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Dynamics and Time Scales of Higher-Order Correlations in Supercooled Colloidal Systems
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Dynamics and Time Scales of Higher-Order Correlations in Supercooled Colloidal Systems
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  • Nele N. Striker*
    Nele N. Striker
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
    *[email protected]
  • Irina Lokteva
    Irina Lokteva
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
    The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
  • Michael Dartsch
    Michael Dartsch
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
    The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
  • Francesco Dallari
    Francesco Dallari
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
  • Claudia Goy
    Claudia Goy
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
    More by Claudia Goy
  • Fabian Westermeier
    Fabian Westermeier
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
  • Verena Markmann
    Verena Markmann
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
  • Svenja C. Hövelmann
    Svenja C. Hövelmann
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
    Institut für Experimentelle und Angewandte Physik, Christian-Albrechts-Universität zu Kiel, Leibnizstraße 19, 24098 Kiel, Germany
  • Gerhard Grübel
    Gerhard Grübel
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
    The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
  • Felix Lehmkühler*
    Felix Lehmkühler
    Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
    The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
    *[email protected]
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The Journal of Physical Chemistry Letters

Cite this: J. Phys. Chem. Lett. 2023, 14, 20, 4719–4725
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https://doi.org/10.1021/acs.jpclett.3c00631
Published May 12, 2023

Copyright © 2023 The Authors. Published by American Chemical Society. This publication is licensed under

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Abstract

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The dynamics and time scales of higher-order correlations are studied in supercooled colloidal systems. A combination of X-ray photon correlation spectroscopy (XPCS) and X-ray cross-correlation analysis (XCCA) shows the typical slowing of the dynamics of a hard sphere system when approaching the glass transition. The time scales of higher-order correlations are probed using a novel time correlation function gC, tracking the time evolution of cross-correlation function C. With an increasing volume fraction, the ratio of relaxation times of gC to the standard individual particle relaxation time obtained by XPCS increases from ∼0.4 to ∼0.9. While a value of ∼0.5 is expected for free diffusion, the increasing values suggest that the local orders within the sample are becoming more long-lived for larger volume fractions. Furthermore, the dynamics of local order is more heterogeneous than the individual particle dynamics. These results indicate that not only the presence but also the lifetime of locally favored structures increases close to the glass transition.

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Copyright © 2023 The Authors. Published by American Chemical Society
When a supercooled liquid approaches the glass transition, its relaxation time increases by several orders of magnitude while the structure remains liquid-like. (1−5) Although there has been extensive research into the nature of the glass transition, its mechanisms are mostly unclear. One of the key concepts appears to be dynamical heterogeneity, i.e., the existence of regions with different relaxation times in the system, which occur frequently upon supercooling. (6−9) This is often discussed together with the appearance of locally favored structures (LFS), such as icosahedral structures forming in this regime. (10−12)
Studies of the structure and dynamics of liquids and glasses show that a local order develops when approaching the glass transition. (13−18) Many studies investigate these phenomena using colloidal systems, as their structure can be analogous to that in atomic systems and their phase behavior, including approaching the glass transition, can be controlled by different parameters such as volume fraction ϕ. (19) A recent study showed a link between the number of tetrahedral clusters in hard sphere mixtures and their diffusivity, introducing a way to predict dynamics within the system. (20) This again demonstrated the important role that the formation of LFS plays in the glass transition. However, although the structural properties of LFS are the subject of many studies, little is known about the time scales and lifetime of LFS and other multiparticle clusters. These can be investigated by higher-order correlations in space and time that, e.g., have been used to reveal a two-step crystallization process in hard sphere colloidal systems via detecting local order. (21−24) One way to access such higher-order correlations is coherent X-ray scattering.
In general, coherent X-ray scattering methods such as X-ray photon correlation spectroscopy (XPCS) (25−28) and X-ray cross-correlation analysis (XCCA) (29−31) are well suited to studying the dynamics and structure of supercooled liquids and glasses because they allow investigation of intrinsic time and length scales, down to molecular dimensions.
Recently, a combined XPCS and XCCA study revealed correlations between the slowing of the dynamics and an increase in structural higher-order correlations preceding the glass transition of colloidal hard spheres. (32) Because of the increase in the brilliance of current large-scale X-ray sources, new correlation functions can be probed that will enable studies of structure–dynamics correlation over wide time and length scales. (33) In this work, we use such a newly defined higher-order correlation function combining XPCS and XCCA, which gives access to the lifetime of local order in a colloidal liquid. In the fluid state, the dynamics of the colloids resembles Brownian motion without any influence from higher-order correlations. With an increasing degree of supercooling, the diffusivity measured by XPCS decreases, so that D0/D(qnn) follows the expected Vogel–Fulcher–Tamann behavior. This is accompanied by a larger increase in the time scales of higher-order correlations by a factor of almost 2. This indicates that higher-order correlations increase as shown previously in refs (13), (32), and (34), and their lifetime increases while approaching the glass transition.
To study the structure and dynamics of supercooled colloidal hard spheres, we performed coherent X-ray scattering experiments at beamline P10, PETRA III (see the Experimental Section for details), for two sample sets of different radii R and particle volume fractions ϕ between 0.4 and 0.57. This enables us to access the average static structure as well as dynamics by XPCS and higher-order correlations by XCCA. Both sample sets showed the same behavior with an increasing volume fraction, independent of their individual radii, and will be discussed together below. The static structure of the samples was characterized using structure factors S(q). The extracted structure factors for some of the samples are shown in Figure 1a as a function of qR for the sake of comparison. Modulus of the wave vector q is defined as q=|q|=4πλsin(θ/2), where λ is the wavelength of the X-rays and θ is the scattering angle. With increasing volume fractions ϕ, the position of the first structure factor peak qnn, which corresponds to the next-neighbor distance, shifts toward larger q values, and its amplitude decreases slightly, indicating a deviation from the Percus–Yevick model. Such a deviation, especially in the peak height, has been observed before in other studies for this range of volume fractions in the vicinity of the glass transition and may indicate that the sample is no longer in a simple liquid state. (35) As a consequence, S(q) is no longer suitable to describe the state of the sample, and therefore, information about the dynamics and structural higher-order correlations are needed.

Figure 1

Figure 1. Structure and dynamics. (a) Structure factors S(q) for samples of HS1 (solid lines) and HS2 (dashed lines), shown as a function of qR for R of each sample. The corresponding volume fractions ϕ are shown in the legend. For the sake of visibility, not all structure factors are shown. (b) Relative diffusivity D0/D(qnn) as a function of volume fraction ϕ for HS1 (circles) and HS2 (triangles). The line represents a fit of the VFT law to the data with a ϕVFT of 0.65 ± 0.04.

The dynamical properties of the samples were studied using XPCS. (25−28) In XPCS, intensity I(q, t) is correlated via the g2 function at time t and wave vector q. The g2 function is defined as
g2(q,Δt)=I(q,t)I(q,t+Δt)t,ωI(q,t)t,ω2
(1)
where ω is the azimuth angle representing all detector pixels within the same q ring.
It gives access to intermediate scattering function f(q, Δt), which can be described by the Kohlrausch–Williams–Watts (KWW) function for many soft matter systems:
g2(q,Δt)=1+β|f(q,Δt)|2=1+βexp[2(ΓΔt)γ]
(2)
where Γ is the relaxation rate and γ the KWW exponent. Speckle contrast β depends on the coherence properties of the X-rays and the experimental setup and is 0.4 for this experiment. Characteristic relaxation time τ of the system is directly related to the relaxation rate via τ = 1/Γ and shows a q dependence τ ∝ qp with p < 0. Together, exponents γ and p specify the dynamics of the system; for example, in the case of free diffusion, γ = 1 and p = −2.
Relaxation times τ were determined by fitting eq 2 to the measured g2 for the probed q values. For the sake of comparison of the two sample systems, which have different radii R, the relative diffusivity D0/D = D0τq2 was calculated. Here, D0 is the free diffusion coefficient and is given by the Stokes–Einstein representation D0 = kBT/(6πηR), where kB is the Boltzmann constant, T the temperature, and η the solvent viscosity. The results obtained at the first structure factor peak are shown in Figure 1b. Three different samples of HS2 had a volume fraction (ϕ) very close to 0.4 and showed the same behavior, so only one of the samples is discussed further. Relative diffusivity D0/D increases with volume fraction ϕ. This slowing of the dynamics with an increasing volume fraction is typical for the transition from a liquid to a glassy state and has been observed also for soft colloids. (1,11,36,37) The increase in D0/D with volume fraction can be modeled empirically by the Vogel–Fulcher–Tammann (VFT) law
D0/D(qnn)=τ0exp(mϕVFTϕVFTϕ)
(3)
where τ0 is the scaling time, m the fragility index, and ϕVFT the VFT volume fraction, where D0/D. (34) As this VFT model is fully empirical, many approaches have been performed to connect the deceleration to structural properties. For instance, Tong and Tanaka found a VFT relation between the relaxation time and an order parameter of supercooled liquids. (38) In our previous XCCA study, we could show that structural higher-order correlations represent such an order parameter. (32) An alternative approach is the recent model of long-lived neighbors. (39,40) Here, the decrease in the relaxation time could be modeled by observable quantities only, which could be extracted from microscopy experiments on micrometer-sized charged colloids. However, as we do not have access to all of these parameters in the scattering experiment, they cannot be disentangled (see the Supporting Information). A fit of the VFT law to the data is shown in Figure 1b, resulting in a ϕVFT of 0.65 ± 0.04. This almost matches the random close packing concentration of ϕRCP = 0.68 for hard spheres with a size dispersity of δ = 15% and has been observed previously. (41−44)
The dynamics is further characterized by exponents γ and p (see the Supporting Information). We find three dynamic domains. (i) When ϕ < 0.45, the dynamics characterizes a subdiffusive liquid (p ≈ −2, and γ < 1). (ii) When 0.45 < ϕ ≲ 0.55, a transition toward ballistic dynamics (−1.5 < p < −1, and γ ≈ 1) is observed. (iii) When ϕ > 0.55, a continued transition toward ballistic and correlated dynamics is observed as found in many colloidal or metallic glasses (γ ≥ 1.5, and −1.5 < p < −1). This is often discussed to be a fingerprint of stress-dominated dynamics. Note that in the latter case the relaxation time or diffusivity is expected not to follow the VFT model anymore. (30) In addition, the dynamics becomes non-isotropically expressed by a direction dependence in the speckle pattern as reported in glasses and gels. (45,46) However, in the data shown here, the dynamics is isotropic with only very weak modulations with ω (see the Supporting Information), proving that we probe structural dynamics in the sample.
The crossover from the stretched to compressed exponential relaxation, i.e., from γ < 1 to γ > 1, is a current subject of research. It has been found in many different systems undergoing the glass transition, including metallic glasses (47) and colloidal systems. (32,36,37) Simulations could connect this transition to changes in local order in metallic glasses. (48) Furthermore, a recent theory study could model experimental results on the glass transition by connecting stretched exponentials to slowing the dynamics of local relaxation events, while compressed exponentials are related to avalanche-like dynamics in the glass state. (49) This model reproduced experimental results of metallic glasses well; however, an open question remains whether it can be extended to other glass-forming systems such as colloids, where, e.g., exponents of <0.5 are reported in the supercooled liquid (32,36,37) (see also the Supporting Information).
The time scales of higher-order correlations were investigated using an approach similar to XPCS. Instead of tracking I(q, t) over time, we defined a correlation function gC(q, Δt) that quantifies the time evolution of the cross-correlation function C(q, Δ). C(q, Δ) is the convential correlation function used in XCCA and is a measure for higher-order correlations within a sample. It has been successfully used to study orientational order, e.g., local symmetries in colloidal systems (30,32,50,51) or the structure and in situ self-assembly of colloidal crystals. (52−55) A recent theoretical study used cross-correlation analysis of XPCS data to determine rotational diffusion coefficients. (33) Therein, a numerical algorithm for predicting rotational diffusion coefficients has been designed and verified by using simulated data. C(q, Δ) correlates the scattered intensity at two points separated by angle Δ and is defined as
C(q,Δ)=I(q,ω)I(q,ω+Δ)ωI(q,ω)ω2I(q,ω)ω2
(4)
where ω is the azimuth angle on the detector and q = |q1| = |q2|. The schematics of the time scales of higher-order correlation analysis along with a simulated scattering pattern and the geometry used in the definition of C(q, Δ) are shown in Figure 2.

Figure 2

Figure 2. Schematics of the XPCS and time scales of higher-order correlations. A sample is illuminated with coherent X-rays, and the resulting scattering patterns are recorded at time intervals Δtmin. Via calculation of intensity correlation function g2 and the time scales of higher-order correlation function gC, more details about the local order, its time scale, and the heterogenity of the system can be obtained.

A correlation function gC(q, Δt) that measures the time evolution of C(q, Δ) is then defined similar to the g2 function as
gC(q,Δt)=C*(q,Δ,t)C*(q,Δ,t+Δt)t,ΔC*(q,Δ,t)t,Δ2
(5)
where C*(q, Δ, t) = C(q, Δ, t) – min[C(q, Δ, t)], so that C*(q, Δ, t) ≥ 0 and thus gC(q, Δ, t) ≥ 0. Compared to g2, the average in eq 5 is over both t and Δ, while it is done over t and ω (pixel of the same q) in eq 1.
To relate the gC function to the g2 function, we performed two simulations: (a) a two-dimensional (2D) simulation for particles undergoing Brownian motion and (b) a 2D simulation of particles in a square symmetry rotating around its center (see the Supporting Information). The first simulation showed that if the sample undergoes free diffusion so that there are no higher-order correlations present, gC(g2)2 and as a consequence τgC/τg2=21/γg2, i.e., 0.5 for γg2=1 as in the simulation case. In the second simulation, two cases were investigated: (i) a fixed square symmetry that rotates by a fixed angle around its center per time step and (ii) the particles forming the square being subject to additional Brownian motion in each time step, thus slowly reducing the degree of 4-fold symmetry. The second simulation shows that if orientational order is present and long-lived, gC shows a much larger characteristic relaxation time. In particular, the square relation gC(g2)2 is not valid and the ratio τgC/τg2 can become large. Thus, gC is sensitive to the lifetime of local order independent of dynamics on the single-particle level.
The normalized gC functions were calculated for each sample and are shown alongside the intermediate scattering functions from the XPCS analysis in Figure 3. The shapes of both correlation functions are very similar for a given sample; both show either a stretched or compressed exponential decay that was modeled by a KWW function. The gC function typically decreases at slightly shorter times Δt than the intermediate scattering function.

Figure 3

Figure 3. Correlation functions. (a) Intermediate scattering functions and (b) gC functions at qnn for the various volume fractions ϕ of HS1 (circles) and HS2 (triangles). The respective fits are shown as solid lines.

To compare the relaxation times of the gC and g2 functions, the quotient of the relaxation times, τgC/τg2, was calculated and is shown as a function of volume fraction ϕ in Figure 4a. For the smaller volume fractions, τgC/τg2 is between 0.4 and 0.6. This matches the expected results for diffusing particles without a particular structural local order and suggests that the sample is in a liquid state. When ϕ > 0.55, however, τgC/τg2 increases from 0.6 to ∼0.9 when ϕ = 0.569. This threshold coincides with the regimes found in the analysis of the dynamics; above ϕ > 0.55, the samples showed signs of ballistic dynamics that are found in many glasses.

Figure 4

Figure 4. Comparison of the correlation functions. (a) Quotient of the relaxation times, τgC/τg2, obtained from the gC and g2 functions as a function of volume fraction ϕ. The dashed line signifies τgC/τg2=0.5. The inset shows τgC/τg2 as a function of relaxation time τg2. (b) Exponents γgC and γg2 plotted against each other. The solid line signifies γgC=γg2 and is a guide to the eye, and the dashed line is a linear fit γgc=aγg2+b to the data with a = 0.73 and b = 0.17. The inset shows τgC/τg2 as a function of γg2 with the solid line representing 21/γg2.

The increase in the ratio τgC/τg2 indicates that C(q, Δ) is slowing more quickly than g2(q, Δt) when the volume fraction is increasing. Because C(q, Δ) is a measure for higher-order correlations that are connected to the local order within the sample, our data indicate that the local order becomes more and more long-lived in the supercooled state. The increase in the ratio of the relaxation times is highlighted even more when τgC/τg2 is plotted as a function of τg2, as shown in the inset of Figure 4a. Here, the experimental accuracy of determining ϕ is excluded by focusing on the dynamical properties only. The figure shows the same behavior; the slower the dynamics of the sample, the more long-lived the local order.
As mentioned above, the shape of both functions g2 and gC seems to be similar at the same volume fractions. However, taking a closer look, we can find remarkable differences (see Figure 4b). While we find a linear increase in γgC as a function of γg2, the slope of this linear increase is <1 and γgCγg2 for all cases (except for the smallest volume fraction studied). A smaller value of the KWW exponent γ corresponds to a wider distribution of relaxation times within the exposed sample volume, which can be interpreted as a higher degree of dynamical heterogeneity. Thus, we conclude that the dynamics of the higher-order correlations expressed by gC are slightly more heterogeneous than the individual particle dynamics, especially in the vicinity of the glass transition. The analysis of two-time correlation functions (25,56,57) for both XPCS and dynamics of local order shows a small degree of dynamical heterogeneity, but a direct comparison between both is limited by different statistical accuracies (see the Supporting Information).
The inset of Figure 4b shows the ratio τgC/τg2 as a function of γg2 with the solid line representing the behavior expected for gC(g2)2 of τgC/τg2=21/γg2. The lower values of γg2 that represent the samples at small volume fractions follow this expected behavior for (sub)diffusion. The higher values deviate from this diffusion expectation. Instead, τgC/τg2 is larger and thus well above the solid line in the inset of Figure 4b and increases further with γg2. This indicates that the gC values of these samples cannot be explained by the diffusion relation gC(g2)2 and in fact exhibit long-lived local order.
In conclusion, we studied the dynamics and time scales of higher-order correlations in hard sphere systems using a novel combination of spatial and temporal correlation functions. By combining the correlation functions from XCCA and XPCS, we were able to gain a unique insight into the correlation between structure and dynamics in colloidal systems. For samples with a volume fraction above ϕ ≈ 0.55, we found that the ratio τgC/τg2 increases to 0.9. These values are significantly higher than the values for diffusive samples without any preferred local order of τgC/τg2=0.5, evidencing that the local order becomes more long-lived when the glass transition is approached. Furthermore, the dynamics of local order are slightly more heterogeneous than the individual particle dynamics.
Our results are fundamental for understanding supercooled colloidal fluids in the vicinity of the glass transition. The experimental approach not only shows that local structures evolve but also provides access to their lifetime. This could help in understanding dynamical heterogeneities and the formation of locally favored structures. Additionally, as our approach combines studies of structure and dynamics using coherent X-rays, it will benefit from diffraction-limited storage rings (DLSR) such as PETRA IV, ESRF-EBS, or APS-U. (58−61) Most importantly, our approach is not limited to studying correlations in colloidal systems. Because of the short wavelength of X-rays, it can be extended to detect structure–dynamics correlation down to molecular length scales and will thus be a valuable tool for studies of glass and phase transitions of different materials. Furthermore, it may provide access to crystallization kinetics, e.g., during self-assembly of nanocrystals, (55) or detect transient structures in molecular liquids such as water.

Experimental Section

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We performed X-ray scattering experiments on hard sphere poly(methyl methacrylate) (PMMA) particles dispersed in decalin at beamline P10 of PETRA III, DESY. This sample system is known to be an excellent experimental representation of a hard sphere system. (62) Two samples, HS1 with R = 94 ± 10 nm and a size dispersity of δ = 17% and HS2 with radius R = 73 ± 3 nm and δ = 10%, were investigated. To study supercooled liquids, highly concentrated PMMA stock samples were diluted with known quantities of decalin to produce volume fractions (ϕ) ranging from 0.4 to 0.57. The samples were studied in ultra-small-angle X-ray scattering (USAXS) geometry with a beam size of 30 μm × 30 μm and a photon energy of 8.5 keV, corresponding to a wavelength of 1.46 Å. An Eiger X 4M instrument was used as a detector with a sample–detector distance of 21.2 m. The volume fractions were determined by modeling the static structure factors with the Percus–Yevick model, also taking into account the size dispersity of the particles. (63,64) The absolute error of the determined volume fractions is estimated to be ≲3%. (65) Note that the relative error between different concentrations for each sample is much smaller because they were obtained by dilution from the same stock solution.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpclett.3c00631.

  • 2D simulations for particles undergoing free diffusion and additional analysis of particle dynamics (PDF)

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Author Information

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  • Corresponding Authors
  • Authors
    • Irina Lokteva - Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, GermanyThe Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, GermanyOrcidhttps://orcid.org/0000-0002-2286-477X
    • Michael Dartsch - Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, GermanyThe Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
    • Francesco Dallari - Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, GermanyPresent Address: Dipartimento di Fisica e Astronomia, “Galileo Galilei”, Università degli Studi di Padova, Via F. Marzolo 8, 35131 Padova, Italy
    • Claudia Goy - Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, GermanyOrcidhttps://orcid.org/0000-0001-5771-8564
    • Fabian Westermeier - Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
    • Verena Markmann - Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, GermanyPresent Address: Department of Physics, Technical University of Denmark, Kgs. Lyngby 2800, Denmark
    • Svenja C. Hövelmann - Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, GermanyInstitut für Experimentelle und Angewandte Physik, Christian-Albrechts-Universität zu Kiel, Leibnizstraße 19, 24098 Kiel, GermanyOrcidhttps://orcid.org/0000-0001-5533-6714
    • Gerhard Grübel - Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, GermanyThe Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, GermanyPresent Address: European X-ray Free-Electron Laser, Holzkoppel 4, 22869 Schenefeld, Germany
  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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The authors thank Walter Kob for the discussion, Andrew Schofield for providing the samples, and Wojciech Roseker for the proofreading. The research was carried out at light source PETRA III at DESY, a member of the Helmholtz Association (HGF). The authors also thank Michael Sprung for assistance in using beamline P10. This work is supported by the Cluster of Excellence Advanced Imaging of Matter of the Deutsche Forschungsgemeinschaft (DFG) (EXC 2056, Project 390715994). The authors also acknowledge the scientific exchange and support of the Centre for Molecular Water Science (CMWS).

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  1. Zixi Hu, Jeffrey J. Donatelli. Multitiered computational methodology for extracting three-dimensional rotational diffusion coefficients from x-ray photon correlation spectroscopy data without structural information. Physical Review B 2024, 110 (21) https://doi.org/10.1103/PhysRevB.110.214305
  2. C. Patrick Royall, Patrick Charbonneau, Marjolein Dijkstra, John Russo, Frank Smallenburg, Thomas Speck, Chantal Valeriani. Colloidal hard spheres: Triumphs, challenges, and mysteries. Reviews of Modern Physics 2024, 96 (4) https://doi.org/10.1103/RevModPhys.96.045003

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  • Abstract

    Figure 1

    Figure 1. Structure and dynamics. (a) Structure factors S(q) for samples of HS1 (solid lines) and HS2 (dashed lines), shown as a function of qR for R of each sample. The corresponding volume fractions ϕ are shown in the legend. For the sake of visibility, not all structure factors are shown. (b) Relative diffusivity D0/D(qnn) as a function of volume fraction ϕ for HS1 (circles) and HS2 (triangles). The line represents a fit of the VFT law to the data with a ϕVFT of 0.65 ± 0.04.

    Figure 2

    Figure 2. Schematics of the XPCS and time scales of higher-order correlations. A sample is illuminated with coherent X-rays, and the resulting scattering patterns are recorded at time intervals Δtmin. Via calculation of intensity correlation function g2 and the time scales of higher-order correlation function gC, more details about the local order, its time scale, and the heterogenity of the system can be obtained.

    Figure 3

    Figure 3. Correlation functions. (a) Intermediate scattering functions and (b) gC functions at qnn for the various volume fractions ϕ of HS1 (circles) and HS2 (triangles). The respective fits are shown as solid lines.

    Figure 4

    Figure 4. Comparison of the correlation functions. (a) Quotient of the relaxation times, τgC/τg2, obtained from the gC and g2 functions as a function of volume fraction ϕ. The dashed line signifies τgC/τg2=0.5. The inset shows τgC/τg2 as a function of relaxation time τg2. (b) Exponents γgC and γg2 plotted against each other. The solid line signifies γgC=γg2 and is a guide to the eye, and the dashed line is a linear fit γgc=aγg2+b to the data with a = 0.73 and b = 0.17. The inset shows τgC/τg2 as a function of γg2 with the solid line representing 21/γg2.

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