X-Ray Photon Correlation Spectroscopy (XPCS)

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1 Introduction

One of the outstanding properties of third generation synchrotron radiation sources is their capability of producing coherent X-ray beams several orders of magnitude more intense than previously available [1]. The access to coherent X-rays opens up a variety of possibilities for new techniques such as X-ray photon correlation spectroscopy [2], coherent X-ray diffraction [3–7], speckle-mapping and coherent diffraction imaging [8–17] and X-ray interferometry [18, 19]. Coherent X-rays have also a major impact on imaging techniques such as topography [20], phase-contrast and holographic imaging [21–23]. Much of the excitement about scattering with coherent X-rays, however, arises from the perspective to perform atomic resolution correlation spectroscopy and this paper shall focus on the progress towards studying the complex dynamics of disordered systems on length- and time scales inaccessible by other techniques.

If coherent light is scattered from a disordered system it gives rise to a random diffraction or “speckle” pattern. Speckle patterns, long known from laser light scattering [24] but also observed recently with coherent X-rays [25], are related to the exact spatial arrangement of the disorder. Such information is not accessible with incoherent light because the diffraction pattern observed in an ordinary diffraction experiment is typically an ensemble average containing information on the average correlations in the sample only. If the spatial arrangement of the disorder changes with time the corresponding speckle pattern will also change and the intensity fluctuations of the speckles can provide information on the underlying dynamics. X-ray Photon Correlation Spectroscopy (XPCS) probes the dynamic properties of matter by analyzing the temporal correlations among photons scattered by the studied material. It can measure the low frequency dynamics (10⁷–10⁻³ Hz) in a Q range from typically 10⁻³ Å⁻¹ up to several Å⁻¹, depending on the sample and the scattering geometry.

Figure 18-1 shows the Frequency-Scattering vector range accessible by this technique compared to other methods frequently used to study the dynamics in disordered systems. X-ray Photon Correlation Spectroscopy is in particular complementary to Dynamic Light Scattering (DLS), also denoted Photon Correlation Spectroscopy (PCS) with visible coherent light which probes also slow dynamics (ω < 10⁶ Hz) but can cover only the small Q (<4·10⁻³ Å⁻¹) regime [24]. XPCS is furthermore not subject to multiple scattering, a phenomenon frequently complicating the analysis of PCS data in optically opaque systems [26]. Neutron based techniques (inelastic and quasi-elastic neutron scattering, neutron spin-echo), as well as Inelastic X-Ray Scattering (IXS) can access the same Q range but probe the dynamic properties of matter at high frequencies from above 10¹⁴ Hz down to about 10⁸ Hz.

Figure 18-1

Frequency–Scattering vector space covered by X-ray Photon Correlation Spectroscopy (XPCS) and complementary techniques (PCS-Photon Correlation Spectroscopy, INS-Inelastic Neutron Scattering, IXS-Inelastic X-ray Scattering, NFS-Nuclear Forward Scattering)

XPCS is a young technique but has already shown the potential to impact many areas of statistical physics and provide experimental access to a variety of important dynamic phenomena. Among them are equilibrium critical fluctuations [27] and the low frequency dynamics in disordered hard (e.g., non-equilibrium dynamics in phase separating alloys or glasses [28]) and soft condensed matter materials, in particular complex fluids (e.g., hydrodynamic modes in concentrated colloidal suspensions [29–34], dynamics near the colloidal glass transition [35, 36], dynamics in polymer systems [37, 38] or glassy and jammed dynamics [39–41]). The study of capillary wave dynamics at liquid surfaces has proven to be particularly successful [42–44] because these studies take advantage of two specific X-ray beam properties, coherence and surface sensitivity. These experiments have furthermore shown that it is possible, similar to the DLS case, to operate in heterodyne detection mode [45]. Correlation times ranging from hundreds of seconds (measured in aging gels and glasses [46] or intermetallic phases [47]) down to nanoseconds (fluctuating films [48] and membranes [49, 50]) have been detected, thus showing in particular that the time gap to the Neutron Spin Echo (NSE) technique has been bridged. It is expected that coherence based techniques will continue to prosper in the future since corresponding experimental activities are not only implemented at the pioneering third generation storage rings ESRF, APS and Spring8 [51] but are also foreseen for the new facilities SLS, Diamond, Soleil, Petra III and NSLS II [52]. Finally, the X-ray Free Electron Laser (XFEL) sources at the horizon in Germany, USA and Japan [53] are not only characterized by their ultra short pulses but will in particular also provide the most intense coherent X-ray beams ever produced.

The format of the article is as follows: The coherence properties of undulator radiation will be introduced in Section 2 . Disordered systems under coherent illumination and the characteristics of X-ray speckle will be discussed in Section 3 , followed by a description of the XPCS technique in Section 4 . The experimental set-up is described in Section 5 . Applications to colloidal systems will be discussed in Section 6 , followed by an overview on liquid surface dynamics in Section 7 . Experiments in hard-condensed matter systems are summarized in Section 8 . Conclusions and a short outlook are presented in Section 9 .

2 Coherent X-Rays from a Synchrotron Source

Storage-ring based synchrotron radiation sources are chaotic sources since the emission processes of the individual relativistic electrons (or positrons) are independent and spontaneous. The coherence properties of synchrotron radiation can be described by the wavelength spread Δλ/λ of the photons, and the phase-space volume (Σ Σ’)² in which the photons are contained, where Σ is the source size and Σ’ is the divergence of the photon beam [54]. For the beam to be spatially coherent it is required that Σ_xΣ_x’Σ_zΣ_z’ ≤ (λ/4π)², which is usually not fulfilled for wavelengths λ in the X-ray regime, but can be achieved by introducing collimating apertures in the beam. This however also reduces the photon flux and the intensity I_c of coherent photons per 0.1% bandwidth is given by

$${\rm{I}}_{\rm{c}} = ({{{\rm \lambda} /2)}}^{\rm{2}} {\rm{B}} $$

((1))

where B is the brilliance of the source given in units of photons/s/mrad²/mm²/0.1% bandwidth. Today’s third generation undulator sources can provide brilliances of order 10²⁰ or higher. This became feasible through the development of undulator insertion devices, i.e. periodic magnet structures installed in the straight sections of a synchrotron storage ring, that produce a discrete spectrum of intense synchrotron radiation at a fundamental energy E₀ and higher harmonics n·E₀ (n = 2, 3, 4,…). The size of the collimating aperture for producing a coherent beam is determined by the transverse coherence length ξ_t of the photon beam which can be defined via the visibility of interference fringes. The fringe visibility for a uniform monochromatic disk source of size Σ and an aperture of size d at a distance R from the source is [55]:

$$ {\rm{V(d) = 2|J}}_{\rm{1}} ({\rm{\pi d}} {\rm \Delta} {\rm \Theta} {\rm{/{\rm \lambda} )/(\pi d{\rm \Delta} {\rm \Theta} /{\rm \lambda} )|}} $$

((2))

where ΔΘ = Σ/R is the angular (FWHM) source size and J₁ is the first-order Bessel function. A visibility V > 50% for a disk source of size Σ yields

$$ \xi _t \approx (1/2) ({{{\rm \lambda} /}}{\rm \Delta} {\rm \Theta} {\rm{) = ({\rm \lambda} /2)}} ({\rm{R/ {\rm \Sigma})}} $$

((3))

Typical transverse coherence lengths at third generation sources range between 10 and 100 μm for λ = 1 Å and a distance R ≈ 50 m from the source. The spatial coherence properties of a synchrotron X-ray beam can be monitored by Fraunhofer diffraction from a collimating aperture. Figure 18-2 shows Airy fringes from a 5 × 5 μm² slit located 46 m from the ID10A undulator source at ESRF. Circular pinhole apertures produce Fraunhofer diffraction patterns where the central maximum has an angular width that depends upon the pinhole size d and is typically of order λ/d [55, 56].

Figure 18-2

Airy fringes from a 5 × 5 μm² slit, recorded with λ = 1.54 Å radiation at 1.5 m from the slit. The visibility V of the fringes can be quantified by V = (I_max−I_min)/(I_max + I_min), where I_max is a fringe maximum and I_min is an adjacent minimum

The temporal coherence of the beam can be described by the longitudinal coherence time τ₀ which defines a longitudinal coherence length ξ_l = cτ₀ over which the phase of the field amplitude undergoes no fluctuations. τ₀ is a measure of the monochromaticity of the beam and is related to the bandwidth Δν of the light by

$$ {\rm{\tau }}_{\rm{0}} = 1/{\rm \Delta} {\rm \nu} $$

((4))

A well monochromatized X-ray beam has a bandwidth of Δλ/λ = 10⁻⁴ and hence a longitudinal coherence time τ₀ ≈ 10⁻¹⁵ s. The longitudinal coherence length is given by

$$ \xi _1 = {\rm{c\tau }}_{\rm{0}} = {{\rm \lambda} ({\rm \lambda} / {\rm \Delta} {\rm \lambda} )} $$

((5))

The intrinsic relative bandwidth Δλ/λ ≈ 1/nN for the n-th harmonic of an undulator with N periods. Thus, the longitudinal coherence length ξ_l = λnN ≈ 100 Å for the third harmonic of a 35 period undulator at 1 Å wavelength. Larger longitudinal coherence lengths can be achieved by monochromatizing the photon beam at the expense of flux. Table 18-1 gives values for the relative bandwidth, longitudinal coherence lengths and the intensity of coherent photons for different experimental configurations.

Table 18-1 Bandpass Δλ/λ, longitudinal coherence length ξ_l at 1 Å wavelength and coherent intensity Ic (B = 10²⁰ ph/s/0.1%bw/mm²/mrad²) for Si(220), Si(111) and a configuration using the intrinsic bandwidth of the third (n = 3) harmonic of a N = 35 period undulator

There are several methods to characterize the coherence properties of synchrotron X-ray beams [21–23, 57–59]. The statistical analysis of static speckle patterns [5] allows one to measure and optimize the coherence parameters in exactly the same experimental configuration that is used for a dynamical XPCS measurement and permits therefore to include e.g. the effects of the sample environment.

3 Disorder under Coherent Illumination

When coherent light is scattered from a disordered system it might give rise to a random diffraction or “speckle” pattern. For illustration consider a coherent beam of cross section ξ_t ² and incident wave vector k, scattered with outgoing wave vector k′ from a disordered sample. The instantaneous intensity at a given point in the far field can be written as the square of a total field E(Q, t) and

$$ {\rm{I({\bf Q},t) = |}}{\bf E}{\rm{(}}{\bf Q}{\rm{,t)|}}^{\rm{2}} = |\sum_{\rm n} {\rm{b_n(}}{\bf Q}{\rm{)exp[i\bf Q}} \cdot {\bf r}{\rm{_n(t)]|}}^{\rm{2}} $$

((6))

Here, b_n(Q) is the scattering amplitude of the n-th scatterer located at position r _n(t), and Q = k′−k is the momentum transfer. The sum is taken over scatterers in the coherence volume, spanned by the transverse and longitudinal coherence lengths, and the beam is assumed to be fully coherent. For clarity the Thomson scattering length $\rm r_o= ({{e^2}\over4{\rm \pi}{\rm \varepsilon} _0mc^2})$ and polarization factors are omitted in (6). The field E(Q,t) in (6) may also be expressed in terms of an electron density function ρ(r′) and

$$ \rm {\bf E}{\rm{(}}{\bf Q}{\rm{,t}}) = \int {{\rm{d}}{\bf r}} {\rm{' {\rm \rho} ({\bf r'}) exp [i}} {\bf Q} \cdot {\bf r}'(t)] $$

((7))

A measurement of the intensity will naturally be a time average <I(Q, t)>_T taken over the acquisition time T, but does not involve any statistical ensemble average. 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 variations in intensity, known as “speckle.” If, on the other hand, the system is ergodic, with fluctuation time scales very short as compared to the counting time, the measured time average is equivalent to an ensemble average and <I(Q, t)>_T can be replaced by the usual ensemble average, denoted by <I(Q, t)>. The observed scattering is then featureless apart from time averaged correlations in the sample similar to a regular scattering experiment with incoherent radiation. Figure 18-3 shows a static speckle pattern from a porous glass (vycor) taken with a CCD detector. A 8 keV partially coherent X-ray beam from the ID10C branch of the ID10 beamline at ESRF [60] was used. The “random” nature of the scattering is illustrated by the solid line in Figure 18-4 (top) showing a cut through the pattern along the radial direction with intensity variations clearly beyond counting statistics. The open symbols indicate the ensemble averaged scattering <I(Q,t)>, calculated from the speckle pattern by azimuthally averaging. The observed maximum in Figure 18-3 (shaded area) and Figure 18-4 at Q = 0.0225 Å⁻¹ is due to pore-pore correlations in the glass.

Figure 18-3

Speckle pattern from porous silica glass (vycor) recorded with a 20 × 20 μm² pixel size CCD detector located 3.3 m from the sample. A 20 μm partially coherent 8 keV monochromatic X-ray beam was used. Dark (blue) colors correspond to regions of low intensity. The shaded area (highest intensity) corresponds to the region of interest analyzed to produce the results shown in Figure 18-5 (bottom)

Figure 18-4

(top): Intensity profile through the speckle pattern along the radial direction showing strong spatial intensity fluctuations compared to circular averaged scattering from the sample, normalized to give detected “photons” per second (open circles). (bottom): Contrast β(Q) (open symbols) as a function of the wavevector Q

Figure 18-5

(top): Intensity probability distribution calculated from (9) for a single (M = 1), and the sum of 5 (M = 5) and 20 (M = 20) speckle patterns with a mean intensity <I> = 1. (bottom): Normalized intensity probability distribution I/<I> from Figure 18-3 at Q = 0.024 Å⁻¹ ± 6·10⁻⁴ Å⁻¹

Static X-ray speckle has been observed in a variety of different systems ranging from porous materials [5], modulated bulk and surface structures [61, 62], surfaces [8], magnetic materials [63–65] to systems that exhibit (microscopic) domain disorder [25, 27, 66].

3.1 Statistical Properties of Speckle Patterns

A static speckle pattern contains not only information about the disorder of the scattering sample but also on the radiation that produced it, including the degree of coherence and the illuminated volume size. For a fully coherent beam it can be shown [67] that the intensity at a point in the speckle pattern obeys negative exponential statistics if the amplitudes b_n(Q) and the phases Q·r _n (see (6)) are statistically independent and if the phases are statistically distributed over 2π.

The probability distribution P(I) of the intensity is then given by

$$ {\rm{P(I) = (1/ < I > ) exp [ - I/< I > ]}}$$

((8))

with mean intensity <I> and a standard deviation σ = (<I²> − <I>²)^1/2 = <I>. Figure 18-5 (top) shows the calculated probability density function for a single (M = 1) speckle pattern with <I> = 1, predicting in particular the existence of dark (P(I = 0) = 1) spots in a corresponding speckle pattern. The ratio β = σ²/<I>² is a measure of the contrast of a speckle pattern and shows that the contrast under fully coherent conditions is unity.

When the illuminated sample volume is bigger than the coherence volume (partially coherent or incoherent illumination) one can imagine the sample to be divided into small volume elements (of size of the coherence volume) so that the total intensity is arising from M such volumes. If an observed pattern is in fact the intensity sum of M independent speckle patterns one finds

$$ {\rm{P}}_{\rm{M}} {\rm{(I)}}\;{\rm{ = }}\;{\rm{M}}^{\rm{M}} {\rm{(I/ < I >) }}^{{\rm{M - 1}}} \rm \exp (- MI/<I >)/[{\rm \Gamma} (M) < I > ] $$

((9))

where Γ is the gamma function, σ = <I>/M^1/2 and β = (1/M). The probability densities for M = 5 and M = 20 are also shown in Figure 18-5 (top) and one notes in particular the elimination of dark spots (P(I = 0) = 0). The contrast β is reduced and reaches zero in the incoherent limit (M→∞). Figure 18-5 (bottom) shows the measured probability distribution for intensities in an annulus about Q = 0.024 Å⁻¹ ± 6·10⁻⁴ Å⁻¹ (indicated by the shaded area in Figure 18-3 ) plotted versus the intensity divided by the mean. The dashed line in Figure 18-5 (bottom) shows the result of a fit of (9) to the data, yielding M = 8.32. The agreement is moderate and a close examination of the measured intensity distribution shows that there is almost no scaled intensity below about 36% of the mean. This indicates [5] that there is a constant part of the incident beam that does not participate in the coherent interference leading to the speckle pattern. This incoherent fraction, (1−α), reduces the measured contrast by a factor α². The solid line in Figure 18-5 (bottom) is the gamma distribution that results if the mean and standard deviation are calculated after subtracting (1−α) = 0.365 from the scaled intensities. This procedure yields M = 2.81 and a contrast β = α²/M = 0.143. The same treatment was applied for the whole Q-range of the speckle pattern. The contrast or degree of coherence appeared to be constant for all investigated Q as shown by the open symbols in Figure 18-4 (bottom). This result supports the validity of the applied simple procedure and indicates in particular that the pore-pore correlations in the sample (responsible for the peak at Q = 0.0225 Å⁻¹) do not influence the analysis. The result also shows that the used experimental configuration provided a partially coherent beam with a “coherent fraction” α of about 2/3.

It was shown [5] that M is Q-dependent for a partially coherent beam and the contrast decreases with increasing Q. This effect is however weak for a monochromatic beam and small Q-values.

An analysis of a static speckle pattern [5] in terms of intensity probability functions can give a reasonable first estimate of the coherence properties of the set-up. Quantitative information, in particular concerning the size of a speckle or the Q dependence of the speckle size is however not easily accessible. A more elaborate approach [5] describes the statistical properties of a speckle pattern with help of a normalized two-point correlation function

$$ {\rm{C(}}\rm {\bf Q}_{\rm{1}} {\rm{,}}{\bf Q}_{\rm{2}} {\rm{)}}\;{\rm{ = }}\;{\rm{[ < I({\bf Q}_1) I(}}{\bf Q}_{\rm{2}} {\rm{) > / < I(}}{\bf Q}_{\rm{1}} {\rm{) > < I(}}{\bf Q}_{\rm{2}} {\rm{) > ] - 1}}{\rm{.}} $$

((10))

Here, I(Q ₁) and I(Q ₂) describe the scattered intensity at two positions on a two-dimensional detector image relative to the beam-zero position on the detector. C(Q ₁, Q ₂) is a function that peaks for Q ₁ = Q ₂ and falls to zero at larger separations. Its width is a measure of the speckle size while the contrast of a speckle pattern can be defined by its peak value

$$ {\rm{{\rm \beta} ({\bf Q})}}\;{\rm{ = }}\;{\rm{C({\bf Q},{\bf Q})}}{\rm{.}} $$

((11))

We may write

$$ {\rm{C({\bf Q}}}_{{\rm{1,}}} {\rm{{\bf Q}}}_{\rm{2}} {\rm{)}}\;{\rm{ = }}\;{\rm{|\mu }}_{\rm{o}} {\rm{ ({\bf Q}}}_{\rm{1}} {\rm{,{\bf Q}}}_{\rm{2}} {\rm{)|}}^{\rm{2}} $$

((12))

where μ_o (Q ₁,Q ₂) is the complex coherence factor. If the incident beam is constant within a given aperture and zero outside, then

$$ {\mu }_{\rm{o}} ({\bf Q}_1, {\bf Q}_2) \sim { \int{ {{\rm{d}} {\bf r}}' \exp[i({\bf Q}_1 - {\bf Q}_2 ) {\bf r}^\prime]}}$$

((13))

and within the small-angle scattering limit the autocorrelation function of the speckle pattern follows the shape of the Fraunhofer diffraction pattern from an aperture. For the monochromatic case and a circular pinhole with diameter d the (angular) FWHM is Δ = 1.03 (λ/d). Using this approach it has been shown [5] that the shape of a speckle can be anisotropic and dependent on Q. This is the consequence of the interplay between finite monochromaticity and the size of the illuminated volume. An equivalent approach [68] evaluates the intensity I(Q) via integration over cut-off functions. These might be interpreted as detector resolution and/or as defining coherence volumes for the radiation in the sample. Both effects couple to the degree of ensemble averaging and thus influence the visibility or the contrast.

3.2 Reconstruction of Static Speckle Patterns

A speckle pattern is related to the exact spatial arrangement of the disordered system but the inherent phase problem of crystallography also applies and thus additional, supplementary information is necessary. It had been proposed [69] to oversample data to exceed the spatial Nyquist frequency so that the inversion of (6) becomes mathematically over-determined. Iterative procedures, first proposed by Gerchberg and Saxton [70] and further developed by Fienup [71] can then be used to determine the phases and finally the real phase structure. This approach has been shown to work in simulations for inherently non-periodic objects and thus continuous diffraction patterns [72] and also in experimental work [11]. Figure 18-6 (left) shows a SEM image of a specimen consisting of 100 nm diameter gold dots deposited on a silicon nitride membrane. This sample was illuminated with a soft, 17 Å wavelength coherent X-ray beam and the resulting speckle image was recorded with a high resolution CCD camera as shown in Figure 18-6 (middle). The reconstructed image is shown in Figure 18-6 (right).

Figure 18-6

SEM image of a sample consisting of gold dots (0.1 μm diameter, 80 nm thick) on a SiN membrane (left). Speckle pattern taken with a λ = 17 Å coherent beam (1.3^.10⁹ ph/s through a 10 μm pinhole) with a 24 × 24 μm² pixel CCD (middle). Result of the reconstruction via the “oversampling” technique (right); from [11]

An extension of this approach to diffraction from single crystals, small enough to generate a continuous diffraction pattern, which can be oversampled has recently been demonstrated [12]. Coherent, λ = 1.65 Å X-rays have been used to successfully reconstruct the shape of small, 1 μm size gold nanocrystals. The same concept was used in a simulation exploring the possibility to determine three-dimensional structures of biomolecules from single-molecule diffraction images [13]. Here the information from 10⁶ calculated speckle images arising from 10⁶ single biomolecules of known orientation was used. Each molecule was subject to a (simulated) 10 fs pulse of 2 · 10¹² coherent photons of 1.5 Å wavelength from a future X-ray free electron laser. The simulation indicated the possibility to reconstruct the real space structure with 2.5 Å resolution. Coherence based imaging techniques have been developing rapidly in recent years. Among the most relevant achievements are soft X-ray fourier transform holography (FTH) for magnetic domain imaging [14], strain mapping in nanocrystals [15] by Coherent Diffraction Imaging (CDI), single shot CDI imaging [16] and hard X-ray ptychography [17].

4 X-Ray Photon Correlation Spectroscopy (XPCS)

If the spatial arrangement of the scatterers changes with time the corresponding speckle pattern will also change and a measurement of the intensity fluctuations of the speckles can reveal information on the dynamics of the system. The measurement of temporal intensity correlations by X-ray Photon Correlation Spectroscopy (XPCS) is identical to a Photon Correlation Spectroscopy (PCS) experiment with visible coherent light, which is well documented in the literature [24]. Temporal correlations can be quantified with help of the normalized (second order) intensity correlation function g₂(Q,t). If the field E(Q,t) is a zero mean, complex Gaussian variable the (time averaged) intensity correlation function can be written in terms of the (ensemble averaged) time correlation functions of the scattered field:

$$ \eqalign {{{\rm{g}}_{2} ({\bf Q},\ {\rm t})} & = { < {\rm I}({\bf Q},0){\rm I}({\bf Q},{\rm t}) > / < {\rm I}({\bf Q})> }^{2} \cr & = {1 \ + \ {\rm \beta} ({\bf Q}) < {\rm E}({\bf Q},0) {\rm E}^*({\bf Q},{\rm t})>}^2 / < I({\bf Q})>^2}$$

((14))

where β(Q) is the contrast of the set-up (see Section 3 ). The time autocorrelation function g₂(Q, t) is frequently expressed in terms of the normalized intermediate scattering function f(Q, t) with

$$ {\rm{g}}_{\rm{2}} {\rm{(}}{\bf Q}{\rm{,t)}}\;{\rm{ = }}\;{\rm{1}}\;{\rm{ + }}\;{\rm{{\rm \beta} (}}{\bf Q}{\rm{) |f(}}{\bf Q}{\rm{,t)|}}^{\rm{2}} $$

((15))

where

$$ \rm{f(}{\bf Q}{\rm{,t)}}\;{\rm{ = }}\;{\rm{F(}}{\bf Q}{\rm{,t)/F(}}{\bf Q}{\rm{,0)}} $$

((16))

and

$$ {\rm{F(}}{\bf Q}{\rm{,t) = [1/N\{ b}}^{\rm{2}} {\rm{(}}{\bf Q}{\rm{)\} ]}}\sum _{{\rm{n}}\;{\rm{ = }}\;{\rm{1}}}^{\rm{N}} \sum _{{\rm{m}}\;{\rm{ = }}\;{\rm{1}}}^{\rm{N}} {\rm{ < b}}_{\rm{n}} {\rm{(}}{\bf Q}{\rm{)b}}_{\rm{m}} {\rm{(}}{\bf Q}{\rm{)}} \cdot {\rm{exp\{ i}}{\bf Q}{\rm{[}}{\bf r}_{\rm{n}} {\rm{(0) - {\bf r}}}_{\rm{m}} {\rm{(t)]\} > }} $$

((17))

Here, N is the number of scatterers and {b²(Q)} is the square of the scattering amplitude averaged over the particle size distribution. The brackets (<>) denote an ensemble average and F(Q, 0) can be identified with the static structure factor.

To illustrate, consider in the following the simple example of monodisperse particles undergoing Brownian motion [24, 73]. In the absence of interactions between the particles their positions are uncorrelated and cross (n≠m) terms in (17) average to zero and F(Q, 0) = 1.

The mean square value of the displacement [r(0) – r(t)] for a free Brownian particle is

$$ {\rm{ < [}}\rm {\bf r}{\rm{(0) - }}{\bf r}{\rm{(t)}}]^{\rm{2}} {\rm{ > }}\,{\rm{ = }}\;{\rm{6D}}_{\rm{0}} {\rm{t}} $$

((18))

where D₀ is the free particle (Stokes-Einstein) diffusion coefficient of a particle with radius R and

$$ {\rm{D}}_{\rm{0}} \;{\rm{ = }}\;{\rm{k}}_{\rm{B}} {\rm{T/6\pi \eta R,}} $$

((19))

where η is the shear viscosity of the surrounding medium. Thus (16) reduces to

$$ {\rm{f(}}{\bf Q}{\rm{,t)}}\;{\rm{ = }}\;{\rm{exp (}} - {\rm{D}}_{\rm{0}} {\rm Q}^2{\rm{t)}}{\rm{.}} $$

((20))

In the presence of particle interactions (20) is no longer valid and one frequently considers a time- and wavevector dependent diffusion coefficient D(Q,t) [74]. A useful quantity is the initial (t→0) slope Γ(Q), or the first cumulant of the “measured” intermediate scattering function f^M(Q, t), and it can be shown that [74]

$$ {\rm{{\rm \Gamma} (}} {\bf Q}{\rm{)}}\;{\rm{ = }}\lim \nolimits _{t \to 0} {\rm{d/dt }}[\ln {\rm{f}}^{\rm{M}} {\rm{(}}{\bf Q}{\rm{,t)]}} = - {\rm D}{\bf (}{\bf Q}{\bf )} \cdot {\rm{Q}}^2 $$

((21))

and

$$ {\rm{f(}} {\bf Q}{\rm{,t)}}\;{\rm{ = }}\;{\rm{exp(}} - {\rm{D(}}{\bf Q}{\rm{){\rm Q}}}^{\rm{2}} {\rm{t)}}{\rm{.}} $$

((22))

Correlation functions (14) can be measured by coupling a point detector to a digital autocorrelator giving access to the correlation function over a wide range of correlation times at a single Q value. Figure 18-7 shows a typical intermediate scattering function taken on a concentrated suspension of colloidal PMMA particles. The measured data have been fitted with the cumulant expansion [74] up to the fourth order term (solid black line). The behavior at short times is described by the first cumulant Γ(Q) (solid red line).

Figure 18-7

Intermediate scattering function f^M(Q, t) measured on a concentrated suspension of colloidal poly(methylmethacrylate) (PMMA) particles in cis-decalin. The solid (black) line shows a cumulant fit to the data using terms up to fourth order. The (red) line shows the contribution of the first cumulant Γ(Q) only

5 Experimental Set-Up

Scattering experiments with coherent X-rays require the sample to be illuminated coherently, implying that the maximum path-length difference (PLD) for rays in the sample has to be equal to or smaller than the longitudinal coherence length ξ_l of the beam. In addition, the lateral size of the illuminated sample volume must be smaller than or comparable to the transverse coherence length ξ_t. The maximum pathlength difference is thus given by

$$ {\rm{PLD}} \approx {\rm{2{\rm \mu} }}^{ - {\rm{1}}} \sin ^2 {\rm{{\rm \theta} \ or \ PLD }} \approx 2{\rm{Wsin}}^{\rm{2}}{\rm{{\rm \theta} }}\; + \;{\rm{d}}\sin 2{\rm{{\rm \theta} }} $$

((23))

in reflection or transmission geometry, respectively. Here, μ is the linear absorption coefficient, d the beam diameter, W the sample thickness and θ the scattering angle. Equation (23) sets a limit for the maximum wavevector transfer Q_max = (4π/λ) sinθ_max that is compatible with coherent illumination. For a Si(111) monochromator (ξ_l = 0.7 μm and λ = 1 Å) the maximum wavevector transfer in reflection geometry Q_max ≈ 2 Å⁻¹ for 1/μ ≈ 15 μm. Shorter longitudinal coherence lengths are acceptable for small angle scattering experiments. If a momentum transfer of Q ≈ 0.01 Å⁻¹ is sufficient and assuming W ≈ 1 mm and d ≈ 10 μm one finds that PLD ≈ 100 Å. This allows one to relax the monochromaticity of the photon beam to the intrinsic bandwidth of an undulator harmonic with ξ_l typically around 100 Å. A wide bandpass beam can be produced by replacing a crystal monochromator by a mirror. A schematic set-up of the ID10A branch of the TROIKA beamline [75] at ESRF for experiments with coherent X-rays is shown in Figure 18-8 . Similar beamline set-ups are described in [76, 77]. A perfect crystal monochromator or a short mirror are located 44.2 m from source in horizontal reflection geometry. A second mirror is installed in vertical reflection geometry about 0.8 m downstream of the first mirror (monochromator) in order to reduce the harmonic content.

Figure 18-8

Schematic set-up of the ID10A beamline at ESRF for coherent X-ray scattering. Typical distances are: Source–Slits: 27 m, Source–Monochromator/Mirror: 44.2 m, Monochromator/Mirror-Mirror: 0.8 m, Mirror-Pinhole: 0.5 m, Pinhole-Sample: 0.1 m, Sample-Detector: 2 m

Collimating pinholes of different diameters d are installed downstream of the second mirror. The asymmetric (FWHM) source size (values given for a high-β section of the ESRF storage ring) of 928 μm (h) and 34 μm (v) causes different transverse coherence lengths in the horizontal (ξ_t = 3 μm) and vertical (ξ_t = 98 μm) directions (λ = 1 Å, R = 45 m). It is desirable to tune the transverse coherence lengths such that they are matched to the correlation lengths in the sample under investigation. ξ_t can be increased by reducing the effective source size with the help of slits (e.g., at the 27 m position in Figure 18-8 ).

Smaller transverse coherence lengths can be produced by increasing the beam divergence and i.e., focusing the beam (e.g., with a vertically focusing mirror) and thereby increasing the flux. Laterally coherent X-ray beams with transverse coherence lengths between typically 4 and 20 μm in both directions have experimentally been produced. The integrated coherent flux through a 12 μm pinhole is 10¹⁰–10¹¹ photons/s at 100 mA storage ring current in a 1.3% bw configuration at ID10A (ESRF). The coherent flux achieved with a perfect crystal Si(111) monochromator is about 10⁸–10⁹ photons/s at 100 mA ring current and 1 Å. The sample is kept at a distance <d²/λ in the near field region of the pinhole.

The experimentally observed angular size of a Fraunhofer maximum or an individual speckle is approximately given by D_s ≈ [(λ/d)² + (Δθ)²]^1/2, where Δθ is the effective angular source size. Fraunhofer or speckle patterns can be recorded by either scanning a point detector combined with an analyzer pinhole (diameter <D_s) in the scattered beam or by employing a one- or two-dimensional position sensitive detector with the appropriate spatial resolution.

The combination of a point detector (scintillation counter or avalanche photodiode) and a hardware correlator yields a fast access to the intensity autocorrelation function at the selected Q value. Typically, the count rate determines the fastest time scales accessible by XPCS and as a rule of thumb one can assume that a measurement of 100 Hz dynamics also requires at least a 100 Hz countrate. The detector, however, also sets a limit (by its deadtime) for the fastest dynamics accessible by XPCS, and hence classical scintillation counters can only be used up to about a few hundred kHz. For faster dynamics it is necessary operate with e.g. avalanche photodiode (APD) detectors with an intrinsic dead-time of about 1 ns or less [78]. In this case the hardware correlator might actually be the limiting factor, or, as is often the case, the time structure of the synchrotron storage ring. The uniform filling mode of the ESRF ring is particularly advantageous for fast XPCS since the electrons here are gathered in 992 equidistantly spaced bunches around the ring. This yields a very strong time-correlation in the beam with a characteristic time of approximately 2.8 ns which is detectable by XPCS but apart from that the correlation function is featureless. It is in this mode that the fastest correlation functions to date have been recorded [50, 79].

If the aim is to measure dynamics on the timescale of a few seconds or longer, it is possible to use a photon-counting two-dimensional detector. The advantage is that a broad range of Q-values is covered within one frame; the disadvantage that the read-out of such devices is usually rather slow and can only be used for slow dynamics. The 2D detector of choice for XPCS has been a deep-depletion CCD camera with 1,200 × 1,200 pixels and a pixel size of about 20 μm. The camera has low dark current and read out noise and is used in direct illumination mode without any additional optics. Each detected photon results in a certain number of Analog Digital Units (ADU) and hence the event (one incident photon) is well defined in time and space within the given constraints (spatial/temporal resolution). The CCD camera can be used in a so-called “kinetics mode” which allows faster read-out at the expense of field-of-view [80]. Recently another CCD camera with a readout speed of 60 Hz (full frame), has become available. In any case, for 2D detection the intensity autocorrelation function is subsequently calculated pixel by pixel over all the acquired 2D frames and afterwards averaged over pixels corresponding to the same Q [80, 81]. The first XPCS compatible 2-D pixel detectors have become available more recently [82].

6 XPCS in Soft Condensed Matter Systems

Slow, complex relaxations in soft condensed matter materials have successfully been addressed by (coherent) visible light scattering techniques as described and summarized in a number of distinguished textbooks (see e.g., [24, 83, 84]). Prominent examples comprise in particular studies of the dynamics of colloidal particles in suspension [73] or surface dynamics [85]. It however also becomes apparent that there are limitations in the accessible lengthscale and/or complications due to multiple scattering effects, both well known intrinsic obstacles caused by the usage of visible light. Coherent X-rays overcome both limitations and were naturally first used to revisit questions related to the dynamics and hydrodynamics of colloidal systems, slow dynamics in polymers and capillary wave motion in complex fluids. For the examples discussed in this section all relevant quantities are of scalar character.

6.1 Static and Dynamic Properties of Colloidal Suspensions

Colloidal suspensions consist of particles (clusters of atoms or molecules) with typical diameters 2R between 50 and 5,000 Å that are suspended in a solvent (e.g., a molecular fluid). Agglomeration of the particles due to van der Waals attraction is prevented by introducing a repulsive (steric or coulombic) interaction between the particles. The interparticle interactions are then described [73] by either a short-range pair potential (hard-spheres) or by a long-range Yukawa type potential (soft-spheres), respectively. The time-averaged structure of a colloidal suspension can be described by the static structure factor

$$ {\rm{S(Q)}}\;{\rm{ = }}\;{\rm{1}}\;{\rm{ + }}\;{\rm{4}}\pi {\rm{n }}\int _{\rm{0} } ^\infty[{\rm{g(r)}} - 1](\sin ({\rm{QR}} )/\rm QR){\rm{r}}^2 {\rm{dr}} $$

((24))

where n is the particle number density and g(r) = exp [−V(r)/k_BT] is the radial distribution function related to the (mean field) interaction potential V(r) between two spheres separated by a distance r. In the dilute case (n << 1) and in the absence of interparticle correlations (S(Q) = 1) the colloidal particles migrate driven by the thermal fluctuations of the solvent with the Stokes-Einstein (free particle) diffusion coefficient D₀ (cf. (19)).

At larger concentrations direct interparticle interactions as well as indirect, hydrodynamic interactions, mediated by the solvent become important. Based on the Smoluchowski (many-particle diffusion) equation it has been shown that the short-time (t < R²/D₀) behavior of the intermediate scattering function f(Q, t) can be described by an effective diffusion coefficient

$$ {\rm{D(Q)}}\;{\rm{ = }}\;{\rm{D}}_{\rm{0}} {\rm{H(Q) /S(Q),}} $$

((25))

where H(Q) is the hydrodynamic function. The limiting cases, D(Q→0) = D_c and D(Q→∞) = D_s are referred to as the collective diffusion coefficient D_c (describing long-wavelength, collective dynamics) or the self diffusion coefficient D_s, respectively.

In the absence of hydrodynamic interactions (H(Q) = 1) (25) reduces to D(Q) = D₀/S(Q) and the slowing down in the vicinity of the structure factor peak is well known experimentally and referred to as de Gennes narrowing [86]. In this case the self diffusion coefficient D_s = D₀ since S(Q→∞) = 1. The determination of D(Q) and S(Q) by photon correlation spectroscopy (PCS) with visible light is a formidable task. A consistent picture has nevertheless emerged for hard-sphere fluids on the basis of PCS work and analytical expressions for the structure factor (within the Percus-Yevick approximation [87]) and the hydrodynamic functions [88]. The situation for charge stabilized colloidal systems is considerably more complicated [89–91]. The static structure factor can only be calculated within approximations (Rescaled Mean Spherical Approximation–RMSA [89, 90], Rogers-Young or Hypernetted Chain [91]), and predictions for the hydrodynamics exist only for the dilute case where the interactions can be regarded as pairwise-additive [92]. Quantitative agreement between experiment and theory could only be established for the ultra dilute case [93]. At intermediate concentrations there are only few experimental data [26, 94, 95] and unfortunately no rigorous theoretical framework to compare to. Here, XPCS can offer two unique advantages compared to visible light scattering techniques: The short X-ray wavelength allows one to access larger momentum transfers, and the technique is not subject to multiple scattering effects since the refractive index of X-rays is always very close to unity. This allows one to study turbid and optically opaque samples on all relevant lengthscales.

6.2 XPCS and SAXS Measurements in Colloidal Suspensions

The static and dynamic behavior of colloidal suspensions can be fully determined by a combination of Small Angle X-ray Scattering (SAXS) and XPCS. The cross section for the scattering of X-rays from a suspension of monodisperse colloidal particles can be written as

$$ {\rm{(d{\rm \sigma} /d{\rm \Omega} )/V}}\;{\rm{ = }}\;{\rm{r}}_{\rm{o}}^{\rm{2}} {\rm{n({\rm \rho} }}_{\rm{c}} {\rm{ - {\rm \rho} }}_{\rm{s}} {\rm{)}}^{\rm{2}} {\rm{v}}^{\rm{2}} {\rm{P(Q)S(Q)}} $$

((26))

where r₀ is the Thomson radius, n is the particle number density, v is the particle volume and ρ_c, ρ_s are the electronic densities of the colloid and the solvent, respectively. P(Q) is the single particle form factor which for a homogenous sphere with radius R is given by

$$ {\rm{P(Q)}}\;{\rm{ = }}\;{\rm{[3/(QR)}}^{\rm{3}} {\rm{]}}^{\rm{2}} {\rm{ \cdot [sin(QR) - QRcos(QR)]}}^{\rm{2}} $$

((27))

and S(Q) is the static structure factor, describing interparticle correlations.

The first X-ray time correlation functions from a suspension of colloidal gold particles were reported by Dierker et al. [29] followed by the observation of translational diffusion of colloidal Pd agglomerates in glycerol [30]. Static and dynamic data taken on a model system, a dilute suspension of colloidal silica in a 1:1 mixture of water/glycerol (1% volume concentration), are shown in Figure 18-9 [96].

Figure 18-9

(top): X-ray small angle scattering intensity as a function of the momentum transfer Q for a 1 vol%. suspension of colloidal silica (R = 2,427 Å) in a glycerol/water mixture at 293 K. (middle): Q dependence of the relaxation rate Γ as determined from intensity correlation functions taken at various Q values. The solid line is the result of a fit to the data with D₀ as a fit parameter. (bottom): Inverse, normalized diffusion coefficient D₀/D(Q) as a function of Q [96]

The top panel shows the scattering intensity as a function of the momentum transfer indicating pronounced oscillations that are perfectly described by the particle form factor (27), convoluted with a Shultz size distribution function yielding R = 2,427 Å and ΔR/R = 0.045. X-ray time correlation functions were taken over the whole Q range and fits to a single exponential (20) yield the relaxation rate Γ, plotted in the middle panel. The solid line reveals the characteristic Q² dependence of a simple translational diffusion process with D₀ = 1.31·10⁻⁹ cm²/s. This is also illustrated in the bottom panel where the inverse, normalized diffusion coefficient D₀/D(Q) is plotted.

Both, the static and dynamic behavior changes dramatically when the colloidal particles interact. This is illustrated for a hard-sphere model system, colloidal Poly(methylmethacrylate) (PMMA) particles (R = 1,112 Å) suspended with a volume fraction Φ = 0.37 in cis-decalin [97].

Figure 18-10a shows the SAXS scattering intensity as a function of the momentum transfer Q. The data at low Q can no longer be described by the particle form factor (solid line) but contain the effects of the direct interparticle interactions. These can be quantified by the static structure factor S(Q) derived by dividing the measured data I(Q) by the form factor P(Q) as shown in Figure 18-10c by the open squares. The hard-sphere character of the interactions is confirmed by the fact that S(Q) is well described by a Percus-Yevick [87] structure factor shown by the solid line. Important differences to the dilute case are also observed in the dynamic behavior. Figure 18-10b shows the correlation rates in the concentrated sample (open circles) compared to the diffusive behavior in the dilute case (solid line). The inverse, normalized diffusion coefficient D₀/D(Q) is shown in Figure 18-10c (open circles). There is a pronounced maximum at Q ≈ 0.003 Å⁻¹ that coincides with the peak in the static structure factor showing that the most likely density fluctuations decay the slowest. It is furthermore evident that D₀/D(Q) is similar but not equal to S(Q). That shows that indirect, hydrodynamic interactions are important for the system and the corresponding hydrodynamic function H(Q) was derived using (25) with the result shown in Figure 18-10d .

Figure 18-10

(a) SAXS intensity as a function of Q for a concentrated hard sphere suspension of polymethylmethacrylate (PMMA) particles in cis-decalin with volume fraction φ = 0.37 [97]. The solid line describes the single particle form factor. (b) Q dependence of the relaxation rate Γ(Q) (open symbol) where the solid line shows the predicted behavior for a dilute (non-interacting) colloidal suspension undergoing free Brownian motion (Γ(Q) = D₀Q²). (c) Inverse, normalized effective diffusion coefficient D₀/D(Q) as a function of Q, compared to the static structure factor S(Q). The solid line is the calculated static structure factor using the Percus-Yevick expression for φ = 0.37. (d) Hydrodynamic function H(Q) = S(Q)/[D₀/D(Q)]. The solid line is the result of the model calculation based on the δγ-expansion (see text)

The hydrodynamic behavior of hard-sphere systems can be modeled within the δγ-expansion developed by Beenakker and Mazur [88]. Hydrodynamic functions, that depend only on the volume fraction φ and the static structure factor S(Q) can be derived analytically. The solid line in Figure 18-10d is the calculated hydrodynamic function H(Q) using the measured S(Q) as an input for the δγ-expansion The agreement with the experimental data is excellent.

Similar investigations were performed on concentrated charge–stabilized systems for which the dynamic behavior is considerably more complicated and far from being understood. This is illustrated by an investigation [33] of charge-stabilized poly-octafluoropentylacrylate colloidal particles suspended in a mixture of water and glycerol. Figure 18-11 shows the results for a volume concentration of Φ = 0.18. Small Angle X-ray Scattering (Figure 18-11a ) reveals a particle radius of 625 Å and size polydispersity ΔR/R = 0.048. A comparison of the scattered intensity and the single particle form factor (solid line in Figure 18-11a ) allows one to extract the static structure factor (open symbols) displayed in Figure 18-11c . A comparison with a calculated Percus-Yevick structure factor (dashed line in Figure 18-11c ) unambiguously shows that the system under study is not behaving hard-sphere like. A fit to the data within the Rescaled Mean Spherical Approximation (RMSA) is shown by the solid line. This model describes the data well and yields an effective number of charges per particle Z = 205e⁻ and a reduced screening length κσ = 3.14, where σ = 2R and κ is the inverse Debye-Hückel screening length.

Figure 18-11

(a) SAXS intensity (open symbol) for a concentrated charge-stabilized sample of poly-octafluoropentylacrylate at φ = 0.18 [33] compared to the corresponding form factor represented by the solid line. (b) Measured relaxation rates Γ(Q) as function of QR compared to free particle Brownian motion as indicated by the solid line. (c) Static structure factor S(Q) (open symbols). The solid line is a fit to the data using the RMSA approximation. The inverse, normalized effective diffusion coefficient D₀/D(Q) is shown for comparison (closed symbols). (d) Hydrodynamic function of the system compared to different models (dotted line: Pairwise Additive approximation; dashed line: δγ-expansion; solid line: modified δγ-expansion. See text for details.)

Intensity autocorrelation functions were measured with XPCS in the Q-regime surrounding the S(Q) peak. The functions were fitted by a single exponential and the resulting decay rate Γ is plotted (open symbols) as a function of QR in Figure 18-11b . It clearly deviates from the free particle diffusion coefficient (D₀ = 4.13·10⁻⁹ cm² s⁻¹) indicated by the solid line. The effective diffusion coefficient D(Q) is obtained using (22) and the inverse, normalized diffusion coefficient D₀/D(Q) (closed symbols) is displayed in Figure 18-11c .

The discrepancy between S(Q) and D₀/D(Q) can be attributed to the presence of indirect hydrodynamic interaction and their characterization requires the extraction of the hydrodynamic function H(Q) = S(Q)/[D₀/D(Q)] which is presented in Figure 18-11d . Modeling the hydrodynamics is difficult in this case. The Pairwise Additive approximation [92] (which describes the hydrodynamics of charge-stabilized colloidal systems at low concentrations) clearly fails as illustrated by the dotted line in Figure 18-11d . The possible use of the δγ-expansion, originally developed for hard sphere systems, was explored, as indicated by the dashed line with prior incorporation of the measured static structure factor S(Q). It equally fails to describe the experimental data. A phenomenological approach based on the “modified” δγ-expansion appears more promising if one allows for an increased effective viscosity as compared to a corresponding hard sphere system of identical volume fraction. The result of this treatment is shown by the solid line in Figure 18-11d . The resulting increase of the effective viscosity is attributed to the presence and strength of long-range repulsive coulomb interactions [89]. This study clearly points towards a need for more detailed data and a theoretical framework for describing the hydrodynamic interactions in concentrated charge-stabilized colloidal suspensions.

A variety of other combined XPCS and SAXS studies can be found in the literature. Among them are investigations of charge-stabilized Sb₂O₅ [98], suspensions of colloidal silica [26, 99] and colloidal latex particles [31, 100]. Furthermore there are studies of (aggregated) magnetic particles (ferrofluids) [101] and core-shell particles comprising a magnetic core [102, 103].

6.3 Slow Dynamics in Polymer Systems

The concepts of colloidal dynamics can in some cases be directly applied to polymer systems as has been shown in a study of spherical (polystyrene-polyisoprene: PS-PI) block copolymer micelles dispersed in a (polystyrene) polymer matrix and occupying about 30% of the volume [37]. This sample is solid below the glass transition temperature of PS (about 360 K) and shows static speckle in the SAXS experiment. At elevated temperatures the micelles are mobile and undergo Brownian motion. Figure 18-12 shows the Q-dependence of the static structure factor (solid line) with a peak close to 2π/d, where d is the diameter of the micelles. XPCS data were taken at two different temperatures (393 K and 398 K) and the inverse diffusion coefficient D(Q) normalized to temperature and viscosity is plotted for comparison in the figure. One observes again a slowing-down of the diffusion coefficient on the length scale defined by the dominating interparticle correlations.

Figure 18-12

Wavevector dependence of the measured diffusion coefficient at T = 393 K (open symbols) and T = 398 K (closed symbols), plotted as kT/(Dη) and as DηS(Q)/(kT) in the insert. The solid line is the calculated S(Q) and the dashed lines are guides to the eye [37]

XPCS allows one also to probe other types of dynamics occurring in polymer systems. Applying XPCS to polymer blends is more challenging than applying it to a colloidal system because of the much smaller scattering cross section in these systems. This is illustrated in an XPCS study of the dynamics of a homogeneous, binary blend of monodisperse, highly entangled chains of poly(ethylene oxide) (PEO) and poly(methyl-methacrylate) (PMMA) [38]. Here the non-diffusive relaxation of compositional fluctuations on length scales smaller than the extent of individual polymer coils was investigated on timescales encompassing the disentanglement time.

The reptation model depicts the primary motion of each polymer in such an entangled system as a creep along the length of a tube delimited by temporary entanglements with neighboring chains. XPCS thus provides a direct measure of reptative features in the diffusion of polymer blends. The dynamics was characterized via intensity autocorrelation functions of sequential two-dimensional speckle patterns, each obtained during a 5 s exposure of a CCD detector. The data were modeled by a single exponential decay from which the decay rate Γ was obtained, as shown in Figure 18-13 , as a function of Q²R² at 70°C. Here R denotes the radius of gyration of the polymer. The most striking feature of Figure 18-13 is that Γ varies only weakly with Q²R², in marked contrast to a variation of at least Q³ that would occur in this wavevector range in the absence of entanglements [104]. The data are thus consistent with a crossover from an approximately Q² variation at smaller wavevectors to a more “constant” behavior at larger wavevectors. This is commensurate with the predictions of the classical reptation model, as shown with the solid line in Figure 18-13 .

Figure 18-13

Relaxation rates (open symbols) at 70°C versus Q²R² for the binary entangled polymer blend PEO-PMMA [38]. The solid line corresponds to the predictions of the reptation model [104]

7 Liquid Surface Dynamics Studied by XPCS

It is well known that the surface of every liquid is covered by thermally excited capillary waves [105]. Their characteristics are fully determined by temperature, the surface tension γ, mass density ρ and the viscosity η of the fluid. The frequency of the capillary wave ω = iΓ + ω_p is an imaginary number where the propagation frequency is assigned to the real part ω_p while the imaginary part Γ yields the damping. Capillary waves have been studied by dynamic light scattering (DLS) [85, 106] but also by static X-ray scattering from various liquid surfaces [107–109]. Recent studies have covered the regime of small length scales (parallel to the surface) where continuum hydrodynamic theory fails to describe the observed diffuse surface scattering profiles [110, 111]. X-ray scattering, being a time-averaged technique, cannot reveal the dynamic behavior of the surface waves. This has changed since it was shown that XPCS can probe dynamics in grazing incidence surface scattering geometry [79, 112, 113]. XPCS has the advantage (compared to DLS) of the shorter wavelength which makes it possible to probe dynamics on shorter length scales. In addition, the refractive index for X-rays is smaller than unity and XPCS hence becomes extremely surface sensitive when the beam is impinging at a grazing angle of incidence α<α_C, where α_C is the critical angle for total external reflection.

7.1 Homodyne versus Heterodyne Detection

If the intensity scattered from the sample contains a static component, heterodyne mixing between the static and the dynamic part of the signal is possible. In grazing incidence geometry the momentum transfer parallel to the surface q _|| is zero at the specular reflection and this means that no lateral surface dynamics (e.g. capillary waves) is probed here. Hence, for liquid surfaces the specular reflection can be used as the required static reference signal for heterodyne mixing. If the scattered field E _s is brought to interfere with a static reference field E _r the intensity correlation function G takes the form

$$ {\rm{G(}} {\bf Q}{\rm{,\tau }})\;{\rm{ = }}\;{\rm{ < I(}}{\bf Q}{\rm{,t)I(}}{\bf Q}{\rm{,t}}+{\rm{\tau )}} > _{\rm{T}} \approx 2{\rm{I}}_{\rm{r}} {\rm{I}}_{\rm{s}} {\mathop{\rm Re}\nolimits} \{ {\rm{g}}_{\rm{1}} {\rm{(}}{\bf Q}{\rm{,}}\tau )\}\; + \;{\rm I}_s^2 {\rm g}_2 ({\bf Q},\tau )\; + \;{\rm c}_0 $$

((29))

Here I_r = |E _r|² and I_s = |E _s|² are the intensities of the reference signal and the scattered signal respectively, c₀ is a positive, time-independent term depending on I_r,I_s and the speckle contrast β, and g₁, g₂ are the first and second order correlation function of the scattered intensity [45]. The brackets <>_T denote time averaging over the acquisition time T and from (29) it is obvious that if I_r>>I_s (heterodyne detection) the intensity correlation function will depend mostly on the real part of g₁ given by:

$$ {\rm{g}}_{\rm{1}} {\rm{(}}{\bf Q}{\rm{,\tau ) = < }}{\rm E}_{\rm{s}} {\rm{(}}{\bf Q}{\rm{,t)}}{\rm E}_{\rm{s}} {\rm{(}}{\bf Q}{\rm{,t + \tau ) > }}_{\rm{T}} {\rm{/< I}}_{\rm{s}} {\rm{ > }} $$

((30))

For homodyne detection I_s>>I_r and then the intensity correlation function is dominated by g₂ given (see (14)) by:

$$ {\rm{g}}_{\rm{2}} {\rm{(}}{\bf Q}{\rm{,\tau )}}\;{\rm{ = }}\;{\rm{ < I}}_{\rm{s}} {\rm{(}}{\bf Q}{\rm{,t)I}}_{\rm{s}} {\rm{(}}{\bf Q}{\rm{,t}}{\rm{ + }}{\rm{\tau ) > }}_{\rm{T}} {\rm{/ < I}}_{\rm{s}} {\rm{ > }}^{\rm{2}} $$

((31))

For capillary waves Re{g₁(τ)} = cos(ω_pτ)exp(−Γt) and g₂(τ) = cos²(ω_pτ)exp(−2Γτ) + 1 [45] and hence there is a factor two in difference between the characteristic frequencies and relaxation rates obtained in homodyne and heterodyne mode. While heterodyne mixing can increase the number of recorded photons during the experiment from I_sT to I_rT, the normalized correlation function obtained in the experiment G(Q, τ) = βC(Q, τ)/I² will inevitably loose contrast. This can be seen from (29) where the contrast G(Q, 0) in heterodyne detection (I_r >> I_s) will go down by a factor of I_r/(2I_s) compared to the homodyne case (I_s >> I_r). Both, lack of statistics (too few recorded photons) and too low contrast can render the correlation function G undetectable, hence depending on the actual experimental conditions it can be favorable to be in either homodyne or heterodyne [45] detection mode. In the following examples ( Sections 7.2 –7.4) the relevant part of the momentum transfer is the component parallel to the surface denoted q _||.

7.2 Dynamics of Thin Polymer Films

The surface dynamics of thin polymer films depends on the thickness h of the film, which marks an important difference to the dynamic behavior of free liquid surfaces (c.f. Section 7.3). The dynamic behavior of polystyrene (PS) thin films (h = 800–3,000 Å) on silicon substrates has been investigated [42] with the aim to study the extent to which the polymer films develop inhomogeneities that may affect the viscosity deduced from the XPCS measurements. PS is a glass forming polymer and for the chain length used (molar mass 123 kg/mol) the glass transition temperature T_g is less than 150^°C. Three temperatures 150, 160 and 170^°C, were used during the experiment which was carried out in grazing incidence geometry with a horizontal scattering plane. The incidence angle was kept constant at about 90% of α_C, the critical angle for total external reflection (∼0.16^° for 7.66 keV). A direct illumination, deep depletion CCD camera was used in kinetics mode [80] to record the off-specular scattering images from which the intensity autocorrelation functions could be determined at various q_||, the momentum transfer parallel to the surface. In the experiment q_|| ranged from about 2·10⁻⁴ to 1·10⁻³ Å⁻¹. The autocorrelation functions were all found to exhibit a strong exponential decay in good agreement with the expression β exp(−t/τ) + 1 where β is the speckle contrast and τ is the relaxation time. The dependence of τ on q_|| is illustrated in Figure 18-14 for different temperatures (a) and film thicknesses (b).

Figure 18-14

The relaxation time versus the momentum transfer q_|| parallel to the surface. (a) For a 177 nm thick film at three different temperatures. (b) For three different film thicknesses at the same temperature, 160°C [42]

The dynamic behavior of viscoelastic thin films including the effects of finite thickness has been calculated [114] and it was found that τ/h is a function of q_||h and is directly proportional to the ratio of the viscosity over the surface tension η/γ. To test this prediction, the quantity τ/h was plotted versus q_||h for the various film thicknesses and temperatures as shown in Figure 18-15 . The data points can be modeled as shown in the figure. The lines are fits to the predicted behavior [42] with only one fit parameter, namely η/γ. The surface tension γ of a film can be measured simultaneously by static X-ray scattering and hence the viscosity η determined for various temperatures. The results indicate that the capillary wave theory with its wave vector and thickness independent viscosity η is well suited for describing the PS films in the range of temperatures and thicknesses studied here.

Figure 18-15

Plot of τ/h versus q_||h for the investigated PS films: h = 84 nm (circles), ∼170 nm (triangles) and ∼320 nm (diamonds). The data points fall onto three curves and can be modeled with one adjustable parameter, the viscosity η, which depends on the temperature [42]

7.3 Dynamic Cross-Over Behavior of Liquid Mixtures

On less viscous liquids a transition from over-damped to propagating behavior can be observed in the surface dynamics. This transition has been quantified for the first time on a mixture of water and glycerol which was investigated by XPCS at beamline ID10A at the ESRF [44]. The experiments were carried out in vertical scattering geometry with a coherent beam impinging on the liquid surface. The angle of incidence was chosen below the critical angle α_C of total external reflection in order to be strictly surface sensitive and in this way only the topmost ∼100 Å of the sample was probed. In Figure 18-16 three correlation functions taken at identical q_|| but at different temperatures are shown. The data were recorded with help of a fast avalanche photodiode detector that allowed for count rates of several MHz without the need for any dead time correction. The signal from the detector was sent to a fast hardware correlator for computation of the intensity auto-correlation function. The data taken at 30^°C clearly shows the fingerprint of a propagating wave while the correlation function measured at 5^°C indicates over-damped behavior. This distinction is not obvious for the correlation function recorded at the intermediate temperature (Figure 18-16 ).

Figure 18-16

Three autocorrelation functions taken at different temperatures (curves at 12 and 30°C shifted by 0.05 and 0.1 respectively along the ordinate). The momentum transfer parallel to the surface q_|| is 6·10⁻⁶ Å⁻¹ and the shape of the correlation function indicates a transition from an oscillating (propagating) to a simple exponential (over-damped) behavior as a function of temperature [44]

The dynamic response of a simple liquid surface to an external force may be evaluated by linear response theory which allows the calculation of [115] the dynamic structure factor S(Q, ω) and one can show that the cross-over from propagating to over-damped behavior is expected to happen at a critical wave vector k_C = 4γρ/(5η²) [44]. Capillary modes with k > k_C are overdamped while k < k_C result in propagating waves.

In Figure 18-17 the results of the measurements at 12^°C are shown. The determination of ω_p (open circles) and Γ (open squares) relies on an inverse Fourier transformation of the correlation functions, as described in Ref. [44], assuming that the detection is heterodyne. The data indicate that the dynamic behavior of the surface is determined by propagating capillary waves at low q_|| because ω_p ≠ 0 and the dashed lines in Figure 18-17 illustrate the predictions of the classical models ω_p = q_|| ^3/2(γ/ρ)^1/2 and Γ = 2ηq_|| ²/ρ valid in the limit of small damping. Obviously there is only poor agreement with the data because the small-damping model assumes a Lorentzian shape of S(q, ω) which is not the case in this cross-over region. The dispersion relation for the propagation frequency ω_p shows a maximum at 3k_C/4 after which it curves over and decreases rapidly to 0 at q_|| = k_C (solid line, Figure 18-17 ). These data are the first experimental verification of the predicted behavior [44]. Furthermore at the same time, the damping changes from Γ∝q_|| ² to a Γ∝q_|| behavior (solid line, Figure 18-17 ) which is a further indication of the transition from propagating to over-damped modes.

Figure 18-17

Propagation frequency (circles) and damping constant (squares) versus momentum transfer q_|| for the liquid mixture at 12°C. The data are well modeled by linear response theory (solid line) and from the measurements a transition from propagating to over-damped capillary wave behavior is evident at q_|| = k_c = 8·10⁻⁶ Å⁻¹ [44]

7.4 Critical Dynamic Behavior of a Liquid Crystal Surface

For non-isotropic molecular liquids the dynamic properties of the surface may deviate significantly from the above described cases. This is for instance true for liquid crystals where the second order nematic-to-smectic phase transition is affecting the free surface dynamics. For the liquid crystal compound 8OCB the static ordering associated with the smectic phase has been investigated by classical grazing incidence X-ray scattering from the free surface. The occurrence of smectic surface layers in the nematic phase is a second order process and a power-law behavior of the associated correlation length has been found in several studies [116, 117]. The static ordering, however, has also implications for the dynamic behavior of the surface and X-ray reflectivity and XPCS have been applied to investigate the simultaneous change in static layering and dynamic behavior of the 8OCB surface [43]. 8OCB is an elongated liquid crystal molecule that tends to form a layered (smectic) structure with the long axis n of the molecules perpendicular to the layers. For such an anisotropic system the viscosity is described by three viscosity coefficients η₁, η₂ and η₃ depending for instance on the orientation between n and the wave vector q of the excitation (q parallel to the surface).

The fact that the molecules stand upright at the free surface of 8OCB [117] implies that the effective viscosity η_eff for capillary wave flow is rather high and contains a component η₃ which diverges at the nematic-smectic phase transition. Typical XPCS data are shown in Figure 18-18 and for all probed momentum transfers q_|| and temperatures, the recorded correlation functions could be well fitted assuming simple over-damped capillary wave dynamics. The slope of the straight line in Figure 18-18 is inversely proportional to the effective viscosity η_eff. Upon approaching the critical temperature T_NA of the transition, η_eff diverges resulting in a critical slowing down of the dynamics. This is illustrated in Figure 18-19 where η_eff is plotted versus t = (T−T_NA)/T_NA, the reduced temperature. The critical behavior of η₃ can be extracted using η₃ = 4(η_eff ²/η₁−η₁) [43], and is also shown in Figure 18-19 . A power law fit to the data At^−x + B gives the critical exponent x = 0.95(5). This represents the first experimental evidence for the theoretical prediction x = 3ν_||−2ν_⊥ [118] where ν_|| (0.70) and ν_⊥ (0.58) are the critical exponents associated with the correlation lengths of the smectic domains parallel and perpendicular to n, respectively, as determined by static X-ray scattering [43, 116]. This example illustrates the power of combining static and dynamic (XPCS) X-ray scattering to obtain new insight into the correlations between ordering phenomena and dynamic behavior of condensed matter.

Figure 18-18

XPCS data demonstrating the linear relation (solid line) between relaxation rate and momentum transfer for over-damped capillary waves. The slope of the line is inversely proportional to the effective viscosity η_eff. The inset shows a typical correlation function which is well described by an exponential decay g₂(τ)∼βexp(−τ/τ₀) + 1 (solid line) [43]

Figure 18-19

Critical behavior of η₃ (squares) extracted from the measured behavior of η_eff (circles) as described in the text. The non-critical Arrhenius-type behavior of η₁ is also shown (dash-dotted line). The solid line is a fit to a power-law behavior and yields a critical exponent x = 0.95(5) as indicated [43]

8 Slow Dynamics in Hard Condensed Matter Systems

Coherent X-rays can probe disorder and dynamics in hard condensed matter systems on the same lengthscale range that is accessible in a conventional X-ray scattering experiment. Phenomena of interest include the dynamics accompanying domain formation in phase separating alloys or glasses, the dynamics of glass forming systems and critical dynamics or disorder in magnetic materials.

The time dependence of equilibrium critical fluctuations has been studied in the binary alloy Fe₃Al close to the continuous order disorder transition at T_c = 824 K [27]. In the ordered state (below the transition) the sample consists out of a random arrangement of long range order domains and the characteristic (1/2,1/2,1/2) superlattice reflection is thus a static speckle pattern when recorded with a coherent X-ray beam. The time correlation functions taken just below the transition are constant g(t) − 1 = 0 indicating the absence of fluctuations. The scattering from the superlattice reflection persists just above the transition due to short range order fluctuations and the time correlation functions taken 0.15 K to 0.55 K above T_c show g(t)-1 > 0 with correlation times of about 1,000 s, which is consistent with the expectations from theory. These results illustrated the potential of XPCS to study critical dynamics even if the data did not yet allow to extract a value for the dynamic critical exponent. More recently, measurements with coherent X-rays were furthermore performed on the Co₆₀Ga₄₀ intermetallic alloy to study antiphase domain dynamics [47] and in AlPdMn quasicrystals to study phason fluctuations [119].

Non-equilibrium fluctuations during domain coarsening were studied in a phase separating sodium borsilicate glass [28]. The sample with a critical composition for phase separation was allowed to equilibrate at high T in the single phase state, then quenched to a lower temperature and allowed to separate isothermally into B₂O₃-rich and SiO₂-rich domains.

These domains grow as a function of time. The theory of dynamic scaling predicts that the average structure factor does not change if the length scales are measured in units of the average domain size. This is in fact observed and Figure 18-20 shows speckle patterns (1 s acquisition time) during the phase separation process at 963 K, taken 100, 400 and 1,200 s after the quench from 1,033 K. Fluctuations about the average intensity can be quantified by means of a two time (t₁, t₂) intensity correlation function

$$\eqalign{ {\rm{C(Q,t_1 ,t_2)}} = & {\rm < I(t_1)I(t_2) > - < I(t_1) > < I(t_2)>} \over {\rm[< I^2 (t_1) > - <I(t_1) >^2 ]^{1/2} [< I^2(t_2) > - < I(t_2 ) > ^2 ]^{1/2}}} $$

((32))

Figure 18-21 shows a contour plot of a two-time correlation function for the example discussed in Ref. [28]. Values of C for each contour and the directions of the alternative coordinates t_mean = (t₁ + t₂)/2 and Δt are indicated. It is readily observed that the correlation time τ = Δt/2 increases as a function of increasing absolute time t_mean thus showing that the dynamics is getting slower as a function of t_mean. It was shown in the experiment that the correlation times actually obey a scaling law in agreement with the model expectations [120]. Similar experiments were carried out in phase-separating AlLi alloys and analyzed successfully in terms of two-time correlation functions [121]. Ordering kinetics [122] and coarsening dynamics [123] were studied in metallic alloys.

Figure 18-20

Speckle patterns produced by a coherent X-ray beam scattered from a sodium borsilicate glass sample undergoing phase separation at 963 K. One-second exposures at 100, 400, and 1,200 s after the quench from 1,033 K are shown [28]

Figure 18-21

Contour plot of two-time correlation function at Q = 0.01 Å in borsilicate glass after a quench to 963 K. Δt refers to the correlation time and t = t _mean = (t ₁ + t ₂ )/2 [28]

Coherent X-rays have also been used to explore disorder in magnetic systems. The scattering amplitude b_n(Q) as derived within second order perturbation theory [124] contains contributions sensitive to the magnetization of each scatterer which gives rise to X-ray magnetic scattering. Two limiting cases of magnetic scattering can be discerned from such a derivation: Resonant scattering (with the incident photon energy tuned near an absorption edge of a magnetic species in the sample) and non-resonant scattering (with the photon energy far from an absorption edge). The scattering amplitude may thus be written as

$$ {\rm{b}}_{_{\rm{n}} } {\rm{(}}{\bf k}{\rm{,}}{\bf k}{\rm{',{\rm\hbar}\omega )}}\;{\rm{ = }}\;{\rm{b}}_{\rm{n}} ^{{\rm{charge}}} {\rm{(}}{\bf Q}{\rm{)}}\;{\rm{ + }}\;{\rm{b}}_{\rm{n}} ^{{\rm{non - resonant}}} {\rm{(}}{\bf k}{\rm{,}}{\bf k}{\rm{', }}{\rm\hbar} {\rm{\omega ) + b}}_{\rm{n}} ^{{\rm{resonant}}} {\rm{(}}{\bf k}{\rm{,}}{\bf k}{\rm{',{\rm\hbar}\omega )}} $$

((33))

where the first term describes non-resonant charge scattering given by b_n ^charge = −ρ_n(Q) ε·ε’. Here Q = k′–k is the momentum transfer, ħω is the energy of the photons, ε,ε’ are the incident (scattered) photon polarization states and ρ_n(Q) is the Fourier transform of the electronic charge density. The non-resonant scattering amplitude may be written as

$$ {\rm{b}}_{\rm{n}} ^{{\rm{non - resonant}}} {\rm{(}}{\bf k}{\rm{,}}{\bf k}{\rm{',}}{\rm\hbar} {\rm{\omega )}}\;{\rm{ = }}\;{\rm{ - i({\rm\hbar}\omega /mc}}^{\rm{2}} {\rm{)[1/2 \cdot L}}_{\rm{n}} {\rm{(}}{\bf Q}{\rm{) \cdot }}{\bf A}\;{\rm{ + }}\;{\rm{S}}_{\rm{n}} {\rm{(}}{\bf Q}{\rm{) \cdot }}{\rm B}{\rm{]}} $$

((34))

where L_n(Q) and S_n(Q) are respectively the Fourier transform of the orbital and spin magnetization densities and the vectors A and B contain the polarization dependencies. The square of the pre-factor (ħω/mc²) is about 10⁶ times smaller than the pure charge scattering intensity at typical X-ray energies (5–10 keV). Non-resonant scattering experiments are thus challenging and, with a coherent beam, very difficult.

The resonant scattering amplitude b_n ^resonant(k,k′,ħω) is a complicated object describing both resonant charge and (resonant) magnetic scattering. The latter occurs if the wave-vector transfer is set to a magnetic wave-vector, the energy is tuned close to a specific absorption edge and at least one of the levels involved in the resonance is subject to spin-orbit correlations and/or exchange effects. Resonant magnetic scattering has been observed in the rare earths, the actinides and the transition metals at the L and M absorption edges, respectively. Dramatic enhancements of the scattering intensity, up to factors of 10⁴ have been detected experimentally and all magnetic scattering experiments with coherent beams have up to now been carried out in resonant mode.

Figure 18-22 shows a magnetic small angle scattering pattern (first order magnetic diffraction annulus) from meandering magnetic stripe domains in a 350 Å thick film of GdFe₂ illuminated by a 15 μm diameter beam of circularly polarized X-rays. The annulus is only observed if the incident beam energy is tuned to the Gd M_V resonance at 11 Å wavelength (1,183.6 eV) thus supporting the pure magnetic character of the observed scattering [65]. Similar magnetic speckle patterns had been observed in FePd stripe domain alloys [125], in actinide compounds [64] and manganite samples [126]. Common to all these experiments is the use of low energy X-rays tuned to the M absorption edges of the magnetic species.

Figure 18-22

Magnetic speckle pattern on the first order magnetic diffration annulus from meandering magnetic stripe domains in a 350 Å thick film of GdFe₂ illuminated by a 15 μm diameter beam of circularly polarized X-rays tuned to the Gd M_V resonance at 1,183.6 eV [65]

The reconstruction of soft X-ray speckle pattern is discussed in several publications [127] and applications to the magnetic case are found in refs. [63, 128]. Quasi-static speckle metrology is described in [129]. Slow antiferromagnetic domain fluctuations in Cr have recently been reported [130]. Most relevant dynamic phenomena in magnetic systems such as demagnetization processes or magnetization reversal however are fast (fs to ps) and thus very difficult to access with today’s partially coherent sources and 2-D detectors. The study of magnetization processes will however become feasible when intense free electron laser sources become available.

9 Conclusions and Outlook

Scattering with coherent X-rays and X-ray photon correlation spectroscopy (XPCS) in particular have created considerable interest in the community of synchrotron users and start to impact several scientific fields. Progress in reconstructing X-ray speckle patterns has encouraged scientists to work on magnetically disordered systems and fostered dreams to study magnetization dynamics. Biologists are dreaming about resolving the structure of small macromolecular assemblies. XPCS is progressing in both the hard and soft condensed matter domain and correlation times from tens of nanoseconds to hundreds of seconds have been recorded. The longterm success of the technique will crucially depend on the development and use of fast two-dimensional detectors to exploit fast dynamics also at large Q. Appropriate detectors will also be crucial for the study of non-equilibrium phenomena and for soft condensed matter materials that are sensitive to radiation damage effects. XPCS has successfully been extended to the soft X-ray region. Yet unexploited features of coherent beams are the polarization, polarization tunability and the possibility to tune the energy of the beam. Finally, one can anticipate new, revolutionary applications [131] from novel X-ray sources such as the X-Ray free electron laser (FEL) sources [53] that are expected to start operation in this decade.