[MUSIC] Dear students, from today's lecture, we are going to have our discussions on the defects of crystals. From previous lectures, we know the structure of perfect crystal lattices and in reality, nothing is actually perfect. And in the discussions in the follows, we will talk about zero dimensional defects of point defects and then we will move on to one dimensional defects such as dislocations. And then we will talk about two dimensional defects such as grain boundaries and all kinds of interfaces. And then we will talk about the three dimensional defects such as precipitates and phase aggregations. So, defects are important because it is actually very good for a lot of material properties, for example mechanical properties. Here I show two typical curves for pure aluminum, which is this one and aluminum copper alloy called duralumin, this one. And this is a typical stress and displacement curve, which means the higher the curve, the higher the mechanical strength of the alloy. So, essentially you can see that you have much higher strength in alloy as compared with the pure aluminium. So, this is actually the result of all kinds of defects in alloy. So, you can have solid solution defects such as interstitials, and substitutional elements and you can also have in your alloys, three dimensional defects such as precipitate. And these defective metal alloys have a lot of applications in your daily life. For example, for your portable electronic devices such as cell phones and iPads, you always have these metal frame or cases made of defective metal alloys. So, we start our first topic on defects, equilibrium concentration of point defects. So, point defects include vacancies and interstitials. Vacancies means you have an empty site in your lab lattice forming a vacancy. And interstitial size is where you have additional item occupying empty space between. Neighboring items which are called interstitial sites. And you have this equilibrium point de FIDE concentration in this case it's a vacancy equilibrium concentration at a finite temperature above the zero K or absolute zero temperature. And this Is because of thermodynamic considerations. For those of you who haven't really take a thermodynamic course, here I need to review a couple of slides on the basic concepts of thermodynamics. The first concept is the Maxwell-Boltzmann statistics. Where, if you have a total of n items, and the number of items with a free energy higher than a value q0 is proportional to the total number of items times an exponential term of a minus sign, the energy term divided by the Boltzmann constant KB and absolute temperature T. So from this expression we know that with increasing energy term to zero you will have a much smaller number of items having this particular energy. And with increasing temperature T, you will increasingly have more items having this high energy of Q zero. So, and at absolute zero k temperature essentially, this n number will be zero which means that all your items are in the ground state. So no items are having this high energy. So according to the Boltzmann statistics, we can write the concentration of items or the fraction of the items in the total number of items that have this energy which have this expression with the exponential term of minus energy divided by kt. So this is the basics of Maxwell-Boltzmann constant statistics. So, for thermodynamics,a system always want to minimize is free energy G and there are two contributions to this free energy. One is the internal energy U. The other one is the temperature times an entropy term. Entropy is a thermodynamic parameter which describes, the degree of disorder in the system, so essentially the configurational entropy is proportional to the number of micro space in a system, so the more disorder of the system, the higher the configurational entropy. With these thermodynamic knowledge, we can now calculate the actual equilibrium vacancy concentration. So essentially, we have two contributions. One is internal energy U, okay? This comes from the chemical bonding energy of the crystal. For example, if you have one item jumping from the lightest size out and leaving this vacancy behind. The way you achieve this is to break the chemical bonds. Between the item and its neighbors. So essentially you will have a linear dependence on this internal energy term, so this one, where the internal energy change is proportional to the total number of your vacancy size times energy term called a vacancy formation energy. This end is actually the coordination number of your items in the crystal structure which means how many chemical bonds you have for each of these items. You have to break all the chemical bonds in order to form this vacancy and this is the contribution of the internal energy part. And with the creation of your vacancy you effectively have an increase you effectively you have an increase in this entropy term as well because you have more microstates in your system. So essentially, the change in the entropy term is described by this curve, and effectively the total change of your free energy can be described by, the summation of the internal energy contribution and your entropy contribution. And eventually, you will end up with one minimum value for the free energy change, which corresponds to that your equilibrium state. Now we look at the details of calculation. So suppose you have a total number of items of capital N, and you have a total number of vacancies, which is the lowercase n. And you assume that the number of vacancies is much smaller than the total number of items. So that your total number of lattice sites is the number of items plus the number of vacancies. This one keeps as a constant. Then from knowledge of mathematics, you can write the number of configurations as this one. So, this is pronounced as N plus N chooses N. So from the total N bigger N plus smaller insights you choose the number N number of vacancies. And this can be calculated as the factorial, of the total number lattice size divided by the factorials of your number of vacancies, and the number of items. An example is shown here, if you have three items, and one vacancy site and you have to occupy this four lattice sites, with just a one we can see. And the way to calculate is to have this, C41 which equals four, which means you have four ways, to fulfill this requirement, so that you have four states corresponding to this. One, vacancy all of four lattice sites. So, this is how you calculate the total states of the system with the presence of vacancies. Then you can calculate the configurational entropy through. This equation. So this is the derived from the definition of your configurational entropy. And this is fairly complex because you are dealing with quite a few factorial values and fortunately. So fortunately, we can simplify this relatively complex factorial calculation by this Stirling approximation, on condition that you have sufficiently large number of lattice sites. Especially when you have lattice sites more than about 100 times and then you can simply find this factorial calculation to the simple analogical solution. And just for your knowledge, you can have your maximum. Configurational entropy when the number of your vacancy size is about half of your total number of lattice sites. So where's the Stirling's approximation, you can then put all your numbers in. And finally, you get an expression for your free energy change as a result of the generation of a certain number of vacancies. And you have this extra term of vibrational entropy as well, which is relevant was the vibration of the lattice. And this one is usually much smaller or negligible as compared to this configurational entropy. So, with this equation, you can gather minimum value of your free energy by doing the differential of your free energy change over your total number of vacancy sites. So if you equal this to 0. Eventually, you can get the number of vacancy sites at equilibrium. So eventually you can end up with the exponential term, like expression for the equilibrium vacancy concentration, as we have discussed at the beginning of this lecture. So essentially. The concentration of vacancies, can be expressed by a constant times this exponential term times of this minus energy formation, energy for vacancy formation over Boltzmann constant times the temperature. So, this is a quantitative example showing how the different parameters will affect your equilibrium vacancy concentration. So suppose we have a total number of lattice size of 10 to the orders of 12. And we have a vacancy formation energy of 0.58e vote and the temperature is 300 K and your vibrational entropy is much smaller than the vacancy formation energy. And you end up with a relatively large number of vacancies at equilibrium of several 10s of vacancies. And however, if you choose to slightly increase the vacancy formation energy from this 0.858 eVolts to 0.6 eVolts, you will have a much, much fewer vacancies at equilibrium of almost zero just to several vacancies at equilibrium. So this means your vacancy formation energy have a great impact on the number of equilibrium vacancy concentration. This can be exemplified by these two simple examples, and this case with relatively low vacancy formation energy. You have a large vacancy equillibrium concentration, it is like you have a relatively empty bus, right you have a lot of vacant seat and with high vacancy formation energy, you have very few vacancies at equilibrium. And this is like you have a very crowded bus with almost zero seats available. So similarly, we can talk about the equilibrium interstitial concentration from our previous lectures on crystal structure. We know that there are interstitial sites in crystals. For example, you can have these octahedral interstitial sites in your FCC structure. And you can also have an Octahedral interstitial sites in your BCC structure. However, the equilibrium interstitial concentration is usually much, much smaller than a vacancy concentration. The reason is that to form an interstitial. You need to distort the lattice which means your interstitial item kind of push or pull your neighboring items. And this is a string energy cost will increase your interstitial formation energy. So effectively, you have a much larger term here for interstitials formation energy. So eventually you will end up with a relatively small equilibrium concentration of interstitials. So subsequently, we'll talk a little bit about the kind of point defects in real metals. For pure metals. The defects are so called intrinsic defects, which are vacancies and interstitials. And as we have already discussed Interstitial is much more difficult to form as vacancies. So, essentially the vacancies are the main type of intrinsic defects in metals. And if you have an alloy, then the defects will be the alloying elements. For example, if you are working with an interstitial alloy for example, if you put interstitial B items into the e-lattice, you will have this B interstitial items as a point defect. Alternatively, you may have substitutional alloys where you are defect is substitutional solute. So these details will be provided later when we study the mechanical properties of metals. So a few general comments on the motion of point defects. So they can be generated by thermal activation, as we have I just discussed when you have a finite temperature there will be equilibrium point defect concentration. And they can also be generated by non equilibrium processes such as rapid quenching or ion irradiation. And they can be annihilated by being absorbed by services or interfaces or recombination at elevated temperature. And the motion of point defects is very important especially for a lot of diffusive processes such as nucleation and growth, recrystallisation crib, which will be covered in our future chapters. So today we have talked about the point defects in crystals. And thank you very much, goodbye. [MUSIC]
