On the use of two-time correlation functions for X-ray photon correlation spectroscopy data analysis

Salt induced slowdown of kinetics and dynamics during thermal gelation of egg-yolk

We investigated the effect of the NaCl concentration (0.3-2M) on the structure and dynamics of hen egg yolk at room temperature and during thermal gelation at temperatures in the range of 66-90 °C utilizing low-dose x-ray photon correlation spectroscopy in ultra-small angle x-ray scattering geometry. With an increase in the salt concentration, we observe progressive structural and dynamic changes at room temperature, indicating the disruption of yolk components such as yolk-granules and yolk-plasma proteins. Temperature- and salt-dependent structural and dynamic investigations suggest a delay in the gel formation and aggregation of yolk low-density lipoproteins with increasing ionic strength. However, the time-temperature superposition relationship observed in all samples suggests an identical mechanism underlying protein aggregation-gelation with a temperature-dependent reaction rate. The sol-gel transition time extracted from kinetic and dynamic information follows Arrhenius's behavior, and the activation energy (460 kJ/mol) is found to be independent of the salt concentration.

Transport coefficient approach for characterizing nonequilibrium dynamics in soft matter

Nonequilibrium states in soft condensed matter require a systematic approach to characterize and model materials, enhancing predictability and applications. Among the tools, X-ray photon correlation spectroscopy (XPCS) provides exceptional temporal and spatial resolution to extract dynamic insight into the properties of the material. However, existing models might overlook intricate details. We introduce an approach for extracting the transport coefficient, denoted as [Formula: see text], from the XPCS studies. This coefficient is a fundamental parameter in nonequilibrium statistical mechanics and is crucial for characterizing transport processes within a system. Our method unifies the Green-Kubo formulas associated with various transport coefficients, including gradient flows, particle-particle interactions, friction matrices, and continuous noise. We achieve this by integrating the collective influence of random and systematic forces acting on the particles within the framework of a Markov chain. We initially validated this method using molecular dynamics simulations of a system subjected to changes in temperatures over time. Subsequently, we conducted further verification using experimental systems reported in the literature and known for their complex nonequilibrium characteristics. The results, including the derived [Formula: see text] and other relevant physical parameters, align with the previous observations and reveal detailed dynamical information in nonequilibrium states. This approach represents an advancement in XPCS analysis, addressing the growing demand to extract intricate nonequilibrium dynamics. Further, the methods presented are agnostic to the nature of the material system and can be potentially expanded to hard condensed matter systems.

AI-NERD: Elucidation of relaxation dynamics beyond equilibrium through AI-informed X-ray photon correlation spectroscopy

Understanding and interpreting dynamics of functional materials in situ is a grand challenge in physics and materials science due to the difficulty of experimentally probing materials at varied length and time scales. X-ray photon correlation spectroscopy (XPCS) is uniquely well-suited for characterizing materials dynamics over wide-ranging time scales. However, spatial and temporal heterogeneity in material behavior can make interpretation of experimental XPCS data difficult. In this work, we have developed an unsupervised deep learning (DL) framework for automated classification of relaxation dynamics from experimental data without requiring any prior physical knowledge of the system. We demonstrate how this method can be used to accelerate exploration of large datasets to identify samples of interest, and we apply this approach to directly correlate microscopic dynamics with macroscopic properties of a model system. Importantly, this DL framework is material and process agnostic, marking a concrete step towards autonomous materials discovery.

Real-time swelling-collapse kinetics of nanogels driven by XFEL pulses

Stimuli-responsive polymers are an important class of materials with many applications in nanotechnology and drug delivery. The most prominent one is poly-N-isopropylacrylamide (PNIPAm). The characterization of the kinetics of its change after a temperature jump is still a lively research topic, especially at nanometer-length scales where it is not possible to rely on conventional microscopic techniques. Here, we measured in real time the collapse of a PNIPAm shell on silica nanoparticles with megahertz x-ray photon correlation spectroscopy at the European XFEL. We characterize the changes of the particles diffusion constant as a function of time and consequently local temperature on sub-microsecond timescales. We developed a phenomenological model to describe the observed data and extract the characteristic times associated to the swelling and collapse processes. Different from previous studies tracking the turbidity of PNIPAm dispersions and using laser heating, we find collapse times below microsecond timescales and two to three orders of magnitude slower swelling times.

X-ray-induced piezoresponse during X-ray photon correlation spectroscopy of PbMg1/3Nb2/3O3

X-ray photon correlation spectroscopy (XPCS) holds strong promise for observing atomic-scale dynamics in materials, both at equilibrium and during non-equilibrium transitions. Here an in situ XPCS study of the relaxor ferroelectric PbMg1/3Nb2/3O3 (PMN) is reported. A weak applied AC electric field generates strong response in the speckle of the diffuse scattering from the polar nanodomains, which is captured using the two-time correlation function. Correlated motions of the Bragg peak are also observed, which indicate dynamic tilting of the illuminated volume. This tilting quantitatively accounts for the observed two-time speckle correlations. The magnitude of the tilting would not be expected solely from the modest applied field, since PMN is an electrostrictive material with no linear strain response to the field. A model is developed based on non-uniform static charging of the illuminated surface spot by the incident micrometre-scale X-ray beam and the electrostrictive material response to the combination of static and dynamic fields. The model qualitatively explains the direction and magnitude of the observed tilting, and predicts that X-ray-induced piezoresponse could be an important factor in correctly interpreting results from XPCS and nanodiffraction studies of other insulating materials under applied AC field or varying X-ray illumination.

A phenomenological model of the X-ray pulse statistics of a high-repetition-rate X-ray free-electron laser

Many coherent imaging applications that utilize ultrafast X-ray free-electron laser (XFEL) radiation pulses are highly sensitive to fluctuations in the shot-to-shot statistical properties of the source. Understanding and modelling these fluctuations are key to successful experiment planning and necessary to maximize the potential of XFEL facilities. Current models of XFEL radiation and their shot-to-shot statistics are based on theoretical descriptions of the source and are limited in their ability to capture the shot-to-shot intensity fluctuations observed experimentally. The lack of accurate temporal statistics in simulations that utilize these models is a significant barrier to optimizing and interpreting data from XFEL coherent diffraction experiments. Presented here is a phenomenological model of XFEL radiation that is capable of capturing the shot-to-shot statistics observed experimentally using a simple time-dependent approximation of the pulse wavefront. The model is applied to reproduce non-stationary shot-to-shot intensity fluctuations observed at the European XFEL, whilst accurately representing the single-shot properties predicted by FEL theory. Compared with previous models, this approach provides a simple, robust and computationally inexpensive method of generating statistical representations of XFEL radiation.

Exploring non-equilibrium processes and spatio-temporal scaling laws in heated egg yolk using coherent X-rays

The soft-grainy microstructure of cooked egg yolk is the result of a series of out-of-equilibrium processes of its protein-lipid contents; however, it is unclear how egg yolk constituents contribute to these processes to create the desired microstructure. By employing X-ray photon correlation spectroscopy, we investigate the functional contribution of egg yolk constituents: proteins, low-density lipoproteins (LDLs), and yolk-granules to the development of grainy-gel microstructure and microscopic dynamics during cooking. We find that the viscosity of the heated egg yolk is solely determined by the degree of protein gelation, whereas the grainy-gel microstructure is controlled by the extent of LDL aggregation. Overall, protein denaturation-aggregation-gelation and LDL-aggregation follows Arrhenius-type time-temperature superposition (TTS), indicating an identical mechanism with a temperature-dependent reaction rate. However, above 75 °C TTS breaks down and temperature-independent gelation dynamics is observed, demonstrating that the temperature can no longer accelerate certain non-equilibrium processes above a threshold value.

Investigation of the yielding transition in concentrated colloidal systems via rheo-XPCS

We probe the microstructural yielding dynamics of a concentrated colloidal system by performing creep/recovery tests with simultaneous collection of coherent scattering data via X-ray Photon Correlation Spectroscopy (XPCS). This combination of rheology and scattering allows for time-resolved observations of the microstructural dynamics as yielding occurs, which can be linked back to the applied rheological deformation to form structure-property relations. Under sufficiently small applied creep stresses, examination of the correlation in the flow direction reveals that the scattering response recorrelates with its predeformed state, indicating nearly complete microstructural recovery, and the dynamics of the system under these conditions slows considerably. Conversely, larger creep stresses increase the speed of the dynamics under both applied creep and recovery. The data show a strong connection between the microstructural dynamics and the acquisition of unrecoverable strain. By comparing this relationship to that predicted from homogeneous, affine shearing, we find that the yielding transition in concentrated colloidal systems is highly heterogeneous on the microstructural level.

Dynamic behaviours of epoxy resin thin films during the curing process

Epoxy resin thin films are widely used in applications such as coating materials, insulator films, and adhesives; accordingly, investigations of their physical properties have garnered increasing importance. Although the physical properties of thermoset epoxy thin films are strongly affected by the curing conditions, such as the heating temperature and curing time, the dynamic properties during the curing process have not been studied thoroughly. In this study, we investigated the thermal fluctuations on the surface of epoxy resin thin films using grazing-incidence X-ray photon correlation spectroscopy, to elucidate the dynamic behaviours during the curing process. We thus succeeded in observing the freezing of capillary waves during the thermal curing process. These results are expected to facilitate a deeper understanding of the curing mechanisms of various thin films.

Effects of temperature and ionic strength on the microscopic structure and dynamics of egg white gels

We investigate the thermal gelation of egg white proteins at different temperatures with varying salt concentrations using x-ray photon correlation spectroscopy in the geometry of ultra-small angle x-ray scattering. Temperature-dependent structural investigation suggests a faster network formation with increasing temperature, and the gel adopts a more compact network, which is inconsistent with the conventional understanding of thermal aggregation. The resulting gel network shows a fractal dimension δ, ranging from 1.5 to 2.2. The values of δ display a non-monotonic behavior with increasing amount of salt. The corresponding dynamics in the q range of 0.002-0.1 nm^-1 is observable after major change of the gel structure. The extracted relaxation time exhibits a two-step power law growth in dynamics as a function of waiting time. In the first regime, the dynamics is associated with structural growth, whereas the second regime is associated with the aging of the gel, which is directly linked with its compactness, as quantified by the fractal dimension. The gel dynamics is characterized by a compressed exponential relaxation with a ballistic-type of motion. The addition of salt gradually makes the early stage dynamics faster. Both gelation kinetics and microscopic dynamics show that the activation energy barrier in the system systematically decreases with increasing salt concentration.

Automated matching of two-time X-ray photon correlation maps from phase-separating proteins with Cahn-Hilliard-type simulations using auto-encoder networks

Machine learning methods are used for an automated classification of experimental two-time X-ray photon correlation maps from an arrested liquid-liquid phase separation of a protein solution. The correlation maps are matched with correlation maps generated with Cahn-Hilliard-type simulations of liquid-liquid phase separations according to two simulation parameters and in the last step interpreted in the framework of the simulation. The matching routine employs an auto-encoder network and a differential evolution based algorithm. The method presented here is a first step towards handling large amounts of dynamic data measured at high-brilliance synchrotron and X-ray free-electron laser sources, facilitating fast comparison with phase field models of phase separation.

A multiscale time-Laplace method to extract relaxation times from non-stationary dynamic light scattering signals

Dynamic Light Scattering (DLS) is a well-known technique to study the relaxation times of systems at equilibrium. In many soft matter systems, we actually have to consider non-equilibrium or non-stationary situations. We discuss here the principles, the signal processing techniques we developed, based on regularized inverse Laplace transform, sliding with time, and the light scattering signal acquisition, which enable us to use DLS experiments in this general situation. In this article, we show how to obtain such a time-Laplace analysis. We claim that this method can be adapted to numerous DLS experiments dealing with non-equilibrium systems so as to extract the non-stationary distribution of relaxation times. To prove that, we test this time-Laplace method on three different non-equilibrium processes or systems investigated by means of the DLS technique: the cooling kinetics of a colloidal particle solution, the sol-gel transition and the internal dynamics of a living cell nucleus.

Real-space observation of fluctuating antiferromagnetic domains

Magnetic domains play a fundamental role in physics of magnetism and its technological applications. Dynamics of antiferromagnetic domains is poorly understood, although antiferromagnets are expected to be extensively used in future electronic devices wherein it determines the stability and operational speed. Dynamics of antiferromagnets also features prominently in the studies of topological quantum matter. Real-space imaging of fluctuating antiferromagnetic domains is therefore highly desired but has never been demonstrated. We use coherent x-ray diffraction to obtain videos of fluctuating micrometer-scale antiferromagnetic domains in Ni₂MnTeO₆ on time scales from 10^-1 to 10³ s. In the collinear phase, thermally activated domain wall motion is observed in the vicinity of the Néel temperature. Unexpectedly, the fluctuations persist through the full range of the higher-temperature helical phase. These observations illustrate the high potential significance of the dynamic domain imaging in phase transition studies and in magnetic device research.

Microsecond hydrodynamic interactions in dense colloidal dispersions probed at the European XFEL

Many soft-matter systems are composed of macromolecules or nanoparticles suspended in water. The characteristic times at intrinsic length scales of a few nanometres fall therefore in the microsecond and sub-microsecond time regimes. With the development of free-electron lasers (FELs) and fourth-generation synchrotron light-sources, time-resolved experiments in such time and length ranges will become routinely accessible in the near future. In the present work we report our findings on prototypical soft-matter systems, composed of charge-stabilized silica nanoparticles dispersed in water, with radii between 12 and 15 nm and volume fractions between 0.005 and 0.2. The sample dynamics were probed by means of X-ray photon correlation spectroscopy, employing the megahertz pulse repetition rate of the European XFEL and the Adaptive Gain Integrating Pixel Detector. We show that it is possible to correctly identify the dynamical properties that determine the diffusion constant, both for stationary samples and for systems driven by XFEL pulses. Remarkably, despite the high photon density the only observable induced effect is the heating of the scattering volume, meaning that all other X-ray induced effects do not influence the structure and the dynamics on the probed timescales. This work also illustrates the potential to control such induced heating and it can be predicted with thermodynamic models.

Interplay between Kinetics and Dynamics of Liquid-Liquid Phase Separation in a Protein Solution Revealed by Coherent X-ray Spectroscopy

Microscopic dynamics of complex fluids in the early stage of spinodal decomposition (SD) is strongly intertwined with the kinetics of structural evolution, which makes a quantitative characterization challenging. In this work, we use X-ray photon correlation spectroscopy to study the dynamics and kinetics of a protein solution undergoing liquid-liquid phase separation (LLPS). We demonstrate that in the early stage of SD, the kinetics relaxation is up to 40 times slower than the dynamics and thus can be decoupled. The microscopic dynamics can be well described by hyper-diffusive ballistic motions with a relaxation time exponentially growing with time in the early stage followed by a power-law increase with fluctuations. These experimental results are further supported by simulations based on the Cahn-Hilliard equation. The established framework is applicable to other condensed matter and biological systems undergoing phase transitions and may also inspire further theoretical work.

From Femtoseconds to Hours—Measuring Dynamics over 18 Orders of Magnitude with Coherent X-rays

X-ray photon correlation spectroscopy (XPCS) enables the study of sample dynamics between micrometer and atomic length scales. As a coherent scattering technique, it benefits from the increased brilliance of the next-generation synchrotron radiation and Free-Electron Laser (FEL) sources. In this article, we will introduce the XPCS concepts and review the latest developments of XPCS with special attention on the extension of accessible time scales to sub-μs and the application of XPCS at FELs. Furthermore, we will discuss future opportunities of XPCS and the related technique X-ray speckle visibility spectroscopy (XSVS) at new X-ray sources. Due to its particular signal-to-noise ratio, the time scales accessible by XPCS scale with the square of the coherent flux, allowing to dramatically extend its applications. This will soon enable studies over more than 18 orders of magnitude in time by XPCS and XSVS.

Microscopic Dynamics of Liquid-Liquid Phase Separation and Domain Coarsening in a Protein Solution Revealed by X-Ray Photon Correlation Spectroscopy

While the interplay between liquid-liquid phase separation (LLPS) and glass formation in biological systems is highly relevant for their structure formation and thus function, the exact underlying mechanisms are not well known. The kinetic arrest originates from the slowdown at the molecular level, but how this propagates to the dynamics of microscopic phase domains is not clear. Since with diffusion, viscoelasticity, and hydrodynamics, distinctly different mechanisms are at play, the dynamics needs to be monitored on the relevant time and length scales and compared to theories of phase separation. Using x-ray photon correlation spectroscopy, we determine the LLPS dynamics of a model protein solution upon low temperature quenches and find distinctly different dynamical regimes. We observe that the early stage LLPS is driven by the curvature of the free energy and speeds up upon increasing quench depth. In contrast, the late stage dynamics slows down with increasing quench depth, fingerprinting a nearby glass transition. The dynamics observed shows a ballistic type of motion, implying that viscoelasticity plays an important role during LLPS. We explore possible explanations based on the Cahn-Hilliard theory with nontrivial mobility parameters and find that these can only partially explain our findings.

de Gennes Narrowing and Relationship between Structure and Dynamics in Self-Organized Ion-Beam Nanopatterning

Investigating the relationship between structure and dynamical processes is a central goal in condensed matter physics. Perhaps the most noted relationship between the two is the phenomenon of de Gennes narrowing, in which relaxation times in liquids are proportional to the scattering structure factor. Here, a similar relationship is discovered during the self-organized ion-beam nanopatterning of silicon using coherent x-ray scattering. However, in contrast to the exponential relaxation of fluctuations in classic de Gennes narrowing, the dynamic surface exhibits a wide range of behaviors as a function of the length scale, with a compressed exponential relaxation at lengths corresponding to the dominant structural motif-self-organized nanoscale ripples. These behaviors are reproduced in simulations of a nonlinear model describing the surface evolution. We suggest that the compressed exponential behavior observed here is due to the morphological persistence of the self-organized surface ripple patterns which form and evolve during ion-beam nanopatterning.

20 µs-resolved high-throughput X-ray photon correlation spectroscopy on a 500k pixel detector enabled by data-management workflow

The performance of the new 52 kHz frame rate Rigaku XSPA-500k detector was characterized on beamline 8-ID-I at the Advanced Photon Source at Argonne for X-ray photon correlation spectroscopy (XPCS) applications. Due to the large data flow produced by this detector (0.2 PB of data per 24 h of continuous operation), a workflow system was deployed that uses the Advanced Photon Source data-management (DM) system and high-performance software to rapidly reduce area-detector data to multi-tau and two-time correlation functions in near real time, providing human-in-the-loop feedback to experimenters. The utility and performance of the workflow system are demonstrated via its application to a variety of small-angle XPCS measurements acquired from different detectors in different XPCS measurement modalities. The XSPA-500k detector, the software and the DM workflow system allow for the efficient acquisition and reduction of up to ∼10⁹ area-detector data frames per day, facilitating the application of XPCS to measuring samples with weak scattering and fast dynamics.

Emergence of anomalous dynamics in soft matter probed at the European XFEL

Dynamics and kinetics in soft matter physics, biology, and nanoscience frequently occur on fast (sub)microsecond but not ultrafast timescales which are difficult to probe experimentally. The European X-ray Free-Electron Laser (European XFEL), a megahertz hard X-ray Free-Electron Laser source, enables such experiments via taking series of diffraction patterns at repetition rates of up to 4.5 MHz. Here, we demonstrate X-ray photon correlation spectroscopy (XPCS) with submicrosecond time resolution of soft matter samples at the European XFEL. We show that the XFEL driven by a superconducting accelerator provides unprecedented beam stability within a pulse train. We performed microsecond sequential XPCS experiments probing equilibrium and nonequilibrium diffusion dynamics in water. We find nonlinear heating on microsecond timescales with dynamics beyond hot Brownian motion and superheated water states persisting up to 100 μs at high fluences. At short times up to 20 μs we observe that the dynamics do not obey the Stokes-Einstein predictions.

Anisotropic and heterogeneous dynamics in an aging colloidal gel

We investigate the out-of-equilibrium dynamics of a colloidal gel obtained by quenching a suspension of soft polymer-coated gold nanoparticles close to and below its gelation point using X-ray Photon Correlation Spectroscopy (XPCS). A faster relaxation process emergent from the localized motions of the nanoparticles reveals a dynamically-arrested network at the nanoscale as a key signature of the gelation process. We find that the slower network dynamics is hyperdiffusive with a compressed exponential form, consistent with stress-driven relaxation processes. Specifically, we use direction-dependent correlation functions to characterize the anisotropy in dynamics. We show that the anisotropy is greater for the gel close to its gelation point than at lower temperatures, and the anisotropy decreases as the gel ages. We quantify the anisotropic dynamical heterogeneities emergent in such a stress-driven dynamical system using higher order intensity correlations, and demonstrate that the aging phenomenon contributes significantly to the properties evaluated by the fluctuations in the intensity correlations. Our results provide important insights into the structural origin of the emergent anisotropic and cooperative heterogeneous dynamics, and we discuss analogies with previous work on other soft disordered systems.

Anti-Aging in Ultrastable Metallic Glasses

As ultrastable metallic glasses (UMGs) are promising candidates to solve the stability issues of conventional metallic glasses, their study is of exceptional interest. By means of x-ray photon correlation spectroscopy, we have investigated the stability of UMGs at the atomic level. We find a clear signature of ultrastability at the atomic level that results in slower relaxation dynamics of UMGs with respect to conventional (rapidly quenched) metallic glasses, and in a peculiar acceleration of the dynamics by near T_{g} annealing. This surprising phenomenon, called here anti-aging, can be understood in the framework of the potential energy landscape. For all samples, the structural relaxation process can be described with a highly compressed shape of the density fluctuations, unaffected by thermal treatments and regardless of the ultrastability of the glass.