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 Equation \(10.9,\) 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 \(\tilde{\theta}_{n},\) introduced in (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 \(\langle\cos X\rangle=\mathrm{e}^{-X^{2} / 2},\) which holds for a Gaussian distributed random variable \(X,\) to obtain the equation for the tangent-tangent correlation function, \((\mathbf{t}(s) \cdot \mathbf{t}(0)\rangle=\mathrm{e}^{-\theta(s)^{2} / 2} .\) Then compute \(\left\langle\theta(s)^{2}\right)\) by using the Fourier series representation of \(\theta(s)\) and the average values of the Fourier amplitudes \(\bar{\theta}_{n}\) obtained in (c). Convince yourself either by plotting or Fourier analysis that on the \\[\text { interval } 0

Short answer

The persistence length in two dimensions is found to differ from that in three dimensions, with a slower decay rate in the tangent-tangent correlation function for two-dimensional situations. This is due to the lower degrees of freedom available in the two-dimensional case. The one-dimensional case, being constraining to motion in only one axis, will have the quickest decay of the tangent-tangent correlation function.

Step-by-step solution

Rewrite beam bending energy

Taking the given equation \(10.9\), replace \(\mathbf{t}(s)\) with \((\cos \theta(s), \sin \theta(s))\) enabling us to express the equation in terms of the polar angle.

Expanding Into Fourier Series

Expand the polar angle \(\theta(s)\) into a Fourier series. The boundary conditions \(\theta(0)=0\) and $\frac{d \theta }{d s}=0$ at \( s=L \) will be considered during this expansion.

Rewrite Bending Energy and Calculate Averages

Rearrange the bending energy to feature the Fourier amplitudes \(\tilde{\theta}_{n}\). Consequently, prove that it takes the form of multiple independent harmonic oscillators. Then, apply the equipartition theorem to calculate the thermal averages of each of the Fourier amplitudes.

Obtain Correlation Function and Compute the Average Value

Use the Gaussian identity to calculate the correlation function \(\langle \mathbf{t}(s) \cdot \mathbf{t}(0)\rangle\). Proceed by computing the average value of \(\langle\theta(s)^{2}\rangle\) using the Fourier series representation of \(\theta(s)\) and the averages of the Fourier amplitudes obtained in step (c).

Compare Persistence Length In Different Dimensions

Using the results from the previous steps, do a comparison between persistence lengths in two dimensions and three dimensions. Finally, explain why the tangent-tangent correlation function decays slower in two dimensions and give an assessment of the situation in one dimension.

Key concepts

Persistence Length

Persistence length is a critical concept in understanding the stiffness of a polymer chain. It is a measure of how far a polymer chain remains directional correlated before the random thermal motion takes over and it bends flexibly. In simple terms, it tells you how long a segment of the chain will appear straight or rigid. This property is heavily influenced by the polymer's material properties and its environment.

In two dimensions, observations indicate that the persistence length tends to be longer compared to three dimensions. This is because there's less room for the polymer to wiggle around in just two dimensions. Hence, the chain maintains its direction over a greater length. The persistence length in three dimensions is shorter as the polymer can explore more spatial pathways.

From this analysis, you understand that the longer the persistence length, the stiffer the polymer appears. In one dimension, theoretically, the concept of persistence length isn't as applicable, since there's no room to "bend" the chain at all.

Fourier Analysis

Fourier analysis is a mathematical tool used for breaking down functions, such as the polar angle \(\theta(s)\), into a sum of simpler sinusoidal functions like sine and cosine. This technique is powerful because it transforms complex oscillatory motions into a series of simpler components, which makes analyzing them more manageable.

In the context of polymer physics, Fourier analysis helps express the bending of a polymer in terms of its basic oscillatory modes. By expanding \(\theta(s)\) into a Fourier series, we can identify each "mode" of oscillation contributing to the polymer's shape. This expansion accommodates the boundary conditions, ensuring the mathematical model remains physically accurate.

• Fourier amplitudes like \(\tilde{\theta}_{n}\) represent the strength or influence of each mode.
• This representation helps in computing averages using the equipartition theorem, linking statistical mechanics with structural deformations.

Using Fourier analysis is akin to putting on special glasses that filter the complex shape of a polymer into understandable vibrations at various scales.

Tangent-Tangent Correlation Function

The tangent-tangent correlation function \( \langle \mathbf{t}(s) \cdot \mathbf{t}(0)\rangle \) is a mathematical expression that provides insights into how the orientation of a polymer changes along its length. Simply put, it measures how similar the tangent direction at one point on the polymer is to another point, as we move along the chain.

In polymer physics, this function is used to understand how quickly the polymer "forgets" its initial direction as it fluctuates due to thermal motion. If the correlation function decays rapidly, the polymer bends or twists easily. If it decays slowly, the polymer segment remains relatively straight.

• This function is derived using statistical mechanics principles and mathematical identities like \( \langle\cos X\rangle=e^{-X^{2} / 2} \) for Gaussian variables.
• Analyzing this function allows us to quantify the flexibility and stiffness of polymers, dictated by environmental conditions and material properties.

The correlation function is essential for predicting polymer behavior, being an indicator of how thermal fluctuations influence polymer structure over different dimensions.