 Okay, so today we continue the overview of electrostatics. I will very rapidly review statistical mechanics. I mean, not review, but just write a few useful formulas. And then we'll start Poisson-Boltzmann theory. But before I want to come back to yesterday to the electrostatic energy. So if you have charges, qi, ri, a certain number of charges, you have a potential energy, which is one-half sum over i not equal to j of 1 over 4 pi epsilon 0, qi, qj, divided by ri minus rj. And this is just, if you remember that if you define the potential phi of r as sum over i of 1 over 4 pi epsilon 0, qi divided by ri minus rj. This is just one-half of sum over j of, or sum over i of qi phi of ri, or phi i, where phi i is the electrostatic potential created by all charges except the charge i. So the generalization of this, as we saw, if you go to the continuum with the density of charge rho r equals sum over i of qi delta of r minus ri, then this u is equal to one-half integral d3r. So you see that if you put this here and if you put phi of r, this is just rho of r, phi of r. So the electrostatic energy of a system of charges is one-half the integral of the charge density times the electrostatic potential. And now if you remember the Poisson equation, the Poisson equation tells you that Laplacian of phi equals minus rho over epsilon 0. So if I replace here rho by its expression, this is minus, so rho is minus epsilon 0 Laplacian. So this is minus epsilon 0 over 2 integral d3r Laplacian phi times phi. And if you integrate by part, so I will show you this integration by part because it happens, it comes in all the time. So the integration by part of Laplacian phi, sorry, is, you can write it as integral d3r of gradient phi times phi minus, so if you write it like this, the first term is equal to this. And then there is an additional term, which is gradient phi times gradient phi. So it's gradient phi squared, so you have to subtract it, so it's minus integral gradient phi squared. And then you have the Gauss theorem. The Gauss theorem tells you that it's a theorem about integral, right? It's not Gauss law. Gauss law, which we saw, which is a divergence. So there's two things. There is Gauss theorem, which is that, and there is Gauss law. So Gauss law is just that divergence of E equals rho over epsilon, epsilon 0, where rho is the charge density, right? That's Gauss law, that's one of the basic, that's one of the Maxwell equations actually, it's one of the basic law. This equation in fact is derived or is equivalent to the Gauss theorem, and the Gauss theorem, or it has other names also, tells you that the integral over a certain volume d3r of a certain function, divergence of a certain vector f is equal to the integral on the surface of f times the normal outside the surface ds. So if you have a volume v on the surface s, you define the unit vector n on the surface, and this is a theorem of differential calculus. OK, so from this theorem you see that this quantity is exactly of this type, and therefore this quantity is just, so if you have your system in a whole volume, this is just, so I don't know if you can see it, but if we, I will write it here. So we have something like integral d3r over the volume where all the system is confined of gradient of divergence let's say of phi phi. This is the integral ds on the surface outside of n phi gradient phi. And now if the system, and so this is on the external surface of the system, now if your system is at equilibrium and if there is no external field applied to the surface, then this quantity on the surface outside is zero, therefore this integrated term in general is zero. And if this integrated term is zero, we have this nice property that this integral phi, Laplace in phi, is equal to plus epsilon zero. So there is only a contribution of this term, and this is gradient phi square, which you know from electrostatic, which you can write as the electrostatic energy is just the integral of e squared dr. This is the contribution of the electrostatic energy. It's an important result. Okay. Yes? This one. Because if your system, if you're in a large system, the electric field on the boundary, if you have the electrostatic field is zero, if it was not zero, then the charges would not be at equilibrium at infinity. Okay. So if you have boundaries, if you have solid boundaries. Yes? The reasoning about the boundaries is without using the, just because we have an integral. Yes, it's just, this is a mathematical theorem. Yes. Right? This is Gauss theorem. That's a math theorem. That's not physics. That's just purely math. It's just the integration by part. You know, it's, when you do integration by part, you have a term which comes from the boundary minus, so this is exactly what I write here. Now, this term is equal to that from this theorem, and this is equal to zero. It's an integral of a gradient. Yeah, but the integral of a divergence, it's not necessarily zero. It depends what you have on your surface. You said that you put the boundaries in infinity, so. Yes, so if every, if the potential goes to zero, if the electric field is zero at infinity, then it is zero. But you can have situations where it's not zero. It can be, but in general it is zero. It's nothing, it's not a big deal. I mean, it's, okay. Okay. Last thing I want to do in this review of electrostatics, maybe I will do it when we, I was going to talk about dielectric, what happens when you have a dielectric medium. But maybe I will do that. Okay, I will do it later in the course of the class. Okay, so let's say we have done electrostatics in the vacuum, and at some point I will come back to electrostatics for the electric media, especially when we'll study water, dipolar fluids, and the surface tension in electrolyte solutions. So now I come to a very rapid overview of statistical physics, and I will just review what I need in the beginning for the course. Okay, so essentially I will work, and everything we do will be in the canonical ensemble. So canonical ensemble is an ensemble of particles which are in thermal equilibrium with a heat bath at a certain temperature T. And so you know there are several ensembles which can describe the thermodynamics of systems. So there is the micro canonical where the energy is fixed. There is the canonical, et cetera. And in general, these ensembles are equivalent in the thermodynamic limit for large systems. And this relies very much on the fact that the interactions between the particles are short-ranged. So here the interactions of the particles are not short-ranged because it's cool-on interaction. So it decays like 1 over r. But we will see that for a neutral system, in fact, when the system is globally neutral, effectively the interaction between particles is short-ranged, and there is a exponential decay of the interaction at large distance. So this is not true. So if you think about gravity now, gravity has the same kind of interaction. It goes like 1 over r. But the difference between gravity and electrostatics is that in gravity there is no negative mass. There are only positive masses, and they attract. So there is no repulsion. So there is no such thing like screening or charge compensation like you have. And therefore, when you have a gravitational system, there is no equivalence between micro-canonical ensemble, canonical ensemble, between all the thermodynamic ensemble. And you have some very weird things which can happen depending on the ensemble where you look at. OK. So if the Hamiltonian of the system is, if you have n charges, each one has a momentum pi, position r i. So the Hamiltonian is sum over i of pi squared over 2m, plus a certain u of r i, where u of r i is the interaction energy of the particles. So if it's Coulomb, so I don't specify, but if it's a Coulomb energy, u would be 1 half of sum over i j of v Coulomb of r i minus r j times q i, q j, or something like that. But it's a function of all the r i's. So then the Boltzmann weight, so the probability to find a system at equilibrium, the probability to find a system in a certain configuration r 1, r n, which I write p of r i, is given by the Boltzmann weight, which is e to the minus beta h of t i r i divided by z, where z is the partition function and is defined, it's a normalization, if it's a probability, sorry, it's p of, I forgot, there is all the p i's, the r i's and p i's. It's p, the probability as function of the momenta and the partition function is the integral d p i over all p i over all r i of e to the minus beta h of p i r i, so that the integral of p over r i and p i is 1. Thanks. And as a side note, you know that, so this is the Boltzmann weight, and Boltzmann has 2n, and Boltzmann committed suicide not far from here in Duino, which is very nearby, and you can go and there is no problem with the formula for the entropy. OK, so in most cases the Hamiltonian has this form, which means the kinetic energy of the particles is quadratic, it's p i square over 2m. In that case, you see that all this is very simple because the integral, so in general the integral over the r i is very complicated, you cannot do it. And the distribution of p i is just Gaussian, it's a so-called, so the distribution of p i is essentially proportional to e to the beta p i square over 2m. And this is just a Maxwell distribution. So in general one is not really interested in the distribution of velocities because it's very simple. There is no surprise, it's always like this. And it's decoupled. It's not mixed with the position. It's independent because the weight e to the minus beta h is the product of this Maxwell distribution by e to the minus beta u. So rather than this, in the following we will mostly use the p of r i, the probability to find a certain configuration of the r i, which is the integral over p i of this distribution here, right? It's the probability to find a particular at r i with any p i, with any momentum. And since the distribution on p i and r i are independent, the distribution factorized and you can integrate. So it's integral e to the minus beta p i square over 2m. So it's always sum over i d3 p i times e to the minus beta u of r i divided by integral d3 p i e to the minus beta sum over i p i square over 2m. Times e to the minus beta u of r i. And therefore, we come to the result that the probability to find the system in a configuration r1. So when I write like this, it means all the particles r1, rn. This is just given by the reduced Boltzmann weight e to the minus beta u of r i divided by z, where z is the reduced partition function, which is product over i d3 r i e to the minus beta u of r i, where u of r i is the potential interaction energy of the particles. Now, the great properties of this is that z is related to the free energy by the standard relation, z is e to the minus beta f. So f is the free energy. So f equals minus kT log z. k is the Boltzmann constant. And beta i, OK, I think I said it, beta is 1 over kT. And you see that, for instance, the internal energy, which I write u, but it's a bit. So the internal energy is the expectation value of u of r i. And so you see that if you take, so it's related to the free energy by the following way. If you take the log of z and if you take the derivative of log of z with respect to beta, so if you calculate d by d beta log z, so if I take the log of z, it's 1 over z. And the derivative of this with respect to beta will be integral product over i d3 r i. And I bring down minus u of r i e to the minus beta u of r i. And therefore, you have this property that the expectation value of u is minus d by d beta of log z. The expectation value of u is the integral of u times the Boltzmann weight divided by this. Is it OK? Clear? So you have this property u. So we have this property that u equals minus d by d beta log z. And since z is minus log z is beta f, so it's d by d beta of beta f. So this is an important relation which relates the internal energy. The internal energy u is related to the partition function or to the free energy by this standard relation. And so from this you can get also the entropy by writing that u. So if you expand the derivative, u is therefore f plus beta df by d beta. And I remind you that f is u minus ts. So you see that ts equals beta df by d beta, which means that s, if I divided by t, it's beta square df by d beta. And this, of course, you can check by going from beta to t that this is minus df by dt. So these are a few thermodynamic relations which you can deduce from statistical mechanics and which we will use very often in the next of the course. OK, to conclude this section before starting on Poisson Boltzmann theory, I want to go to define what's called to go to the Grand Canonical Ensemble. So any question about this? This is, I guess, very standard. You have seen this probably many times, but I prefer to have the notations straight. So last thing is the Grand Canonical Ensemble. So in a Canonical Ensemble, the energy is not fixed. The average energy is given. But what is fixed is the temperature of the bath, of the heat bath. And the energy fluctuates. The average energy of the system is the internal energy, which is given like this. And the fluctuation of the internal, the fluctuation of the energy is related to the specific heat, which we'll see later. OK, so now you can have a system which is in equilibrium. So in a Canonical Ensemble, your system, is at temperature T, and it's exchanging energy with a heat bath, which is a thermostat at temperature T. So it exchanges energy so that the temperature of the system is fixed at temperature T. In the Grand Canonical Ensemble, the system is cannot not only exchange energy with the system to be at temperature T, but it can also exchange particles. So there is an exchange of particles so that the average number of particles is fixed. When it's Canonical, the total number of particles is exactly fixed. You work in an ensemble with a given fixed number of particles. Here in the Grand Canonical Ensemble, you don't work with a fixed number of particles. So the system can be in contact with a reservoir of particles and can exchange particles with it. What is fixed is N, the average number of particles. And the equivalent of the temperature here, in that case, is mu, which is the chemical potential. OK, so the Grand Partition function now, so the Boltzmann weights, everything can be defined in a similar way. The difference is that we now define a Grand Partition function. So this Grand Partition function is a function of temperature, of course, and mu, let's say mu. OK, let's assume there is only one species. But if you have many species of particles in the system, you can have many different mu's. And this Grand Partition function is equal to sum, N equals zero to infinity, so N is the number of particles. So the chemical potential enters as e to the beta mu N over factorial N, and then integral product over i equals one to N, d3 r i, e to the minus beta u of r i. So this is the partition, the canonical partition function with fixed N, like this one. And the difference is that you sum over all possible number of particles with a chemical potential, which is here. And the chemical potential is adjusted in such a way. So first of all, this partition function, this Grand Partition function is equal to the exponential of the Grand Potential, which plays the role of the free energy. So it's e to the minus beta omega of t and mu. So this omega is the equivalent of the free energy, which is here, but it depends on the chemical potential mu. And you adjust mu in such a way that the average number of particle is fixed. And you see that in order to get the average number of particle, if you take a derivative with respect to mu, you bring down beta N. So you have that average N is equal to d by d mu log z of t mu. And the d by d mu brings the beta, so it's 1 over beta. And therefore, it is minus d by d mu of omega. And I don't know if we will use it, but omega is directly related to the pressure in the system. And omega is minus pv in the Grand Canonical Ensemble. Omega is minus pv, where p is the pressure, v is the volume. These are standard thermodynamic relations, which we will use in order to calculate, for instance, the osmotic pressure in the Coulomb group. OK, so this is all I wanted to tell you about statistical physics, thermodynamics. And now we can start specifically the Coulomb electrolytes and charge. Is there any question about all these? No? Is it two? Yes? Sorry? Yes? I think K is missing. Yes, there is a K missing, absolutely. So we'll take K equals 1. K is missing, yes, because, yes, definitely K is missing. The last spot. Yes, so there is a K, yes. So here, 1 over T, so 1 over T is K beta. So there is a K here. OK, thank you. Yes, I always forget the case. Sorry? No, beta omega is dimensionless. Beta omega is dimensionless. Omega is a free energy. Yes, so beta omega is dimensionless, so omega has a dimension of an energy. It's a free energy. It's a grand potential. It's called a grand potential. Yes? Thank you. Any other comment, mistake, question? OK, so this is it for now I come to Poisson Boltzmann theory, and I will do simple Poisson Boltzmann. So this is really the simplified approach to Poisson Boltzmann, the simplest form, and you will see how one can derive in a few lines the Poisson Boltzmann equation. So the idea is the following. Assume you have N1 particle of charge Q1 equals Z1E. So Z1 is the valence, is the number of charges. Everything is measured in number of electronic charges. So Q1 you have N2 Z2E, et cetera, NM, OK, NM QM. And then we can have, so these charges are mobile, so they can move around, they're floating, they're at thermal equilibrium, and you may have some solid surfaces which are charged with some thick charge density, rho F of r. So it can be walls, charge walls, charged objects, and you have a solution floating around, and so the charged object, you have fixed charges which are there, and you have these mobile charges which are there. OK, so what you can write is that the charge density, rho of r, is rho F of r plus rho M of r, where rho M of r is the charge of mobile ions. OK, so we have the Poisson equation, and I will assume that there is a mean, we are, all this is working in a medium with dielectric constant epsilon, constant dielectric constant epsilon. So the Poisson equation is that Laplacian phi equals minus 1 over epsilon rho of r. So Laplacian phi equals minus rho F over epsilon. Minus 1 over epsilon rho mobile of r. Now, the second thing is if your mobile charges are at equilibrium in the bath, so what they see, the external potential phi of r, so their energy is qk or qk phi of r. That's their potential energy, right? If phi of r is the potential, their energy is qk of r. So the probability to find the specific k at point r is 1 is given by e to the minus beta qk phi of r divided by zk. This is the probability to find a particle of specific r, of specific k at point r. And therefore, you see that the Boltzmann equation is, so it's minus rho F of r divided by epsilon, minus 1 over epsilon. And the density of mobile ion is a sum over k of the probability to find the particle k at point r. And I assume that there are nk such particles, so it's minus nk qk. This is the charge, the total charge times the probability e to the minus beta qk phi of r divided by zk. Is it clear? Yes. OK, so this is the probability for a particle of type k to be at point r. If I have nk such particle, the average number of particles of type k which will be at point r is nk times this. And since they each carry a charge qk, the charge density, the total charge density, is given by this. Let me continue just one second. So here I didn't specify zk. It's just, of course, the local, the partition function of this species. So it's just the normalization factor for this. So it's just integral d3r of e to the minus beta qk phi of r. Yes? Sorry? It's the partition function for, OK, so in this approximation if you want the total energy of the system would be sum over k of qk phi of r. And therefore you see that there is a factorization for particle, if you have a particle at point, if you have your particles at point r1, r2, rk, they live in a potential which is phi of r1, phi of r2, phi of rk. So the true, the full partition function would be integral product over k drk e to the minus beta sum over k of qk phi of rk. But if you look for each particle if you want, for a single particle the probability to find a single particle of type k at a point r is given by this divided by zk because it factorizes each particle they don't, I mean in this approximation in this Poisson Boltzmann kind of framework the particles live in an external potential phi which is created by themselves but it's the same potential for everybody. So each particle sees the same potential phi of r. And now the big thing which is the further simplification is that if the potential goes to zero at infinity then this is integrated over a certain volume. If the potential goes to zero and you take the volume to go to infinity this is essentially equal to the volume. So if your particles, if the system is confined in a certain volume and you take this volume to infinity and the potential goes to zero at infinity then zk is essentially equal to zero and we get therefore nk over v. So the equation becomes minus rho f over epsilon minus one over epsilon sum over k of qk ck zero e to the minus beta qk phi of r. So where ck zero is the average uniform density nk over v so it's the concentration of particles of type k. If there were uniformly spread over the system. So this is the concentration, the bulk concentration of your ions of type k in the system. And this is the, so this equation is the so-called Poisson Boltzmann equation. Yes? Potential in the boundary condition goes to zero and so zk goes to v but as I mentioned I don't understand that. Well, a partition function has the dimension of a volume because the exponential is dimensionless. A partition function has always the dimension of a volume. In fact, so if you want to be really, there is a subtlety which I avoid but if you are interested, when you derive all this, you have a question? Yes, okay. So in fact, yes, but that's okay. So you integrate it over the volume and you get one. It's a density of probability. It's not a probability. It's a probability distribution function so it's a density. But, okay, so the question of the dimension of partition function. In fact, if you remember what you learned probably in the beginning of statistical physics is that the real partition function, you start in quantum mechanics and when you do in quantum mechanics you have to make cells in the phase space. And the partition function in fact is product over i, d3pi, d3ri over, so I don't remember if it's h bar cubed or 2pi or h cubed. And this is dimensionless. Is it h? It's h bar cubed, I think. Is it h? Well, the dimension is the same anyway. And then e to the minus beta h of pi rn. So, of course, if you write it like this and that's what comes out naturally from quantum mechanics, the partition function is dimensionless and then that's okay, et cetera. But you see that when you deal with Boltzmann weights or things like that, you take always ratios. And so if you take ratios, this constant, which is here, it disappears and nobody bothers to write explicitly this constant there. The price is, of course, that partition function has the dimension. So if you have n particles, it has the dimension of a volume to the n. If you omit the h bar cubed thing, it doesn't matter because it's always ratio, quantities are always ratios. Sorry, the n? No, no. The n comes if you have n particles. So you integrate over, no, in an ideal, okay. So let me, is it okay? Well, if you take, so let me do in one second the ideal gas, the ideal gas. So if I take the reduced partition function without its integral product over i d3 r i, e to the minus beta times zero, because there is no interaction between the particles. So the partition function of the ideal gas is just, so the Boltzmann weight is one, it's uniform, and therefore it's just v to the n. That's why it's right in here. And from this, of course, you can easily get the equation of state, the p equals k, pv equals kt, et cetera. You can write a few. So exercise, derive all the laws of ideal gas from this. You can get it very easily. And, okay, that's fairly trivial. Okay, so this is, again, the Poisson-Boltzmann equation, which is a mean field kind of, so mean field in the sense that you assume that all the particles are creating a potential phi, and they leave all in the same potential phi, which is created by them. So you see that it's a very highly non-linear equation, which is a kind of self-consistent equation, which enters in phi. So the density, the concentration of charges is created by the potential, and the potential creates itself, so it's a kind of self-consistent non-linear equation. So this has to be supplemented by some boundary conditions. And in electrostatics, there are essentially two types of boundary conditions. So one is fixed potential. So if you have some systems, some fixed charges in the system like a wall, like whatever, and you fix the potential phi equals phi zero, you fix the potential on the surfaces, so then this is called the Dirichlet boundary condition, or Dirichlet condition. Another frequently used, so this can be phi one, you fix the potential here, so essentially you fix potential on some surfaces. The other boundary condition, which is very often used, and we will see, because we will see how to solve this equation in a few simple cases, so it's a fixed charge density. So fixed charge density is you fix, on the boundary, you fix sigma. But fixing sigma, you know that as we saw, if, for instance, if it's a conductor, if you fix sigma, then close to the surface, the electrostatic, the electric field, E at the surface, is sigma over epsilon, and therefore it is minus, so it's sigma times n, where n is a normal to the surface, and therefore this is minus gradient phi n at the surface. So this is the other boundary condition, that the normal gradient of phi, so the normal component of phi near the surface, is fixed and determined by the surface charge density. Okay, so a few comments, so these are the two, and this is called the Neumann boundary condition, Neumann Dirichlet. And it changes, it can change a lot. What else did I want to say? So let me give you some example of the Poisson-Boltzmann equation in some simple cases. So any question about boundary conditions? So it's a second order equation, you have boundary conditions which are everywhere, and so I give you an example because of course this can simplify a lot, for instance, zz-salt. So what is a zz-salt? It's a symmetric salt, where you have two types of charges, plus z, plus ze, and minus ze. So you have cations, anions, anions, z can be one in NaCl, but it can be different. And of course if the system is neutral, then you have the same concentration for both. So the Poisson-Boltzmann equation takes the form Laplacian phi equals minus rho F over epsilon, and the interesting term is this one. So you have two species, so here of course it's one to m, when you have all the m species, now m is equal to two. They both have the same concentration, so you have a factor of c, which is this one, and one comes with a charge plus ze, so it's e to the minus beta ze i, and then the other one comes with a charge minus ze, so it's minus ze, and with a plus here, and this is just minus 2 cz sinh beta ze phi. So the Poisson-Boltzmann equation takes the form, so there is a sin minus, so it's plus 2 cz over epsilon sinh beta ze phi. So this is the Poisson-Boltzmann equation, so you can forget the rho F because usually the rho F is just a boundary condition. So this is the form of the Poisson-Boltzmann equation for a system, for a symmetric plus ze minus ze solved. C and c are the concentration of the plus ion and of the minus ion, so they have the same concentration because of charge neutrality. I will always assume charge neutrality in the system, so because they have the same charge, of course the concentration. Okay, so you can work out more complicated non-symmetric cases, but this is an example and we'll see how one can solve it in certain simple geometries or simple cases. What else did I want to say? Okay, so in general the Poisson-Boltzmann equation, yes, of fixed charges, not the mobile charges. This, the rho F, what, can you repeat? I don't understand. Yes, there are fixed charges like walls, like surfaces which don't move. Okay, so they are not mobile and then you have charges which move around. It's like if you take here in the room, right, you have walls, the walls don't move and you have air particles which are floating around and which are a thermal equilibrium. So these are the fixed charges which are in the system, so you can have, for instance, you can have a wall, a charged wall, or you can have a sphere, a charged sphere, whatever you want. And then this is the contribution, or this if you want, is the contribution of the ions which are floating around in the solution, which are a thermal equilibrium and they're moving around and sampling the space, et cetera. If it's pure salt, you don't have this. But if you have salt, for instance, we'll see one example we'll study next time. This, for instance, you have a wall, a charged wall with a certain charge density and you have here a salt around, then this will be, and usually this term is a boundary condition, right? It tells you that at the surface you will have a charge density or something like that. So you can put it or not, I mean keep in mind that there is some surface charges that can be present. So either you write it explicitly or you take it into account as a boundary condition. So just a few comments about the Poisson-Boltzmann equation. It's a very complicated equation because it's non-linear. If you have general, so for instance, people use it to describe the profile of ions near molecules like in biology. You have a protein, you want to see how the ions are surrounding the protein, what is the profile, the concentration of ions around the protein and things like that. So in this case, you cannot solve analytically the equation. So there are very few cases where this Poisson-Boltzmann equation can be solved analytically. We will see a few. One example is the one-dimensional case when you have a wall. I mean you can write it. So in 1D you can do quite a few calculations. In particular, you can do wall with counter ion, wall with salt, et cetera. So it's a charged surface. And of course it is used to model a biological membrane in a physiological condition where you have in presence of physiological fluid which is essentially a salt at 0.1 molar concentration. In 2D it's the cylinder. And the cylinder, so it's again a very simple geometry. It can be solved only for the case of counter ion. I will explain what's the difference between counter ion and salt. And that is a model. So charged surface is membrane. This is polyelectrolytes. So polyelectrolytes, as I said, are polymers which are charged. DNA is extremely charged, so it's very rigid because of the Coulomb repulsion. Locally it looks like a cylinder. So you can ask what is the profile or what is the distribution of ions around the DNA in a solution. And this is more or less given by solving the Poisson-Boltzmann equation. So there is an analytic solution for that. And in 3D, no analytic solution, but there are some approximate analytic solutions. Now, if you want to go beyond, you have to go to a numerical solution. So there are many packages which are commercial open source or so which solve numerically the Poisson-Boltzmann equation because chemists, biologists really use it in order to, as we will see, the Poisson-Boltzmann equation allows you also to calculate the force between the objects to see if they bind together or not due to, so in pharmaceutical companies, biologists, I mean biochemists and all kinds of people use the Poisson-Boltzmann equation and therefore there are a lot of work on solving the numerically the Poisson-Boltzmann equation. The way you do it is the following. You start, so if I take, for instance, this equation, so what you do is you start with a guess. So first you guess. You guess a phi zero. So you guess a certain profile of electrostatic potential in your system. So you discretize your system. First you discretize. So you make a lattice with a certain lattice spacing, whatever. And on this lattice, first you put all the fixed charges. So for instance, you can have an object like this. So your lattice, of course, there is big discussions what the lattice spacing should be. This all is discussed. And you put the charges, you spread the charge. So for instance, if this is a charged, a fixed charge object like a macro molecule or any molecule or whatever you have in your system, you spread the charges on the lattice points. So you have this. Two, you guess initial phi zero of r. So you guess a certain potential everywhere on your lattice. Then three, solve Laplacian phi equals V of phi zero. So what I mean by that is that you put phi zero here. So this is what I call V of phi zero. So you solve this equation Laplacian of phi equals V of phi zero. So this equation, this is quite easy to do because I mean numerically, if you give yourself the boundary conditions, either Dirichlet or Neumann or whatever, then this equation is easy to solve. You solve it, you get a new solution which is phi one by solving this equation. And then you use it as an initial solution and you iterate the process. And by doing that, it converges and you get finally a solution for the Poisson-Boltzmann equation which satisfies this. Of course it converges if your initial phi zero is not too far from the final state that you're looking at. So it's a kind of fixed point method which is completely widely used to solve the Poisson-Boltzmann equation numerically. Yes? Yes? You see that there is a complete, I mean the, okay, I can show you. You see, I write that rho k of r for the species is equal to e to the minus beta qk phi of r divided by zk. But the real rho k of r, what should it be? So it's 1 over z, the real z integral dr1 drk minus 1 drk plus 1 drm or dr, I don't know, okay. Let me e to the minus beta qk sum over i not equal to j of v of r, v Coulomb of r i minus rj. Right? So it's something like that. So it's much more complicated than this expression because it mixes all the particles. So it's a kind of, so then this is a mean field equation, a self-consistent mean field equation which ignores completely all the correlation and the correlations that exist between the particles. You see, here, if I look at rho k and rho l, they are separated. They don't, whereas here they are not separated. For instance, in Poisson-Boltzmann, the probability or, yeah, the probability to have r1 rk is e to the minus beta sum over k of qk phi of rk divided by z1, z2, right, by the product of the z. But in reality, it doesn't factorize. Right? The probability is given by this Boltzmann factor divided by the total z. So this is the approximation. So when you formulate it like this, it sounds like it's exact, but it's completely not exact. We will see in which limit it is correct when we go to the field theoretic local answer. What do you mean local? Yes. You mean local minima? Yes. Local solution. No. I tell you why, because what you can show is that, so it's a second-order differential equation, and it has some, okay, so there is some convexity property that can be formulated, which makes that if you satisfy the boundary conditions which are fixed, it's a unique solution. So if your initial guess satisfies the boundary condition and at each step of iteration you satisfy the boundary condition, if in the end it converges, you are sure that it has converged to the right solution. There is a unicity of the, so, okay. So now I will leave the equation here. So I will now show one of the most used approximations. So before going to the actual analytic solution of the equation, I will go to the so-called Debye-Huckel one second. I just finished writing this. Yes? Our initial guess is based on which observation? Sorry, can you read it? Our initial guess is based on which observation? That's a good question. So of course you have to satisfy, look, you have to satisfy the boundary condition, and, okay, I'll take it over. There is a unique solution to this equation. Sorry? Sorry? There is a unique solution to this equation. Yes? So that's the same question, right? Yes. Okay, I'll look it up. I will think about it. The initial guess. So the point is that if you look at this equation, if you satisfy the boundary condition, then the solution is unique. But of course, if you solve from a completely wrong solution, even if it has the right boundary condition, it will not converge. So you will, so you have to be close enough to the, or you have to be in the basin of attraction of the, okay, I will think about it and try to answer tomorrow. Okay, so I would forget the row F because it's, or I can keep it, actually, it doesn't matter. Okay, so one of the interesting approximations is what happens in weak, weak what? Weak field, or weak charge. So weak charges means, essentially, that you are in a situation where beta qk phi is much smaller than 1. So I can give you numbers. I have calculated it. So if you are at t equals 300k, phi equals 25 millivolts, then beta e phi is equal to 1. Okay, so let's give you a scale of, so qk, I remind you, is beta zk. So it's beta e phi times zk. That's what you want. So if you have a model valence system, for instance, you want to have essentially potentials which are weak, smaller than 25 millivolts, which in biology is very, very frequent. It's typically less than 10 millivolts, so it's a fairly good approximation. So in that case, we can linearize to first, we can expand, yes? It's not in this. No, not in this. Here I am, well, indirectly because here I assume that this is smaller than, that this is much smaller than 1. But of course the value of phi is determined by this equation which has, so we will see that, you will see. Okay, so anyway, what I want to say is that if I take this equation, if I am in this condition of weak charge, I can expand to first order in phi. And if I expand in first order in phi, so let me write it as minus, so I put the minus sign on the other side, minus Laplace in phi equals rho f over epsilon plus 1 over epsilon sum over k of, and of qk, say ck0 times 1. So I expand this, so 1 minus beta qk. So the first term is sum over k of qk ck0. And this quantity is equal to 0, right? If the system is neutral, if the system is globally neutral, then this quantity is just 0. And therefore the Poisson-Boltzman equation takes the form of the Debye-Huckel equation, which I write, so this term, the 1, this term disappears. And I can write it as minus Laplacian plus kappa d square phi equals rho f of r over epsilon, where kappa d square is beta over epsilon, right? Sum over k of ck0 qk square. So it's a linear equation, and there is an external source if there is fixed charges outside with minus Laplacian. And kappa d square is this quantity. So kappa d square has the dimension of the inverse of a length square, and it is what I call the Debye length. So kappa d square, because it's homogeneous to Laplacian, so it goes like 1 over l square. So it's lambda d to the minus 2, so it's beta epsilon sum over k. So if you remember qk is the valence zk times e. Sorry, it's not beta. So it's beta e square over epsilon sum over k of ck0 zk square. Yes, I use qk in terms of this. And if you remember what was the Berm length, the Berm length is beta e square over 4 pi epsilon. So we have lambda d to the minus 2 equals 4 pi lb sum over k of ck0, zk square. And there is something which the chemists have introduced, which is called, so this is lambda d. So lambda d is the Debye length. I will show you how it enters in the system. Chemists introduce what they call the ionic strength, which is a kind of measure of the total number of charges present in the system. So this ionic strength is just 1 half of sum over k of ck0 zk square. And in which case lambda d to the minus 2 is 8 pi lb times the ionic strength. So what's interesting is that the so for instance, if you have a salt, if you have a salt, a zz salt as we had before, for a zz salt, then you can find that lambda d to the minus 2 is 4 pi lb. And the two, so the two concentrations are the same, plus c minus, and there is plus z minus z. So essentially it's 8 pi lb cz square where c is the concentration of the salt. So you see that lambda d for a salt scale like 1 over z square root of c. So the Debye length scales like the inverse of the square root of the concentration and like 1 over the valence of the ions. So we will see that this lambda d essentially is the screening length. It's the distance over which the electrostatic potential in the solution decreases exponentially. I will show this next. And to give some units, if you put numbers, lambda d is essentially, so 3.05 angstroms divided by z square root of c in molar. To that you just do the numerics. And so lambda d is 3.05 angstrom divided by z square root of the concentration of salt in molar. And I remind you that in physiological condition, typically the Debye length is about one nanometer. Yes? C is not a pure number. No, c is a concentration. So it's a number divided by a volume. So it's 1 over l cube. So c is 1 over l cube and l is the distance, so it's 1 over l square. No, it's a number of concentrations, so it's a number divided by a volume. OK, so now I have a choice. Either I let you go and start practicing and rehearsing for your exam or I do. So the next thing, let me tell you what I will do next because Matteo will not get too angry. OK, I just tell you what I want to do. So now I want to see what happens if I have one fixed charge in the system. So I have at that point charge q at, let's say at r equals 0 at the origin. And I want to solve this equation. So I want to see what is the electrostatic potential. So I have my particle here. I have a salt all around. And I want to solve minus laplacian plus kappa d square phi equals q over epsilon delta of r. OK, so what is the potential created by one charge screened, one charge in a bath of plus and minus ions? And so what I want to show, which I will show probably next time, is that phi of r is equal to q over 4 pi epsilon e to the minus kappa d r divided by r. And this is called the Debye-Huckel potential. And this is very interesting because this shows that, so of course you see that if kappa d is 0, then this is just the Poisson equation for pure Coulomb. If kappa d is 0, you just recover the standard Coulomb law. When there is screening, when there is this ions in the solution, there is a non-zero kappa. And the electrostatic potential instead of decaying algebraically like 1 over r, it decays much faster like an exponential e to the minus kappa d over r. And kappa d, I remind you, is 1 over lambda d. So that's the by length. So the by length, as we will see, it's essentially the length over which the potential, electrostatic potential is not too much affected and beyond which the electrostatic potential essentially can be neglected. OK, so maybe I stop here, is that OK? And so the next time we'll see how to derive this property. Any questions before you run away?