All right. The second example will be colloidal particles in an electrolyte. So, here, a phenomenon where the locations of charges governed by a potential that arises in part from the same charge, but also from Boltzmann distribution which is derived from thermodynamics. Let me ask Melodie one more time. What is a colloid? So, a colloid is a lump of charge. Okay. So, that's what's her guess. Here, we define a type of mixture in which one substance is dispersed evenly throughout another. So, we can think of three phases: gas, liquid, solid for the disperse phase and as well as continuous medium. So, we have nine combinations, nine possible combinations. So, one good example in our daily life is milk. In case of milk continuous medium is liquid and dispersed phase is also liquid, so this is liquid colloid. So, milk is an emulsified colloid of liquid butterfat globules dispersed within a water-based liquid. But of course in case of battery electrolyte, for example, then continuous medium is liquid but the dispersed one might be solid. That's also found in pigment ink or blood as well. So, those are kind of colloid. Now, so electrolyte, where any substance containing free ions that make the substance electrically conductive such as ionic solution that is electrolyte, we have electrolytes inside our body as well, has liquid and ions inside. In electrolyte, if there is some salt dissolved in the water in addition to positively charged colloids, it will be dissociated into positive and negative charge ions, and negative ions will be attracted to colloid particles, and the positive ions will be repelled. So, usually, if we don't want to agglomerate the particles, we charge the particles to same polarity, for example, plus charge. So they can have cooling repulsion to stay away from each other. But if we put salt like NaCl, then what happened is the negative ions will shield this positive charge. So, there are chances that they can stick together to agglomerate. So, we are going to study that phenomenon using the equation that we learned. Again, let's think the charged particle as a huge particle in this world. So, if I think about the interface right between the surface of the colloid particle and the electrolyte, then we can assume it is one dimension. As we just mentioned, the minus charge will be attracted, plus charge will be repelled from the surface. With that knowledge, we are going to calculate what would be the electrostatic potential distribution from the charged colloid surface. So, now supposing there were such a potential Phi of x, how would the ions distribute themselves in it? So, Melodie, what is your guess? How do you think those minus charge will line up? I think in one of my classes, we learned it has something like this where they're closer to the charged surface and then they distribute all. So, closer to the surface, you expect to have more minus ions. As you go further away from the surface, there will be decrease of the amount, and there might be a lot of reasons for that. But let's take a look at what happens. So, according to statistical mechanics, particles in thermal equilibrium in a force field are distributed as follows. It's following the Boltzmann distribution. As you can see, the number of particles depend on the driving force as well as the temperature. If the temperature is given in constant, then it is exponentially dependent on the driving force. So, further away from the surface, the driving force will decrease, not only because the distances is increasing, but also because more and more minus ions will shield the driving force. So, we will indeed see the trend will be something like Melodie has graphed here. Okay. So, as you can see, we're going to use the Poisson's equation in one dimension using only x-axis. So, d squared Phi over dx squared equals minus Rho over Epsilon naught. We will assume the ions carry one electronic charge like in the case of sodium chloride, positive or negative. Then, the driving force, which is the internal potential here, energy is here, q_e, which is charge, times Phi X, which is electrostatic potential. So, we can separate out for two cases, plus charge and minus charge, and spell them out like here, and then because we have to think about both of them, we add them up. So, when we add them up, we will have the net charge density Rho, and adding with this Poisson equations and Boltzmann distribution, we'll have this equation. Of course, you can solve this analytically with advanced math, but here we're going to do a further approximation, first order. So, we were going to assume that this is very small. If that's very small, we can change it by a linear function like here. If we do that, then it will give you a very simple form of equation here. For this one, you know the solutions for Phi are not oscillatory but exponential, so we have those two. For the boundary condition where you have infinite distance from the charge particle, you know the potential should go to zero. So, this term is not appropriate. So, B coefficient should be zero, and we only have A times e to the minus x over D, where D is the Debye constant which defines the shielding links of the minus ions on the charged particles. So, here, we're going to just map out the potential outside the charge particle, which follows this exponential curve as mentioned by Melodie, and as you can see here, one Debye, two Debye, three Debye length, the curve is dramatically decreasing following the exponential function. So, when the potential reaches is one over E, E is the natural E value that we call Debye length. Here are some interesting facts that you can check. So, the potential at the surface Phi of zero is equal to Sigma D over Epsilon naught. Sigma D over Epsilon naught. So, Debye length, if you think of this as the length or distance between two plates of capacitors, then it really looks similar to that equation, where the potential over D will be the electric field is equal to Sigma over Epsilon naught. We learned that. So, as mentioned briefly, you can think of this as plus Sigma minus Sigma and parallel plate capacitor, and you have distance of D between them. In that case, it follows the same equation. So, the potential of the colloid, the particle at its surface is the same as a Delta Phi across the capacitor with a plate-spacing D and a surface charge density Sigma. In that way, you can memorize or easier about the problem and solution of this question. All right, so let's discuss some interesting insights into colloidal particles. So Melody, have you ever tried to put some salt into, let say, tofu or salt in milk? No, unfortunately not. Never. So I'm asking the same question to the students, if you have ever done that. If you do that, you will see some agglomeration inside colloidal particle. The reason why we have agglomeration when we put more salt into the electrolyte is because the Debye length depends on the concentration of the salt. If we increase it in naught, then Debye length becomes smaller. Now, as I told you, if you want to disperse particles, we are using the cooling repulsion between the particle. So make sure when they get closer, they're repelled. However, if you have a shield, Debye shield like this, then all of the repulsive force will be shielded by the presence of this minus charge around them. So when they are sticking, it is too late when they see the repulsive force. So in this way, you can have more chance to have agglomeration. So that we called salting out. As you can see here, another interesting example is the fact that a salt solution has an protein molecules. So molecule has various charge on it, as we know. All DNA also have charged. Sometimes happens that there is a net charge distributed along the chain. So you can, as you see here, if the sheath, the Debye shielding, is thin enough, the particles have a good chance of knocking against each other. So they will then stick and the colloid will coagulate and precipitate out of the liquid, and we call this process salting out a colloid. So we can now understand quantitatively why that's the case and how we can control it, okay? So the last but not the least example will be the electrostatic field of a grid. So Melody, in our daily life, where do you find grid? I think like in the front of a microwave. Yes, in front of a microwave, there's a grid inside the glass that protects you from being boiled if you look into the food inside, right? We wonder how this grid is protecting us from the microwave damage. So in this example, we are going to study that. So the character of the electric field near a grid of charge wires is depicted in this picture. As you can see, you can imagine there is a pull like this, and you're moving the ocean wave through the poles. Then you can imagine this poles will create some ripples but as you go further away, it will have a planner feature, very similar. Likewise, if you have grids like these, and these are charged with plus a potential or the same potential, you will have ripples of potential as you move away from this grid, but at some point, it will behave like a parallel plate. So that's the beauty of the grid. So how do we know that? First, how do we know that? We can know that using electrostatic equation that we learned and also we can use Fourier transform, which deals with the harmonic oscillations or periodic function using sinusoidal and cosine function to make it equivalent. So as mentioned in the previous slide, if you look at the field a large distance above the plane of the wires, we see a constant electric field just as if the charge were uniformly spread over a plane. As we approach the grid of wires, the field begins to deviate from the uniform field. Then our question would be, how close should we approach the grid to see appreciable variations in potential? In the example of microwave oven, how close can I approach my eyes to the glass? But to our safety, as you can see, if this is a microwave glass, this is the food you're cooking, and this is your eyes. Then they put the grid inside the glass to make sure the thickness of the glass will protect you. So the thickness of your glass will be somewhere like this. So your eyes can never approach the variations of the field, and we will see what would be the distance. So as we travel parallel to the grid, we observe that the fields fluctuate in a periodic manner. That we can deduce. Any periodic quantity can be expressed as a sum of sine waves, this is Fourier's theorem. Phi of x and z, where z is the direction upward, x is this way, is F of n of z times cosine two Pi nx over a. Where n is the integer, harmonic numbers starting from zero and a is the parameter between those two wires, like lattice parameter. As you can see, if I increase n, the frequency becomes higher or the period becomes lower, smaller. So if this is to be a valid potential, it must satisfy Laplace equation, the region above the wires, where there are no charge. So we have this Laplace equation. If I put this into this equation, you have this equation. You see that? So with respect to x and with respect to z. If I solve this, you will have this component d square fn over dc squared equals four Pi square n square over a square fn. For this derivatives, we know the solution is f of n equals a sub n times e to the minus z over z naught and z naught is a over two Pi n. Now, this is exponential function, it decreases exponentially. Now Melody, z over z naught, if it is larger, you have more decrease. Am I right? So what would be the largest contribution if you think of this fact. Knowing n is starts from zero to positive numbers. I'm sorry. It starts from one to positive numbers. Which number will be the biggest contributor to the F? I think n. N equals? One? Yes, one. Because if n equals one, the z naught will be maximized. If z naught is maximized, then this becomes the smallest. So the parameter in front of z is smallest makes the decay the smallest, right? So we have found that if there's a Fourier component of the field of harmonic n, that component will decrease exponentially with characteristic distance z naught equals a over two Pi n. As n goes further larger and larger, that characteristic distance will be smaller and smaller and smaller. So the most important distance will be n equals one, which is the largest distance. That distance that for the first harmonic n equals one, the amplitude falls by the factor of e to the minus two Pi, a large decrease each time we increase z by one grid spacing a. So the other harmonics fall off even more rapidly as we move away from the grids so they are less important. We see that if we are only a few times the distance a away from the grid, which is the spacing of the grids. If it is few times distance from there, the field is nearly uniform. Okay, nearly uniform. So you can calculate, if I'm one distance away from the grid, how small this variation is. There will always remain the zero harmonic field which is Phi naught equals minus e naught z. This is from parallely. The method that we have just developed can be used to explain why electrostatic shielding by means of a screen is often just as good as with a solid metal sheet. So that's why we can use grid inside the microwave oven and with the thickness close to the grid spacing, you're safe. All right, there is another example, the sources from material science and engineering in 1997. Where we put the grid in between the cathode and anode and sputter deposition chamber. The reason why we did that was because we found if we do sputter deposition of chromium oxide, there are particles agglomerated on top of the films due to the high energy particle flux. So we wanted to sieve them out using the grid, where the potential near the anode will be perturbed. So the high-energy particle will be sieved out through the grid. So putting a grounded grid in the sputtering chamber, can change electric field nearer of it and control the energetic species. So that was an example we applied from this knowledge to our real research. So with that, we're going to leave here and leave you here, and teach you again. Bye.