Persistence Length of Semiflexible Conjugated Polymers using Transfer Matrix Method

Introduction

This repository is an update of Persistence Length Using Monte Carlo Sampling. The original method uses the Monte Carlo sampling. However, it seems that the correlation length ($N_p$) is irrelevant to the bond lengths. Here, I re-implemented the method using the transfer matrix method.

Transfer Matrix Method

Statistical Averaging of Dihedral Angles — Single-Step Average Rotation Operator Dihedral angles are random (according to a known potential energy distribution), so for a given position i, we define the single-step average rotation operator as:

$$ A_i \equiv \langle Q_i(\phi)\rangle_{p_i} = \int_0^{2\pi} Q_i(\phi)p_i(\phi)d\phi, $$

where $p_i(\phi)=\dfrac{e^{-V_i(\phi)/k_B T}}{Z_i}$, $Z_i=\int_0^{2\pi}e^{-V_i(\phi)/k_B T} d\phi$.

Since $R_z(\theta_i)$ is independent of $\phi$, the above equation can be written as:

$$ A_i = \bigg(\int_0^{2\pi} R_z(\theta_i)R_x(\phi)p_i(\phi)d\phi\bigg) \equiv R_z(\theta_i) S_i , $$

and

$$ \int_0^{2\pi} R_x(\phi)p(\phi) d\phi =\begin{pmatrix} 1 & 0 & 0\\ 0 & \langle\cos\phi\rangle & -\langle\sin\phi\rangle\\ 0 & \langle\sin\phi\rangle & \langle\cos\phi\rangle\\ \end{pmatrix}, $$

where

$$ \langle\cos\phi\rangle_i=\frac{\int_0^{2\pi}\cos\phi e^{-V_i(\phi)/k_BT} d\phi}{\int_0^{2\pi}e^{-V_i(\phi)/k_BT} d\phi},\quad\langle\sin\phi\rangle_i=\frac{\int_0^{2\pi}\sin\phi e^{-V_i(\phi)/k_BT} d\phi}{\int_0^{2\pi}e^{-V_i(\phi)/k_BT} d\phi}. $$

Since each step is the action of a linear operator (with independent dihedral angles), the average transformation for n steps can be written as a product of operators:

$$ \langle t_n \rangle = A_0 \cdots A_{n-2}A_{n-1}t_0. $$

the autocorrelation is:

$$ C(n)=\langle t_n\cdot t_0\rangle =t_0^{T} \Big( \prod_{i=0}^{n-1} A_i \Big) t_0, $$

where $\prod_{i=0}^{n-1} A_i \equiv A_0 \cdots A_{n-1}$.

If the chain is periodic (a repeating unit has M segments, where $A_{i+M}=A_i$), we shall calculate the transfer matrix for one repeating unit:

$$ \mathcal{M}=\prod_{i=0}^{M-1} A_i, $$

Then, the correlation for $r$ repeating units decays as $\mathcal{M}^r$. Let $\lambda_{\max}$ be the maximum eigenvalue (in modulus) of $\mathcal{M}$, then the persistence length in repeating units ($N_p$) is:

$$ N_p = -\frac{1}{\ln\lambda_{\max}}. $$

Definition of bond length and deflection angle in the script

• T-bond-DPP-bond-T-bond-T-bond-E-bond-T-bond
• l = [2.533, 1.432, 3.533, 1.432, 2.533, 1.432, 2.533, 1.433, 1.363, 1.433, 2.533, 1.432] # in Angstrom
• 2.533 is the bond length of Thiophene (l[0])
• 1.432 is the bond length of first linker (l[1])
• Angle = np.deg2rad(np.array([-14.92, -10.83, 30.79, -30.79, 10.83, 14.92, -14.91, -13.29, -53.16, 53.16, 13.29, 14.91])) # convert degree to radian
• labels = {1: {'label': 'T-DPP', 'color': 'b'}, 2: {'label': 'T-T', 'color': 'm'}, 3: {'label': 'T-E', 'color': 'c'}}
• rotation = np.array([0, 1, 0, 1, 0, 2, 0, 3, 0, 3, 0, 2])
• l[1] rotated by rotation_type 1 with a deflection angle Angle[1]

Update

• Add the rotational isomeric state (RIS) model and the mixed HR-RIS model using the same logic