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Complex Modulus
So, welcome back everybody to the last lecture on introduction to soft matter. (Refer Slide Time: 00:48) And in the last class we were discussing the idea of the complex modulus and we had stopped at a particular problem and the problem was the idea of the Maxwell fluid where we took the governing equation in the format of (Refer Slide Time: 00:57) And we were asked to, the idea is to find out the complex modules for this particular situation. So, we have already established that this formula is correct, where So, we should go ahead and just complete this integration in that case. So, let us just go ahead and do that. So, now you have Now, this is a complex number. You can even keep it in this form or you can go ahead and simplify it further. If you want to simplify it and express it as a proper complex number, then you have to get rid of the denominator, the complex numbers in the denominator. And what you can do is you can simply multiply then the denominator by its complex conjugate. So, you can go ahead and do this, you can multiply it with 0 1p − p . So, you have basically gotten rid of the complex number from the denominator and now you have G the function as a complex number itself. So, this is the solution that we are looking for. (Refer Slide Time: 04:21) Now, for a Maxwell model you obviously know what these 1 p , 0 p , 1 q are and you can go ahead and replace these and you can evaluate that for the model that you have built. Now, there is one more way of getting to the same solution and that for that so, just consider a situation where the transients are died out, your input was a sinusoidal wave right. So, your input was so, an alternate method is we have the input is a sinusoidal function. So, your this is the situation when the transients are died out leaving the steady state response. So, if you consider back the equation of the form that we had taken, so for an equation of the form that we had taken we are taking In this kind of a case, you can just go ahead and replace, since you know the functional forms, you can just go ahead and replace So, this is what we got by evaluating the integral and by a more direct method we got the same result. So, the direct solution suggests an important generalization. So, what you see here, you see that if you have an derivative then you can replace this entire quantity the derivative part by (i) and you can rewrite the original polynomial form in this you can keep the original polynomial form. (Refer Slide Time: 08:20) So, basically what I am saying here is note that if you use the operator form then in the operator form the previous equation can be written . So, in this expression to give get the answer, to get the answer the correct set your D as i and  as 1. So, yes take the previous equation set D as i here and set  as 1 and you can get back the same form of *G () and we have to set one more thing, we have to set the  as *G () . So, in the generalized case you can show. In the general case of a governing equation being So, and this is your answer. And please note that this complex modulus system contains both the storage and the loss module. So, we have come a long way. In this particular course, we have discussed quite a few topics and these two sample problems were meant to wrap up the course. And before we finish the lecture, what I did like to do is I like to quickly go over all the different things that we have discussed. (Refer Slide Time: 12:23) So, what I have done is I have prepared a PPT format, where all the things that we discussed from the beginning of the class, we have sort of summarized here, so that we can quickly go over it and evaluate what we have learned in the class right. So, we started off the beginning of the class by discussing some important definitions, which included definitions of soft condensed matter, colloids, viscoelasticity. We discuss the idea of natural timescales in a historical context. And we discuss the historical context for the for the definition of the Deborah number as well. And we also discuss the differentiation between the Deborah number and the Weissenberg number right. And once we did that, we decided to take macroscopic formulation for viscoelasticity. And we saw that the basis of that can be tested by six important tests including the stress control test, the release of stress, the strain control test and the effect of different histories, energy dissipation and effector sinusoidal oscillations. So, we did not touch upon energy dissipation too much in this particular course that can be done in a more advanced level course, but we did touch upon at the very end, we also discussed the idea of the sinusoidal oscillations, although initially we had started with the stress response and strain control test. (Refer Slide Time: 14:02) So, the next thing we discuss was a step stress test. And we saw that the three important behaviors that of the classical elastic solid for example, here. So, the classical elastic solid, when you input a stress, step stress, then the response is actually rather simple, right and this is something that you have already been familiar with. We also discuss the behavior of the classical viscous fluid, where when you put a step stress, then the viscous fluid continuously strains it so, your, the fluid, this is, this basically encapsulates the idea that a fluid cannot withstand shear. Now, these two together the classical elastic and the classical viscous in some senses, when you put these responses together, becomes the response viscoelastic material. Because it has both the term visco and the elastic, and this is the response to a step stress test where we see that when you apply a stress you have an immediate solid like response from 0 to A, and then you had a continuous straining from A to B and then when you relax, when you when the stress is taken away, then the strain slowly decays and this can go down decay to 0 or it can decay to a nonzero value. So, this can either be 0 or a nonzero value, right. So, all nonzero depending on the situation. (Refer Slide Time: 15:25) Next, the other important test was the step, step strain test, where again we saw and we understood the response of a viscoelastic material in terms of the responses of the classical elastic solid and classical viscous fluid. And the important thing here to note was that when you have a viscoelastic fluid and you apply a step, strain, then the stress decreases. And this is the very classical stress relaxation phenomena and these two tests together the fact that you can have a material that is continuously straining under applied constant stress and the fact that you have you can have a decay in stress when you apply a constant strain. (Refer Slide Time: 16:15) This led us to develop two important concepts, which is the concepts of the creep function and the stress relaxation function. Now, the creep function here J and these two here and brackets stress implies that J is a function of the applied stress and is also a function of time. When linear when the idea of linear scaling or the principle of linear scaling is valid, then you can divide this function J by  and you get a function that is only dependent upon time and this became the creep modulus. Similarly, for the stress relaxation phenomena, if the principle of linear scaling is valid, then you can divide the function G here the function G is a function of two important parameters, the strain and the time and when you divide it and leave assuming that linear scaling is valid, you end up getting a function which is only a function of time now. So, this was, this is one of the singular important aspects that we covered in this particular course. (Refer Slide Time: 17:21) Once we did that we try to familiarize ourselves as to why this sort of happens and we discussed as because we have all we already know that soft material refers to a collective of many different types of materials, we try to discuss the different types of materials that fall under this category. So, we discuss polymers we discuss surfactants and liquid crystals. To discuss these we also discuss the, in the preliminary ideas of the inter atomic bonds and how they are relevant here. Though this does not obviously reflect a exhaustive list, but polymers are one of the most widely used materials nowadays in industry. So, we discuss that a little bit more detail. So, we for example, discuss the idea molecular weight calculations of polymers for example, the weight average molecular weight, this is the viscosity average molecular weight and other important parameters are associated with it. (Refer Slide Time: 18:22) After that we moved on to distinguish an important set of equations which is called the constitutive equation. So, we first started off with understanding that they are different than the conservation laws. And then we enumerated many of the important constitutive equations that we find ourselves commonly in solid and fluid mechanics for example, the perfect fluid and then the Hookean solid. And the obviously the Newtonian fluid and here once you will use this constitutive equation you will end up with the Navier stokes equation once you do the proper formulation for the conservation of linear momentum. The two important models here listed here the Carreau model and the Power law fluid. Now, these were two inelastic models that we discussed and they were really important introductions into this behavior of different types of soft materials such as polymers. They are called inelastic because they do not have the elastic term in them, but still, the fact that the viscosity varies as a function of shear rate was an important understanding and these models are very, very useful. From that, we went on to understand the concept of simple shear flows and related concepts of normal stress differences, right. So, we saw that in certain fluids you can have in the stress sensor you can have normal stress differences and we encountered these particular these particular functions, where N1 is the first normal stress difference and N2 is a second normal stress difference. We familiarize ourselves with the expert we saw some experimental results on the rod climbing effect. And this is a very, very classical effect whenever it comes to polymers and viscoelastic materials. With this constant introduction to constitutive equations, we started our journey into the viscoelastic domain. (Refer Slide Time: 20:52) And we started it with the simplest of the possible models, which is called the Maxwell model. And in the Maxwell model, you we, in fact looked at the original paper of Maxwell. And it is the model is it's rather simple because it only has a spring and a dashpot and the spring constant and the dash and the parameters of the dashpot, they take into account together they make up the viscoelastic model. Here the spring and the dashpot have placed in series and we saw a similar other another model which is Kelvin Meyer Voigt body. But here the spring and the dashpot are now in parallel. So, these were the two of the simplest possible viscoelastic models. And then we built out using more complex models using this. For example, various three parameters models were built, for example, Jeffrey's fluid and then we went on to understand models which have many different springs and dashpots together to make a more complex constitutive equation. Now, please note that all these specified models are linear viscoelastic models and these are on one dimensional models. (Refer Slide Time: 21:55) So, with that, so with the last part that we are discussing towards that the, the use of n springs and n dashpots that helped us arrive at a general constitutive equation, where we saw that the stress and the strain can be related by a in a polynomial type of form, where this coefficient is basically consists of the multiplier of the stress here consists of many coefficients and the differential and powers of the differential operator. Now, these exact values of pN , N 1 p − or 0 q etc. depends upon the kind of model that you have used and you can use and we saw how to derive this particular equation in different contexts. We also saw the response to the arbitrary stress history is given by this particular equation right here. Similarly, the response to the arbitrary strain history is given by this particular equation right here. We also familiarize ourselves with different variations of these equations, because this term here is represents a convolution, you can have different variations which are equivalent in nature. I also discussed that in some books, instead of 0 plus you have the lower bound is sorry the lower limit starts from minus infinity and in this case this first term is not required. (Refer Slide Time: 23:27) Then, to cement our understanding of these constitutive relationships, we move into a slightly more advance topic of what is called objectivity. And we saw that different scalar quantities like density etc. Those are frame invariant right and they are objective by themselves. But vectors are objective if they transform according to this particular law, where this is these are the transformed components and Q is a rotational tensor. So, basically this is an orthogonal tensor and what if vector transforms according to this particular law it is called objective or framing indifferent. Similarly, a tensor has to obey this particular rule, if it is to be frame invariant, frame indifferent. Now, the reason we are looking at this is because we realize that the time derivative of the stress tensor is not objective and the time derivative of the stress tensor occurs in the in the some of the models that we had discussed previously. So, that so this basically serves as an introduction to an idea, the ideas of continuum mechanics where there are objective derivatives. And we did not go into depths of that but Oldroyd was one of the pioneers of this and the Oldroyd AB fluids, use objective derivatives and other there are other types of viscoelastic models, which also employ objective derivatives. We did not do this in this particular course, but this is the logical conclusion of what we had discussed and you can take up this in a more advance course on this particular topic. After that, as in this course, in the very beginning of this course, we had said that, we should understand material properties not only as being time dependent variables, but also as frequency dependent variables. (Refer Slide Time: 25:40) So, we looked at response to sinusoidal oscillations and we asked ourselves the question that if a viscoelastic material is subjected to a sinusoidal oscillation, where the input of strain is a sinusoidal function, then what will be the corresponding output and we found out that the corresponding output occurs in the form of this equation here equation 4 where this term ere, the fact that we have omega in the in brackets refers to the idea that G'() , G''() are functions of the imposed frequency omega. (Refer Slide Time: 27:02) The same formulation that we did previously can be written in the complex form. And if we choose to write the input in this particular form, where i is the complex number, So, 0 ( ) i s s e  = , this is now a complex number with a complex number has cosine and sine forms which are so, basically you still have the time dependent oscillations. And the corresponding output can be written in this particular form. It is a very simple equation here, but G'() now is a complex number sorry, this is *G () not G'() , *G () is a now complex number and this complex number is also called the complex modulus and this is given as Once again please, i here is the complex number or i = −1 . Now, this G'() for the complex modulus is a material property. And we also looked at two other material properties, which are dynamic viscosity defined as , and the complex viscosity, which is defined as And so that is sorry. So, that sums up about almost all the things that we covered in this particular course. And as I said, this is a beginning level course. So, you are probably, some of you who are taking this course are probably going to take courses on rheology, etc. And hopefully that will be very helpful. (Refer Slide Time: 28:55) This, the contents of this course will be very helpful to you. Now, I like to just include two more important notes for clarification and that is we had initially used the symbol  for viscosity and a few lectures later we change it to  for viscosity. So, this slight discrepancy is there and in the integration of the stress relaxation function and the creep function, we have many times we have used s as a dummy variable for integrations and the dummy variable s is used in the time sense, but it is different than the variable using Laplace transforms. So, we use the Laplace transforms to solve some of the equations and there are also an s appeared but the two s are different one is in the Laplace domain and the other is in the time domain. So, with this note, I would like to conclude and what we have done is, in this entire course, we have gone through quite a lot of material. Our biggest focus was on understanding the phenomena of creep, and this phenomenon of stress relaxation that we kept seeing in different forms, even when we are discussing the different viscoelastic models. Our focus was still on understanding the creep function and the stress relaxation function for those models. So, this course is said add said earlier also this is a beginning level course. And it is meant to familiarize yourself with the various terms that are used in this area. And hopefully you are going to take more advance level courses in this, for example, forces of rheology or courses of condensed matter physics or viscoelasticity, etc. And I hope that this course the contents of this course, will be very useful to you in understanding the area of soft materials. And if you decide to take advanced courses, the material covered hopefully would be extremely helpful and advantages to you. So, with this, we are ready to end this course. And thank you very much for your time and good luck.