Polymer Chain in Statistical Mechanics

• Silviu
In summary, the problem at hand involves finding the mean distance ##\textbf{r}## between the ends of a polymer chain consisting of N segments of length d each, at a temperature T. The segments are able to rotate freely relative to each other and a force f is applied at the ends of the chain. The partition function is given by $$Z=(2 \pi \int_0^\pi d \theta sin (\theta) e^{\beta f d cos (\theta)})^N = 4 \pi \frac{sinh(\beta f d)}{\beta f d}$$ and the probability distribution function for the states of the chain is $$P(\theta_1, \phi_1, ... Silviu Homework Statement A polymer chain consist of a large number N>>1 segments of length d each. The temperature of the system is T. The segments can freely rotate relative to each other. A force f is applied at the ends of the chain. Find the mean distance ##\textbf{r}## between the ends. Homework Equations The Attempt at a Solution So when there was no force the result was 0. Now I am a bit confused. I wrote the partition function as:$$Z=(2 \pi \int_0^\pi d \theta sin (\theta) e^{\beta f d cos (\theta)})^N = 4 \pi \frac{sinh(\beta f d)}{\beta f d}$$. Now the probability of a given state is$$p_{state}=\frac{e^{\beta f d \sum_1^N cos(\theta_i)}}{Z}$$and I am thinking to write$$<\textbf{r}> = \frac{(2 \pi \int_0^\pi f(\theta) d \theta sin (\theta) e^{\beta f d cos (\theta)})^N}{Z}$$. But I am not sure what this ##f(\theta)## should be. I want it to represent the vectorial distance between 2 neighboring points as a function of the angle between the direction of the force and d, but I am not sure how to write it. Should I do it separately for x, y and z and add them up or is there a way to write it directly? Also the solution I have uses:$$F=-NTlog(Z)$$and then:$$L = -\frac{\partial F}{\partial f}$$where L is the answer required. I am not sure I understand why. First, it looks like a scalar not a vector and why would the distance be given by that derivative? Thank you! Silviu said: Now I am a bit confused. I wrote the partition function as:$$Z=(2 \pi \int_0^\pi d \theta sin (\theta) e^{\beta f d cos (\theta)})^N = 4 \pi \frac{sinh(\beta f d)}{\beta f d}$$. OK, this looks right. It might be helpful to note that ##Z## can be expressed alternately as$$Z = \int_0^{2\pi} d \phi_1 \int_0^\pi d \theta_1 sin \theta_1 \int_0^{2\pi} d \phi_2 \int_0^\pi d \theta_2 sin \theta_2 ... \int_0^{2\pi} d \phi_N \int_0^\pi d \theta_N sin \theta_N \, e^{\beta f d \sum_1^N cos(\theta_i)}$$##\phi_i## is the azimuthal orientation of the ith segment. This expresses ##Z## as an integration over the microstates of the entire chain. A microstate of the chain is the specification of all of the individual ##\theta_i## and ##\phi_i##. Thus, the probability distribution function for the states of the chain is$$P(\theta_1, \phi_1, ... \theta_N, \phi_N) = \sin \theta_1 ... \sin \theta_N e^{\beta f d \sum_1^N cos(\theta_i)}$$Regarding ##\langle \textbf{r} \rangle##, I would follow your suggestion of doing the components separately.$$\langle \textbf{r} \rangle =\langle r_x \rangle \mathbf{e_x} +\langle r_y \rangle \mathbf{e_y} + \langle r_z \rangle \mathbf{e_z}$$Note that ##r_x = \sum_1^N \Delta x_i## where ##\Delta x_i## is the x-displacement of the ith segment. You can express ##\Delta x_i## in terms of ##\theta_i## and ##\phi_i##. You can then evaluate ##\langle r_x \rangle## using the probability distribution function ##P(\theta_1, \phi_1, ... \theta_N, \phi_N)##. Similarly for ##\langle r_y \rangle## and ##\langle r_z \rangle##. Also the solution I have uses:$$F=-NTlog(Z)$$and then:$$L = -\frac{\partial F}{\partial f} where L is the answer required. I am not sure I understand why.
If you use the alternate expression for ##Z## as I gave above, what expression do you get for ##\frac{\partial F}{\partial f}##? Do you get something that is related to ##\langle r_x \rangle##, ##\langle r_x \rangle##, or ##\langle r_z \rangle##?

1. What is a polymer chain?

A polymer chain is a molecule made up of repeating units of smaller molecules called monomers. These monomers are connected by covalent bonds to form a long chain structure. Examples of polymer chains include plastics, proteins, and DNA.

2. How does statistical mechanics apply to polymer chains?

Statistical mechanics is a branch of physics that studies the behavior of systems with a large number of particles, such as polymer chains. It uses statistical methods to describe the properties of these systems, such as their energy, entropy, and temperature.

3. What is the role of entropy in polymer chains?

Entropy is a measure of the disorder or randomness in a system. In polymer chains, entropy plays a crucial role in determining their physical properties, such as their flexibility, elasticity, and solubility. As the temperature increases, the entropy of the polymer chains also increases, leading to changes in their behavior.

4. How do external factors affect the behavior of polymer chains?

Polymer chains are sensitive to external factors such as temperature, pressure, and chemical environment. These factors can influence the motion, conformation, and interactions of polymer chains, ultimately affecting their physical properties. For example, increasing the temperature can cause polymer chains to become more flexible and less ordered.

5. What are some real-world applications of understanding polymer chains in statistical mechanics?

Understanding polymer chains in statistical mechanics has many practical applications. It is essential in the development of new materials, such as polymers with specific properties for different industries. It is also crucial in understanding biological processes, such as protein folding and DNA replication. Additionally, statistical mechanics plays a role in the study of phase transitions and self-assembly, which have applications in drug delivery, nanotechnology, and more.

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