In the previous video, we talked a little viscoelasticity
and specifically looked at this case
where you put a heavy table onto a carpeted floor.
That's a carpet.
So we've put a heavy table into a carpeted floor
and then remove it.
And you see a little divot.
You see this flat spot where the leg of the table was.
If you put it there for a short period of time,
you see the divot for less, less pronounced divot
and it doesn't last as long.
If table's there for a long time, like a year or something,
the divot might actually last for a long time,
even after you move the table away.
And so what we're observing in this case here
is you've applied a fixed load, that is,
the force doesn't change with time.
But over time, what actually happened
is-- it's difficult for me to sketch it,
but the table actually with time-- I mean,
if you left it there for a couple years,
the table actually would have pushed into the carpet
even further.
So in this example, we have a fixed load.
And we observed the strain.
So that is a nice example to start with for describing
visocelasticity.
But one way, at least, that it's often quantified
is by doing this, by applying a fixed strain,
epsilon 0, and then observe the stress decreasing
as a function of time.
So that will be sigma in parentheses,
t as a function of time.
That's decreasing or when fibres elongate.
We could also say it's relaxing.
The stress is relaxing as a function of time, with time.
A little more difficult to think of everyday examples
where you might see that.
But one example would be perhaps tuning a guitar.
And if you tune in the tension in the string,
the length of the guitar doesn't change with time.
But with enough time, the stress will
decrease so that the guitar string, a nylon
string, at least, would go out of tune, even
without any kind of a humidity or temperature
change in the guitar.
Anyway, if you think of some other examples, let me know.
But in a lab, it's quite easy to do.
You can put it into a tensile testing machine
and stop the strain.
So the strain is fixed at a certain value
that you determine.
And all you need to do is use the load cell
to give you a load and you can calculate stress.
So it's quite easy to do.
And in fact, if you do that for most polymers,
and we were to plot the-- I didn't introduce that.
Let me introduce this.
We want to describe, essentially,
the stress decreasing.
So we want to see-- in fact, what I'll do
is I'm going to go like this.
I'm going to put stress function of time here vs time.
And what would you expect to happen?
So you apply a fixed load.
You elongate a sample.
You stretch it out.
What might it look like?
So you take a sample like this and then you stretch it
out by applying some load.
And you fix, though, this final length,
so that you've fixed the strain at some value,
delta l over l0 is fixed.
You hold it there.
And then all you need to do is observe
the stress that is needed to maintain that length.
That's the stress.
So what's that going to look like with time?
Well, it's stress relaxation.
Polymer, you remember, is made up of all these little fibres,
these-- not fibres, correction-- molecules.
And with enough time, they can overcome
those weak secondary interactions
and move past each other.
So the stress, which we would expect
some kind of a downward trend in the stress.
So the stress decreases with time.
Now what we can do, we define this new term.
It's a special number, so we're going
to call it a modulus, because that's all a modulus means,
special number.
So let's define a special number that describes
relaxation of a polymer.
So let's call it the relaxation modulus.
So that's going to be E with a subscript R.
And that's just defined as that time dependent stress divided
by the strain that we impose on the sample.
So that's relaxation modulus.
So now what I can do is actually,
I could go back over to this little-- so erase that.
What we can actually do is replace that stress
as a function of time with this new term, relaxation modulus.
So we're going to put in here the relaxation modulus
with a function of time.
Now technically, we can be careful here.
This relaxation modulus covers many orders of magnitude-- 10,
100, 1000, 10,000.
So it covers a huge range of data.
Now, if we want to cover a lot of data one way
that we can deal with that is to plot on a logarithmic axis
or take the log of the value so that now
we're just working in orders of magnitude steps.
We're compressing the data, if you will.
Do the same thing to the time to cover lots of range of time.
And take a look at the curves and we'll
see that we get a decrease in the essential stress, function
of time.
But the other thing I want to show you
is, if you give the system more energy, that
is you apply a higher temperature-- so T1 less
than T2 less than-- I'll write T3
in a moment-- you find that the slope decreases even more
rapidly as you get to higher and higher temperatures.
And hopefully again, that makes some sense when you consider
the structure of a polymer.
There's more kinetic energy.
It's going to be more likely to overcome
all these little weak, secondary interactions between chains
with that increasing thermal energy at higher temperatures.
So that's all well and good, but what
if we'd like to look at relaxation
modulus versus temperature?
In fact, this is quite interesting.
What we could do is just define a time, say-- some time,
I don't know, 10 seconds perhaps--
and fix that time and then observe
what the stresses relax to you for each of the temperatures.
And then we can plot now relaxation modulus
versus temperature.
So it's quite interesting.
And if we do that, again, we want
to plot a large range of data for relaxation modulus.
So take a log of that.
And then we'll plot it now versus temperature.
But temperatures are not covering
a huge range, because these are terrestrial temperatures,
temperatures close to room temperature-- temperatures
that humans are interacting with,
or close to that, at least-- Freezing to a couple
degrees C or something like that.
And we're going to plot now the relaxation modulus
after 10 seconds for these different temperatures.
If you did that and you did a number of tests,
you'd find for a lot of polymers,
it's something kind of like this.
Now technically, this is actually
a semi-crystalling polymer.
We'll get to that in a moment.
But at the low temperatures-- here, this is low temperature.
This is high.
At the low temperatures, as you might expect,
polymers are kind of brittle.
There's not enough kinetic energy or thermal energy
in the vibrations of these molecules
to overcome these secondary interactions between chains.
And so the polymer molecules essentially
are locked in place.
And so it's kind of hard and brittle.
And so we use this common term.
It's a bit of a misuse of the term,
really, but glassy in this case here,
referring to the mechanical properties, not
the microstructure.
Because you may know that-- mechanical properties only.
That's what we're referring to, not the structure.
Because glassy, in every strict sense in material science
refers to the absence of crystal structure, no long range order.
But in this case, glassy is not used in that sense.
It's used to refer to hard and brittle.
Molecules can't slide past each other, so they're held in place
and then it's hard and brittle.
And then out here, in this region, the polymer--
it's a high temperature.
It's melted.
And so we describe this region as
viscous flow, flow of a liquid.
And in that region, the molecules
are all free to move past one another.
But there's some other interesting things
that happen in between that.
And so around here, we define this melting temperature.
And that's pretty easy to wrap your head around, I think.
It becomes a liquid.
And that's fine.
But the interesting thing is that we also
have to define another temperature, which is the glass
transition temperature.
And to really fully understand this,
we need a little bit of a better model
for the microstructure of polymer than the one
that I've shown you there.
So this polymer, which I said is a semi-crystalline polymer,
is the case for a lot of polymers-- not all, but many.
And their microstructure looks something like this.
There's some regions of crystallinity,
where the polymer molecules are organised.
And there's other regions-- this is
a cartoon sketch of the polymer microstructure,
polymer molecules.
So there are some disorganised regions
and then there's some organised regions.
So it's a nice little cartoon sketch
of polymer microstructure.
And so we've got these disorganised regions,
or amorphous, amorphous meaning lack of crystal
structure, lack of a long range order.
And then we've got these other ones, regions,
that are highly organised.
And so name for that is crystalline.
So there's both crystalline amorphous regions.
And again, if we think about these secondary interactions
that are established between molecules,
well, in the crystalline religion,
the density is actually higher.
I mean, a great example of this is high density polyethylene--
just for example, high density polyethylene.
The high density is achieved by designing the polymer structure
such that it more readily crystallises and choose
a higher density-- it's got higher temperature
resistance-- versus low density polyethylene.
Chemistry is the same, but the microstructure is different.
I don't know why I wrote a second line underneath the E.
But there's less crystallinity in low density polyethylene.
And that's because-- I'm sorry-- and that leads to the density
difference.
So this is higher density.
And as a product of that, the molecules are closer together,
of course, and so there's more opportunities
to establish secondary interactions
in the crystalline domain.
And that's what those little red lines are supposed to be.
They're supposed to be secondary interactions,
versus in the amorphous region-- well,
there's fewer places where the molecules are relatively close
together.
And so there's fewer of those secondary interactions
in the amorphous regions-- higher density here
for crystalline and more numerous secondary bonds
or interactions.
And by contrast, the amorphous has
fewer secondary interactions.
So now what we can do is we'll scroll back up
here to our relaxation modulus curve
and think about what's happening upon heating this.
So we start off at low temperature
and all these molecules here, the crystalline and amorphous,
they're all locked in place.
And we continue to increase the temperature
so we're giving more thermal energy, more
movement of the molecules.
And the first thing that we're going
to find happens is the molecules in the amorphous region
get enough energy to overcome the secondary interactions
between molecules.
And that's what happened at the TG,
at the glass changes in temperature.
And we see a rapid decrease in the relaxation modulus.
So upon heating, then the first thing that happens
is we free up these amorphous regions.
Amorphous regions have enough energy
to overcome or break the secondary interactions
between the molecules.
And that's at what point?
At the TG, the glass transition temperature.
And then continue to heat it up, continue to heat up.
And then you hit the TM.
And the TM, the melting temperature,
is where you've got enough kinetic energy,
or thermal energy, for the molecules
to overcome all those little secondary interactions
in the crystalline domains.
So similarly, upon heating at the TM,
the same thing happens in the crystalline regions
so they can overcome the secondary interactions
between molecules.
Excellent.
So now we understand why there is
a need with polymers for both a TG glass transition and a TM
melting temperature.