Crystals of large molecules exhibit diverse structures. In other words, the geometric arrangements of the molecules relative to one another are complex. Given a specific molecule, theoretically predicting its crystal structure is a challenge and is the basis of a competition.
One of the reasons the structures are rich and the theoretical problem is so challenging is that there are typically many different interactions between different molecules, including electrostatic, hydrogen bonding, pi-pi,...
Another challenge is to understand the elastic and plastic properties of the crystals.
Some of my UQ colleagues recently published a paper that highlights some of the complexity.
Origins of elasticity in molecular materials
Amy J. Thompson, Bowie S. K. Chong, Elise P. Kenny, Jack D. Evans, Joshua A. Powell, Mark A. Spackman, John C. McMurtrie, Benjamin J. Powell, and Jack K. Clegg
They used calculations based on Density Functional Theory (DFT) to separate the contributions to the elasticity from the different interactions between the molecules. The figure below shows the three dominant interactions in the family of crystals that they consider.
Note the purple vertical bar, which is the value of the coordinate in the equilibrium geometry of the whole crystal. The width of the bar represents variations in both lengths that occur in typical elastic experiments.
What is striking to me is the large difference between the positions of the potential minima for the individual interactions and the minima for the combined interactions.
This is an example of frustration: it is not possible to simultaneously minimise the energy of all the individual pairwise interactions. They are competing with one another.
A toy model illustrates the essential physics. I came up with this model partly motivated by similar physics that occurs in "spin-crossover" materials.
The upper (lower) spring has equilibrium length a (b) and spring constant k (k'). In the harmonic approximation, the total elastic energy is
The equilibrium separation of the two molecules is given by
which is intermediate between a + 2R and b. This illustrates the elastic frustration. Neither of the springs (bonds) is at its optimum length.
The system is stable provided that k + k' is positive. Thus, it is not necessary that both k and k' be positive. The possibility that one of the k's is negative is relevant to reality. Thompson et al. showed that the individual molecular interaction energies are described by Morse potentials. If one is far enough from the minimum of the potential, the local curvature can be negative.
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