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Abstract
The ability of pulsed nature of synchrotron radiation opens up the possibility of studying microsecond dynamics in complex materials via speckle-based techniques. Here, we present the study of measuring the dynamics of a colloidal system by combining single and multiple X-ray pulses of a storage ring. In addition, we apply speckle correlation techniques at various pulse patterns to collect correlation functions from nanoseconds to milliseconds. The obtained sample dynamics from all correlation techniques at different pulse patterns are in very good agreement with the expected dynamics of Brownian motions of silica nanoparticles in water. Our study will pave the way for future pulsed X-ray investigations at various synchrotron X-ray sources using individual X-ray pulse patterns.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Speckle-based techniques are well established to investigate sample dynamics in soft matter. The high coherence beam leaves an imprint of the spatial distribution of the subjected system on a diffraction image as a grainy interference pattern at the detector plane, which is typically called “speckle”. The movement of disordered particles shifts the relative phases of the scattered field and induces temporal variations in the speckle patterns. Consequently, these fluctuating speckle intensities reflect the information of the system dynamics, which can be quantified by a correlation function. For many decades, optical laser systems have been widely employed to generate speckle patterns for photon correlation spectroscopy (PCS) [1–3]. Thanks to the improvement of the coherent X-ray flux from synchrotron sources, X-rays have been employed for speckle-based measurements of dynamics on the atomic scale [4] or optically turbid system [5].
One of the most robust techniques, called X-ray photon correlation spectroscopy (XPCS), is a powerful and convenient tool for measuring dynamics. The temporal resolution of the technique is defined by the number of images taken in time. In the past decades, considerably slow systems have been in the focus of XPCS studies [6–14]. Recently, new advanced detector technology has begun to offer 2D detectors with the microsecond temporal resolutions in XPCS measurements (e.g., VIPIC [15], UFXC32k [16], Tristan detector [17], and the XSPA-500k [18]). Recently developed X-ray detectors offering MHz frame rates, such as adaptive gain integrated pixel detector (AGIPD), enable to investigate of nanosecond colloidal dynamics at 3$^{\text {rd}}$ generation X-ray synchrotron sources [19] and X-ray free electron laser (XFEL) sources [20,21]. In order to measure sample dynamics faster than accessible with XPCS, X-ray speckle visibility spectroscopy (XSVS) has been developed, which evaluates a speckle contrast as a function of illumination time. The longer illumination time compared to system dynamics leads the more blurred speckle images and a loss in contrast of the speckle images. This concept was introduced in optical laser systems [22] and extended to X-ray regime [23–27].
An alternative speckle technique is double pulse XPCS (DP-XPCS). The technique evaluates a speckle contrast of two summed speckle patterns separated with a delay time, which corresponds to the temporal resolution. DP-XPCS has been demonstrated with sequentially obtained speckle patterns [28] and an optical pulse laser scheme [29]. More recently, DP-XPCS has been employed in a special operation mode of linear accelerators of XFEL [30,31] or X-ray split-and-delay unit [32–42] to investigate nanosecond dynamics beyond detector frame rate limitations.
In this paper, we demonstrated XPCS measurements using single and multiple X-ray pulses from nanoseconds to milliseconds time scales. Thanks to the MHz frame rate of the detector, we investigated the sample dynamics with 192 ns temporal resolution in a single pulse mode. In addition, we covered the gap between nanoseconds to milliseconds using the multiple X-ray pulse modes in the correlation function. The obtained sample dynamics are in excellent agreement with the expected dynamics of the system.
2. Materials and methods
2.1 Experiments
The experiment was conducted at the coherent application beamline P10 of PETRA III [43]. The monochromatic 8 keV X-ray beam was focused vertically and horizontally down to $d_b$ = 2.5 $\mu$m at the sample position. The speckle size $d_s$ = 310 $\mu$m was estimated at the detector plane located at a distance $L$ = 5 m, according to $d_s = \lambda L / d_b$, where $\lambda$ is the X-ray wavelength. The silica colloidal nanoparticles (NPs) with a core radius $R$ of 65 nm were dispersed in water and filled in a 700 $\mu$m thick quartz capillary. The SPARTA detector, which was developed by the X-Spectrum GmbH with AGIPD technology [44–46] was employed to measure the sample dynamics. The detector is composed of 128 $\times$ 512 pixels with a pixel size of 200 $\mu$m $\times$ 200 $\mu$m. It can collect up to 352 memory cells (i.e., frames) at a frame rate in the MHz range in a single burst (i.e., measurement). The images have to be read out until the next MHz acquisition burst can be triggered 0.1 s later. Note that the number of X-ray pulses ($N$) between two consecutive frames adjusts the time resolution and the achievable time window in correlation functions. In this work, we selected $N =$ 1, 5, and 40 to capture the sample dynamics (see Supplement 1 and Table 1). The detector signal was carefully extracted using a background subtraction method that is elaborated elsewhere [19,21] and subsequent conversion of the cleaned signal to an integer number of photons. An additional pixel mask was applied in the analysis to exclude detector pixels with high electronic noise. The depicted speckle patterns after the data conversion process to photons are shown in Fig. 1(b).
Fig. 1. (a) Schematics of the speckle measurement experiment employing the intrinsic pulse structure of PETRA III. Coherent X-ray pulses are delivered at the sample position with 192 ns temporal separation. SPARTA collects 352 images in a single measurement, and the delay time between each frame is set by the number of X-ray pulses between two consecutive frames $N$. (b) Speckle images of XSVS and DP-XPCS for $N_{40}$ with varying number of frames $n$.
Table 1. Data collection modes of SPARTA. $N$ and $\Delta t$ denote the number of X-ray pulses between two successive frames and the time range of correlation functions, respectively.
2.2 X-ray photon correlation spectroscopy
Three complementary XPCS techniques (i.e., XPCS, XSVS, and DP-XPCS) were employed to evaluate the sample dynamics obtained from different collection modes.
2.2.1 XPCS
The intensity fluctuations in time are caused by the spatial rearrangement of colloidal particles. The sample dynamic properties are traced by the intensity autocorrelation function together with the Siegert relation [47,48]
The obtained correlation function provides the contrast $\beta ^{C}(Q,\Delta t)= \beta _0\exp (-2\Gamma (Q) \Delta t)$ that starts with $\beta _0$ and decreases as a function of $\Delta t$ due to the sample dynamics.
2.2.2 X-ray speckle visibility spectroscopy
In order to investigate system dynamics using XSVS from sequentially obtained speckle patterns, a speckle image with an exposure time $t_e$ is defined as $I^{V}(Q,t_e) = \int _{t}^{t+t_e}I(Q,t')dt' = \sum _{n}^{n+n_e} I(Q,n)$, where $t_e = 190$ ns $\times ~n_eN$, and $n_e$ is the number of frames corresponding to $t_e$. The extending of $t_e$ does not affect the speckle contrast until the sample characteristic time $\Gamma ^{-1}$ is sufficiently smaller than $t_e$. In contrast, when $t_e > \Gamma ^{-1}$, the speckle image is washed out, and consequently, the contrast decreases. In Fig. 1(b), the degrading of speckle image clarity as a function of $n$ is clearly observed, and the normalized variance $\beta \equiv M^{-1} = \sigma ^2(I^{V})/\langle I^{V} \rangle ^2$, where $M$ is a number of speckle mode in the intensity is given by [22,51,52]
In case of the simple exponential decay function $f(Q,\Delta t)$ = $\exp (-\Gamma \Delta t)$, Eq. (4) can be solved analytically as
Note that the Eq. (5) shows more moderate decay in contrast compared to the $\beta ^C$.
2.2.3 Double pulse XPCS
Double pulse speckle contrast $\beta$ from the sum of two speckle patterns $S(Q,\Delta t) = I(Q,t) +I(Q,t+\Delta t)$ enables to quantify the level of displacements of colloidal particles within the delay time $\Delta t$. The contrast is given by the normalized variance of intensity distribution at a wavevector $Q$ evaluated as [28]
For weak speckle patterns, the photon counting statistics should be accounted for. The probability distribution of the number of photons per pixel $k$ in a speckle pattern is described by the negative binomial distribution function [53]
In the negative binomial distribution, the normalized variance can be expressed as $M^{-1} = \sigma ^2(k)/\langle k \rangle ^2-1/\langle k\rangle$ [4].
3. Result and discussion
Figure 2(a) shows the averaged intensity collected at the $N_1$, $N_5$, and $N_{40}$, respectively. A total of 4700 bursts were employed to obtain a sufficient signal-to-noise ratio for $N_1$ and $N_5$, and 470 bursts were used for $N_{40}$. The azimuthally integrated intensity profiles are shown in Fig. 2(b). The intensity profiles fall off as a function of $Q$. The particle formfactor ring is well pronounced at $Q =$ 0.09 nm$^{-1}$ for $N_{40}$, indicating a system of sufficiently dilute and non-interacting silica particles. A single-sphere formfactor fit to the intensity profile obtained with $N_{40}$ yields $R$ = 65.2$\pm$0.3 nm and size polydispersity $\Delta R/R$ = 7.41 $\%$.
Fig. 2. (a) 2D speckle pattern taken by SPARTA for $N_{\text {1}}$, $N_5$, and $N_{40}$. (b) Corresponding azimuthally integrated intensity profiles as a function of $Q$. The black dotted line represents the formfactor fit, taking into account the sample polydispersity ($\Delta R/R$).
We divided the detector plane into regions of interest (ROI) that are limited by contours of equivalent wavevector $Q$ for further analysis.
Figure 3 shows $g^{(2)}$ obtained from $N_1$, $N_5$, and $N_{40}$ using Eq. (1). The dashed lines represent expected dynamics according to Eq. (3) where the initial contrast at $\Delta t \rightarrow$ 0 is $\beta _0$ = 0.2. A considerably larger error in $N_1$ is due to the lower photon count rates. Nevertheless, resolving a single X-ray pulse in XPCS offers a 192 ns temporal resolution in this pulse pattern [19].
Fig. 3. Intensity correlation functions as a function of delay time obtained from $N_1$ (green-square), $N_5$ (orange-triangle), and $N_{40}$ (blue-circle) for $Q = 0.02$ nm$^{-1}$ (a) and 0.028 nm$^{-1}$ (b). The dashed lines indicate the expected correlation functions using Eq. (3).
Figure 4 shows a square of ISF obtained from XPCS ($g^{(2)}$: filled markers) and DP-XPCS ($c^{(2)}$: empty markers) in $N_{40}$ for various $Q$ values. The ISF from $g^{(2)}$ extracted from Eq. (1) decays as a function of delay time representing the motion of silica particles. The obtained initial contrast $\beta _0$ from $g^{(2)}$ was employed to Eq. (6) to extract ISF and $\alpha$ from $c^{(2)}$. The black dashed lines represent dynamics of Brownian motion in freely diffusing silica particles in water with the relaxation rate $\Gamma$ of 1315, 2578, and 4262 $s^{-1}$ at $Q =$ 0.02, 0.028, and 0.037 nm$^{-1}$, respectively.
Fig. 4. The obtained intermediate scattering functions from XPCS (circle) and DP-XPCS (square) for varying $Q$. The black dashed lines represent the expected curves of a single exponential decay curve from Eq. (2).
We calculated speckle contrast as a function of exposure time to obtain the X-ray speckle visibility using the normalized variance. In contrast to the XPCS and DP-XPCS, the image $I^V(Q,t_e)$ for XSVS is a result of the summation of all the memory cells between the corresponding delay time. Figure 5 shows speckle visibility contrast as red squares for $Q$ = 0.02 (a), 0.028 (b), and 0.037 (c), respectively. Also, XPCS results are shown as blue circles at corresponding $Q$ in $N_{40}$. The dotted and dashed lines for the XPCS and XSVS experiment are the fit using Eq. (3) and Eq. (5), respectively. The extracted $\Gamma$ as a function of wavevector transfer is in excellent agreement for all three approaches in Fig. 6. This result shows that all the methods are well suited for pulsed-based studies.
Fig. 5. Combined XPCS $g^{(2)}$-1 and speckle visibility contrast $\beta _{V}$ at the wavevector transfer $Q$ of (a) $0.02$ nm$^{-1}$, (b) $0.028$ nm$^{-1}$, (c) $0.037$ nm$^{-1}$. The dashed and dotted lines are the fit result using Eq. (3) and Eq. (5), respectively.
Fig. 6. Extracted $\Gamma$ as a function of $Q$. The lines are the fitting results using $\Gamma (Q) = D_0 Q^2$.
Dynamics measured via pulsed-based XPCS in a model colloidal model system of silica particles in water mark an important step in adopting various pulse patterns of storage rings in XPCS measurements. A higher coherent flux available with diffraction-limited storage rings (DLSR) will extend the access to study weakly scattering, but very relevant systems [54]. The possibility of employing a time structure of DLSRs paves the way to studying dynamics on nanosecond to microsecond timescales bridging the experimental gap between 3rd generation synchrotron sources and XFEL sources in investigating more complex systems, in particular, fast relaxation processes in glasses [11], atomic diffusion in metal alloys [55], soft matter [20] and biological systems [56].
4. Conclusions
We have demonstrated pulsed XPCS at a synchrotron source by employing various pulse patterns of PETRA III. The diffusion coefficients obtained from the three speckle correlation techniques, i.e., XPCS, XSVS, and DP-XPCS, show very good agreement with the expected dynamics of the colloidal system. The SPARTA detector capability of MHz frame rates has provided a way of conducting such experiments with very short time resolutions, spanning the range from 190 ns over microseconds to milliseconds to cover a fast but also wide range of times in correlation functions. These studies will provide great potential for X-ray speckle-based investigation for all synchrotron facilities operated by individual X-ray repetition rates and, in particular, diffraction-limited storage rings.
Funding
Deutsche Forschungsgemeinschaft (SFB 925 170620586.) Cluster of Excellence "CUI: The Hanburg Centre for Ultrafast Imaging," (ECX 1074 194651731), Cluster of Excellence "CUI: Advanced Imaging of Matter," (EXC 2056 390715994); Hamburgishe Investitions und Forderbank (IFB); XSpectrum GmbH (51090010.);
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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