 Welcome to the course of nanostructured materials, synthesis, properties, self assembly and applications. We are going to start the first lecture of module 3. We earlier had module 1 comprising 2 lectures and module 2 comprising 12 lectures. So, totally we have completed 14 lectures and today will be our 15th lecture of for the whole course and this is the first lecture of module 3. And the first lecture we will start with fullerines and carbon nanotubes and we will have three lectures on fullerines and carbon based nanotubes. Today we have the first lecture of this. Now, fullerines as some of you must have heard discovered in 1985, but previous to 1985 also there were reports about it. So, we will discuss the historical development of this structure nanostructure called fullerine. You must have all known that normally carbon has three allotropic forms one is diamond the other is graphite and we have amorphous type of or non crystalline carbon. Today we know of many other forms of carbon and among them the first to be discovered among these new forms of carbon is the C 60 molecule which is also called the bucky ball and is also called fullerine. Today there are many other carbon based molecules which are related to this bucky ball with large number of carbon atoms per molecule and there are more than 30 forms of such fullerines. Now, why this is called fullerine we will come to that it basically looks like a soccer ball which has got 6 membered carbon rings and also 5 membered carbon rings. So, this typical molecule has 60 carbon atoms which is arranged like in a soccer ball or a football as we know in India and if you replace the vertices of each of those with carbon then this becomes a C 60 molecule and this was discovered in 1985 and a Nobel Prize was given for this discovery. So, historically was this discovery of C 60 in 1985 the first time people thought of this molecule no people have thought about such molecules much earlier for example, it was predicted by Osawa in 1970 that there can be a molecule like C 60. In 1970 there was another proposal of a model of C 60, but the experimental evidence was not very strong and hence this structure was not accepted at that time. These results have been later acknowledged for example, in the journal carbon in 1999 much later that Henson proposed this structure in 1970 today we know that it is correct and it is known experimentally. Apart from that in 1973 in USSR earlier USSR it was calculated using quantum chemical calculations that a molecule like C 60 would be stable and the electronic structure of the molecule was calculated. The energy levels of the molecule in molecular orbital was calculated for a molecule like C 60 theoretically and a paper was published in proceedings of the USSR Academy of Sciences. So, there were several such studies which were kind of indicating that there can be a molecule like C 60 a very symmetrical molecule a molecule which looks like a football or a soccer ball and this molecule was predicted theoretically and some experiments was done, but not clinching evidence was not there. But in 1985 a team of people from England and USA contributed to the actual discovery of C 60. The main players in the discovery of C 60 were Harold Croto, Robert Curle and Richard Smalley and James Heath was a student at that time and these four people worked in Rice University. Though Harold Croto who was visiting a Rice University from UK and together they discovered C 60 in 1985 and the idea behind their discovery was they were looking for molecules and which can be synthesized in the upper atmosphere or in space where plasma exists and they were trying to create a plasma or through an electric discharge where such molecules which are probably formed in outer space can be recreated in the laboratory. During that arc discharge of using graphite electrodes they found a black colored material getting deposited and then it was analyzed using several techniques including NMR or nuclear magnetic resonance and they confirmed that it is a C 60 molecule a molecule which has 60 atoms of carbon and the structure was just like the structure we showed like a soccer ball. So, this was the discovery of C 60 by Croto, Curle, Smalley and their co-workers at Rice University using electric discharge and then analyzed by several techniques gave the structure which was exactly which was kind like a soccer ball with 60 carbons having hexagonal rings and pentagonal rings. Later such C 60 or fullerene kind of materials have been found naturally occurring in Russia and it has also been discovered in cosmic dust in a distant star several light years several thousand light years away. So, C 60 molecules are present in the in space they are also found in minerals and of course it was made in the laboratory and for their work Harold Croto, Robert Curle and Richard Smalley were awarded the 1996 Nobel Prize in chemistry though the discovery they made was in 1985 they were awarded the Nobel Prize in chemistry in 1996. It was a great discovery a new allotrope of carbon was discovered and it brought about several new molecules related to C 60 like C 70, C 80, C 82 and several large cage like structures or ring with cluster like structures were discovered after these fullerene or C 60 was discovered. So, the type of fullerene that we know today are like the common C 60 molecule we just discussed apart from that we have now made several nano tubes which are made up of carbon. These hollow tubes which have very small dimensions can have single walls or multiple walls based on carbon these nano tubes are very important for several applications as we will discuss in the coming lectures. Then you can have mega tubes much larger in diameter than the nano tubes and can be prepared with walls which have varying thicknesses. Then you can have fullerene rings you can also have fullerene which are linked by carbon chains. So, those are called ball and chain kind of dimers. So, two bucky balls or two C 60 C 60 molecules which are linked by carbon chain will then be called a ball and chain dimer. You can have two bucky balls which are connected to each other to form fullerene rings. Then you can have what are called nano onions spherical particles based on multiple carbon layers surrounding a bucky ball core. So, you have a C 60 core and there are layers of carbon surrounding that C 60 molecule. So, you may have five six seven layers of carbon which is surrounding a C 60 molecule. So, if you take out one layer another layer will be there. So, that is why the structure will be like an onion and hence it is called nano onion because the dimension of the C 60 molecule is nanometer in size less than one nanometer it is around 0.7 nanometers. Now, the name fullerene comes from the name of an architect Buckminster Fuller who first made domes which are like the fullerene structure. So, these geodesic domes as they are called have been made in several places in Canada and many other places which have domes which are in the shape of this kind of hexagonal rings and pentagonal rings fused together. And if you take half of it then it forms a dome which is very similar to domes which Buckminster Fuller an architect made in the 1970s in several countries. And so, this molecule is called fullerene after the name of Buckminster Fuller. So, fullerene have three dimension network of carbon atoms they contain pentagonal and hexagonal rings in which no two pentagons share an edge. So, hexagons can share an edge one hexagon can share an edge with one pentagon, but two pentagons cannot share an edge then each atom is connected to exactly three neighbors three other carbon atoms. And each atom is bonded to two single bonds and one double bond example in C 82. So, again going through the main characteristics of C 60 molecule there are 60 carbon atoms there are 5 membered rings and there are. So, this is a 5 membered ring as you see here there are 5 carbons and there are 6 membered rings here. And this 6 membered ring can be connected to another 6 membered ring in this C 60 molecule, but 2 5 membered rings cannot be connected to each other. The van der Waals diameter that means the distance between the electron clouds from one end to another end is about 1.1 nanometers. So, that is like 11 angstrom, but if you take the nucleus of the carbon here and the nucleus of the carbon then that distance is around 0.7 nanometers which is much less than 1 nanometer. So, it is on an average you can just say that the fuller in molecule is of the order of 1 nanometer, but of course, it depends on what kind of diameter are you defining if you take the van der Waals diameter it is 1.1 nanometer. And if you talk about the nucleus to nucleus diameter then it is 0.71 nanometer which is equal to 7 angstroms 7.1 angstroms. Then you can have other than C 60 which has 60 carbon atoms you can have C 70 which has 70 carbon atoms you can have C 72, C 76, C 84. So, you can have fullerines with different number of carbon atoms, but the shape will change slightly. Now, your number of hexagons and number of pentagons will change because the number of hexagons and the pentagons are related always you will have 12 pentagons the hexagons will change and that number will be equal to V by 2 minus 10 where V is the number of vertices. So, this is given by Euler's polyhedron formula. So, if you have C 82 if you have 82 vertices then you need to know how many edges are there and how many faces are there and you can use this formula V minus E plus F equal to 2 where V E F are the vertices edges and faces. And you can find out that there will be exactly 12 pentagons and V by 2 minus 10 hexagons. So, if you have C 84 if you have 12 pentagons then you if you the number of vertices is 84. So, 84 by 2 is 42 and 42 minus 10 is 32 hexagons. So, you can say that C 84 that means the cluster or the molecule C 84 will have 32 hexagons and 12 pentagons. So, in all these fullerines you will have variable number of hexagons and you will have 12 pentagons. And the number of hexagons you can find out if you know the number of vertices and the number of vertices you can get from the number of carbon atoms you have on this cluster. So, you can have a large variety of these fullerines all related through hexagons and pentagons of carbon. Now, recently in 2007 instead of carbon a boron have at the vertices has also been created to lead to a bucky ball kind of structure. However, the formula which has been obtained is B 80 and with each boron forming 5 or 6 bonds and it is predicted that it will be more stable than C 60. So, this kind of boron bucky ball has not been isolated it has been predicted like theoretically and described. And it is suggested that if it is made then this V 80 molecule will be more stable than C 60. So, there are lot of new things related to fullerines which are still under research. How do you synthesize these fullerines the C 60, C 70 etcetera. So, the technique which is used is either you take a graphite graphite is basically carbon one allotrope of carbon where which has got sheet like structure or layers of carbon forming hexagonal rings and each layer separated from another layer by van der Waals distances of around 3.3 angstrom. And this graphite if you evaporate by shining laser then you can get some suit some carbon material deposited from where you can extract fullerines. Another method is what is called the arc discharge method. This is a method where you use two graphite electrodes. So, again graphite is being used, but you have two graphite electrodes. So, you know in the electrodes you apply a potential and you generate a discharge between the two electrodes. So, there is a small gap between the two electrodes. And between that arc has to be formed. So, when you apply a very high potential on the two electrodes then and an arc is created then some suit is deposited in the chamber around these electrodes. And this technique was developed in 1990 by Crashmer and Huffman. And this also leads to several fullerines. So, you can use the technique of shining laser on graphite and collecting the suit or you can use a arc discharge method to using graphite electrodes by which you can generate suit which collects on the inner walls of the chamber and which you can collect and from that you will get a variety of C 60, 70 type of fullerines and then you have to separate them. Then you can use other techniques like where electron beam evaporation where you can produce the higher fullerines to a larger amount or you can take some aromatic compounds and then heat them. So, you take aromatic compounds like normally heat them and you can get a suit from that suit also you can get some fullerine type of compound. Of course, the proportion of which fullerine you get depends on the method and to get pure fullerines by one method is quite a tough job. Normally you have to separate these fullerines once you get the suit through different chemical processes. Now, what would fit inside a bucky ball or a C 60 molecule? So, you see this is a C 60 molecule and if can we put something inside that that has to be smaller in size than this diameter of around 1 nanometer or 0.7 to 1 nanometer depending on how you define the radius the distances. So, either you take the van der Waals distances or you take the distance between the nucleus to the nucleus at the end of these diameter. So, what would fit inside a bucky ball? Of course, it has to be much smaller than 7 angstrom or much smaller than 0.7 nanometers and such materials where there is something inside the cage are called endohedral compounds. These are also called cage compounds or endohedral compounds. So, nitrogen can you put an atom of nitrogen? Sure you can put an atom of nitrogen because what is the diameter of the nitrogen atom? It is of the order of 1.2 angstrom or 0.12 nanometers which is much smaller than 0.7 nanometers is the diameter of the fullerine. So, definitely you can put nitrogen can you put hydrogen? Hydrogen also we can put because it is of the order of 1.5 angstroms or 0.15 nanometers. So, we can put a molecule of hydrogen in this cage. Now, can we put a larger molecule say a molecule of sulphuric acid? So, molecule of sulphuric acid has a diameter of approximately 7 angstroms which is like 0.7 nanometers and you know this whole thing is of the order of 0.71 nanometers. So, this will be very difficult or more or less impossible to put a molecule of sulphuric acid inside a C 60 cage. So, not likely. So, this is not likely to be the case. Now, such molecules like if you put nitrogen inside this C 60 molecule then these are called endohedral compounds and these endohedral compounds are given by this formula where M is whatever you have put inside the cage. So, if it is nitrogen then you write N at C 60 that means N is inside C 60. If you put lanthanum a rare earth element and it is known that it goes in inside this cage, but lanthanum goes in to a larger fullerine C 82 which has a larger diameter than C 60 and lanthanum is a bigger atom than nitrogen. So, lanthanum goes in a C 82 kind of a fullerine, but not a C 60 fullerine whereas a small atom like nitrogen can go inside C 60. So, both of them are endohedral fullerines because they are lying inside the cage and the formula is M at C 60 where M is inside the cage. Now, can we have exohedral this one was endohedral means inside can we have exohedral yes we can have. So, this is the C 60 molecule and you have got molecule on top of it outside. So, this is called an exohedral compound. So, we can have endohedral compound and we can have exohedral compound in the exohedral compound you can have either inorganic groups. So, you may have a metal with some ligands. So, you can have say platinum or some nickel or gold some metal with some accompanying ligands on the surface of the C 60 or C 70 molecule and this is another projection. So, the C 60 or C 70 molecule is inside and you have got these atoms outside. So, this kind of exohedral compounds have also been made in the laboratory and these give you lot of applications because you can modify the properties of C 60 C 70 using these molecules which are attached outside on the periphery of the surface of C 60 or C 70 or other fullerine type of compounds. Now, you can also have atoms bound to fullerines as salts for example, you can have a salt where you have a positive metal and a negative anionic C 60. So, when you have anionic C 60 then it is called a fulleride. So, if it is neutral say C 60 molecule then we call it a fullerine when you have a anionic C 60 then we call it a fulleride. So, this metal cation is like a typical cation that you study in chemistry cations are formed from elements which donate electrons easily. So, you have these alkali metals alkaline earth metals like lithium sodium potassium or alkali metals and you have elements like calcium barium strontium which are alkaline earth materials they like to donate electrons these elements. So, when they donate electrons the electrons go to the C 60 and C 60 can accept those electrons because there are lot of orbitals in C 60 pi orbitals and you can transfer electrons and then C 60 becomes negatively charged. So, depending on how many electrons you donate this charge will change from 1 minus 2 minus 2 n minus. So, you have a cation which is outside and you have x the number to balance the charge over here. Now, what happens when you add these electrons when you add these electrons then the electrical conductivity or the resistivity changes because now you are adding electrons to this fullerene moiety which becomes a fulleride and the electrical resistivity decreases by several orders of magnitude in the case when you are adding alkali metals ions. As x increases you reach a minimum in the metallic resistivity for x equal to 3. So, the typical formula that you can generate in these fullerides are something like m 3 C 60 that means 3 moles of potassium or rubidium or cesium per 1 mole of C 60 that is the maximum it goes. So, in that case you will have 3 electrons transferred to C 60. So, the charge here will become 3 minus and actually several of these materials like k 3 C 60 or rubidium 3 C 60 becomes superconducting at low temperature. So, at 30 Kelvin which is minus 243 degree Celsius for the metal being rubidium which is a alkali metal down the group this compound rubidium 3 C 60 becomes a superconductor at low temperatures of 30 Kelvin which is minus 243 degree centigrade and has the superconducting properties which means 0 resistance perfect diamagnetism and will show levitation and other properties that any superconductor shows. So, in C 60 and fullerene kind of materials also you can see superconductivity. Now, you can also recently it has been shown that an organic compound like C H B R 3 it is like bromoform it is called bromoform like you have chloroform for C H C L 3 you have bromoform and this can be added to C 60 to increase the conductivity or lower the resistivity. So, if you have metal ions like potassium rubidium etcetera also you can lower the resistivity by transferring electrons. Similarly, it has been recently shown that organic compounds can also be added to C 60 to show increase in conductivity or decrease in resistivity. Now, both combination of endo and exo-heatrol compounds that means you have gadolinium G D it is a lanthanide that means it belongs to the lanthanide series and gadolinium is inside the C 82 moiety. So, this is the formula for an endohedral compounds. So, gadolinium is inside the cage of C 82 and outside C 82 you have got hydroxyl groups. So, this is the exo-heatrol part and this is the endohedral part. So, you can have a combination of endo and exo-heatrol compounds and this is a classic case. So, this is what I just mentioned the gadolinium is inside the cage and outside is covered with hydroxyl groups and it is possible material very good material for magnetic resonance imaging that is what the research has shown and there is lot of potential in this material. It has also been shown that this material can be used for anti cancer therapy which is very important and this kind of material based on gadolinium can be made as a endohedral compound and can also be made as a endo and exo-heatrol compound. Now, there are several other properties of fullerins. Now, fullerins if you apply pressure. So, you apply pressure from outside external pressure very high pressure like 3000 atmospheres. So, we are at one atmosphere now imagine 3000 times that pressure is falling on an object. So, very high pressure the fullerins get deformed, but as soon as you remove the pressure the fullerins get back to their original shape. So, this is a very interesting property of the fullerins that after being subjected to very high pressure like 3000 atmospheres. If you release the pressure the fullerin molecule again comes back to its normal or original shape the then the fullerins do not bond to each other through chemical bonding. So, they if you take two fullerins they bond to each other through weak van der Waal forces and like in graphite where you have got layers of rings of carbon atoms which do not bond to each other through covalent bonds, but through van der Waal bonds and hence graphite is a good lubricant. Similarly, fullerins also do not bond through covalent bonds to each other and hence they are also used as good lubricants. There are catalytic properties of fullerins which has been shown for example, a very important reaction industrial process which is one of the ten most important industrial processes in the world is to convert ethyl benzene to styrene and C 60 has been shown the fullerins have been shown to be good catalyst in this conversion of ethyl benzene into styrene. There are other properties like electrical conductivity data which can be used in data storage devices in solar cells and in fuel cells. This fullerins also show large non-linear optical response. So, non-linear optical response means that if you have a frequency omega then you can generate a frequency 2 omega or 3 omega that is non-linear kind of behavior is observed when you use a fullerin type of materials and this is important for telecommunications. Now, there have been many applications as drugs and also as vesicles for drug delivery. So, C 60 or other fullerins have or their derivatives have been used as drugs and the C 60 has been used to make vesicles that means channels through which drugs can be delivered inside the body. So, there are lots of properties of fullerins. These are some commercial and biological applications like sunscreens which is due to the photo physical properties of fullerins. They are used as antibacterials due to their chemical reactivity and redox properties and superconducting properties like in the alkaline alkali metal doped fullerides like K 3 C 60 or rubidium 3 C 60 which shows superconducting properties. So, you can have photo physical properties, antibacterials and superconducting properties in these fullerins or the derivatives of fullerins and fullerides. Now, in fullerins or their larger congeners like C 82 etcetera, you had a kind of a spherical molecule like a spherical cluster. But, if you go towards a cylindrical object then we get what are called carbon nanotubes. So, a cylindrical fullerin was discovered in 1991 or was exactly understood in 1991 by Ijima in electron microscopic studies. And this nanostructure has diameter in the nanometer range like 1 nanometer or so, but the lengths can be very large, they can be 100 nanometers or they can even be much longer. So, the internal diameter can be varied from 1 to 15 nanometers and length can be much larger up till several microns. So, several thousands of nanometers you can extend and these carbon nanotubes also called CNTs have tremendous applications. They can be made of a single layer of graphene sheet that means, only one layer of carbon is present and rolled together to form to a tube or there are multiple layers. So, if it is made up of only one layer then it is called single walled nanotube. If it is made up of multiple layers then it is called multi walled nanotube. So, as you see here there are 1, 2, 3 and 4 layers. So, this is a 4 layer multi walled nanotube of course, all made of carbon, but you can also make single walled nanotube double walled nanotube etcetera. They have lots and lots of properties very interesting properties. Many of these are semiconducting in nature, but you can also make conducting nanotubes. So, typical room temperature resistivity is given here it is around 108 ohms and it should be ohms centimeter which is the resistivity of simple carbon nanotubes at room temperature. Now, these carbon nanotubes are made of one atom thick sheet of carbon. So, if you take graphite which has got layers of carbon hexagonally oriented carbon. So, you have got rings of carbon and if you have one layer of carbon only then it is called graphene. But, if you have several layers of carbon one below the other which are connected through weak van der Waals forces then that is graphite. So, graphene is only one layer of graphite. Now, if you take that one layer of graphite which is called graphene and you roll it up in a cylinder then you get the carbon nanotube. And if you have only one layer then you get single walled carbon nanotube. And if you have several layers you will have multi walled carbon nanotube. Now, these sheets if you are rolling the graphene sheet this how are you rolling the graphene sheet will change the nature of the tube which you will get ultimately. So, the sheets which are rolled at specific and discrete angles will give rise to different types of nanotubes some of them will be chiral the others will be called zigzag or armchair as we will discuss. So, you can get single walled nanotubes multi walled nanotubes chiral nanotubes and several other kind of nanotubes. And individual nanotubes align themselves and are weakly held by van der Waals forces. So, if you have a several nanotubes then you get a bundle of nanotubes and these nanotubes interact with each other through van der Waals forces and they kind of form ropes in one direction. Now, the chemical bonding of nanotubes inside the carbons are all sp2 hybridized carbons. So, that is true for all these carbons in these nanotubes. So, to have a look at these nanotubes this is a single walled nanotubes and this diameter is of the order of one nanometer and this is a multi walled nanotubes. So, you have one nanotube here and then this is the second nanotube and then you have a third nanotube. So, this is a multi walled nanotube it was observed first by Endo in 1975, but was really highlighted by Ijima in 1991 and the world came to know about carbon nanotubes through Ijima's work in 1991. Now, these are some of the real pictures the transmission electron micrographs of multi walled nanotubes and you can see some of them are broad and wide and this scale is of hundred nanometers and so this diameter is this is a very thick nanotube. These are thin nanotubes so this may be of the order of may be 5 or 10 carbon layers are there in this tube. So, this may be a 10 nanometer or 5 nanometer tube this may be a 15 nanometer tube and none of them are single walled nanotubes because single walled nanotubes the diameter will be of the order of one nanometer in general. Now, so nanotubes can be straight they can be spiral and this they can be of the type of springs they depend how you grow these nanotubes then you can control then you need to know how to control the shapes of these nanotubes how to get them straight how to get them in the spring fashion for certain applications. So, that depends on the growth conditions how you are doing the discharge or if you are doing using a laser how are you creating these nanotubes are you using a metal catalyst many times metal catalysts are used to grow carbon nanotubes. So, all these things matter to ultimately control the shape of these nanotubes now here you can see a transmission electron micrograph of bundles of single walled carbon nanotubes. So, these are a single walled nanotubes, but there are many such nanotubes. So, this one nanotube second nanotube is the third nanotube like that there are several nanotubes which are forming a bundle right. So, this ability of these nanotubes to come together is through van der Waals forces and these are again pictures of nanotubes this is a curved nanotube and you can see the scale is of 5 nanometers. So, this diameter is of the order of 1 to 1.5 nanometers typically for a single walled nanotubes. So, these are all single walled carbon nanotubes. Now, how to roll the nanotube as we were discussing if you have one layer of graphite which is called graphene. So, this is a graphene sheet. So, you have all 6 membered carbons forming the sheet and how do you roll this sheet, because if you roll this sheet in one way you get one kind of nanotube if you roll it in another way you get another type of nanotube. So, there are certain definitions. So, in this hexagonal lattice you define what is called a chiral vector. So, in this two dimensional lattice you define a chiral vector C H which is dependent on two vectors A 1 and A 2. So, these are the two vectors A 1 and A 2. So, in any hexagonal lattice you can define these two vectors and what are the numbers or the coefficients of these two vectors. So, if you take a very large A 1 and a very small A 2 that means N is very large and M is very small you will get one type of rolling. If you take N and M both same then it will not result in a chiral it will result in something else. If you take N some number and M you make it 0 then you get another kind of nanotube again it will not be chiral. So, the chirality of the tube is dependent on this formula and from this formula you can define the chiral angle theta and so these vectors and their coefficients are important. The coefficients are very important and also what is how do you end the nanotube toward if you want to close the nanotube at the end and not leave it open then what how do you cap it. So, these are certain things which give flexibility to the various kinds of nanotubes that you can generate using a simple graphene sheet, but just based on the angle or the chiral angle at which you are rotating this planar structure or rolling the planar structure into a cylindrical structure. So, that is what I said if you take a vector a 1 and a 2 such that the coefficient of a 1 is N and a 2 is 0 that means the vector that you are taking is in this direction because a 2 is 0 and only a 1 exists. So, you are looking at this vector and that means this is the N 0 vector. So, this is called the zigzag nanotube the nanotube that you will get if N has a value and M is 0 then if you roll the graphene sheet in that manner then you ultimately end up in a nanotube whose direction this is this will be the direction of the nanotube. And it is called a zigzag nanotube. So, the coefficient M is 0 however you can choose any other coefficient N and M if you choose N equal to M that means N and M both are same then your direction will be in this manner because you have the same magnitude the coefficient of this vector and coefficient of this vector both are same. So, you will have a resultant like this and that is what it is being shown this is parallel to that result the resultant which you get here and you will get the N N carbon nanotube which is called commonly as the armchair nanotube. So, the zigzag nanotube and the armchair nanotube are two special cases all other nanotubes what if you take any other value of N and M you will get chiral nanotubes but N 0 and N N that means the same value for N and M will give you zigzag and armchair nanotubes. So, this is the N comma N is a naming scheme it tells you about the vectors which you have chosen these coefficients tell you about that and it will tell you about the chirality of the nanotube which will result if you roll the tube in such a manner that the coefficients which you have chosen are N and M and this is very important in finding out the property because the properties of the nanotube will depend on this factor. So, the integers N and M as I discussed tell you about the unit vectors along two directions in the planar graphene layer or the carbon layer and if M equal to 0 the nanotubes are zigzag if N and M are same then they are called armchair any other value of N and M they are called chiral nanotubes and the diameter of an ideal nanotube can be calculated using N M and A where A is this value 0.246 nanometer and actually it comes from this distance. So, it is the distance between this carbon and this carbon the carbon which are 1 and 3 positions if you measure this distance that is equal to 2.46 angstrom or 0.246 nanometers. So, that is the value of A if you use that value and you know what is N and what is M you can calculate the diameter of any nanotube. Now, when you look at these nanotubes which result as you roll them in a particular fashion for example, when M is equal to N then you get the armchair type of nanotube and the angle that which you have is 30 degrees chiral angle of course, this here 5 5 means M is 5 and N is 5 and it gives you a nanotube like that and at the end if you want to close it this part will look exactly as half of the C 60 molecule. So, if you have a armchair nanotube the capping part will be C 60 molecule so, exactly like the C 60 molecule. However, if you have a zigzag nanotube a zigzag nanotube has a theta value of 0 and if you come to the end of the nanotube you cannot close it with a C 60 molecule you have to close it with a C 70 molecule and that is what is shown here. So, if you take half of the C 70 that will exactly fit at this part so, what how to cap the end of a nanotube is also dependent on what kind of nanotube it is if it is a armchair nanotube only half of C 60 can cap it if it is a zigzag nanotube then half of a C 70 molecule can cap it if it is any other kind for example, this is chiral molecule it is not zigzag it is not armchair and the angle theta is neither 0 nor 30 it is in between 0 and 30 then that chiral nanotube will have an end which is neither C 60 nor C 70, but is a half of C 80 molecule. So, as half of C 80 molecule will cap it that means for different types of nanotubes you need different types of caps the properties of nanotubes change significantly with the n and m values and one like electrical conductivity is a very important property and it shows drastic difference you can have metallic nanotubes semiconducting nanotubes and these properties have been used for applications one of the first applications was a molecular field effect transistor and this was made in 2001 by IBM who showed that if use these carbon nanotubes you can make a molecular field effect transistors which is a great invention because you are lowering the size of the transistor from a normal transistor one molecule is being used as a transistor and this was done in 2001. So, there are several other applications of these nanotubes look at the mechanical properties of these nanotubes so carbon nanotube compared to stainless steel carbon fiber glass and Kevlar Kevlar you know is a polymer and it is used in bullet proof vests if you compare even with Kevlar most of these numbers you see for carbon nanotubes are much higher than these numbers right. So, look at the strength the strength is 10 to 60 for carbon nanotube where are none of these are of the order of 10 they are all less than 5 whether it is steel whether it is carbon fiber or whether it is this polymer which is used for bullet proof vests. So, today we have discussed several of the features of carbon nanotubes and fullerins and we will continue our study of carbon based nanostructures in the subsequent lectures. Thank you very much.