Cross-correlation analysis of X-ray photon correlation spectroscopy to extract rotational diffusion coefficients

Coefficients for translational and rotational diffusion characterize the Brownian motion of particles. Emerging X-ray photon correlation spectroscopy (XPCS) experiments probe a broad range of length scales and time scales and are well-suited for investigation of Brownian motion. While methods for estimating the translational diffusion coefficients from XPCS are well-developed, there are no algorithms for measuring the rotational diffusion coefficients based on XPCS, even though the required raw data are accessible from such experiments. In this paper, we propose angular-temporal cross-correlation analysis of XPCS data and show that this information can be used to design a numerical algorithm (Multi-Tiered Estimation for Correlation Spectroscopy [MTECS]) for predicting the rotational diffusion coefficient utilizing the cross-correlation: This approach is applicable to other wavelengths beyond this regime. We verify the accuracy of this algorithmic approach across a range of simulated data.

The analysis of Brownian motion of different types of particles is a classic problem in research fields such as molecular biology and materials science. For a dilute suspension, collisions from the solvent particles lead to random reposition and reorientation that can be decomposed into translational and rotational diffusion. These two types of diffusion can be characterized by coefficients, namely, the translational diffusion coefficient Dt and rotational diffusion coefficient Dr in two dimensions and the corresponding 3×3 tensor in three dimensions. Knowledge of these parameters provides insight into the structure and dynamic properties of the particles, opening the door for understanding functions and transport process of proteins (1), synthesis and stability of materials (2), biomolecular reactions (3), etc.As an emerging X-ray scattering technique, X-ray photon correlation spectroscopy (XPCS) is able to probe length scale down to nanometers and time scale from below microseconds to hours and is well-suited for studying dynamics of disordered systems, of which analyzing diffusion is a critical informative characteristic. In XPCS experiments, samples are illuminated by partially coherent X-ray beams, and time series of scattering images are collected by detectors. (See Fig. 1 for an illustration of the XPCS experiments.) The inhomogeneity of the electron density leads to spatial variation of the brightness of the images, which is called a ”speckle pattern,” and the fluctuation of the electron density leads to the temporal variation of the speckle patterns that contains valuable information of the sample dynamics, which can be revealed by analyzing these collected, time-dependent images.Open in a separate windowFig. 1.Schematic illustration of the XPCS experiments. The translation and rotation of the particles within the scattering volume leads to variation of the speckle patterns shown on the right. (While the grainy, noise-like texture makes these images appear visually similar, the proposed algorithm is able to detect and analyze the contained variations.)One of the most well-known tools for analyzing the images is the temporal autocorrelation function g2, which depends on the second-order degree of coherence (4); see, for example, refs. 5–7. While translational diffusion coefficients Dt can be determined through g2, to the best of our knowledge, there is no current algorithm to extract Dr from XPCS data.The objective of this paper is to present a methodology for calculating the rotational diffusion coefficient Dr in the two-dimensional (2D) case using XPCS data. By exploiting the more detailed angular-temporal cross-correlation function, we are able to discover more information than the autocorrelation function g2. Information about this cross-correlation function can be related to the static electron density of individual particles and the rotational diffusion coefficient Dr. To estimate Dr from the proposed cross-correlation, we first introduce a mathematical model and related set of equations, whose solution corresponds to the rotational diffusion. We then design a numerical algorithm, “Multi-Tiered Estimation for Correlation Spectroscopy (MTECS),” based on an approach introduced here, which solves the relevant equations by following a multitiered iterative projection (M-TIP) philosophy, first introduced in ref. 8. In particular, we construct several operators and apply them in an iteration to efficiently map the cross-correlation data into a form that can then be processed by the algorithm to converge to the solution of the underlying equations.While we mainly discuss XPCS in this paper, the developed methodology does not rely upon the specific range of the wavelength essentially, so that it can be generalized to other experiment techniques, such as dynamic light scattering. Though dynamic light scattering has been applied to measure the rotational diffusion (9, 10), these methods are limited by their requirements of the type and structure of the particles. The MTECS algorithm requires almost no prior assumption about particle structure, but can take advantage of such information if a priori known. MTECS is robust against noisy data, as it includes additional filtering methods applied to the input cross-correlation data.