# Polymer forces

## Contents

## Polymer ordering at surfaces

PEO has an “inverse” water solubility – it becomes less soluble at higher temperatures.

A general principle:

The less soluble the adsorbed polymer – the less stable the dispersion.

PEO has a nonpolar part but also an oxygen atom that allows for hydrogen bonding. Thus, at lower temperatures, these molecules can interact with water, but as temperature increases, hydrogen bonding weakens, and the nonpolar part of the polymer chain begins to dominate. The molecules become more hydrophobic. And when molecules are hydrophobic and not soluble, they want to form clumps and come closer together. That is why the equilibrium energy decreases with temperature.

## MW and temperature effects

Polystyrene polymers on mica. Polystyrene is a hydrocarbon chain with phenyl group rings attached to every second carbon, and is therefore hydrophobic.

(a) End-grafted in toluene, (b) Adsorbed from cyclohexane.

Both solubility and bridging effects are possible in (b)

So, when the polymers are put in toulene (a), they are soluble and repel each other more. When they are put in cyclohexane, they are not soluble, and thus minimize energy by clumping together more.

## Polymers at surfaces

- (a) In solution
- (b) End-grafted
- (c) Adsorbed
- (d) Adsorbed at low &thetha;
- (e) Adsorbed at high &thetha;
- (f) Bridging

### Free energy of weekly adsorbed polymer chain

A charming scaling argument by de Gennes ('Scaling concepts in polymer physics' - 1979), to calculate the free energy of a weekly adsorbed polymer chain, corresponding to image (d). Let us first consider a polymer chain confined within a tube of diameter D. Let's call the unperturbed size of the polymer <math>R_0</math>, in a solution of temperature T. For a tube of diameter D << <math>R_0</math>, the free energy of the chain will be:

<math>F \approx T \frac{{R_0}^2}{D^2}</math>

The argument is dimensional and based on the assumptions that the tube walls repel the chain strongly and confinement in the y-direction will not affect the polymer's random walk in the x-direction.

Next, let's move to the slightly more complex case of a polymer lightly adsorbed to a wall. In this problem, D is the average distance that polymer loops extend from the wall. We build on the previous problem by assuming that the free energy will feature a confinement term as well as a polymer-wall contact interaction term:

<math>F \approx T \frac{{R_0}^2}{D^2} - T \delta f_b N</math>

Here, <math>T \delta</math> is the effective attraction seen by a monomer adsorbed at the surface , <math>f_b</math> is the fraction of bound monomers and N is the number of steps in the random walk.

## Polymer Elasticity

### Length Scales

In polymer theory, there are three types of filaments, characterized by the persistence length and the contour length. The persistence length is a measure of stiffness: for pieces of polymer shorter than the persistence length, the molecules can be approximated as a flexible rod; for pieces longer than the persistence length, a random walk is more appropriate to describe the mechanics of the polymer. The contour length is simply the length of the polymer if stretched to its full length. The first two types of filaments are filaments for which the persistence length dominates the contour length, and vice verse. The third is one for which the two length scales are comparable. These filaments do not form loops and knows, but they are flexible enough to have thermal bending fluctuations.

### FJC Model

Rubber elasticity, also known as hyperelasticity, describes the mechanical behavior of many polymers, especially those with crosslinking. Invoking the theory of rubber elasticity, one considers a polymer chain in a crosslinked network as an entropic spring. When the chain is stretched, the entropy is reduced by a large margin because there are fewer conformations available. Therefore, there is a restoring force, which causes the polymer chain to return to its equilibrium or unstretched state, such as a high entropy random coil configuration, once the external force is removed. This is the reason why rubber bands return to their original state. Two common models for rubber elasticity are the freely-jointed chain model and the worm-like chain model.

Polymers can be modeled as freely jointed chains with one fixed end and one free end (FJC model) where <math>b \,</math> is the length of a rigid segment, <math>n \,</math> is the number of segments of length <math>b \,</math>, <math>r \,</math> is the distance between the fixed and free ends, and <math>L_c \,</math> is the "contour length" or <math>nb \,</math>. Above the glass transition temperature, the polymer chain oscillates and <math>r \,</math> changes over time. The probability of finding the chain ends a distance <math>r \,</math> apart is given by the following Gaussian distribution:

- <math>P(r,n)dr = 4 \pi r^2\left( \frac{2 n b^2 \pi}{3}\right)^{-3/2} \exp \left( \frac{-3r^2}{2nb^2} \right) dr \,</math>

Note that the movement could be backwards or forwards, so the net time average <math>\langle r\rangle</math> will be zero. However, one can use the root mean square as a useful measure of that distance.

- <math>\langle r\rangle = 0 \,</math>
- <math>\langle r^2\rangle = nb^2 \,</math>
- <math>\langle r^2\rangle^{1/2} = \sqrt{n} b \,</math>

Ideally, the polymer chain's movement is purely entropic (no enthalpic, or heat-related, forces involved). By using the following basic equations for entropy and Helmholtz free energy, we can model the driving force of entropy "pulling" the polymer into an unstretched conformation. Note that the force equation resembles that of a spring: F=kx.

- <math>S = k_B \ln \Omega \, \approx k_B \ln ( P(r,n) dr ) \,</math>
- <math>A \approx -TS = -k_B T \frac{3 r^2}{2 L_c b} \,</math>
- <math>F \approx \frac{-dA}{dr} = \frac{3 k_B T}{L_c b} r \,</math>

The worm-like chain model(WLC) takes the energy required to bend a molecule into account. The variables are the same except that <math>L_p \,</math>, the persistence length, replaces <math>b \,</math>. Then, the force follows this equation:

- <math>F \approx \frac{k_B T}{L_p} \left ( \frac{1}{4 \left( 1- \frac{r}{L_c} \right )^2} - \frac{1}{4} + \frac{r}{L_c} \right ) \,</math>

Therefore, when there is no distance between chain ends (r=0), the force required to do so is zero, and to fully extend the polymer chain (<math> r=L_c \,</math>), an infinite force is required, which is intuitive. Graphically, the force begins at the origin and initially increases linearly with <math>r \,</math>. The force then plateaus but eventually increases again and approaches infinity as the chain length approaches <math>L_c \,</math>.

### Some musings on elasticity

At small enough length scales, the concept of elastic stretching does not make sense, and every system strain can be thought of as geometric unfolding. But if we go even smaller, to the level of the atom, the concept of geometric unfolding does not seem to make sense anymore, and an elastic model making use of the attraction between the protons and the electron cloud seems like a better description. But what really is the rearrangement of particles, than some simple geometric unfolding... So maybe we have shown that when we talk about geometric unfolding and elastic stretching, we are really talking about the same thing, it just depends on what length scale we want to look and which hat we are wearing at the time.

Not getting ahead of ourselves too quickly, there does seem to be an important difference between geometric unfolding and elastic stretching: a restoring force. In pure geometric unfolding, the system does not have a "favorite" state. Elastic stretching involves taking a system from its preferred state to another state. A force will need to be maintained on the system to keep it in the other state.

Now, no system exists without forces on it, so it is an academic exercise to consider purely geometric systems. This is not to say the exercise is useless. I only mean to point out that when discussing systems with reversible strains, it may be sufficient to discuss *Geometric Unfolding with Restoring Forces*.

We seem to have left out systems with more than one stable state. But this is nothing more than different restoring forces trying to bring the system to a number of different stable states. The particulars of the forces acting on the system will determine which stable state is chosen.

Intuitively, it makes sense that the only way to get something to reach from point A to point B, is to rearrange it in some way. When these rearrangements are small enough so we don't care about the details of the process, we call it elastic stretching. This concept is easier to work with than thinking about a rubber band as geometrically unfolding with restoring forces.

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