Inverse transformation: unleashing spatially heterogeneous dynamics with an alternative approach to XPCS data analysis

Abstract

X-ray photon correlation spectroscopy (XPCS), an extension of dynamic light scattering (DLS) in the X-ray regime, detects temporal intensity fluctuations of coherent speckles and provides scattering vector-dependent sample dynamics at length scales smaller than DLS. The penetrating power of X-rays enables probing dynamics in a broad array of materials with XPCS, including polymers, glasses and metal alloys, where attempts to describe the dynamics with a simple exponential fit usually fails. In these cases, the prevailing XPCS data analysis approach employs stretched or compressed exponential decay functions (Kohlrausch functions), which implicitly assume homogeneous dynamics. In this paper, we propose an alternative analysis scheme based upon inverse Laplace or Gaussian transformation for elucidating heterogeneous distributions of dynamic time scales in XPCS, an approach analogous to the algorithm widely accepted in the analysis of DLS from polydisperse and multimodal systems. Using XPCS data measured from colloidal gels, we demonstrate the inverse transform approach reveals hidden multimodal dynamics in materials, unleashing the full potential of XPCS.

CONTIN XPCS: software for inverse transform analysis of X-ray photon correlation spectroscopy dynamics

“…We demonstrated the extension of the inverse transform technique to XPCS data analysis by including the inverse Gaussian transformation for faster than exponential dynamics (Andrews et al ., 2017). The examples showed that inverse Gaussian transform analysis can reveal dynamics hidden by conventional Kohlrausch exponential analysis.…”

“…Figure based upon our earlier work (Andrews et al ., 2017). Techniques for analysis of DLS and XPCS temporal autocorrelation functions g 2 ( t ).Γ shows relaxation rate in its conventional definition.…”

“…In most cases, while we find the automatic selection the best intuitive choice, one should investigate the solution behavior in the vicinity of ${\mathcal{P}}_{\text{rej}}=\frac{1}{2}$. Here, ${\mathcal{P}}_{\text{rej}}$ represents a rejection probability metric as defined in (Andrews et al ., 2017). exports the distribution solution and inverse transform parameters into text files.…”

“…Traditional DLS analysis commonly interprets nonexponential relaxation in terms of underlying dynamic heterogeneity using cumulant (Frisken, 2001) or analysis (Provencher, 1979; Provencher, 1982 a ; Provencher, 1982 b ), whereas XPCS analysis uses a Kohlrausch function. (Kohlrausch, 1854 a ; Kohlrausch, 1854 b ; Madsen et al ., 2010; Berberan-Santos et al ., 2008) We demonstrated the use of modified with XPCS measurements (Andrews et al ., 2017) from XPCS data sets showing classic simple diffusive kinetics, compressed exponential dynamics and bimodal compressed exponential dynamics. The mathematical functions commonly used for DLS and XPCS analysis, as well as our notation for - and -XPCS analysis, are summarized in Fig.…”

“…As an alternative to Kohlrausch exponential analysis of heterogeneous dynamics, we demonstrated the use of modified (Andrews et al ., 2017) with XPCS data sets showing simple diffusive and compressed exponential dynamics. In the final example presented there, inverse transform analysis revealed a bimodal distribution of ballistic relaxation rates not obvious based upon classical Kohlrausch function analysis.…”