Cage reactions in sodalites – A phenomenological approach using cellular automata

Journal Pre-proof Cage reactions in sodalites – A phenomenological approach using cellular automata L. Robben PII:

S1387-1811(19)30731-0

DOI:

https://doi.org/10.1016/j.micromeso.2019.109874

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MICMAT 109874

To appear in:

Microporous and Mesoporous Materials

Received Date: 3 July 2019 Revised Date:

1 November 2019

Accepted Date: 4 November 2019

Please cite this article as: L. Robben, Cage reactions in sodalites – A phenomenological approach using cellular automata, Microporous and Mesoporous Materials (2019), doi: https://doi.org/10.1016/ j.micromeso.2019.109874. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Inc.

Cage reactions in sodalites Cage reactions in sodalites – a phenomenological approach using cellular automata

Author: L. Robben

Journal: Microporous and Mesoporous Materials

Abstract: Three-dimensional cellular automata (CA) models for cage template reactions in sodalites are developed. The CA’s basic structure of cells and their neighborhood relations in sodalites is defined by their body-centered cubic net of cages. They are connected via four- and six-ring windows, with transport mechanism only allowed via the six-ring windows. Cage reactions are encoded in the CA’s transformation rules. Two reactions are modelled: The decomposition of the permanganate ion in the sodalite cage at elevated temperatures and the transformation of the nitrite sodalite to the carbonate nosean in a CO2 atmosphere. In both cases parameters are extracted which can be compared with experimental results, showing a good agreement. Advantageous features of such CA are: The possibility to extract the contributions of single reaction steps, easy examination of the influence of crystallite shapes. Furthermore, the influence of the reaction’s confinement on the cages and their connections are naturally included in these models.

Keywords: sodalites; cage reactions; cellular automata

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Cage reactions in sodalites 1. Introduction Models of nature’s systems could be classified into three groups following Vichniac [1]: a) models that are solvable exactly (by analytical means), b) models that are solvable approximately (by numerical means), c) Models that are exactly computable. Cellular automata (CA) are members of the last group, they are discrete in space and time, and their behavior follows a deterministic set of rules. Although the construction of a CA can be very simple regarding states and rules, their resulting behavior can be very complex and even chaotic. The basic idea of cellular automata was formulated by John von Neumann and Stanislaw Ulam in 1948 [1,2] and focused on the simulation of biological systems. The idea behind such automata was to simulate the human brain and its functions (i.e. to understand the mind) and to construct automata, which are computational universal [2]. Computations and operations to do so were reduced to very simple calculations carried out by many simple cells. In his book “Theory of self-reproducing automata” [3] von Neumann describes the necessities of a cellular automaton in detail (Chapter 2: “A system of 29 states with a general transition rule” p. 132): 1. a crystalline lattice 2. a finite set of states, which can be attributed to the lattice points 3. a defined neighborhood 4. and unambiguous transition rules for the state change of a lattice point based on the states of the neighbors Besides the specialist’s interest in cellular automata a broader public interest was created by J.H. Conways “Game of Live”, a 2-dimensional CA with a statistical rule set, presented by Gardner [4], which can show a very complex behavior. Stephen Wolfram presented detailed

2

Cage reactions in sodalites examinations on 1-dimensional CA in 1983 [5], of these simple 1-dimensional CA some can reproduce patterns, which are observed in nature, e.g. in the shells of certain mussels. Cellular automata became tools in such diverse areas such as e.g. social sciences [6,7], traffic flow simulations [8] or ecology (e.g. [9]). In material sciences cellular automata are e.g. used for the simulation of the recrystallization processes [10,11]. A large collection of chemical effects and situations, like particle behavior in solutions, surface – water boundaries behavior and chemical kinetics were described by Kier, Seybold and Chang in a book accompanied by the simulation program CASim [12]. In crystallography, Krivovichev used CA as a dynamic topological tool to describe the crystal growth of metal sulfides based on fundamental building blocks [13] or to model the crystal growth in heteropolyhedral layered complexes of uranyl arsenates [14]. For zeolitic materials, like the sodalites considered in the present work, only lattice gas CA (LGCA) modelling of the adsorption of gas molecules are published [15,16]. LGCA simulate the dynamics of gas molecules including elastic collisions (see e.g. [17] p. 360). Structure and properties of sodalites Sodalites are zeolite-type materials , their framework forms only one cage type (sod) and is one of the simplest zeolitic frameworks observed [18]. Naturally occurring and synthetic sodalites show a great variability of elemental combinations in the framework and ions in the cages. A comprehensive overview was collected by Fischer and Baur [19,20]. The framework is build up by corner-sharing TO4 tetrahedra, where T could be Si4+ [21], Al3+ [22] or typical equivalent substitutions of tetra- and trivalent ions like Al3+/Si4+, Al3+/Ge4+, Ga3+/Si4+ or Ga3+/Ge4+ [23–26]. But also non-equivalent mixtures [27] or combinations like Zn2+/P5+ are feasible [28]. Each unit cell contains two sod cages, one in the center and 1/8th at each corner. The cages are connected via 4- and 6-ring windows, due to their small diameter diffusion and reactions via the 4-ring windows can be excluded. The cages form a body-centered cubic

3

Cage reactions in sodalites (bcu) net (the dual net of the sod net), which defines the grid for the CA used here (more details are given in the section “Computational details”).

Figure 1: The topological net sod (left), its dual net bcu (right) and their combination (middle).

Of special interest in this study are the changes in the cage fillings due to cage reactions, their effects on structural parameters as well as the phenomenological kinetics of such reactions to get models for the real kinetic effects observed. The present work will focus on the following two reactions as suitable systems for the development of the CA model: a) the thermal decomposition of |Na8(MnO4)2|[AlSiO4]6 b) the transformation of |Na8(NO2)2|[AlSiO4]6 to |Na8(CO3)□|[AlSiO4]6 at elevated temperatures in CO2 atmosphere These reactions were examined in detail by Petersen et al. [29] and Šehović et al. [30], respectively. For details and previous works on these materials see these references and the references therein. For the comparison of the CA calculations with experimental data, a few key parameters are useful. For |Na8(MnO4)2|[AlSiO4]6 (case a) the nosean formation causes a distinct difference in the halfwidth of reflections being symmetry forbidden for the description of the sodalite structure in space group

43

but are allowed for the Nosean phases described in space

group 23. This was interpreted as different domain sizes (i.e. size of homogeneous ordered regions) of the framework and the cage-filling ions. This different domain sizes are fixed by 4

Cage reactions in sodalites the system during the reaction; in long-time heating experiments they do not change after completion of the reaction [29]. The reflection halfwidth can be used to calculate the mean Lorentzian scattering volume for both groups of reflections. The ratio of these is defined here as the degree of order. For the phase transformation of |Na8(NO2)2|[AlSiO4]6 to |Na8(CO3)□|[AlSiO4]6 (case b) there are two main experimental observations. The first one is regarding the domain sizes of framework and template ordering, analogous to a) but in this case a time-dependent change of the degree of order can be observed [30]. The second one is the reaction kinetics, which was observed in [30] by time-dependent X-ray powder diffraction at different temperatures and used to evaluate the activation energies. As in case a) also here a successful CA should reproduce the experimentally observed domain sizes as well as the principal shape of the observed Avrami plots.

2. Computational Details A very formal definition of a cellular automaton starts with a non-deterministic finite automaton , which is a tuple =

, ,

, ,

. eq 1

Here, ( :

is a finite set of states, ×

→ 2 ) and

considered,

the input alphabet,

the set of final states

the initial state, ⊆

a transition function

[31]. For the cage-reactions here

is a set containing all cage-fillings (i.e. educts, products and intermediate steps)

of a certain reaction. The set of final states

should ideally contain only the reaction product,

but it is possible, that a cell could not react in the calculation, so that in the worst case The input alphabet

= .

is here of minor interest, it is in this context the set containing all

possible cage fillings (that is reaction educts, products and intermediate cage-fillings) of all

5

Cage reactions in sodalites possible cage-reactions or it can be reduced to chemical reaction. Starting from state of the automaton. Thus

. The transition function

describes the

, the initial cell state, via intermediate steps to the final can be subdivided into partial reaction steps

, when

necessary. Let the cellular automaton CA be defined as a set of =

,…,

∈ ℕ elements: !

eq 2

where N# are tuples:

N# = x# , y# , z# , v# , r# eq 3

With x# , y# , z# ∈ ℤ being coordinates in a three-dimensional space, v# ∈ Q ⊆ ℕ is the state attributed to the ith cell and r# = + x# − x-

.

+ y# − y-

.

+ z# − z-

.

eq 4

is the distance of the ith cell to an arbitrary chosen cell x- , y- , z- making this CA an ordered

set. Regarding the CA definition by von Neumann (see introduction), the N# define now the so-called crystalline lattice in three dimensions with discrete cells and each cell has a cell state

v# . Dealing with CA, such a regular and isotropic 3-dimensional grid is often called “universe” (Figure 2).

6

Cage reactions in sodalites

Figure 2: The CA universe (left) with a plane cut with a normal parallel to the space diagonal. Right: The same universe with initial cell states colour coded: Green cells: Outside gas environment, white cells belong to the spherical particle.

On this grid a subset of cells is selected, forming a spherical crystallite with radius 01 . All

cells with 0 < 01 receive the cell state “inside” and all cells with 0 > 01 the cell state “outside” (Figure 2, right panel). With an unit cell length of 900 pm a particle with 01 =

5 56778 corresponds to a crystallite diameter of 4.5 nm (values for typical 01 values used here

are collected in Table 1). Table 1: 01 , number of cells in the the crystallite and the corresponding estimated crystallite diameter

9: / # cells

# cells in volume

crystallite diameter /nm

5 10 15 20 25

531 6081 22995 57313 115603

4.5 9.0 13.5 18.0 22.5

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Cage reactions in sodalites

Figure 3: The neighborhood of the cell (x,y,z): cells connected via the sodalites 6-ring windows

In the second step, the bcu-net representing sodalite cages is defined within the crystallite, so that only cells with the neighborhood relations shown in Figure 3 receive the initial cell state representing the sodalites template molecule before the reaction takes place, all other cells in the crystallite are ignored and do not change. The topological connection shown in Figure 3 enables only material flow via the cages six-ring-windows. The secondary connection via the four-ring-windows is much too small to allow an effective exchange of ions. This defines the neighborhood relations demanded by von Neumanns definition. The transition rules describe the cell-state change due to the cage-reactions and are described in detail in the respective chapters. The intrinsic basic time or iteration unit of a cellular automaton is called a “generation”: One generation is the time needed to check for all cells once, if one of the transition rules could be applied. For the subsequent change of the cell state there are two modi possible: Either synchronous, the transition rules are applied on all cells and all cells change the state at the same moment, or asynchronous, the transition rules are applied on a cell and the cell changes its state upon this. When modelling physical systems the asynchronous change of cell state is

8

Cage reactions in sodalites advantageous because it preserves mass-conservation [12,32]. The cell states and transition rules are described for each modelled reaction in the respective chapter. CA modelling has the advantage that at each step certain characteristic parameter values can be extracted from the model. The conversion rate ; for a reaction at the ith generation could be defined as: ; =

1<=>?@A B>?@A

eq 5

with

B>?@A

being the number of cells containing the educts at the start of the calculation

(generation 1) and

1<=>?@A

the number of cells containing the reaction products in the ith

generation. The cells can also be counted with respect to their status and their neighbors’ states. A cell with state a is called ordered with respect to a state b if all its neighbors have the cell status b. To get a parameter that is comparable to the average scattering volume size (Lorentzian scattering volumes) obtained from Rietveld refinements, an equivalent ordered radius 0=<>C could be calculated using the total amount of ordered cells in the ith generation =<>C :

3 0=<>C = D F

=<>C

4E

eq 6

This value can be related to the crystallite size by the quotient =

0=<>C , 0GH
which is comparable to the respective ratio of Lorentzian scattering volumes obtained from the Rietveld refinements.

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Cage reactions in sodalites Furthermore, it is easily possible to extract the number of cell changes caused by a rule in each generation. This is especially useful in the second example to examine the number of CO2 molecules flowing from the outside environment into the crystallite, the number of reactions or jump processes inside the crystallite. The calculations were realized using a code written in Python, 3-dimensional data visualization was carried out using the software ParaView [33,34].

3. Results and Discussion 3.1. The thermal decomposition of |Na6+x(MnO4)x(H2O)8-4x|[AlSiO4]6 (0 ≤ x ≤ 2) The template-framework interaction and the thermal decomposition of MnO4- were examined by Petersen et al. [29]. The permanganate ion decomposes via 2 J KLM → J KL.M + J K. + K. This decomposition is only possible, when two neighboring cages react, and it needs no diffusion from gaseous species from the outside of the particle into it. The difference in the electron density between manganate and the manganese oxide gives rise to a reduction of the symmetry from space-group 43 to space-group 23, due to the non-statistical distribution of the two cage filling ions. Accordingly, new reflections appear in the diffractogram as the reaction proceeds. As it was observed in this reaction [29] and the carbonate nosean formation, described later on [30], the upcoming reflections show much higher halfwidths than the ones belonging to the initial sodalite, which is caused by different domain sizes of ordered framework and template [30]. Contrary to the carbonate-nosean formation the degree of order of the MnO4-/MnO2 nosean does not increase with longer reaction times or in longtime annealing experiments, indicating that diffusion processes of these large ions in the framework are not possible. It was experimentally determined that 97.0(7) % of the J KLM ions decompose in the aforementioned reaction and that the best

10

,CNGC< OC AHI

value is

Cage reactions in sodalites approximately 0.18 [29]. As a benchmark, the cellular automaton should reproduce these experimental values. Figure 4 shows the cell states and the transition rules schematically. The occurring cell states are: = K. , ∅, J KLM , J KL.M , J K. ! eq 8

Where K. are the cells outside the particle, ∅ unalterable cells inside the particle to build the

bcu-net (and in a second step, some randomly selected J KLM cells are converted to ∅ to

represent water). J KLM , J KL.M , J K. are the bcu-net cell states. The CA rules are

graphically exemplified in Figure 4: The initial cell states are shown in the left column (K.,

J KLM and ∅). The half-circled arrows at the O2 and ∅ designate that there is no change in these cell states. For the J KLM cells there is the transition function

transforming one cell

to J KL.M and one to J K. . Of course, certain cells must be selected for the cell state change (i.e. the reaction) to happen, this selection procedure can have influence on the degree of order observed as well as the overall water content. The water content can be adjusted by setting statistically a certain number of bcu-net J KLM cells to the state ∅.

Figure 4: Scheme of cell states and transitions of the Mn-SOD decomposition

Random vs. density cell selection scheme

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Cage reactions in sodalites Two different cell selection schemes were devised giving two different models for this cagereaction. The first model is based on a random cell selection (designated “random” below) : If a cell with state J KLM is found and a randomly chosen neighbor cell as well, then

can be

applied. The second model is based on a selection scheme, that selects the neighboring cell, that itself has the most J KLM filled cells in its neighborhood (i.e. that space has the highest density of

J KLM cells). If there are several cells with the same number of J KLM neighbors a random one of these is chosen (this scheme is called “density” below).

Due to the statistical nature of the cell selection process, it must be controlled how strong the variances in the parameters 0=<>C and ; depend on the system size, thus model calculations of 20 generations with a varied crystallite size and J KLM content x were repeated

1000 times. The obtained average values of

and ; and their variances are shown in Figure

5. It is obvious that for a system size 9: ≥ RS the variance of the obtained parameters becomes neglectable.

Figure 5: average values of ;.T (left) and 20 (right) after 1000 repetitive calculations of 20 generations in variable system sizes (rp = 4, 5, 10) and MnO4- content x to estimate the standard deviations

12

Cage reactions in sodalites The examination of the final stable CA status and its properties was carried out with a particle radius 01 = 25, corresponding to 115603 cells in the calculation. 100 generations were used, which was in every case enough to lead to a stable CA status at the end. Visualizations of the CA end states for both models described above (random and density) and different J KLM contents are shown in Figure 6.

Figure 6: Stable CA configurations after 100 generations for different initial J KLM contents x and the two models (left column: density; right column: random; The density mechanism shows a star-like pattern, which is a computational artefact and less pronounced when the system size is increased)

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Cage reactions in sodalites The benchmark values

TT ,

the domain size ratio after 100 generations, and ;

TT ,

the

transformation rate after 100 generations, are shown in Figure 7 for different J KLM contents x.

Figure 7: TT (left) and ; TT (right) depending on the initial J KLM content x for the two cell selection schemes. The lines are the least squares refinements (details see text). Grey shaded areas designate the value range obtained from the experiments (i.e. in the temperature range in which the values were quantifiable in temperature-dependent X-ray diffraction experiments).

A simple second order polynomial TT

V = W + XV + 5V . eq 9

was used to describe the

TT

V behaviour and to calculate the J KLM content for the two

models using the experimental value

,CNGC< OC AHI

= 0.18. The random cell selection

scheme gives an unreasonable x larger than two, whereas the density cell selection scheme can reproduce the experimental x of 1.7(2) very well. Table 2: Results of the polynomial fitting of

parameter values TT

V = 0.18

TT

for the two models and the calculated MnO4- content

Random a = 0.06(5) b = -0.15(7) c = 0.10(2) V ≈ 2.08

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Density a = 0.57(7) b = -0.99(10) c = 0.43(3) V ≈ 1.79

Cage reactions in sodalites On the other side for ;

TT

the density model gives lower values than the experimentally

observed ones, whereas the random model fits better to the experiment. Thus, the decision, which model is describing the results better, is quite arbitrary. In the real experiment other energy contributions may influence the choice which two cages are taking part in the reaction. The two models are in this sense idealized and in the real experiment the mechanism may switch between the two according to the local situation. 3.2. The transformation of |Na8(NO2)2|[AlSiO4]6 to |Na8(CO3)□|[AlSiO4]6 at elevated temperatures in K. atmosphere The sodalite cage reaction | W`

K. . |[ 7bcKL ]e + K. → | W`

Kf □|[ 7bcKL ]e + K + K. eq 10

was examined in detail by Šehović et al. [30] (the small box symbol indicates an empty cage in the unit cell), the main results of this work are: The reaction kinetics can be described by a simple Avrami equation and the obtained temperature-dependent time constants were used to determine an activation energy of 8.7(10) kJ/mol. The average crystallite size ratio of framework and template domain sizes tends to a value of 2 at long times, and opposite to the first example, tends to lower values at longer times. According to the reaction equation the following cell states are needed in the CA: =

K. , ∅, K.M , Kf.M , K.M / Kf.M ,□! eq 11

The experimental results indicate K. diffusion into the crystallite and subsequent ordering processes of Kf.M and empty cages. The transition rules must allow such processes.

15

Cage reactions in sodalites

Figure 8: Scheme of cell states and transitions of the nitrite sodalite reaction in a CO2 atmosphere

Cell states and their state changes according to the rules are exemplified in Figure 8. The rules are: •

iR : If a cell contains K.M + K. follows.

•

ij : The intermediate state containing

•

K.M or □, K. can enter that cell. An intermediate state K.M + K. can react to

Kf.M if there is another cell

K.M in the neighborhood. The second cell changes it state to □.

ik : Diffusion of K.: Neighbouring cells can exchange their state with K. if their state is either □ or Kf.M .

The cell state changes according to the different rules per generation can be counted and give a deeper insight into the system behavior. For this reaction the influence of the crystallite shape was also examined: Besides the spherical crystallite (115603 cells) a cubic crystallite (89531 cells) was constructed. The CA with the described cell states and transition rules were set up and run until equilibrium (here 4000 generations). Figure 9 pictures these CA at the start, in progress and at the end of the calculation.

16

Cage reactions in sodalites

Figure 9: The CA at different time steps between start (generation 0) and end of the calculation (generation 3999), left column: spherical crystallite, right column: cuboid crystallite (an animation is available in the supplementary information).

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Cage reactions in sodalites More important than this figurative result are derived values, which can be used to compare the CA behavior with the experimental results. From the number of cells with states

K.M and

Kf.M the transformation rate ; can be calculated and plotted in an Avrami-like plot of

ln − ln ;

versus ln n6 60Wocp

(Figure 10). As in the experimental paper [30] a linear

regression (black line) is included. Comparing these theoretical reaction kinetics with the measured ones (Figure 6 in [30], especially 948 K), similar deviations from linearity can be observed: At short reaction times (between 1.5 and 3 in the CA, between 3.25 and 3.75 in the 948 K experimental curve) the linear is lower than the experimental values, for intermediate times both, the CA and the experimental conversion rates, are lower than the linear and to the end of the reaction they are higher again. The deviation is stronger for the cubic crystallite. In the experimental work it was speculated that some deviations at the start of the reaction are caused by experimental effects (temperature equilibration of the sample) [30], but the CA produces qualitatively exact the same kind of deviations.

Figure 10: Avrami-like plot of the conversion rate obtained from the CA for a spherical (left) and cuboid crystallite (right). The black lines are linear fits. Noteworthy are the deviations of linearity of the conversion rate, which were also observed in the experiments.

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Cage reactions in sodalites The CA model allows a detailed look on the single reaction steps (i.e. the transition rules). Figure 11 shows (from left to right) the flow of K. from the outside environment into the particle (rule graph gives

), internal cell state jumps (rule

f)

and the reaction (rule

. ).

The rightmost

, the crystallite size ratio.

Figure 11: The number of rule applications in each generation: Flow: CO2 enters the particle; jump: Movement of CO2 inside the particle; reaction: Formation of CO32-. The right plot shows the development of q. Upper row: Spherical particle, lower row: Cuboid particle.

The flow and the reactions show in both cases a decrease with time, the number of reactions naturally shows a small lag to the flow. The intra-crystallite jump-processes start at 1900, decrease to 1100 (at ln n6 60Wocp ln n6 60Wocp

= 3) and increase to their maximum of 4200 at

= 7.2. Regarding the conversion rate (Figure 10) the deviations from the

linear behavior occur at the times the jump-rate shows its local maxima. The conversion rates first positive deviation is stronger than the last one, which could be explained by the high reaction and flow rates as the reaction starts, whereas the last one (ln n6 60Wocp

= 7)

happens, when the reaction rates are already very low and the jump rate start to drop. Initially, the ratio of jump to flow transitions is slightly lower for the cubic particle (≈0.5) than for the spherical crystallite (≈0.66), which may explain the initial differences in the Avrami-like plot (Figure 10).

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Cage reactions in sodalites The CA’s time development of

matches the experimentally obtained values (Figure 9 in

[30]) very well. But it also falls below the experimentally observed apparent lower boundary of two, regardless of the crystallite shape. However, in real powder samples the reactions take place at particle surfaces, whereas the X-ray diffraction crystallite sizes are defined as volume elements in which coherent scattering occurs, so several crystallites can build a particle. In the CA model a crystallite equals a particle and the final limit value of

seems to be one. This

could lead to the speculation that in the experiment each particle consisted of at least two crystallites.

4. Summary and Conclusions The application of simple CA models on selected cage reactions in the sodalite framework has been examined. Cellular automata are ideally suited for this task, because their structure closely resembles the sodalite’s cages and their connectivity. They naturally implement the geometrical constraints that the cages impose on the diffusion and motion of molecules in the sodalite. The first examined reaction, the decomposition of the permanganate ion, is a very simple reaction, because it does not need any external reaction partner, but it strongly depends on the topological arrangement of the cages. This case was used to examine the statistical validity of the parameter values, used for comparison with experimental values. A purely random cell selection scheme and a second one based on the local density of permanganate ions were implemented and the latter explains the experimental results for the domain size ratio best, especially the experimentally determined water content can be deduced from the calculated degree of order. The random model provides better agreement with the conversion rate. Thus, a clear decision, which of the two idealized models is the predominant one in the experiment, is difficult. However, in the experiment other factors may locally induce a switch from one to the other mechanism. The second case, the carbonate-nosean formation from the

20

Cage reactions in sodalites nitrite sodalite, depends on the diffusion of K. into the cages. As an additional parameter the influence of different particle shapes was considered. The CA reproduces important experimental observations, like the deviations from linearity in the Avrami-plots, which are a result of different ratios of K. flow into the sodalite and the internal diffusion and reaction rates. Here a slight influence of the particle shape can be observed. The time-dependent development of

is reproduced by the CA, although with a different limit value, which can

be explained by the highly idealized particle in the model. To conclude: Such simple CA models are capable to reproduce experimental observations. They are easy to implement and can be used to directly check different assumptions about the reaction mechanism, which is encoded in the transition rules. The influence of the single reaction steps can be observed in detail. One of the most important advantages is that the influence of the cage topology on the reaction is naturally included in the model. In the present state the biggest disadvantage is the missing connection to real time and consequently to real energy values. The main cause for this obstacle is the missing experimental data for ion jump rates etc. in different sodalites.

21

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Cage reactions in sodalites [30] M. Šehović, L. Robben, T.M. Gesing, Carbon dioxide uptake in nitrite-sodalite: reaction kinetics and template ordering of the carbonate-nosean formation, Zeitschrift Für Krist. - Cryst. Mater. 230 (2015) 263–269. doi:10.1515/zkri-2014-1815. [31] M. Hoffmann, M. Lange, Automatentheorie und Logik, Springer, Heidelberg, 2011. [32] J. Schiff, Cellular Automata - A discrete view of the world, John Wiley & Sons, Hoboken, 2008. [33] ParaView 5.3.0-RC1, 2017. http://paraview.org. [34] J. Ahrens, B. Geveci, C. Law, ParaView: An End-User Tool for Large-Data Visualization, in: Vis. Handb., Elsevier, 2005: pp. 717–731. doi:10.1016/B978012387582-2/50038-1.

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Cellular automata (CA) for the simulation of cage reactions in sodalites are presented

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The thermal decomposition of |Na6+x(MnO4)x(H2O)8-4x|[AlSiO4]6 is simulated

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The high-temperature reaction nitrite sodalite to carbonate nosean is simulated

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The CA models reproduce experimental observations

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Easy verification of reaction models and reaction steps is possible

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: