Capillary waves at the transition from propagating to overdamped behavior

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.

Measuring capillary wave dynamics using differential dynamic microscopy

The interface between two fluids is roughened by thermally excited capillary waves. By using colloid-polymer systems which exhibit liquid-gas phase separation, the time and length scales of capillary waves become accessible to optical microscopy methods. Here, we study such a system using bright-field optical microscopy combined with a novel extension of differential dynamic microscopy. With differential dynamic microscopy, we analyze images in order to determine the decay time of interfacial fluctuations spanning wavevectors from 0.1 to 1 μm-1. We find capillary velocities on the order of 0.1 μm s-1 that depend on the sample composition in expected ways and that match values from the literature. This work demonstrates the first application of differential dynamic microscopy to the study of interfacial dynamics.

Distributed X-ray photon correlation spectroscopy data reduction using Hadoop MapReduce

Multi-speckle X-ray photon correlation spectroscopy (XPCS) is a powerful technique for characterizing the dynamic nature of complex materials over a range of time scales. XPCS has been successfully applied to study a wide range of systems. Recent developments in higher-frame-rate detectors, while aiding in the study of faster dynamical processes, creates large amounts of data that require parallel computational techniques to process in near real-time. Here, an implementation of the multi-tau and two-time autocorrelation algorithms using the Hadoop MapReduce framework for distributed computing is presented. The system scales well with regard to the increase in the data size, and has been serving the users of beamline 8-ID-I at the Advanced Photon Source for near real-time autocorrelations for the past five years.

Anisotropic Superattenuation of Capillary Waves on Driven Glass Interfaces

Metrological atomic force microscopy measurements are performed on the silica glass interfaces of photonic band-gap fibers and hollow capillaries. The freezing of attenuated out-of-equilibrium capillary waves during the drawing process is shown to result in a reduced surface roughness. The roughness attenuation with respect to the expected thermodynamical limit is determined to vary with the drawing stress following a power law. A striking anisotropic character of the height correlation is observed: glass surfaces thus retain a structural record of the direction of the flow to which the liquid was submitted.

Foundation of correlation ellipsometry

An experimental strategy for the detection of fluctuation dynamics at interfaces based on a combination of photon correlation spectroscopy (PCS) with a nulling ellipsometry scheme is investigated theoretically. The intensity description of ellipsometry measurements is generalized to PCS time correlation functions. The nulling ellipsometry procedure is applied for every lag time t of the correlation functions, to extract the dynamics connected to the coherent signal which contains the interface dynamics. The classical ellipsometry parameters Δ and tanΨ are generalized to functions and tan [capital Psi, Greek, tilde]_Q(t). A suitable Siegert relation is derived and employed to show that either field correlation functions or intensity correlation functions after baseline subtraction can be used as the starting point for the nulling ellipsometry procedure.

Frequency dispersion of small-amplitude capillary waves in viscous fluids

This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wave number of critical damping based on a harmonic-oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers the entire underdamped regime, which allows an accurate evaluation of the frequency at a given wave number, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic time scales of dispersive and dissipative mechanisms are balanced. In addition, the presented results suggest that the widely adopted hydrodynamic theory for damped capillary waves does not accurately predict the dispersion when viscous damping is significant, and an alternative definition of the damping rate, which provides consistent accuracy in the underdamped regime, is presented.

Effect of tethering on the surface dynamics of a thin polymer melt layer

The surface height fluctuations of a layer of low molecular weight (2.2k) untethered perdeuterated polystyrene (dPS) chains adjacent to a densely grafted polystyrene brush are slowed dramatically. Due to the interpenetration of the brush with the layer of "untethered chains" a hydrodynamic continuum theory can only describe the fluctuations when the effective thickness of the film is taken to be that which remains above the swollen brush. The portion of the film of initially untethered chains that interpenetrates with the brush becomes so viscous as to effectively play the role of a rigid substrate. Since these hybrid samples containing a covalently tethered layer at the bottom do not readily dewet, and are more robust than thin layers of untethered short chains on rigid substrates, they provide a route for tailoring polymer layer surface properties such as wetting, adhesion and friction.

X-ray photon correlation spectroscopy studies of surfaces and thin films

The technique of X-ray Photon Correlation Spectroscopy (XPCS) is reviewed as a method for studying the relatively slow dynamics of materials on time scales ranging from microseconds to thousands of seconds and length scales ranging from microns down to nanometers. We focus on the application of this technique to study dynamical fluctuations of surfaces, interfaces and thin films. We first discuss instrumental issues such as the effects of partial coherence (or alternatively finite instrumental resolution) and optimization of signal-to-noise ratios in the experiments. We then review what has been learned from recent XPCS studies of capillary wave fluctuations on liquid surfaces and polymer films, of nanoparticles used as probes to study the interior dynamics of polymer films, of liquid crystals and multilamellar surfactant films, and of metal surfaces, and magnetic domain wall fluctuations in antiferromagnets. We then discuss studies of non-equilibrium dynamics described by 2-time correlation functions. Finally, we briefly speculate on possible future XPCS experiments at new synchrotron sources currently under development including studies of dynamics on time scales down to femtoseconds.

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.

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.

Interfacial roughening in nonideal fluids: dynamic scaling in the weak- and strong-damping regime

Interfacial roughening denotes the nonequilibrium process by which an initially flat interface reaches its equilibrium state, characterized by the presence of thermally excited capillary waves. Roughening of fluid interfaces has been first analyzed by Flekkoy and Rothman [Phys. Rev. Lett. 75, 260 (1995)], where the dynamic scaling exponents in the weakly damped case in two dimensions were found to agree with the Kardar-Parisi-Zhang universality class. We extend this work by taking into account also the strong-damping regime and perform extensive fluctuating hydrodynamics simulations in two dimensions using the Lattice Boltzmann method. We show that the dynamic scaling behavior is different in the weakly and strongly damped case.

Tuning kinetics to control droplet shapes on chemically striped patterned surfaces

The typically elongated shape of droplets on chemically microstriped surfaces has been suggested to depend strongly on the kinetics during deposition. Here, we unequivocally establish the importance of impact kinetics by comparing the geometry of pico- to microliter droplets deposited from an inkjet nozzle with those obtained by conventional deposition from a syringe. For large Weber numbers, the strongly enhanced spreading during the impact in combination with direction-dependent pinning of the contact line gives rise to more spherical droplets with a low aspect ratio. The impact energy can be minimized by the prolonged firing of small picoliter droplets to form larger droplets or, as shown in the past, by using high-viscosity liquids. In the first case, the impact energy is absorbed by the liquid already present, therewith reducing the impact diameter and consequently forming markedly more elongated droplets.

Acceleration of membrane dynamics adjacent to a wall

The dynamics of an induced lamellar microemulsion adjacent to a planar hydrophilic surface (45 ns) were found to be three times faster compared to the bicontinuous bulk structure (133 ns). For these investigations the grazing incidence technique for neutron spin echo spectroscopy has been developed to resolve the depth dependent near surface dynamics. The observation is rationalized in terms of membrane hydrodynamics, where the flow fields reflected by the surface lead to a crossover from classical to confined fluctuations, and faster dynamics on large length scales (also known as "lubrication") are predicted.

Magnetic domain fluctuations in an antiferromagnetic film observed with coherent resonant soft x-ray scattering

We report the direct observation of slow fluctuations of helical antiferromagnetic domains in an ultrathin holmium film using coherent resonant magnetic x-ray scattering. We observe a gradual increase of the fluctuations in the speckle pattern with increasing temperature, while at the same time a static contribution to the speckle pattern remains. This finding indicates that domain-wall fluctuations occur over a large range of time scales. We ascribe this nonergodic behavior to the strong dependence of the fluctuation rate on the local thickness of the film.

Dynamics at the air-water interface revealed by evanescent wave light scattering

A new tool to study surface phenomena by evanescent wave light scattering is employed for an investigation of an aqueous surface through the water phase. When the angle of incidence passes the critical angle of total internal reflection, a high and narrow scattering peak is observed. It is discussed as an enhancement of scattering at critical angle illumination. Peak width and height are affected by the interfacial profile and the focusing of the beam. In addition, the propagation of capillary waves was studied at the surface of pure water and in the presence of latex particles and amphiphilic diblock copolymers. The range of the scattering vectors where propagating surface waves were detected is by far wider than standard surface quasi-elastic light scattering (SQELS) and comparable with those of X-ray photon correlation spectroscopy (XPCS).

Entanglement effects in capillary waves on liquid polymer films

Overdamped surface capillary wave relaxations on molten polymer films were measured using x-ray photon correlation spectroscopy. We found a transition from a single through a stretched to another single exponential regime as the temperature is decreased from well above to near the bulk glass transition temperature. A universal scaling of the dynamics was discovered over a wide range of film thicknesses, temperatures, and molecular weights (except in the multiple relaxation regime). These observations are justified by hydrodynamic theory and the time-temperature superposition principle by considering an effective viscosity instead of the bulk zero shear viscosity.

Dispersion relation of capillary waves on ionic liquids: observation of the fast overdamped mode

We measured the capillary wave spectra on the surface of ionic liquids using dynamic light scattering techniques. Empirical dispersion relations deduced from the time domain analysis clearly reveal the transition from oscillating to overdamped behavior with changing temperature and the wave vector. In particular, not only slow but also fast modes were observed in the overdamped regime. The empirical dispersion relations coincide with the theoretical dispersion relations deduced from the linearized Navier-Stokes equation when experimentally determined density, viscosity, and surface tension are assigned to the latter.

Hydrodynamics of nanoscopic capillary waves

The dynamics of nanoscopic capillary waves on simple liquid surfaces is analyzed using molecular dynamics simulations. Each Fourier mode of the surface is obtained from the molecular positions, and its time behavior compared with the hydrodynamic prediction. We trace the transition from propagating to overdamped modes, at short wavelengths. The damping rate is in very good agreement with the hydrodynamic theory up to surprisingly small wavelengths, of about four molecular diameters, but only if the wave number dependent surface tension is considered. At shorter scales, surface tension hydrodynamics break down and we find a transition to a molecular diffusion regime.

Dynamics and critical damping of capillary waves in an ionic liquid

The dynamics of thermal capillary waves (CWs) on an ionic liquid's surface are studied at the transition from propagating to overdamped CWs by x-ray photon correlation spectroscopy. The analysis considers both homodyne and heterodyne contributions, and yields excellent full line-shape experiment-theory agreement for the structure factor. The CWs' Brillouin scattering becomes extinct at a critical temperature Tc JK approximately 10 K above Tc LL , the propagating modes' hydrodynamic limit, in agreement with linear response theory. Surprisingly, the same power law applies at both Tc. The results rule out the presence of a suggested surface dipole layer.

Diffraction with a coherent X-ray beam: dynamics and imaging

Methods for carrying out coherent X-ray scattering experiments are reviewed. The brilliance of the available synchrotron sources, the characteristics of the existing optics, the various ways of obtaining a beam of controlled coherence properties and the detectors used are summarized. Applications in the study of the dynamics of speckle patterns are described. In the case of soft condensed matter, the movement of inclusions like fillers in polymers or colloidal particles can be observed and these can reflect polymer or liquid-crystal fluctuations. In hard condensed-matter problems, like phase transitions, charge-density waves or phasons in quasicrystals, the study of speckle fluctuations provides new time-resolved methods. In the domain of lensless imaging, the coherent beam gives the modulus of the sample Fourier transform. If oversampling conditions are fulfilled, the phase can be obtained and the image in the direct space can be reconstructed. The forthcoming improvements of all these techniques are discussed.

Spectra of the liquid cluster surface thermal fluctuations

Classification of cluster particles is proposed that introduces three particle types: the internal particles, surface particles, and virtual chains of particles. Thermal fluctuations of a surface passing through the surface particles of a Lennard-Jones liquid cluster are studied using a molecular dynamics simulation. It is shown that for large clusters, the Fourier spectral amplitude of these fluctuations decays faster than 1q, where q is the wave number. The frequency Fourier spectrum shows an overdamped system behavior, which is the evidence for the absence of thermal capillary waves for clusters comprising less than 10(5) particles. The time-averaged cluster density profile is given by an error function with the width parameter diverging as the logarithm of the cluster size.

The surface structure of ionic liquids: Comparing simulations with x-ray measurements

The surface-normal electron density profile of an ionic liquid, [bmim][PF6], derived from x-ray reflectivity measurements, is compared with two independent molecular-dynamics simulations. It is shown that a meaningful comparison requires a detailed accounting for both thermal and nonthermal surface roughening effects. The former is due to thermally excited capillary waves, and the latter is due to the molecular zero-point motion and form. These quantities influence very significantly, but differently, the simulated and measured density profiles. Stripping off these effects from both types of profiles yields the intrinsic structure factor of the surface. The simulated intrinsic structure factors are found to deviate considerably from the measured one. The introduction of additional ad hoc surface roughness to the simulated profiles greatly reduces the deviation, however, no physical origin for this effect can be identified. The method employed in this study should prove useful for simulation-experiment comparisons of other liquid surfaces, provided they obey capillary-wave theory, as do almost all liquid surfaces studied to date by x-ray reflectivity.

Dynamics of bulk fluctuations in a lamellar phase studied by coherent x-ray scattering

Using x-ray photon correlation spectroscopy, we studied the layer fluctuations in the lamellar phase of an ionic lyotropic system. We measured the relaxation rate of in-plane (undulation) fluctuations as a function of the wave vector. Static and dynamic results obtained during the same experiment were combined to yield the values of both elastic constants of the lamellar phase (compression and bending moduli) as well as that of the sliding viscosity. The results are in very good agreement with dynamic light-scattering data, validating the use of the technique in ordered phases.

Dynamics of fluctuations in smectic membranes

We present a comprehensive account of the dynamics of layer-displacement fluctuations in smectic liquid-crystal membranes as studied by x-ray photon correlation spectroscopy (XPCS) and neutron-spin echo (NSE). Combining these two techniques at fast relaxation times, three distinct relaxation regimes can be distinguished. For thin membranes, at the specular Bragg position oscillatory relaxation occurs, which transforms for thicker samples into exponential decay. Above a critical off-specular angle, in XPCS exponential relaxation is observed that does not depend on the scattering angle. This indicates relaxation times that are independent of the wavelength of the fluctuations. In this regime the relaxation of the fluctuations is dominated by the surface tension. Using NSE larger off-specular angles can be reached than by XPCS, for which the relaxation time decreases with the scattering angle. This regime is dominated by the bulk elasticity of the smectic membrane. The results are compared with theoretical models for the fluctuation behavior of smectic membranes, in which effects of the mosaic distribution and of the center of mass movement of the smectic membranes must be incorporated.

Fluctuation dynamics of block copolymer vesicles

X-ray photon correlation spectroscopy was used to characterize the wave-vector- and temperature-dependent dynamics of spontaneous thermal fluctuations in a vesicle (L4) 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 stretched-exponential behavior versus time, with a stretching exponent slightly larger than 2/3. The corresponding relaxation rates show an approximate q(3) dependence versus wave vector. Overall, the experimental measurements are well described by theories that treat the dynamics of independent membrane plaquettes.

Surface roughness of supercooled polymer melts

We report on in situ x-ray reflectivity measurements of the surface roughness of supercooled glass forming polymers in a temperature range from 190 to 330 K. The experimentally determined rms roughness has been found to obey the capillary wave model of a single liquid/vapor interface over the entire temperature range. An expression for the surface roughness below the bulk glass transition (T(G) approximately equal to 200 K) is deduced from the viscoelastic theory of surface fluctuations; however, no indication of a frozen-in surface roughness has been observed in the experiment. Additionally, it is shown that precise values of the surface tension of highly viscous liquids in the supercooled state can be determined by x-ray reflectivity.