A mathematical model of one-dimensional inclusion compounds: a new approach towards understanding commensurate and incommensurate behaviour

Abstract
Many inclusion compounds are known in which `guest' molecules are spatially confined within crystallographically well-defined cavities in a crystalline `host' solid. In this paper, we focus upon those systems (typified by urea inclusion compounds) in which the host solid contains one-dimensional tunnels (channels) heavily loaded with guest molecules. Crystallographically, it is convenient to consider the inclusion compound to be composed of distinct host and guest substructures. Of particular relevance to this paper are the repeat distances (denoted c$_{\text{h}}$ and c$_{\text{g}}$ respectively) of the host and guest substructures along the channel axis, since the ratio c$_{\text{g}}$/c$_{\text{h}}$ is conventionally used as a basis for dividing such one-dimensional inclusion compounds into two categories: commensurate and incommensurate systems. Classically, this division has been applied by considering a system as commensurate if c$_{\text{g}}$/c$_{\text{h}}$ is rational and as incommensurate if c$_{\text{g}}$/c$_{\text{h}}$ is irrational. However, since c$_{\text{g}}$ and c$_{\text{h}}$ can never be measured with absolute precision, it is more useful from the practical viewpoint to define an inclusion compound as commensurate if and only if c$_{\text{g}}$/c$_{\text{h}}$ is sufficiently close to a rational number with low denominator. In this paper, we construct a mathematical model that allows the structural properties of one-dimensional inclusion compounds to be investigated in detail. In summary, our model considers a single channel of an inclusion compound containing strictly periodic host and guest substructures. Since the commensurate versus incommensurate classification should reflect a division in the `behaviour' of the inclusion compounds within each category, we proceed to develop a comprehensive mathematical understanding of commensurate and incommensurate behaviours within the confines of this model. In particular, the inclusion compound is considered to exhibit incommensurate behaviour if the interaction between the host and guest substructures is insensitive to the position of the guest substructure along the channel axis, and to exhibit commensurate behaviour if this is not so. We translate these ideas into strict mathematical definitions, and then derive and apply various theorems which allow an understanding of how commensurate or incommensurate behaviour depends upon the value of c$_{\text{g}}$/c$_{\text{h}}$. The consequences of this approach are then considered, particularly in comparison to the more traditional dichotomy in which rational and irrational c$_{\text{g}}$/c$_{\text{h}}$ are taken to represent commensurate and incommensurate systems respectively. Indeed, we find that the practical definition of commensurability (based upon commensurate systems having c$_{\text{g}}$/c$_{\text{h}}$ close to a low denominator rational) is a better reflection of the distinction between commensurate and incommensurate behaviours than is the more traditional definition. Having developed this understanding of commensurate and incommensurate behaviours, we develop a methodology which allows the optimal value of c$_{\text{g}}$ to be determined (for fixed c$_{\text{h}}$) from known potential energy functions for a particular one-dimensional inclusion compound. The behaviour of the inclusion compound with this optimal guest periodicity can then be assessed by applying concepts discussed earlier in the paper. Finally, we discuss briefly the consequences of relaxing some of the conditions imposed within our mathematical model.

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