# User:Nefeli

## About me

I'm a G1 in applied physics with a biopolymer past. I'm originally from Greece, although for the past 2 years I was living in Amsterdam, where I completed a masters in biophysics.

## Fun facts on soft matter

## Final Project

For my final project I want to explore the fundamental interactions between charged polymer chains in confinement. This is a tricky subject partly because at the level of entangled polymer gels, fundamental polymer-polymer interactions do not come up in relevant literature. Past research has instead focused on the collective behavior of these polymer melts, i.e. orientational phase transitions, viscosity and relaxation scaling laws. This is reasonable, since in such dense and entangled systems fundamental interactions are maybe too complex and it is unclear how they relate to macroscopic gel behavior. So, as a first step, I decide to avoid the complexity of a confined polymer gel and instead focus on the interactions (attractive and repulsive) between two polyelectrolyte chains immersed in a good solvent. As a second step, I want to introduce a single wall ( a confining barrier) perpendicular to the chains and see how the confining potential influences the polymer-polymer interaction. Possibly, a third step would be to include a crowding agent, meant just to occupy space in the solvent, to tip over the repulsion between the polymers into attraction and watch aggregation take place. Below is a first draft:

1) Polymer-polymer interactions

In this regime three interactions (at least!) are keeping the chains apart:

- Entropic repulsion

Entropic repulsion is inherent to the term 'polymer', since by polymer we mean a large chain of repetitive segments N which perform a thermal, self-avoiding random walk further modified by excluded volume effects and solvent condition. We assume that our polymers are swollen since they are immersed in a good solvent (<math>\chi < 1/2</math>). The end-to-end distance for these polymers will consequently depend exponentially with N, and the value of the exponent is the typical Flory value of 3/5. I will not dwell on the free energy of the single chain, since we are looking for an interaction energy between the two chains.

To get an estimate of that interaction I resort to the prediction for the osmotic pressure between polymer blobs in a semi-dilute polymer regime of concentration <math>\phi</math>. According to our Witten textbook, the osmotic pressure pushing the blobs apart will be:

<math>\frac{\Pi}{k_BT} = (3.2A)^{-3}\phi^{9/4} \Rightarrow \Pi = k_BT(3.2A)^{-3}\phi^{9/4}</math>

Here A is an experimental parameter depending on the type of polymer and solvent, yet independent on polymer chain length. Thus we obtain an estimate for entropic repulsion in units of <math>k_BT</math>. But am I right to neglect the physics of the single polymer chain and opt for a scaling argument for a semi-dilute polymer solution?

- Electrostatic repulsion

To make this discussion relevant to biopolymers (such as cytoskeletal polymers or DNA chains, which invariably posses a surface charge), we have to consider the electrostatic repulsion arising from the surface charge of the two polymer chains. Before we launch into this, it is worth mentioning that charge -much like favorable solvent conditions- further stretches out a polymer chain and partially suppresses thermal fluctuations. This is intuitively understood if we think back to the self avoiding potential (amplified by charge-charge repulsion along chain segments) that stretches out the polymer chain.

To get an estimate of the screened Coulomb interaction between two polyelectrolyte chains immersed in an ionic solvent, we have to solve a Poisson-Boltzmann equation. I don't despair just yet, since a problem of the sort was already cleverly solved by Brenner & Parsegian back in 1974. The publication actually addresses the problem of two rigid, cylindrical polyelectrolytes (as opposed to two cylindrical wiggling polyelectrolytes) at mutual angle <math>\theta</math>, but I figure that this approximation is good enough at this point and decide to live with that discrepancy. (I secretly wonder whether it matters a lot and intend to come back to this delicate assumption later in the game). So, Brenner & Parsegian solve the following equation:

<math>\nabla^2\Psi = - \frac{4\pi e}{\epsilon} \sum n{_i}^0z_i e^{\frac{-z_ie\psi}{kT}}</math> <math>\Rightarrow</math> <math>\nabla^2\Psi = \kappa^2 \Psi</math>

This first step is achieved by expanding the exponentials in the first equation and keeping only the leading term of the expansion (it seems that back in 1974 they already knew a few Applied Math 201 tricks...). <math>\kappa^2</math> is defined as:

<math>\kappa^2 = \frac{8 \pi n e^2}{\epsilon kT}</math>

Here <math>\epsilon</math> is the dielectric constant of the bathing medium and <math>n = \frac{1}{2} \sum n{_i}^0z_i</math> designates the concentration of the ionic species (ions of valence <math>z_i</math> having concentrations <math>n{_i}^0</math>. The potential <math>\Psi</math> varies with the distance r from the rod body as:

<math>\Psi = \frac{2 \nu_h}{\epsilon}K_0(\kappa r)</math>,

where <math>\nu_h</math> is the line-charge density of the rod and <math>K_0</math> is a zeroth order Bessel function. Now this is lightly puzzling since traditionally we assume the electric potential to be a simple exponential charge distribution, but for now I trust the trick. Finally, at angle <math>\theta = 180</math> , the electrostatic repulsion between rods at a distance r from each other is:

<math>U_C(r) \approx C_1 sqrt{\frac{k_BT}{C_2r}}e^{-C_2r/k_BT}</math>.

where <math>C_1 = C_1(R_A, \epsilon, \sigma) and C_2 = C_2(n,\epsilon)</math>

Here the constants are dependent on the radius of the cylinders <math>R_A</math>, the dielectric constant of the medium <math>\epsilon</math>, the rod surface charge <math>\sigma</math> and the solvent ionic concentration n. Again, after ploughing through the constants we are left with an energy estimate in terms of <math>k_BT</math>. This is encouraging.

2) Polymer-polymer interactions in the presence of confining potential

3) Polymer-polymer interactions in the presence of confining potential and crowding agent