Computational modelling, structural dynamics and vibrational entropy

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Dr Jonathan Skelton Department of Chemistry, University of Manchester ([email protected]) Computational modelling, structural dynamics and vibrational entropy

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Structural dynamics in solids Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 2

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Structural dynamics in solids Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 3

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Statistical thermodynamics At finite temperature, thermal energy is partitioned over phonon modes 𝜆 through to the vibrational partition function 𝑍vib (𝑇): 𝐴 𝑇 = 𝑈latt + 𝐴vib 𝑇 The phonons contribute to the constant-volume (Helmholtz) free energy 𝐴(𝑇) through the bridge relation: Adding 𝐴vib 𝑇 to the lattice energy 𝑈latt gives us a model for the temperature- dependent free energy 𝐴(𝑇): 𝐴vib 𝑇 = 𝑘B 𝑇 ln 𝑍vib (𝑇) 𝑍vib 𝑇 = ෑ 𝜆 exp Τ −ℏ𝜔𝜆 2𝑘B 𝑇 1 − exp Τ −ℏ𝜔𝜆 𝑘B 𝑇 𝐴 𝑇 = 𝑈latt + 𝑈vib 𝑇 − 𝑇𝑆vib 𝑇 Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 4

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Phase stability: sulfamerazine A. R. Pallipurath et al., Mol. Pharmaceutics 12 (10), 3735 (2015) Form-1 Form-2 Form-3 Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 5

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Phase stability: sulfamerazine A. R. Pallipurath et al., Mol. Pharmaceutics 12 (10), 3735 (2015) Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 6

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Phase stability: sulfamerazine A. R. Pallipurath et al., Mol. Pharmaceutics 12 (10), 3735 (2015) 𝑇 = 293 K Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 7

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Phase stability: SnS Pnma Rocksalt 𝜋-cubic SnS2 Sn2 S3 Cmcm Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 8

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Phase stability: SnS J. M. Skelton et al., J. Phys. Chem. C 121 (12), 6446 (2017) Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 9

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Dynamic disorder: soft modes I. Pallikara and J. M. Skelton, ChemRxiv preprint - DOI: 10.26434/chemrxiv.14187689 Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 10

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Dynamic disorder: soft modes Low 𝑇: Pnma High 𝑇: Cmcm (Average structure) I. Pallikara and J. M. Skelton, ChemRxiv preprint - DOI: 10.26434/chemrxiv.14187689 Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 11

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Dynamic disorder II: MAPbI3 A. N. Beecher et al., ACS Energy Lett. 1 (4), 880 (2016) Orthorhombic (𝑇 < 165 K) Tetragonal (𝑇 =165-327 K) Cubic (𝑇 > 327 K) Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 12

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Dynamic disorder II: MAPbI3 L. D. Whalley et al., Phys. Rev. B 94, 220301(R) (2016) c-MAPbI3 Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 13

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Dynamic disorder II: MAPbI3 A. N. Beecher et al., ACS Energy Lett. 1 (4), 880 (2016) Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 14

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Statistical thermodynamics II Using the harmonic approximation, we can calculate the Helmholtz free energy 𝐴(𝑇): It we also take into account the volume dependence of 𝑈latt and the phonon frequencies, we can calculate the Gibbs free energy 𝐺(𝑇) (the quasi-harmonic approximation): (𝐺 is arguably a more experimentally-relevant quantity, and we can also explore the effect of pressure through the 𝑝𝑉 term.) 𝐴(𝑇) = 𝑈latt + 𝑈vib (𝑇) − 𝑇𝑆vib (𝑇) 𝐺(𝑇) = min 𝑉 𝐴(𝑇; 𝑉) + 𝑝𝑉 𝐺(𝑇) = min 𝑉 𝑈latt (𝑉) + 𝑈vib (𝑇; 𝑉) − 𝑇𝑆vib (𝑇; 𝑉) + 𝑝𝑉 Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 15

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The SnSe phase diagram Average: Cmcm Local: Cmcm Average: Cmcm Local: Pnma Average: Cmcm Local: ??? I. Pallikara and J. M. Skelton, ChemRxiv preprint - DOI: 10.26434/chemrxiv.14187689 Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 16

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Pnma Sn(S1-x Sex ) alloys J. M. Skelton, J. Phys.: Energy 2 (2), 025006 (2020) 𝑈latt 𝐴vib Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 17

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Summary The theory of lattice dynamics describe the natural thermal motion of atoms in crystalline solids (phonons) Phonons contribute to the temperature-dependent Helmholtz free energy through the vibrational partition function The 𝐴(𝑇) - most importantly the 𝑆vib (𝑇) term - can have an important impact on the relative stability of different material phases at finite 𝑇 Materials with imaginary harmonic modes in their dispersion are expected to show a divergence between the local (short-range) and average (long-range) structure Using the quasi-harmonic approximation, we can model 𝐺(𝑇) and study pressure effects Recent work has shown that the vibrational free energy can skew an alloy phase diagram away from ideality Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 18