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Computational modelling, structural dynamics and vibrational entropy

Computational modelling, structural dynamics and vibrational entropy

Presented at the 2021 British Crystallographic Association (BCA) Spring Meeting.

Jonathan Skelton

March 14, 2021
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  1. Dr Jonathan Skelton
    Department of Chemistry, University of Manchester
    ([email protected])
    Computational modelling, structural dynamics
    and vibrational entropy

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

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

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  4. 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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  5. 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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  6. 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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  7. 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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  8. Phase stability: SnS
    Pnma
    Rocksalt 𝜋-cubic
    SnS2
    Sn2
    S3
    Cmcm
    Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 8

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  9. 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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  10. 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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  11. 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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  12. 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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  13. 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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  14. 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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  15. 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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  16. 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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  17. 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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  18. 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

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  19. Acknowledgements
    Dr Jonathan Skelton | 2021 BCA Spring Meeting | Slide 19

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  20. These slides are available on Speaker Deck:
    http://bit.ly/2OVggoZ

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