Intrinsic Dynamics of Amorphous Ice Revealed by a Heterodyne Signal in X-ray Photon Correlation Spectroscopy Experiments

Unraveling the mechanism of water’s glass transition and the interconnection between amorphous ices and liquid water plays an important role in our overall understanding of water. X-ray photon correlation spectroscopy (XPCS) experiments were conducted to study the dynamics and the complex interplay between the hypothesized glass transition in high-density amorphous ice (HDA) and the subsequent transition to low-density amorphous ice (LDA). Our XPCS experiments demonstrate that a heterodyne signal appears in the correlation function. Such a signal is known to originate from the interplay of a static component and a dynamic component. Quantitative analysis was performed on this heterodyne signal to extract the intrinsic dynamics of amorphous ice during the HDA–LDA transition. An angular dependence indicates non-isotropic, heterogeneous dynamics in the sample. Using the Stokes–Einstein relation to extract diffusion coefficients, the data are consistent with the scenario of static LDA islands floating within a diffusive matrix of high-density liquid water.

W ater is the key component for the existence of life on Earth and plays a central role in a wide range of scientific disciplines such as biology, chemistry, atmospheric chemistry, geophysics, food science, and electrocatalysis.It is also one of the most unusual liquids with many anomalous properties, such as the density maximum at 4 °C, a lower density in the solid phase than in the liquid phase, and many more. 1,2−7 The two liquid states are thought to be the counterparts of high-and low-density amorphous ices (HDA and LDA, 1,8,9 respectively).Recently, amorphous ice's glassy nature has been challenged by experimental findings on fast compression pathways that suggest that HDA is not connected to a liquid state but rather a disordered crystalline state. 10On the contrary, current highpressure studies on the very-high-density (VHDA) state show that both VHDA and HDA are connected to an ultraviscous liquid at high pressure but reach the same HDL state at intermediate pressures of <0.3 GPa. 11−12 As of today, X-ray photon correlation spectroscopy (XPCS) has been used as a powerful technique to investigate dynamics of different systems, like colloidal 13−18 and polymeric 19−25 materials, in wide-and small-angle scattering (WAXS and SAXS, respectively) regimes.Similar in principle to dynamic light scattering, 26 XPCS may readily be used to probe structural dynamics throughout different length scales from atomic to hundreds of nanometers.Most of these studies are focused on the dynamics of systems in the equilibrium state or under different flow conditions, where no phase transition or structural change happens.For example, the advective and diffusive dynamics of colloidal particle dispersion was studied under homogeneous shear flow. 27However, physiochemical processes often also happen under non-equilibrium conditions or involve phase transitions.Few dynamic studies have focused on phase transition or phase ordering processes.For example, the dynamics of a protein solution during liquid−liquid phase separation upon low-temperature quenches was determined and revealed distinctly different dynamical regimes. 28Heterogeneous dynamics during the transformation of the facecentered cubic phase to the hexagonal close-packed phase of cobalt was investigated. 29Recently, the dynamics and structural changes of amorphous ice during the HDA−LDA transition were tracked by combining wide-angle X-ray scattering (WAXS) and XPCS at small scattering angles (SAXS). 4Two dynamic processes were found during the transition; one was related to a liquid-like motion.The nature of the second process, however, remained unclear.
Here, we study the phase transition of amorphous ices but use a distinct preparation method for ice samples.Instead of a powdered sample, we use a compact, free-standing piece of ice, which is 10 times thinner than that in the previous study. 4We now observe a previously hidden heterodyne signal in a onecomponent system, which allows us to shed light on the nucleation and growth process during the phase transition from HDA to LDA. 5,30−33 Upon warming, HDA transforms into LDA through an exothermic transition 32 accompanied by a volume change of 20%. 33Figure 1a shows five representative two-dimensional (2D) WAXS patterns recorded at different temperatures, starting from equilibrated HDA (eHDA) at 90 K to LDA at 120 K (T cryostat ).The corresponding azimuthally integrated intensities 1b.The characteristic diffraction maximum of eHDA at Q = 2.17 Å −1 was observed at temperatures of ≤90 K.At 100 K, a weak shoulder of LDA at Q = 1.7 Å −1 appears, and its intensity increases with temperature.Simultaneously, the scattering intensity recorded in SAXS geometry increases between 90 and 115 K (see Figure S1 for details).Note that the reported temperatures are not measured directly at the sample position; therefore, we expect the actual sample temperature to be slightly higher [T cryostat + 5 K (see the Supporting Information)].The sample was held for 30 min at the desired temperature, before the measurement was begun.The bimodal scattering intensity in the WAXS pattern implies phase coexistence of eHDA and LDA and is consistent with our previous results. 4,30,34The complete transition to LDA happens above 120 K (T sample ≈ 125 K).This is slightly lower than the values from other studies of eHDA 30,35 but can be explained by a different sample preparation (see also the Supporting Information).We note that intense Bragg spots can be found on the scattering patterns (Figure 1a).These spots appear as sharp Bragg peaks from crystalline ice in our WAXS curves (Figure 1b).We attribute them to the presence of a small amount of hexagonal ice that condensed on the sample during sample transfer. 34As the amount of these hexagonal ices is expected to be small and not embedded in the high-pressure ice matrix, it is reasonable to neglect its effect on the dynamics.
Figure 1c shows the calculated intensity autocorrelation functions g 2 (Q, Δt) at different values of Q and temperatures.For all temperatures, the speckle contrast slightly decreases with an increase in Q.This is seen as the short time plateau value of the correlation function decreases.Such an effect was also observed in other studies 13,28,36,37 and might be related to a faster decay process that could not be resolved in the experiment.From the decay of speckle contrast, we determined the length scale at which such a localized process occurs to be ∼45 Å (see Figure S2 for details).Upon heating to 120 K, it is evident that the dynamics become faster.Surprisingly, an additional oscillation was observed in the g 2 curves at 115 and 117.5 K.This phenomenon was not observed previously and might have been hidden in the powder samples. 4The oscillation may be explained by heterodyne mixing of a diffusive and static component.We discuss this in the following.
To investigate whether the sample was in equilibrium during the measurements, we additionally calculate the two-time correlation function (TTC), 38 which is defined as where I(Q, t 1 ) and I(Q, t 2 ) denote the intensity of a pixel at distinct times t 1 and t 2 , respectively.The subscript "pix" implies that the averaging is, in this case, solely performed over pixels

The Journal of Physical Chemistry Letters
within the same Q-bin and not over time.The TTCs for temperatures ranging from 90 to 120 K at Q = 0.0032 Å −1 are shown in Figure 2. A narrow intensity on the diagonal line for higher temperatures (115, 117.5, and 120 K) indicates rather fast dynamics, while a broad and smeared out intensity marks "slower" dynamics, as the correlations perpendicular to the diagonal line decay more slowly.Additionally, the oscillatory behavior is visible for temperatures of 115 and 117.5 K, which is consistent with the results depicted in Figure 1c.The TTC line shape for all temperatures reveals that the sample is stable  The Journal of Physical Chemistry Letters over time and does not exhibit a change in dynamics during the measurements.
To quantitatively evaluate the dynamics at the different temperatures, in particular, once approaching the state of coexistence, a fit to the g 2 curves including the oscillations is necessary.Although a sum of Kohlrausch−Williams−Watts (KWW) functions 39 is successfully applied for systems with multiple dynamics, 4,28 it is not suitable for our case (see Figure S3 for details).We applied different models to our data, including shear flow as reported in the literature, 27,40 but noticed that none of them could describe the data (see Figures S4 and S5 for details).−44 Inspired by this idea, we obtain the following intensity− intensity autocorrelation function: where I(Q, t) is the intensity at the modulus of the momentum transfer Q = 4π sin(θ)/λ, Δt denotes the correlation time delay, the brackets ⟨•••⟩ indicate averaging over all detector pixels within the same Q interval as well as all frames at different times t, I 1 and I 2 are the scattering intensities of the two different components, I total is the total scattering intensity, Γ 1 (Q) and Γ 2 (Q) are the Q-dependent relaxation rates (1/τ 1 and 1/τ 2 ), ω is the oscillation frequency and relates to the heterodyne period (2π/ω), and γ 1 and γ 2 are the Kohlrausch− Williams−Watts (KWW) exponents.Stretched correlation functions (γ < 1) are typically found in supercooled liquids, while Brownian diffusive processes are characterized by a γ = 1.
Glasses usually show compressed exponentials (γ > 1), but for different reasons (stress relaxation).The heterodyne equation (eq 2) does not require necessarily that either of the dynamical processes be diffusive.Only if the relaxation rate Γ(Q) has a linear relationship with Q 2 , namely Γ(Q) = D 0 Q 2 , is a diffusive motion present, where D 0 represents the Stokes−Einstein diffusion coefficient. 19,36quation 2 was fitted to all of the data in this work.Panels a and d of Figure 3 represent the fitting results for the g 2 curves at Q = 0.0099 Å −1 for 115 and 117.5 K, respectively.More fitting results are displayed in Figures S6 and S7.The contribution of the first exponential part I 1 2 exp{−2[Γ 1 (Q)-Δt] γ 1 } in eq 2 to the total g 2 curve is small (blue dashed curve in panels a and d of Figure 3).The Q dependence of 1/τ 1 (Figure S14) at two different temperatures indicates that the fast decay process is very robust and undergoes a hyperdiffusion or ballistic motion, which would be related to γ 1 = 2.We therefore kept γ 1 = 2 as a constant value for all of the later fitting procedures.However, we noticed that the fitting results would not be affected by changing the value for γ 1 to 1, as the contribution of this part to the total g 2 is small.In the following, we focus on the dominant second exponential part and the third oscillatory part.Panels b and e of Figure 3 summarize the characteristic times obtained from the fitting of the second exponential component (black dashed curve in panels a and d of Figure 3) at different Q values at 115 and 117.5 K, respectively.An almost Q 2 dependence of the extracted relaxation rates is observed with an additional small offset. 34The green solid line depicts the result of a fit 1/τ = D 0 Q 2 + c with a diffusion coefficients (D 0 ) of 12 and 63 Å 2 /s for 115 and 117.5 K, respectively, and an The Journal of Physical Chemistry Letters offset c of 0.0004 s −1 for both temperatures.A linear Q dependence 1/τ ∼ Q (red dashed−dotted line) and a power law fit 1/τ ∼ Q p (blue solid line) are plotted for comparison but fail to describe the data.It is evident that the dynamics become faster and diffusive when 117.5 K is approached.This is accompanied by an increasing velocity of the sample changing from 2.9 to 14.5 Å/s, which is extracted from a linear fit of the heterodyne period 2π/ω as a function of Q −1 (green solid line in panels c and f of Figure 3, left y-axis).We further investigate the KWW exponent, γ 2 , of the second exponential process.Except for the last four data points at small values of Q, at 115 K, which have large error bars due to limited statistics, we observed that γ 2 is ∼1 (red dashed line in Figure 3c).In contrast, when the temperature is increased to 117.5 K, γ 2 becomes <1 (Figure 3f, blue axis); hence, a crossover to a stretched exponential relaxation behavior with a strong Q dependence takes place.This broad distribution of relaxation times is not in full agreement with the finding of a diffusive liquid-like behavior. 4Moreover, γ 2 decreases with an increase in Q, indicating that the liquid is more rigid at smaller length scales.We further investigated potential heterogeneities by calculating g 2 for different azimuthal directions.
We additionally calculated the g 2 functions at different azimuthal angles for each temperature and Q as depicted in Figure 4a, hence now treating Q as a vector.g 2 curves at different values of ϕ for 115 K and Q = 0.0041 Å −1 are plotted in Figure 4b, and appearing streaks have been masked out before integration.The g 2 curves are shifted vertically for the sake of clarity.More results can be found in Figures S9, S10, and S12.The observed oscillation occurs at different times for different ϕ angles.For some specific angles, like 45°and 225°, the oscillations are no longer visible.The same fitting procedure (eq 2) was performed on all of the g 2 curves at different Q values, angles, and temperatures.We determined the velocity by fitting the heterodyne period as a function of Q −1 , as shown in the example in Figure 4c, where a velocity of 13.7 Å/s was extracted for 270°at 117.5 K (Figure 4c). Figure 4d summarizes the velocities at different angles for sample A at 115 and 117.5 K and sample B (two positions) at 117.5 and 120 K.The maximum velocity was observed at ∼135°and ∼315°for both temperatures and both samples, and the minimum velocity was observed orthogonal from the angle of the velocity maximum.The angular dependence of the velocity, and therefore also the heterodyning, shows a similar trend in different samples.A similar behavior of the angledependent velocity was reported by Dallari et al. 45 when they investigated the stress relaxation in repulsive colloidal glasses.As a coincidence, in our measurements streaks appear on the detector (Figures S9 and S10) at ∼45°.Streaks have also been observed in other directions in other samples and temperatures, but here they appear mostly at this angle.We assume that the streaks are related to grain boundaries or stress. 44This is, the reason for the angle-dependent velocity observed here is most probably related to the heterogeneity of the macroscopic morphology inside the free-standing amorphous ice film during the sample preparation process.From optical microscopy images, 34 it is known that the sample contains cracks and macroscopic grain boundaries.More detailed studies are required to fully understand this phenomenon.However, we can exclude a gravitational flow due to the sample characteristics, and lateral flow, which in other experiments 27,40 has been used to describe the oscillations, fails to describe our data (see Figures S4 and S5).We also exclude the possibility of the signal originating from surface roughness, as concluded from the ratio between factors I 1 and I 2 from eq 2 in Figure S8.
An oscillatory behavior of the g 2 curves is typically observed in a heterodyne detection scheme of dynamic light scattering (DLS) 46 and XPCS, 25,47−49 where an external static reference must be applied.The phase shift in the scattering due to the motion of the sample with respect to the fixed reference yields the oscillation.However, in this work, the oscillatory behavior was observed in a homodyne detection scheme without adding any external static reference.This phenomenon is rare but has been observed in flowing media, such as sedimentation. 50−52 A similar oscillatory behavior was discovered by Gutt et al. two decades ago using XPCS in reflection geometry, 53 where they investigated the capillary waves on liquid water in grazingincidence small-angle X-ray scattering (GISAXS) geometry.This heterodyne XPCS signal in reflection geometry was later extended to understand the surface dynamics of thin films during sputter deposition by Ulbrandt et al. 54 and Ju et al. 55 They demonstrated that the variable mixing of surface and bulk scattering signals causes oscillation, which suggests that an external static reference is not necessary for heterodyne detection.This idea was supported by the homodyne DLS work in supramolecular self-assembly of triarylamine molecules with tailored side groups by the Buhler group, 56,57 where a similar oscillation behavior was observed and was attributed to two components in the system: a dangling fiber and microgel with different dynamics.Inspired by the work of Ulbrandt et al. 54 and Buhler et al., 56,57 we propose that the oscillatory behavior (heterodyne signal) observed in our work is due to the coexistence of two components, which have different dynamics and distinct glass transition temperatures. 32The two components could be HDL and LDA, as the oscillations appear simultaneously with the observed transformation in the WAXS signal.Calorimetry and dielectric spectroscopy suggest the glass transition onset temperature to be T g (eHDA) = 115 K and T g (LDA) = 136 K, which is consistent with the previous XPCS data on powder samples. 4The measurements presented here are taken below the glass transition temperature of LDA.Some sample spots also show crystalline ice peaks (Figure S13) at the temperatures at which the oscillatory component is observed, which suggests that the heterodyne mixing could also be related to HDL and crystalline ice.Interestingly, this oscillatory behavior was not found for the powder amorphous ice samples, 4 which might be caused by the bulk sample being confined between two diamond windows, thus not allowing for a fast expansion.Alternatively, the oscillations might be hidden by other effects caused by the grainy structure of the powdered bulk samples.In the work presented here, the XPCS experiments were performed on free-standing amorphous ice films, which allows for a macroscopic expansion with a volume change of 20% during the HDA−LDA transition, which is another possible source of the heterodyne signal.We ensure annealing of the sample for 20 min before each measurement, hence allowing thermally activated motion to relax; after 20 min, the WAXS signal remains stable.Still, it is a metastable system, and a fraction of water molecules will, also after equilibration, be displaced when high-density domains expand to low-density regions.The velocity obtained from the heterodyne signal can be related to this displacement (expansion).The liquid-like motion of the high-density matrix and the expansion to low-density are two correlated processes that are difficult to disentangle; the heterodyne signal now allows a first estimation.The free- The Journal of Physical Chemistry Letters standing amorphous ice film provides us with a new model sample system for investigating the intrinsic dynamics of the glass transition of amorphous ice's glass transition.
Our previous data from eHDA-grid samples 34 show that eHDA enters a hyperdiffusive regime around T cryostat = 110 K, determined by a diffusion coefficient of D 0 = 3.1 Å 2 /s, a compressed exponential behavior (γ > 1.5), and a slight offset in the Q 2 dependence of the relaxation time, which could be related to stress relaxation inside the glassy matrix.Here we investigated temperatures above 110 K, showing the additional oscillatory behavior of g 2 .The calculated diffusion coefficient increases to D 0 = 12 and 63 Å 2 /s for 115 and 117.5 K, respectively.Considering γ 2 ∼ 1 (Figure 3c), the previous finding indicates the formation of a high-density liquid (HDL) at ∼115 K (T sample ≈ 120 K), while the stretched exponential behavior γ 2 < 1 (Figure 3f) at higher temperatures could be related to heterogeneities caused by stress relaxation within the sample.However, the heterodyne signal must be due to a second static component, as discussed above.This might be related to small domains of HDL transforming into LDA, as depicted in Figure 5. LDA seeds can act as a heterodyne static component as its T g is higher (136 K) than that of eHDA (115 K).The LDA islands grow and float inside the HDL matrix, which is detected in our small-angle geometry.Alternatively, the appearance of crystalline ice can act as a static component, formed inside the sample or condensed on the surface.However, we consider this scenario to be less likely, as we see the oscillations always appearing simultaneously with the coexistence of the two amorphous states.In addition, the angular dependence as well as the stretched exponential behavior indicates stress-related heterogeneities, potentially due to the preparation process.If the signal originated from condensed ice, it should not exhibit such a strong angular dependence.
On the basis of the discussion presented above, we propose a picture for the dynamic and structural evolution of freestanding amorphous ice films (Figure 5).Equilibrated HDA is prepared along the pathway depicted in the phase diagram (Figure 5, right), it remains metastable at ambient pressure and temperatures of ≤90 K. Upon being heated, HDA transforms into a high-density liquid (HDL), which dynamically is slow but diffusive (D 0 = 12 Å 2 /s at 115 K) and is therefore called "ultraviscous".Within the HDL matrix, small domains then transform into low-density amorphous ice (LDA), as seen in the WAXS signal.Due to the density difference of the two states, expansion takes place when crossing the boundary between HDA and LDA (red arrow in the right ppanel of Figure 5).The appearance of a heterodyne signal (oscillation) must be related to the static component.Possible sources are the slow or almost static LDA domains, condensed crystalline ice, or creep due to expansion.We exclude the possibility of the signal originating from mechanical motion, as this cannot be described by eq 2. The heterodyne signal appears in the g 2 curves at 115 and 117.5 K, hence simultaneously indicating the observation of two coexisting components in WAXS.Once the sample fully transforms into LDA, the heterodyne signal disappears.
In summary, we determined the dynamics during the eHDA−LDA transition by conducting XPCS experiments in transmission geometry (using thin grid samples).A heterodyne signal was surprisingly observed at 115 and 117.5 K (cryostat temperature) when HDL and LDA coexisted within the sample.One of the novelties of our study is the demonstration that such a heterodyne signal can be observed in a homodyne detection scheme in a one-component system and using XPCS in the transmission geometry.Quantitative analysis was performed on this heterodyne signal to extract the intrinsic dynamics of amorphous ice during the HDA−LDA transition.The Q 2 dependence of the extracted relaxation rates and a γ value of ∼1 suggest that a liquid state was formed at relatively high temperature of 115 and 117.5 K (T sample values of 120 and 122.5 K, respectively).The higher diffusion coefficient and velocity observed with an increase in temperature further support this conclusion.This leaves us with the scenario depicted in Figure 5 and is consistent with the hypothesized glass transition in HDA at T > 110 K and LDA as a static component "swimming" within the liquid matrix.Interestingly, an angular dependence of the heterodyne signal was observed, which might be due to the heterogeneity of the macroscopic morphology inside the free-standing amorphous ice film during the sample preparation process.Overall, the heterodyne signal observed in a homodyne XPCS detection scheme sheds light on the glassy nature of amorphous ices and their phase transitions.This work paves the way for the fundamental studies of other systems where different dynamics and phase transition occur under different conditions.

■ EXPERIMENTAL METHODS
Details for sample preparation and X-ray measurements can be found in the Supporting Information.In brief, HDA was prepared inside a Cu grid, following our previous protocol, 31 as depicted in Figure 5. XPCS measurements were performed at beamline P10 and PETRA III at DESY.

Figure 1 .
Figure 1.In situ WAXS/XPCS experiments using two detectors simultaneously.(a) Representative 2D WAXS patterns recorded at different temperatures.Note that all given temperatures refer to the measured T cryostat ; the sample temperature is estimated to be 5 K higher.(b) Corresponding azimuthally integrated intensity I(Q) in the momentum transfer range of 1.3 Å −1 < Q < 3.0 Å −1 .Details can be found in the Supporting Information.(c) Intensity autocorrelation functions g 2 (Q, Δt) − 1 calculated at different values of Q for the corresponding temperatures.

Figure 2 .
Figure 2. Two-time correlation function (TTC) at different temperatures indicated in the panel tiles at momentum transfer Q = 0.0032 Å −1.Note that at 120 K the data were collected for only ∼600 s.All given temperatures refer to the measured T cryostat ; the sample temperature is estimated to be 5 K higher.

Figure 3 .
Figure 3. Quantitative analysis of the g 2 curves.(a and d) Representatives of experimental data (black dots) showing oscillations obtained at Q = 0.0099 Å −1 for 115 and 117.5 K (T cryostat ), respectively.The data are fitted with eq 2. The green solid, black dashed, blue dashed, and red dashed lines indicate the total, first and second exponential components, and oscillatory component of the model, respectively.(b and e) Characteristic time obtained from fitting of the second exponential component at different Q values at 115 and 117.5 K, respectively.(c and f) Heterodyne period 2π/ω and linear fit (green solid line) to calculate the velocity (left axis).KWW exponent γ 2 of the second exponential component (blue, right axis).

Figure 4 .
Figure 4. Angle-dependent dynamics.(a) 2D small-angle X-ray scattering pattern.The radial/azimuthal bins indicate the regions where the correlation functions of the individual pixels were averaged.(b) Intensity autocorrelation functions g 2 (Q, Δt) calculated for different angles at 117.5 K and Q = 0.0041 Å −1 .The g 2 curves are shifted vertically for the sake of clarity.(c) Heterodyne period 2π/ω as a function of Q −1 at 117.5 K and ϕ = 270°.From the linear fit (green solid line), a velocity of the sample of 13.7 Å/s is obtained.(d) Velocity as a function of azimuthal angle ϕ for sample A at 115 and 117.5 K and sample B (two positions) at 117.5 and 120 K.Note that all given temperatures refer to the measured T cryostat ; the sample temperature is estimated to be 5 K higher.

Figure 5 .
Figure 5. Schematic representation of the structural and dynamic evolutions, including a phase diagram (right) and a preparation pathway (gray dashed).Schematic (left) of sample evolution upon heating at ambient pressure [along red arrow "XPCS" in the phase diagram (right)].The hypothesis is that HDA first transforms into HDL upon heating to 110 K (T sample ≈ 115 K) and further changes to LDA at ∼115 K (T sample ≈ 120 K), until it fully transforms at 120 K (T sample ≈ 125 K).Oscillatory behavior in the g 2 functions can be explained by static domains that are "swimming" in the HDL matrix.