thicker on the bottom. I figure I am going to

give this a really long run time, so I am going to make it 500 years.

I want to show you how to put this problem to the test.

What I am going to do is derive a quantity that I can say yes or no to.

What I am going to try to do is pull out a value of the diffusion

coefficient, so I know that x squared equals Dt,

if I just square both sides of that thing. That means that D,

the diffusion coefficient should be x squared divided by t.

I am going to do this in centimeters. This is ten squared on

the top and time is 500 years, so that is 500. And how many

seconds in a year? Pi times ten to the seventh.

Pi times ten million is good to within about 1%.

That is it. Who cares? This is a diffusion calculation.

You don't need anymore. You are not going to go 365 days and how

many leap years were there? Come on. Get with it. 6.4 times

ten to the minus nine centimeters squared per second.

I would need a diffusion coefficient of a silicate glass of

ten to the minus nine. That is the diffusion coefficient of

a close-packed metal at its melting point. I went and looked up some

data, and here is what I found for the diffusion coefficient of sodium

in a soda lime glass such as the type that would be used in those

windows. And, using those data,

at 10 degrees C, the diffusion coefficient would be ten to the

minus nineteen centimeters squared per second. It is off by a mere ten

orders of magnitude. I don't care if you want to make

this one centimeter or if you want to make it one millimeter.

There is no way that is the explanation. We know what the

reason is. This proves that it cannot be. What I want to show you

is that you can come at it from both sides. The other thing is to figure

out how they make the class. You know how they made the glass?

They pull it out of a layer and put it on a spinning table.

That is how they could flatten it. They didn't have the float glass

process where you pour silicate on top of liquid tin.

And liquid tin has a perfectly flat surface, the silicate glass is less

dense and is insoluble, and so it floats on the liquid tin

surface and is perfectly flat. They did not do that back in

Medieval times. They spun cast it.

It was like a spin coating technique. This is top view.

You have some table spinning, and you plop some glass on it and it

goes whoosh by centrifugal force. Now the cross-section of the glass,

what does it look like? Let's start from the center and go out.

Guess what? Not to scale. It is ever so slightly thicker on the

outside than the inside. And people would actually be very

careful about choosing the pieces. And then the glazers said when you

mount the windows, let's put all the glass in with the

thick side down. Wouldn't that make sense?

And that is why the glass is thicker, because it

was made that way. Simple answer.

But you can do the calculation. I don't know how many times, but I

get asked this at least once a year. Someone says is it true that the

cathedrals, blah, blah, blah. And I go does this

person know Fick's Law or not? Don't know, so we have to give a

different explanation. Now I want to finish the story of

the automobile exhaust catalyst and show how we can bring a lot of what

we have been learning in the last several days together.

Just to review. This is an old sketch because it

was taken from an early General Motors publication from the ‘80s.

Basically, what you have is - we have seen the catalytic converter.

And it is taking the exhaust gases and trying to convert some of the

toxins into something less offensive. And I had reported to you that we

need to control in order to achieve both oxidation and reduction.

And how do we control? We control with the aid of an oxygen

sensor. Could we switch to the document camera,

please, for a second? Take that input. What I am showing

you is an oxygen sensor.

What you are looking at is this is the metal sleeve on top of the

oxygen sensor, and you can see veins here.

This is inserted just past the engine, before the catalytic

converter. And what I want to do is to describe the solid state

chemistry that is going on inside there. May we go back to

the computer, please?

What I just showed you is sitting right here just as the exhaust gases

leave the engine. And what they are doing is

measuring the oxygen pressure in the exhaust gases so that we can control

the air-to-fuel ratio. Because it turns out that to convert

the CO and the hydrocarbons to the desired products,

in other words, carbon dioxide, water vapor and carbon dioxide, you

need to oxidize. And the catalyst is pretty good at

very, very high air ratios. And to reduce NOX to nitrogen you

need to have rich fuel ratios. And mercifully nature was kind.

And there is this tiny window at about 14.6 to one where you get

fairly high yields on both catalytic reactions. This is where you want

to park yourself. Now you need to have a sensor,

and that sensor looks like this. It is zirconia. It is an ionic

oxide, and it is doped with calcium oxide to create oxygen vacancies.

We are going to get defects involved and we are going to talk

about diffusion. When we put calcium oxide into

zirconia, we have two plus sitting on a four plus site.

And so we need to make an oxygen vacancy to compensate.

As we add calcium oxide, we make more oxygen vacancies which

means that the diffusion coefficient goes up. And that shortens the

response time. It is no good having a sensor that

responds on a time scale of minutes when you're driving conditions are

changing on a time scale of seconds. And so we dope with a subvalent

oxide in order to improve the performance.

And here is a cartoon that shows this is what is underneath the vein.

This blue that I have drawn is a closed one end zirconia tube.

There are electrodes on the inside. This is the exhaust. And on the

outside it is open to the air. And this measures a voltage across

the zirconia sensor. And that voltage goes to the CPU.

Every car has several computers on it, but the first cars to have

computers had them in order to control air to fuel ratio to make

the catalytic converter effective. And then that goes to control either

the carburetor or nowadays the fuel injectors, which then keeps the

fuel-to-air ratio so that this keeps measuring 14.6 to one.

And if you change altitude, if you get stuck in the Callahan

Tunnel and the ambient conditions are changing, this can respond to

changes in temperature. What do you think the partial

pressure of oxygen is? About 20% in air. But how much

oxygen per unit volume do you have at 100 degrees Fahrenheit versus

zero degrees Fahrenheit? This thing can figure all that out

and, thereby, run the system at the optimum fuel-to-air ratio to

maximally get the output from the catalytic converter.

And so in 3.091 we have learned about the doping of silicon to make

what goes into here. We have learned about the ceramics

that go into the catalytic converter and the catalyst that goes on it.

And now we have learned about diffusion and everything that goes

into the oxygen sensor. That is why I tell you chemistry is

the central science to energy efficiency and environmental

conservancy. Learn your chemistry. It is the most important subject.

And with that I will say have a nice day and we will see on Wednesday.