Kinetics of Decelerated Melting

Abstract Melting presents one of the most prominent phenomena in condensed matter science. Its microscopic understanding, however, is still fragmented, ranging from simplistic theory to the observation of melting point depressions. Here, a multimethod experimental approach is combined with computational simulation to study the microscopic mechanism of melting between these two extremes. Crystalline structures are exploited in which melting occurs into a metastable liquid close to its glass transition temperature. The associated sluggish dynamics concur with real‐time observation of homogeneous melting. In‐depth information on the structural signature is obtained from various independent spectroscopic and scattering methods, revealing a step‐wise nature of the transition before reaching the liquid state. A kinetic model is derived in which the first reaction step is promoted by local instability events, and the second is driven by diffusive mobility. Computational simulation provides further confirmation for the sequential reaction steps and for the details of the associated structural dynamics. The successful quantitative modeling of the low‐temperature decelerated melting of zeolite crystals, reconciling homogeneous with heterogeneous processes, should serve as a platform for understanding the inherent instability of other zeolitic structures, as well as the prolific and more complex nanoporous metal–organic frameworks.

which, once conveniently plotting (ln(-ln(ϕ(t)) as a function of ln(t)) should yield a straight line with β as the angular coefficient.
For the reaction from LSX to liquid, we assume a model reaction path as depicted in Figure  S1. Following a simple reaction rate scheme for sequential reactions, we have for the effective rate of LDA formation that (SE3) , where is the appearance rate of LDA due to the reaction LSXLDA, and is the disappearance rate of LDA due to the reaction of LDAHDA.
From our hypotheses, we now assume that the non-transformed fraction of LSX follows a KWW dependence given by , therefore, We argue in the manuscript that  = 2 for the first reaction, and  = 1 for the second reaction.
When solely the reaction of LDAHDA is considered, it is further and so ( ).
Eq. SE7 is the disappearance rate of LDA, not the effective (overall) rate. Substituting into Eq. (1) provided in the manuscript, we find that the effective rate of change of LDA is given by , or, rearranging, Integrating Eq. SE9, In the context of this study, a reasonable analytical solution of this integral is available only for the cases of β = {1,2}. In accordance with the previous arguments, here, we analyze the experimental data for β = 2. This yields the function for g(t) as stated in Eq. 2. The extracted reaction timescales are provided in Table S1. Table S1: Reaction times  1 and  2 as obtained from fitting the experimental data to Eq. 2. The effective reaction time  eff is read from the data shown in Figure 1c of the manuscript for  = 1/e after linear interpolation between sequential data points. The temperature-dependence of the reaction times as extracted from fitting the XRD data to Eq. 2 (Table S2) of the manuscript was framed within a simple Arrhenius equation, , with intercept  0 , energy barrier H and the universal gas constant R. This assumes constant H in all three cases. For  1 , this is taken as resulting from the super-strong nature of the LSX  LDA transition [2]. For  2 and  eff , it is a reasonable approximation within the relatively narrow viscosity range of consideration [ 3 ]. The resulting Arrhenius plots are provided in Figure 1e of the manuscript. Slopes of ln() vs. 1/T are given in Table S2.
In order to judge the observed timescale, classical data of surface pre-melting was considered. Due to the lack of data on carnegieite, albite was used for reference. Albite premelting has been described by Greenwood and Hess [4]. In their report, surface melting rates are provided for a range of temperatures close to the melting temperature of albite, i.e., 1120 °C. In order to approximate reaction times from these melting rates, we calculated the melting progress of a spherical particle with diameter d, and extracted the time  SPM after which the melted volume has reached a fraction of (1-1/e) as shown in Figure S3. This situation best resembles the experimental set-up shown in Figure 1. In particular, it avoids the notorious divergence of other approaches for translating reaction rates into characteristic times such as the Stokes-Einstein equation.

Fig. S3
. Surface melting progress of crystalline albite for spherical particles with diameter of 10 µm. Rates according to [4].

Reproducibility of isothermal experiments
The following graph gives an example of reproduction. Here, the measurement at 798 °C was conducted two times, on individual samples, using different X-ray detection angular bandwidth for differently fast data acquisition.

Photoluminescence probing
Progress of XRD and PL emission spectra with annealing for LSX:Eu at 830 C and 850 C is shown in Fig. S5. Asymmetry data is shown in Fig. 2b.

NMR and IXS Analysis
NMR and IXS data are provided in the following Figs. S6-8.

INS Analysis
INS data are provided in Fig. S9.

Reproducibility of MD simulation
The following figure shows a reproduction of MD simulations.  capillaries. The LSX crystal shows the typical powder diagram of a crystalline material but the Bragg reflections appear poorly resolved. The detector setting is optimized for covering a large range of scattering angles but that is accompanied with a poor angular resolution. The S(Q) of the collapsed LSX shows an amorphous structure with only tiny relicts of the LSX crystal. Fig. S11 compares the S(Q)'s of the LSX collapsed and the LSX glass. The differences are very small. Moreover, the LSX glass was measured in capillary and as slabshaped sample. Both these S(Q)'s are nearly identical. The drop of intensities close to 200 nm -1 for all three S(Q)'s seems to have an unphysical origin.  The parameters of the Al-O and Na-O first-neighbor distances are determined by Gaussian fitting of the smooth T(r) functions that were obtained with damping. The effects of damping and truncation (Q max ) in the FT of the experimental T(r) data are simulated for the model T(r) functions by the convolution techniques as described in [6,7]. For fitting the T(r) data, the Marquardt algorithm [8] is used where coordination numbers, N ij , mean distances, r ij , and full widths at half maximum (fwhm), r ij , are the parameters of the model Gaussian functions. The final parameters are listed in Table S3 where each line corresponds to a Gaussian function. Some parameters are marked with asterisks in case of the first sample. For illustration of the quality of the fits also those model T(r) functions are calculated and compared with the experimental data which were obtained without use of damping functions in FT (Fig. 4). Here, the agreement is excellent for the LSX crystal, as well, just not perfect for the other samples. Distinct peaks are visible at 0.21 nm and 0.24 nm which could be The results of the fits do not show any new sensations or discoveries but they follow the expectations for the samples and confirm the usefulness of the assumptions. The N AlO values are a little larger than four but that is not sufficient to conclude a significant AlO 6 fraction. The N NaO value of the LSX crystal is close to six which is significantly larger than the other N NaO of the collapsed LSX or the LSX glass. The larger N NaO could be attributed to the crystalline order and the presence of water.

Computational details
Cell parameters for MD simulation: