Open Access Article

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Evan
Kiely
^{a},
Reabetswe
Zwane
^{bc},
Robert
Fox
^{bc},
Anthony M.
Reilly
^{bc} and
Sarah
Guerin
*^{ac}
^{a}Department of Physics, Bernal Institute, University of Limerick, V94 T9PX, Ireland. E-mail: sarah.guerin@ul.ie
^{b}School of Chemical Sciences, Dublin City University (DCU), Glasnevin, D09 C7F8 Dublin, Ireland
^{c}SSPC, Science Foundation Ireland Research Centre for Pharmaceuticals, University of Limerick, V94 T9PX, Ireland

Received
4th April 2021
, Accepted 22nd May 2021

First published on 2nd July 2021

The mechanical properties of crystalline materials are crucial knowledge for their screening, design, and exploitation. Density functional theory (DFT), remains one of the most effective computational tools for quantitatively predicting and rationalising the mechanical response of these materials. DFT predictions have been shown to quantitatively correlate to a number of experimental techniques, such as nanoindentation, high-pressure X-ray crystallography, impedance spectroscopy, and spectroscopic ellipsometry. Not only can bulk mechanical properties be derived from DFT calculations, this computational methodology allows for a full understanding of the elastic anisotropy in complex crystalline systems. Here we introduce the concepts behind DFT, and highlight a number of case studies and methodologies for predicting the elastic constants of materials that span ice, biomolecular crystals, polymer crystals, and metal–organic frameworks (MOFs). Key parameters that should be considered for theorists are discussed, including exchange–correlation functionals and dispersion corrections. The broad range of software packages and post-analysis tools are also brought to the attention of current and future DFT users. It is envisioned that the accuracy of DFT predictions of elastic constants will continue to improve with advances in high-performance computing power, as well as the incorporation of many-body interactions with quasi-harmonic approximations to overcome the negative effects of calculations carried out at absolute zero.

Of all the molecular modelling tools, density functional theory^{28–30} remains one of the most efficient methods of calculating the mechanical properties of a material.^{31–33} While DFT can be used to study almost any material of suitable size, its strength lies in predicting the properties of crystals.^{34–36} Herein we discuss the various DFT methodologies that can be used to calculate the mechanical properties of crystals, as well as exploring other computational chemistry methods that are used today in this endeavour. The limitations of and challenges facing DFT-based predictions mechanical properties are discussed, as well as the exciting future that lies ahead if these challenges are overcome.

Research into the mechanical behaviour of a new class of solid-state materials is central to both the design and optimal performance of potential technological applications. Take for example metal–organic frameworks (MOFs) where theorists and experimentalists can examine the elasticity of these hybrid frameworks by examining their Young's modulus, Poisson's ratio, bulk modulus and shear modulus.^{39} Also crucial are discussions on their hardness, plasticity, yield strength and fracture behaviour. For these materials predicted elastic properties such as compressibility and bulk moduli can be compared to high-pressure X-ray crystallography.^{40} Spectroscopic ellipsometry has also been used to estimate the elastic moduli of MOF nanoparticles and deposited films.^{41} Nanoindentation has emerged as a key technique for quantifying the mechanical properties of crystalline materials (Fig. 1),^{42–44} and recently nanoindentation data has been used to train machine learning algorithms.^{45}

Fig. 1 Schematic of the nanoindentation process and measurement (a) diagram showing the working principles of indenting the sample via loading and unloading, (b) a corresponding load–displacement curve showing the effect of the loading and unloading process. S is the contact stiffness of unloading. Reproduced with permission from Wiley.^{46} |

In the pharmaceutical industry, it is of utmost importance to understand the elastic and mechanical properties of active pharmaceutical ingredients (APIs). For the API, mechanical properties govern physicochemical properties such as solubility, tabletability, stability and the bioavailability of a drug substance. It is especially important as roughly one in two APIs can exist in multiple solid forms, with each form markedly showing different physicochemical and mechanical properties.^{47,48} Unwanted phase transformations during the development stages (the handling, manufacturing, processing and even the storage) of the API and drug can occur.^{49–51} In the case of polymorphism, the solid–solid transformations can also cause formulation problems. Since polymorphs of the same molecular crystal have differences in interaction energies, other polymorphs tend to transform into the polymorph with the least free energy, and therefore the most stable polymorph. The resulting polymorphs can have undesirable properties, such as in the case of ritonavir and rotigotine.^{52,53} Unwanted phase transformations can affect the drug stability during its lifespan and in the handling of the drug, particularly if a shearing stress is applied. The piroxicam–succinic acid co-crystal for example is formed via the application of mechanical stress to the two components, but undergoes decomposition when shearing occurs. These process-induced transformations are difficult to predict and control due to lack of understanding of the mechanochemical process at an atomistic level. DFT calculations can be a crucial tool to understand and quantify polymorph stability, and can be used to study the interactions between APIs, co-formers, and excipients in both amorphous and crystalline environments.

The term DFT comes the fact that the functional used in DFT calculations is the electron density,^{56} which is itself a function of space and time (mathematically, a functional takes a function and gives a resulting scalar value). The Hohenberg–Kohn theorem^{57} tells us that the total ground state energy of a many-electron system is a functional of the density. The total energy of the system is written in terms of a number of individual energy contributions,^{58} each of which are functionals of the charge density:

• ion–electron potential energy

• ion–ion potential energy

• electron–electron energy

• kinetic energy

• exchange–correlation energy.

The most computationally challenging energy contributions are the kinetic energy and the exchange–correlation energy. The kinetic energy is calculated using the Kohn–Sham orbitals.^{59} Generally, these do not correspond to actual electron orbitals – they are orthonormal orbitals. The Kohn–Sham orbitals map the system of interacting electrons on to a system of non-interacting electrons moving in an effective potential. The exchange–correlation energy accounts for the exchange interaction due to repulsion between electrons with parallel spins, and the correlation interaction, which is the correlated motion between electrons of anti-parallel spins due to their mutual coulombic repulsion. In its simplest implementation, exchange–correlation effects are treated via the Perdew, Burke, and Ernzerhof (PBE)^{60} implementation of the generalised gradient approximation (GGA).^{61} GGA builds on what is known as the local density approximation (LDA), by considering both the local electron density and its gradient, as the electron density can vary rapidly over a small region of space.

A vital aspect to the dispersion correction is dampening, with methods that lack an adequate dampening failing to give consistent results for crystal structure and energies.^{65} The dampening function within the dispersion correction determines the range at which the dispersion correction acts^{66} as well as the steepness of the cut off of the dispersion correction.^{67} The dampening function means that the dispersion effects approach 1 at long distances, this meaning that it is purely a dispersion interaction, but at as the distance between the dipoles shortens the dispersion correction gets dampened eventually going to zero.^{68} This means that there is no dispersion effect at shorter distances where the XC functionals perform better. The dampening effects are also intended to reduce double counting effects.^{69} Further studies are needed on which dampening function performs best with each dispersion correction and exchange functional.^{68} However with increasing numbers of atoms in the unit cell the difficulty arises in deciding where dampening should take place, as this needs to be symmetric.^{70}

Dispersion corrections can be calculated in several ways. This can be with a pairwise approach, a three-body approach or a many body approach. The pairwise approach which is chosen in dispersion corrections schemes such as Grimme-D2 (ref. 71 and 72) and Tkatchenko–Scheffler (TS)^{73} summates over the C_{6}R^{−6} potential where R is the atomic distance and the C_{6} is the dispersion coefficient. For the Grimme-D2 method these are multiplied by a global scaling factor.^{71} where the TS scheme calculates the pairwise dispersion energy using the formula:

where θ represents the internal angles formed by R

The many body dispersion method, as used in the many body dispersion scheme (MBD), builds on the pairwise TS approach and addresses the fact that the nature of long-range energy is many-body in nature.^{74} The main drawback of the MBD scheme is the high computational cost^{70} which is due to the fact that within this method it involves having to calculate both pairwise and three-body dispersion energy utilising the formula:

d_{ik} = e_{ij}/c_{kj} |

Using this methodology, we have calculated to a high accuracy versus experiment the elastic constants of amino acid^{90,98–100} and peptide crystals,^{101,102} co-crystals,^{103,104} and biominerals^{105} (Fig. 2). We have also recently calculated the elastic properties of large protein crystals using classical force fields. Elastic constants of the transmembrane protein ba3 cytochrome c oxidase, as well as lysozyme, and aldehyde dehydrogenase were predicted using the classical CHARMM forcefield model for the protein, ions and water with structures calculated using the CP2K modelling software augmented with homemade subroutines to impose crystal symmetry. In any calculation of elastic properties if crystal symmetry is not preserved then stiffness and compliance tensors will be incorrect. To evaluate the mechanical stability of the crystal we can apply the Born–Huang elastic stability criteria for the appropriate crystal system. As an example for the cubic crystal system, it is required that c_{11} − c_{12} > 0, c_{11} + 2c_{12} > 0, c_{44} > 0. If one or more of these criteria is violated, one or more of the elastic tensor eigenvalues is negative and the crystal is mechanically unstable.

Fig. 2 Crystal structures and mechanical properties of different classes of crystal calculated using a PBE-only DFT methodology as published in previous literature^{90,103,105} and summarised in the previous section a. inorganic piezoelectric crystals quartz (SiO_{2}), aluminium nitrate (AlN), and zinc oxide (ZnO) b. the biomineral calcite (CaCO_{3}) c. molecular crystals 4,4′-bipyridine (BPY), N-acetyl-L-alanine (AcA), and their combined BPY/AcA cocrystal. Experimental values are shown in brackets. |

Fig. 2 shows the high quantitative accuracy that can be obtained using the above PBE-only methodology for individual stiffness tensor components and derived Young's modulus values for a small sample of the different classes of crystal that we have studied. As the piezoelectric response is inversely proportional to the elastic stiffness if the material is predicted to be more flexible than it is this will lead to an overestimation of the electromechanical coupling. Full mechanical testing of piezoelectric materials is always recommended as properties such as fracture limit and hardness ultimately determine the specific applications and environments the material can be used in.

Pei & Zeng^{106} computed structural and elastic properties of nine phases of piezoelectric polyvinylidene fluoride (PVDF) crystals using DFT with and without a variety of dispersion corrections. In addition to the four known crystalline forms the mechanical properties of five theoretically predicted crystalline forms of PVDF were also investigated. The DFT/PBE calculations show that the cell parameters of four known crystalline forms are in good agreement with experiment. However, they identified that including empirical van der Waals corrections, specifically the Grimme-D2 method, led to a large error in the calculated unit cell lattice parameters. By comparing conventional PBE (without dispersion corrections) and DFT-D2 calculations the authors could highlight that the PBE method provides a better description of the structural and mechanic properties of PVDF crystals.

Another excellent screening of DFT functionals for elastic constant prediction was carried out by Rego & de Koning in their recent study on hexagonal proton-disordered ice using the Quantum ESPRESSO software package.^{110} They evaluated nine different exchange–correlation functionals, four of which include long-range dispersion interactions through the non-local van der Waals (vdW) approach. While we have observed that dispersion corrections can over-bind crystals and induce polymorphic transitions in small crystals,^{90,106} they are known to play an important role in the condensed phases of water. The authors utilise the well-established energy-strain approach to calculate the elastic constants, in which one exploits the relation between the energy of a crystal and its state of deformation, using the equation

Fig. 3 Relative difference of DFT values with respect to experimental data for the structural parameters of ice.^{110,111} Reproduced with permission from AIP Publishing. |

Fig. 4 Comparison of experimental and calculated bulk moduli for a selected set of systems, with calculated Pearson correlation coefficient r and Spearman correlation coefficient ρ reported.^{19} |

For this study the elastic constants were calculated using a stress–strain methodology. Starting from a relaxed structure for each compound, a set of distorted structures is generated. For each of the applied strains ∈_{ij}, the full stress tensor is obtained from a DFT calculation in which ionic positions are relaxed. One row (or equivalently, column) of the elastic matrix is obtained from a linear fit of the calculated stresses over the range of imposed strains. Repeating this procedure for each of the 6 independent strain components, all elements of the elastic modulus tensor can be calculated. The result is a calculated set of c_{ij} values that can be used to calculate properties such as the bulk modulus K and the shear modulus G as in previously mentioned studies.

Fig. 5 DFT-calculated intermolecular interactions that contribute to the Hirshfeld surfaces of polymorphic form-II (top) and form-I (bottom) of a 1D Ni(II) polymer with iminodiacetic acid (IDA), namely diaquaiminodiacetatonickel(II). Reproduced from ref. 117 with permission from the Royal Society of Chemistry. |

The work of Reilly and Tkatchenko shows that extensive characterization of polymorphic forms can be achieved from the determination of elastic tensor using DFT and other in silico methods. A strong interconnect between molecular modelling and experiment is preferable, as even with structure–property correlations, a polycrystalline lens is needed to fully understand properties like plasticity and tabletability. The tabletability of a polycrystalline material is influenced by factors such as the relative movements of grains and grain size, which are difficult to extrapolate from single crystal DFT. Karki et al. have successfully used the eigenvalues of the compliance tensor to rationalize the differences in mechanical behavior of paracetamol polymorphic forms – which show significant differences in structural features – as well as paracetamol form II cocrystals.^{48} They used the value of the highest compliance eigenvalue to draw conclusions about the compliance of a crystal and the relative strengths of shear planes. From that, they establish that form II shows a higher compliance eigenvalue compared to form I, because it is compliant to shearing, which leads to plastic deformation during tableting. This result is consistent with the layered structure of form II which grants it preferable compaction properties.^{127} Despite these milestones of linking mechanical behavior to structure using DFT, crystal engineering of pharmaceuticals with desired mechanical behavior is still to be achieved. With the maturity of DFT, crystal structure prediction, and ever-increasing computational power, there is continued opportunity for new insights.

The elastic stiffness tensor predicted by some DFT methods is calculated as a 6 × 6 matrix that naturally describes the elastic anisotropy of a crystalline system. However, it is only recently that elastic anisotropy has begun to be considered in experimental measurement of mechanical properties. Mishra and co-workers systematically examined the mechanical properties of dimorphic forms, forms I and II, of a 1:1 caffeine–glutaric acid cocrystal on multiple faces. Here nanoindentation was used to fully understand the co-crystal mechanical anisotropy and mechanical stability under an applied load.^{133} The higher hardness and elastic modulus of stable form II was rationalized on the basis of its corrugated layers, higher interlayer energy, lower interlayer separation, and the presence of more intermolecular interactions in the crystal structure compared to metastable form I. The results show that mechanical anisotropy in both polymorphs arises due to the difference in orientation of the identical 2D structural features, namely, the number of possible slip systems and the strength of the intermolecular interactions with respect to the indentation direction. It is hoped that studies like these will influence future experimental investigations, where directional elastic stiffness can be correlated with DFT-predicted tensors to rationalise mechanical properties from the nanoscale up.

Thermal contributions to the elastic constants can be incorporated by treating the lattice dynamics of a crystal structure within the quasi-harmonic approximation (QHA), as opposed to the more commonly adopted harmonic approximation (HA) approach.^{135} In the QHA, the lattice dynamics of the structure are modelled within the HA at several unit-cell volumes, therefore incorporating the volume dependence missing in the pure HA. It follows then that with the volume dependence, elastic constants now depend on temperature. The incorporation of temperature effects into elastic constant predictions from QHA calculations accounts for approximately 30% of the disagreement observed^{134,136} compared to elastic constants at 0 K (Fig. 6).

Fig. 6 Three-dimensional plot in GPa showing the magnitude and anisotropy of the material's Young modulus a. experimental measurement^{137} for deuterated ammonia (ND_{3}) b. DFT-calculated Young's modulus with no dispersion corrections (PBE), and many-body dispersion corrections (PBE + MBD). c. DFT-calculated Young's modulus using the same two methods but with the inclusion of the quasi-harmonic approximation (QHA) to simulate the Young's modulus at a temperature of 194 K. Adapted from ref. 134 with permission from Wiley. |

Treating lattice dynamics in this way is known to increase computational expense and effort when considering complex systems, and even more so with the inclusion of dispersion corrections and many-body effects. Lower-level density-functional based methods can be considered in efforts to reduce computational expense.^{138} Density-functional tight binding (DFT-B) is one such method,^{139} being an approximate treatment of the Kohn–Sham DFT formalism with less empirical parameters compared to classical force fields. It therefore lies between ab initio methods and classical force fields in terms of time scales and attainable system sizes. Despite its computational efficiency, DFT-B is still to be fully explored as a tool for predicting the elastic constants of materials.

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