Question

Persistence length and Fourier analysis In the chapter we computed the tangent-tangent correlation function for a polymer, which we modeled as an elastic beam undergoing thermal fluctuations. The calculation was carried out in the limit of small fluctuations and it led to an expression for the persistence length in terms of the flexural rigidity of the polymer. Here we reexamine this problem for a two-dimensional polymer, but without the assumption of small fluctuations. (a) For a polymer confined to a plane the tangent vector \(\mathbf{t}(s)\) can be written in terms of the polar angle \(\theta(s)\) as: \(\mathbf{t}(s)=(\cos \theta(s), \sin \theta(s)) .\) Rewrite the beam bending energy eqn 10.9 . in terms of the polar angle \(\theta(s)\) (b) Expand the polar angle \(\theta(s)\) into a Fourier series taking into account the boundary conditions \(\theta(0)=0\) and \(\mathrm{d} \theta / \mathrm{d} s=0\) for \(s=L\) The first boundary condition comes about by choosing the orientation of the polymer so that the tangent vector at \(s=0\) is always along the \(x\) -axis. Convince yourself that the second is a consequence of there being no force acting on the end of the polymer. (c) Rewrite the bending energy in terms of the Fourier amplitudes \(\bar{\theta}_{n},\) introduced in \((\mathrm{b}),\) and show that it takes on the form equivalent to that of many independent harmonic oscillators. Use equipartition to compute the thermal average of each of the Fourier amplitudes. (d) Make use of the identity \((\cos (X))=\mathrm{e}^{-X^{2} / 2}\), which holds for a Gaussian distributed random variable \(X,\) to obtain the equation for the tangent-tangent correlation function: \(\langle t(s) \cdot t(0)\rangle=\mathrm{e}^{-\theta(s)^{2} / 2},\) Then compute \(\left\langle\theta(s)^{2}\right\rangle\) by using the Fourier series representation of \(\theta(s)\) and the average values of the Fourier amplitudes \(\theta_{n}\) obtained in (c). Convince yourself either by plotting or Fourier analysis that on the interval \(0 < s < L,\left(\theta(s)^{2}\right)=s / \xi_{p}\) (e) How does the persistence length in two dimensions compare with the value obtained in three dimensions? Explain why the tangent-tangent correlation function decays faster in two dimensions. What is the situation in one dimension?

Short answer

The two-dimensional persistence length ends up to be L²/(π²⟨θ²⟩), which deviates from the three-dimensional case due to the increased freedom (2 dimensions) for thermal motions. This explains the faster decay of the correlation function in 2D.

Step-by-step solution

Rewrite energy equation

Rewrite the beam bending energy equation, such that it's represented in terms of the polar angle, θ(s). The rewritten equation would take the form: E=½k∫(dθ/ds)² ds.

Expand into Fourier series

Expand the polar angle, θ(s), into a Fourier series while respecting the boundary conditions {θ(0)=0, dθ/ds=0 for s=L}. The Fourier series could look like θ(s) = ∑ θn sin(nπs/L).

Rewrite bending energy in terms of Fourier amplitudes

Rewrite the bending energy as a function of the Fourier amplitudes. The energy after substitution will be E = ½K∑(nπ/L)² θn².

Compute thermal average

Use the equipartition theorem to calculate the thermal average of each of the Fourier amplitudes. Using equipartition theorem ⟨θn²⟩ = kT / K (nπ/L)² .

Compute the tangent-tangent correlation

Using the provided identity and the thermal average of the Fourier amplitudes computed, the tangent-tangent correlation takes on the following form: ⟨t(s) . t(0)⟩ = exp[-L²/(4π²ξ_p²)]⟨sin²(πs/L)⟩.

Obtain ⟨θ(s)²⟩

Use the Fourier series representation of θ(s) and the average values of the Fourier amplitudes θn obtained in Step 4 to compute ⟨θ(s)²⟩. The relation ⟨θ(s)²⟩=s/ξ_p can also be established.

Compare the persistence length

Compare the obtained persistence length in two dimensions with the known value for three dimensions and provide an explanation for any discrepancy. In two dimensions, the correlation function decays faster due to thermal motions occurring in two dimensions beside the chain.

Key concepts

Tangent-Tangent Correlation Function

The tangent-tangent correlation function plays a vital role in understanding how segments of a polymer correlate with each other over a distance. It is mathematically expressed as \( \langle \mathbf{t}(s) \cdot \mathbf{t}(0) \rangle \), where \( \mathbf{t}(s) \) is the tangent vector at position \( s \) along the polymer. This correlation decays as the distance between the points on the polymer increases.
In two-dimensional scenarios, the correlation function can take the form \( \mathrm{e}^{-\theta(s)^2/2} \), assuming that the angle \( \theta(s) \) is Gaussian distributed. This means the function provides a measure of how quickly the polymer's configuration loses its initial directionality.
Keeping in mind the effects of thermal fluctuations, understanding this function is crucial for predicting how flexible or rigid a polymer might behave in various conditions.

Fourier Series

A Fourier series is a tool used to express a periodic function as a sum of sines and cosines. This approach simplifies complex periodic functions by decomposing them into their fundamental frequencies. For a polymer's angular displacement \( \theta(s) \), it can be represented as:

• \( \theta(s) = \sum \theta_n \sin \left( \frac{n\pi s}{L} \right) \)

This series respects the boundary conditions set in the exercise, providing a compact way to describe the thermally induced bending of a polymer.
By utilizing the Fourier series, researchers and students can make complex calculations more manageable and extract meaningful physical insights from them, such as understanding the distribution of beam bending energy across different modes.

Beam Bending Energy

Beam bending energy in the context of polymers is crucial for understanding its mechanical properties. The energy is associated with the deformation of the polymer when it is subjected to bending forces. The beam bending energy for a polymer was expressed in terms of the polar angle \( \theta(s) \) and takes the form:

• \[ E = \frac{1}{2}k\int \left( \frac{d\theta}{ds} \right)^2 ds \]

This expression tells us how much energy is stored in a polymer as it bends.
Transforming this into a Fourier series makes it clearer how different modes contribute to the bending energy. Each mode’s contribution can be analyzed separately, which simplifies understanding the overall energy distribution. By relating the bending energy to thermal fluctuations, one can predict how polymers behave under different conditions, such as changes in temperature or external forces.

Thermal Fluctuations

Thermal fluctuations are the random deviations in a system due to thermal energy. In polymers, these fluctuations lead to structural changes that impact properties like flexibility and rigidity. At the molecular level, polymers are never at complete rest but constantly vibrate due to thermal energy present in their environment.
This constant motion impacts the structural configuration of polymers, causing fluctuations that are often approximated using statistical mechanics principles, such as the equipartition theorem. This theorem helps in calculating the average energy stored in each vibrational mode of the system.
Such detailed understanding can be beneficial in designing materials with desired physical properties. For instance, knowing how thermal fluctuations affect a polymer can inform how it might perform in practical applications like packing solutions or flexible electronics.