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Planning time-resolved experiments: kinetic modelling

Planning time-resolved experiments: kinetic modelling

Presented at the Time-Resolved I19 Workshop at Diamond Light Source.

Jonathan Skelton

January 24, 2023
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  1. Dr Jonathan Skelton
    Department of Chemistry, University of Manchester
    ([email protected])
    Planning time-resolved experiments:
    kinetic modelling

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  2. Acknowledgements
    Dr Jonathan Skelton i19 TR Workshop, 24th Jan 2023 | Slide 2
    ... plus many others, too numerous to
    mention

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  3. Overview
    i19 TR Workshop, 24th Jan 2023 | Slide 3
    Dr Jonathan Skelton
    o Solid-state linkage isomerism
    o Photocrystallography
    o Kinetics: JMAK theory
    • The decay process
    • The excitation process
    • Steady-state behaviour
    • Numerical simulations
    o Summary: key questions to ask
    o Time-resolved experiment
    https://bit.ly/3ZWTayo

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  4. Solid-state linkage isomerism
    i19 TR Workshop, 24th Jan 2023 | Slide 4
    Dr Jonathan Skelton
    https://chem-is-you.blogspot.com/2013/05/chemistry-of-d-and-f-block-origin-of.html

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  5. Solid-state linkage isomerism
    i19 TR Workshop, 24th Jan 2023 | Slide 5
    Dr Jonathan Skelton
    SO2
    (sulphoxide)
    NO (nitrosyl)
    NO2
    - (nitrite)
    N2
    (“dinitrogen”)

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  6. Solid-state linkage isomerism
    i19 TR Workshop, 24th Jan 2023 | Slide 6
    Dr Jonathan Skelton
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J

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  7. Photocrystallography
    i19 TR Workshop, 24th Jan 2023 | Slide 7
    Dr Jonathan Skelton
    Images: L. E. Hatcher and M. R. Warren

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  8. Photocrystallography
    i19 TR Workshop, 24th Jan 2023 | Slide 8
    Dr Jonathan Skelton

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  9. Photocrystallography
    i19 TR Workshop, 24th Jan 2023 | Slide 9
    Dr Jonathan Skelton
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J

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  10. Kinetics: JMAK theory
    i19 TR Workshop, 24th Jan 2023 | Slide 10
    Dr Jonathan Skelton
    Slow initial rate: waiting
    for nuclei to form
    Fast transformation: existing nuclei
    grow and new nuclei form
    Slow final rate: little untransformed
    phase for nuclei to continue to
    grow
    https://www.tf.uni-kiel.de/matwis/amat/iss/kap_8/illustr/s8_4_3b.html

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  11. Kinetics: JMAK theory
    i19 TR Workshop, 24th Jan 2023 | Slide 11
    Dr Jonathan Skelton
    Can be described by the JMAK
    equation:
    𝛼 𝑡 = 𝛼∞
    + (𝛼0
    − 𝛼∞
    )𝑒−𝑘𝑡𝑛
    where:
    𝛼 𝑡
    𝛼0
    /𝛼∞
    𝑘
    𝑛
    =
    =
    =
    =
    fraction transformed
    initial/final fractions
    rate constant
    Avrami exponent
    https://www.tf.uni-kiel.de/matwis/amat/iss/kap_8/illustr/s8_4_3b.html

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  12. JMAK: the 𝒌 and 𝒏 parameters
    i19 TR Workshop, 24th Jan 2023 | Slide 12
    Dr Jonathan Skelton
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    0 10 20 30 40 50 60
    α(t)
    t [s]
    k = 0.01 k = 0.1 k = 1
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    0 10 20 30 40 50 60
    α(t)
    t [s]
    n = 1 n = 2 n = 3 n = 4
    𝛼0
    = 1, 𝛼∞
    = 0, 𝑛 = 1 𝛼0
    = 1, 𝛼∞
    = 0, 𝑘 = 10-2 s−𝑛

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  13. JMAK: the Avrami exponent 𝒏
    i19 TR Workshop, 24th Jan 2023 | Slide 13
    Dr Jonathan Skelton
    Initial
    𝑛 = 1 (0D growth) 𝑛 = 2 (1D growth)
    𝑛 = 4 (3D growth)
    𝑛 = 3 (2D growth)

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  14. The decay process
    i19 TR Workshop, 24th Jan 2023 | Slide 14
    Dr Jonathan Skelton
    Decay corresponds to 𝛼𝑡=0
    = 1 and 𝛼𝑡=∞
    = 0:
    𝛼 𝑡 = 𝑒−𝑘dec𝑡𝑛
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J

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  15. Temperature dependence
    i19 TR Workshop, 24th Jan 2023 | Slide 15
    Dr Jonathan Skelton
    The 𝑘dec
    as a function of temperature usually follows the Arrhenius law:
    𝑘dec
    𝑇 = 𝐴exp −
    𝐸A
    𝑅𝑇
    → ln 𝑘dec
    𝑇 = ln 𝐴 −
    𝐸A
    𝑅
    1
    𝑇
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J

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  16. Metastable state lifetime
    i19 TR Workshop, 24th Jan 2023 | Slide 16
    Dr Jonathan Skelton
    Substitute 𝑘dec
    (𝑇) into JMAK equation, set 𝛼 = 0.5 and solve for 𝑡:
    𝑡
    𝛼 = 0.5 = −
    1
    𝐴
    ln 0.5 exp
    𝐸A
    𝑅𝑇
    1
    𝑛
    1 s
    10 s
    1 min
    1 hr
    1 day
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J

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  17. The excitation process
    i19 TR Workshop, 24th Jan 2023 | Slide 17
    Dr Jonathan Skelton
    Excitation corresponds to 𝛼𝑡=0
    = 0 and 𝛼𝑡=∞
    = 1:
    𝛼 𝑡 = 1 − 𝑒−𝑘exc𝑡𝑛
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J

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  18. The excitation process
    i19 TR Workshop, 24th Jan 2023 | Slide 18
    Dr Jonathan Skelton
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J
    MS occupation after
    120 s illumination w/
    four different xtals

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  19. Steady-state behaviour
    i19 TR Workshop, 24th Jan 2023 | Slide 19
    Dr Jonathan Skelton
    For a typical linkage isomer system, we have:
    o A strongly temperature-dependent decay rate 𝑘dec
    o A weakly temperature-dependent (i.e. approximately constant) excitation
    rate 𝑘exc
    Competing processes result in three temperature regimes:
    o Low 𝑇: 𝑘dec
    << 𝑘exc
    → 𝛼 = 1 (complete excitation)
    o High 𝑇: 𝑘dec
    >> 𝑘exc
    → 𝛼 = 0 (no excitation)
    o Intermediate 𝑇: 𝑘dec
    ≈ 𝑘exc
    → 0 < 𝛼 < 1 (steady state)
    Can measure the steady-state occupation as a function of temperature by
    collecting structures under continuous illumination

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  20. Steady-state behaviour
    i19 TR Workshop, 24th Jan 2023 | Slide 20
    Dr Jonathan Skelton
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J
    “metastable limit”

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  21. Numerical simulations
    i19 TR Workshop, 24th Jan 2023 | Slide 21
    Dr Jonathan Skelton
    Cannot predict steady-state behaviour analytically -> need numerical simulations
    𝑡 = 0
    𝛼 = 𝛼0
    𝑡dec
    ′ = Τ
    −ln𝛼 𝑘dec
    Τ
    1 𝑛
    ∆𝛼 dec
    = exp −𝑘dec
    𝑡dec
    ′ + ∆𝑡 𝑛 − 𝛼
    Excitation active?
    𝑡 = 𝑡 + ∆𝑡
    𝛼 = 𝛼 + ∆𝛼 dec
    + ∆𝛼 exc
    ∆𝛼 exc
    = 0
    𝑡exc
    ′ = Τ
    −ln 1 − 𝛼 𝑘exc
    Τ
    1 𝑛
    ∆𝛼 exc
    = 1 − exp −𝑘exc
    𝑡exc
    ′ + ∆𝑡 𝑛 − 𝛼
    𝑡 = 𝑡max
    ?
    𝑡 = 𝑡
    𝛼(𝑡) = 𝛼
    Update params?
    Predicted 𝑡, 𝛼(𝑡)
    Y
    N
    N
    Y
    N
    Y

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  22. Numerical simulations 1
    i19 TR Workshop, 24th Jan 2023 | Slide 22
    Dr Jonathan Skelton
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J

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  23. Numerical simulations 2
    i19 TR Workshop, 24th Jan 2023 | Slide 23
    Dr Jonathan Skelton
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J
    Each simulation started with 𝛼 = 0 and run
    in three segments:
    1. 𝑡 = 0-2 mins: no excitation → nothing
    happens
    2. 𝑡 = 2-4 mins: excitation switched on →
    excites towards steady state
    3. 𝑡 = 4-24 mins: excitation switched off
    → steady-state population decays

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  24. Numerical simulations 3
    i19 TR Workshop, 24th Jan 2023 | Slide 24
    Dr Jonathan Skelton
    L. E. Hatcher et al., Phys. Chem. Chem. Phys. 20, 5874 (2018), DOI: 10.1039/C7CP05422J
    180 s pulse

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  25. Summary: key questions to ask
    Dr Jonathan Skelton
    o Can we measure the kinetics of the process we want to study?
    • Performed low-𝑇 (“slow”) photocrystallography experiments on a lab
    machine and extrapolated to higher 𝑇
    o Can we estimate a “ballpark” lifetime for the excited state?
    • Can derive from JMAK kinetic fits; ranges from ~days at 𝑇 = 200 K to ~1s at
    300 K
    o Can we use our data to plan any other aspects of the experiments?
    • Can use fairly simple numerical simulations parameterised by kinetic
    measurements to:
    1) predict behaviour during a simulated pump/probe experiment; and
    2) suggest experimental parameters e.g. excitation time and
    measurement temperature
    i19 TR Workshop, 24th Jan 2023 | Slide 25

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  26. Time-resolved experiment
    i19 TR Workshop, 24th Jan 2023 | Slide 26
    Dr Jonathan Skelton

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  27. Time-resolved experiment
    i19 TR Workshop, 24th Jan 2023 | Slide 27
    Dr Jonathan Skelton
    Our workflow:
    1. Preliminary experiments:
    o Decay curves at 𝑇 = 240-270 K
    o Excitation curve at 𝑇 = 150 K
    o Steady-state occupation between 𝑇 = 250-300 K
    2. Kinetic fitting to derive 𝐴 and 𝐸A
    for decay and estimate 𝑘exc
    3. Numerical simulations to select 𝑡exc
    /𝑡dec
    and estimate 𝑇 for given 𝑡cyc
    4. Time-resolved measurement at estimated 𝑇 + automatic processing to
    determine rough 𝛼 𝑡 - inspect result and raise/lower 𝑇 as required
    5. Data fitting using numerical simulations to extract 𝑘exc
    and 𝑘dec
    from each
    TR dataset

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  28. Preliminary measurements
    i19 TR Workshop, 24th Jan 2023 | Slide 28
    Dr Jonathan Skelton

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  29. Numerical simulations
    i19 TR Workshop, 24th Jan 2023 | Slide 29
    Dr Jonathan Skelton
    𝑡cyc
    = 170 s 𝑡cyc
    = 34 s
    𝑡cyc
    = 14 s
    𝑡cyc
    = 22 s
    𝑡cyc
    = 108 s
    L. E. Hatcher et al., Nature Comms. Chem. 5, 102 (2022), DOI: 10.1038/s42004-022-00716-1

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  30. TR results 1
    i19 TR Workshop, 24th Jan 2023 | Slide 30
    Dr Jonathan Skelton

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  31. TR results 2
    i19 TR Workshop, 24th Jan 2023 | Slide 31
    Dr Jonathan Skelton
    Data from lab (Ph-SCXRD), DLS (Ph-SCXRD) and DLS (TR)
    L. E. Hatcher et al., Nature Comms. Chem. 5, 102 (2022), DOI: 10.1038/s42004-022-00716-1

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  32. TR results 3
    i19 TR Workshop, 24th Jan 2023 | Slide 32
    Dr Jonathan Skelton
    L. E. Hatcher et al., Nature Comms. Chem. 5, 102 (2022), DOI: 10.1038/s42004-022-00716-1

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  33. Follow up: Timepix tests
    i19 TR Workshop, 24th Jan 2023 | Slide 33
    Dr Jonathan Skelton
    L. E. Hatcher et al., Nature Comms. Chem. 5, 102 (2022), DOI: 10.1038/s42004-022-00716-1
    Single module,
    (-2 1 0) reflection

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  34. Thankyou for listening!
    Any questions?

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