Introduction

Most natural mineral groups exist over a range of chemical compositions and solid solutions play a major role in determining mineral stability and chemical behaviour (Putnis 1992). Non-ideality of such solutions and the resulting tendency to unmix often depends on the atomic ordering. For example, the enthalpy of mixing of disordered dolomite is positive, but negative for ordered dolomite. Similar behaviour is found in the diopside–jadeite solid solution—it is non-ideal, yet an intermediate ordered phase forms on cooling. Non-ideality is fundamental also to the interpretation of any processes involving partitioning between phases. Solid solutions continue to pose considerable challenges for computation, as does non-ideality in particular. In this paper, we report the use of genetic algorithms to predict the existence of an ordered phase ab initio and discuss why this phase is more stable than the disordered form.

We have developed a number of computational methods for the study of solid solutions and grossly non-stoichiometric compounds. Any technique must be able to sample many arrangements of the atoms, allowing for the exchange of ions located at different positions. It is also crucial to allow for the relaxation of the local environment of each ion, i.e. local structural movements which can reduce considerably the energy associated with ion exchange. Local effects due to ion association or clustering must not be averaged out. We have previously used basin-sampling approaches, such as configurational Boltzmann averaging (Purton et al. 1998a, b; Allan et al. 2001; Todorov et al. 2004; Mohn et al. 2005) and also exchange Monte Carlo methods (Purton et al. 1998a, b; Todorov et al. 2004; Purton et al. 2007, 2013).

This paper is primarily concerned with basin-sampling and exploring the energy landscape of a strongly non-ideal system, MgCO3–CaCO3. Such marked non-ideality poses a number of difficult problems. Unlike the energy landscapes of the binary oxide mixtures considered in our previous work (Purton et al. 1998a, b), where configurations with distinct cation arrangements were generated randomly, the energy landscape for MgCO3–CaCO3 is very different. It contains a small number of deep minima, some of which correspond to the formation of ordered dolomite at 50% Mg, 50% Ca. Generating cation arrangements at random is almost bound to fail to discover such deep minima with very small weights (degeneracies). Since only a few minima are thermally accessible, simply generating a random selection of starting configurations will not probe sufficiently the low-energy parts of the landscape, and so any averaged thermodynamic property is likely to be highly inaccurate. Monte Carlo techniques are also likely to fail to locate such minima due to “basin trapping”; the large mismatch in ionic radii [all ionic radii are for sixfold coordination and taken from Shannon (1976)], between Ca2+ (1.00 Å) and Mg2+ (0.72 Å) is such that changes in local environments associated with any exchange are large, and therefore, the acceptance rate of exchanges is very low even at high temperatures.

The development of tools for locating low-lying minima for such situations is of considerable importance. Ordering patterns may be extremely complex and somewhat counter-intuitive, such as in the garnet solid solution, where third-, fourth- and even fifth-nearest neighbour cation orderings are energetically more important than first and second neighbours (van Westrenen et al. 2003; Sluiter et al. 2004; Freeman et al. 2006; Lavrentiev et al. 2006; Lyakhov et al. 2010, 2013; Oganov et al. 2009, 2010, 2011; Zhu et al. 2012). Before tackling systems where ordering is not fully established, in this paper, we take a well understood system and show how a genetic algorithm (GA) together with energy minimisation can be used to find low-energy minima in the energy landscape (Holland 1975; Goldberg 1989). We then examine these deep minima in some detail.

Most applications of GA carried out at the atomic level have involved application to and optimisation of a range of nanoclusters (Deaven and Ho 1995; Johnston 2003; Chen et al. 2007; Ferrando et al. 2008). Nevertheless, there are an increasing number of successful applications to other materials science problems, such as the prediction of crystal structures (Woodley et al. 1999; Woodley and Catlow 2008; Oganov et al. 2006; Woodley 2009), and modelling reconstruction and construction of surfaces (Chuang et al. 2004) and grain boundaries (Zhang et al. 2009; Chua et al. 2010) as well as prediction of ordering in disordered alloys (Johnston 2003; Smith 1992; Mohn and Kob 2009, 2015, Mohn et al. 2011), and searching for alloys with desired physical properties (Dudiy and Zunger 2006). Only a very few examples have demonstrated how GAs can be used to understand ordering patterns or local structure in grossly disordered ceramics. Mohn and Stølen (2005) used a GA to map low energy minima for a binary oxide solid solution but their simulation box was restricted to just 64 ions. Often much larger cells are needed to model gross non-stoichiometry in ceramics (Taylor et al. 1997a, b; Todorov et al. 2004; Bakken et al. 2003). Here we extend the earlier study of Mohn and Stølen (2005) to the more complex carbonate system.

The CaCO3–MgCO3 solid solution itself is one of the most well-examined solid solutions in mineralogy. The essential features are two asymmetric miscibility gaps separated by a narrow stability field for the dolomite (50:50) composition. Calorimetric studies (Navrotsky and Capobianco 1987; Chai et al. 1995; Chai and Navrotsky 1996; Navrotsky et al. 1999) yield a negative enthalpy of formation for ordered dolomite relative to the end-members MgCO3 and CaCO3. In contrast, the enthalpy of formation of a disordered solid solution with the same 50:50 composition is positive (Burton and Kikuchi 1984; Burton 1987). Theoretical work has been substantial (Burton and Kikuchi 1984; Burton 1987; Davidson 1994; Burton and Van de Walle 2003; Purton et al. 2006) and is consistent with experiment. For example, in Refs. (Vinograd et al. 2007, 2009) the fully optimised energies of a large set of randomly varied structures were used to parameterise a cluster expansion of 12 pair-wise effective interactions to obtain the activity–composition relations and a phase diagram in good agreement with experiment.

In the next section, we discuss the theoretical methods and the genetic algorithms. Results follow. We then examine the energy landscape in detail, concentrating on the link between the enthalpies of formation and structures of individual configurations. Of particular interest is the local environment of individual Ca2+ and Mg2+ ions, and how the carbonate ions adjust to accommodate cation neighbours with very different sizes, leading to the observed stability of the ordered dolomite structure. Some brief remarks about extracting thermodynamic properties for such systems from the energy landscape follow, and we also consider the consequences of the form of the energy landscape for the kinetics of transitions between different orderings and hence the difficulty in preparation of ordered dolomite (the so-called “dolomite” problem or paradox).

Theoretical methods

Energy minimisation using interatomic potentials

For the structural optimisations within the GA algorithm and also for the molecular calculations we have used the set of interatomic shell-model potentials and atomic charges as Fisler et al. (2000). Energy minimisations involved full structural optimisation (Taylor et al. 1997a, b, 1998) in the static limit (thus ignoring vibrational contributions) of all lattice parameters and atomic positions with no symmetry constraints, and were carried out with the GULP code (Gale 1997; Gale and Rohl 2003; Gale 2005). Calculated lattice parameters for MgCO3, CaCO3 and ordered dolomite are in good agreement with experimental values (Table 1).

Table 1 Calculated and experimental lattice parameters for magnesite, calcite and dolomite
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Since preliminary runs indicated that inclusion of the shell model (Dick and Overhauser 1958) in this potential set only affected absolute energies (and then only slightly) but not the relative energies of different arrangements, shells were omitted from the GA runs described below. Shells were included for the molecular mechanics studies described in the final sections of this paper. In addition, since we found that a few high-energy configurations minimisations failed because the carbonate ion became non-planar, we increased the four body O–C–O–O torsion constant to 0.7510 eV; again this has no effect on any relative energies.

Ab initio optimisations

For selected very low and very high energy arrangements, we also carried out structural optimisations using the ab initio all-electron periodic Hartree–Fock method, as implemented in the CRYSTAL09 code (Dovesi et al. 2005; Dovesi et al. 2009). Previously published basis sets were used (Catti et al. 1991, 1993; McCarthy and Harrison 1994; Towler et al. 1994) with a Monkhorst–Pack k-point grid of 8 × 8 × 8. Once again, no symmetry constraints were applied and calculated lattice parameters are in good agreement with experiment (Table 1).

Genetic algorithms

The GA used here consists of five steps:

  1. 1.

    Setting up an initial population (of arrangements/configurations): An initial population is selected at random. Typical population sizes used in GA studies range from a few 100 to several 1000 members; here the initial population is 1000. Each member of the initial population is generated by distributing 48 Ca and 48 Mg ions at random over the cation positions in a hexagonal supercell containing in total 480 atoms and six cation layers. The energy of each configuration is calculated by a static energy minimisation with full structural relaxation (optimisation) of all basis atom positions and unit cell parameters. These structural optimisations thus permit distortion of the cell shape from hexagonal, depending on the symmetry of the particular cation configuration. We shall see later that while it is computer-time intensive, this full optimisation is crucial.

  2. 2.

    Selection: Parents with high fitness are preferentially selected using an appropriate scheme. We used a Boltzmann selection where two parents are chosen with a Boltzmann probability, given by exp(− ΔE/kT) where ΔE is the energy difference of a configuration relative to the lowest energy so far found and, after a number of test calculations, the value of T was set to 7000 K. The temperature controls the amount of selection, i.e. the probability of choosing two parents.

  3. 3.

    Mating: We used a real-space crossover where slices and small clusters from the Parent 1 structure are randomly selected and combined with the complementary structure of Parent 2, maintaining the correct composition. We also used a uniform crossover where a random set of cations are selected from Parent 1 and the complementary set of cations is selected from Parent 2, with the constraint that the child has the correct composition. Far less bonds are broken when a real-space crossover is applied compared to that of a binary uniform crossover and the child inherits more local structural information from its parents. On the contrary, a uniform crossover ensures more diversity in the population since no such structural constraints are imposed on the crossover. Results below are reported using a uniform crossover but we compare with calculations carried out using a real-space crossover.

  4. 4.

    A full structural optimisation of the child structure is completed, as described above, and the child structure is then added to the population if it has a lower energy than the worst (highest-energy) member in the population, which is itself removed. An important modification that avoids the slow convergence associated with “conventional” GA is the incorporation of the symmetry of the underlying lattice within the GA operators using a randomly chosen symmetry crossover operation, i.e. we simply replace the child with a random structure which is symmetrically equivalent (Mohn and Kob 2009). In this work, only translational symmetry operations along the c-axis (i.e. the direction in which the cation layers are stacked) were used and these create sufficient diversity to locate the global minima configuration. A periodic translation along the c axis, T(n/6), where n is a randomly chosen integer (0 ≤ n < 6), generates a new symmetrically equivalent child, e.g.

$$T({\text{2}}/{\text{6}}){\text{Ca}}{{\text{t}}_{{\text{lay1}}}}{\text{Ca}}{{\text{t}}_{{\text{lay2}}}}{\text{Ca}}{{\text{t}}_{{\text{lay3}}}}{\text{Ca}}{{\text{t}}_{{\text{lay4}}}}{\text{Ca}}{{\text{t}}_{{\text{lay5}}}}{\text{Ca}}{{\text{t}}_{{\text{lay6}}}} \to {\text{Ca}}{{\text{t}}_{{\text{lay5}}}}{\text{Ca}}{{\text{t}}_{{\text{lay6}}}}{\text{Ca}}{{\text{t}}_{{\text{lay1}}}}{\text{Ca}}{{\text{t}}_{{\text{lay2}}}}{\text{Ca}}{{\text{t}}_{{\text{lay3}}}}{\text{Ca}}{{\text{t}}_{{\text{lay4}}}}$$

which has the same energy as the child before the symmetry operations were applied. Replacing the child with the symmetry operated child drastically increases the diversity of the population since different symmetrically equivalent regions in the energy landscape are explored simultaneously.

  1. 5.

    Mutation: This involves the exchange of a pair of different cations, chosen at random, in the child. In this work, the mutation probability is 0.1.

Steps 2–5 are repeated until there are no further changes in the lowest energy arrangements, i.e. we carried out the GA simulations for an additional 5N steps, where N was the number of steps required to find the ordered dolomite arrangement.

It is straightforward to adapt this procedure to search for high-energy rather than low-energy arrangements, provided the method used for the energy calculation is accurate for the different interatomic distances often sampled in such arrangements. In this case, step 2 is modified by replacing ∆E with − ∆E so that the members in each population with high energies are selected to mate preferentially, and a child is only added to the population if it has higher energy than the lowest-energy member.

Results

GA

Figure 1 shows the evolution of the lowest-energy member of the population as a function of the number of generations, N. The three dotted lines show the progress of typical calculations each starting from a different initial population of 1000 randomly-generated structures. The red curve is the average lowest energy after N generations of the results obtained from 100 such initial populations. The crystal structure is shown at three different stages of one of the calculations, showing the emergence of intact layers along the c-axis containing single cation types and finally the emergence of ordered dolomite as the lowest-energy structure. This global minimum structure contains alternating layers of Mg and Ca ions along the c-axis, and each individual cation layer contains only one type of ion. 94% of all runs reached the global minimum within 20,000 generations and 80% of all runs within 10,000 generations.

Fig. 1
figure 1

Progression plots (lowest unit cell energy vs. number of generations N) of the genetic algorithm for Mg0.5Ca0.5CO3 using a uniform crossover. The energy value is relative to the global energy minimum. The unit cell contained 480 atoms (96 cations). The dotted lines show the progress of the calculations starting from three different initial populations of 1000 randomly-generated structures. The red curve shows the energy obtained from averaging the results at each generation for 100 such starting populations. In the accompanying crystal structures, the a-axis points out of the plane of the paper, Mg ions are orange, Ca blue, C grey and oxygen red. The black arrows show the step and the GA run to which these structures relate

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The speed at which the GA can find orderings very similar in structure and energy to ordered dolomite with layers of one only cation type along the c-axis is striking. Searching for low-energy structures is generally a very challenging problem for unit cells with, as here, at least several 100 atoms. There are about 6.4 × 1027 arrangements of the 48 Ca and 48 Mg cations in the supercell used in this work. The speed and reliability of finding the global minimum structure demonstrates the effectiveness of the GA methods for searching for low-energy structures in systems of this type.

The benefit of including the symmetry crossover operations is evident. The global minimum structure was only found in 3% of all runs after 20,000 steps, in contrast to 94% when the symmetry operations were used. When a real-space crossover is applied instead of a uniform crossover, the success rate was slightly lower because the diversity in the population is larger when an uniform crossover is used. However, when using a real-space crossover, the convergence to the global minima was slightly faster since fewer bonds are broken after crossover and the child, therefore, inherits more structural information from its parents (the crossover is more efficient). It is worth bearing in mind that when full relaxations of the structures generated in each generation are not included, the GA algorithm fails to find the ordered dolomite structure. This emphasises the necessity of including full relaxations of all the atom positions and unit cell parameters, despite the considerable extra expense in computer time. The reasons at the atomic level for this will become clearer in the next section.

The GA algorithm is readily reversed to generate states which are high, rather than low, in energy. There is no one global maximum, but rather a large number of structures close in energy in which each layer along the c-axis contains both Mg and Ca ions.

Structure analysis

We have probed the important features of low- and high-energy structures through molecular mechanics calculations, carried out with full structural optimisations and interatomic potentials as described earlier. For comparison, geometric optimisations and total energies were also calculated using the periodic ab initio Hartree–Fock method. We wish to examine the structural changes that accompany the differences in energy between different cation orderings and ultimately give rise to the strong non-ideality of the CaCO3–MgCO3 solid solution. Four different orderings of Mg0.5Ca0.5CO3 have been investigated (Fig. 2), selected as those of particular interest based on the GA results.

Fig. 2
figure 2

a Ordered dolomite with stacking sequence Mglay1Calay2Mglay3Calay4Mglay5Calay6 along the c axis (Ordering 1), representing the final structure (global minimum) found from the GA, b Ordering 2 contains intact layers of Ca and Mg with Mglay1Calay2Calay3Mglay4Calay5Mglay6 stacking-sequence along the c-axis, c Ordering 3 with stacking sequence Calay1Calay2Mglay3Mg lay4Mg lay5Calay6 and d Ordering 4 in which all layers contain a mixture of both Ca and Mg ions. Corresponding formation energies are listed in Table 2. The blue atoms are Ca, the orange atoms Mg, the brown atoms carbon and the red atoms oxygen. The black lines show the unit cell boundaries. The c-axis direction is the vertical (stacking) direction

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Table 2 Optimised volumes and formation energies, in the static limit, calculated using molecular mechanics and (in parentheses) periodic Hartree–Fock theory, of some different cation orderings in Mg0.5Ca0.5CO3
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For these studies we have used a unit cell containing 12 cations (six cation layers with two cations per layer). Energies (in the static limit) are reported as the energy of formation of the mixed system relative to that of the pure end-members:

$${E_{{\text{formation}}}}=E({\text{M}}{{\text{g}}_{0.{\text{5}}}}{\text{C}}{{\text{a}}_{0.{\text{5}}}}{\text{C}}{{\text{O}}_{\text{3}}}) - 1/2E({\text{MgC}}{{\text{O}}_{\text{3}}}) - 1/2E({\text{CaC}}{{\text{O}}_{\text{3}}})$$

The Hartree–Fock formation energies are all slightly less negative (or more positive) than those calculated with the interatomic potentials but both follow the same trends. For comparison, the formation energy reported by Chan and Zunger for ordered dolomite (Chan and Zunger 2009) using density functional theory in the generalised gradient approximation is − 39 meV per cation pair, while the experimentally determined value is approximately − 100 meV per cation pair (Navrotsky 1987). Our calculated values of − 50 meV (molecular mechanics) and − 82 meV (Hartree–Fock) are in closer agreement with experiment.

Discussion

The results from the molecular mechanics calculations show that, in the structures in which each layer contains cations of only a single element (Orderings 1, 2 and 3), the carbonate groups lie flat between the cation layers (Fig. 2a–c), as they do in the end-members CaCO3 and MgCO3.

In CaCO3 and MgCO3, the carbonate group is oriented to give identical Ca–O/Mg–O distances to cations in the layers both above and below the anion layer (Fig. 3a, b). In ordered dolomite, clearly the cation–O distances should not all be identical, but Ca–O distances should be significantly longer than Mg–O. If the same orientation of the carbonate groups were maintained in ordered dolomite as in CaCO3 and MgCO3, the only degree of freedom allowing optimisation of the Mg–O and Ca–O distances would be the interlayer spacing. However, an additional degree of freedom is obtained through rotation of the carbonate group. Thus, when there are alternating layers of Ca and Mg ions along the c-axis (as in ordered dolomite, Ordering 1, Fig. 2a), the carbonate group rotates within the ab-plane so that the oxygen atoms are closer to the Mg ions in one neighbouring cation layer and further from the calcium ions in the other neighbouring cation layer (Fig. 3c). Hence, the rotation of the carbonate groups is crucial since it allows both the Mg–O and Ca–O distances to be optimised without introducing significant strain into the C–O bonds. The rotation of the carbonate group in ordered dolomite relative to that in CaCO3 (and MgCO3) is shown in Fig. 3d.

Fig. 3
figure 3

Two cation layers (ab-plane) and one anion layer of a CaCO3, b MgCO3, c ordered dolomite (Ordering 1, shown in Fig. 2a), viewed along the c-axis (cations in the layer below the anion layer are labelled “1”; cations above the anion layer are labelled “2”). d Ordered dolomite with the positions of the oxygen atoms in the end members MgCO3 and CaCO3 overlaid (pink atoms show the position of the oxygen atoms in MgCO3 and CaCO3). Note that, in ordered dolomite, the carbonate group rotates clockwise to move the oxygen atoms closer to Mg ions in the layer above and further from Ca ions in the layer below, hence optimising both the Mg–O and Ca–O distances. e Ordering 3 (the calcium layer shown is the middle of the three adjacent layers in Fig. 2c). Blue atoms are Ca, orange atoms are Mg, brown atoms are carbon and red atoms are oxygen. The black lines show the unit cell boundaries. Some important bond lengths are shown; note that the Ca–O and C–O bond lengths in ordered dolomite (c) are closer to those of pure calcite (a) than in Ordering 3 (e), in which these bond lengths are reduced

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Like ordered dolomite, Orderings 2 and 3 also have intact Mg2+ and Ca2+ layers (Fig. 2b, c), but are higher in energy. In Ordering 3, for which Table 2 shows that the formation energy is positive, there are three adjacent layers of cations of the same element (Fig. 2c). The lattice parameters are intermediate between those of CaCO3 and MgCO3, therefore, the CaCO3 layers are compressed in the ab-plane relative to pure calcite (Fig. 3e), while the MgCO3 layers are stretched relative to pure magnesite. The Ca–O distances are thus shorter than those in the end-member CaCO3, while Mg–O distances are longer. Some strain is also found in the C–O bonds within the carbonate groups, with these bonds being significantly compressed in anion layers adjacent to Ca (Fig. 3e) and stretched in anion layers adjacent to Mg.

Except at the interfaces between the Ca and Mg layers, these strains in the bond lengths cannot be relieved by rotations of the carbonate groups, in contrast to ordered dolomite. In these structures with two or more adjacent layers with cations of the same element (e.g. Orderings 2 and 3, Fig. 2b, c), no advantage can be gained through rotation of the carbonate groups. For example, rotation to decrease the Mg–O distances in one direction (e.g. to the layer below) would increase them in the other direction (e.g. to the layer above). Hence, the carbonate groups between two layers of the same cation remain in the same orientation as in pure magnesite and calcite; the bond lengths are compromised because the lattice parameters are in between those of the end-members and there are insufficient degrees of freedom for bond length optimisation.

Thus, rotation of the carbonate groups to optimise bond lengths is only effective when the anion layer has Mg ions in one neighbouring layer and Ca ions in the other, and therefore, the lowest energy ordering for Mg0.5Ca0.5CO3 is that with alternating layers of Ca and Mg ions. Ordering 2 has a formation energy intermediate between those of ordered dolomite (Ordering 1) and Ordering 3 (Table 2); the number of anion layers with Mg ions in one neighbouring layer and calcium ions in the other is also intermediate between Ordering 1 and Ordering 3. This again indicates the energetic favorability of the cation species alternating between layers, so that the carbonate groups can rotate to achieve optimal O–Ca and O–Mg separations.

In Ordering 4, in which all layers contain a mixture of Mg2+ and Ca2+ (Fig. 2d), the carbonate groups are distorted and no longer lie flat between the cation layers. There is considerable variation in C–O bond lengths in this particular ordering and O–C–O bond angles range from 117.2° to 122.5°. In contrast, in the ordered dolomite structure, all C–O bond lengths are the same and all O–C–O bond angles are 120.0° (as they are in the CaCO3 and MgCO3 end-members). Similarly, in the ordered dolomite structure, all Mg–O and Ca–O distances are the same, whereas there is significant variation in these distances for Ordering 4. Several different orderings with layers containing a mixture of Mg2+ and Ca2+ have been investigated in addition to Ordering 4; all have positive energies of formation, with Ordering 4 the maximum. This indicates that layers containing a mixture of cations are unfavourable, consistent with the GA results. Chan and Zunger (2009) also found that a random cation distribution is high in energy and accompanied by carbonate ion distortions and strain. All the same qualitative conclusions and trends are also evident in the results from the Hartree–Fock calculations.

Ordered dolomite has a small but negative enthalpy of mixing (Table 2). The rotation of the carbonate group also allows more volume-efficient stacking of the layers than in either MgCO3 or CaCO3 end-members. The separation along the c-axis of a layer of Mg2+ ions and the neighbouring anion layer is 1.249 Å in MgCO3 and 1.228 Å in ordered dolomite; the separation in the c-axis direction of a layer of Ca2+ ions and the neighbouring anion layer is 1.421 Å in CaCO3 and 1.409 Å in ordered dolomite. Thus, the volume of ordered dolomite is ~ 1% smaller than the average volume of MgCO3 and CaCO3 and at fixed temperature the thermodynamic stability of ordered dolomite with respect to the end-members increases with increasing pressure.

The nature of the low-energy excited states as revealed by both the GA and the molecular mechanics calculations also provides some insight into the difficulty of formation of ordered dolomite (Deelman 1999) If we “funnel” down in energy so we end up in local minima with structures in which each cation layer contains ions of only one element but the Ca2+ and Mg2+ are not alternating, there are substantial kinetic barriers in proceeding further to the global minimum, since this would require interchange of cations between layers, producing intermediate high-energy structures with a mixture of Ca and Mg cations in the layers. The activation energies for such cation interchanges are large.

We end with a few remarks about the consequences of non-ideality for calculation of thermodynamic properties, in particular by configurational averaging (often referred to as basin sampling). In principle, a solid solution can assume any state, i.e. each atom can be at any position and each will have a different probability. However, the only states of practical importance away from the melting point lie at the bottom of K local minima in the energy landscape, so the thermodynamic averaging is carried out over results from a set of optimisations of different cation arrangements within a given supercell. The configurational averaging approach to solid solutions commonly uses the isobaric–isothermal (NPT) ensemble; therefore, for example, the enthalpy of the solid solution H is given by:

$$H=\frac{{\mathop \sum \nolimits_{k}^{K} {H_k}\exp ( - \;\beta {G_k})}}{{\mathop \sum \nolimits_{k}^{K} \exp ( - \;\beta {G_k})}},$$
(1)

where Gk and Hk are the free energy and enthalpy of each local minimum, respectively. All vibrational entropy terms are usually neglected so Gk is replaced by Hk calculated in the static limit.

For other than the smallest supercells it is impractical to sum over all K configurations, and therefore, both summations in Eq. (1) are restricted to K′ configurations chosen at random. In previous work on non-ideal solid solutions showing much smaller deviations from non-ideality than magnesite–calcite, we demonstrated convergence with a manageable value of K′ configurations, chosen at random, and good agreement with Monte Carlo simulation (Todorov et al. 2004 and; Allan et al. 2001). For a 32-atom supercell of composition 50% MgO/50% MnO, convergence of the formation enthalpy of the solid solution to 0.04 kJ mol−1 is typically obtained with only ≈ 150 out of a total of 12,870 configurations (Allan et al. 2001).

This procedure requires adaption for very strong non-ideal systems such as MgCO3–CaCO3. For the so-called “disordered” dolomite, a random selection of configurations can be used, excluding any with negative heats of formation. In any case, in any reasonably sized simulation cell the chance of selection of an ordering with a negative heat of formation is extremely small due to the small weightings of the highly-ordered states which are the only states with such exothermic heats of formation.

For ordered dolomite a modified procedure is needed. A working procedure is to select a random set of configurations and combine this set with the lower energy configurations found by the GA during evolution. The GA is able to sample efficiently a large fraction of low energy minima as it “funnels” down the potential energy landscape but does not of course, by construction, converge to the Boltzmann distribution (except at T = 0 K). Nevertheless, after using symmetry arguments to find the correct weighting of the states found by GA with the use of symmetry arguments, we can represent the low-energy tail of the parent distribution of minima. This “tail” can be glued together with the distribution from a random selection of minima as explained in detail by Mohn and Stølen (2005) to generate a “combined” distribution for the calculation of ensemble averages. Such a “combined” distribution does not sample the high-energy tail of the parent distribution of minima, but in principle the GA can easily be modified to preferentially select for such high-energy configurations. We stress that this procedure requires a careful map of low-energy distributions; including only a few states fully weighted tends to underestimate the partition function in the denominator of Eq. (1) and therefore, overestimates the final result. In previous work (Purton et al. 2006), we did not follow such a procedure and consequently our results for the enthalpy of mixing were overestimated. The modified method gives values for the enthalpy of formation for dolomite very close to those obtained using the cluster variation method by Vinograd et al. (2007).

Conclusions

In this paper, we have shown that a GA incorporating symmetry is a particularly computationally efficient method of establishing non-ideality and any preferential ordering in a solid solution. Combined with configurational averaging (basin sampling), it thus provides a very powerful tool for modelling solid solutions and non-stoichiometry in general. It readily provides information as to low-lying and higher excited states. Extension to high pressures would be straightforward as would extension to Mg:Ca ratios other than 50:50. Here, the generated energy landscape provides atomistic insights into why dolomite forms—the rotation of the carbonate ions between adjacent layers of Ca and Mg relieves strain—and also into the dolomite problem. The presence of a polyatomic ion is thus crucial for the formation of an ordered mixed phase—the same reduction of strain is impossible in the MgO–CaO binary solution despite the similarity of the rock salt and calcite structures. The understanding of local order in substitutionally disordered materials is important in fields as diverse as the development of new materials with improved mechanical or electrical properties and the understanding of fundamental geochemical processes in the deep Earth, and we hope the techniques and the encouraging results presented in this paper will assist in such investigations.