A quantitative analysis of guest periodicity in one-dimensional inclusion compounds

Abstract

We consider those crystalline inclusion compounds that can be modeled as one-dimensional systems, and develop a mathematical model in which the one-dimensional inclusion compound comprises a linear, infinite host channel (with periodic repeat distance ch along the channel axis) containing a regular, periodic arrangement of n guest molecules (with periodic repeat distance cg ). The principal aim of this paper is to develop a method for the quantitative prediction of cg, and a theoretically justified procedure is described for using known potential energy functions for host–guest and guest–guest interactions as a route towards deriving this information. Fundamental to the analysis is the definition of an appropriate energy expression—denoted the ‘‘characteristic energy’’ of an inclusion compound Ê(α,n)—that directly indicates the relative energetic favorability of inclusion compounds with different guest periodicities α(α=cg/ch). Then, via a graphical method—the characteristic energy diagram, which is essentially a means of constructing a bounded region that will contain the points (α,Ê(α,n))—a restricted range containing the optimal value of α can be determined. A procedure is provided for constructing the characteristic energy diagram using computed potential energy functions for ‘‘real’’ inclusion compounds, and the method for interpretation of the diagram is discussed in full. Furthermore, our method of analysis not only permits an assessment of the optimum guest periodicity for the inclusion compound, but also addresses, from energetic considerations, the question of whether ‘‘lock-in’’ of the host and guest substructures will occur. The applicability of our method of analysis is illustrated using potential energy functions computed for the hexadecane/urea inclusion compound. The characteristic energy diagram is constructed for this system and used to deduce that the optimal guest periodicity is in the range cg=22.6 ű0.1 Å, in excellent agreement with recent experimental results. At this optimal periodicity, the interaction between adjacent guest molecules is repulsive; any naive approach to predict cg by finding the minimum of the guest–guest interaction energy curve is blatantly incorrect.