 In this lecture, we will learn about surfaces and interfaces. Surfaces are the ones which interact with directly with the environment whereas, interfaces are the ones which interact with another media. So, it can be solid solid interface, solid liquid interface, solid vapor interface, liquid liquid liquid vapor interface or it can be totally vapor it may not be a interface, but it can be related to the interacting with a particular surface. So, we can see that surfaces and interface they are highly critical in dictating the properties mainly for the engineering material. So, that we can adopt them specifically for certain applications. So, surface and interfaces between different grains and faces are highly important in predicting the properties and also controlling certain processes. Surfaces can also be treated or interfaces can also be treated as 2D imperfections or defects in the ideal structures of lattices. So, we can see that surfaces and interfaces they are highly important in predicting certain properties and these properties can be either mechanical, chemical, biological, electrical, there can be so many properties. So, boundaries are of very specific importance in influencing these properties. Also, these surface and interfaces can be helpful in defining ideal crystal structure for a set of specific environment. So, we can see that surface and interfaces they play a very important role in dictating certain properties which are related to the electrical, magnetic or even the bulk, chemical, biological properties. So, these are highly critical. So, only once you understand what are surfaces, interfaces then only we will be able to engineer them for a certain application. And why these surface and interfaces they are highly critical specifically in nano materials why because the overall surface or the surface of the grain boundary region to the bulk volume it is very high that ratio between the grain boundary and the bulk of a particular grain in a nano material is very, very high. So, that is why we have a very specific importance for surface and interfaces because the overall volume of surface or interfaces very much comparable to that of bulk. So, what happens is the entities of grain boundary and the contribution from the surfaces or the grain boundaries or the grain they are both equivalent or there is much more contribution of grain boundary in nano materials. Whereas, in bulk structures which are which are micro or macro grain structures in that case we have very large amount of grain bulk in comparison to that of grain boundary. So, in that case it is mainly dictated by the all over the grain. So, in nano materials it becomes very essential that we consider surface and interfaces separately you can go next. So, these surface and interfaces they are mainly given by three terms which are surface tension, surface stresses and surface energy. So, we will come to them as we go along. So, let us see first surface tension. First tension is a force which is required to extend a liquid surface. So, if you have a liquid surface we require certain force to extend this liquid surface to something else or it is also being defined as reversible work required to increase the surface of a liquid by a unit area. So, we need to do a certain work or certain work has to be done to increase the surface area of a liquid. So, we have we have to increase the surface of this particular liquid by a unit area. So, we can see that this term is dictated by surface tension. Surface tension is the reversible work required to increase the surface of a liquid by a unit area. So, gamma is equal to d w by d a and surface tension is mainly responsible for the solidization of a liquid particle. So, we will realize that the surface tension is equivalent to surface energy for liquid whereas it is not so in the case of solids. When surface tension is the one which makes the deformation of a surface because of this additional force which is existent which does work on the liquid. So, we can see that the Gibbs free energy of a system can be defined as basically the variation in the Gibbs free energy of a system is in changing from one equilibrium state to another. So, if you have say area one a 1 to going to a 2. So, we can see that the overall free energy Gibbs free energy of the system is being given by this particular equation which is dependent on the entropy that the temperature volume the pressure the surface tension the area and also the chemical potential. So, we can realize that once we keep all other entities constant the pressure temperature and the number of moles of an entity. We can see that we can get surface tension as a function of free energy Gibbs free energy with respect to change in the area. So, at equilibrium what will happen work of expansion that is given by P d V will equal the increase in the surface energy this gamma multiplied by d A. So, we can realize that delta P d V will equal gamma into d A. So, essentially what is happening we have phase A we have phase B and there is always some surface layer and that it takes the overall generation of the enhancement in the free energy. How these two interact will give out the overall surface layer which will dictate the overall free energy of the new system and this term can again be expanded that delta P into d V in gamma is equal to gamma into d A. So, we can realize that d V the d V can be given as 4 pi r square the overall volume of a of a of a globular entity will be 4 by 3 pi r cube. So, we can see that d V by d r will equal 4 pi r square similarly the d A can be given as A A is nothing but 4 pi r square. So, d A will be 8 pi r multiplied by d r here also we have. So, what we can see that delta P into d V. So, what we can see that delta P into d V is 4 pi r square d r which equals gamma multiplied by 8 pi r into d r or we can see that delta P is equal to 2 gamma by r. This is the case for a spherical entity. So, in case when we do not have a spherical entity we can always say delta P is equal to gamma multiplied by 1 by r 1 plus 1 by r 2 where r 1 and r 2 are different curvatures. So, basically what surface tension can be defined as that liquids they cannot support shear stresses. So, what is happening is reversible stretching or creation of new surfaces can provide surface tension. So, surface tension is being generated because of the stretching of the reversible stretching of the new surfaces. So, for liquids we can see that surface energy is equal to surface tension. So, there is no change in the surface tension because of the change in the because of application of stresses. So, in that case we will realize that liquids for liquids the surface energy is equal to surface tension. But what is happening in solids is that solids can sustain very high shearing stresses. So, what is happening is they are also opposing the contraction in the surface area. So, one surface tension is acting on the surface it will tend to restrict or change the deform the surfaces and that will be opposed in case of solids. And also what is happening depending on the orientation of a particular crystal the surface energy also will be very different. So, in case of solids the it can sustain the shearing stresses and the opposed contraction of surface area which which arise because of surface tension. So, we can define that surface tension in solids is equal to reversible work in creating new surface. So, as we can see that in this case what is happening surface energy is equal to surface tension plus some extra term in in terms of changing the surface tension with respect to the strain with respect to the strain which is being generated. So, we can see that in case of solids the surface energy is not equal to surface tension. So, that is that is a disparity between liquids and solids because in liquid we can see that surface energy is equal to surface tension because what is happening is as soon as we are stretching the surface film the relaxation is much more rapid whereas, in solids the relaxation is not that rapid. So, that is why it can oppose that contraction in the surface area and that in turn this resulting a some opposition of contraction on surface area in solids. And so we can define the surface tension in solids is the reversible work in creating a new surface also the addition of the surface stresses which act on which act on it. So, how we can define surface energy is it is a increase in free energy per unit area when a new surface is being formed. So, surface energy is always a additional term which is kind of incorporating why it is happening is because let us consider that we have a particular material. So, we take a particular material with certain atoms which are arranged in certain fashion and let us say if you consider one atom out here and we want to bring it to the surface. So, what is happening now this atom has to break the bonding with the other atoms which are nearby and then in turn it can come to the surface, but with that it has broken the bonds which should have been on the top as well. So, in turn it has gotten some energy it has because of breaking the bonds. So, in this case it has to break the bonds and then come to the surface. So, that increase in the free energy per unit area when the new surface is being formed is called as surface energy. So, that is the typical nature of surface energy whereas, surface tension we can see that surface energy is equal to surface tension in liquids because liquids can easily sustain they can easily relax as soon as the work is done on them whereas, in solid they can oppose the contraction of surface area when surface tension is acting on it. So, surface energy is not equal to surface tension there is some additional work which is also been incorporated and how that really occurs we will see in the next few slides. So, we can see that the energy is a scalar quantity whereas, surface tension is a vector quantity. So, how we can really relate them. So, we can see that surface tension is equal to reversible work to create new area that is d w by d a. So, gamma is equal to other quantities of pressure temperature and your number of moles or chemical potential is kept constant. Whereas, what is happening in the second cases that if we consider that surface energy we need to break the bonds. So, for surface energy we are we are breaking the bonds and we are getting some enhancement in the free energy per unit area of the new surface which is being formed. So, we can see that delta G can be equal to when a 1 and a 2 are the two equilibrium positions. So, two equilibrium states we have d is equal to gamma a 2 minus a 1 basically while keeping everything the pressure temperature and the n 1 is constant. Now, second concept can be that gamma is a surface tension is a function of orientation of crystal as well. So, in solids we have plane of 1 1 1 which is being stretched or plane 1 1 0 which is being stretched or plane 1 0 0 which is being stretched. The overall energy or the surface tension required for stretching those surface will be very very different. So, we require a surface stress tensor that has to do work because of the strain which is to be induced on a particular surface orientation. So, that that basically helps and helps helps us in controlling the energy to the surface tension which is a vector quantity. So, we require certain energy to deform a certain orientation of a crystal for assisting the surface tension which does work on a particular surface. And seeing that d w is the reversible work that is being done while keeping the temperature volume and the chemical potential constant to increase the area unit area d a without changing any volume or state of each phase. So, gamma as we saw that gamma is equal to d w by d a while keeping temperature volume and the chemical potential is the constant. And the total surface work s can be given by gamma or the omega s that is the function again of the gamma into d a. So, what we can see if you have phase 1 or phase a to phase b the entire layer between them is called a interface that can be seen as a 2 d defect. So, equilibrium shape of this interface will be given by the minimum value of the integral. So, we have the integral out here. So, equilibrium shape can be given by the minimum value of this particular integral. So, that no work is done on the bulk phase. So, bulk phase will not do any work the work is done by mainly by the interface. So, that will dictate the overall curvature of this particular interface. And as we saw earlier that depends on the pressure difference which can again be given by gamma 1 by r 1 plus 1 by r 2. So, if you want to induce a relationship which is between the surface stresses and the surface energy in solids. So, what we can do we can try to do a small experiment virtual experiment to see the difference between what is happening when the how the surface stresses and surface energy are basically being related in the current work. So, let us take a crystal. So, let us take a crystal assuming all the all the lengths, breadth and heights are unity. So, we can see that we have a unit crystal. So, we can take as x z and along y. So, what we do what we can do here is we can stretch. So, first step is we can stretch and split whereas, in second case what we can do we can split and then stretch. So, once we are stretching it. So, let us say we stretch it along the x direction while keeping y and z as constant and then do the splitting. So, what we have done? So, what we have done instead of x now we have made it delta x. So, now what we have done we have now increase it. So, along x we have made it 1 plus d x in this case it is z it is again 1 and then we have again 1 on the y. So, we have y as 1 z is 1, but in x we have 1 plus d x. So, now we have some additional surface along the x direction and then we what do we then then we split it. So, that I can get a crystal. So, in this case what I am getting. So, I have 1 plus d x and then 1 plus d x this is length is 1. Now, in this case I have z as half. So, this is the first case. So, we took a crystal of 1 1 1 each unit length along x along y and along z. So, we have unit length along x y and z what we are doing in the first case is we are stretching and splitting it means we are stretching along the x direction. So, instead of 1 unit distance I am I am extending it to 1 plus d x and then I am splitting it along the z direction. So, it has been split along z. So, we can see along z it has been split it has been stretch along x and it has been split along z direction. In second case what we what we can do in second case what we can do we can split it first along z direction. So, in this case we achieve the splitting first. So, in this case we have 1 we have half again this is 1. So, along x I have 1 I have z along y also I have travelled along 1 1 unit length but, along z I have now split it. So, I can get half of 2 2 cells which have height of half and the second step is to stretch it then along the x direction. So, I can get the similar crystal as I got in the first case. So, in this case what I am getting now I am getting this crystal the top crystal top half crystal and now I am extending it I am extending it to half. So, I can extend it to d x. So, I add 1 and then d x and the same crystal I will try to get d x out here. So, I have now half half 1 and then 1 plus d x along the x side. So, what do we have with the similar final conditions, but in first case what is happening we are first stretching it and then splitting it, but in the second case what we are doing we are first splitting it then stretching. So, in this case I can get to see the work which is done by the extra surfaces. So, in first case we are stretching and then splitting. So, in the once we are stretching it then I have some additional surface available and then we can see what is the work which is done by the additional surface, but in second case I am first splitting it and then it is being stretched. So, we can see the corresponding part of this can be. So, when I am stretching it and splitting it. So, in stretching what I was getting I was getting overall new surface energy which was gamma plus d gamma because now because I have now a new surface. Also the work done on stretching was now considering to be w naught. So, w naught is the work done on stretching what is the strain which is being developed on stretching is it is the extra work this is d x divided by total length initial length it is equal to d x. So, d x is the strain which is being developed on the stretching. So, strain on stretching is given by d epsilon along x direction. So, we can see the overall work done the total work is given by work done to stretch that is w naught plus work done to split. So, now we can see the total work is given by work done to stretch plus work done to split was 2 gamma plus d gamma because we are creating 2 new surfaces it is not 1, but 2 crystals which have generated because of stretching and then splitting. So, once we split we are seeing 2 surfaces. So, once the stretching work due to stretching is w naught, but for splitting we are creating 2 new surfaces which are gamma plus d gamma there is a new surface energy because of the stretching and now it is been multiplied by the strain that is being developed that is 1 plus d x multiplied by the term. So, what we can see it is equal to w naught plus 2 gamma plus d gamma this gamma plus d gamma multiplied by 1 plus d x and the second case what is happening the total work done is equal to work done to split and work done to stretch. So, work done to split work done to stretch is say w 1 because in this case I have only 1 1 1 along all the 3 direction and that is being stretched. So, the work done in stretching is different than what was the work done in the earlier case which was stretching in this case the stretching was done first. So, that resulted in the surface energy and the overall length to which it was extended, but in second case when I am splitting it I am getting a total surface energy of 2 gamma multiplied by 1. So, I can see that the total work done is given by 2 gamma plus w 1. So, I have equation 1 and equation 2 and they have to be equal because the final condition is the same for both of them. So, we can see the work done on stretching the split halves is different from the work done in stretching the unsplit halves because of some surface stresses. The overall change or difference between them is arising because of the work done which is being done by the surface stresses. So, we can see w 1 minus w naught is equal to work done by the surface stresses because in this case I had only the surface of 1 1 that was being stretched. In first case what is happening first we stretch it and then we split it. So, we can see that once we subtract the splitting and stretching minus stretching and then splitting we are allowing some work done which is to be done by the surface stresses. So, we can see that because w 1. So, in this case we are seeing the generation of additional surface when 2 surfaces are being stretched. So, we can see that w 1 minus w naught which is equal to w naught plus 2 times gamma plus g gamma 1 plus d x. So, now w 1 w 1 minus w naught will equal 2 gamma plus 2 d gamma and this term also equals the work done which is done by the surface stresses this also equals 2 times sigma x d epsilon x. So, we can see that is equal to 2 d gamma this term will be very small. So, we can ignore this term. So, now we can see then that we can see that d epsilon can be taken away. So, we can see that sigma x is equal to gamma plus. So, we can see that the work which is being done by the surface stresses equal to surface tension plus the change in the surface tension because of the stretching which is being due to the strain which is being generated in the material. So, that part we can see that surface stresses and surface tension they are not equal. So, we can see that surface stresses it does not equal the surface tension in case of solids. So, in this case we can clearly see that the work which is being done by the surface stresses is very different when we split and stretch. So, in this case what we are doing we are now allowing any additional surface to act on the material and that is resulting the enhancement in the surface overall energy which is being generated and in comparison to the stretching and splitting and that part we can see that additional work is being done by the surface stresses. So, that is equal to 2 sigma x into d epsilon and that is being related to the difference in the work then which is done because of the stretch and split minus split and stretch. So, that part we did see that and from that we can see surface stress equals the surface tension plus change in the surface tension because of the strain and that actually results that surface stresses they do not equal surface tension in case of solid because there is some additional term which arises which raises the deformation of the solid surfaces. And similar in a similar manner we can also see it for the shearing process that shearing and then splitting and second case is splitting and then shearing. So, we can see the similar thing while incorporating the shear stresses. Similarly, for shearing process assuming that area does not change during the shearing we can have shearing followed by splitting and then splitting followed by shearing. So, we can see that the total work done in shearing and then splitting. So, once we are shearing it we can get a overall work done which is W naught plus once you split it we have now enhancement in the change in the surface tension that is given by d gamma gamma plus d gamma multiplied by 1. So, what do we get is W naught dash which is different from the work which was done by the surface stresses and this is now the shearing stresses. So, we have we see W naught dash plus 2 gamma plus 2 d gamma, but once we split and shear what do we see that we have to do work for splitting at first. So, we can directly see that the total work done in this case is equal to work done to split that is the 2 gamma plus work done to shear. So, that is W 1 dash and again this is different from what we obtained on the earlier case. So, in this case we have material and then we split it first and then we shear it, but in first case we are first shearing it and then we are cutting it down splitting it. So, we can see that we get a similar kind of ships before and after, but overall thing is that we start with the same entity and then we finish it with the. So, in this case also we can see the overall work which can be done by the work which is being done by the surface shear stresses. So, that is again W 1 minus W naught dash that is given by 2 tau x y d epsilon x y. So, what we can see here is we can see 2 gamma 2 gamma plus 2 d gamma minus 2 gamma which is equal into 2 tau x y d epsilon x y we can see that tau x y is equal to d gamma y. So, we can see the similar fashion that this term does not appear for the liquids. So, again for solids we can see that the shearing stress is basically dependent on the change in the surface tension due to the shearing strains that is basically changing the overall surface energy of the system. So, we can see the surface energy is not equal to surface tension in case of solids and we can see that the surface shearing stress is also doing some work in case of solids. So, what is happening here in this case is we can see that the surface atoms they show an increased separation as compared to the bulk. So, in case of liquids we have some separation, but the surface will show some increased separation compared to the bulk that is some sort of a negative pressure which is parallel to the surface and that is called surface tension because that actually what it is doing it is now constructing the liquid to form a contour shape. So, that is why we can see that the rain droplets they come out to be spheres because now it is trying to stretch this is a negative pressure it tries to stretch them along the surface try to compress them. So, in that stretch them. So, that is why we can see here because of surface tension we see a globular or the spherical nature of the liquids and the atomic displacement of surface atoms such that they are perpendicular to the surface or it means that sigma z equal to 0 or the liquid surface the surface of the liquid is in the plane stress condition there is no strain which are acting beneath it or the stresses along the directions are 0. So, that is why we can see all the stress are confined to x y plane and the stress along the z is 0. So, there is no stress along that. So, there is no all the stretching is occurring along the x and y direction only and that makes a plane to become kind of a globular in case of liquids when it is freely available. So, that is very nice finding of this one that the surface atom surface atoms of the liquid that show an increased separation as compared to that of bulk and that is more or less like a negative pressure that is building on a surface and that pressure is makes it to be in a plane stress condition it or it means the stress along the z directions are 0. So, in summary we can see that the overall stresses which are acting the surface stresses they are equivalent to surface tension plus an additional term. So, we can see surface energy will equal surface tension surface stresses only for liquids because this d sigma by d epsilon term is absent for liquids, but in case when we have this particular term the surface stresses are basically being dictated in this case. So, we can see we have additional term of change in the surface tension with respect to the strain that is being present on that. So, equality of gamma and sigma depend on the ability of surface to maintain its configuration while it is being stretched because in liquid there is no strain in the third direction. So, what is happening the mobility of atoms is very very faster in liquids that makes it equivalent that sigma becomes equivalent to gamma, but in certain cases like in grain boundaries or disorder grain boundaries we have two relaxation approximately equal to two stretching. So, in those cases it can behave as a liquid film, but in liquids as such the relaxation time is much much smaller than the stretching time. So, what is happening in that case is the films are already in equilibrium even when the stretching is being done. So, in that case we can see that d sigma by d epsilon is basically not present and in that case we can see sigma x equal to gamma or surface energy equal to surface stresses when gamma is not changing with the stretching process. That is the overall deal with the surface tension and surface energy or the surface stresses relation to the surface energy. So, we can see that the origin of gamma is attribute to the broken bonds. So, if you have a surface we can see that the roughness in the surface can be described by a terraces or the ledges. So, we have this terraces which are like a stepped structure and there is some vertical broken bonds which are the ledges in a system. So, we can see the total length of the ledges is given by the total length. So, this is a 1 plus a 2 plus a 3. So, we can see that the ledge or the terraces. So, terraces can be given t 1 plus t 2 plus t 3. So, the total number say we can say to t and then we have total area of terraces and then area of ledges can be given by ledge 1, ledge 2, ledge 3. So, l 1 plus l 2 plus l 3 and so on. So, we can see the total length of this one is around a l and seeing the similarity we can see that cosine theta if we consider it like this. We can see that cosine theta is equal to b by h that is the base by a hypotenuse that means it is a t divided by total length that which is s sin theta is equal to perpendicular distance that is the a l divided by s. So, we can see that the tan theta can be given as e by b that is a l divided by a t. So, the total gamma theta or the origin of gamma is arising because of the total energy per unit area. So, this tension is given by area energy per unit area that is equal to now gamma t a t plus gamma l a l divided by the total length. So, in similarity we can see that gamma a t is a t can be given as s cosine theta. So, what we can see here is gamma t s cosine theta plus gamma l a l is given as s sin theta we can see it out here that a t is equal to s cosine theta and a l is equal to s sin theta. So, in this term we can see divided by s we can cancel out s from here and now we can see that gamma theta the total gamma which is given by gamma of the terraces multiplied by cosine theta plus gamma l multiplied by sin theta. So, now we can see the overall origin of gamma is arising because of the broken bonds as we can see the broken bonds are given by the total length along this side this side. So, the overall contribution which is arising from these broken bonds is now being captured by this surface tension or surface energy for this particular case. So, in this case we can see the overall surface energy is dependent on the surface energy of the terraces and the links or the ledges which are basically being present in a material because of broken bonds. So, as soon as we are starting to break the bonds we can see the change in the surface energy. So, in summary we can basically we say that the surface tension is given by a reversible work which is done to increase the surface of liquid by a unit area and that creates the attraction between the liquids molecules. So, surface tension is being given as a reversible work which is done to increase the surface of liquid by unit area. So, we need to do some work of the reversible work which is done by the surface to increase the surface of a liquid by a unit area whereas, surface stresses are the reversible work per unit area is needed to elastically stretch a preexisting surface. So, surface stresses are doing some work to elastically stretch already existing surface energy is given by increase in surface energy when we are increasing when we are creating a new surface. So, increase in the increase in free energy per unit area when a new surface is being created. So, that causes the disruption of intermolecular bonds when a surface is being created. So, in summary we can see we can say that the surface tension it is nothing but a reversible work to increase the surface of a liquid by a unit area. So, once we are increasing the surface by a unit area then we see some additional work or some work which is required to be done by that for increasing the surface area and that is nothing but the surface tension. And when surface tension starts depending on the straining of the material then it becomes essential to denote that by surface stresses. So, we can see the surface tension is equal to surface energy for liquids, but in solids what is happening the stretching is basically being dependent on the the overall surface tension is dependent on the orientation of that and that results the difference in the strain which arises in the different orientation. So, once we take a plane of 1 0 0 1 1 1 1 1 0. So, depending on the orientation of atoms out there the stretching will be very very different and the surface energy of those regions will be very very different. So, what is happening is we define them we define the solids via a term called surface stresses and that surface stress equals surface tension plus change in the surface tension due to the change in the surface tension due to with respect to change in the straining of the material. So, we can say that a surface stress equal to gamma plus d gamma by d epsilon and that is nothing but the surface stress. And surface tension is basically equivalent to surface energy when we talk about liquids, but in because the liquid d gamma by d epsilon term is totally absent. So, that we can see the reversible work per unit area is needed to elastically stretch a preexisting surface. So, we are taking a preexisting surface and then we are trying to elastically stretch it that gives rise to the surface stresses. Whereas, surface energy and surface tension they are equivalent for liquid whereas, in case of solids it is a creation of new surface is being dictated by also change in the also change in the surface tension with respect to the strain that is being generated out there. So, in this case we can see that surface energy or the surface stresses are nothing but surface tension plus change in the surface tension with respect to the strain change in the strain that results the overall surface stresses. So, that is the difference between surface tension, surface stresses and the surface energy. And as we also see that we can evolve the surface stresses via splitting and then stretching and then first stretching and then splitting. So, once we are splitting it now we are creating two surfaces and then we are stretching it. So, that is that means that now we are letting surface stresses to act on it because once we split we now have one additional surface or two additional surfaces which are which have been generated. Whereas, in stretching and splitting we take only the deformed surface and then we are splitting it. So, we can see that we can allow surfaces to do additional work in reaching the final entity. So, from that we can always evolve what are of stresses are being generated and how they can evolve in terms of adding a new term that is called d gamma by d epsilon to achieve the final thing. So, then that is now being collated to the work which is being done. So, we can relate 2 sigma x d epsilon with respect to the w 1 minus w naught and that reveals the overall dependence of surface stresses with respect to surface tension. And with this we can see the complications that are arising. So, in simplicity surface tension is the reversible work which is required which is going to increase the surface of a liquid by unit area. So, surface tension is directly related to the liquids. Surface stresses are the reversible work which are needed to elastically deform or stretch a preexisting surface. Whereas, surface energy is the increase in the free energy per unit area when a new surface is being created. So, that basically distinguishes the three categories of surface tension, surface stresses and surface energy. Surface energy is a very generic term and that needs to be used for both surface tension as well as surface stresses. Whereas, surface tension is limited only to liquids, surface stresses more for the solids. So, that is how we can see that how these three terms are very essential and depending on them we can again tailor them for certain processing requirements or also determining the biological or electrical or mechanical properties based on these features. So, with this I end my lecture. Thank you.