Deborah Number
Welcome to the second lecture on Introduction to Soft Matter. We were discussing in last class the various time scales or the natural time scales that were described in some of the indict texts and we have just discussed the for example Nimesa is the duration taken for a twinkling and a twinkling of an eye in this case then, there was a another time scale called the Truti and the Truti is the time taken by a sharp needle in piercing a lotus petal.
Many other time scales were also discussed obviously like days, lunar months, years, etc. and these are some of the Ancient time scales that were discussed in many different texts one of the foremost text being the Suryasiddhanta which was the text on astronomy, you can find many of these reference materials in book called the Ancient Indian Leaps into Mathematics which was edited by Yadav and Mohan.
The edited books has many different chapters contributed by other authors but, the very point that, we have discussing natural time scales was to ensure that we understand that the different phenomena which are associated with their own natural time scales.
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Along with this comes the idea that natural systems have a response time and in order to understand or observe that system we have to naturally be able to first and foremost appreciate what the time scale associated with that system is, a very important experiment in this context is called the very, it is a very famous experiment called the pitch drop experiment.
And in this manuscript which is called the pitch drop experiment was published in the European Journal of Physics, we have authors Edgeworth, Dolton and Parnell, and they describe this, we will go to the pdf in a moment but I just want to describe to you what the pitch drop experiment was.
So, in the foyer of the department of physics at the University of Queensland in Brisbane is an experiment to illustrate, for teaching purposes, the fluidity and the very high viscosity of pitch, set up in 1927 by Professor Thomas Parnell the first professor of physics there.
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Now, when we talk of something of pitch which is usually a buy product, a natural product, we think of it in terms of a as a solid with this very famous experiment challenged our view point and is often looked upon as a one of the pioneering experiments in trying to understand viscoelasticity.
So, here is, in the beginning I already read it out to you that we have at the department of physics a very high viscosity of pitch, which was professor Parnell wanted to illustrate and what he did is, he setup a very simple experiment you have a funnel shaped here, let just a minute to show that so, this is the apparatus and then you have pitch that is contained within it.
And what happens is much like water or much like a fluid you have the pitch starting to form droplets and then this falls down into this beaker now, obviously pitch has a very very high viscosity and because, it has a very high viscosity this is not going to be fast in human terms.
So, this experiment was set up and it was one of the very long running experiment and this experiment, they found that the drop, actually, a single drop usually comes out at around 8 to 9 years of time, so they went around and you are welcome to more read this paper at your leisure, which is again the pitch drop experiment by Edgeworth Dolton and Parnell, it was published in the European Journal of physics in 1984.
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And the other point thing I wanted to quickly point out is the record of the pitch drops or the so 1930 when the first it was started so 1938 the first drop fell, 1947 is when the second drop fell so almost a gap of a 9 years there then the third drop falls in 1954 almost a gap of 7 years here, then 54 to 62 so that is about 8 years I am not counting the months but, approximately that is the time scale. So, imagine you are a creature which would only view something at a interval of 6 years or 7 years.
For you with every, if you are twinkling of your alien creature so, let us imagine for a second that there is an alien creature where his twinkling of the eye, or their twinkling of the eye, is takes 6 to 7 years of human time for them the moment he blinks eye the next moment that drop has fallen, has fallen. In a sense for him the material would not seem like a solid it would seem more like a liquid quit which is, you have droplets coming out.
So, but for humans our twinkling or the blinking of an eye is much faster so for us it seems as a rather long time to wait for a drop to form and it is appropriate in the mean time if you are using pitch or a material like that in your laboratory for an experiment that is only lasting 10 minutes, it is appropriate to discuss that as a solid body. What this experiment or what the framework, what we are trying to understand is the idea that. Just a second I will open up the other document that I want to now discuss.
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So, I hope you sort of understand that, when you are discussing the material properties or the material response there are two things that are important and that is why I discussed the two issues, one was the time scale associated with that phenomena and then the other is the time of your observation, and these two become very very important in understanding the phenomena itself.
Now, this idea was proposed that this idea can lead to a non-dimensional important number and this was proposed by Reiner in a paper that was published in 1964 January in physics today. And in the beginning he is so, this is after dinner talk a so, there is a prelude to this manuscript.
And in the introduction he starts describing situation which sort of lead to this entire, to this discussion but, the important thing that here, he wants to say is that he was inspired by the story of Prophetess Deborah and in a famous song after the victory over the Philistines she sang. The mountains flowed before the Lord.
When over 300 years ago the bible was translated in to English the translators who had never heard of Heraclitus translated the passage as the mountains melted before the Lords, before the lord. But, Deborah knew of two things first, that the mountains flow and as everything flows. But, secondly that they flowed before the Lord and not before the man for the simple reason that man in his short lifetime cannot see them flowing. Whereas, the time observation of God is infinite.
So, using so from what I understand these lines are from the old testament of bible and this is the story from there, but he is using it is as an inspiration to go ahead and define a very very important number for us which is going to become the Deborah number and this Deborah number he defines as the time of relaxation divided by the time of observation.
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So, you are more than welcome to read this paper also your leisure but for us the important thing is, the definition of the Deborah number, which is defined right now, as time of relaxation divided by the time of observation. What is this time of relaxation? So, I will come to that but, for the time being I will suffice to say that we are going to use λ as a symbol for time of relaxation and for the time of observation we are going to use tobs.
There is another small change that we are going to do is that we are going to use the letters De to denote Deborah number rather than just D as the original definition so, let us go back to our writing pad and I will come back top this.
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So, we are discussing a number called the Deborah number, and we just said that we will write it as De this non-dimensional number and this is the ratio of two time scales and this is the time of relaxation and this is divided by the time of observation, so this should account for, it accounts for two things, one is response of materials and the whole issue of human observation or experimental observation.
Now this, so the question is what is the first question that should come to you is, what is lambda, so λ is a relaxation time scale and it is usually taken as the time scale of the viscoelastic material. I have not yet discussed what viscoelasticity is, so in some sense I am having to use that but, for the time we will come back to this obviously at a later course, at a later stage so this is usually taken as when we say the time scale of viscoelastic materials it usually implies the largest time constant describing molecular relaxations, or.
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Or, it could be some average time scale, some averaged value of response time, of response as determined by linear viscoelasticity or, it could also be taken as some time scale, some time constant that appears in a constitutive relationship. So, I must point out here that there are some terms that I am using that we have not yet come across.
So, λ is going to be something that we are just familiarizing ourselves with or we are going discuss it in more detail in the course so I have not discuss what linear viscoelasticity is I have not discuss what a constitutive relationship is, or what a molecular relaxations is okay we will come back to that.
But more importantly we had another term just quickly going back to this there is tobs so this term is also there, this term is slightly easier to understand and tobs can either be a time scale
associated with experimental measurements, or it can also be the time scale associated with the flow.
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If you take it as a time scale associated with the flow often people tend to write the Deborah number as
Now, the question is what is the time scale of flow? The best way to understand that is
through some examples but we will like to note one thing here that is the reciprocal of the
average shear rate can also be taken as a time scale of the flow. So, we note here that
reciprocal of the shear rate which we will write here as
can be taken as a flow time scale.
And this is usually done for steady state flows okay, so you are make another note here which means a usually done for steady state flows. But, if you do that, then the equation for Deborah number
is usually a another number that we use often in a viscoelasticity or soft matter which is although Weissenberg number (Wi) which is usually defined as
So, if you start using the shear rate or the reciprocal of the shear rate as the appropriate time scale then the number the Deborah number of that, in that case becomes identical with the Weissenberg number so, this is an important point. Now, so I will just quickly summarize this, or before we summarize there is another couple of things I would like to mention. So, now see, in the very very beginning, I will just go back to this definition.
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In the very beginning, we had said that soft matter or soft condensed matter is convenient term for materials in states of matter that are neither simple liquids nor crystalline solids. So, what happens to relaxation time scale in case of simple liquids? Now, we know, if you recall some of your undergraduate courses, we used to say that a liquid is something that cannot stand shear at all. Which means that it starts to response instantaneously.
So, if it response instantaneously, then this flow time scale or lambda becomes 0 in that case so, it becomes infinitely small which means that the material is responding almost immediately to a force and that is why it starting to flow. So, it is useful to understand that if, Deborah number tends to 0, we say that the material is fluid or like very much like a fluid.
On the other hand, despite the application of force from our common understanding we see that the solid never flows so the time scale associated with flow of a solid body is almost infinitely large compared to that of a liquid. So, the other extreme that we can consider here is the crystalline solid.
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So, if Deborah number tends to infinity, we say that the material behaves like a solid, behaves like a solid. So, the advantage of considering these two extreme cases is, we are now ready to
in a sense, began discussing what a viscoelastic material or soft matter would be and you can probably guess that when we say viscoelastic materials or soft materials the Deborah number associated with them will be a finite quantity. So, it will neither be somewhere between 0 and infinity and that where it lies between 0 and infinity gives you an idea of whether it is behaving more like a liquid or more like a solid.
So, some of the classes of materials we had discusses and we will see later are examples of so like polymers will have a finite Deborah number and the Deborah number becomes important similarly, for other types of soft materials. So, now we see that although the original definition here appealed to intuition in some senses.
So, although the this here it is not really stated what a simple liquid or crystalline solid means with the use of Deborah number, we can start quantifying our understanding of soft materials in a sense.
Now, what I had like to do is I would like to consider a couple of problems okay, so we go ahead and have a sample problem, so let say you have a sphere of radius R, and some fluid is coming with the velocity of U or U∞ , the freestream velocity of U∞, then in this system, and it is given to you that this fluid has a relaxation time of λ so this is given okay, so I will just write it down here, given fluid has relaxation time scale of λ.
So, the question that we want to ask is, what is Deborah number in this case? So, this is just go back to our formal definition we write Deborah number as
λ is already given to you and this is the more appropriate definition to use in this case, because, we are not stating the this is an experiment or anything, so there should be a time scale associated with the flow. So, then your question simply becomes, what is this time scale of flow?
So, the time scale of flow in this case
this is a time scale that is associated is appropriate to describe this flow. R is the radius of sphere and U∞ is the freestream velocity which means that we can write our Deborah number as
(Refer Slide Time: 26:51)
Okay, we will do one quick other example which is you have a sphere of radius R and that is rotating in a fluid with an angular velocity of ω and again the fluid here has relaxation time scale given by λ so this number is given to you. So, in this case again you have Deborah number as
and the natural time scale here, is actually given by ω itself, because ω is angular velocity.
So,
So, Deborah number becomes
So, I will give you one problem that you might want to think over and we will discuss that in the next class okay, and you have let say two spheres, one has R1 radius, the other one has R2
radius. So, these are concentric cylinders think of this as problem of two concentric cylinders. Sorry, I had said spheres in the beginning sorry. So, let us think of this as two concentric spheres and then one of them the inside one is rotating at an angular velocity of omega.
So, what I would like you to is to, given today’s lecture material tries to think as to what would be an appropriate Deborah number in this case.
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So, what we did today is that we discussed how natural systems have their own time scale and how your observation of that also has an effect of your understanding of the material itself. And, the, we discussed the famous pitch drop experiment and how a pitch drops. So, if you are an alien being you would not necessarily understand it in terms of a solid material.
(Refer Slide Time: 29:51)
So, that brought us to the very important concept of the Deborah number and we saw that Deborah number in the original definition as given by Reiner and his paper in the 1964 paper if I am correct, he gave it us time of relaxation divided by the time of observation. Where lambda is the time scale of the viscoelastic material and t observation is the duration of the experimental observation.
In frequent usage, the definition of or the format of the Deborah number that is typically used is we replace this t observation by the time scale of the characteristic time scale of the flow. We saw some of the examples and there is a problem for you to also think about before the next class. So, we will stop here for today’s lecture. Thanks.
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