Machine Learning Inversion from Small-Angle Scattering for Charged Polymers

TL;DR

This work tackles the inverse problem of extracting conformational and electrostatic parameters from small-angle scattering data of semiflexible polyelectrolytes. It builds an off-lattice Monte Carlo model with bending stiffness $\kappa$ and a screened Yukawa interaction of strength $A$ governed by Debye length $\lambda_D$, generates a large $S(q)$ dataset, and then uses principal component analysis to assess inversion feasibility, followed by Gaussian process regression to map $\log S(q)$ to $(\kappa, A, R_g^2/L^2, R^2/L^2)$. The results show that $R^2/L^2$, $R_g^2/L^2$, and $\kappa$ are recoverable across a broad range of $\lambda_D$, while $A$ is recoverable only when $\lambda_D$ is sufficiently large, with low relative errors in the recovered parameters. This framework provides a practical route to quantify bending stiffness and charge density from SAS measurements, with potential extensions to patterned or zwitterionic polymers and applications to SANS experiments.

Abstract

We develop Monte Carlo simulations for uniformly charged polymers and machine learning algorithm to interpret the intra-polymer structure factor of the charged polymer system, which can be obtained from small-angle scattering experiments. The polymer is modeled as a chain of fixed-length bonds, where the connected bonds are subject to bending energy, and there is also a screened Coulomb potential for charge interaction between all joints. The bending energy is determined by the intrinsic bending stiffness, and the charge interaction depends on the interaction strength and screening length. All three contribute to the stiffness of the polymer chain and lead to longer and larger polymer conformations. The screening length also introduces a second length scale for the polymer besides the bending persistence length. To obtain the inverse mapping from the structure factor to these polymer conformation and energy-related parameters, we generate a large data set of structure factors by running simulations for a wide range of polymer energy parameters. We use principal component analysis to investigate the intra-polymer structure factors and determine the feasibility of the inversion using the nearest neighbor distance. We employ Gaussian process regression to achieve the inverse mapping and extract the characteristic parameters of polymers from the structure factor with low relative error.

Figures (8)

• Figure 1: Radius of gyration $R_g^2$ and end-to-end distance $R^2$ of the charged polymer versus various bending stiffness $\kappa$, charge interaction strength $A$ and screen length $\lambda_D$. (a) Normalized end-to-end distance $R^2/L^2$ versus screen length $\lambda_D$ for various bending stiffness $\kappa$. (b) $R^2/L^2$ versus screen length $\lambda_D$ for various charge interaction strength $A$. (c) and (d), similar to (a) and (b), respectively, but for normalized radius of gyration $R_g^2/L^2$
• Figure 2: Different length scales of the charged polymer, fitted using both single length scale and double length scale model. (a) Bond angle correlation $\left<\cos\theta(s)\right>$ for various screening length $\lambda_D$ with $\kappa=30$, $A=5$, solid lines are fitted using single length scale Equ. \ref{['equ:lam0']}. (b) similarly, but fitted using double length scale Equ. \ref{['equ:lam1_lam2']}. (c) Three persistent length $\lambda_0$ for solid line, $\lambda_1$ for dashed line and $\lambda_e$ for dotted line, versus screening length $\lambda_D$ for various $\kappa$ with $A=5$. (d) Similar to (c), but for various $A$ with $\kappa=30$.
• Figure 3: Variation of the structure factor of the charged polymer. (a) Structure factor $S(q)$ for various bending stiffness $\kappa$ with $\lambda_D=3$, $A=5$ and rod effectively representing the $\kappa=\infty$ case. (b) Structure factor $S(q)$ normalized by the rod's structure factor $S_{rod}(q)$ for various $\kappa$. (c) $S(q)/S_{rod}(q)$ for various charge interaction strength $A$ with $\kappa=30$, $\lambda_D=3$. (d) $S(q)/S_{rod}(q)$ for various screening length $\lambda_D$ with $\kappa=30$, $A=5$.
• Figure 4: Singular Value Decomposition of the structure factor data set $\vb{F}=\{\log{S(q)}\}$. (a) Singular value $\Sigma$ versus Singular Value Rank (SVR), with top 3 rank highlighted in red circle. (b) First 3 singular vectors $V_0$,$V_1$ and $V_2$. (c) Decomposition of the $\log{S(q)}$ with $\kappa=10$, $A=5$, $\lambda_D=3$, $\log(S_0),\log(S_1)$ and $\log(S_2)$ are projection of $\log{S(q)}$ on to the $V_0$, $V_1$ and $V_2$, respectively.
• Figure 5: Distribution of the polymer parameters $(R^2/L^2, R_g^2/L^2, \kappa, A)$ in the SVD space spanned by $(V_0, V_1, V_2)$. (a) End-to-end distance divided by length square $R^2/L^2$, (b) Radius of gyration square divided by length square $R_g^2/L^2$. (c) Bending stiffness $\kappa$. (d) Charge interaction strength $A$.