[MUSIC] What do you crave for at the end of a long run or workout at the gym? You crave for those electrolytes. What are electrolytes? Electrolytes is simply solutions that contain salt in it. Now, electrolytes behave in very different ways from the solution that makes up, or just the solvent that makes up the electrolyte. Electrolytes are important in a wide variety of applications. They're important biology. They're important in laptop batteries that we carry around everyday. Now, how is it that the electrolyte behaves differently from the solvent, from the pure solvent? It turns out that the electrostatics in an electrolyte are very different from those of the solution. What we will do in this lecture is to understand these interactions. And by the end of this module, you should be able to develop a thermal dynamic model to describe the electrotype solutions, and the electrostatics that are involved in electrolyte solutions. And in the end you should be able to rationalize this phenomenon called Manning Condensation, which governs the behavior of poly electrolytes. [SOUND] Electrostatic interactions in an aqueous environment dramatically influence its thermodynamic behavior. Polyelectrolytes, charged polymers, are soluble in water, in contrast to most hydrocarbon-based polymers which are insoluble. Most biopolymers are polyelectrolytes, and in many instances, their biological function hinges on the role of electrostatic interactions. We'll begin by reviewing some basic electrostatics. Now remember, the force on a point particle with charge Q that's kept in electric field E is simply given by the charge times the Electric Field. Now, the Potential Energy can be written as q times an electrostatic potential site. The Electric Field is simply defined as a gradient of a scalar, the electrostatic potential. Now, the Electric Field itself satisfies Gauss's Law which simply says, that the integral of the electric field over a surface equals the total charge contained inside that surface. Note that in this equation, epsilon is the Dielectric Constant at the medium and Rho is the local charge density. Now, there's another important theorem in electrostatics which relates the divergence of a quantity, and it's known as the Divergence Theorem. Now, this theorem converts a surface integral into a volume integral through the emergence of a quantity called a divergence. Now, this leads to an important relation for the case that we are considering, that the electrostatic potential must satisfy a well-known equation known as the Poisson Equation. The Poisson Equation simply states that the Laplacian of the Electrostatic Potential can be related to the local charge density rho divided by the dielectric constant of the medium. Now, let's consider a simple example. A sphere with radius a, which has a total charge q now, due to spherical symmetry of the problem, the electrostatic potential around the sphere is radium. That is, Psi is simply a function of r. Now, let's write down the Poisson Equation outside the sphere. Now, this equation can be easily integrated, leading to 2 unknown constants, A1 and A2. Now, evaluating A1 and A2 requires 2 boundary conditions, now 1 boundary condition is simply that as the distance from the sphere tends to infinity the electrostatic potential should 10 to 0. Now, this immediately yields the relation that A2 = 0. Now, we can apply Gauss's law and this gives us a way to evaluate the unknown A1 to the charge Q. Now, for a finite sized sphere of radius A, where does the derived expression of the potential valid? Now, the derived potential is strictly valid, outside the radius of the sphere A, that is R is greater than A. Now, for a point particle, that is when A turns to 0, this description is valid throughout space. Now, the electrostatic potential provides all the necessary information to find the energy of the system. Now remember, the energy interaction between two ions would charge E. That is the charge of an electron separated by distance R. Is simply given by Cullom's law. Now, from this we can actually define a length scale a term as the Berum Length. The Berum Length is simply the distance at which the two electrons need to be separated when the energy associated with this interaction is equal to the thermal energy given by the Bolsman constant times the temperature. This length for water is around seven angstrom at room temperature. Hence, the energy of interaction for two monovalent ions in a dielectric medium can be rewritten in the following way. Now, let's shift gears and consider a second example, this time a negatively charged cylinder of radius A with the surface charge such that the average distance between the surface charges is B. Now, the Electrostatic Potential in an uncharged medium, that is, Rho is equal to 0, can be derived using the Poisson equation. Now, in this case, what coordinate system should we use? Now, given the Cylindrical Symmetry of the problem, it is useful to think about this problem in cylindrical coordinates. Now, expressing the Laplacian in ylindrical coordinates yields a solution of the electrostatic potential. Now, once again, we will have two unknown constants, and this requires two boundary conditions to evaluate them using the Divergence Theorem we can write down one boundary condition. This gives the boundary condition on the surface of the cylinder. Now, the boundary condition on the cylinder's surface can be rewritten in this following way. This gives us a way to evaluate a1. Now, for simplicity we can set that the electrostatic potential goes to 0 as R tends to aid, that is, when the distance is very close to the radius of the cylinder. Now, this fixes the second unknown constant a2 as well. Now, this gives the complete solution for the Electrostatic Potential. Now, unlike the point charge case, the Electrostatic Potential outside a charged cylinder diverges logarithmically which has dramatic consequences on the counter ion distribution. Now, lets consider the simple case of adding salt to the solution. The nature of the electrostatic interactions is dramatically altered. Now, due to the strength of the electrostatic interactions. Mobile counter ions have strong spacial correlations with their opposite charge. Now, let's consider a polyanion, a negative polyelectrolyte which has a flurry of positive counter ions swarming around the polymer. Now, in the case of the on the other hand which consists of positive and negative ions. The interaction between the charges are dramatically reduced due to the screening of the electrostatic interactions by the medium. Now, consider the interaction between two isolated ions in a monovalent salt solution with concentration NS. For example, sodium chloride that dissociates into sodium plus ions and chloride minus anions. Now, the Electrostatic Potential side around the tagged ion is going by the Poisson equation. Now, it's very tricky to evaluate the charge density in the solution. Now, let's take a simple picture where the local concentration of counter ions is given by a simple Boltzmann weight of the ions in the Electrostatic Potential site. The energy associated with an Electrostatic Potential site is simply the charge multiplied by the Electrostatic Potential. Hence, the density can be easily evaluated in this simple picture. Now, in this equation, the left term is due to the local concentration of the sodium plus ions, and the right term is that of the chloride anions. Now, in the example that we are considering, we can now derive an equation that the Electrostatic Potential must satisfy. Now, this equation is known as the Poisson-Boltzmann equation. Even using this simple model, this highly non-linear equation does not have a simple closed form solution. Now remember, implicit in this theory is the neglect of local concentration fluctuations Akin to a mean field approximation. Now, let's make a few more approximations to make this model analytically tractable. For sufficiently large lengths, when the electrostatic potential tends to zero, we can expand the hyperbolic sine function to lowest order. Now from this equation, we can define a useful quantity called the Debye length. This equation now becomes analytically tractable. In the limit of large length separation r, and dilute solution of counterions the Electrostatic Energy between two ions in a salt can be written down analytically. Now, this is known as the Debeye Huckel Theory of charge screening. The Debeye Huckel theory adds a simple correction to the bare Electrostatic Potential to account for counter ion interactions. Now, in the absence of the salt what is the Electrostatic Energy? While in the absence of the salt the electrostatic interaction is extremely long ranged and scaled as one over R. Now, salt screening leads to a short range Electrostatic Energy that decays exponentially with the lists in separation. Now, in the discussion of we neglected all the nonlinear terms in the Poisson-Boltzmann equation. This leads to qualitatively incorrect behavior for distances very close to a charged object. For example, the surface of a polyelectrolyte. For a dilute solution, the average distance between the counterions scales inversely to the concentration raised to the one third power. Now, our distance is closer than the average distance, the mean filled approximation that you've invoked breaks down and we must address the interactions explicitly. Now, let's consider a Polyelectrolyte chain to be a charged cylinder with radius a. We'll assume that this radius a is small, that is, a tends to 0 in our analysis. The charged cylinder has negative charges smeared all over its surface such that the average distance between the negative charges is. We want to find the behavior of an isolated ion interacting with the cylinder at distances less than the average inter ion spacing are not. Now, earlier in this lecture we derived the electrostatic potential around a negatively charged cylinder in a dialectic medium. This was found to depend logarithmically on the distance away from the cylinder. Now, the Electrostatic Energy between the cylinder and a positively charged Monovalent ion is given by the charge, e, multiplied by the Electrostatic Potential, Psi. Now, the partition function for an isolated ion interacting with the job cylinder within the distance are not for a small cylinder radius that is a dense to 0 scales in the following way. Now, immediately one can not that the partition function scales as a power of r raise to 2- 2q, now remember this q is given by the length divided by the average difference between the charges. That is LB divided by. Now, when q is greater than 1, the partition function actually diverges. Now, how do you reconcile this? Well, in order to reconcile this, Manning proposed the following picture. The divergence of free energy is something that is rather interesting. Something very, very unusual happens. What happens is that there are mobile ions that condense onto the cylinder surface to neutralize the negative surface charges that are present. Now, what happens as a result, the effective queue goes down and saturates out at a critical value of one. Now, at short length scales, the result of this phenomenon is known as Manning condensation, and this leads to an overall reduced surface charge. Now, at long length scales, the interaction is the normal screened electrostatic cooling potential. Now, what applications do these theories have? The Dbye-Huckel Theory is the simplest description for describing electrolyte solutions. There are electrolyte solutions used, well the electrolyte solutions are extremely important in battery systems. It's also crucial In describing electrostatic interactions in biological systems. Systematic improvements to these theories are still an active area of pursuit research. Now, to summarize in this lecture, using the basis of electrostatics, we discussed a model for the treatment of electrostatic interactions in a salt solution, that is an electrolyte, we developed the Debye-Huckel model which showed that the Electrostatic Energy decays exponentially with the distance of separation. Finally, we discussed a phenomenon of Manning condensation, which occurs in a polyelectrolyte where counterions condense and reduce the surface charge density below a certain critical value.