Global $q$-dependent inverse transforms of intensity autocorrelation data

TL;DR

This work addresses the inversion of intensity autocorrelation data $g_2(q,\tau)$ from DLS and XPCS by introducing a nonlinear, regularized global inverse transform that jointly accounts for $q$-dependent kernels. The method formulates a constrained nonlinear optimization problem solved via a multi-start SQP approach, with regularization strength $\\lambda$ chosen using Provencher's 50% rejection criterion and generalized degrees of freedom estimated by Ye's method. It enables robust decompositions into diffusive, ballistic, or other dynamical components across $q$, demonstrated on colloidal suspensions and amorphous ice, and supported by analytical simplifications (e.g., log-normal density) and global stretched/compressed exponential fitting (KWW). The accompanying open-source MATLAB tools implement the full framework, provide example data, and aim to facilitate widespread adoption in XPCS/DLS studies of soft and disordered matter. $g_2$ data, nonlinear modeling, and global $q$-dependent inversion are central to extracting physically meaningful relaxation spectra from complex dynamics.

Abstract

We present a new analysis approach for intensity autocorrelation data, as measured with dynamic light scattering and X-ray photon correlation spectroscopy. Our analysis generalizes the established CONTIN and MULTIQ methods by direct nonlinear modeling of the $g_2$ function, enabling decomposition of complex dynamics without a priori knowledge of experimental scaling factors. We describe the mathematical formulation, implementation details, and strategies for solution, as well as demonstrate decompositions of soft matter dynamics data into distributions of diffusion rates/velocities. The open-source MATLAB implementation, including example data, is publicly available for adoption and further development.

Figures (5)

• Figure 1: Example $g_2$ curves from a measurement on amorphous ice taken at 110 K temperature. In this example, the very long delay times ($>10^3$ seconds) are missing, giving a truncated tail, and a priori unknown values for baseline $b$ and contrast $\beta$.
• Figure 2: Example decompositions into diffusive modes, ${f=\int\Phi\mathrm{e}^{-Dq^2\tau}\;\,\mathrm{d} D}$, with different regularizer weightings. The data is from an XPCS measurement of dilutely suspended colloidal particles. Top: Intensity autocorrelation data and lines of fit (dashed) from four select $q$ values (4.4, 6.5, 9.5, 15$\cdot10^{-4}$ Å-1). Middle: Solution distributions of diffusive modes. Bottom left: Degrees of freedom estimated by Ye's algorithm. Bottom right: Residual sum of squares. The blue and purple curves/points ($\lambda=10^8$, $10^{10}$) show results from lightly regularized optimization. The green curve ($\lambda=10^{15}$) is close to optimal according to Provencher's $F$-test, eq. \ref{['eq:f-test']}. The orange curve ($\lambda=10^{17}$) is overly regularized. The purple, blue and green $g_2$ lines of fit are virtually identical at this scale, and all fall on top of each other.
• Figure 3: Two-component decomposition, ${f=\int\Phi\mathrm{e}^{-q^2D\tau}\,\mathrm{d} D + \int\Psi\mathrm{e}^{-(qv\tau)^2}}\,\mathrm{d} v$, of XPCS data. A dilute suspensions of colloidal particles is measured at two temperatures frenzel_2021. The plots show six select $q$ values out of 12 in the data-set. Left column: Low temperature measurement. Right column: High temperature measurement. Top row: Intensity autocorrelation data plotted against $q$-scaled time axes. Second row: Best fit intermediate scattering function (solid lines) and data-points corrected by best fit contrast and baseline (${\sqrt{(g_2-1-b)/\beta}}$). For the low temperature measurement, the ballistic and diffusive components are plotted as separate lines for the highest $q$ (${3.0\cdot10^{-3}}$ Å-1). Bottom two rows: Solution distributions of diffusive and ballistic modes. Rectangles are plotted with height $\Phi_{m}w_m$ and width $w_m$, such that their areas appear correct on the paper when plotted against a logarithmic $x$-axis.
• Figure 4: XPCS measurement on amorphous ice decomposed into a diffusive and a ballistic part, ${f=\int\Phi\mathrm{e}^{-q^2D\tau}\,\mathrm{d} D+\int\Psi\mathrm{e}^{-(qv\tau)^2}\,\mathrm{d} v}$. Shown are six select $q$ values out of 12 in the data-set (0.0017 to 0.018 Å-1). Top left: Intensity autocorrelation data and lines of fit. Top right: The model intermediate scattering function (three select $q$ curves), broken up by component (diffusive/ballistic). Bottom: Solution distributions of diffusive and ballistic modes. Rectangles are plotted with height $\Phi_{m}w_m$ and width $w_m$, such that the areas appear correct on the paper when plotted against a logarithmic $x$-axis.
• Figure 5: Comparison of the three methods discussed in this paper. A diffusive model, ${f=\int\Phi\mathrm{e}^{-Dq^2\tau}\,\mathrm{d} D}$, is fitted to the high temperature data from figure \ref{['fig:PRE2021_2comp']}. The CONTIN-like fit (blue) solves for an arbitrary density function $\Phi(D)$, discretized into fitting coefficients. The log-normal fit (green) fits a log-normal distribution, equation \ref{['eq:simplified_dist_model']}. The KWW-fit (orange) is a global stretched exponential fit, equation \ref{['eq:global_KWW']}. The median, $\mathrm{med}(D)$, of each distribution is printed in the legend.