Deep Learning the Small-Angle Scattering of Polydisperse Hard Rods

TL;DR

The paper develops a simulation-informed deep learning framework to analyze small-angle scattering from polydisperse hard rods. A variational autoencoder–based generator maps system parameters $(\phi,L,\sigma_D)$ to the scattering function $I(Q)$, trained on $N=20{,}000$ rod MCMC data generated with HOOMD-blue, and validated against four datasets including uniform, normal, and lognormal polydispersity. The approach achieves higher accuracy than the Percus–Yevick approximation, enables direct least-squares fitting of experimental-like spectra, and reveals a low-dimensional latent structure that captures the essential physics of rod polydispersity and concentration. This method provides a scalable, generalizable tool for SAS analysis of anisotropic colloids, with potential extensions to more complex mixtures and electrostatic interactions.

Abstract

We present a deep learning framework for modeling and analyzing the small-angle scattering data of polydisperse hard-rod systems, a widely used models for anisotropic colloidal particles. We use a variational autoencoder-based neural network to learn the mapping from the system parameters such as the volume fraction, rod length, and polydispersity, to the scattering function. The dataset for training and testing such neural network model is obtained from Markov chain Monte Carlo simulation of 20,000 hard spherocylinders using the hard particle Monte Carlo package from the HOOMD-blue. Four datasets were generated, each with 5,500 pairs of system parameters and corresponding scattering functions. We use one of the dataset to investigate the feasibility of the learning, and three additional datasets with different polydisperse distribution to demonstrate the generality of our approach. The neural network model transcends the fundamental limitations of the Percus-Yevick approximation by accurately capturing anisotropic interactions and high-concentration effects that analytical models often fail to resolve. This framework achieves significantly higher accuracy in reproducing scattering functions and enables a least-square fitting routine for quantitative data analysis.

Figures (9)

• Figure 1: Illustration of scattering function of polydisperse hard rods. (a) Snapshot of the system configuration with volume fraction $\phi=0.15$, mean rod length $L=3$, length polydispersity $\sigma_L=0$, and diameter polydispersity $\sigma_D=0.1$. Color bar is the volume of each rod. (b) Scattering function $I(Q)$ with different $\phi$, $L$, and $\sigma_D$. $I(Q)$ are shifted vertically for better visualization.
• Figure 2: Overview of the neural network architecture for learning the mapping between the system parameters $(\phi, L,\sigma_D)$ and the scattering function $I(Q)$. The neural network consist of three parts, in which an encoder with 2 convolutional layers and the symmetric decoder compress the input scattering function into a low dimensional latent space, a multilayer perceptron learns the mapping between the system parameters and the latent variables.
• Figure 3: Examples of the scattering function $I(Q)$ for polydisperse hard rods with uniform distributed length and diameter size $L_i\in U(1-\sigma_L,1+\sigma_L), D_i\in U(1-\sigma_D,1+\sigma_D)$, for various volume fraction $\phi$, mean length $L$ and polydispersity $\sigma_L,\sigma_D$. Default values are $(\phi,L,\sigma_L,\sigma_D)=(0.15,2,0,0)$. (a) Scattering function $I(Q)$ for different volume fraction $\phi$ with other parameters fixed. (b) $I(Q)$ for different mean length $L$. (c) $I(Q)$ for different length polydispersity $\sigma_L$. (d) $I(Q)$ for different diameter polydispersity $\sigma_D$.
• Figure 4: Principal component analysis of the scattering function dataset for $\vb{F} = \{\log_{10}{I(Q)}\}$ of uniform-distribution polydisperse hard spheres. (a) Decay of the singular value entry in $\Sigma$ in $\vb{F}=\vb{U}\vb{\Sigma}\vb{V}^T$ versus the singular value rank (SVR). (b) First 3 singular vectors $(V0,V1,V2)\in\vb{V}$ . (c) Distribution of the volume fraction $\phi$ in the singular value space $(FV0,FV1,FV2)$ in which each $\log_{10}(I(Q))\in \vb{F}$ is projected to the $(V0,V1,V2)$. (d)-(f) Distribution of the mean rod length $L$, length polydispersity $\sigma_L$, and diameter polydispersity $\sigma_D$, respectively.
• Figure 5: Distribution of latent variables $\mu$ and $\log(s^2)$ for the scattering function of rods with uniformly distributed diameter. (a), (c) and (e) are for the distribution of volume fraction $\phi$, mean length $L$, and diameter polydispersity $\sigma_D$ in the $\mu$ space, respectively. (b), (d), (f) are for the corresponding system parameters in the $\log(s^2)$ space.