#### 6.1. Methylamine-Templated Uranyl Selenates: Ordered and Disordered Structures

Gurzhiy et al.26 investigated phase formation in thesystem UO_{2}(NO_{3})_{2}/H_{2}SeO_{4}/methylamine/H_{2}O and found at least eight different crystalline uranyl selenates with U/Se = 1:2, 2:3, 3:5, and 5:8, some of them metastable or unstable in air. Of particular interest is the family of structures with general formula (CH_{3}NH_{3})[(UO_{2})(SeO_{4})_{2}(H_{2}O)](H_{2}O)_{n}, where *n* = 0, 0.5, 1. The structures are based upon complex units with composition [(UO_{2})(SeO_{4})_{2}(H_{2}O)]^{2–}, but with different topologies of linkage of U and Se polyhedra for different values of *n*. For *n* = 1, the structure contains 1D chains (Figure 13a), for *n* = 0 the structure is based upon 2D layers (Figure 13b). For *n* = 0.5, a disordered structure was observed that represents a superposition of the 1D and 2D structures (Figure 14). Transition between two overlapping configurations can be achieved by a 1/2 translation of the [1+4+1] unit (see above). It is important to note that all three structures shown in Figures 13 and 14 can be obtained by successive polymerization of the [1+4+1] cluster, which supports our suggestion concerning its existence as a prenucleation building block in aqueous solutions. Thus, in the (CH_{3}NH_{3})[(UO_{2})(SeO_{4})_{2}(H_{2}O)](H_{2}O)_{n} family, the appearance of disordered structure with *n* = 0.5 can be represented as an oscillation of the growth process between the structure types with *n* = 0 and 1. This process can be modeled by construction of abstract dynamic systems known as cellular automata.

#### 6.2. Cellular Automata: Basic Concepts

Cellular automata (CA) have been introduced for simulation of self-reproductive biological systems27 and have attracted considerable attention as a possible environment for modeling of a broad range of physical objects and processes,28 in particular, of periodic growth of complex chemical structures.29

From the formal point of view, CA is defined as a collection of five basic components:

where *Z* is a lattice (discrete working space of the CA consisting of cells; the simplest example is a 2D plane filled with square cells); *S* = {0, 1, 2, ...} is a finite number of values that the cells may take [usually, these values are associated with colors, e.g. *S* = (0, 1) characterizes a binary (2-color) CA]; *N* = {–*k*_{1}, –*k*_{1} + 1, ..., –1, 0, 1, ..., *k*_{2} – 1, *k*_{2}} is a neighborhood of CA action for 1D CA; the value of cell *x*_{0} at time *t* = 1 is determined by the values of *k*_{1} and *k*_{2} cells on the left and right sides at the time *t* = 0 [in the simplest case, the neighborhood is symmetric *k*_{1} = *k*_{2} = 1 and has a radius *k*_{0} = 1, that is, it consists of three cells (*k* = 3): *x*_{–1}, *x*_{0}, *x*_{1}; the value of cell *x*_{0} at time *t* = 1 is determined by the values *x*_{–1}, *x*_{0}, *x*_{1} at time *t* = 0]; *f* is a local transition function that works for a certain neighborhood (usually written as a set of rules of the form 0101); and *B* is a boundary condition.

The example of a CA is shown in Figure 15. According to the Wolfram classification,28c the CA has a number of 90. Its local transition function can be written as *f* = {1110, 1101, 1010, 1001, 0111, 0100, 0011, 0000}. Using one black cell as an initial condition, this CA results in the formation of a branching pattern known as the Serpinsky triangle. It is obvious that, for modeling of periodic structures (that may work as diffraction lattices), one needs a specific class of CA.

#### 6.3. CA Model of Uranyl Selenate Structures and Their Growth

In order to construct CA that reproduce the formation of uranyl selenate structures with U/Se = 1:2, we topologically (i.e. without breaking of bonds) transform the graphs shown in Figure 13c and d into the graphs shown in Figure 16a and b, respectively. In turn, these graphs may be replaced by a 2D cellular structure, where black and white vertices are replaced by black and gray cells, respectively (white cells are reserved for “empty” space regions) (Figure 16c and d). The cells that share common edges correspond to the vertices linked by an edge. As a result, we have two tricolor cellular structures shown in Figure 16e and f. Analysis of these structures indicate that they can be successively constructed by using ternary CA with transition rules shown in Figure 17. By assignment of values of 0, 1, and 2 to white, gray, and black cells, respectively, the set of transition rules can be written down as *f* = {0000, 0011, 0020, 0102, 0110, 0122, 0201, 0211, 0220, 1001, 1011, 1021, 1100, 1110, 1120, 1201, 1211, 1220, 2001, 2011, 2021, 2100, 2111, 2120, 2200, 2210, 2220}.

Most of these rules are excessive in our case; however, they are used for the reasons of generality. This CA can be used to obtain the cellular structures shown in Figure 16e and f. This means that *the same* CA generates *different* structural topologies. The resulting topology is therefore determined by the initial conditions, that is, by the structure of the first row. In the language of chemistry, this could suggest that the molecular-level growth mechanism of uranyl selenate layers is the same and the topology of the structure is controlled by the structure of the nucleus spontaneously formed in solution.

In order to investigate the dependence of the structure topology on the structure of initial conditions, a computer experiment was performed by means of the Mathematica 6.0 program package.30 The structure of the input row was given as an infinite periodic sequence of numbers 0, 1, and 2. Some results of the modeling are shown in Figure 18. The chain topology (Figure 18a) can be produced by using the sequence [0121], whereas the layer topology (Figure 18b) is the result of an input of the sequence [01012121]. The cellular structure shown in Figure 18c and resulting from input [12] deserves special attention. It corresponds to the structural unit shown in Figure 19b (graph depicted in Figure 19a), which has the composition [(AnO_{2})(TO_{4})_{2}(H_{2}O)] and is observed in a large number of actinyl oxysalts, including uranyl selenates.7 It is of interest that this unit can also be produced by using the CA described above.

The results of computer modeling may shed some light into the process of self-assembly in uranyl selenate systems. In the case of ordered nuclei, either chain or layer topology is formed, whereas, in the case of disordered nuclei or growth faults (in computer language, errors of the CA function), a disordered topology is formed that represents a superposition of chain and layer topologies.

In general, rather simple examples of CA modeling of self-assembly processes in actinyl-based systems indicate that the theory of CA and finite automata may provide a coherent framework for the description of the dynamics of chemical systems, provided that an appropriate abstract model is constructed. Moreover, it seems feasible that, in the deep sense, growth of periodic structures is similar to the growth of cellular structures during the development of a CA, and crystal growth can therefore be viewed as a computation.

The parallels between chemical structures and such formal objects as CA can also be extended further. For instance, the CA theory contains the concept of a “Garden-of-Eden” configuration, which is a configuration that can never appear as a result of the specific CA work. In the case of a 1D automaton, this is the row of cells, which has no predecessors. There could be an infinite number of “Garden-of-Eden” configurations for a particular CA, and there are specific elementary configurations that any “Garden-of-Eden” contains. These configurations are called orphans. By inspection of the CA rules shown in Figure 17, it is easy to conclude that, for the given transition functions, sequences [11] and [22] are orphans, that is, they may not form as the result of the CA work under any circumstances. Translating these sequences into the language of chemistry, this could imply that, in our system, no uranyl dimers or diselenate groups Se_{2}O_{7} may form. Indeed, formation of uranyl dimers is generally possible (with bridging hydroxy of fluoride ions), but not under acidic conditions employed in our particular experiment. Formation of diselenate groups is also rather impossible, taking into account their instability relative to the monoselenate groups.

Application of CA allows the prediction of possible topologies that may form in a specific system. It may also provide a computational basis for studying complexity and dynamics of topologically and chemically similar structures forming under similar physico-chemical conditions.