Correlations between chemical and structural complexities of minerals were analysed using a total of 4962 datasets on the chemical compositions and 3989 datasets on the crystal structures of minerals. The amounts of structural and chemical Shannon information per atom and per unit cell or formula unit were calculated using the approach proposed by Krivovichev with no H-correction for the minerals with unknown H positions. Statistical analysis shows that there are strong and positive correlations (R2 > 0.95) between the chemical and structural complexities and the number of different chemical elements in a mineral. Analysis of relations between chemical and structural complexities provides a strong evidence that there is an overall trend of increasing structural complexity with the increasing chemical complexity. Following Hazen, four groups of minerals were considered that represent four eras of mineral evolution: “ur-minerals”, minerals from chondritic meteorites, Hadean minerals, and minerals of the post-Hadean era. The analysis of mean chemical and structural complexities for the four groups demonstrate that both are gradually increasing in the course of mineral evolution. The increasing complexity follows an overall passive trend: more complex minerals form with the passage of geological time, yet the simpler ones are not replaced. The observed correlations between the chemical and structural complexities understood in terms of Shannon information suggest that, at a first approximation, chemical differentiation is a major force driving the increase of complexity of minerals in the course of geological time. New levels of complexity and diversification observed in mineral evolution are achieved through the chemical differentiation, which favours local concentrations of particular rare elements and creation of new geochemical environments.

mineral evolution, structural complexity, chemical complexity, Shannon information, chemical differentiation

The idea that complexity increases in evolution of the Universe has been widely discussed in different contexts, including biological evolution and the evolution of matter in general (e.g., Chaisson, 2001; McShea & Brandon, 2010; Lineweaver et al., 2013). The concept of mineral evolution first proposed in Russian mineralogical literature (e.g., Zhabin, 1981; Yushkin, 1982) was recently developed from a general viewpoint (Hazen et al., 2008, 2011; Hazen, 2013, 2014) and applied to Hg minerals (Hazen et al., 2012), clay minerals (Hazen et al., 2013a), carbon minerals (Hazen et al., 2013b), Be minerals (Grew & Hazen, 2014) and B minerals (Grew et al., 2016), leading to the formulation of mineral ecology as a field of mineralogy dealing with the study of mineral distributions in space and time (Hazen et al., 2015a and b; Hystad et al., 2015). The consideration of mineral evolution as a new avenue of research in mineralogy raises several important questions concerning complexity of minerals and its behaviour in the course of the evolution of the Universe.

There is no widely accepted definition of complexity of biological systems, although different measures of complexity of organisms have been proposed, including the number of cells and the number of cell-types, the gene number, the number of different interactions of parts of an organism (McShea, 1996; Carroll, 2001). However, no universal quantitative complexity measure exists that can be applied to all biological species and their systems, even though the problem of a general trend of complexity in biological evolution has been discussed since the XIXth century (Lamarck, 1809). There is little doubt that minerals are much less complex than biological organisms, but the problem of complexity has not been considered in mineralogy as it was in biology. As minerals are crystalline chemical compounds of natural origin, it would be appropriate to apply to them complexity measures proposed for crystals in general. However, even in crystallography, until recently there were no universal complexity measures that could be applied to all crystalline compounds, though many proposals have been put forward by various crystallographers, starting with Pauling (1929). Krivovichev (2012) proposed to estimate complexity of crystals on the basis of Shannon information content of a reduced unit cell, taking into account the number of atoms, the number of independent sites and the ratio of their multiplicities. Krivovichev (2016) demonstrated that the Shannon information content per atom correlates to a configurational entropy of crystals. Information-based complexity measures can be applied to study general trends in structural complexity versus chemical composition, its behaviour in the course of crystallization (Krivovichev, 2013; Cempírek et al., 2016; Krivovichev et al., 2016, 2017) and micro- and macroscale mineral evolution (Krivovichev, 2013; Grew et al., 2016). It is important to note that Shannon information measures were first applied to the investigations of complexity of geochemical systems in general (i.e., including minerals) by Petrov (1970) and Bulkin (1972a and b) (see also Yushkin, 1977).

The aim of the present study is to investigate the relations between chemical and structural complexity of minerals, both understood as amounts of Shannon information, and to apply these measures to mineral evolution, using the lists provided by Hazen et al. (2008) and Hazen (2013).

For the investigation of chemical and structural complexity of minerals, a total of 4962 datasets on chemical compositions and 3989 datasets on crystal structures of minerals was considered. The amounts of structural Shannon information per atom (strIG) and per unit cell (strIG,total) were calculated using the approach developed by Krivovichev (2012, 2013, 2014, 2015, 2016) according to the following equations:
strIG=−∑i=1kpilog2pi (bits/atom),
(1)
strIG,total=−vIG=−v∑i=1kpilog2pi (bits/cell),
(2)
where k is the number of different crystallographic orbits (independent crystallographic Wyckoff sites) in the structure and pi is the random choice probability for an atom from the ith crystallographic orbit, that is:
pi=mi/v,
(3)
where mi is a multiplicity of a crystallographic orbit (i.e., the number of atoms of a specific Wyckoff site in the reduced unit cell), and v is the total number of atoms in the reduced unit cell.

The calculation of the information-based structural complexity parameters using equations (1)–(3) implies that positions of all atoms in the crystal structures are known. However, it is not always the case, as in crystal structures of hydrated oxysalt minerals very frequently positions of H atoms could not be determined due to the experimental problems associated with crystal quality and the low scattering power of H for X-rays, most frequently used in crystal-structure analysis. For the crystal structures of hydrated minerals with no H positions determined, Pankova et al. (2018) proposed to use the procedure of H-correction by introducing “dummy” H atoms into structural datasets. According to our general estimates (see also Pankova et al., 2018), the H-correction for the crystal structures with unknown H positions may result in a significant increase of their structural information measures, depending upon their H content. However, no H-correction was applied to the datasets under consideration, and therefore the results we obtained can be considered only as a first approximation.

By analogy with structural complexity, chemical complexity was evaluated by the amount of chemical information per atom (chemIG) and per formula unit, f.u. (chemIG,total), as suggested by Siidra et al. (2014). Following this approach, for the idealized chemical formula of a mineral or inorganic compound, Ec1(1)Ec2(2)…Eck(k)⁠, where E(i) is an ith chemical element in the formula and ci is its integer coefficient, the chemical information can be calculated as follows:
chemIG=−∑i=1kpilog2pi (bits/atom),
(4)
chemIG,total=−eIG=−v∑i=1kpilog2pi (bits/f.u.),
(5)
where k is the number of different elements in the formula and pi is the random choice probability for an atom of the ith element, that is:
pi=ci/e,
(6)
where e is the total number of atoms in the chemical formula:
e=∑i=1kci.
(7)

The ideal chemical formulas of minerals used for the calculations are those approved by the International Mineralogical Association (IMA) and contained in the continuously updated lists published by Pasero (2016) at the website of Commission on New Minerals, Nomenclature and Classification IMA (CNMNC IMA). Only essential chemical elements were taken into account, without consideration of isomorphic substitutions (see also Krivovichev et al., 2017).

Complexity and mineral systems

Information-based chemical and structural complexity parameters for minerals calculated following equations (1)–(7) have been separated into groups according to the number N of different chemical elements present in the chemical formulae (i.e., into different minerals systems) (Krivovichev & Charykova, 2013, 2015, 2016; Krivovichev et al., 2017). The mean chemical and structural complexities and associated statistical parameters are given in Table 1. Figure 1 shows distribution of Shannon information among different types of mineral systems. The dependencies of different IG values from N were best approximated by the use of the following functions (corresponding curves are plotted as dash-and-dot lines in Fig. 1):
chemIG=−2.9+2.2×[1−exp(−N/2.8)]+2.8×[1−exp(−N/0.7)] (R2=0.999),
(8)
strIG=−6595+5.85×[1−exp(−N/4.2)]+6595×[1−exp(−N/0.1)] (R2=0.988),
(9)
chemIG,total=2.2+10.8×exp[(N−1.2)/2.5] (R2=0.959),
(10)
strIG,total=−605+26.7×exp[(N+35)/11.6] (R2=0.974).
(11)

The observed relations indicate that there is a strong and positive correlation between the chemical and structural complexities and the number of essential chemical elements in a mineral.

Correlations between chemical and structural complexity

Figure 2 shows the relations between chemical and structural complexities per atom (Fig. 2a) and per unit cell or formula (Fig. 2b) calculated for different groups of minerals with the same number of chemical elements in a chemical formula. The best fitting for the chemIGvs.strIG and chemIG,totalvs.strIG,total relations was obtained by means of the exponential and allometric functions, respectively:
strIG=−0.97+2.3×[exp(chemIG−0.646)/1.63] (R2=0.988),
(12)
strIG,total=30.2×[chemIG,total]0.544 (R2=0.957).
(13)

The observed relationships provide strong evidence that there is an overall trend of increasing structural complexity with the increasing chemical complexity of minerals.

Evolution of complexity in geological time

According to Hazen et al. (2008, 2013a and b), mineral evolution can be subdivided into four partially overlapping stages, each of which saw the expansion of mineralogical diversity and/or variation in relative mineral abundances. The starting point of mineral evolution is that of the “ur-minerals”, the twelve earliest mineral phases to appear in the pre-solar nebulae (1). Chondritic meteorites incorporate about 60 primary mineral phases, which constitute the second phase (2). For the Hadean Eon, Hazen (2013) estimated 425 mineral species (3), whereas post-Hadean processes were responsible for the appearance of more than 5000 mineral species known today (4). The mean chemical and structural complexities for the four groups of minerals mentioned above provided in Table 2 and depicted in diagrams in Fig. 3 demonstrate that both chemical and structural complexity are gradually increasing in the course of mineral evolution.

The obtained statistical correlations between information-based chemical and structural complexities of minerals and the general trend of increasing complexity in the course of mineral evolution show that minerals become more and more complex with the flow of geological time. This agrees well with the general trend of chemical differentiation of matter as indicated by Yushkin (1982) and Hazen et al. (2008). In fact, the observed relations between chemical and structural complexity suggest that, at a first approximation, chemical differentiation is a major force driving the increase of chemical and structural complexity in the course of geological time. Indeed, in a companion paper, Krivovichev et al. (2018) showed that the average number N of chemical elements in a mineral increases for the different eras of mineral evolution, in agreement with the results of information-based considerations reported herein. It should, however, be taken into account that the list of minerals of the present era contains all minerals formed in the post-Hadean time. On the example of boron minerals, Grew et al. (2016) demonstrated that the variation of complexity in minerals follows a passive trend: more complex minerals arise with the passage of time, yet the simple ones are not replaced. The current work shows that this conclusion is valid for all minerals as well, with the global trend being passive due to the general increase in variance. New levels of complexity and diversification observed in mineral evolution are achieved through the chemical differentiation, which favours local concentrations of particular rare elements and creation of new geochemical environments.

We thank Jakub Plašil, J. Hao and Associate Editor Edward Grew for useful comments on the manuscript. This study was supported for S.V.K. and V.G.K. by the Russian Foundation for Basic Research (Grant No. 16-05-00293) and for S.V.K. by the President of Russian Federation grant for leading scientific schools (NSh-10005.2016.5). This study was supported for R.M.H. by the Deep Carbon Observatory, the Keck Foundation, and the Carnegie Institution for Science.

© 2018 E. Schweizerbart'sche Verlagsbuchhandlung, D-70176 Stuttgart
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