Semi-flexible polymer in two dimensions: configurations of a...

Semi-flexible polymer in two dimensions: configurations of a model polymer can be described by either a set of vectors $\left\{\mathbf{t}_i\right\}$ of length $a$ in two dimensions (for $i=1, \cdots, N$ ), or alternatively by the angles $\left\{\phi_i\right\}$ between successive vectors, as indicated in the figure below. The polymer is at a temperature $T$, and subject to an energy $$ \mathcal{H}=-\kappa \sum_{i=1}^{N-1} \mathbf{t}_i \cdot \mathbf{t}_{i+1}=-\kappa a^2 \sum_{i=1}^{N-1} \cos \phi_i, $$ where $\kappa$ is related to the bending rigidity, such that the probability of any configuration is proportional to $\exp \left(-\mathcal{H} / k_B T\right)$. (a) Show that $\left\langle\mathbf{t}_m-\mathbf{t}_n\right\rangle \propto \exp (-|n-m| / \xi)$, and obtain an expression for the persistence length $\ell_p=a \xi$. (You can leave the answer as the ratio of simple integrals.) (b) Consider the end-to-end distance $\mathbf{R}$ as illustrated in the figure. Obtain an expression for $\left(R^2\right)$ in the limit of $N \gg 1$. (c) Find the probability $p(\mathbf{R})$ in the limit of $N \gg 1$. (d) If the ends of the polymer are pulled apart by a force $\mathbf{F}$, the probabilities for polymer configurations are modified by the Boltzmann weight $\exp \left(\frac{F \cdot R}{k_a T}\right)$. By expanding this weight, or otherwise, show that $$ \langle\mathbf{R}\rangle=K^{-1} \mathbf{F}+\mathcal{O}\left(F^3\right) $$ and give an expression for the Hookian constant $K$ in terms of quantities calculated before.

Semi-flexible polymer in two dimensions: configurations of a model polymer can be described by either a set of vectors $\left\{\mathbf{t}_i\right\}$ of length $a$ in two dimensions (for $i=1, \cdots, N$ ), or alternatively by the angles $\left\{\phi_i\right\}$ between successive vectors, as indicated in the figure below.
The polymer is at a temperature $T$, and subject to an energy
$$
\mathcal{H}=-\kappa \sum_{i=1}^{N-1} \mathbf{t}_i \cdot \mathbf{t}_{i+1}=-\kappa a^2 \sum_{i=1}^{N-1} \cos \phi_i,
$$
where $\kappa$ is related to the bending rigidity, such that the probability of any configuration is proportional to $\exp \left(-\mathcal{H} / k_B T\right)$.
(a) Show that $\left\langle\mathbf{t}_m-\mathbf{t}_n\right\rangle \propto \exp (-|n-m| / \xi)$, and obtain an expression for the persistence length $\ell_p=a \xi$. (You can leave the answer as the ratio of simple integrals.)
(b) Consider the end-to-end distance $\mathbf{R}$ as illustrated in the figure. Obtain an expression for $\left(R^2\right)$ in the limit of $N \gg 1$.
(c) Find the probability $p(\mathbf{R})$ in the limit of $N \gg 1$.
(d) If the ends of the polymer are pulled apart by a force $\mathbf{F}$, the probabilities for polymer configurations are modified by the Boltzmann weight $\exp \left(\frac{F \cdot R}{k_a T}\right)$. By expanding this weight, or otherwise, show that
$$
\langle\mathbf{R}\rangle=K^{-1} \mathbf{F}+\mathcal{O}\left(F^3\right)
$$
and give an expression for the Hookian constant $K$ in terms of quantities calculated before.

Key Concepts

Force-Extension and Linear Response

When an external force is applied to a polymer, the chain responds by extending in the direction of the force. The force-extension relationship, often linear in the small-force regime, can be characterized by a Hookean constant. This response emerges from the modification of the Boltzmann weight in the presence of an external force and is a key concept in understanding how mechanical forces couple to thermal fluctuations in polymer materials.

End-to-End Distance

The end-to-end distance of a polymer is a macroscopic measure of its conformation, obtained by summing up all the segment vectors along the chain. In the limit of a long chain (N ? 1), statistical methods allow one to calculate the mean-square end-to-end distance, which provides insights into the polymer’s size, shape, and flexibility. This is a fundamental metric in polymer theory linking microscopic interactions with macroscopic observables.

Probability Distribution of Polymer Configurations

The probability distribution p(R) describes the likelihood of the polymer assuming a particular end-to-end vector R. In the context of thermal equilibrium and for long chains, this distribution often approaches a Gaussian form by virtue of the central limit theorem. This distribution is essential for understanding the statistical ensemble of polymer configurations and for predicting macroscopic properties like elasticity.

Persistence Length

In polymer physics, the persistence length is the length scale over which correlations in the direction of the tangent vectors of a polymer decay. It quantifies the stiffness of the polymer: a larger persistence length indicates a stiffer chain that maintains its directional orientation over a longer distance. The concept usually arises from the exponential decay of the correlation ?t_m ? t_n? as a function of the segment separation, and it plays a central role in describing the statistical mechanics of semi-flexible polymers.

Exponential Decay of Tangent Correlations

The correlation between successive segments along a polymer chain often decays exponentially, reflecting the influence of bending energies and thermal fluctuations. This decay is typically expressed as ?t_m ? t_n? ? exp(?|n ? m|/?), where ? is a dimensionless measure of the decay length. This concept is critical in describing how quickly the memory of the initial direction is lost along the polymer's contour.

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