On the heterogeneous character of water's amorphous polymorphism

In this letter we report {\it in situ} small--angle neutron scattering results on the high--density (HDA) and low-density amorphous (LDA) ice structures and on intermediate structures as found during the temperature induced transformation of HDA into LDA. We show that the small--angle signal is characterised by two $Q$ regimes featuring different properties ($Q$ is the modulus of the scattering vector defined as $Q = 4\pi\sin{(\Theta)}/\lambda_{\rm i}$ with $\Theta$ being half the scattering angle and $\lambda_{\rm i}$ the incident neutron wavelength). The very low--$Q$ regime ($<5\times 10^{-2}$ \AA $^{-1}$) is dominated by a Porod--limit scattering. Its intensity reduces in the course of the HDA to LDA transformation following a kinetics reminiscent of that observed in wide--angle diffraction experiments. The small--angle neutron scattering formfactor in the intermediate regime of $5 \times 10^{-2}<Q<0.5$ \AA$^{-1}$ HDA and LDA features a rather flat plateau. However, the HDA signal shows an ascending intensity towards smaller $Q$ marking this amorphous structure as heterogeneous. When following the HDA to LDA transition the formfactor shows a pronounced transient excess in intensity marking all intermediate structures as strongly heterogeneous on a length scale of some nano--meters.

Throughout recent years different concepts have been introduced to explain the phenomenon of amorphous polymorphism, i.e. the existence of more than one amorphous structure in a single substance (Mishima, 1998;Stanley, 2000;Debenedetti, 2003). The most intriguing scenario is based on the existence of two distinct liquid states, that has been established as a possibility in molecular dynamics simulations (Poole, 1993). The liquid polymorphism was supposed to account for the formation of the high-density amorphous (HDA, ρ ≈ 39 molecules/nm 3 ) and the low-density amorphous (LDA, ρ ≈ 31 molecules/nm 3 ) ice structures as the quenched liquid phases. This scenario has been successfully extended towards other systems, indicating that amorphous polymorphism, as a manifestation of distinct liquid states, could be a general feature of condensed matter (Kurita, 2004(Kurita, , 2005. However, it has been equally questioned by recent computer experiments that introduced bandwidth of transformation scenarios spanning between the extrema of a pure relaxation phenomenon of an amorphous matrix and a multiple-phase transition scheme (Guillot, 2003;Brovchenko, 2003;Martonak, 2004).
The experimental proof or counterevidence of the twoliquid scenario in water is a subtle task, as any attempt to directly access the hypothetical two-liquid regime is bound to fail. Fingerprints of liquid polymorphism, thus, are looked for in the amorphous states. One crucial indicator is the presence of a first-order transition between HDA and LDA. However, any experimental approach towards a classification of the HDA-to-LDA transformation is severely hampered by the non-ergodic nature of the amorphous structures (Koza, 2005a).
Hence, an experimental search for characteristic features that might help to discern between the proposed thermodynamic concepts is the only approach to shed some light on the origin of the amorphous polymorphism of ice. One characteristic feature of the HDA-to-LDA transformation is an enhancement of the elastic signal in the small-angle scattering regime. Already the very first in situ studies of the HDA-to-LDA transformation have shown that despite the continuously changing static and dynamic structure factors of the amorphous ice there is also a transient excess signal at low scattering angles (Schober, 1998;Schober, 2000). The enhanced smallangle neutron scattering (SANS) and small-angle X-ray scattering (SAXS) signals could be understood as the response of frozen-in density variations, i.e. spatial heterogeneities, occurring transiently during the structural transformation of HDA into LDA.
In this manuscript we report on in situ neutron diffraction experiments in the small-angle scattering regime monitoring structures of typically 10-1000Å. From a set of time and temperature dependent experiments, we are able to substantiate the transient, thoroughly heterogeneous character of the intermediate transition stages on a spatial scale of some nano-meter and to describe qualitatively the temperature dependence and time evolution of the transient heterogeneities. We will equally show, that beyond the Q regime of the transient enhanced scattering there is a signal due to interfaces whose kinetics follows the wide-angle response (Koza, 2003).
All samples were prepared by slow compression of D 2 O (purity 99.8 %) ice I h at T ≈ 77 K up to pressures of 1.8 GPa in a piston cylinder apparatus (Koza, 2003). Each preparation run resulted in a sample volume of 3 ml. To meet the condition of a 10 % scattering sample for the suppression of multiple scattering, the samples were carefully crushed into milli-meter size chunks and placed within a cadmium diaphragm into a standard flat aluminium sample holder. The diaphragm optimised the size of the sample, leaving a free space of 12 mm in diameter and about 2.2 mm thickness, at its centre with respect to the homogeneous neutron beam.
The purity of the HDA samples had been confirmed at the spectrometer IN6 (0.3Å −1 ≤ Q ≤ 2.7Å −1 ) before they were mounted for measurements in a standard cryostat at the small-angle diffractometer D22. Q is the modulus of the scattering vector defined as Q = 4π sin (Θ)/λ i with Θ and λ i being half the scattering angel and the wavelength of the incident radiation, respectively. Both instruments are situated at the Institut Laue Langevin in Grenoble, France. The transmissions of the samples were determined experimentally in the HDA state at D22 to 89 ± 2% corresponding with an effective sample thickness of 1.5 ± 0.3 mm. Samples had been kept for about 30 min at 78 K for a complete removal of liquid Nitrogen, before any high accuracy measurement was started. During the experiments an atmosphere of 200 mbar of helium was applied.
The measurements were carried out with an incident neutron wavelength of 6Å and additional test measurements were carried out with a wavelength of 24Å in order to access the largest structural units. A lowest limit of Q ≈ 5 × 10 −4Å−1 could be reached. The exploitation of such a large Q range requires in practice the modification of the instrumental setup. High accuracy measurements in the structurally stable states HDA and LDA were carried out with three detector to sample distances, namely 1.4 m, 5 m and 18 m. Data acquisition time was 10 min at each position. The setup choice for in situ studies of the transforming structures was contingent upon the transformation kinetics at the chosen temperatures since a dead time of 1.72 min. was due to the setup changes. Consecutive measurements were performed with a detector to sample distance of 1.4 m and 10 m for 3 min. each. At the temperature 105 K only a single detector to sample distance of 2.5 m could be applied due to the fast transformation kinetics of the sample.
In this paper we can only present a subset of our experimental data. The four samples reported here were followed in situ at the nominal temperatures of 100 K, 101.5 K, 103 K and 105 K. Please note that throughout this paper we refer to the sample states measured prior to the heat treatment as HDA and after having followed the transformation and an annealing procedure at 127 K as LDA. No changes of the LDA structure factor could be found upon cooling of the samples back to ≈ 78 K.
Standard data corrections for empty can and background scattering were applied for the setups with 1.4, 2.5, 5 and 10 m. The calibration of the detector and normalisation to absolute units were accomplished with a water (H 2 O) standard of 0.1 mm thickness. The effective scattering power of the water standard at λ = 6Å was taken into account (Lindner, 2002). All corrections and the azimuthal averaging of the two-dimensional data were done with the software package GRASP (Dewhurst, 2003). For a clear presentation the data sets were normalised to unity with respect to an LDA baseline. The normalisation factor onto an absolute scale is 4.8(1) × 10 −2 cm −1 .
For the readers' convenience we report in figure 1 a FIG. 1: Diffraction data recorded in the course of the HDA to LDA transformation. The small-Q intensity indicates a transient intensity excess and a state of strongest heterogeneity (SSH) can be identified.
selected set of wide-angle diffraction data recorded in the course of the HDA to LDA transformation at the diffractometer D20 at Institut Laue Langevin. The grey shaded area stresses the small-Q regime in which intensity changes indicate that the sample passes through a state of strongest heterogeneity (SSH). Please note that already the HDA structure displays an intensity higher than the one of LDA which is in agreement with prior publications for neutron and X-ray scattering (Schober, 1998;Schober, 2000;Koza, 2005b;Koza, 2006). The small-Q signal of the other intermediate tranformation stages has been suppressed for a clear presentation.
Small-angle scattering data taken at D22 are shown in figure 2. In the left panel we report the intensities of all samples measured in the HDA and the LDA states. Data sets of different samples are shifted for clarity. Two features dominate the signal, an apparently flat background at Q > 5 × 10 −2Å−1 , comprising contributions from the incoherent scattering from D 2 O and from density variations that can be associated with the compressibility of the amorphous matrix, and a power law scattering towards smaller Q. The pronounced power law dependence S(Q) ∝ Q −4 is the so called Porod-limit scattering (PLS). It is the final slope of a SANS form factor that appears due to a sharp boundary between two phases in a sample and depends only on the scattering contrast and the interface area, but not on the shape of the structures or particles present in the sample (Glatter, 1982;Lindner, 2002). Note that the intensity of the PLS was well reproduced in all our samples following the same preparational procedure. At low Q, our data do not cover the range necessary to observe a Guinier-limit scattering. Furthermore, comparison of results obtained with 6 and 24Å show a clear influence of multiple scattering on the data (Lindner, 2002). The intensities of the PLS were determined to 4.8(1) × 10 −15Å−5 and 3.7(1) × 10 −15Å−5 for HDA and LDA, respectively. Please see the Appendix  (Schober, 1998). The data have been shifted for clarity.
for more information.
Let us focus in the following on the momentum range Q > 0.1Å −1 , i.e., at the apparently flat background beyond the PLS. Right panel of figure 2 reports the intensities of the four samples in comparison to prior results obtained on the spectrometer IN6 (Schober, 1998). As it has been shown before in X-ray and neutron scattering experiments (Schober, 1998;Schober, 2000;Koza, 2005b;Koza, 2006) an excess of the SAS signal indicates a pronounced heterogeneous character of the initial HDA structure. Whereas the LDA modification shows a constant signal. These features are entirely reproducible. Figure 3 reports the in situ SANS formfactors I(Q, t, T ) of samples #2 (left hand side) and #1 (right hand side) in a double-log plot. As it is indicated by the vertical arrows the increase of the transient signal is plotted in the top figures, its downturn is shown at the bottom. Equally indicated is the time t after which the data have been recorded.
It is evident that I(Q, t, T ), in the presented SANS regime, is a characteristic measure of the HDA-to-LDA transition, In analogy to the features of the wide-angle diffraction (WAD) signal elaborated in reference (Koza, 2005a), we can find for example to any I(Q, t, T = 103K) a matching signal observed at a different T after a welldefined but different evolution time t. As a consequence, it is not only the mere presence of a SANS signal that is characteristic of the intermediate structures, but it is in particular its intensity and the details of its profile that discern and, basically, define distinct transition stages.
A similarity with the wide-angle signal can be also found in the kinetic properties computed from I(Q, t, T ). For simplicity we restrict our consideration here to intensities integrated over two different Q-ranges. One range (10 −2 ≤ Q ≤ 2 × 10 −2Å−1 ) stresses the PLS evolution (I P (t, T )) and the other (0.1 ≤ Q ≤ 0.15Å −1 ) represents the Q regime of the transient excess scattering (I I (t, T )). Figure 4 shows the time dependence of I P (t, T ) (top panel) and I I (t, T ) (bottom panel) of the two samples. The data have been normalised in accordance to reference (Koza, 2005b). I P (t, T ) takes on values between unity, representing the HDA state, and null, representing the LDA state. I I (t, T ) is defined as null for the LDA state and unity for the SSH. Note that the relaxed statistics of the I P (t, T = 105 K) signal is due to the limited Q-range given by the detector to sample distance of 2.5 m.
I P (t, T ) displays a dependence well comparable with the evolution of the WAD signal reported in detail in reference (Koza, 2003). Two features are dominating the time response. First, a sluggish transformation process is observable, and second, a sigmoid shaped (Avrami-Kolmogorov type) step is detectable. Although, I P (t, T ) bears fingerprints of an Avrami-Kolmogorov type transformation the entire process cannot be understood as a simple nucleation and growth scenario (Doremus, 1985). This has been discussed in detail in ref. (Koza, 2003).
We may find a simple explanation for the equivalent behaviour of the PLS and WAD kinetics. Taking into account that the intensity of the PLS is proportional to the square of the difference in scattering density and the specific surface ∆ρ 2 ∼ S/V , it is an index of the density of the sample. In an equivalent way a dependence has been established between the density of amorphous samples and the relative position of the structure factor maximum, however, determined by structural changes on a local length scale of someÅ (Elliott, 1991(Elliott, , 1995. It has been indicated recently based on diffraction experiments (Koza, 2005b) that the kinetics of the SANS FIG. 4: Left: Kinetic response of sample #2 in the Porodlimit scattering IP(t, T = 103 K) and the intermediate Q regime II(t, T = 103 K). Right: Corresponding response of sample #1 at 105 K. Please note that the fall off the signals at the end of data sets is due to the annealing process of the samples to LDA at 127 K. signal in the intermediate Q range is equally closely related with the WAD. Here we show that this relation applies as well to the PLS kinetics. The SSH can be found always close to the centre of the transformation between HDA and LDA. This behaviour is independent of T . However, one has to take some care when interpreting I I (t, T ) in detail. The decisive and accurate observable constitutes the Porod-invariant q = The present SANS data establish unequivocally the thoroughly heterogeneous nature of the amorphous ice structures when following the transformation from HDA to LDA. This characteristics exclude the possibility of a homogeneous relaxation process of an amorphous ice matrix. Hence, to explain the transformation behaviour we may think of two other simple scenarios.
First, taking the non-ergodicity of the amorphous ice structures into consideration we may think of the sample as being composed of sub-ensembles each of which is governed by distinctly different relaxation dynamics, i.e. relaxing for a given T on different time scales. This heterogeneity in relaxation behavior translates into a strong spatial heterogeneity of the system while going through the transition, the reason being the large density differences between the still present high-density and already relaxed low-density sub-ensembles.
Second, a first-order transition may not be excluded as a process underlying the HDA to LDA transformation. Since this transformation is accomponied by an appreciable density change of almost 30 % the kinetics of the transformation is expected to be strongly perturbed by the additional elastic energy contribution as discussed in reference (Tanaka, 2000). In particular we may expect that early transition stages encountering molecules within a low-density environment surrounded by a highdensity matrix will be strongly stressed. On a local scale, the sample is influenced by a non uniform pressure distribution leading to departures from the properties of a non-stressed bulk low-density amorphous structure. Irrespective of the scenario, i.e. a heterogeneous relaxation or a real phase transition, underlying the transformation between a high-density and a low-density amorphous structures it is obvious that HDA as it is prepared by compression at 77 K is a heterogeneous structure on a nano-meter scale. Hence, it is tempting to consider the very-high density amorphous ice modification as the initial stage of the transformation (Mishima, 1996;Loerting, 2001;Koza, 2005b;Koza, 2006). As a consequence, the heterogeneous character of HDA has to be properly accounted for when structural properties are computed or modelled in real space from experimental data. The pronounced small-angle signal should in general be a help in descerning between different models trying to explain the phenomenon of amorphous polymorphism.
The overall behaviour reported here on amorphous ice modifications is not unlike the properties reported on a different system showing apparently amorphous polymorphism, namely triphenylphosphite (TPP). A thorough heterogeneous character of the TPP sample passing through a phase transition between two homogeneous states has been established by nuclear magnetic resonance, light-scattering and SANS experiments (Senker, 2005). TPP SANS data show a pronounced PLS and an excess signal at intermediate Q in the amorphous state (Alba-Simionesco, 2000). Light scattering data confirm the transient heterogeneous nature of the sample on a micro-meter scale and indicate a complex kinetics of the transition which can deviate from an Avrami-Kolmogorov nucleation and growth scenario when the transition happens via a spinodal decomposition (Kurita, 2004). Moreover, properties of intermediate stages cannot be reproduced by a superposition of properties of the initial and the final transition states, i.e. the superposition principle fails.
The features established during the spinodal decomposition in TPP signify the complexity of a transition between amorphous structures, which might be equally the case for amorphous ice. This findings force us to conclude that superposition principles, isosbestic point criteria or classical nucleation and growth scenarios are of no particular significance when trying to account for the real origin and nature of the transformation between amorphous ice structures.
We have applied small-angle neutron scattering (SANS) techniques to study the structural properties of amorphous ice modifications on mesoscopic lengthscales. It has been shown that the high-density amorphous (HDA) ice produced by compressing crystalline ice is a heterogeneous structure on a spatial scale of some nano-meters. When following the transformation of HDA into the low-density amorphous modification (LDA) the SANS signal displays a contrast maximum at about the center of the transformation. Thus, the sample passes through a state of strongest heterogeneity.
As it has been reported earlier and shown here in detail the transient SANS signal is a characteristic feature of the HDA to LDA transformation, and it is intrinsic to structures intermediate with respect to the very-highdensity amorphous (vHDA) modification and LDA.
When following the HDA-to-LDA transformation in situ the evolution of the Porod-limit scattering shows a time dependence reminiscent of the WAD signal (Koza, 2003). Its kinetics cannot be described by a pure Avrami-Kolmogorov time dependence, that characterises a plain nucleation and growth scenario. We have pointed out and discussed in detail that the nonapplicability of a nucleation and growth scenario does not exclude a real phase transition of first order between two amorphous ice structures. We may only draw the conclusion that a homogeneous relaxation of an amorphous matrix is to be excluded as a possible transformation scenario between high-density and low-density amorphous ice.
An obviously important question is the origin of the strong PLS in the samples. The PLS persists on the explored Q and time scales of the experiments not only beyond their recrystallisation to ice I c (Koza, 2005a; SANS data not shown here) but also upon annealing HDA into the very-high density modification (Koza, 2006). We have undertaken efforts to reduce the PLS intensity, e.g. by different sample treatments. For example we have measured HDA disk samples of about 1 mm thickness and 12 mm diameter before and after crushing them into mmsized chunks. The effect of the sample consistency did not effect the PLS intensity appreciably. If we consider a scenario of uniform, spherically shaped heterogeneous domains as the source of the PLS and approximate the scattering densities by the sample-to-vacuum contrast we may estimate the size of the domains to 1-10 µm. Hence, they are well separated by at least two orders of magnitude from the transient structural changes on the intermediate scale and sufficiently large to accomodate crystallites of ice I c after a recrystallisation of LDA.
The consistent reproducibility of the PLS indicates that it is either a generic feature of the amorphous ice samples or is created by the compression process of the crystalline ice matrix. For this reason we have examined crystalline samples that had been precompressed to different pressures. Figure 5 reports example data from three different runs. The first crystalline sample has been formed within the pressure device as for the preparation of the amorphous structure at 77 K, however, no pressure was applied. The second sample has been precompressed FIG. 5: Contrast plot of the two-dimansional signal measured with three samples having been precompressed to 0 GPa (a), 0.9 GPa (b) and 1.8 GPa (c), respectively. to 0.9 GPa which corresponds to a pressure close to the formation of HDA. Figure 5 c reports the signal measured with one of the HDA samples having been compressed to 1.8 GPa.
All our test runs showed a pronounced presence of impurities, dislocations and stacking faults already within the uncompressed crystalline samples. This is visualized in figure 5 a by the anisotropic scattering characteristic. By applying pressure to the samples the signal from the perturbed crystalline matrices indicated a trend towards isotropic scattering (figure 5 b) whereby a complete isotropic characteristics was reached in the HDA structures (figure 5 c). Although, this observation is based on ex situ compression runs it indicates that the PLS is a generic feature of the compressed ice samples and might be of essential significance for the formation of the amorphous matrix. It is interesting and important to note that the pressure at which amorphous ice can be formed is depending on the consistency and grain-size of the initial sample state (Johari, 2000). The lowest formation pressure of HDA is observed when compressing the LDA matrix, i.e. when the PLS scattering gives evidence of a strong and isotropic distribution of interfaces within the amorphous matrix.