 Now, this is another concept where somebody would like to correlate soil suction with conductivity measurements or in other words you have seen that it is very difficult to measure soil suction and to develop soil water characteristic curves. So, then the question comes in mind by indirectly can we measure suction and if we can measure the suction then we can complete the SWCC or the soil water characteristic curve. So, this is where we were working in this direction to show that if you measure the electrical conductivity of the soil mass or the porous media, you can easily obtain its suction value. And this paper has recently been accepted by ASC it has not come in the print yet by Dr. Anumand Rao. The phasic philosophy here is that metric suction of the soil is a function of volumetric moisture content. So, this is nothing, but your you know classical SWCC the relationship between metric suction and volumetric moisture content. At the same time we have seen that sigma is a function of theta is it not. So, conductivity is also a function of volumetric moisture content this is generalized Archie's law. So, if you mix these two together when we say psi m is a function of theta and function of theta is psi a sigma what it ends with is that psi m is a function of sigma. So, what it indicates is I can measure suction of the soil if I can measure its conductivity. So, this was in short the thesis of Anumand Rao and he has published and quite extensively in different journals by using this philosophy. So, this is the classical SWCC curve where you are plotting volumetric moisture content with respect to suction. So, you get some experimental data point and these experimental data points are fitted by using a standard fits which are available in literature. Effects corresponds to Fredlund and Zing, Brooks and Corey these are the scientists or the researchers who have given the equations for the best fits when connection VG and volum and mu. So, by using different fitting functions you can obtain the soil water characteristic curve even if you have a little bit of data point. For all these points you can find out what is the regression coefficient and the best regression coefficient can be used as the most suitable as the WCC. Now, look at this situation by following this law that is psi m is a function of theta and sigma is a function of theta. If you plot sigma with respect to psi n what you get is the nature of the relationship which again is equivalent to SWCC, but we should appreciate a point here that measuring conductivity is much more easy as compared to measuring suction of the soil. So, it is very easy as well as a very rapid method to determine SWCC. So, if you go through the if you go through the works published by N 1000 say some more time may be 6, 8 months then you will notice that we have shown that how these philosophies are working perfectly in order. Now, based on the complete impedance analysis based on the complete impedance analysis some equations can be derived. Below the question you asked in the previous lecture now this is the starting point of the answer to your question about the multiphase material properties and how to determine them if you know the dielectric constant. There are several equations which are available in literature which correlate K is a dielectric constant of the porous media with its volumetric moisture content. So, you can see it easily that under root of K is this term under root of the entire term that means K is nothing, but some coefficients multiplied by theta value. Now, this is the famous Topps equation. If you know theta you know the dielectric constant you can calibrate a mineral and when mineral comes in contact with water how you are going to use this in real life is if you know the K value you can measure the K value and you can determine theta and this is how the TDR or FDR probes work. So, this is where you are measuring K value and you get the value of volumetric moisture content. The second equation is given by Roth et al which again is a function of volumetric moisture content theta and the porosity. So, this is the mapping of two phenomena one is the matrix of the soil porosity and the volumetric moisture content theta alright. This was given by Roth the third equation was given by you et al where the dielectric constant is a function of volumetric moisture content of the soil mass. Then some efforts have been made to correlate dielectric constant with rho d, rho d is nothing, but dry density of the soil mass. Now, this equation was further refined by my student Rohini where the philosophy was that is difficult to measure the dry density of the soil mass is it not. The way you measure it is you know the total volume which you measure physically and where the sample get the moisture content and say total total unit weight divided by 1 plus w is the gamma d. It is a very very non precise way of getting the gamma d rather if you use this concept of measurement of K and if you know theta very precisely which can be measured with the help of any advance instrumentation what you will end up is with knowing the value of rho d. So, that will be a very accurate measurement and this philosophy was proposed by Gardner at all in 1998. So, this was the contribution of Rohini who published this paper in methodology of determination of electrical properties of soils in journal of testing and evaluation ASTM and we extended this relationship for determining the in situ dry density of the soil mass. The reason is theta can always be correlated with w and gamma d that is theta equal to moisture content gravimetric multiplied by gamma d divided by gamma w alright. So, gamma d upon gamma w will be a non dimensional term multiplied by gravimetric moisture content and that will be the value of theta. So, if you know the dielectric constant of the soil system or a porous media you can obtain its dry density very easily. All these equations are valid for single phase minerals assume to be alright, but when you talk about the multi phase minerals then what happens I think this is what your question was been ill if I remember correctly. So, this is where we talk about the mixing models. So, certain weightage is given to certain weightage is given to each mineral of known dielectric constant. So, I will come back to this equation slightly later. If you know the phases of the minerals which are present in the soils defined as M 1 and M 2. So, these are composition let us say 80 percent quartz 20 percent a light or whatever. So, if you know the phases of the minerals which are present in the soil and if you know dielectric constants of these minerals that is K M 1 and K M 2 and if you know the dielectric constants of the pore fluids. So, this could be a situation dealing with only water then the pore fluid is only one it could be partially saturated soil that the pore solutions would be to air and water it could be soil contaminated with some organics. So, if you know the dielectric constant of the organics you may have 2 or 3 phases of the liquid itself. So, one more term will come over here it becomes a 3 phase combination for dielectric constant determination. Eta is the porosity of the porous media and SR is the saturation of the porous media. Now, the question is how would you find out dielectric constant of the minerals take the mineral in the purest form put it in impedance cell and you can measure its dielectric constant. How would you find out the dielectric constant of the pore fluid? Extract the pore solution by using the pressure membrane extractor again put the solution in a impedance cell and measure its dielectric constant. M 1 and M 2 how would you find out how would you find out the percentage of the minerals? Excellent that is right. So, that is the reason why X R D analysis is so important. So, that you know what are the minerals which are present and then one is quantitative analysis for X R D. So, you can quantify the percentage of the mineral which is present in the soil this is difficult to achieve and difficult to work on very difficult. Quantification of minerals which are present in the soil mass takes big amount of time. Porosity is a macro term which can be obtained very easily by any of the methods which you adopt and saturation can also be find out very easily. Now, you look at this equation. So, the equivalent dielectric constant of a multi phase system would be these are the equivalent terms that is under root of k of first mineral under root of k of the second mineral multiplied by their mass phases or the percentages this multiplied by 1 minus porosity. So, 1 minus porosity term will give you what is the solid phase of the material which is present in the matrix when you multiplied by the porosity this becomes the liquid phase is it not you agree or not V V upon capital V is the porosity term. So, volume of voids would be porosity multiplied by total volume. So, if you multiply the porosity with total volume this is the volumetric form in terms of saturation of water. So, this is where your saturation term comes for the pore flit 1 and pore flit 2 and then this is nothing, but the superimposition effect of pore flit 1 on pore flit 2. So, this is how you have created a two phase system. Now, if you are working with the minerals with the soil which has let us say multi phase of the minerals and multi phase of pore flits. What will happen to this term? This term will get further expanded by taking into account k m 1, k m 2, k m 3 and so on. Similarly, k p f 1, k p f 2, k p f 3, k p f 4. So, you think of a system where you have a frozen soil which is contaminated that is right, that is right. That means, then what that is what I was telling you just now. If you think about a multi phase system of soil what will happen to the saturation term? Right now the way you have defined saturation is only volume of voids with respect to volume of solids. Now, your SR term it still will get bifurcated that means, the saturation with respect to the vapor phase, the saturation with respect to the liquid phase. So, that means, you will be having SR prime which will be equal to SR 1 and 1 minus SR 1. So, those terms get multiplied into this equation. I think this is what precisely he was asking in the previous lecture that how would you talk about the material in homogeneity. I thought that this will be a good example to show to you that how equivalent models can be developed and the state of the art is that most of the minerals are known for their dielectric constants, most of the pore fluids are known for their dielectric constants. So, if you use let us say remote sensing equipment or imagery. So, what essentially you get is k value is it not, how would you get k value by remote sensing device. You can transmit electromagnetic waves and then you can find out how much time they take to reflect on the surface and that time lag between the waves will give you the dielectric constant. So, once you know the dielectric constant k you know the mineralogy of the entire soil and you can find out how much saturated the system is. So, depending upon this concept you can locate a reservoir which is full of water underground. It will be a good source of fresh water supply or whether the water is a brain solution or brackish water or whether the pore solution happens to be a oil reserve or whether this soil mass happens to be having more crude oil or some soil contamination and so on. So, these type of models which are known as mixing models are utilized for ascertaining different compositions of the soils. This is a very big subject and we lot of people are still working, but this must give you an idea about how small small concepts can be put together to characterize the entire soil mass alright. Some efforts were done by one of my masters students Azaz Bhatt and Hanuman Thrao in publishing this work which came up as generalized relationship for estimating dielectric constants of soils. This was also published in journal of ASTM International in 2007. Is this partly or something else you are asking that is right. So, here the saturation is it is with respect to the fluid 1 because that is why you are noticing here the moment you talk about the fluid 2 it becomes 1 minus S R. So, here the intention is let us say we are talking about partially saturated soil. So, still the saturation term is air saturation or water saturation. So, if you define S R as the saturation which is water saturation then 1 minus S R is nothing, but air saturation. So, K P F 2 becomes the dielectric constant of air and K P F 1 become dielectric constant of water. The perception changes the moment your profession or the requirements change. As a geotechnical engineer we are more interested in water in the soil, but somebody is more interested in let us say oil rather than water. Another person would be more interested in vapors present in the soil rather than water or the liquid form and so on. A geologist will be more interested in no saturation, but only minerals. So, for him everything is submerged in water let us say. So, he does not talk about any of these phases, but he is more interested in the values of the percentages of the minerals so that he can then ultimately what is that you are trying to do. If you have this type of model you can match the left hand side value of K with the right hand side value of the parameter and see where you are going to converge and that type of model is going to describe whether this location is having a resource or not so that later on intense activities can be taken up. I suppose this is how the whole mining process or the identification of the minerals must be getting guided by your imagery satellite imagery. Do you want to add something Sangeeta ok yeah what happened fever ok please take first, but then reflectance will also depend upon the directly constant of the material of the minerals. So, the basic concept has to be same another good application would be in GPR whether your concrete is cracked or not or some microfruits are there micro cracks are there. So, what happens in the regular concrete phase the movement cracks take place then the air impregnates into the system and the dielectric constant of the air comes to the picture. So, this becomes a two phase system of concrete which is cracked and hence air is present in it. You can use this equation again for finding out the composition of the composites which you are developing each phase will be having its own peculiar property all right. Now, this is another interesting relationship which shows how Etterberg limits change as a function of dielectric constant. So, for a given soil if you contaminate it with different organic and inorganic materials xylene, propylene, methanol, water what you will notice is that the placity index does not change much, but with dielectric constant it will change a lot however you can find out easily that the soil is contaminated with what type of chemical if you talk about its liquid limit. So, most of the time we talk about liquid limit with water or soils which are contaminated with methanol other organic components for them the liquid limits are going to be much more higher. The question is then how would you identify whether the soil is contaminated or not. Take the soil sample, check its electrical property, take out its pore solution. So, now you have divided in two parts one is the pore solution characterization and another one is soil matrix characterization and then come follow complete analysis again and see what fraction of contamination was present in the soil mass. This is part clear. So, then the question is that what are the basic concepts of impedance spectroscopy and how to utilize the response spectra which you get for engineering application. The basic idea here is that what we are trying to do is we are trying to study the response of the material under alternating current. So, you apply any electrical stimulus on the material or a substrate, substrate is nothing but the pore solution and you try to get its response spectra. So, for that matter if any sinusoidal wave is passing through this of frequency f the material of the substrate property would be specific gravity in case of soils volumetric moisture content, unit weight, void ratio and its saturation. So, we want to get the answers to all these parameters, we want to identify these parameters. So, what you are going to get is you are going to get response spectra in the form of impedance plot. What are the impedance plots? The real part of the impedance would be z cos of phi, imaginary part of z would be z double prime modulus of z sin phi, where phi is the angle between the two components that is imaginary part and imaginary part and the real part of z. So, this is what basically you are trying to do. Now, using this phi parameter which is tan inverse of z double prime divided by z prime or the imaginary to real part, when you plot it with respect to the frequency whatever pattern you get is known as a Bode's plot. So, we have talked about Nyx plot, this is another representation of the data which you get from impedance spectroscopy which is known as Bode plot, it is a relationship between phi angle the end frequency. So, you plot the results wherever you get the minimum value of phi alright. Now, this corresponds to the value of z double prime divided by z prime, but still most of the time people try to use or they stick to Nyx plot rather than Bode's plot. This is one of the ways of getting the information related to the material properties. So, this is where the concept of equivalent circuits comes into the picture. What you have done is you have taken a material put it in impedance cell and then you try to develop the equivalent circuit. What is the meaning of equivalent circuit? You are trying to map the response of the materials in terms of its resistance, capacitance and inductance. And if you are working on soils you ignore the component of inductance, you are only interested in considering resistive and capacitive part. So, this is basically fitting of the data which you get from the impedance analysis by using Z view software. So, to start with we always assume soil mass or a porous media to be consisting of a resistance and capacitor in parallel, Saturday yes please. So, for this circuit when you are starting with you have some R 1 and C 1 values whatever impedance data you get it is plotted as a red line. This portion you remember is electrode polarization followed by the main part of the circuit. So, this is where if you superimpose the response of RC parallel circuit you will end up with this type of a semi-circle. What is the meaning of this? This much is the total resistance of the circuit and by the time you come over here the resistance is 0 the capacitive part is also 0 the maximum capacitance capacitive resistance would be somewhere here which corresponds to this value. So, what is your feeling is this a good match of the phenomena which is happening in the soil mass or not? No, because your equivalent circuit could not capture the real response of the material. The real response is far away from the theoretical response which you have assumed to be a simple R and C circuit. Now, in the process what you get is from this graph you can easily measure what is the value of R and what is the value of C 1. So, you know the value of R 1 and you know the value of C 1. So, you are not happy with this what we should do is we should refine this further. So, if you refine it further you have to try so many combinations. So, starting with this circuit what you should do is what is your feeling when there will be a good match when this green circle shifts on the left hand side or right hand side your right hand side my right hand side also. So, if shifts on the right hand side then at least this circle green circle will be embracing the red one. So, what we should do we should translate the green circle on the x axis how to do that by adding a resistance. So, you have picked it up now. So, you add a resistance over here to this circuit what happens? You end up getting this circle shifted on the right hand side, but are you happy with this circle because this circle is still not capturing the entire response of the material. What we should do now? We should slightly spread the diameter of the green circle how this will be done for the initial circuit you have added a resistance you add more capacitive part to this. So, the moment you do that what happens to the response now this response is very much close to the real response of the material. You can see there is almost a near overlap between the green circle and the red circle and still you are not happy let us say. So, what I should do add another circuit like this and then this is a trial and error and comes by more and more practice and intuition sometimes. So, you take two R and C circuits put them in resistance in series and what you will notice is the discrepancy is almost disappear, but to refine the results further what you can do is you can go on with another compartment of RC circuit and what you will notice is this is a perfect match of the results and the best possible fit which you get is when you adopt a circuit like this. Now, this is where a philosophy is if you remember in yesterday's discussion I was showing you that the impedance cell contains two electrodes and the soil mass in between. So, unless you give due weightage to the electrode and the soil mass contact. So, this circuit of R and C basically depicts the contact between the electrode and the soil mass. So, this is one electrode this is another electrode and in between the soil is represented by this. I have omitted lot of steps in between this was achieved after at least three months of everyday trial and may be we tried more than 5600 circuits this was done by Azaz my m tech is today. So, we simply cut short all the steps in between and you can see it is just like building blocks. So, you keep on playing with the circuit and one fine day it may result in a good fit and that is the fit which has been shown over here. So, here the R square values are perfect to very close to unity and physically also this circuit seems to be a good circuit where you are giving enough weightage to the electrodes and you are giving enough weightage to the soil mass which is encapsulated between the electrodes. So, this became the standard circuits for the impedances which are being used for analysis of electrical properties of soils. So, Azaz contribution is he has made impedance cell he has derived a methodology to get these type of components and he has come up with a generalized circuit which represents the response of the material. Now the question is how we are going to use this response in geotechnical engineering is this part clear water weightage this has been achieved by people like you only know you find it trivial or interesting you know. So, now I will be talking about further interpretation of the components of the circuits is the basic philosophy clear. The basic philosophy is that we are trying to capture the response of a geomaterial by conducting some test in the laboratory using an impedance analysis and then each component of the circuit should be giving you some information about the material. Now if you plot the values of the different components in the circuit if I define it as the resistance and if I plot resistance with respect to volumetric moisture content what you will notice is as the moisture content increases the resistance drops is perfectly alright. I will have to go once again back to this figure in yesterday's class we were talking about grain to grain contact. If you put two grains together it is a sort of a capacitor formation there is some fluid in between you look at the circuit over here this is a depiction of grain to grain contact. The resistance offered by this system is associated in the form of R4 the resistance offered by the pore fluid is nothing, but this resistance clear. So, the issue is it is a sort of a philosophy by which you are trying to study what contacts between the grains is causing in physical sense the contact between the fluid and the grain alright is a sort of a grain boundary effect. So, if you plot these type of relationships and if you show RGB, RGB is nothing, but the resistance of grain boundary and Rg the resistance offered by the grain itself these are two things different things one is the resistance offered by the grain boundaries another one is the resistance offered by the macro system the grain itself and if you put them together to get the total resistance of the circuit then what happens to the total resistance of a circuit this is what people would like to identify. So, the message which you get from this analysis is you can characterize the soil as a granular material if RGB is negligible what is the significance of this if in this figure the value of R4 becomes very very small. So, this shows it is a granular material if R4 is very appreciable it is going to be a fine grain material is this clear it is basically a philosophy being put in practice to show yes there is an analogy and this analogy captures a real phenomenon the form of some equivalent models for these soils Rg value will be very very high. So, if you train a circuit and if you get the components of the circuit and if you find that Rg is extremely high it is understood that this material is going to be a coarse grain material R grains fine grain materials will show you very very less resistance because of the surface charge. So, the soil can be characterized as a fine grain soil if both Rgb and Rg are present in the equivalent circuits, but their values are less. However, values of these resistance should be quite low as compared to the granular soils. So, the basic idea of giving this information to you is just by looking at the values of the components which are appearing in the circuit, you can identify whether the circuit belongs to a fine grain material or a coarse grain material. One step ahead of this would be a certain where whether the pore fluid is contaminated or not that means the circuit components corresponding to R3 you know which is or R2 which is shunting this process if value of R2 is very very small most of the current will be passing through the pore solution, but if R2 is very high it becomes a dry soil clear. So, based on R2 value you can find out the saturation state of the material. So, ultimately what was the whole idea? The whole idea was to characterize the soil mass or the porous media and when you say characterization the basic attributes were grains, how they are located, what type of pore fluid is present, what is the density of the system and so on. So, the efforts are on in this direction to you know master this subject in such a way that you just take one measurement of the soil and then you analyze the results and you should be able to diagnose what this material is clear. Any questions here? Of course, this work is not still complete, we are working in this direction fully one day this part should be in a position to be handed over to the you know professionals, but I am sure that you must have got some idea about what are the applications of impedance spectroscopy and why people are working in this area and ultimately what do they achieve out of it.