PROFESSOR: All right, now for
something completely different.
Before beginning though, I would like to raise a
procedural question.
It was my intent that quiz number two, which is like a
little more than a week away, was going to be part symmetry
and part tensors.
I don't want to have your fortunes based by weight of
2:1 on symmetry theory as opposed to properties.
What I would suggest, and you can express your opinion
either way, is that we postpone quiz number two by
maybe as much as 10 days, so that it can be half symmetry
and half the introductory discussion of tensors.
And if that does not create a conflict with some of your
other classes, I would propose that as a suggestion.
If not, we can just have on the quiz what we're going to
cover of tensors in the next couple days and
have the rest symmetry.
So what I am suggesting is the quiz was scheduled for a week
from November 8, which is a Tuesday.
What I would suggest is putting that off to the 18th,
and that will give us four extra lectures--
six extra lectures-- on tensors and we can make the
quiz 50-50.
AUDIENCE: What does that do to quiz number three?
PROFESSOR: Quiz number three is going to stay in place
because it's right at the end of the term.
AUDIENCE: What about the subject in three?
PROFESSOR: It will be all tensors.
It will be all tensors and properties.
And the third quiz is set up for longer than you think.
It's December 8, so it'll be about three weeks.
AUDIENCE: So everything from here on?
PROFESSOR: Yeah.
Does that make sense?
Or is somebody going to have a big, traumatic responsibility
at that point?
AUDIENCE: I don't know, but I think it would be probably be
easiest for us not to have it three days
before close of exams.
PROFESSOR: OK, OK, that's what I wanted to hear.
How about if we went just a little bit further then and
did it not on the--
when was the suggestion?
The 18th.
Let's say we did it on Tuesday the 15th?
Does that conflict with anything?
AUDIENCE: That's a little bit.
AUDIENCE: We were doing it on the 18th.
AUDIENCE: Is that Friday?
[INTERPOSING VOICES]
PROFESSOR: I'm sorry, 18th.
I said it would be 17th.
Excuse me.
AUDIENCE: That definitely would be a little better.
PROFESSOR: OK, let me allow you to think about that, and
that would be pushing it back one week and maybe it'd be--
we'll see how much material on tensors we cover.
So let's table it for now but let's say, tentatively, let's
consider seriously moving it to the 15th.
OK?
But you guys have final say if there's a
preference that you have.
OK let's talk about properties in first a general and rather
philosophic way.
When we discuss properties we very commonly lump together a
set of behaviors which have some common phenomenology.
So for example, we will talk about mechanical properties.
And that's a basket in which we place many different sorts
of behavior.
We place things like fracture toughness, yield strength,
elastic constants, a whole variety of things involving
strength deformation and so on.
Or another thing we very often lump together in a basket is
something called electrical properties.
And we talk about electrical conductivity, ionic
conductivity, electronic conductivity.
We talk about dielectric constants, we talk about
permittivity and permeability, magnetic properties--
a whole bunch of other classes of behavior, which have some
sort of root or description in terms of electromagnetism.
There are though a lot of strange aspects to properties.
There are some properties that are determined for material
which no longer exists when you're through
determining the property.
You have a sample you want to know what the
yield strength is.
So you put the gradually increasing load on it until
finally, POP, it breaks.
And now you know what the yield strength is for a
material that no longer exists.
Another example is an old one.
Back before the days of X-ray diffraction when people would
try to characterize material, anything that is very
intensely colored usually looks black, and that's not a
very definitive property.
The color of something is a very prominent characteristic.
So for mineralogists, in particular, anybody going out
into the hills and wanting to be prepared to determine what
a particular rock that they tripped over was had something
on a rope around their neck called a streak plate.
And what the streak plate was was a little
rectangle of porcelain.
And they would take this black mineral and rub it on this
piece of porcelain and it would leave a mark--
that was called the streak.
And what this was was actually little bunch of fragments that
rubbed off on to the porcelain because the porcelain was
white and the fragments were very small.
A black piece of rock could give you a streak that was
brown or green or deep orange.
And so you really were determining the color of the
material when it was in a form finely divided enough that
light could pass through it.
So the streak test was a very important diagnostic for
determining minerals.
There are some properties that we refer to
as structure sensitive.
In the sense that their value--
conductivity is a good example--
the value of ionic conductivity depends on the
impurities and point defects that are
present in the material.
So to say that the ionic conductivity of a certain
material is such and such, you have to specify the purity and
perfection of the material or the property's meaningless.
There are some properties that are not even single-valued.
And the best example there are dielectric or magnetic
properties.
If you plot the magnetization of a material, the magnetic
moment per unit volume as a function of the applied
magnetic field B, the material is initially non-magnetized.
When you apply B, you get a magnetic moment per unit
volume that eventually saturates.
And then if you remove the magnetic field, the material
keeps some of its magnetization.
It doesn't go to zero when you reduce the field to zero.
You have to reverse the field to get the property to go to
zero and then magnetize it in another direction.
And if you continue to cycle the magnetic field, you get a
behavior like this.
This is hysteresis, and this type of behavior is called
generally hysteretic.
Same thing would be a relation between the displacement
vector and an applied electric field.
But clearly you can have any value of the magnetization in
this range.
And if you relate the magnetization as a function of
the applied field, you can get any value of the
proportionality constant you like between a negative
maximum and a positive maximum including zero.
So there's a property that's not even single-valued.
Depends on the past history of the sample.
And then there are even more peculiar properties.
There are properties that are--
we call them composite properties and very often
qualitative.
What do they mean by a composite property.
Let me give you one example--
The property fuzzy.
If I say fuzzy, you know exactly what I mean.
It means something that has a diffuse reflectivity,
something that has a surface texture that's yielding.
It's not like rubbing a wire brush.
It's soft and giving, a whole collection of different
properties.
But yet when I say fuzzy, you know exactly what I mean.
Let me not be so facetious as talking
about a fuzzy property.
People such as ceramists or powder metallurgists very
often will take a powder of a material and consolidate it by
compacting it and heating it, very often subject to a
compacting stress.
And when you do that little necks grow between the
particles and they hold together and
the material is centric.
So you refer to a property of a powder as being centerable
or the centerability of a powder.
You know exactly what somebody means by that.
But what does it depend on?
It depends on the surface energy, it depends on vapor
pressure, depends on bulk diffusion coefficients.
It depends on surface diffusion coefficients.
And all of these things have to be just right to make the
powder something that easily densifies upon heating in the
application or not of pressure.
So you know exactly what I mean by centerability, but it
is a very complex property.
And was one that really was not understood until probably
the late 1950s.
Until then, it was an art that was entirely empirical.
And it was somebody here at MIT, a fellow named Robert
Coble, who developed a theory of centering that was the
first really workable theory that described densification
by heating and compaction.
OK, composite property that depends on many different
individual properties of the material.
OK, what we are going to examine here are equilibrium
properties that can be rigorously defined and
measurable.
So we're going to leave out of the picture things like fuzzy
and centerability.
And let me give a very nice, obscure definition of what I
mean by a property.
And what I mean by a property, in terms of a formal
statement, is the response of a material to a specific
change in a given set of conditions.
So the response of the material to a specific change
in a set of conditions that relates independent properties
and dependent properties for a particular process.
So I'll state that again because I think
it's terribly elegant.
So a property is the response of the material to a specific
change in a given set of conditions that relates
independent and dependent quantities in
a particular process.
Now there are a lot of properties that have their
roots solidly embedded in thermodynamics.
So let me give you a few of those.
What we'll talk about when we specify a property is
something that we will refer to as a displacement.
We'll talk about a generalized displacement in response to a
generalized force.
So some of the thermodynamic quantities that are related in
this fashion--
if we list some forces and some displacements, the thing
that happens as a result of that stimulus that's applied
to the material.
Temperature can be regarded as a force that results in all
sorts of processes as a result--
thermal energy flow, thermal expansion,
all sorts of things.
But one of the things that will happen in response to a
temperature change
thermodynamically is an entropy.
So you can view entropy as a generalized displacement
resulting from the application of temperature as a
generalized force.
Another example is electric field and the thing that
happens there, and electromagnetism, is the
quantity, D, which is displacement.
Still another example, stress, and the result of applying a
stress, among other things, is a strain.
What is special about these forces in this displacement is
that their product is, in each of these cases, energy, the
change in internal energy of the particular body.
And when that is the case, these are said to be
conjugate--
a conjugate force and displacement.
And there are other examples that one can come up .with.
Now to talk about a specific set of properties.
Very often, and this crept into the earlier discussion,
very often the thing that we do to a material, as a
generalized force, is a vector.
So very often the thing that we do to the material, apply
an electric field, apply a magnetic field, apply a
temperature gradient, apply a tensile stress, it has the
characteristics of a vector, magnitude and direction.
And in many cases, the thing that happens as a generalized
displacement, we may call this in general,
q, is also a vector.
An example is if we apply an electric field as a
generalized force, one thing that might happen is a current
flow, which is also a vector.
And we are accustomed to writing, q, the generalized
displacement, as a proportionality constant,
sigma, which is in the case of electric field and current
flow, the electrical conductivity.
Let me call it, in general terms, proportionality
constant, a, times the applied vector, p.
Is this something we would like to stick with as a
general relation?
Well let me submit that writing an expression of this
form makes an inherent assumption.
Namely that this will be true for small, and we'll have to
define for each property what we mean by a
small, applied vector.
Let me give you an example.
If p were--
the applied vector was electric field, and the
resulting vector was current flow, and the relation between
those is the conductivity--
a relation of this sort says that if you double the field,
you double the conductivity.
You triple the field, you triple the conductivity.
Obviously this can't go on indefinitely because all a
sudden, POW, dielectric breakdown.
The sample evaporates at the smoke.
And again, you have the property of a material which
no longer exists.
So a lot of properties, we inherently assume that the
applied vector is small in order to write something in an
expression of this form.
So for conductivity, dielectric breakdown is going
to destroy the nice linear relation between current flow
and electric field.
That's not always the case.
Let me give you an example of another property.
And this is a property, magnetic susceptibility, which
relates the magnetization, which is magnetic moment per
unit volume, and relates that to an applied electric field--
an applied magnetic field, H. And the proportionality
constant, represented by a Greek chi, is the magnetic
susceptibility.
So let me give you a specific example.
Suppose we have a chunk of glass and the glass contains a
dilute concentration of iron.
And iron carries a permanent magnetic moment.
And the reason I want to make it a dilute concentration of
iron is I don't want these magnetic moments close enough
that they can interact with one another.
I want this to, therefore, be a dilute system.
So we have different iron atoms in this.
Each iron atom has a magnetic moment.
And then we put this in a magnetic field, H. And the
magnetic field acts on each of these little magnetic moments
just as though it were a compass needle.
And so it will try to take each of these moments and drag
it into coincidence with the magnetic field.
But at a finite temperature, temperatures making these
magnetic moments jiggle around, so at a finite
temperature, the magnetic moments will just not simply
zing into coincidence with the magnetic field.
You'll have to increase the magnetic
field to make it larger.
When that happens, more and more of the magnetic moments
will come into alignment.
And if you put on a really strong magnetic field, then
every single magnetic moment will be dragged into exact
alignment with the magnetic field and
the system has saturated.
There's no way you could squeeze further magnetic
moment per unit volume out of it.
So again, you would expect, for this particular property,
magnetic susceptibility of a material, if you plot it as a
function of the applied magnetic field--
linear maybe be at low applied fields--
but eventually if you make the field strong enough, the
system is going to saturate.
And then again, no longer will the direction of the
magnetization, and that magnetic moment, be parallel
to H. But the property becomes nonlinear if the applied
vector is strong enough.
This is an example of a property where you would not
go wrong at all by stating direct proportionality.
For ordering to occur, temperature and applied field
go hand in hand.
Increased temperature tends to create more disorder.
Increased magnetic field tends to align the moments.
In the days when MIT had a national magnet laboratory,
the magnet laboratory held the record for the strongest
magnetic field ever produced artificially by man.
And if you took that magnetic field and applied it to this
system of dilute iron in a glass, you would have to lower
the temperature of the sample to about 3 Kelvin before you
would begin to see saturation.
So even the strongest field that you could produce in a
laboratory environment would not succeed in producing
non-linearity until you cooled the sample down almost to
absolute zero.
So here would be a case where under any practical
consideration whatsoever, assumption of strict
proportionality would be right on the money.
You'd be absolutely correct.
But there's another assumption built into this statement.
P, the generalized force, is a vector in the cases we're
discussing now, and q is also a vector.
So when we write an expression of this form you're making
another assumption.
And that is that the vector displacement that results is
always exactly parallel to the vector that you apply.
Do I make a big deal out of this?
Isn't that always going to be the case?
I mean whoever heard of taking a piece of metal and putting
an electric field on it in this direction and having the
current run off in this direction.
It's absurd.
Or maybe it isn't so absurd.
So let's think of some of the atomistics of this process.
Now, since I know I'm among friends, I will not hesitate
to display my ignorance, total
ignorance, of polymer chemistry.
So suppose we had a piece of polymer.
That's what a polymer molecule looks like.
It's a more or less linear molecule, and so these might,
in a very highly ordered polymer, be chains that lined
up like this.
And suppose we now put an electric field on this polymer
and asked how the current will flow.
Again, it would not be absurd to say that an electron
sitting on this polymer chain in response to this field
would be constrained only to flow in the direction of the
chain and would find it rather difficult to hop from one
chain to another.
So maybe, just maybe, we could apply an electric field to a
polymer and it would have a flow in this direction, that
would be pointing in this direction, and the current
flow, J, would be in that direction.
So should we maybe rethink this idea of the direction of
the generalized displacement being not parallel to the
direction of the generalized force.
OK, these are hypothetical examples.
Let me now give you an example of a real property for a real
material, for which the thing that happens, the vector that
happens, is decidedly not parallel to the direction of
the applied vector.
OK, thermal conductivity is something that relates a heat
flux, usually represented by the symbol K, and relates that
to a temperature gradient, dt dx, which is a vector.
The thing that gives rise to thermal conductivity can be
either propagation of radiation as in propagation of
light traveling through a transparent material.
But the other mechanism for thermal conductivity is modes
of lattice vibration.
When a material is hot, the atoms are jiggling around and
you can make the displacement of an individual atom be
represented by a sum of waves moving in all different
directions with a variety of wavelengths in a variety of
amplitudes.
Take all those waves, add them together at a particular time,
and you get the displacement of the atom.
Propagation of heat by this mechanism, by modes of thermal
vibration, can be very, very anisotropic and probably the
best example of this is--
get a single crystal of graphite.
And have the layers, the graphite sheets, which are
hexagonal rings in which each carbon has three neighbors,
and have the sheets be parallel to the surface of an
extended two-dimensional slab.
Now we can't do that.
Single crystals of graphite don't occur
in sizes like that.
But what you can do is make a material called pyrolytic
graphite that you make by having a reaction in the vapor
phase and having the soot that's formed settle down onto
a substrate.
And what happens is you nucleate a graphite crystal in
one orientation, and that grows preferentially with its
layers parallel to the surface on which it's nucleated.
And these nuclei occur at random.
So you get a bunch of single crystals of graphite, which
all have their layers, their three coordinated layers at
parallel, but they are oriented at
random about their c-axis.
And that's a material that's very easy to prepare.
And it's called pyrolytic graphite, and it has
interesting applications and properties.
OK, now I'm going to suggest an experiment to you and I'll
caution you, please do not try this at home.
Get yourself a piece of pyrolytic graphite.
Put a Bunsen burner underneath it.
Play the Bunsen burner on the bottom of the sheet of
pyrolytic graphite.
And to determine temperature, we'll take a ball of cotton
and put it directly above the flame.
And then take another ball of cotton and put it down at the
end of the sheet.
If you do that and bring up the Bunsen burner, this piece
of cotton will sit there and sit there and sit
there and sit there.
And this piece of cotton will instantly burst into flame.
So here's a case where dt dx, the thermal gradient, points
in this direction.
And the heat flow goes off in a direction at right angles to
the temperature gradient, almost
exactly at right angles.
It turns out that the thermal conductivity in this
direction, the value of K in the--
I want to put crystallographic directions on this.
But K parallel to the layers is equal to 10 to the third
times K, perpendicular to the layers.
So there's an anisotropy of the property that amounts to a
factor of 1,000.
Very dramatic.
So what I'm leading up is that in a general relation between
a generalized displacement, p, and a generalized force--
I'm sorry, I'm using q-- p and q--
we just can't simply put some constant in front.
Because that assumes inherently that the direction
of what happens, the vector that happens, is exactly
parallel to the direction of the applied vector and that
generally is not true.
The difference may be small but it has to be taken into
consideration.
All right, let me now suggest a way of patching this up.
What I'm going to do is assume--
and it is an assumption--
we're going to assume that each component of the vector
that happens is given by--
in fancy terms, the vector that results as the
generalized displacement is given by a linear combination
of every component of the vector that's applied.
And that's what we've defined as the generalized force.
So how would we express this analytically?
I am going to use as my coordinate system, not x, y,
and z as we usually do, I'm going to call
it x1, x2, and x3.
The reason is we'll see some unique properties of these
indices that are very useful algebraically.
So I'm going to now take my applied vector, p, and I'm
going to assume that it has three components, p1, p2, and
p3 along x1, x2, x3, respectively.
My coordinate system, although I did not state it explicitly,
is going to be a Cartesian coordinate system, not a
crystallographic coordinate system.
So what I'm proposing here is that we take each component of
the resulting vector q--
let's say q1--
and we'll assume that that's given by a linear combination
of all three components of the vector p.
So it'll be some number times p1 plus some number times p2
plus some number times p3.
And those numbers in general will be different.
So let's say I call the coefficient a, and now I'm
going to define a convention that will stay with us.
I'm going to define each of these coefficients, a, in
terms of the index of the component of the generalized
displacement which is being computed, and the coefficient
modifies the component of the generalized force for that
particular term.
So I'm going to call this a11, where the 1 goes with this and
the 1 goes with this.
I'm going to call this a12, so that the 1 again says it's a
contribution to q1.
The 2 says that this term modifies p2. .
And this similarly would be a13.
For the term component of the generalized displacement, q2,
I'll use the coefficients a21 times p1 plus a22 times p2,
plus 23 times p3.
And q3 similarly will be a31 times p1, plus a32 times p2
plus a33 times p3.
So I've got three simultaneous equations then, one for each
component of the vector that results, the generalized
displacement.
So I can sum up this set of three equations by saying that
the i-th component of q, where this is some particular
component, q1, q2, or q3, is given by the sum over j from 1
to 3 of aij times p sub j.
So I've got ai1 times p1, ai2 times p2 plus ai3 times p3.
So this is in a nice, compact little nugget the expression
that we are assuming will apply for all sorts of
physical properties in which this is a vector
and this is a vector--
electrical conductivity, magnetic susceptibility,
thermal conductivity, and so on.
OK, this is a bad point to introduce something as hairy
as what comes next.
But I've got one minute left, and I think I can do it.
I'm going to introduce something called the Einstein
convention after old Albert Einstein himself.
I'm sorry, but nobody said everything this
term had to be easy.
So let me introduce, if you're ready for it, the Einstein
convention.
Old Albert, I think, was just as lazy as anybody else.
And he said it is going to be a bloody pain in the butt--
I don't know if he put it exactly this way.
It's going to be a pain in the butt to write this summation
every time we want to combine three terms in a linear
combination.
So the Einstein convention is let us throw out the sigma and
write this expression just as qi times aij p sub j.
And whenever we see a subscript repeated, summation
over repeated subscript is implied.
OK so that's the Einstein convention.
It will save us the trouble of writing in a lot
of summation signs.
But I would caution you that this convention, compact and
convenient as it is, can define some polynomials that
are nightmares.
So let me give you one example of this, and this is actually
a physical property.
Let me say that Cijkl is given by ai capital I aj capital J
ak capital K, al capital L times let's say D, capital I,
capital J, capital K, capital L. That actually, believe it
or not, means something physically and we're going to
get to that in due course.
But what this is, taking the Einstein convention into
account, this is a quadruple summation over capital I,
capital J, capital K, and capital L. These are very
often referred to in this business as dummy indices,
meaning that they don't really mean anything physically.
They're just indices of summation.
I would caution you that this is a term
that does not permute.
If I say those are dummy indices, it means one thing.
If I say those are indices, dummy, it means something
completely different, and you're apt to get
a poke in the nose.
OK so these are dummy indices.
Now, what does this represent?
I won't say what it represents physically, but this consists
of terms in four variables times a coefficient.
So there are five quantities in each term.
If I sum over capital I, J, and K from 1 to 3, there are
going to be 81 such terms, each with five
elements in each term.
So it's going to be on the order of 405 terms in this
summation, and we've collapsed the whole thing down, 405
terms in this nice summation.
So it is a great facility for writing down expressions
explicitly, but these terms can hide a terribly, terribly
complex polynomial.
All right, I think that's a good place to quit.
It continues to amaze me, though, how some absolutely
trivial convention, when first proposed by a great man, will
carry that man's name no matter how stupid it is.
So this is called the Einstein convention by everybody who
works in this field.
Another one, just briefly to finish up, when you discuss
dislocations, you talk about a circuit of steps around the
dislocation.
That's called a Burgers circuit.
And if you do the same circuit in a corresponding piece of
material that doesn't have a dislocation, if this circuit
closes, this circuit fails to close by something that's
called the Burgers vector.
Every book on dislocation theory says, Shockley called
this good material.
I'm sorry, Shockley called this good material, and
Shockley called this bad material.
Big deal--
good material, bad material because it's got a
dislocation in it.
But that is identified with Shockley's name in every
discussion of dislocation.
I'm just jealous because nobody has ever said according
to me, such and such is--
all right, see you later in the week
for more great things.