It’s not all about u: the role of volume and entropy in weakly bound crystal structures

Abstract

Crystals play a vital role in everyday life, from optical electronics and memory storage, to pharmaceuticals and flavourings. Many crystalline compounds can exist in a variety of interchangeable forms that can have remarkably different properties. To safely and reliably use crystalline compounds, it is vital to understand the relative stabilities of these forms and the conditions under which they can interconvert.

For a crystal form to exist it must represent a minimum in free energy, 𝐺, and relative phase stability can be rationalised using the Gibbs equation (Δ𝐺=Δ𝑈+𝑃Δ𝑉−𝑇Δ𝑆). Attention is often focused on calculations of relative internal energies (Δ𝑈), for example through ab initio and force field methods, and descriptions of the enthalpy of individual bonds dominate descriptions of crystal packing. However, at high pressure the volume term (𝑃Δ𝑉) becomes increasingly significant with the need to minimise volume driving almost all high-pressure phase transitions. The entropy term (𝑇Δ𝑆) is often neglected from both high pressure and ambient pressure studies. Justified by a claim that its magnitude is too small to ever cause a re-ranking in polymorph stability, in the absence of strong structure directing interactions, this too can become significant.

Chapters 1 and 2 are the introductory chapters of this thesis. Chapter 1 starts by introducing the concepts of polymorphism and the classification of different types of phase transitions. The chapter then describes the origins of the terms of the Gibbs equation and their varying importance under different conditions. Chapter 2 describes the experimental and computational methods used in this thesis and outlines how different terms of the Gibbs equation were determined.

Chapter 3 aims to address the role of volume at pressure by introduction of the CellVol code, for calculation of occupied and void space in crystal structures. The code partitions crystal structures into occupied and unoccupied space based on van der Waals radii to define the limits of inter and intramolecular contacts around atom centres. High pressure crystallographic and spectroscopic studies often show misleading correlations but by studying the contribution of these volume elements individually we have uncovered a highly sensitive method for detecting subtle phase behaviour at high pressure. This method has been used to characterise phase transitions in polymorphs of L-histidine and to identify a crystallographic signature for a transition in naphthalene which has been debated for over 80 years. It is also shown to reveal premonitory behaviour before a reconstructive phase transition. The method was applied to the high-pressure entries in the Cambridge Structural Database (CSD) to reveal similarities in the way molecular compounds use void space to accommodate increased pressure, which explains their narrow range of bulk moduli.

Chapter 4 describes a high-pressure single crystal study on the herbicide glyphosate (N-(phosomethyl)glycine) to 5.17 GPa using a 4:1 mixture of methanol and ethanol as a pressure-transmitting medium. The CellVol code is used to show that two transitions previously suggested to be substantial first order structural reorganisations on the basis of high pressure Raman spectroscopy are both more subtle second order processes. The crystal structure of glyphosate consists of strongly hydrogen-bonded layers, and the first transition corresponds to the onset of a greater compressibility within these layers. The second is an intramolecular feature, corresponding to deformation of the molecular backbone. The previous Raman study and our own crystallographic data show that glyphosate undergoes a more substantial transition above 5.6 GPa, which destroys the integrity of the single crystal. The nature of this transformation is explored further in Chapter 6.

Chapter 5 considers the role of entropy in stabilising weakly bound crystal structures through studies of the C—H⋯F intermolecular bond. Although C—H⋯F interactions are enthalpically weak, they occur much more commonly in crystal structures than would be predicted by chance. In the case of a series of five fluorobenzene derivatives, heat capacity data show that the entropic contribution to free energy at 150 K is of the order −10 to −15 kJ mol⁻¹, which is similar to the stabilisation afforded by N—H⋯N and O—H⋯O hydrogen bonds in ammonia and water, respectively. The origin of entropy in ordered, crystalline solids lies in access to low energy vibrational modes (phonons) which can be observed in inelastic neutron scattering spectra and simulated by periodic density functional theory (DFT) calculations. DFT calculations on the fluorobenzenes, which reproduce experimental entropies and inelastic neutron scattering data, demonstrate that the lowest energy phonons are dominated by motions of the F atoms. The low energies of these vibrations stem both from the mass of the F atom and the highly deformable nature of C—H⋯F interactions, revealed by relatively flat potential curves calculated using symmetry adapted perturbation theory. These data show that the stabilising influence of C—H⋯F interactions can be explained at least as much by the entropic contribution of their deformability as by the enthalpic contribution of the contact energies.

Chapter 6 describes formation of potentially four new high-pressure polymorphs of glyphosate. Although the integrity of a single crystal of glyphosate is lost on slow compression above 5.6 GPa, rapid compression to 5.9 GPa in a 4:1 mixture of methanol and ethanol and to 7.0 GPa in neon as pressure-transmitting media generates two new single-crystal phases. The crystal structure of the first of these phases was determined from in-house single crystal diffraction data, that of the second from synchrotron data obtained on the P02.2 beamline at DESY. The phase at 7.0 GPa, named phase-II, has a volume 1.023 Å3 per molecule higher than the phase obtained on rapid compression in methanol-ethanol at 5.9 GPa, named phase III. Formation of phase II from phase III with increasing pressure is therefore thermodynamically forbidden. The volume difference corresponds to an estimated 𝑃Δ𝑉-stabilisation of phase III relative to phase II of 17.3 kJ mol−1 at 7.0 GPa. In an attempt to recover the destructive transition seen previously and to potentially drive its transition to phase II or III, neutron powder diffraction data on glyphosate-d4 was completed at ISIS neutron and muon source. This revealed a far more complex structural landscape and the formation of two further phases, named IV and V, which did not fit the structural parameters of phases I, II or III. So far, the structures of phases IV and V have not been characterised further and it is the aim of future work to attempt to recover structural parameters for these phases, possibly through crystal structure prediction. Free energy calculations involving internal energy, volume and entropy are used to establish the driving forces of the single crystal transitions. Extrapolation of the parameters of phase I to the pressures both phase II and III were observed reveals that both transitions are driven by volume as the only significant stabilising contribution.