5-1 Volume and shear viscosities, heat capacity and glass transition. Today we're going to delve a little bit into the details of the relaxation process. And we're going to discover the main attributes of relaxation in silicate melts, one of those, which we'll see very quickly at the outset, is the so called equivalence, which means that no matter how the relaxation time of the system is determined, which properties we use or which experimental apparatus, if we're careful and we compare the results, in terms of the timescales of the experiment, then we discover that we get the same value, regardless of property investigated. Very important at the outset for amorphous materials is to point out that when we're dealing with stress-strain relationships in such systems, free elasticity or for viscosity, that we should be careful to observe that there are two tensorial components of the deformation of the system. There is a volume component, which essentially relates to changes in density, as a [reaction] reaction to changes in the volumetric stress, or the pressure. And there is a shear component which reacts to changes in the shear stress in the system. Now, luckily, numerically, for purposes of evaluating the relaxation time, we will discover in a moment that these have the same value. In other words, the viscosity and the elasticity, either in a shear or a volume sense, are very, very similar in magnitude, which means that it makes it relatively easy to compare the relaxation times, regardless of these two tensorial components. We can define a longitudinal value of either the elasticity or of the viscosity simply by adding the volume component to the shear component times four thirds. If we do a test of the relative values of the volume and the shear viscosity, the viscous response to changes in pressure, for the volume, then the changes in shear stress, for the shear viscosity. Then, we can see very quickly by comparing the longitudinal value of the viscosity, which is measure using longitudinal wave propagation, the damping of longitudinal waves and silicate liquids on the vertical axis here, and the shear viscosity along the horizontal axis, which is determined by common techniques for determining shear viscosity in the laboratory involving very large strains, that the one to one relationship that we would expect, in other words volume viscosity equal to shear viscosity, would obtain such that seven thirds of the longitudinal viscosity is equal to the shear viscosity, such that seven thirds of the shear viscosity is equal to the longitudinal viscosity.
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5-1 Volume and shear viscosities, heat capacity and glass transition.
Today we're going to delve a little bit into the details of the relaxation process. And we're going to discover the main attributes of relaxation in silicate melts, one of those, which we'll see very quickly at the outset, is the so called equivalence, which means that no matter how the relaxation time of the system is determined, which properties we use or which experimental apparatus, if we're careful and we compare the results, in terms of the timescales of the experiment, then we discover that we get the same value, regardless of property investigated.
Very important at the outset for amorphous materials is to point out that when we're dealing with stress-strain relationships in such systems, free elasticity or for viscosity, that we should be careful to observe that there are two tensorial components of the deformation of the system.
There is a volume component, which essentially relates to changes in density, as a [reaction] reaction to changes in the volumetric stress, or the pressure.
And there is a shear component which reacts to changes in the shear stress in the system.
Now, luckily, numerically, for purposes of evaluating the relaxation time, we will discover in a moment that these have the same value.
In other words, the viscosity and the elasticity, either in a shear or a volume sense, are very, very similar in magnitude, which means that it makes it relatively easy to compare the relaxation times, regardless of these two tensorial components.
We can define a longitudinal value of either the elasticity or of the viscosity simply by adding the volume component to the shear component times four thirds.
If we do a test of the relative values of the volume and the shear viscosity, the viscous response to changes in pressure, for the volume, then the changes in shear stress, for the shear viscosity.
Then, we can see very quickly by comparing the longitudinal value of the viscosity, which is measure using longitudinal wave propagation, the damping of longitudinal waves and silicate liquids on the vertical axis here, and the shear viscosity along the horizontal axis, which is determined by common techniques for determining shear viscosity in the laboratory involving very large strains, that the one to one relationship that we would expect, in other words volume viscosity equal to shear viscosity, would obtain such that seven thirds of the longitudinal viscosity is equal to the shear viscosity, such that seven thirds of the shear viscosity is equal to the longitudinal viscosity.
And what you see here isn't a comparison of the experimental sets of data from the longitudinal and the shear viscosity and you see that they cluster around the one to one line.
So shear viscosity is approximately equal to volume viscosity.
And we can go forward now without being overly concerned about the relative contributions of shear and volume in our stress-strain relationships.
On the next graph, you see a plot of the two main thermal analysis determinations that are made on silicate liquids plotted one over the other.
On the top half of the diagram, you see a plot of the heat capacity versus the temperature on the horizontal axis.
And below it, you see a plot of the thermal expansivity of the liquid versus the same temperature scale.
And you will probably note immediately that there is a difference between these two curves.
We see, for example, in the case of the heat capacity curve on the left-hand side below the peak, below the glass transition, that there is a slight increase in the value of the heat capacity in the glassy state, whereas in the case of thermal expansion, we see no significant increase on the left hand side on the bottom.
You will also see that on the far right the curves behave differently.
You see that the heat capacity curve appears to be coming to an equilibrium value, on the right hand side, beyond the peak, where it hits the metastable equilibrium liquid state, whereas the more ragged, more noisy thermal expansion curve appears to fall and fall and fall and not reach equilibrium.
This particular attribute above the glass transition temperature is an experimental artifact of the way that thermal expansion is measured in these systems.
These are free standing cylinders, which, as they reach the glass transition and begin to have the ability to flow viscously and to relax, are no longer capable of yielding their absolute value of thermal expansion using this technique.
And, therefore, at the extreme far right of the expansivity curve, the results are falsified by this artifact.
Doesn't need to concern us at the moment, as you might have suspected.
And what we see in the middle is the most important point.
We see the peak value of the thermal expansion on the bottom and the peak value of the Cp, the heat capacity on the top, at precisely as accurately as we can measure it at exactly the same value of temperature.
And this is the first demonstration that I give you of the so-called equivalence of this glass transition.
These are two completely different experiments.
In the case of the heat capacity experiments, when measuring heat fluxes, as we heat up a sample and see how much it's going to cost us, in terms of the heat which goes in.
And then, the second case, we're measuring length of a standing cylinder, in this case, of a glass initially, which is being heated up and crossed the glass transition, and we're measuring the height of the cylinder in a sub-micron accuracy with a variable displacement transducer.
So these three experiments would normally be found in very different labs in any institution, and yet they give us the same number, the same result, the same peak value.
And, therefore, they both identify a lot is appearing to be a unique value of that temperature.
Now, very important is in order for these two curves to be comparable, in order for us to observe that peak occurring at the same position on both cases, we must have had identical materials, of course, in both experiments that doesn't only involve their chemistry, it also involves their thermal history.
In other words, the glass which we use at room temperature to initiate the thermal run in both of these cases, is the same chemistry, the same mix if you will, the same melt.
And it has been also prepared in the same manner by cooling it to the glassy state.
And then, when we reheat this, a structurally identical material in both of these experiments, we're allowed to observe the fact that the glass transition peak expresses itself similarly in both cases.
We'll see many more examples of equivalence.
But this first one can be brought home by normalizing the comparison of those two curves that you've seen.
What you see here now is the same data, but it is plotted now as a normalized plot.
And you can see on the vertical axis, dimensionless units going from zero at the bottom, to a peak height, equal to 1.0 at the top.
And if I normalize both of them, what have I done?
I've taken the glassy value of the relative property of the derivative property, be it the heat capacity or be in the thermal expansivity, and I've clearly set that equal to zero.
That's why you see these two lines now not deviating from a value of zero until I begin to deviate from the glassy state.
And as I go to the peak, you see I've set, arbitrarily, the peak value of both of those curves to be equal to one and, therefore, they meet again at 1.0.
Most importantly, is they meet at the same temperature, and secondly, if you follow the trace of the curve, as you rise up from the glassy state to the peak value at the glass transition, you see that the two curves overlap and overlay each other quite well.
So it's showing that, actually, the path, in terms of the change in property in driving from a glassy to a liquid state, is also the same if I have the same chemistry.
And if I have the same thermal history of the material that I'm working with.
You do see that on the right hand side in the liquid state the two curves deviate from one another.
No question.
The curves are not similar when we reach the right hand side.
The reason we just mentioned, it is an experimental artifact in this case because the extreme right of the noisier curve here, which is the dV⁄dT curve, is flawed because the cylinders are beginning to viscously collapse.
And thermal expansion can't be measured using that technique so easily.
So it's an experimental detail.
And no, and in no way, impacts on the equivalence statement that we've made.
5-2 Glass transition, hydrous granitic systems.
Now look at this next plot.
It seems rather benign.
It's a plot of the glass transition temperature.
In this case, the glass transition temperature has been defined how?
It's been defined implicitly as a time scale.
Remember that I told you that we have a time-temperature relationship, which we call
the glass transition.
And in order to talk about a glass transition temperature, either everyone in the room
has to agree on a certain convention, implicitly, or it has to be stated explicitly.
The latter is perhaps better.
And so, what you see on the vertical axis here is an explicit statement about what glass
transition temperature am I speaking of.
I'm speaking of the one, which corresponds to a particular relaxation time, which, in
turn, corresponds to a particular shear viscosity, through the Maxwell relation as we've
seen previously.
And the shear viscosity that we've chosen is stated here very precisely at 10 to the 12.38
Pascal seconds.
Now if you look on the horizontal axis you see water content.
Now this is one of the first mentions that we make of significant water.
Water is the most common volatile substance in these melts.
We've mentioned that at the beginning of the course, and you can see here that the water
contents we're talking about are very, very substantial.
These are weight percents of water and they run up to many, many weight percent.
So it means that it is relevant in volcanic systems to concern ourselves with water
contents in silicate liquids, which run up to 3, 4, 5, 6 weight percent of water.
And, as we'll see later, the details of how these things oversaturate, when they're rising
to the surface before an eruption, is a large, large control on how volcanic eruptions
actually take place.
And one of the reasons that it's so sensitive that eruptive behavior really is sensitively
dependent on water content, is in essence given by this plot.
You see the glass transition temperature has a function of dissolved water content in the
melt.
And what you see is a very, very large vertical scale in temperature.
You see it running, in this case, it's temperature in degrees Kelvin over approximately
400 degrees, 400 degrees Kelvin or 400 degrees Celsius.
Now, that implies that we have started from a very very dry melt at the beginning at
zero weight percent of water.
In nature, nothing is completely dry, but nevertheless in a laboratory it's relatively easy
for us to bring the water contents down to a few ppm, which we can describe
approximately as dry.
And you can see the data points, which run from that dry value at the extreme upper left
of the diagram, where we're up near 1100 Kelvin, at the glass transition temperature.
That's a very high glass transition temperature.
And then, it runs down, it runs down fairly fast, fairly steeply, as we add the first couple
8% of water to the system.
And then, it begins to flatten out, such that, if I go beyond 5 weight percent of water in
this diagram, I will more or less have a very, very subtle drop in the further variation of
Tg.
Now, all of the data points that have been taken to produce this curve have come from
very different experiments.
And that's why I show it to you at this point in time.
This is a triumph, if you will, of the equivalence principle.
Because there are four different techniques from five different studies, which have been
plotted on this curve.
And as you see very clearly, they all lie fairly tightly on this curve.
There have been volumetric measurements of the relaxation of this system.
There have been measurements of the viscosity of this system, through the Maxwell
relationship to determine the relaxation time.
There have been measurements of the heat capacity to determine the peaks on the heat
capacity curve and, and we'll see more if it in a moment, there have actually been
measurements of the structural role of water through spectroscopy in these liquids,
which also can be used to give a reaction time or a relaxation time of spectroscopic
results of structure.
And all of these things: volume, enthalpy, viscosity and structure, completely different
experiments done in different labs by different groups around the world for different
purposes, not even for the purpose of determining Tg explicitly, when they are corrected
for the experimental timescales, the details of how long these experiments took to
perform, what temperature-time steps were involved, the details of this, when they are
made comparable, when you bring the timescales to a common value and you collapse
these points on a single curve as you see described here.
So two points here.
A demonstration that the equivalence runs across a wide range of techniques to
determine the glass transition temperature, four different properties, if you will.
And secondly, that water has an enormous effect on this value.
So water content in our samples is going to be critical for us to be able to parameterize
what is going on in relaxation time.
Here, a further example of this equivalence.
You see a very simple plot, where you have, vertically, the glass transition temperature,
which has been determined by picking the peak of calorimetric traces of a series of
glasses and you see on the horizontal diagram once again a determination of the glass
transition, which has been chosen based upon a constant value of viscosity.
And you see that these two curves more or less lie not only on a one to one slope, these
two sets of data, when I plot them versus each other lie on a one to one slope but you
also see that they, more or less, lie at equivalence.
And, that can actually be adjusted for any empirical comparison of calorimetric versus
viscous results for relaxation time, or for glass transition, simply by adjusting the
absolute value of viscosity to bring it into equivalence on the line.
So, to sum up, now, and to bring perhaps the last key feature into the equivalence
conversation, look at the following diagram.
You have to study it for a few moments, it has four axes.
It has two vertical axes and it has two horizontal axes.
The two horizontal axes are different temperatures expressed as the reciprocal or the
inverse of absolute temperature. an Arrhenius plot, as we call it.
And the vertical axes are, on the left-hand side. the (negative) or (the), the inverse minus
the log base 10 of the quench rate in degrees Celsius per second.
And you take that one and put it together with the bottom axis, which is the glass
transition temperature, as a reciprocal value.
And what you're looking at on the left and on the bottom is the variation in the glass
transition temperature observed, when I chance the cooling rate or the heating rate of
the observation.
So now we're bringing in a further parameter.
We're bringing in the heating rate during an experimental determination or the cooling
rate during a preparation or an experimental determination.
And we're plotting the glass transition that we observe as an absolute value of
temperature from the peak.
Okay.
Now, the upper scale, temperature, reciprocal temperature, and the right-hand scale,
which is the log of the viscosity, is simply a plot of the temperature dependence of the
viscosity of the system.
So, by overlaying these two curves, you see, clearly, that the temperature dependence of
the viscosity is identical in slope to the dependence of Tg, the glass transition
temperature on the heating-cooling rate of an experimental determination using
calorimetry.
So we simply slide those two plots into equivalence.
We overlay the two curves.
And it allows us to read from one scale on the left-hand side vertically, to the other scale
on the right-hand side.
And that gives us a relationship, which is expressed algebraically here.
Between the effective cooling rate or heating rate of an experiment to determine the
glass transition temperature and the viscosity, which is obtained at that glass transition
temperature.
Now that may not yet sound dramatic, but remember, if we have a relationship between
the viscosity of the glass transition temperature, and the cooling or heating rate to obtain
it, we also have the Maxwell relationship, which means we're one step away from, and
we take that step now.
We convert the shear viscosity into a relaxation time through Maxwell.
And if we performed this, then we have simply, linearly, a relationship now between a
very powerful relationship between the cooling effect of cooling rate or heating rate of
an experiment and an absolute value of the relaxation time available during the heating
or during the cooling.
This means that we will be able to plot heating and cooling rates on diagrams of
absolute relaxation time.
And we can compare scanning experiments, dynamic experiments, with static
experiments held at a particular temperature for a particular time.
And you see in the algebra presented here, that you simply transform from a logarithmic
sense, the relationship between the heating and cooling rate, and the viscosity at the
glass transition to one involving the relaxation time at the glass transition.