 se prišli od pomečnji, iz neko mostjoj pridi, na domenju, in in zvojne interakcije v biologijnenih kickinga in, tako čestno, sot mater, je innovno zelo tako, konec, tako, na rečetnih štošku istotumgam. As far as I heard, another book, Sunka, will be Sunka. Ok, so what do I do with this first? They know. Ok. So hello, so as you, as Andrea said, well first of all I want to thank the organizers of the school for inviting me to give this course. So as you see, it's a course about electrostatic interaction in biological systems. And as we will see, it's the main interaction which drives the behavior of mostly all biological systems. And I will try to make it understandable and simple to you. So actually the outline of the course will be the following, and I will do according to the time. Of course you can ask as many questions as you want. You can see me after if there is things which are not clear, although I doubt it. Ok, so the course will be organized as follows. So today it will be the first part of the course will be very qualitative. I will just try to motivate and to explain to you what is soft matter and what is biological matter and how this interaction comes into play. So I will give you a lot of examples to try to motivate this study. Ok, so I don't know exactly how it will go, but I will make a review of electrostatics, what you learned in school the previous years, but it turns out when I started working in this field I had completely forgotten my electrostatics, so I will assume that you also forgot a lot of it. And that you forgot a lot also about thermodynamics and statistical physics. So I will give you the concepts and all the things which are necessary at least to understand the rest of the course. Then I will start really in the heart of the matter, which is starting with Poisson-Boltzmann theory, which is a simple mean field theory to describe ions in electrolyte solutions in biological systems. And I will discuss, so it's Poisson-Boltzmann equation and it's linearized version, which is the so-called Debye-Huckel theory. And then I will discuss the problem in various geometries. So the planar geometry, which corresponds to the case of membranes in biology. So when you have a charged membrane in contact with the ions in the cell, so you know that as we will see cellular membranes are charged kind of planar objects in the cell is full of ions, so one can ask about that. Then the cylindrical case, the cylindrical case corresponds to polyelectrolytes, so charged polymers in the cell, so charged polymers like DNA, RNA, proteins. So they are, because of the electrostatic repulsion, they are very cylindrical. They have a large so-called persistent length, and so this is studied in this context. And then the spherical geometry is the study, so it's in the case of indeed the spherical geometry, and then it's what's called colloid, so it's charged spherical particles or non-spherical, but essentially small molecules. Then in the following, I will go to a more advanced part, which is the statistical field theory of charged systems. So if you want to go beyond the Boisson Boltzmann theory, which is a mean field kind of approach, you need to include fluctuation effects, correlations, and for that matter the natural framework is to write a field theory to express the statistical mechanics of this Coulomb system as a field theory and do then what's called a loop expansion, and the loop expansion had zero order will be, as we shall see, the Poisson Boltzmann theory, and then all the corrections are the effect of fluctuations and correlation on the system. And then I will show you some applications of this loop expansion and places where these fluctuation effects are important in these systems. And then last chapter will be further applications. So the first one is application. So as you will see in this case, the only interaction which is present is the Coulomb interaction, but there is one interaction which plays a particular role, which is the steric interaction. So the ions or the molecules have hard cores, they cannot penetrate each other, so there is some steric repulsion, and we shall see how one can modify the Poisson Boltzmann equation at mean field level to take into account the steric effect. Another application is the case of dipolar fluids. So dipolar fluids are the most important because, of course, the most important fluid is water in biology and even in life. And water turns out to be a dipolar fluid, which means the water molecules are dipoles, and all many of the properties of water can be explained in terms of these dipoles. And then, if I have time, I will come to the subject which is quite amazing, which is that in a certain condition, in solve solutions, identical charge can attract instead of having repulsive interactions. So you can have two positive charges which will attract because of the surrounding background. OK, so this is the outline of the course. OK, and there are very essentially no books on the subject, but there are books which, so on the, actually on the last part, on this part there is essentially no book, but on all the rest, of course, on Poisson Boltzman, there are many references, famous one is Israel Ajvili, who passed away recently, unfortunately. There is the book by Sam Safran. This is an old book by Vervet Overbeg, which is very interesting, and this is another book, which is one of the classic, but they are a bit old fashioned and they use older notation than I will. By the way, I will use SI system for electrostatics, whereas most of these books are written in CGS, but I was raised with SI, so that's what we will do. OK, so now I can start, so the first part will be the qualitative description of trying to motivate you. So as you know, in nature you have four kind of forces, strong weak, gravity and electromagnetism. So strong weak are the ones which are relevant at subatomic scales in the nucleus or below the nucleus inside particles in the structure of nuclei, neutrons, protons, etc. This will be completely irrelevant in what we do, because the scale is totally irrelevant. Then there is gravity, which is relevant at macroscopic and astronomical scales. It can be relevant in some cases in the soft matter physics, for instance when you study sedimentation or things like that where the gravity field plays a role, but in all the cases where we will be interested, we will see that this gravity interaction is extremely small compared to the electromagnetic interaction. So the electromagnetic interaction is, of course, the interaction between charged particles, and it's the only relevant interaction between atomic and mesoscopic scales. So whenever you look all the matter that surrounds us, at least the soft matter, it's only dictated by the electrostatic and electromagnetic interaction, plus something which I will mention, which is the Pauli principle, which is the exclusion. But you could describe most of the everyday physics of the materials that you see in terms of the electromagnetic interaction. And, of course, the form of the electromagnetic force is 1 over 4 pi epsilon zero, q1, q2. So q1, q2 are the charges, and r is the distance. And these are the values. So, in fact, this is not what you should remember, but the point is that 1 over 4 pi epsilon zero is 9, 10 to the 9 in SI units. OK, and the charge of the electron is 1.6, 10 to the minus 9. So, as you can see, if you compare gravity to electromagnetic, you see that when you make the ratio, the distance disappears, and the ratio of the forces is always given in terms of the ratio of the constant, of the masses, and of the charges. And when you do the, so as an exercise you can compute the ratio of the two. And if you look at two protons, if my memory is correct, the ratio for two protons, the ratio of electromagnetic force, two gravity force is 10 to the minus 34. So, you can happily forget about it. So, as I said, the only relevant force is the electromagnetic one plus Pauli principle, which prevents the collapse. And all the other forces that we will see that are currently, that are used in soft matter or in statistical physics, are all originating from electromagnetism and Pauli principle. And the unit of energy that we will use, so in physics it's electron volts. In chemistry people use kilo joules per mole, or kilo calories per mole. And the correspondence is that one electron volt is about 100 kilo joule per mole. So, one electron volt is 40 kT, and one kT is 0.025 electron volt. OK, these are numbers you just to keep in mind to have an idea of the order of magnitudes of these quantities. OK, so I see that up to now it's fine. OK, so what are the bonds? So, in fact, all these interactions which are present in these systems can be described in chemistry in terms of several types of bonds. So, there is the chemical bond, the covalent bond. So, the first one is the covalent bond, sorry, and the covalent bond is, as you know, it's the bond which makes up molecules. It's essentially binding of two nuclei by exchange of electrons between the two. So, this is described quantum mechanically. The interaction is really coulomb interaction mediated by electrons, and it's very strong. What's important to know is that the order of magnitude is very large, and so in most cases you can forget it's like an unbreakable bond. It's fixed, and you don't worry about it at room temperature because the energy of these covalent bonds is much larger than kBt, where t is 300k, which is the ordinary temperature. Then you have ionic bond. So, ionic bond is really the pure coulombic interaction between the different atoms in the system. So, the most famous structure is the crystal of NaCl, Na plus Cl minus. So, it's a cubic structure like this, where you have alternating Na plus Cl minus, et cetera. And typically, the temperature of the energy of binding is typically one electron volt, about 100 kJ per mole. So, it's also a very strong interaction, and as a result, the fusion temperature for this object is very high. And then you have another type of interaction, which is much weaker, which is the hydrogen bond, and which plays really a very important role in soft matter, biology, et cetera. And it's called a hydrogen bond. And the hydrogen bond is a bond which is particularly important in water. It's a bond, the value of which is 5 to 20 kJ per mole. So, it's essentially of the order of 1 kT, or a little bit less. And it's a kind of directional dipole-dipole interaction between electron negative oxygens mediated by some H. OK. I'm not going to discuss the exact nature of the hydrogen bond, but it's important in many cases, particularly in biological systems, in proteins, in all kind of things. OK. And this is the energy scale of covalent ionic and hydrogen bonds in electron volts. So, you see the strong, medium, and weak. In addition to that, there is a van der Waals interaction. So, the van der Waals interaction is a long distance induced dipole-dipole interaction between neutral objects. So, if you have, like, a neutral atom, when it's in presence of another atom, it gets polarized because of the charges of the other atom. So, they deform a little bit. By deforming, they get a dipolar moment. And this induced dipolar moment generates van der Waals interaction, which goes as one over R6 at large distances. And it's of the order of one KBT. So, it's responsible for the condensation of gas for adhesion in the glues, for instance. It's van der Waals interaction. Aggregation, floculations in colloids, what else, and many other effects in biology. And to balance this van der Waals interaction, it turns out that there is the so-called steric interaction, which is due to the Pauli principle, which prevents the overlap of electron wave function, so which induces a very strong repulsion at short distance due to the Pauli principle. And very often, this is modeled as a hardcore interaction. So, there is one interaction which captures both the van der Waals large distance interaction and the steric short distance interaction. It's the so-called Lenard-Jones potential. And the Lenard-Jones potential has this form. It goes at short distances. It's a repulsion, which goes like one over R12. And at large distance, it's an attraction, which goes like one over R6. And the shape of this Lenard-Jones potential goes like this. So, it's extremely repulsive at short distance, and weakly attractive at large distance. And so, the minimum is at radius 2 to the 1,6 sigma, which you obtain. I mean, you can study this. It's a very, and it's very frequently used in models, in simulations of atoms, interaction of atoms in molecular dynamics of biopolymers or anything in chemistry. It's a very, very present interaction everywhere. OK. Next comes the question, at what level do we describe soft matter and biological matter? Is it quantum? Is it classical, et cetera? So, as you know, the crossover between quantum and classical is given in terms of the de Broglie length. So, the de Broglie length. So, if you look at the quantum particle, its kinetic energy is essentially h bar square, k square over 2m, where k is the wave vector of the quantum particle. So, this is the energy. Now, the thermal energy of a particle is kBt. So, when the quantum kinetic energy is smaller than the thermal energy, it means that thermal shocks will induce decoherence of the quantum particle, and the particle will behave classical. So, you can define a length, which is called the de Broglie length, by equating the kinetic energy of the quantum particle with the thermal energy that gives this length. And so, if you are looking at lengths, which are smaller than this length, then the system will behave quantum mechanically. But if you are interested at distances or scales, which are much larger than this, then the system will behave classically. So, to give an example, so this is just what I said, the distance between atom and molecule is much smaller than the de Broglie length. So, the thermal fluctuations are not strong enough to induce decoherence in the wave function of the atoms and molecules, and therefore you need a quantum description of the system. So, if you look at room temperature, 300k, for hydrogen atom you can calculate, and you find that the de Broglie length is one angstrom. As you see that the de Broglie length is, it goes like one over square root of m. So, the larger the mass, the larger the mass, the smaller the de Broglie length. So, heavy particles are less quantum than light particles. And of course, so for hydrogen atom it's one angstrom. For oxygen molecules it's 0.16, 0.18 angstroms, sorry, and so since we are looking always at scales which are typically larger than these, we can use classical mechanics to describe the system. The only exception usually is when you look at electrons, so electrons are 2,000 times lighter than protons, so the length is one over square root of 2,000, and the length is 45 angstroms, and therefore if you look at an electron gas in a metal, even at room temperature, it is quantum, and there is no way you can describe an electron gas as classical particles even at room temperature. So, what are the scales involved in soft condensed matter? So, I will go in review of the various materials that are relevant to this study. I guess up to now there is no question. OK, so these are examples of materials, which I will detail a little bit later, but as you can see, so there are polymers, liquid crystals, collies, amplifiers, biomolecules, and typically the scale goes from nanometers to micrometers everywhere. So, of course, polymers can be quite long and they can go to micrometers. Liquid crystals, typically what defines liquid crystals, if you have layers of liquid crystals, it is the interlayer distance, which is typically of the order of a few tens of nanometers. Colloids, the colloidal particles themselves can be from tens of nanometers to micrometers. Colloids make membranes, they make also micelles, and micelles have a time kind of scale. And the only ones, which are a little bit smaller, are biomolecules like DNA, RNA, or proteins, which are really more microscopic, but still in the range of the nanometer, let's say. OK, so now what is characteristic of soft condensed matter and biological system? It's that it's matter that can easily deform by either applying a small or mechanical force or even simply by thermal fluctuation itself. And the hallmark of these systems is that in this system the internal energy or the internal enthalpy of the system is essentially of the same order of magnitude as the entropy. In result, the system is very flexible and very soft, hence the name of soft matter. So soft matter, you have it everywhere, food products, rubber, plastics, gels, paste, emulsions, creams, detergents, paints, you name it. Everything you see essentially except metals and maybe glass, everything is soft matter. And in biological system you have tissues, skin, muscles, blood, DNA, all the biopolymers, et cetera. OK, so OK. OK, so polymers, I remind you what's a polymer. A polymer is a macro molecule consisting of the repeat of a certain unit. So for instance here it's polyethylene. So polyethylene is the chain C2H4. And when you polymerize it, you get something like this and you can make a very long molecule where you repeat a single unit by polymerization and you get a chain which is very long. And there are several types of polymers, so this is a linear polymer, so it's a single chain which are like this. They can be branched if you have some branching around or cross linked if you make a gel or something like that. Then you have phases also of liquid crystals. So liquid crystals are liquids which are made of molecules which are not spherical. So if your molecule has a certain shape, it can have some ordering between liquid and crystal. And these orderings are called liquid crystals. So this type of ordering is called nematic. That's when the axis of the liquid crystal are oriented. But still the molecules can move around. But there is a preferential direction of all the axis of the molecules. If you compress further, you can have a lamellar kind of phase where the, because of the steric repulsion still, not only you have this directional orientation but also you will have a layering of the molecules. And these layers are called smectic phase. And then there is a smectic C, which I will not discuss. Yes? Sorry, I'm sorry. Yes? But the entropy of a polymer, you have a chain and you calculate how many configurations the chain can do. So this is statistics. It's statistical mechanics. It's like a random walk. So it can be either continuous or it can be, if it's continuous, it's called a continuous Brownian motion. If it's, you can also imagine model polymers as models on a lattice, in which case it's a discrete Brownian motion. So in all cases, if there is no interaction, you can calculate exactly the entropy of the polymer. Now, if there are interactions like excluded volume or self avoidance, then it's more complicated. But still there are ways to calculate approximately this entropy. OK. So polymers, liquid crystals, then amphiphiles. So amphiphiles are interesting molecules which are extremely present in biology. Amphiphiles. So amphi means in Greek both or all. And files is to like. So it's molecules which have a head, which likes water. And the tail, which doesn't like water. So the head is hydrophilic. And the tail is hydrophobic. So as a result, if you take these amphiphile molecules and put them in water, they will form certain structures to avoid that the hydrophobic tail will be in contact with water. And so that the heads, which are hydrophilic, will be in contact with water. So by doing that, for instance, if you have oil, oil and water, then the amphiphile molecule will coat the oil droplets so that the tail, the hydrophobic tail, will be in contact with the oil. And the hydrophilic head here will be in contact with the water. And this is the principle of soaps or detergents, right? That's how soaps work and clean up things. And so you can have, this is called the micelle, this self-organization of these amphiphile molecules. So there are several geometries of amphiphiles, but the basic principle is that the head is hydrophilic and the tail is hydrophobic. So you have all kind of phases when you take these amphiphiles and mix them and put them in a mixture of water and oil, you have all kind of very interesting phase diagram with lamellar phases, with hexagonal phases, cubic phases, and these are more exotic phases. And there is even these bi-continuous phases called plumber's nightmare, which are zero curvature. OK. Anyways, complicated phases. In addition, there is something which are called colloids. So colloids are essentially small spherical particles which can aggregate or crystallize and make these kind of structures. So this is a colloid made of polystyrene, polyps, polystyrene spheres. And you can see all kind of phases, ordered phases of these colloids. And also you have biological membranes. So biological membranes are the membranes which surround the cells, and they are made of phospholipids, which are these amphiphilic molecules, which self-assemble to form some kind of micelle. And inside this membrane there is all the apparatus of the cell. OK. So now let's come. I gave you a simple description of ions, sorry, of soft matter in general. So now let's come more generally to the coulombic part of the ions and things like that. So ions in liquid, soft matter, and biology. So the most important, of course the most important part in biology is water, because everything in biology takes place in water. So water, as we know, is a molecule which has this shape, this V shape with a given angle. So the oxygen is slightly, is highly electronegative, and the hydrogen is electropositive. And it has a large dipolar moment. I will come back to this question later. The dipolar moment is 1.85 dB. And one of the characteristic of the water is that it has a very high dielectric constant, which is typically of the order of 80. So the relative dielectric constant of water is 80. The vacuum or air, the dielectric constant is 1, which means that, as we will see, the coulomb interaction in water is 80 times weaker than in the vacuum or in the air. If you take ionic crystal, so ionic crystals or ionic liquid, so which means liquids made purely of ions, with no solvent, just pure plus and minus ions. So as I said before, as we saw, they can make cubic lattices with a high melting temperature because of the high energy, the high binding energy of the bonds, of the ionic bonds. But it turns out that in some cases, these ionic liquids can be liquid at room temperature, and that's used in fuel cells, in batteries and fuel cells. So it's purely a fluid made of plus and minus ions, and it can be liquid. Now, these ion, so these are very important, it's gaining more and more important because technologically these fuel cells are very interesting to develop, and there is many work on trying to find very highly charged fluids to use in fuel cells. OK, another kind of thing is ionic solution. So ionic solution, this will be one of the subjects we will study. You have ions which are dissolved in water, in a dipolar liquid solution, like water. So this is salt. Of course, all the biology takes place in the cell in physiological condition. Physiological condition is essentially salt water. So now as you know, when you put salt in water, the crystal salt will dissolve. The reason is dissolved is due to the large dielectric constant of water. The dielectric constant of water is 80, so which means that the attraction between Na and Cl in water is 80 times weaker than in the air. So if you put your salt in water, the interaction is decreased 80 times, and therefore the salt will dissolve. Of course, until, if you put more and more salt, there is saturation at some point, because there is also a balance with entropy. But that's something like that. So there are several electrostatic length scale, which we will see all over. So one is the so-called bjerom length. So the bjerom length, if you take the interaction in two charges, two electronic charges. So the elementary charge that we will deal with all the time is the electron charge. So if you take the interaction of two electronic charges, the interaction is e squared over 4 pi epsilon zero. Epsilon r is the relative dielectric constant, time r. And if you multiply by beta, so beta v is dimensionless, because v is an energy, so beta v is dimensionless. And therefore if it's dimensionless, you see that since this goes like 1 over r, you can write it as lb over r, where lb, this lb is called the bjerom length. So it's beta, so you just can read it off here. It's beta e squared over 4 pi epsilon zero epsilon r. And it's essentially the distance at which Coulomb energy is equal to the thermal energy. So if you have two charges at temperature, so beta is 1 over kbt by the way I must have said it before. So if you look the distance at which the electrostatic energy is equal to kbt, this distance is called the bjerom length. And it's given like this. And if you look at the bjerom length, so this is the number which will be very present all the time, you have to keep it in mind. The bjerom length in water is seven angstroms. And in the air it's 80 times more. So in the air it's 560 angstroms. A few more slides on this qualitative description. Another length, which will be very important, is the Dubai length. So if you have a solution with a salt, with mobile ions, so the ions are dissolved, so of course what we will study in the following is mobile ions. Because if the ions are not mobiles, it's electrostatics. And these you have studied, I will review that later. But the point is that when you have mobile ions which can move in a solution, then they have a certain entropy and there is a balance between the electrostatic energy and the entropy, which makes the system behave and have a very interesting behavior. So what you can show that if you have in a solution, if you have positive and negative ions, which are mobile, then if you look at the effective interaction between two charges, so if you have a charge, let's say, plus, minus, and you have salt all around, the interaction, the Coulomb interaction is no more, I forgot, 1 over r, by the way. The interaction is no more q1, q2 over 4 pi epsilon 0, epsilon r, 1 over r. That would be the standard Coulomb interaction. There is a misprint here, there is a 1 over r here. But this interaction is screened. There is an effect which is called screening, which we'll study in detail in the coming lectures. And the screening makes, due to the cloud of positive and negative ions in the solution, this will decrease enormously the interaction and make it as e to the minus r over l to the index d. And this length is called the by length, and its expression is given here. So it depends on the dielectric constant. It depends on the density, on the concentration of the salt. N is the concentration of the salt of the positive or negative ions. And q is the valence of the ions in the solution. And if you want to have a evaluation, a simple formula, it's 0.3 nanometer by square root of N, where N is the concentration of the salt expressed in molar. A mole is a number of avogadro molecules per liter. So in physiological conditions in the cell, the concentration of salt is typically 0.1 molar, and the by length is one nanometer, which means that beyond this distance, there is no more coulomb interaction. The coulomb interaction beyond that distance is completely suppressed. So the coulomb interaction exists only within a radius of the order of the by length. And there is another distance, which I will discuss when it comes. OK. So to be complete, there is another thing which is very important in all these systems, which are acids and bases. So acids are proton donors, yes? Yes. No, no, so there is two effects. One comes from here, from this one over epsilon r. When epsilon r increases, so this increases, and of course when epsilon r increases, it increases the range of the coulomb interaction by increasing the by length, but it decreases the strength. The strength of the by-huckel interaction goes like one over epsilon r, so if epsilon r is large, the strength is very small. On the other hand, the range is slightly increased by this square root of epsilon d. The square root, no, I will see that it's, OK, not really, no. OK, so this is the old standard definition of acids and bases, so an acid is something which dissociates into h plus a minus. So it's a donor of protons, and the base is a donor of a hydroxyl group OH minus, so BOH gives this. And of course, in terms of water, what happens is that when you put water, when you put acid in water, the h plus, which comes out of this dissociation, it binds to h2O to make this hydronium complex plus a minus, because h2O is very eager to bind to h plus. And you have all this dissociation that you learned in high school probably, and which I will not discuss. So everything is defined in terms of dissociation constant, and you have strong acids if they are completely dissociated, weak if they are partially dissociated, et cetera. OK. So what are some examples of charged soft and biological matter? So in biological systems, you have, of course, in all cell, in all biological fluids, you have Na plus, Cl minus, which are floating around. You have it in seawater. All the organisms which live in seawater are plunged, are swimming in NaCl. You have also a lot of potassium ions, and the balance between Na plus and K plus between the sodium and potassium ion maintain osmotic balance between the fluids inside and outside the cell. There is magnesium ions, so the magnesium ions are very important to regulate the behavior of DNA and of chromatin. There are divalent ions. Very important. So you have also all kind of other compounds, ions, which are present in biological systems. The examples of polyelectrolytes. So polyelectrolytes, as I said, it's polymers which have charged rubes. Most well-known example is DNA and also RNA protein. So what is important to know is that because you have all these charges along the chain, the chain is very rigid. It has what's called a large persistence length. So locally it is like a tube, and therefore we will study what happens, how the ions around a cylinder behave in a system like this. You have other polyelectrolytes, actin, tubulin, which are making up the skeleton of cells and which are microtubul, for instance, is important in the cell division, and it's a polyelectrolyte. And you have also synthetic polyelectrolyte like PSS, polystyrene, sulfonate. Charged colloids. So colloids are small objects, as I said, spheres, ellipsoid, et cetera. The size ranges from nanometers to micrometers because of van der Waals interactions. So there is a van der Waals attraction between them. They tend to aggregate, and when they aggregate, it's a phenomenon called slokulation, which is a phase transition. So the colloids can be either dispersed or they condense and form aggregates which are fractal aggregates, and this phenomenon is called slokulation. And to prevent this slokulation, there are two options. One is to code the colloids with polymers. So if you have polymers, then the polymer will prevent the aggregation of the colloids. And otherwise, you charge them. You put some charge on the colloids, positive or negative, and then they cannot collapse. And colloids are very important in, for instance, in cosmetics or in paints or things like that. So you don't want them to fluctuate because what's, or milk also, milk is a colloid. So you don't want to have fluctuation because fluctuation is just a separation and then you lose all the nice properties of this solution. OK. Yes, so these are examples of colloids, water-based paints, particles, pigment particles in ink, milk, et cetera. And finally, charge membranes. So charge membranes, so the membranes in the cell is made of two layers of phospholipids. So phospholipids means it's a kind of amphifile molecule with a charged, with a hydrophilic head and hydrophobic tail. So you have the hydrophobic tails which isolate, which bind together on both sides. So that makes a membrane which isolates the outside of the cell from the inside of the cell. OK. And in these biological membranes, they are charged. So 10 to 15% of the lipids, which are the constituents of the membranes, are charged. And this, leading to a charge density of 0.3 electron per nanometer square, which means if you take 0.3, if you take 1 nanometer square, you have 0.3 electron, typically, in this. OK. So these are the use of charged lipids. They assist the binding of charged molecules and charged particles, charged atoms inside the cell. They do exocitosis and doecitosis by absorbing. OK. It doesn't matter. And there are also by these electron, by these Coulombic interactions, you can have ion channels which are fixed on the membranes the traffic of small ions from the inside to the outside of the molecule. This is very important in the function of the cell. These ion channels are responsible for the communication between the inside and the outside of the cell. OK. A last thing that I want to mention is a concept that we will use which is the concept of osmotic pressure. So what is osmotic pressure? So the osmotic pressure is the following. When you have, so you can have a solvent, like water, and assume that you put some particles inside, like colloid, small spheres, or whatever solute so it can be whatever you want. So then you put a membrane here which is a semi-permeable membrane. So the membrane is semi-permeable because it lets the solvent molecule, so it lets the water circulate perfectly easily with no problem, but the solute molecules are bigger and they cannot go through the membrane. So you see that the situation is exactly the same as that of a gas of particle with a piston. The solvent would play the, it's like if you take air, so the solvent, since it's equilibrated on both sides, it doesn't play any role. So you see that the solute molecules which are here, since they cannot go out, they exert a pressure on the cylinder, on the piston here, and this pressure of the solvent on the external cylinder is called the osmotic pressure. And of course for perfect, perfect interactions between these solute molecules, the pressure is given by the ideal gas low, pi equals nkbt. And what I want to emphasize is that this low, in anionic solution, this low is also true. So this low is a low in principle for perfect ideal gases. So gases of particles without interaction. Of course if you have an ionic solution, you have strong interactions because you have a coolant interaction between the particles. However, if you are at thermal equilibrium, there is no flow. So locally the electric field is essentially zero, and therefore the ions don't see any interaction because the local field seen by each ion is essentially zero at equilibrium. And therefore the ions in a solution, in neutral solutions, the ions are essentially free. And therefore this formula applies quite well to ionic solutions. So we will see how one can obtain this in the case of charge solutions, and what are the corrections to this equation. OK. And this is the end for this short presentation of the qualitative part of this. So now I go to the overview. So is there any question about this? OK. It's just a presentation. You can just check that the ratio of gravity to electrostatic is very, very small. OK. Now I will try to go to the overview of electrostatics. OK. OK. So if you have two particles, q1, with charge q1, q2, vector r12, then the force between the two as we saw 4 pi epsilon zero, q1, q2 over r12 times r12, so r12 square, r12. So r12, this is the unit vector along r12. So I give you the values to know. So this constant, as I said, in si unit is 9 10 to the 9. And the electron charge, which we will use, is 1.6 10 to the 19 minus 19 coulomb. OK. So if you look, so the force as we know, the force can be written as q times e. For instance, if I look at the force acting on point, on any point, so since the, see, if I have a charge, if I have a charge q1 here, and I want to see what is the force created here on that point. So q1, the force is f equals qe with e equals 1 over 4 pi epsilon zero q1 divided by r minus r1 to the square times r minus r1, unit vector along this. OK. Now, if I have many charges, if I have charges q1, q2, qi, et cetera, if I look at the electric field created at a certain point r, it's given by 1 over 4 pi epsilon zero, sum over i equals 1 to n, if I have n charges, times qi over, so a way to write it is to write as r minus ri to the cube times r minus ri. Or you can write it also as 1 over r minus ri to the square, and the unit vector r minus ri, but like this. But this is a bit easier to write. OK. So now if we look at the force and we look at the work of the force, we see that the work of the force, that the force can be written as minus gradient, a certain potential energy. And the potential energy is u equals 1 over 4 pi epsilon zero. So this is for f12 between 1 and 2, and this is just q1, q2 over r12. So that's the potential energy of two charges at distance r12 in this configuration. Now, if you remember that the force is qe, then you see that e is minus gradient phi, where phi is the electrostatic potential, and phi, in that case, will be equal to 1 over 4 pi. Sorry, I forgot. No, that's OK. 1 over 4 pi epsilon zero. And if I look at phi at point r, created by a particle at point r1, it will be 1 over 4 pi epsilon zero q1 over r minus r1. That's if I have a particle at r1, particle q1 at r1, if I look at the electrostatic field at point r, and e is given by this. And if I have many particles, many charged particles, then phi of r, like in the case of Coulomb, is 1 over 4 pi epsilon zero sum over i of qi by r minus r1. OK. What else did I want to say? Yes. So, if I define the, and this is something quite important, if I define vc of r minus r prime equals 1 over 4 pi epsilon zero, 1 over r minus r prime, you see that this formula, which is here, tells you that q of r, phi of r, is equal to 1 over 4 pi epsilon zero, is equal, sorry, to sum over i of qi vc of r minus r i. Because this term, 1 over 4 pi epsilon zero r minus r i is just this. So, phi of r is just sum over i of qi vc of r minus r i. And if you see, if you have particles qi at point r i, you can define a charge density rho of r equals sum over i of qi delta of r minus r i, where delta is the Dirac distribution function. Everybody is familiar with Dirac. So, this is the charge density created by particles of charge qi at point r i. OK. So, if you use this notation, then you see that phi of r is the integral d3r prime of vc of r prime of r minus r prime rho of r prime. So, this can be an exercise if you want. So, you just have to replace. So, do the integral and check. It's completely simple. And as a result, from this identity, which is obtained for a set of discrete charge, this identity is also valid for any distribution of charges. So, if you have any distribution of charge rho r, so it can be smooth charge distribution, it can be whatever you want, then the electrostatic potential is given by this expression, where rho of r is the charge distribution of the particles. Now, if you look, how is the electrostatic field, the electric field? So, the electric field is minus e at point r is minus gradient phi of r. And what is gradient vc? So, if I look, what is gradient with respect to r of vc of r minus r prime? Well, essentially it's going to be related to the electrostatic field. So, either you calculate the gradient directly, or you go to the, I mean it's directly related to the electric field. So, it is minus 1 over 4 pi epsilon zero sum over i of r minus r i. Actually, no, sorry, it's only one. So, it's minus r, minus r prime, divided by r minus r prime cube. And, therefore, this is, since it's minus, you just, so it's minus integral d3r prime gradient r vc of r minus r prime, rho of r prime. And, therefore, e of r, you replace the gradient by this expression is 1 over 4 pi epsilon zero integral d3r prime of r minus r prime divided by r minus r prime cube, rho of r prime. So, this is the expression, the equivalent expression of this. Any questions to this point? So, there are three types of geometry that will be, that we will study and which I put here. So, the point geometry, so rho of r is q delta of r. And, for instance, this is, if you have a particle at point r, at point zero, the phi of r is 1 over 4 pi epsilon zero q over r, and the electric field is minus 1, so it's plus 1 over 5 pi epsilon zero q over r square r gradient vector. That's for the point. OK, that's trivial. Then, if you have a line, so the point is like this, right? You have q here, and you look at a certain distance r. Now, you can have a charge line. So, a charge line is defined by a density, by a linear density, rho of r equals lambda delta of x delta of y. So, if the line is along the z axis, here you have x, here you have y. OK, so the density is given by this. Lambda is the so-called linear charge density. Then, the potential, I will write it after, and the electrostatic field. The electrostatic field is perpendicular to the z axis. It's in the parallel to the plane here. Phi is lambda over 2 pi, not phi, e is lambda over 2 pi epsilon zero vector r. So, if you are a distance r from the line, and the corresponding electrostatic potential is logarithmic, it's minus lambda over 2 pi epsilon zero log r over r zero. So, you have to introduce, in that case, you have to introduce, you see, it's an infinite object. The line is an infinite object, so it's extended, so you have to define a reference of potential, because all these expressions make sense when the charges are confined in a finite volume. If the charges are not confined to a finite volume, in the case of an infinite line, you have to define an origin of potentials, and therefore it's this quantity r zero, which is arbitrary. You see that phi is not an observable. The observable is really the electrostatic field. What counts physically is the difference of potentials. It's never the potential value itself, so you can fix its zero value at any scale you want. So, r zero is an arbitrary scale that you choose to fix the potential, the zero of potential. The important thing to think about is that the potential is logarithmic. It goes like log r, and as we will see, this is responsible for a famous phase transition in ionic liquids when you have a line in presence of counter ions, so of ions, a charged line with ions which are oppositely charged. What happens is that the interaction energy of the ions with the line is logarithmic, but the entropy of the ions is also logarithmic. So, you can have a balance, and you have a transition between bound ions and unbound ions, and I will come back to this later. This is called the Manning condensation, which is something we will study. The last geometry is a plane. So, if you have a charged plane with a sigma, charge density sigma. So, the charged plane is something like this. So, how did I? Yes, so if this is the z axis here, the x and y, so rho of r is sigma delta of z. Delta of z is this plane. Then the electrostatic potential is sigma over 2 epsilon zero z. So, where z is the unit vector in the z-direction. So, you know that when you have a charged plane, the electric field is independent of your distance with respect to the plane, which of course makes sense because there is no scale in the problem because this is infinite. And the corresponding phi is sigma over 2 epsilon zero with a minus presumably, and modulus of z. So, the potential is linear as a function of the distance to the plane. So, these are three geometries that we will study in detail in the coming lectures. It's good to remember this. Another case which is very important is the case of dipole. So, I guess all this is familiar, right? You have all studied this in elementary school. So, no question? OK, so, the dipole. OK, so let's say, so dipole is, so this is the z-axis. I put plus q charge here, minus q charge here. Minus d over 2. So, the charge density is rho of r. OK, so it's q times delta of z minus d over 2, minus delta of z plus d over 2. In fact, the potential phi of r is 1 over 4 pi epsilon zero q times 1 over r. So, if I have, if this is point r, so r is the coordinates x, y, z. So, this point has coordinates 0, 0, d over 2. This one has coordinates 0, 0, minus d over 2. So, it's r minus this, so it's 1 over square root of x square plus y square plus z minus d over 2 square, minus 1 over square root of x square plus y square plus z plus d over 2 square. Now, if you look at distances such that r is much larger than d, you can expand this to first order in terms of d over r. And, of course, I leave you to do that as an exercise. It's a classic exercise. And what you find is that phi of r is equal to 1 over 4 pi epsilon zero times p r over r cubed. No, r over r square, if I write it like this. The vector p, the vector p, which enters here, is q times d. Or qd, let me write it as z. So, it's essentially the vector, this is the vector, which connects the negative charge to the positive one, going from the negative to the positive, the unit vector times dipolar moment. And, of course, there is a correction. The correction will be of order 1 over r cubed. From this, you can get the electrostatic field created by a dipole, so point r, so e of r equals 1 over 4 pi epsilon zero. So, 3 r minus r over r, which you obtain by taking the minus gradient of this phi of r. Any questions? So, please try to derive this by doing the expansion. I'm sure you have seen that already. You probably forgot, but I don't know if I could do it here. OK. Now, if instead of one dipole, we have many dipoles, of course it's additive, so the total dipolar moment, if you have small dipoles like this everywhere, so p is going to be sum over i. OK. So, one can define what's called the dipolar density, and the dipolar density, or the total dipole is given by integral d3r of r rho of r, where rho of r is the density of charges. So, actually, you can do also an expansion of these to define first order in d. And that's easy. You see that if you assume that d is small, then you expand to first order. So, if you expand to first order, then q is delta of z minus d over 2d delta prime z of z. So, this is the expansion to order 1 of the delta function. And the other one is minus delta of z minus d over 2 delta prime of z. And therefore, it's equal, so the delta disappear, which is a sign that the system is neutral, where you get minus qd delta prime of z, which in some sense is just minus qd. So, you can write it if this is the dipolar moment. And this, so you can write it as follows as minus p gradient delta of r. I don't know if it is clear, but you can see that as an exercise also. Right? Because p is along the z axis. So, this gives you only a d by dz. It's essentially pz, which is qd, times d by dz of this. OK. OK. OK. Then Gauss theorem. That's an important thing of Gauss law. So, what does Gauss law tell you? Gauss law tells you that if you have a certain volume v surrounded by a certain surface s, and this is the normal to the surface, so Gauss law is that the integral over the surface s n ds, or let me write it e ds. So, you assume that you have some charges inside, and that the normal, so it's a closed surface, and the normal is oriented towards the exterior of the surface. So, n is the unit vector normal to the, yes? Here? Yes? Delta function. You can expand distributions. It's perfectly legal. You do a Taylor expansion of distributions like this, like normal functions. It's itself a distribution, so mathematically it's defined by its action on functions, but it's perfectly legal. OK. We'll come back to that. You will see when we'll study dipolar fluids, we'll come back to that. So, the Gauss law states that this integral of the, so this is called the flux of e through the surface s, the flux is equal to the total charge enclosed by s. So, the total charge divided by epsilon zero, so it's 1 over epsilon zero integral in the volume v d3r of rho of r, where rho of r is the charge density inside the volume v. So, this is a very useful theorem because it allows you, for instance, to derive the electric field near a plane or the electric field created by a cylinder, by a line. You use symmetries and things like that. Maybe, OK, as an exercise, use Gauss law, Coulomb potential for point line that standard exercise. OK. And from this Gauss law, you can deduce the fact that divergence of e of the electric field, which is equal to rho of r divided by epsilon zero. So, these two are strictly equivalent. This is the integral formulation of Gauss law, and this is the differential form of Gauss law. You probably all saw that in your course of electrostatics, right? OK. And since e is equal to minus gradient phi, this relation, so we know that gradient of the divergence of a gradient is the Laplacian. So, if I put a minus, so it means that this is divergence of e, and divergence of e is equal to rho over epsilon zero. So, we get another form of Gauss law, which is Laplacian of phi equals minus rho over epsilon zero. So, there are three quasi identical forms of Gauss law, which are here, and which are very important, and which we will use all the time. And this is called Poisson law, by the way. So, this is Gauss, and this is Poisson. OK. So, another important thing is about boundary conditions. Boundary conditions? Any question about Gauss law? OK. Boundary conditions. Imagine you have a charged surface with charged surface B sigma of r. So, on one side you have e1 very close to the surface. So, let's see. Locally I take a piece of this surface. Here I have the sigma, and here I have e1, here I have e2. So, if I define n the unit vector normal to this, then you can show easily by integrating the Gauss law in this form, or using this on a certain circuit, you can show that the discontinuity. So, it's e1 minus e2. So, this condition is very important. It means that when you have a charged surface, if you look at the electrostatic field on one side, the electrostatic field, if you have a charge, is discontinuous. And the difference between the normal electrostatic field on both sides, so that's the normal electrostatic field, because it's projected on the normal, is equal to the charged density on the surface. So, that's an important consequence of the Gauss theorem. The other boundary condition is that e1 parallel is equal to e2 parallel. There is no discontinuity in the... So, if your field is like this, you have a component like this. So, here this component is the same, but this one can be different. So, the parallel component of the field is conserved at the crossing of the surface, but the perpendicular is discontinuous and the discontinuity of the electric field is equal to the charge density. So, this all comes from the Gauss theorem. OK, what else? Yes, something that we will use, which is the electrostatic energy. So, if you remember that u is the electrostatic potential energy that we saw, it's q5 essentially. So, it's 1 over 1 half of sum over i and j. If I have charges qi at point ri, many charges like this, it's sum over 1 half sum over i not equal to j of qi qj over 4 pi epsilon 0 ri minus rj. The factor 1 half comes from the double counting of the... So, this is the... Each term, 1 over 4 pi epsilon 0 qi qj over ri minus rj is the electrostatic potential energy of interaction between charge i and charge j. If you sum over i and j, you have a double counting because you count per ij and per ji, so you need a factor 1 half to make normal counting. And this you can write as 1 half of sum over i of qi phi i, where phi i is the potential at point ri created by all other charges j not equal to i. Now, if you remember that the charge density is rho of r is equal to sum over i of qi delta of r minus ri, then up to something which I will write. So, u of u, the total interaction energy, is equal to 1 half integral d3r d3r prime of rho of r vc of r minus r prime, rho of r prime. I remind you that vc of r minus r prime is just 1 over 4 pi epsilon 0 1 over r minus r prime. So, you just put this in and you have it. Now, if you do that, of course, you will see that you ink. So, I mean, I can, I will do it in, OK, let me detail it a little bit. So, u, if I replace rho of r by its expression, I will get u equals 1 half integral d3r d3r prime sum over i qi delta of r minus ri 1 over 4 pi epsilon 0 1 over r minus r prime sum over j qj delta of r prime minus rj. I have just rewritten this expression. So, I can do, so I put the sum over i, sum over j outside. I have the qi qj. And then I have this integral times delta times this times delta. So, there is a 1 over 4 pi epsilon 0. And then there is a term, which is 1. Then the delta function tells me to replace r by ri and r prime by rj. So, it's ri minus rj. So, it looks very much like this, except a small problem is that when you write it like this, there is a contribution from the terms i equal j. And the terms i equal j are essentially divergent, and this poses a problem. So, we will write it as minus. So, here, when I wrote the energy like this, I excluded, of course, the interaction of i with itself, the terms i equal j. So, here I have to subtract a term to avoid. So, this expression is not quite correct. There is something which should be subtracted, which are the terms i equal j. So, the term i equal j, it's minus one-half. So, if I put i equal j, so I will have sum over i of qi square over 4 pi epsilon 0. So, I will not write it as 4 pi epsilon 0. I write it qi square, and this will be just vc of 0, which is infinite, but it's a number. So, I can express, this expression is correct, provided I subtract minus one-half sum over i of qi square times vc of 0. But this is an expression that we would use very often. I will skip. So, maybe I will stop at this level. Is there any question? No question. I will do next. So, next time I will finish this rapidly, this electoral statics in continuous media. I will just do a few rapid overview of statistical physics of what will be needed in statistical physics, and then I will start Poisson Boltzmann theory. OK, if it's too fast, too slow, maybe don't hesitate to complain.