# Structural Dynamics of Materials Probed by X-Ray Photon Correlation Spectroscopy

## Abstract

In this chapter we discuss coherent X-ray scattering, photon statistics of speckle patterns, and X-ray photon correlation spectroscopy (XPCS). XPCS is a coherent X-ray scattering technique used to characterize dynamic properties of condensed matter by recording a fluctuating speckle pattern. In the experiments, the time correlation function of the scattered intensity is calculated at different momentum transfers Q and thereby detailed information about the dynamics is obtained. Recently, XPCS applications have broadened to include the study of nonequilibrium and heterogeneous dynamics, e.g., in systems close to jamming or at the glass transition. This is enabled through multi-speckle techniques where a 2D area detector (CCDs or pixel detectors) is employed, and the correlation function is evaluated by averaging over subsets of equivalent pixels (same Q ). In this manner time averaging can be avoided, and the time-dependent dynamics is quantified by the so-called two-times correlation functions. Higher-order correlation functions may also be calculated to investigate questions related to non-Gaussian dynamics and dynamical heterogeneity. We discuss recent forefront applications of XPCS in the study of soft and hard condensed matter dynamics, including phase-separation dynamics of colloid-polymer mixtures, motion of Au nanoparticles at the air-water interface, dynamics of atoms in metallic crystals and glasses, and domain coarsening in phase-ordering binary alloys.

## Introduction

The impressive evolution of X-ray sources over the past half decade is best illustrated by the brilliance parameter B which expresses the number of photons per second emitted in a given solid angle in a given bandwidth per source area. Modern synchrotrons with small emittance and equipped with in-vacuum undulators can provide B = 10^{21} ph/s/(mrad)^{2}/mm^{2}/0.1 %BW or more in average brilliance. This amounts to 10^{9} times the brilliance of first-generation machines where only radiation originating from a single bending magnet was employed. The next generation of X-ray sources, based on self-amplified spontaneous emission (SASE), has recently emerged and is denoted X-ray free-electron lasers (XFEL). Due to the short duration of SASE pulses (about 10 fs), the peak brilliance is up to 10^{12} times higher than for low-emittance synchrotrons. However, being fundamentally different from storage ring sources, to a large extent XFELs require development of new experimental techniques in order to take full advantage of the increase in power.

The coherent intensity a source can deliver is proportional to B and reaches today more than 10^{10} ph/s in the hard X-ray range at forefront synchrotrons. At XFELs the average coherent flux can be up to 10^{5} times higher depending on the repetition rate. Coherent radiation is characterized by a well-defined phase relationship over the wavefront that yields a dependence of the scattered intensity on the exact spatial distribution of scatterers in the illuminated volume and results in the well-known speckle patterns. This is at marked difference with incoherent X-rays scattering where only average sample properties, integrated over the scattering volume, are accessible. This sensitivity beyond averages has lead to two new scattering techniques: X-ray photon correlation spectroscopy (XPCS) (Grübel et al. 2008; Sutton and Physique 2008) and coherent X-ray diffraction imaging (CXDI) (Robinson and Harder 2009; Miao et al. 2012). Over the years coherent scattering has become a standard technique at synchrotron radiation sources but are only in their infancy at XFELs.

In this article XPCS will be reviewed with examples of recent scientific applications. XPCS provides a direct probe of nano- and mesoscale dynamics in condensed matter and is as such an invaluable tool which ultimately could help the transition from technological progress based on observations to one based on control, where materials with specific, targeted, properties are obtained by rational design.

Many materials of profound scientific or technological interest are synthesized via nonequilibrium processes and are often trapped in (transient) out-of-equilibrium states. Thus, achieving a deeper understanding and control of systems which are (far) away from equilibrium is a very important task and probably also one of the biggest challenges in modern condensed matter physics (BESAC Report 2007). For these types of materials, effects such as “aging” and dynamical heterogeneity are becoming increasingly relevant (Berthier et al. 2011). When used in conjunction with modern, high-speed, pixel detectors and powerful X-ray sources, XPCS is ideally well suited to quantify nonequilibrium dynamics and dynamical heterogeneity, as highlighted by the science examples described in this chapter.

As an introduction, in the next section the concept of coherence applied to X-ray scattering experiments will be discussed with emphasis on XPCS.

## Coherent X-Ray Scattering

For a rigorous wave optical introduction to coherence, we refer to the chapter by I. Vartaniants in this handbook. Here we shall only add some corollary comments with special emphasis on X-ray scattering, photon statistics of speckle patterns, and XPCS.

*λ*is the wavelength,

*L*the distance from the source, D

_{h , v}the horizontal(vertical) source size, and

*Δ λ*∕

*λ*the spectral bandwidth of the radiation.

SASE radiation from an XFEL has excellent (laser-like) coherence properties in the transverse directions but poor coherence (chaotic source like) in the longitudinal direction (Geloni et al. 2010; Lee et al. 2012). The spectrum of a single pulse is characterized by several spikes which in turn vary from pulse to pulse. Hence, a single parameter-like coherence length is not sufficient to describe the longitudinal coherence properties of SASE radiation. New schemes based on seeding of the SASE process (Geloni et al. 2011; Amann et al. 2012; Allaria et al. 2012), by which the spectrum can be narrowed considerably, have been successfully developed to improve the longitudinal coherence properties, and a lot of progress is expected in this area over the coming years.

*N*

_{ c}expressed as the number of photons per coherence volume V

_{ c}, where V

_{ c}is spanned by the three coherence lengths introduced above. For a typical optical laser

*N*

_{c}≈ 10

^{7}− 10

^{8}, while \(N_{c} \approx 10^{-3} - 10^{-2}\) for a synchrotron X-ray undulator source. This enormous difference is the reason why the coherent intensity coming from a chaotic X-ray source is so much lower than that of an optical laser and the reason why coherent X-ray scattering is a real experimental challenge. Nevertheless, coherent X-ray scattering is possible at synchrotron sources because a chaotic source can be made (almost) as coherent as a laser source but

*only*at the expense of flux. The goal is to match the illuminated sample volume to the coherence volume V

_{c}, and this is achieved by an adequate beam aperture (approximately l

_{h}× l

_{v}) and by selecting a suitable bandwidth

*Δ λ*around the central wavelength

*λ*. In this manner the coherent part of the radiation is selected, and it amounts to a coherent flux of (Grübel et al. 2008; Grübel and Abernathy 1997)

_{l}; otherwise, incoherent superposition occurs and interference will be lost. In transmission scattering geometry, this leads to the condition

*θ*(or maximum momentum transfer \(Q = \frac{4\pi } {\lambda } \sin \theta\)) up to which interference effects are not smeared out. Typically, l

_{l}is on the order of 1μm if a Si(111) monochromator is employed. Assuming h > d, Eq. 4 describes well the maximum PLD up to 2

*θ*∼ 45

^{∘}, provided that the detector is far away from the sample compared with h and d. However, a large fraction of the scattering volume has a PLD which is significantly smaller, and therefore even if Eq. 4 is not strictly fulfilled, coherence experiments are still feasible but with reduced contrast.

### Time Correlation Functions

*ν*(typically 10

^{−18}s for X-rays), the coherence time

*τ*

_{0}(typically 10

^{−14}s after a monochromator), and the acquisition time T (Lengeler 2001).

*τ*

_{0}is related to the longitudinal coherence length as \(\tau _{0} = l_{l}/c\). This implies that photons from a chaotic source arrive in bunches with a characteristic duration

*τ*

_{0}and hence are governed by Bose-Einstein statistics (Loudon 1991). In practice, the measured number of photons arriving in a given time interval T obeys Poisson statistics, because no X-ray detectors today can resolve the bunch structure of a chaotic source, i.e., T ≫

*τ*

_{0}. Hence, there is no measurable difference between the temporal coherence of chaotic and unimodal sources. This observation can be quantified in terms of the normalized, first- and second-order time correlation functions of the E -field defined by (Loudon 1991)

^{∗}denotes the complex conjugate, and the brackets indicate averaging over the duration T of the acquisition. For a chaotic source g

^{(2)}is completely determined by g

^{(1)}, and the Siegert relation \(g^{(2)}(\tau ) = 1 + \vert g^{(1)}(\tau )\vert ^{2}\) holds (Siegert 1943). For a chaotic source, the first-order correlation function can be approximated by

^{(2)}(0) = 2 according to the Siegert relation. In practice T ≫

*τ*

_{0}and then g

^{(2)}(

*τ*) = 1 for

*τ*≥ T . For one mode laser light g

^{(2)}(

*τ*) = 1 for all

*τ*.

In XPCS the change in g^{(2)}(*τ*) is detected when a sample is inserted into the beam. To characterize the dynamical properties of the sample, g^{(2)}(*τ*) is calculated on I ( Q ), the scattered intensity at a given Q . XPCS is the X-ray counterpart of DLS, dynamic light scattering (Berne and Pecora 2000; Langevin 1992) that is carried out with an optical laser source. As described above, both unimodal laser light and chaotic light can be employed to measure g^{(2)}(*τ*) without influence of the different spectral properties, but the chaotic light must be made at least partially coherent to generate a speckle pattern. In practice, this means for XPCS that the beam size must be limited to ∼ l _{h} × l _{v} through which a flux of I _{c} can be expected. Also, Eq. 4 must be respected; otherwise, contrast is lost.

## Speckle Pattern

When (partially) coherent light is scattered from a disordered system, it gives rise to a random diffraction image known as a speckle pattern (Rigden and Gordon 1962; Ludwig 1988). Speckle patterns were long known from laser light scattering (Berne and Pecora 2000; Goodman 1984; Lehmann 1999), and in 1991 speckles were observed for the first time in coherent X-ray diffraction from Cu_{3}Au (Sutton et al. 1991). Since then, speckle in coherent X-ray diffraction has been observed in a variety of systems; see, for instance, Cai et al. (1994), Robinson et al. (1995), Abernathy et al. (1998), Madsen et al. (2005), and Livet and Sutton (2012).

**Q**is the scattering vector and f

_{n}(

**Q**) is the scattering amplitude of the n th scatterer located at the position

**r**

_{ n }( t ) at time t . The sum is taken over scatterers in the coherence volume which is spanned by the transverse and longitudinal coherence lengths defined in the previous section. For clarity we omit the polarization correction, the Lorentz factor, and the Thomson scattering length

*r*

_{0}

^{2}in Eq. 8 and assume that the scattering volume equals the coherence volume V

_{c}.

An intensity measurement will naturally be a time average 〈 I ( Q , t )〉_{T} taken over the acquisition time T , and if the system is non-ergodic, i.e., has static random disorder, 〈 I ( Q , t )〉_{T} will display, as a function of Q , distinct and sharp (angular size ≈ *λ* ∕ beam size) variations in intensity, known as *speckle*. If, on the other hand, the system is ergodic, with timescales of fluctuations very short compared to T , the measured time average is equivalent to the ensemble average and 〈 I ( Q , t )〉_{T} can be replaced by a spatial ensemble average 〈 I ( Q , t )〉. The observed scattering is then only carrying signatures of the time-averaged correlations (e.g., structure factor) in the sample, similar to what is found in scattering experiments with incoherent radiation. XPCS quantifies the fluctuations in speckle intensity vs. time by calculating g^{(2)}(*τ*). However, a static speckle pattern, through its photon statistics, can reveal a lot about the coherence properties of the setup and hence about the possibility of performing a successful XPCS measurement.

### Photon Statistics of Speckle Patterns

_{n}and the phases

**Q**⋅

**r**

_{ n }in Eq. 8 are statistically independent, and if the phases are evenly distributed over 2

*π*(perfectly random scatterer). The probability distribution P ( I ) of the intensity is then given by

*Γ*( M ) is the gamma function. This gamma distribution has a mean 〈 I 〉 and \(\sigma ^{2} =\langle I\rangle ^{2}/M\), and hence \(\beta = 1/M\), and describes well the statistics of partially coherent speckle patterns with good counting statistics. In a first approximation M can be thought of as the ratio between the scattering volume and the coherence volume. The limiting case of 100 % contrast (

*β*= 1) then corresponds to the case where the scattering volume equals V

_{c}. Loss in contrast ( M > 1) can also happen as a result of too large camera pixels (larger than the speckle size) resulting in a smearing of the speckle pattern. The effect on the photon statistics is equally well described by Eq. 10, so M can be ascribed to loss of coherence both on the incident side (beam) and on the exit side (detector).

*…*photons per pixel, the gamma distribution must be convoluted with the Poisson distribution \(\langle k\rangle ^{k}\exp (-\langle k\rangle )/k!\), and one obtains the so-called Poisson-gamma distribution (aka negative binomial distribution) for integer values of k (number of photons):

An intensity distribution following Poisson-gamma statistics can be very difficult to distinguish from pure Poisson noise, particularly when the number of photons ( k ) is low and the contrast is weak (large M ). Nevertheless, Eq. 11 has been used to quantify the coherence of weak speckle patterns (Hruszkewycz et al. 2012), but it requires that the images are carefully post-processed in order to assign correct photon numbers k to the individual pixels. Charge sharing in a CCD can in some cases be corrected for by a droplet algorithm (Livet et al. 2000; Chushkin et al. 2012), and this is an important ingredient in a successful application of Eq. 11 to determine M .

Figure 2a shows a static speckle pattern taken on a 1 % vol. suspension of spheres that is cooled below the glass transition temperature of the solvent. The image was recorded with a direct illumination deep-depletion CCD camera from Princeton Instruments (22.5 μm pixel size) at the ID10 beamline of ESRF with 8.02 keV X-ray energy, a sample-detector distance of 2.4 m and a beam size of ∼ 10 μm. 200 CCD images were acquired over ≈ 1,000 s, and the speckles were immobile during that period. This is illustrated by the waterfall plot (Fig. 2b) where the vertical stripes indicate persisting speckle intensities. An analysis of the photon statistics on a subset of pixels at Q = 2 . 3 × 10^{−2} nm^{−1} yields M = 2 . 63 (blue data and red curve of Fig. 2c), in excellent agreement with the 38 % (1/2.63) contrast that is observed when a multi-speckle time correlation function g^{(2)}(*τ*) is calculated on the same ensemble of pixels (inset of Fig. 2c). g^{(2)}(*τ*) is constant, confirming that the sample is completely static over 100 frames (500 s).

## X-Ray Photon Correlation Spectroscopy

^{(2)}( Q ,

*τ*) defined in Eq. 6 and

*τ*) is the

*intermediate scattering function*(Berne and Pecora 2000) (similar to g

^{(1)}, the first-order correlation function; see Eq. 5). The intermediate scattering function reflects the time dependence of the density (electronic) correlations in the sample and is related to the temporal Fourier transform of the dynamic structure factor S (

*q*,

*ω*), a well-known quantity that is measured in the frequency (energy) domain, for instance, by inelastic scattering techniques.

The second-order correlation functions can be measured by coupling a point detector (providing high quantum efficiency and small dead time) to a digital autocorrelator device giving online access to g_{2}( Q , *τ*) over a wide range of correlation times (depending on the detector speed, Sikharulidze et al. 2002) at a single Q -value (Fig. 3, Seydel et al. 2003). Alternatively, two-dimensional position-sensitive detectors with appropriate spatial resolution (e.g., a deep-depletion, direct illuminated CCD camera with small pixel size and high quantum efficiency) cover a broad range of Q -values and allow recording a complete speckle pattern. Correlation functions are then calculated for each Q -value by ensemble averaging over equivalent pixels (multi-speckle XPCS, Cipelletti and Weitz 1999; Lumma et al. 2000). The availability of many equivalent pixels permits to calculate g^{(2)} without performing a time averaging, something we shall discuss further in the science examples in the next chapter. However, CCD detectors are subject to limitations in readout speed for large frames and have been used up to now for the study of slow dynamics. It is possible to use only a part of the CCD chip to record the image and utilize the remaining part for fast frame transfer (known as kinetic mode, Lumma et al. 2000). More recently, high-speed pixelated 2D detectors (Ponchut et al. 2011; Radicci et al. 2012) with low noise have appeared bringing the dead time for multi-speckle analysis down into the ms range. Several examples of recent XPCS applications can be found in Madsen et al. (2010) and the references therein.

### Signal-to-Noise Ratio

^{(2)}is calculated by an ensemble averaging over equivalent pixels of the detector, e.g., pixels corresponding to the same momentum transfer within some boundaries

*Δ*Q . This approach greatly improves the signal-to-noise ratio (SNR) compared to the case of a point detector. The SNR in sequential multi-speckle XPCS is approximately given by (Lumma et al. 2000; Falus et al. 2004)

*N*is the number of detector pixels, each of size a × a , m

_{b}is the number of time bins, 〈 I 〉 is the average number of photons registered by a pixel during the time bin, and

*β*is the optical contrast. This formula has the important consequence that one order of magnitude more intensity gives access to two orders of magnitude faster times in XPCS. This is easily realized by noting that \(\langle I\rangle = I_{0}\tau _{m}(a/R)^{2}\), where I

_{0}is the total scattered intensity (photons/s, depends on the sample and the incident flux) at the given momentum transfer Q , ( a ∕ R )

^{2}is the solid angle extended by one pixel with R as the sample-detector distance, and

*τ*

_{m}is the binning time. The total acquisition time is T = m

_{b}

*τ*

_{m}, and hence we get for the SNR

Hence, if I _{0}, for instance, is increased by a factor of 10, *τ* _{m} can be decreased by a factor of 100 keeping the same SNR. In general, the higher the number of counts per pixel, the better the SNR, but there are important limitations to this simple picture. Indeed, the parameter ( a ∕ R )^{2} suggests that large a and small R is increasing the SNR. On the other hand, for a given *Δ* Q ∕ Q , *N* becomes larger if the detector pixels are smaller and the detector is placed further downstream favoring large R and small a . The optical contrast *β* also plays a role for the SNR and points in the same direction where, in order to maximize *β*, a small a and large R is required.

*β*in the experiment, assuming a fully coherent beam of size d × d, is given by the complex degree of coherence convoluted with the detector resolution and can be written as (Lumma et al. 2000)

*λ*is the wavelength. For maximizing the optical contrast, a small w is favored, i.e., both a small beam size d and small pixel size a is better. The dependence of

*β*on w is shown in Fig. 4a. In SAXS geometry, the number of pixels in an annulus on the detector centered on the direct beam and with radius and width defined by Q and

*Δ*Q ∕ Q is

_{0}is proportional to d

^{2}and S ( Q ), the scattering factor of the sample, and we find from Eq. 13

We note the trivial dependence on S ( Q ), typically S ( Q ) ∝ Q ^{− n}, and the linear gain in SNR with d. Moreover, the SNR scales with *λ* ^{2}. The last part of Eq. 17, \(w\beta /\sqrt{1+\beta }\), is shown in Fig. 4b and suggests a maximum in SNR at w ≃ 6. This means that \(a/R \simeq \lambda /d\), i.e., pixel size equals the speckle size, is the optimum working condition in order to maximize the SNR (Madsen 2011).

### Diffusion Dynamics

*τ*) is related to the static structure factor S ( Q ) of the sample and

*N*identical particles in the illuminated volume. The simplest possible dynamics is Brownian motion (Stokes-Einstein free diffusion) of such

*N*particles (Berne and Pecora 2000; Pusey 1991). In the absence of any interactions between particles, S ( Q ) = 1 and all the cross terms (i ≠ j ) in Eq. 18 average out to zero. The mean square value of the displacement of a free Brownian particle is

_{0}denotes the free diffusion coefficient of a particle with radius R

_{ p }in a medium with viscosity

*η*and

Hence, free diffusion is a direct probe of the solvent properties, and, for instance, this has been used in XPCS microrheology experiments to quantify the frequency-dependent viscoelastic moduli of gels and polymer solutions (Papagiannopoulos et al. 2005).

*τ*) (Segre and Pusey 1997). A useful quantity is the initial ( t → 0) slope

*Γ*( Q ), also denoted the first cumulant, of the measured intermediate scattering function f( Q , t ). It can be shown that

## Science Examples

In this section we describe applications of XPCS in studies of condensed matter dynamics with particular emphasis on out-of-equilibrium or aging dynamics in systems close to jamming or the glass transition, or where a phase transition is triggered. Examples include phase-separation dynamics of colloid-polymer mixtures, motion of Au nanoparticle trapped at the air-water interface, dynamics of metallic glasses, atomic diffusion in alloys, and coarsening dynamics of ordered domains in a phase-ordering binary alloy.

### Two-Times Correlation Functions and Dynamics of Phase-Ordering Systems

While several XPCS measurements have focused on studying equilibrium fluctuation dynamics, much fewer attempts have been made to use it as probe of phase transition dynamics. Understanding how long-range order develops out of a disordered state has been studied intensively in materials science and condensed matter physics for decades. The time evolution of mesoscale ordering has especially been the subject of much investigation, but most of the earlier studies, however, have focused on an average description of the process. Here, two XPCS results (Fluerasu et al. 2005; Ludwig et al. 2005) are described where high brilliance synchrotron sources (Advanced Photon Source and European Synchrotron Radiation Facility) enabled the study of ordering dynamics in metallic alloys undergoing first-order phase transitions.

As an ordered state is approached, the time dependence of the relevant length scale R (i.e., the average ordered domain size) follows a power law of the form R ( t ) ∝ t ^{ n}. This scaling behavior can be probed with conventional time-resolved X-ray scattering (Shannon et al. 1992) and applies to a wide range of systems where the microscopic details are different. In general, for first-order transitions in nonconservative systems (such as the ordering in Cu-Au and Cu-Pd alloys studied in Fluerasu et al. (2005) and Ludwig et al. (2005)), the exponent is 1 ∕ 2 (referred to as Model A), whereas for conservative systems such as unmixing in Al-Li alloys (Livet et al. 2001), the exponent is found to be 1/3 (Model B). This scaling describes well the average behavior – i.e., the kinetics – of nonequilibrium systems undergoing phase ordering, but the nature of the dynamical mechanisms and the fluctuations can be quantified only using coherent scattering and XPCS.

In Fluerasu et al. (2005), a specimen of the alloy Cu_{3}Au was chosen for the ordering experiments, as its kinetics is well understood. At high temperatures the alloy has a fcc structure with each site randomly occupied by either copper or gold atoms according to the stoichiometry. When the temperature drops below a critical temperature ( T _{C} ∼ 383 ^{o}C), the alloy exhibits an ordering transition in which the gold atoms occupy the unit cell corners while the copper atoms occupy the face sites. However, since there is a fourfold ambiguity in choosing the “corner,” the ordered state is fourfold degenerate. If the alloy in the disordered state is quenched below T _{C}, ordered regions emerge and grow in any of the four possible states. Eventually, these ordered regions occupy the entire volume of the quenched alloy. At this stage, the domain dynamics enter a slow coarsening regime where the average domain size grows following the power law model with exponent 1 ∕ 2. The underlying mechanism leading to this behavior is dynamic scaling, where correlation times become larger with time – i.e., fluctuations slow down as the system becomes more ordered. This phenomenon was theoretically investigated by Brown et al. (1997) who have used analytical Langevin theory and simulations to examine the behavior of fluctuations quantitatively characterized by *two-times correlation functions* during the coarsening regime.

**q**, t is the normalized intensity fluctuation defined as

Here denotes an ensemble average which is usually calculated by averaging over the fluctuating speckle intensity over a region of “equivalent pixels” on the area detector. This region is typically chosen in the diffuse region around a superlattice peak which is characteristic of the ordered phase. Figure 5a illustrates the growth of the (100) superlattice peak in Cu_{3}Au during the ordering process. Figure 5b shows an example of a two-times correlation function measured in the vicinity of the (100) superlattice reflection. The natural variables for analyzing two-times correlation functions are the average time, which is measured by the distance along the diagonal ( t _{1} = t _{2}) and can be defined by \(\overline{t} = \frac{t_{1}+t_{2}} {2}\), and the time difference \(\delta t = \left \vert t_{1} - t_{2}\right \vert \), which is measured by the distance from the diagonal.

The XPCS experiments on Cu_{3}Au and CuPd showed that the correlation times associated with mesoscale fluctuations increase over time as the solid becomes more ordered. This is evident from Fig. 5b where the contour lines corresponding to equal degree of correlation move further away from the diagonal as time \(\overline{t}\) goes by. Such “broadening” of the two-times plot as one moves from the beginning (lower left) to the end (upper right) of the measurement is a unique fingerprint of dynamical slowdown in the system. From Fig. 5b it is evident that the slowdown mostly happens in the beginning of the measurement (from 0 to ∼ 200 min.), while at later times the changes are much smaller. More specifically, as predicted by the theoretical results of Brown et al. (1997), the fluctuations follow a dynamical scaling law in which the increase in correlation times is described by a power law that was observed to crossover from linear in time to an exponent of 1 ∕ 2 after about 200 min. Such studies reveal the dynamics by which the disordered state gradually disappears as ordered regions grow and coarse due to domain wall dynamics.

### Studies of Nonequilibrium Dynamics and Aging in Colloidal Systems

Aggregation and gelation phenomena in colloidal suspensions are processes of high fundamental interest in soft condensed matter physics. Due to a combination of attractive forces between individual colloids and the overall particle density, percolating networks of particles spanning the entire sample volume can be formed. A number of recent XPCS results have focused on studies related to aging phenomena in various colloidal systems including colloidal gels and glasses; see, e.g., Madsen et al. (2010) and references therein. Delayed sedimentation is a nonequilibrium phenomenon encountered in a variety of colloidal suspensions with sufficiently strong short-ranged attractive interactions, including emulsions, gels, and creams. Due to the attractive potential, the colloidal particles can aggregate at relatively low concentrations (e.g., 20 % volume fraction) to form a gel. However, for some systems this load-bearing structure slowly evolves, until spatial connectivity is lost and the gel suddenly collapses followed by a subsequent sedimentation and phase separation (Pham et al. 2002).

*τ*is the relaxation time (relaxation rate \(\varGamma = 1/\tau\)) and

*γ*is the KWW exponent. The fitting parameters obtained during the delayed sedimentation process are both age ( t

_{a}) and Q -dependent and are shown in Fig. 6 for a single wave vector Q corresponding to a value of Q R = 3. After an initial stage during which the correlation times increase exponentially with age (a), a new region with large fluctuations and a linear increase in the correlation times is entered (b). Such behavior has been observed previously in different systems, and a universal model for aging in soft matter was proposed (Cipelletti et al. 2003). A remarkable crossover from

*γ*< 1 to

*γ*> 1 can be observed during the early stages of the process. Compressed exponential relaxations (

*γ*> 1) have been found in putative jammed states, and the measurements may indicate a jamming transition occurring during the initial stages of the delayed sedimentation process.

### Higher-Order Correlation Functions

In order to further quantify the dynamics of heterogeneous systems, higher-order correlation functions can be calculated. A recent XPCS example investigates the dynamics of a bidimensional (2D) gel formed by a Langmuir monolayer of gold nanoparticles by calculation of the fourth-order time correlation function \(g^{(4)}(t,\tilde{\tau })\) (Orsi et al. 2012). The gold nanoparticles are 7 nm in diameter and trapped at the air-water interface where the interparticle potential is weakly attractive due to hydrophobic interactions. The surface coverage fraction *Φ* is obtained by SEM on transferred monolayers and by in situ null ellipsometry.

The surface dynamics can be probed by XPCS in grazing incidence geometry as demonstrated in previous experiments (Madsen et al. 2005; Seydel et al. 2003; Madsen et al. 2010). The X-ray beam was incident on the surface at an angle of *α* ∼ 0 . 1^{o} which is below the angle for total external reflection. This enhances the sensitivity to the near-surface region with a penetration depth of the evanescent wave of same magnitude as the particle diameter. Hence, only particles at the interface contribute to the scattering. The scattered radiation is measured by a 2D detector (MAXIPIX) consisting of 256 × 256 pixel with 55 × 55 μm^{2} area each. The two-times correlation functions can be calculated as a function of *q* _{| |}, the component of the momentum transfer parallel to the surface by grouping pixels with the same *q* _{| |} and performing a spatial averaging according to the multi-speckle XPCS method and Eq. 24. The dynamics is not aging; therefore, a time averaging of the two-times correlation function is allowed and performed to obtain the usual one-time correlation function g^{(2)}( t ). The correlation functions all obey the compressed exponential form (Eq. 26) with *γ* = 1 . 5 which is a genuine feature of stress-relaxation dynamics (Bouchaud and Pitard 2001).

_{1}, t

_{2}) is shown in Fig. 7a. A calculation of the variance

*χ*of C ( t

_{1}, t

_{2}) along paths parallel to the diagonal of C ( t

_{1}, t

_{2}) in Fig. 7a was performed. The normalized variance (

*χ*divided by the optical contrast

*β*with \(\beta = C(0,0) - 1\)) is plotted in Fig. 7b as a function of the distance \(\tilde{\tau }= \vert t_{1} - t_{2}\vert \) from the diagonal (lag time). The fact that the variance is nonzero and features a peak vs. \(\tilde{\tau }\) indicates that there is temporal heterogeneity in the dynamics. In other words, the correlation function varies as a function of time and the next step is to calculate the

*correlations of the correlations*, also known as the fourth-order correlation function g

^{(4)}, to determine the timescale of the temporal heterogeneity. The fourth-order correlation function is given by

^{(2)}obtained by time averaging of the two-times correlation function. g

^{(4)}decays faster than g

^{(2)}and displays a peak-like shape rather than a monotonous exponential-like decay. For Gaussian fluctuations the fourth-order correlation function is uniquely determined by g

^{(1)}and will mimic an exponential falloff. Hence, the fact that g

^{(4)}features a peak is a clear sign of non-Gaussian behavior of the dynamics.

In Fig. 7d the time *τ* ^{∗} corresponding to the position of the peak in g^{(4)} is plotted together with the relaxation time *τ* obtained by fitting g^{(2)} with Eq. 26, as indicated by the black line in Fig. 7c. One observes that both *τ* ^{∗} and *τ* are proportional to *q* _{| |} ^{−1} but that *τ* ^{∗} ≪ *τ*. Since *τ* ^{∗} is the characteristic time of g^{(4)}, it provides the timescale of fluctuation of the two-times correlation function, also known as the timescale of dynamical heterogeneity. The fact that *τ* ^{∗} ≪ *τ* indicates that the dynamical heterogeneity is characterized by very rapid variations, probably due to the presence of faster non-Gaussian processes which are outside the window of direct detection in the present experiment. In this sense the fluctuations of the correlation function, characterized by g^{(4)}, give hints that there is more to the picture than meets the eye in the two-times plot of Fig. 7a.

### Atomic Dynamics in Hard Condensed Matter

Investigation of dynamics in hard condensed matter at interatomic distances is possible by XPCS, but the studies have been difficult due to insufficient coherent X-ray flux. However, in recent years such experiments became routine for the study of slow dynamics thanks to improvements of third-generation synchrotrons and a more efficient collection of sparse photon signals at wide scattering angles by area detectors (Chushkin et al. 2012; Leitner 2012), covering wave vectors *q* up to \({\sim}4\,\AA^{-1}\). One such experiment was performed by Leitner and coworkers who studied the mechanism responsible for atomic diffusion in a single crystal binary alloy (Leitner et al. 2009).

Atomic diffusion, that is, the migration of atoms from site to site in a lattice, is of crucial importance in many processes occurring in materials, like precipitation and oxidation. Diffusion in solids is usually probed by radiotracer techniques at the macroscopic level and with quasi-elastic neutron scattering or Mößbauer spectroscopy at the atomic scale (Vogl and Sepiol 2005). However, the latter microscopic techniques can be applied only to a limited number of isotopes providing information about random atomic movements in an indirect way, through the evolution of the tracer’s dynamics. XPCS has the advantage that it measures the stochastic diffusive motion of the atoms independently of atomic species or isotopes and can be applied to any material. In addition, measurements of the dynamics along different crystallographic directions give direct information about the mechanisms that control the evolution of lattice site occupancy by atoms of different kinds (Leitner 2012).

_{90}Au

_{10}substitutional alloy at T = 543 K. These data correspond to the smallest bulk diffusivity \(\tilde{D} =\lim _{q\rightarrow 0}\tau (\mathbf{q})^{-1}q^{-2} \sim 10^{-24}\,\mathrm{m}^{2}\,\mathrm{s}^{-1}\) ever measured in a solid at the atomic level. The evolution of

*τ*(

**q**) can be modeled considering both a geometrical contribution due to nearest-neighbor jumps and the short-range ordering, i.e., the influence of neighboring sites on the motion of each atom. The model predicts (Leitner et al. 2009; Sinha and Ross 1988)

*τ*

_{0}is the average time between atomic jumps,

**s**

_{i}are the possible jump vectors with the corresponding probabilities

*p*

_{i}, and I

_{SRO}is the short-range order intensity. The latter parameter contains information about the modulation of diffuse scattered intensity between Bragg peaks, and it affects the dynamics leading to a slowdown at the most energetically favorable configurations, similar to the de Gennes narrowing observed in coherent neutron scattering experiments in liquids (de Gennes 1959).

The agreement between the model and the experimental data in Fig. 8 demonstrates the importance of short-range ordering in the dynamics and opens up the possibility to study a vast range of materials, including polycrystalline and single crystals (Stana et al. 2013).

Atomic-scale XPCS is also a powerful tool to probe out-of-equilibrium dynamics in structural glasses. Indeed, due to the extreme difficulty of investigating the structural relaxation in a glass far below T _{g} in both experiments and simulations, only very little is known about microscopic dynamics in the glassy state (Angell et al. 2000; Berthier and Biroli 2011). Measurements on metallic glasses have revealed the existence of structural rearrangements at the atomic level, contrary to the common expectation of a completely arrested state (Ruta et al. 2012, 2013a,b; Leitner et al. 2012; Ruta et al. 2014a). The left panel of Fig. 9 shows normalized intensity correlation functions measured by XPCS upon cooling a metallic glass former from the supercooled liquid phase down to the glassy state. Upon decreasing the temperature, the dynamics slows down while always displaying a full decorrelation of g^{(2)} to unity. This means that even being out of equilibrium, the glass can rearrange its structure on the atomic level. In addition, the glass transition is found to be accompanied by a marked equilibrium/out-of-equilibrium dynamical crossover where the correlation functions evolve from the typical stretched exponential form (thus described by the KWW model function with a shape exponent *γ* < 1 (see Eq. 26)), in the supercooled liquid, to a compressed decay (thus described by a shape exponent *γ* > 1) in the glass (Ruta et al. 2012, 2013a).

_{w}, and high cooling rates, where

*τ*

_{0}( T ) is the initial value at t

_{w}= 0 of the relaxation time when a given temperature is reached, and

*τ*

^{∗}is the growth rate parameter which is ∼ 6,000–8,000 s for rapidly quenched metallic glasses (see Fig. 9, right panel). For partially annealed system, or after long waiting times, the fast regime abruptly converts to a stationary state where the correlation functions do not evolve with time, at least not on the experimental timescale of several hours.

The observed relaxation dynamics of metallic glasses cannot be explained with the current theories for glasses (Berthier and Biroli 2011; Lubchenko and Wolynes 2004) and suggests a ballistic rather than diffusive atomic motion as in the case of many complex soft materials, such as polymers, colloidal suspensions, emulsions, and gels (Madsen et al. 2010; Fluerasu et al. 2007; Orsi et al. 2012; Cipelletti et al. 2003) also discussed in this chapter. Similar to those cases, the dynamics is possibly related to the presence of internal stresses captured in the system when it is driven out of equilibrium (Ruta et al. 2014a; Yavari 2005).

Albeit at first glance the results on metallic glasses would strengthen the idea of a universal physical mechanism responsible for out-of-equilibrium dynamics in complex materials, a very different scenario emerges in investigations of other systems, like network glasses.

A recent XPCS study of the atomic dynamics in sodium silicate glasses reports the existence of atomic motion in the glassy state, with relaxation times of the order 10^{3} s, even well below T _{g} (Ruta et al. 2014b). In contrast with metallic glasses, the dynamics is characterized by a stretched exponential decay (*γ* < 1) of the correlation functions and the observed timescales are close to literature values of the corresponding supercooled liquid. There is no aging of the dynamics which again is different from the results on metallic glasses and from the usual behavior observed at macroscopic scales in glasses (Angell et al. 2000). A wave vector dependence study shows that the dynamics in network silicate glasses is strongly affected by the amorphous structure with a marked increase of relaxation time at Q -vectors corresponding to the structural pre-peak that is associated with sodium diffusion channels (Ruta et al. 2014b). This is in agreement with numerical simulations of molten silicates (Horbach et al. 2002). A second smaller increase of *τ* is also observed at the Si-O nearest-neighbor distance, again reminiscent of the de Gennes narrowing in liquids (de Gennes 1959).

## Conclusion and Outlook

In this chapter we have described the use of XPCS to study complex dynamics in a variety of different condensed matter systems. The development since the early experiments 20 years ago is impressive both regarding the breadth of applications and the details that can be quantified. The current limitations of the technique are mostly due to the 2D detectors available (speed, size, quality) and the coherent flux offered by forefront X-ray sources. Novel diffraction-limited synchrotron sources are under construction, and it is anticipated that XPCS can benefit from the increase in brilliance hence taking a leap forward toward investigation of milli- and microsecond dynamics even in weakly scattering systems. Free-electron lasers with their unprecedented brilliance (both average and peak) will bring investigations of meso- and atomic-scale dynamics into the femto- to nanosecond region, and also here XPCS can benefit enormously. However, new methods to quantify the correlations must be developed due to the pulsed nature of free-electron lasers (Grübel et al. 2007). Particularly, the combinations of XPCS with pump-probe experimental schemes, multicolor, and split-delay techniques at XFELs will be exciting and are expected to flourish over the next decade.

## References

- D.L. Abernathy et al., J. Sync. Rad.
**5**, 37 (1998)CrossRefGoogle Scholar - E. Allaria et al., Nat. Photonics
**6**, 699 (2012)CrossRefADSGoogle Scholar - J. Amann et al., Nat. Photonics
**6**, 693 (2012)CrossRefADSGoogle Scholar - C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, J. Appl. Phys.
**88**3113 (2000)CrossRefADSGoogle Scholar - B.J. Berne, R. Pecora,
*Dynamic Light Scattering*(Dover, Mineola, 2000)Google Scholar - L. Berthier, G. Biroli, Rev. Mod. Phys.
**83**, 587 (2011)CrossRefADSGoogle Scholar - L. Berthier, G. Biroli, J.-P. Boucahaud, L. Cipelletti, W. Van Saarlos,
*Dynamical Heterogeneities in Glasses, Colloids and Granular Media*(Oxford University Press, Oxford, 2011)CrossRefGoogle Scholar - BESAC Report, The New Era of Science: Directing Matter and Energy: Five Challenges for Science and the Imagination, US Department of Energy (2007)Google Scholar
- J.P. Bouchaud, E. Pitard, Eur. Phys. J. E
**6**, 231 (2001)CrossRefGoogle Scholar - G. Brown, P.A. Rikvold, M. Sutton, M. Grant, Phys. Rev. E
**56**6601 (1997)CrossRefADSGoogle Scholar - Z.H. Cai et al., Phys. Rev. Lett.
**73**, 82 (1994)CrossRefADSGoogle Scholar - Y. Chushkin, C. Caronna, A. Madsen, J. Appl. Cryst.
**45**, 807 (2012)CrossRefGoogle Scholar - L. Cipelletti, D.A. Weitz, Rev. Sci. Instr.
**70**, 3214 (1999)CrossRefADSGoogle Scholar - L. Cipelletti et al., Faraday Discuss. Chem. Soc.
**123**, 237 (2003)CrossRefADSGoogle Scholar - P.G. de Gennes, Physica
**25**, 825 (1959)CrossRefADSGoogle Scholar - P. Falus, M.A. Borthwick, S.G.J. Mochrie, Rev. Sci. Instr.
**75**, 4383 (2004)CrossRefADSGoogle Scholar - A. Fluerasu, M. Sutton, E. Dufresne, Phys. Rev. Lett.
**94**, 055501 (2005)CrossRefADSGoogle Scholar - A. Fluerasu, A. Moussaid, A. Madsen, A. Schofield, Phys. Rev. E
**76**, 0100401(R) (2007)Google Scholar - G. Geloni et al., New J. Phys.
**12**, 035021 (2010)CrossRefADSGoogle Scholar - G. Geloni, V. Kocharyan, E. Saldin, J. Mod. Opt.
**58**, 1391 (2011)CrossRefADSGoogle Scholar - J.W. Goodman, in
*Laser Speckle and Related Phenomena*, ed. by J.C. Dainty (Springer, Berlin, 1984), pp. 9–74; J.W. Goodman,*Speckle Phenomena in Optics: Theory and Applications*(Roberts and Company, Greenwood Village, 2007)Google Scholar - G. Grübel, D.L. Abernathy, SPIE
**3154**, 103 (1997)CrossRefADSGoogle Scholar - G. Grübel, G.B. Stephenson, C. Gutt, H. Sinn, T. Tschentscher, Nucl. Instrum. Methods Phys. Res. B
**262**, 357 (2007)CrossRefADSGoogle Scholar - G. Grübel, A. Madsen, A. Robert, in
*Soft-Matter Characterization*, ed. by R. Borsali, R Pecora (Springer, New York, 2008), pp. 935–995Google Scholar - J. Horbach, W. Kob, K. Binder, Phys. Rev. Lett.
**88**, 125502 (2002)CrossRefADSGoogle Scholar - S.O. Hruszkewycz et al., Phys. Rev. Lett.
**109**, 185502 (2012)CrossRefADSGoogle Scholar - D. Langevin (ed.),
*Light Scattering by Liquid Surfaces and Complementary Techniques*(Marcel Dekker, New York, 1992)Google Scholar - S. Lee et al., Opt. Express
**20**, 9790 (2012)CrossRefADSGoogle Scholar - P. Lehmann, Appl. Opt.
**38**, 1144 (1999)CrossRefADSGoogle Scholar - M. Leitner,
*Studying Atomic Dynamics with Coherent X-rays*(Springer, Heidelberg, 2012)CrossRefGoogle Scholar - M. Leitner, B. Sepiol, L.-M. Stadler, B. Pfau, G. Vogl, Nat. Mater.
**8**, 717 (2009)CrossRefADSGoogle Scholar - M. Leitner, B. Sepiol, L.-M. Stadler, B. Pfau, Phys. Rev. B
**86**, 064202 (2012)CrossRefADSGoogle Scholar - B. Lengeler, Naturwissenschaften
**88**, 249 (2001)CrossRefADSGoogle Scholar - F. Livet, M. Sutton, C. R. Phys.
**13**, 227 (2012)CrossRefADSGoogle Scholar - F. Livet et al., Nucl. Instrum. Methods Phys. Res. Sect. A,
**451**, 596 (2000)Google Scholar - F. Livet et al., Phys. Rev. E
**63**36108 (2001)CrossRefADSGoogle Scholar - R. Loudon,
*The Quantum Theory of Light*, 2nd edn. (Oxford Science, Oxford, 1991)Google Scholar - V. Lubchenko, P.G. Wolynes, J. Chem. Phys.
**121**, 2852 (2004)CrossRefADSGoogle Scholar - K.F. Ludwig, Phys. Rev. Lett.
**61**, 1526 (1988)CrossRefADSGoogle Scholar - K. Ludwig et al., Phys. Rev. B
**72**, 144201 (2005)MathSciNetCrossRefADSGoogle Scholar - D. Lumma, L.B. Lurio, S. Mochrie, M. Sutton, Rev. Sci. Instr.
**71**, 3274 (2000)CrossRefADSGoogle Scholar - A. Madsen, Conceptual Design Report: Scientific Instrument MID, XFEL.EU TR-2011-008 (2011). http://dx.doi.org/10.3204/XFEL.EU/TR-2011-008
- A. Madsen, T. Seydel, M. Tolan, G. Grübel, J. Sync. Rad.
**12**, 786 (2005)CrossRefGoogle Scholar - A. Madsen, R.L. Leheny, H. Guo, M. Sprung, O. Czakkel, New J. Phys
**12**, 055001 (2010)CrossRefADSGoogle Scholar - J. Miao, R.L. Sandberg, C. Song, Sel. Top. Quant. Elect., IEEE
**18**, 399 (2012)Google Scholar - D. Orsi, L. Cristofolini, G. Baldi, A. Madsen, Phys. Rev. Lett.
**108**, 105701 (2012)CrossRefADSGoogle Scholar - A. Papagiannopoulos, T.A. Waigh, A. Fluerasu, C. Fernyhough, A. Madsen, J. Phys.: Condens. Matter
**17**, L279 (2005)ADSGoogle Scholar - K.N. Pham et al., Science
**296**, 104 (2002)CrossRefADSGoogle Scholar - C. Ponchut et al., JINST
**6**, C01069 (2011)CrossRefADSGoogle Scholar - P.N. Pusey, in
*Liquids, Freezing and Glass Transition*, ed. by J.P. Hansen, D. Levesque, J. Zinn-Justin, Les Houches Session LI (Elsevier, Amsterdam, 1991), p. 763–942Google Scholar - V. Radicci et al., JINST
**7**, 02019 (2012)CrossRefGoogle Scholar - J.D. Rigden, E.I. Gordon, Proc. IRE
**50**, 2367 (1962)Google Scholar - I. Robinson, R. Harder, Nat. Mater.
**8**, 291 (2009)CrossRefADSGoogle Scholar - I. Robinson et al., Phys. Rev. B
**52**, 9917 (1995)CrossRefADSGoogle Scholar - B. Ruta et al., Phys. Rev. Lett.
**109**, 165701 (2012)CrossRefADSGoogle Scholar - B. Ruta et al., AIP Conf. Proc.
**1518**, 181 (2013a)CrossRefADSGoogle Scholar - B. Ruta, G. Baldi, G. Monaco, Y. Chushkin, J. Chem. Phys.
**138**, 054508 (2013b)CrossRefADSGoogle Scholar - B. Ruta, V.M. Giordano, L. Erra, C. Liu, E. Pineda, J. Alloy. Compd.
**615**, S45 (2014a)CrossRefGoogle Scholar - B. Ruta et al., Nat. Commun.
**5**, 3939 (2014b)CrossRefADSGoogle Scholar - P.N. Segre, P.N. Pusey, Physica A
**235**, 9 (1997)CrossRefADSGoogle Scholar - T. Seydel et al., Rev. Sci. Instr.
**74**, 4033 (2003)CrossRefADSGoogle Scholar - R.F. Shannon Jr., S.E. Nagler, C.R. Harkless, R.M. Nicklow, Phys. Rev. B
**46**, 40 (1992)CrossRefADSGoogle Scholar - A.J.F. Siegert, Massachusetts Institute of Technology Radiation Labatory Report No. 465 (1943)Google Scholar
- S.K. Sinha, D.K. Ross, Physica B
**149**, 51 (1988)CrossRefGoogle Scholar - I. Sikharulidze et al., Phys. Rev. Lett
**88**, 115503 (2002)CrossRefADSGoogle Scholar - M. Stana, M. Leitner, M. Ross, B. Sepiol, J. Phys.: Condens. Matter
**25**, 065401 (2013)ADSGoogle Scholar - M. Sutton, C.R. Phys.
**9**, 657 (2008)Google Scholar - M. Sutton et al., Nature
**352**, 608 (1991)CrossRefADSGoogle Scholar - G. Vogl, B. Sepiol, in
*Diffusion in Condensed Matter*, ed. by P. Heitjans, J. Kärger (Springer, New York, 2005), pp. 65–91Google Scholar - A.R. Yavari et al., Acta Mater.
**53**, 1611 (2005)CrossRefGoogle Scholar