.. _wlctheory: 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 :math:`\vec{r}(s)` where :math:`s` runs from zero to the contour length :math:`L`. The wormlike chain model has a fixed length (like the freely jointed chain model), which is enforced by setting .. math:: \left( \frac{\partial \vec{r}(s)}{\partial s} \right)^{\! \! 2} = \left( \vec{u}(s) \right)^{2} = 1 at all values of :math:`s` along the chain, where we define the tangent vector :math:`\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 .. math:: 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} .. figure:: figures/wlc-model.pdf :width: 600 :align: center :alt: Schematic representation of the wormlike chain model and definition of geometric parameters. 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 :math:`\mathcal{H}` of the Wormlike chain model is given by .. math:: \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 :math:`\vec{u} = \frac{\partial \vec{r}}{\partial s}` (with constraint :math:`\left| \frac{\partial \vec{r}}{\partial s} \right|=1` for all :math:`s`). Define the Green function :math:`G(\vec{r}, \vec{u}|\vec{r}_{0},\vec{u}_{0}; L)` to be the probability of finding the chain at location :math:`\vec{r}` with orientation :math:`\vec{u}` at length :math:`L` given that the chain begins at location :math:`\vec{r}_{0}` with orientation :math:`\vec{u}_{0}`. The Markov property of the chain is written as .. math:: 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 :math:`s_{A}=0`, :math:`s_{B}=L`, and :math:`s_{C}=L+\delta` with :math:`\vec{r}_{A}=\vec{r}_{0}`, :math:`\vec{r}_{B}=\vec{r}_{1}`, and :math:`\vec{r}_{C}=\vec{r}` to find the governing diffusion equation for :math:`G(\vec{r},\vec{u}|\vec{r}_{0},\vec{u}_{0};L)`. In the limit :math:`\delta \rightarrow 0`, the Green function :math:`G(\vec{r},\vec{u}, N+\delta | \vec{r},\vec{u},N)` will be dominated by a single path whose action is .. math:: 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 .. math:: 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 :math:`A` is to be determined in the limit of :math:`\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 .. math:: 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 :math:`\vec{r}` and :math:`\vec{u}` will contribute to the integral. Expand the Green function :math:`G(\vec{r}_{1}, \vec{u}_{1}, L| \vec{r}_{0},\vec{u}_{0},0)` about :math:`\vec{r}` and :math:`\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 .. math:: 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} where :math:`\Delta \vec{r} = \vec{r} - \vec{r}_{1}` and :math:`\Delta \vec{u} = \vec{u} - \vec{u}_{1}`. The double dot operation between two tensors acts as :math:`\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 :math:`\delta` approaches zero. The following integral determines :math:`A`: .. math:: \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 thus :math:`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 .. math:: \left< ... \right> = \int d \vec{u}_{1} \frac{1}{A} \exp (h \vec{u} \cdot \vec{u}_{1}) (...), where :math:`h = \frac{l_{p}}{\delta}`. We must find the following quantities: :math:`\left< \Delta \vec{r} \right>`, :math:`\left< \Delta \vec{u} \right>`, and :math:`\left< \Delta \vec{u} \Delta \vec{u} \right>`. The other averages ( :math:`\left< \Delta \vec{r} \Delta \vec{r} \right>` and :math:`\left< \Delta \vec{r} \Delta \vec{u} \right>` ) yield terms of quadratic order in :math:`\delta`, thus they do not contribute to the lowest order response. For convenience, define the vector quantity :math:`\vec{h} = h \vec{u}` for use in this derivation. The average quantity :math:`\langle \Delta \vec{u} \rangle` is given by .. math:: \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} ). Thus, in the limit of :math:`h \rightarrow \infty`, the quantity :math:`\langle \Delta \vec{u} \rangle = \frac{\vec{u} \delta}{l_{p}}`. The average quantity :math:`\langle \Delta \vec{u} \Delta \vec{u} \rangle` is given by .. math:: \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}}. The following quantity is found to complete the derivation: .. math:: \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 :math:`h` approaches infinity yields the solution of :math:`\langle \Delta \vec{u} \Delta \vec{u} \rangle = \frac{\delta}{l_{p}} ( \mathbf{I} - \vec{u} \vec{u} )`. Altogether the Green function obeys .. math:: 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 :math:`\vec{u} \cdot \vec{\nabla}_{u} = 0` and :math:`\vec{u} \vec{u} : \vec{\nabla}_{u} \vec{\nabla}_{u} = 0`. The final differential equation for the Green function is given by .. math:: \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), with the initial condition :math:`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 :math:`V=0`. Define the Fourier transform (and inverse transform) of a function :math:`f(\vec{r})` from :math:`\vec{r}` to :math:`\vec{k}` to be .. math:: \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 :math:`\tilde{G}(\vec{k},\vec{u}|\vec{u}_{0};L)` for :math:`V=0` is .. math:: \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 :math:`\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 :math:`\vec{r}` is equivalent to .. math:: 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 :math:`\vec{u}_{0}` and ends with orientation :math:`\vec{u}` regardless of the end positions. The orientation Green function :math:`G(\vec{u}|\vec{u}_{0};L)` satisfies .. math:: \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) :label: gudiffeqn .. \label{eq:gudiffeqn} with initial condition :math:`G(\vec{u}|\vec{u}_{0};L) = \delta(\vec{u}-\vec{u}_{0})`. Here, we note that the operator :math:`\vec{\nabla}_{u}^{2}` has eigenfunctions :math:`Y_{l}^{m}` that satisfy .. math:: \vec{\nabla}_{u}^{2} Y_{l}^{m} = - l(l+1) Y_{l}^{m}. The eigenfunctions :math:`Y_{l}^{m}` are the spherical harmonics [Arfken1999]_ that form a complete basis set for the 3-dimensional angular Laplacian :math:`\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 :math:`l` and :math:`m` are :math:`l \in [0, \infty]` and :math:`m \in [-l, l]`. The spherical harmonics satisfy .. math:: \int d \vec{u} Y_{l}^{m} (\vec{u}) Y_{l'}^{m'*}(\vec{u}) = \delta_{ll'} \delta_{mm'} The solution for :math:`G(\vec{u}|\vec{u}_{0};L)` is constructed as an expansion in the spherical harmonics, such that .. math:: 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] :label: guwlc .. \label{eq:guwlc} where :math:`N=L/(2l_{p})`. This form satisfies the governing Schrödinger equation (Eq. :eq:`gudiffeqn`), and the initial condition is captured by noting that .. math:: \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).