Fluctuation dynamics of block copolymer vesicles

Dynamic structure factor of undulating vesicles: finite-size and spherical geometry effects with application to neutron spin echo experiments

We consider the dynamic structure factor (DSF) of quasi-spherical vesicles and present a generalization of an expression that was originally formulated by Zilman and Granek (ZG) for scattering from isotropically oriented quasi-flat membrane plaquettes. The expression is obtained in the form of a multi-dimensional integral over the undulating membrane surface. The new expression reduces to the original stretched exponential form in the limit of sufficiently large vesicles, i.e., in the micron range or larger. For much smaller unilamellar vesicles, deviations from the asymptotic, stretched exponential equation are noticeable even if one assumes that the Seifert-Langer leaflet density mode is completely relaxed and membrane viscosity is neglected. To avoid the need for an exhaustive numerical integration while fitting to neutron spin echo (NSE) data, we provide a useful approximation for polydisperse systems that tests well against the numerical integration of the complete expression. To validate the new expression, we performed NSE experiments on variable-size vesicles made of a POPC/POPS lipid mixture and demonstrate an advantage over the original stretched exponential form or other manipulations of the original ZG expression that have been deployed over the years to fit the NSE data. In particular, values of the membrane bending rigidity extracted from the NSE data using the new approximations were insensitive to the vesicle radii and scattering wavenumber and compared very well with expected values of the effective bending modulus ([Formula: see text]) calculated from results in the literature. Moreover, the generalized scattering theory presented here for an undulating quasi-spherical shell can be easily extended to other models for the membrane undulation dynamics beyond the Helfrich Hamiltonian and thereby provides the foundation for the study of the nanoscale dynamics in more complex and biologically relevant model membrane systems.

Intrinsic Dynamics of Amorphous Ice Revealed by a Heterodyne Signal in X-ray Photon Correlation Spectroscopy Experiments

Unraveling the mechanism of water's glass transition and the interconnection between amorphous ices and liquid water plays an important role in our overall understanding of water. X-ray photon correlation spectroscopy (XPCS) experiments were conducted to study the dynamics and the complex interplay between the hypothesized glass transition in high-density amorphous ice (HDA) and the subsequent transition to low-density amorphous ice (LDA). Our XPCS experiments demonstrate that a heterodyne signal appears in the correlation function. Such a signal is known to originate from the interplay of a static component and a dynamic component. Quantitative analysis was performed on this heterodyne signal to extract the intrinsic dynamics of amorphous ice during the HDA-LDA transition. An angular dependence indicates non-isotropic, heterogeneous dynamics in the sample. Using the Stokes-Einstein relation to extract diffusion coefficients, the data are consistent with the scenario of static LDA islands floating within a diffusive matrix of high-density liquid water.

Nanoscale Bending Dynamics in Mixed-Chain Lipid Membranes

Lipids that have two tails of different lengths are found throughout biomembranes in nature, yet the effects of this asymmetry on the membrane properties are not well understood, especially when it comes to the membrane dynamics. Here we study the nanoscale bending fluctuations in model mixed-chain 14:0–18:0 PC (MSPC) and 18:0–14:0 PC (SMPC) lipid bilayers using neutron spin echo (NSE) spectroscopy. We find that despite the partial interdigitation that is known to persist in the fluid phase of these membranes, the collective fluctuations are enhanced on timescales of tens of nanoseconds, and the chain-asymmetric lipid bilayers are softer than an analogous chain-symmetric lipid bilayer with the same average number of carbons in the acyl tails, di-16:0 PC (DPPC). Quantitative comparison of the NSE results suggests that the enhanced bending fluctuations at the nanosecond timescales are consistent with experimental and computational studies that showed the compressibility moduli of chain-asymmetric lipid membranes are 20% to 40% lower than chain-symmetric lipid membranes. These studies add to growing evidence that the partial interdigitation in mixed-chain lipid membranes is highly dynamic in the fluid phase and impacts membrane dynamic processes from the molecular to mesoscopic length scales without significantly changing the bilayer thickness or area per lipid.

Methods of Coherent X-Ray Diffraction Imaging

Abstract

Methods of coherent X-ray diffraction imaging of the spatial structure of noncrystalline objects and nanocrystals (nanostructures) are considered. Particular attention is paid to the methods of scanning-based coherent diffraction imaging (ptychography), visualization based on coherent surface scattering with application of correlation spectroscopy approaches, and specific features of visualization using X-ray free-electron laser radiation. The corresponding data in the literature are analyzed to demonstrate the state of the art of the methods of coherent diffraction imaging and fields of their application.

Demonstration of single-frame coherent X-ray diffraction imaging using triangular aperture: Towards dynamic nanoimaging of extended objects

Coherent diffraction imaging (CDI) is a powerful method for visualizing the structure of an object with a high spatial resolution that exceeds the performance limits of the lens. Single-frame CDI in the X-ray region has potential use for probing dynamic phenomena with a high spatiotemporal resolution. Here, we experimentally demonstrate a general method for single-frame X-ray CDI using a triangular aperture and a Fresnel zone plate. Using 5 keV synchrotron radiation X-rays, we reconstructed the object image of the locally illuminated area with a spatial resolution of higher than 50 nm and an exposure time of more than 0.1 s without prior information about the sample. After a 10 s exposure, a resolution of 17 nm was achieved. The present method opens new frontiers in the study of dynamics at the nanoscale by using next-generation synchrotron radiation X-rays/free-electron lasers as light sources.

Split-pulse X-ray photon correlation spectroscopy with seeded X-rays from X-ray laser to study atomic-level dynamics

With their brilliance and temporal structure, X-ray free-electron laser can unveil atomic-scale details of ultrafast phenomena. Recent progress in split-and-delay optics (SDO), which produces two X-ray pulses with time-delays, offers bright prospects for observing dynamics at the atomic-scale. However, their insufficient pulse energy has limited its application either to phenomena with longer correlation length or to measurement with a fixed delay-time. Here we show that the combination of the SDO and self-seeding of X-rays increases the pulse energy and makes it possible to observe the atomic-scale dynamics in a timescale of picoseconds. We show that the speckle contrast in scattering from water depends on the delay-time as expected. Our results demonstrate the capability of measurement using the SDO with seeded X-rays for resolving the dynamics in temporal and spatial scales that are not accessible by other techniques, opening opportunities for studying the atomic-level dynamics.

Future trends in synchrotron science at NSLS-II

In this paper, we summarize briefly some of the future trends in synchrotron science as seen at the National Synchrotron Light Source II, a new, low emittance source recently commissioned at Brookhaven National Laboratory. We touch upon imaging techniques, the study of dynamics, the increasing use of multimodal approaches, the vital importance of data science, and other enabling technologies. Each are presently undergoing a time of rapid change, driving the field of synchrotron science forward at an ever increasing pace. It is truly an exciting time and one in which Roger Cowley, to whom this journal issue is dedicated, would surely be both invigorated by, and at the heart of.

Brownian and advective dynamics in microflow studied by coherent X-ray scattering experiments

Combining microfluidics with coherent X-ray illumination offers the possibility to not only measure the structure but also the dynamics of flowing samples in a single-scattering experiment. Here, the power of this combination is demonstrated by studying the advective and Brownian dynamics of colloidal suspensions in microflow of different geometries. Using an experimental setup with a fast two-dimensional detector and performing X-ray correlation spectroscopy by calculating two-dimensional maps of the intensity auto-correlation functions, it was possible to evaluate the sample structure and furthermore to characterize the detailed flow behavior, including flow geometry, main flow directions, advective flow velocities and diffusive dynamics. By scanning a microfocused X-ray beam over a microfluidic device, the anisotropic auto-correlation functions of driven colloidal suspensions in straight, curved and constricted microchannels were mapped with the spatial resolution of the X-ray beam. This method has not only a huge potential for studying flow patterns in complex fluids but also to generally characterize anisotropic dynamics in materials.

Structure, viscoelasticity, and interfacial dynamics of a model polymeric bicontinuous microemulsion

We have systematically studied the equilibrium structure and dynamics of a polymeric bicontinuous microemulsion (BμE) composed of poly(cyclohexylethylene) (PCHE), poly(ethylene) (PE), and a volumetrically symmetric PCHE-PE diblock copolymer, using dynamic mechanical spectroscopy, small angle X-ray and neutron scattering, and transmission electron microscopy. The BμE was investigated over an 80 °C temperature range, revealing a structural evolution and a rheological response not previously recognized in such systems. As the temperature is reduced below the point associated with the lamellar-disorder transition at compositions adjacent to the microemulsion channel, the interfacial area per chain of the BμE approaches that of the neat (undiluted) lamellar diblock copolymer. With increasing temperature, the diblock-rich interface swells through homopolymer infiltration. Time-temperature-superposed linear dynamic data obtained as a function of frequency show that the viscoelastic response of the BμE is strikingly similar to that of the fluctuating pure diblock copolymer in the disordered state, which we associate with membrane undulations and the breaking and reforming of interfaces. This work provides new insights into the structure and dynamics that characterize thermodynamically stable BμEs in the limits of relatively weak and strong segregation.

X-ray photon correlation spectroscopy

In recent years, X-ray photon correlation spectroscopy (XPCS) has emerged as one of the key probes of slow nanoscale fluctuations, applicable to a wide range of condensed matter and materials systems. This article briefly reviews the basic principles of XPCS as well as some of its recent applications, and discusses some novel approaches to XPCS analysis. It concludes with a discussion of the future impact of diffraction-limited storage rings on new types of XPCS experiments, pushing the temporal resolution to nanosecond and possibly even picosecond time scales.

Direction-dependent freezing of diamagnetic colloidal tracers suspended in paramagnetic ionic liquids

The dynamic behavior of an inverse ferrofluid consisting of diamagnetic, spherical silica particles suspended in the paramagnetic ionic liquid (EMIm)2[Co(NCS)4] is investigated by means of x-ray photon correlation spectroscopy in the presence of an external magnetic field. Dipole-dipole interactions between the diamagnetic holes in the paramagnetic continuum of the suspending medium induce a direction-dependence of the diffusive motion of the colloidal particles: due to a magnetic repulsion perpendicular to the direction of an external field the diffusive motion of the colloidal particles is selectively frozen in this direction.

Structure and dynamics studies of concentrated micrometer-sized colloidal suspensions

We present an experimental study of the structural and dynamical properties of concentrated suspensions of different sized polystyrene microspheres dispersed in glycerol for volume fraction concentrations between 10% and 20%. The static structure, probed with ultrasmall-angle X-ray scattering, shows a behavior very similar to that of hard spheres. The equilibrium dynamics is probed with ultrasmall-angle X-ray scattering-X-ray photon correlation spectroscopy, a new technique that overcomes the limits of visible light-scattering techniques imposed by multiple scattering and is suitable for studies of optically opaque materials containing micrometer-sized structures. We found that the intensity autocorrelation functions are better described by a stretched exponential function and microspheres in a concentrated suspension move collectively. We also found that the inverse of the effective diffusion coefficients displays a peak with respect to the scattering vector that resembles the peaks in the static structure factors, which indicates that a long-lived, low free-energy state exists. The relaxation time is approximately inversely related to scattering vector, a behavior consistent with models that describe the dynamics in terms of random, local structural arrangements in disordered media.

Direction dependent diffusion of aligned magnetic rods by means of x-ray photon correlation spectroscopy

Rodlike hematite particles in suspension align perpendicular to an external magnetic field due to a negative anisotropy of their magnetic susceptibility Δχ. The diffusion tensor consists of two principal constants D(∥) and D(⊥) for the diffusion parallel and perpendicular to the long particle axis. X-ray photon correlation spectroscopy is capable of probing the diffusive motion in optically opaque suspensions of rodlike hematite particles parallel to the direction of the scattering vector Q. Choosing Q parallel or perpendicular to the direction of an external magnetic field H the direction dependent intermediate scattering function is measured by means of x-ray photon correlation spectroscopy. From the intermediate scattering function in both directions the principal diffusion constants D(∥) and D(⊥) are determined. The ratio D(∥)/D(⊥) increases with increasing aspect ratio of the particles and can be described via a rescaled theoretical approach for prolate ellipsoids of revolution.

Capturing dynamics with Eiger, a fast-framing X-ray detector

Eiger is the next-generation single-photon-counting pixel detector following the widely used Pilatus detector. Its smaller pixel size of 75 µm × 75 µm, higher frame rate of up to 22 kHz, and practically zero dead-time (~4 µs) between exposures will further various measurement methods at synchrotron sources. In this article Eiger's suitability for X-ray photon correlation spectroscopy (XPCS) is demonstrated. By exploiting its high frame rate, complementary small-angle X-ray scattering (SAXS) and XPCS data are collected in parallel to determine both the structure factor and collective diffusion coefficient of a nano-colloid suspension. For the first time, correlation times on the submillisecond time scale are accessible with a large-area pixel detector.

The intermediate scattering function for lipid bilayer membranes: from nanometers to microns

A numerical scheme based upon established hydrodynamic and elastic considerations is introduced and used to predict the intermediate scattering function for lipid bilayer membranes. The predictions span multiple wavelength regimes, including those studied by dynamic light scattering (DLS; microns) and neutron spin-echo (NSE) spectroscopy (10-100 nm). The results validate a recent theory specific to the NSE regime and expose slight inaccuracies associated with the theoretical results available in the DLS regime. The assumptions that underlie both our numerical methods and the related theoretical predictions are reviewed in detail to explain when certain results can be applied to experiment and where caution must be exercised.

X-ray photon correlation spectroscopy during homogenous shear flow

We report x-ray photon correlation spectroscopy measurements of advective and diffusive dynamics in a dispersion of colloidal particles subjected to homogeneous shear flow in a rotating-disk shear cell. Intensity autocorrelation functions from scattering data collected using homodyne detection respond to the variation in velocity across the scattering volume when the scattering vector has a component parallel to the flow direction. Theoretical expressions for the impact of homogenous shear flow on the correlation function provide a quantitative prediction of the dependence of correlation functions on the scattering vector and shear rate. Under most circumstances, the applied shear deformation dominates the decay of the intensity correlation function. When scattering data are collected perpendicular to the flow direction, it is possible to measure the diffusive dynamics of the particles free from effects of the superimposed shear flow; however, this approach only works below some upper shear rate limit, beyond which data are affected either by shear effects (caused by the finite width of the detector) or by particle transit through the scattering volume.

Dynamic structure factor of vibrating fractals: proteins as a case study

We study the dynamic structure factor S(k,t) of proteins at large wave numbers k, kR(g)≫1, where R(g) is the gyration radius. At this regime measurements are sensitive to internal dynamics, and we focus on vibrational dynamics of folded proteins. Exploiting the analogy between proteins and fractals, we perform a general analytic calculation of the displacement two-point correlation functions, <[u(−>)(i)(t)-u(−>)(j)(0)](2)>. We confront the derived expressions with numerical evaluations that are based on protein data bank (PDB) structures and the Gaussian network model (GNM) for a few proteins and for the Sierpinski gasket as a controlled check. We use these calculations to evaluate S(k,t) with arrested rotational and translational degrees of freedom, and show that the decay of S(k,t) is dominated by the spatially averaged mean-square displacement of an amino acid. The latter has been previously shown to evolve subdiffusively in time, <[u(−>)(i)(t)-u(−>)(i)(0)](2)> ~t(ν), where ν is the anomalous diffusion exponent that depends on the spectral dimension d(s) and fractal dimension d(f). As a result, for wave numbers obeying k(2)<u(−>)(2)>≳1, S(k,t) effectively decays as a stretched exponential S(k,t)≃S(k)e(-(Γ(k)t)(β)) with β≃ν, where the relaxation rate is Γ(k)~(k(B)T/mω(o)(2))(1/β)k(2/β), T is the temperature, and mω(o)(2) the GNM effective spring constant describing the interaction between neighboring amino acids. The static structure factor is dominated by the fractal character of the native fold, S(k)~k(-d(f)), with negligible to marginal influence of vibrations. The analytical expressions are first confronted with numerically based calculations on the Sierpinski gasket, and very good agreement is found between simulations and theory. We then perform PDB-GNM-based numerical calculations for a few proteins, and an effective stretched exponential decay of the dynamic structure factor is found, albeit their relatively small size. However, when rotational and translational diffusion are added, we find that their contribution is never negligible due to finite size effects. While we can still attribute an effective stretching exponent β to the relaxation profile, this exponent is significantly larger than the anomalous diffusion exponent ν. We compare our theory with recent neutron spin-echo studies of myoglobin and hemoglobin and conclude that experiments in which the rotational and translational degrees of freedom are arrested, e.g., by anchoring the proteins to a surface, will improve the detection of internal vibrational dynamics.

Dissipative curvature fluctuations in bilayer vesicles: Coexistence of pure-bending and hybrid curvature-compression modes

We have studied the relaxation dynamics of shape fluctuations in unilamellar lipid vesicles by neutron spin echo (NSE). The presence of a hybrid curvature-compression mode coexisting with the usual bending one has been revealed in the experimental relaxation functions at high q . Differently to the conventional relaxation approximately q (3) typical for bending modes, the hybrid mode was found to relax as approximately q (2) , which is compatible with a dissipation mechanism arising from intermonolayer friction. Complementary data obtained from flickering spectroscopy (FS) in giant unilamellar vesicles confirm the existence of both modes coexisting together. By combining NSE and FS data we have depicted the experimental bimodal dispersion diagram, which is found compatible with theoretical predictions for reliable values of the material parameters. From the present data two conventional dynamical methods (NSE and FS) have been shown to be suitable for measuring intermonolayer friction coefficients in bilayer vesicles.

Bimodal spectrum for the curvature fluctuations of bilayer vesicles: pure bending plus hybrid curvature-dilation modes

We study thermal undulations of giant bilayer vesicles by flickering spectroscopy. The experimental fluctuation spectra are scrutinized in view of the classical Helfrich theory. Pure bending modes are revealed to be unable to predict the large fluctuations systematically found at a high wave vector. Hybrid curvature-dilational modes are then invoked as a more efficient mode of motion in producing high curvatures. A bimodal spectrum of the thermal undulations has been theoretically developed for the shell-like topology. Reconciliation between experiments and theory is achieved when this bimodal spectrum is considered.

Crossover from stretched to compressed exponential relaxations in a polymer-based sponge phase

X-ray photon correlation spectroscopy was used to characterize the wave vector- and temperature-dependent dynamics of spontaneous thermal fluctuations in a sponge (L3) phase that occurs in a blend of a symmetric poly(styrene-ethylene/butylene-styrene) triblock copolymer with a polystyrene homopolymer. Measurements of the intermediate scattering function reveal a crossover from stretched- to compressed-exponential relaxations as the temperature is lowered from 180 to 120 degrees C.

Relationship between structural and stress relaxation in a block-copolymer melt

The relationship between structural relaxation on molecular length scales and macroscopic stress relaxation was explored in a disordered block-copolymer melt. Experiments show that the structural relaxation time, measured by x-ray photon correlation spectroscopy is larger than the terminal stress relaxation time, measured by rheology, by factors as large as 100. We demonstrate that the structural relaxation data are dominated by the diffusion of intact micelles while the stress relaxation data are dominated by contributions due to disordered concentration fluctuations.

Microrheology of polymeric solutions using x-ray photon correlation spectroscopy

We demonstrate the technique of XPCS microrheology on opaque polymeric solutions (1-20% w/w) using colloidal silica probes. The short time decay of the intensity correlation function provides the mean square displacement (MSD) of the colloidal probes. The MSDs of the probes are subsequently transformed using the generalized Stokes-Einstein equation, allowing the linear viscoelastic spectra of a biopolymer (gellan) and a synthetic polyelectrolyte (polystyrene sulfonate, PSS) to be calculated over two decades of frequency. MSDs can be measured that are two orders of magnitude smaller than those possible with video particle tracking microrheology, with a sensitivity of ∼10 nm s(-1) for displacements of ∼nms. The XPCS data for water, glycerol and PSS combs are in agreement with video particle tracking microrheology experiments performed at lower polymer concentrations.