Wormlike Chain Theory

The wormlike chain model describes the polymer chain as a continuous thread through space that resists bending deformation [Kratky1949]. Subjecting the chain to thermal fluctuations results in a wiggling wormlike polymer chain.

The polymer conformation is defined by a space curve \(\vec{r}(s)\) where \(s\) runs from zero to the contour length \(L\). The wormlike chain model has a fixed length (like the freely jointed chain model), which is enforced by setting

\[\left( \frac{\partial \vec{r}(s)}{\partial s} \right)^{\! \! 2} = \left( \vec{u}(s) \right)^{2} = 1\]

at all values of \(s\) along the chain, where we define the tangent vector \(\vec{u}(s) = \frac{\partial \vec{r}(s)}{\partial s}\). The chain opposes bending deformation, and the bending energy is quadratic in the curvature of the chain. This gives the bending energy

\[E_{bend} = \frac{k_{B}T l_{p}}{2} \int_{0}^{L} ds \left( \frac{\partial^{2} \vec{r}(s)}{\partial s^{2}} \right)^{\! \! 2} = \frac{k_{B}T l_{p}}{2} \int_{0}^{L} ds \left( \frac{\partial \vec{u}(s)}{\partial s} \right)^{\! 2}\]

Schematic representation of the wormlike chain model and definition of geometric parameters.

Wormlike Chain model Green function—Path integral formulation and the Diffusion equation

The governing differential equation for the Wormlike Chain model [Kratky1949] is given through the application of the path integral in the necessary dimensions. The differential Hamiltonian \(\mathcal{H}\) of the Wormlike chain model is given by

\[\beta \mathcal{H} = \frac{l_{p}}{2} \left( \frac{\partial \vec{u}}{\partial s} \right)^{2} +\beta V(\vec{r}, \vec{u}),\]

where the tangent \(\vec{u} = \frac{\partial \vec{r}}{\partial s}\) (with constraint \(\left| \frac{\partial \vec{r}}{\partial s} \right|=1\) for all \(s\)). Define the Green function \(G(\vec{r}, \vec{u}|\vec{r}_{0},\vec{u}_{0}; L)\) to be the probability of finding the chain at location \(\vec{r}\) with orientation \(\vec{u}\) at length \(L\) given that the chain begins at location \(\vec{r}_{0}\) with orientation \(\vec{u}_{0}\). The Markov property of the chain is written as

\[G (\vec{r}_{C}, \vec{u}_{C}, s_{C}|\vec{r}_{A},\vec{u}_{A}, s_{A}) = \int d \vec{r}_{B} d \vec{u}_{B} G (\vec{r}_{C}, \vec{u}_{C}, s_{C}|\vec{r}_{B},\vec{u}_{B}, s_{B}) \times G (\vec{r}_{B}, \vec{u}_{B}, s_{B}|\vec{r}_{A},\vec{u}_{A}, s_{A}).\]

Set \(s_{A}=0\), \(s_{B}=L\), and \(s_{C}=L+\delta\) with \(\vec{r}_{A}=\vec{r}_{0}\), \(\vec{r}_{B}=\vec{r}_{1}\), and \(\vec{r}_{C}=\vec{r}\) to find the governing diffusion equation for \(G(\vec{r},\vec{u}|\vec{r}_{0},\vec{u}_{0};L)\).

In the limit \(\delta \rightarrow 0\), the Green function \(G(\vec{r},\vec{u}, N+\delta | \vec{r},\vec{u},N)\) will be dominated by a single path whose action is

\[S \approx \frac{l_{p}}{2} \delta \left( \frac{ \vec{u} - \vec{u}_{1}}{\delta} \right)^{2} + \delta \beta V \!\! \left( \frac{\vec{r}+\vec{r}_{1}}{2} \right).\]

The Green function for this path is thus

\[G(\vec{r}, \vec{u}, L + \delta| \vec{r}_{1}, \vec{u}_{1}, L) = \frac{\delta (\vec{r} - \vec{r}_{1} - \vec{u} \delta )}{A} \exp \! \left( \frac{l_{p}}{\delta} \vec{u} \cdot \vec{u}_{1} \right) \exp (-\delta \beta V)\]

where the normalization constant \(A\) is to be determined in the limit of \(\delta\) approaching zero, and the delta function is added to the Green function to conserve the path length. The Green function is now given by

\[G(\vec{r}, \vec{u}, L + \delta| \vec{r}_{0}, \vec{u}_{0}, 0) = \int \frac{\delta (\vec{r} - \vec{r}_{1} - \vec{u} \delta )}{A} \exp \left( \frac{l_{p}}{\delta} \vec{u} \cdot \vec{u}_{1} \right)\exp (-\delta \beta V) \times G(\vec{r}_{1}, \vec{u}_{1},L | \vec{r}_{0}, \vec{u}_{0},0) d \vec{r}_{1} d \vec{u}_{1}.\]

Assume only small deviations from \(\vec{r}\) and \(\vec{u}\) will contribute to the integral. Expand the Green function \(G(\vec{r}_{1}, \vec{u}_{1}, L| \vec{r}_{0},\vec{u}_{0},0)\) about \(\vec{r}\) and \(\vec{u}\) to quadratic order in the deviation to find the leading order change in the propagated Green function.

The resulting expansion is given by

\[\begin{split}G(\vec{r}_{1}, \vec{u}_{1}, L| \vec{r}_{0},\vec{u}_{0},0) & = & G(\vec{r}, \vec{u}, L| \vec{r}_{0},\vec{u}_{0},0)- \frac{\partial G}{\partial \vec{r}} \cdot \Delta \vec{r} - \frac{\partial G}{\partial \vec{u}} \cdot \Delta \vec{u} \nonumber \\ & & \hspace{-1in} + \frac{1}{2} \frac{\partial^{2} G}{\partial \vec{r}^{2}} : \Delta \vec{r} \Delta \vec{r} + \frac{1}{2} \frac{\partial^{2} G}{\partial \vec{u}^{2}} : \Delta \vec{u} \Delta \vec{u} + \frac{\partial^{2} G}{\partial \vec{r} \partial \vec{u}} : \Delta \vec{r} \Delta \vec{u}\end{split}\]

where \(\Delta \vec{r} = \vec{r} - \vec{r}_{1}\) and \(\Delta \vec{u} = \vec{u} - \vec{u}_{1}\). The double dot operation between two tensors acts as \(\textbf{A}:\textbf{B} = A_{ij} B_{ji}\). The normalization constant is found by requiring the leading term to approach unity as the time change \(\delta\) approaches zero. The following integral determines \(A\):

\[\begin{split}\frac{1}{A} \int \delta(\vec{r} - \vec{r}_{1} - \vec{u} \delta ) \exp \left( \frac{l_{p}}{\delta} \vec{u} \cdot \vec{u}_{1} \right) d \vec{u}_{1} d \vec{r}_{1} & = & \nonumber \\ \hspace{-0.5in} \frac{1}{A} \int \delta(\vec{r} - \vec{r}_{1} - \vec{u} \delta ) \left[ \int_{0}^{2 \pi} \int_{0}^{\pi} \exp \left( \frac{l_{p}}{\delta} \cos \theta \right) \sin \theta d \theta d \phi \right] d \vec{r}_{1} & = & \nonumber \\ \frac{1}{A} \frac{4 \pi \delta}{l_{p}} \sinh \left( \frac{l_{p}}{\delta} \right) & = & 1\end{split}\]

thus \(A = \frac{4 \pi \delta}{l_{p}} \sinh \left( \frac{l_{p}}{\delta} \right)\). The leading order response of the Green function is found by evaluating the averaged quantities within the integrand. Define the average of a given quantity to be

\[\left< ... \right> = \int d \vec{u}_{1} \frac{1}{A} \exp (h \vec{u} \cdot \vec{u}_{1}) (...),\]

where \(h = \frac{l_{p}}{\delta}\).

We must find the following quantities: \(\left< \Delta \vec{r} \right>\), \(\left< \Delta \vec{u} \right>\), and \(\left< \Delta \vec{u} \Delta \vec{u} \right>\). The other averages ( \(\left< \Delta \vec{r} \Delta \vec{r} \right>\) and \(\left< \Delta \vec{r} \Delta \vec{u} \right>\) ) yield terms of quadratic order in \(\delta\), thus they do not contribute to the lowest order response. For convenience, define the vector quantity \(\vec{h} = h \vec{u}\) for use in this derivation. The average quantity \(\langle \Delta \vec{u} \rangle\) is given by

\[\begin{split}\langle \Delta \vec{u} \rangle & = & \vec{u} - \langle \vec{u}_{1} \rangle \nonumber \\ & = & \vec{u} - \frac{1}{A} \int d \vec{u}_{1} \exp ( \vec{h} \cdot \vec{u}_{1} ) \vec{u}_{1} \nonumber \\ & = & \vec{u} - \frac{1}{A} %\vec{\nabla}_{h} \frac{\partial A}{\partial \vec{h}} \nonumber \\ & = & \vec{u} - \frac{4 \pi \vec{u}}{A} \left( \frac{ \cosh (h) }{h} - \frac{\sinh (h)}{h^{2}} \right) \nonumber \\ & = & \vec{u} (1 - \coth (h) + h^{-1} ).\end{split}\]

Thus, in the limit of \(h \rightarrow \infty\), the quantity \(\langle \Delta \vec{u} \rangle = \frac{\vec{u} \delta}{l_{p}}\). The average quantity \(\langle \Delta \vec{u} \Delta \vec{u} \rangle\) is given by

\[\begin{split}\langle \Delta \vec{u} \Delta \vec{u} \rangle & = & \vec{u} \vec{u} - \vec{u} \langle \vec{u}_{1} \rangle - \langle \vec{u}_{1} \rangle \vec{u} + \langle \vec{u}_{1} \vec{u}_{1} \rangle \nonumber \\ & = & \frac{2 \delta}{l_{p}} \vec{u} \vec{u} - \vec{u} \vec{u} + \langle \vec{u}_{1} \vec{u}_{1} \rangle \nonumber \\ & = & \frac{2 \delta}{l_{p}} \vec{u} \vec{u} - \vec{u} \vec{u} + \frac{1}{A} \frac{\partial^{2} A}{\partial \vec{h} \partial \vec{h}}.\end{split}\]

The following quantity is found to complete the derivation:

\[\frac{\partial^{2} A}{\partial \vec{h} \partial \vec{h}} = 4 \pi \mathbf{I} \left( \frac{\cosh (h)}{h^{2}} - \frac{\sinh (h)}{h^{3}} \right) + 4 \pi \vec{h} \vec{h} \left( \frac{ \sinh (h)}{h^{3}} - \frac{3 \cosh (h)}{h^{4}} + \frac{3 \sinh (h)}{h^{5}} \right).\]

This is evaluated and the limit as \(h\) approaches infinity yields the solution of \(\langle \Delta \vec{u} \Delta \vec{u} \rangle = \frac{\delta}{l_{p}} ( \mathbf{I} - \vec{u} \vec{u} )\). Altogether the Green function obeys

\[G + \frac{\partial G}{\partial s} \delta = G - \beta V G \delta - \frac{\partial G}{\partial \vec{r}} \cdot \vec{u} \delta - \frac{\partial G}{\partial \vec{u}} \cdot \frac{\vec{u}}{l_{p}} \delta + \frac{1}{2} \vec{\nabla}_{u} \vec{\nabla}_{u} G : \frac{\delta}{l_{p}} ( \mathbf{I} - \vec{u} \vec{u} )\]

This differential equation is further simplified by noting that \(\vec{u} \cdot \vec{\nabla}_{u} = 0\) and \(\vec{u} \vec{u} : \vec{\nabla}_{u} \vec{\nabla}_{u} = 0\). The final differential equation for the Green function is given by

\[\begin{split}\frac{\partial G(\vec{r}, \vec{u}, L| \vec{r}_{0}, \vec{u}_{0}, 0)}{\partial L} & = & - \beta V G(\vec{r}, \vec{u}, L| \vec{r}_{0}, \vec{u}_{0}, 0) \nonumber \\ & & - \vec{u} \cdot \vec{\nabla}_{r} G(\vec{r}, \vec{u}, L| \vec{r}_{0}, \vec{u}_{0}, 0) + \frac{1}{2 l_{p}} \vec{\nabla}_{u}^{2} G(\vec{r}, \vec{u}, L | \vec{r}_{0}, \vec{u}_{0}, 0),\end{split}\]

with the initial condition \(G(\vec{r}, \vec{u}, 0 | \vec{r}_{0}, \vec{u}_{0}, 0)= \delta(\vec{r}-\vec{r}_{0})\delta(\vec{u}-\vec{u}_{0})\).

Orientation statistics

Consider the case \(V=0\). Define the Fourier transform (and inverse transform) of a function \(f(\vec{r})\) from \(\vec{r}\) to \(\vec{k}\) to be

\[\tilde{f}(\vec{k}) = \int d \vec{r} f(\vec{r}) \exp \left( i \vec{k} \cdot \vec{r} \right) %= \tilde{f}(\vec{k}) \hspace{0.1in} \mathrm{and} \hspace{0.1in} f(\vec{r}) = \frac{1}{(2\pi)^{3}} \int d \vec{k} \tilde{f} ( \vec{k})\exp \left( -i \vec{k} \cdot \vec{r} \right)\]

The governing differential equation for \(\tilde{G}(\vec{k},\vec{u}|\vec{u}_{0};L)\) for \(V=0\) is

\[\frac{\partial \tilde{G}(\vec{k},\vec{u}|\vec{u}_{0};L)}{\partial L} = \left(i\vec{k} \cdot \vec{u} + \frac{1}{2 l_{p}} \vec{\nabla}_{u}^{2} \right) \tilde{G}(\vec{k},\vec{u}|\vec{u}_{0};L)\]

with initial condition \(\tilde{G}(\vec{k},\vec{u}|\vec{u}_{0};0)=\exp (i\vec{k} \cdot \vec{r}_{0}) \delta(\vec{u}-\vec{u}_{0})\)

The integral of the Green function over the position \(\vec{r}\) is equivalent to

\[G(\vec{u}|\vec{u}_{0};L) = \int d \vec{r} G(\vec{r}, \vec{u}, L | \vec{r}_{0}, \vec{u}_{0}, 0) = \tilde{G}(\vec{k}=\vec{0},\vec{u}|\vec{u}_{0};L),\]

which gives the orientation-only chain statistics, i.e. the probability that a chain begin with orientation \(\vec{u}_{0}\) and ends with orientation \(\vec{u}\) regardless of the end positions. The orientation Green function \(G(\vec{u}|\vec{u}_{0};L)\) satisfies

(1)\[\frac{\partial G(\vec{u}|\vec{u}_{0};L)}{\partial L} = \frac{1}{2 l_{p}} \vec{\nabla}_{u}^{2} G(\vec{u}|\vec{u}_{0};L)\]

with initial condition \(G(\vec{u}|\vec{u}_{0};L) = \delta(\vec{u}-\vec{u}_{0})\).

Here, we note that the operator \(\vec{\nabla}_{u}^{2}\) has eigenfunctions \(Y_{l}^{m}\) that satisfy

\[\vec{\nabla}_{u}^{2} Y_{l}^{m} = - l(l+1) Y_{l}^{m}.\]

The eigenfunctions \(Y_{l}^{m}\) are the spherical harmonics [Arfken1999] that form a complete basis set for the 3-dimensional angular Laplacian \(\vec{\nabla}_{u}^{2}\). This basis set is extended to arbitrary dimensions as the hyperspherical harmonics, and we will make use of this extension in coming chapters. The range of the indices \(l\) and \(m\) are \(l \in [0, \infty]\) and \(m \in [-l, l]\). The spherical harmonics satisfy

\[\int d \vec{u} Y_{l}^{m} (\vec{u}) Y_{l'}^{m'*}(\vec{u}) = \delta_{ll'} \delta_{mm'}\]

The solution for \(G(\vec{u}|\vec{u}_{0};L)\) is constructed as an expansion in the spherical harmonics, such that

(2)\[G(\vec{u}|\vec{u}_{0};L) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{l}^{m} (\vec{u}) Y_{l}^{m*}(\vec{u}_{0}) \exp [-l(l+1)N]\]

where \(N=L/(2l_{p})\). This form satisfies the governing Schrödinger equation (Eq. (1)), and the initial condition is captured by noting that

\[\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{l}^{m} (\vec{u}) Y_{l}^{m*}(\vec{u}_{0}) = \delta \left( \vec{u} - \vec{u}_{0} \right).\]

This development enables the evaluations of average quantities (as discussed in the Average Quantities section).