Soot - Developing anisotropic potentials from first principles for PAH molecules. Tim Totton, Alston Misquitta and 12/11/2009
HRTEM images of soot Some evidence for different soot structures based on different fuels Top: graphitic Bottom: amorphous Pictures from: Uitz, Cracknell, Jansma and Makkee, Impact of diesel fuel composition on soot oxidation Characteristics, SAE 2009-01-0286
Investigating structure How do PAH molecules form clusters? How do these clusters grow? Driven by intermolecular potentials Alston Misquitta, Aron Cohen, Dwaipayan Chakrarti, Mark Miller, David Wales
Atom-atom potential R
What s the use? Study PAH clusters (i.e. soot) How do PAH molecules form clusters? What determines their shape/structure? Interactions with gas-phase molecules? Need to study Potential Energy Surface (PES) created from potential
Comparison of potentials Poor agreement between L-J potential and SAPT(DFT) results Good agreement with W99 potential Potential (kj/mol) 20 10 0-10 -20-30 -40 LJ Gr LJ SP LJ X W99 Gr W99 SP W99 X SAPT Gr SAPT SP SAPT X -50 2.5 3 3.5 4 4.5 5 5.5 6 Dimer Separation (Å) Graphite (Gr) Slipped Parellel (SP) Crossed (X)
Basin hopping Finds stle molecular clusters by searching for minima Based on potential energy landscape Uses Monte-Carlo criterion when jumping between minima Energy
Global minimum clusters 2 Coronene molecules 5 Coronenes molecules 10 Coronene molecules E = -94.90 kj/mol E = -394.35 kj/mol E = -926.42 kj/mol
Experimental comparison Experimental HR-TEM images of an aggregate sampled from a diesel engine. Indicated are length scales of structures within a primary particle (from Mosbach et al., 2009, Combustion and Flame). A TEM-style projection of a computed cluster of 50 coronene molecules
Cluster Density Need to define cluster volume Used scaled van der Waals radii Define Volume in terms of scaling factor, α Determine critical α to determine density
Cluster density Volume varies non-linearly with α Choose critical α at minimum of dv/dα (α crit = 1.7) Corresponds to point where all intermolecular space is covered by overlapping spheres.
Varying density in our models For α crit = 1.7, coronene ρ = 1.1 g/cm 3, pyrene ρ = 1.0 g/cm 3 Standard soot density in models = 1.8 g/cm 3 Density has been shown to be an important parameter
Intermolecular Potentials Use atom-atom potentials Approximated as sum of all pairwise atomic interactions between molecules U = U R, A A< Ba Ab B ( Ω ) (molecule A atoms a and molecule B with atoms b)
Standard atom-atom potentials Lennard-Jones U LJ = 4ε σ R 12 σ R 6 Exponential-6 U exp 6 = B exp A R ( CR ) 6 Specific electrostatic term, e.g. U = elst qaq R b
Isotropic potentials Good agreement between W99 potential and SAPT(DFT) for stacked geometries 20 Poor agreement between either potential and SAPT(DFT) for T-shape geometry 60 10 50 40 0 30 Potential (kj/mol) -10-20 -30-40 LJ Gr LJ SP LJ X W99 Gr W99 SP W99 X SAPT Gr SAPT SP SAPT X -50 2.5 3 3.5 4 4.5 5 5.5 6 Dimer Separation (Å) Potential (kj/mol) 20 10 0-10 -20 LJ T W99 T SAPT T -30 4 4.5 5 5.5 6 6.5 7 7.5 8 Dimer Separation (Å) Graphite (Gr) Slipped Parellel (SP) Crossed (X) T-shape (T)
How accurate are the potentials? Current isotropic atom-atom potentials parameterised to fit stacked (most stle) molecule configurations Really anisotropy in electron density Isotropic atom-atom potential
Ab initio methods In principal could work out high accuracy interaction energies on-the-fly BUT most initio methods are prohibited due to high computational expense. DFT is a possibility but cannot accurately predict dispersion Need to use accurate model potentials (fitted from initio data)
New anisotropic potential Needs Accuracy: energy barriers on PES required to answer questions out morphology and reactivity Transferility: many types of PAH (C 6 -C 400 ) in flames Simplicity: large clusters limit functional complexity
New anisotropic potential Start with benzene generalise to larger PAHs Functional form: U C + [ α ( R ρ ( Ω ))] f ( R ) 6, iso = G exp 6 6 R qaq R b Short-range Split into three parts: Short-range repulsion Long-range dispersion Long-range electrostatics Long-range
Electrostatics Simple point charge model used Ideally would use high-rank multipole model e.g. hexadecapole on atomic sites Unnecessary for PAH molecules as they don t possess strong directional moments (e.g. H-bonding) Fit partial charges to overall molecular electrostatic potential calculated from initio wavefunction.
Benchmark energies SAPT(DFT) to generate high accuracy interaction energy benchmarks Total interaction approximated by taking terms up to second order: int (1) elst (1) (2) (2) (2) (2) ( KS ) + E ( KS ) + E + E + E E E = E + exch ind, exch Calculate dimer energies need uniform coverage of physically important configurations ind disp disp, exch
Dispersion models Calculated from molecular properties (not dimer energies) Model dispersion energies of the form: E (2) disp ) 6 8 R R 6 8 10 ( model) = f6( R) f8( R) f10( R 10 Damping functions (Tang-Toennies): f n ( R C ) = 1 exp( βr ) C ( βr ) n k = 0 k! k C R
Different dispersion models Complex models match SAPT(DFT) energies best Linear deviation of all models leads to scaling factor Λ = ( R ) 2 i f6 C6,iso wi Edisp,tot + ξ 6 i a A, b B R
Short-range energies Define SAPT(DFT) short-range energies: sr (1) exch (1) pen E = E + E + (2) ind,tot Fit parameters of Born-Mayer term to SAPT(DFT) dimer energies G exp E ( ESP) ( (1) (1) (1) E = E E ) pen elst elst [ α ( ρ ( Ω ))] R
Shape function - anisotropy Split shape function into atomic contributions and define in terms of θ a and θ b a ρ ( ) a ( ) b Ω = ρ θ ρ ( θ ) ρ a + a a 1 a ρ + ρ10 cosθa + ρ 2 ( ) ( 2 θ 3cos θ 1) a = 00 20 a b
Fitting short-range energies Important to have physical parameter set Need to partition total interaction energy into atom-atom contributions Use density overlap model to provide good starting estimates and refine using harmonic constraints.
Density Overlap model Assumes short-range interaction is proportional to total electron density overlap between molecules, i.e. E K Esr fit = 0S ρ S ρ ρ = ( ) B r ( r) 3 ρ d r Fit E fit to E sr to get optimised K 0 A e e
Distributed overlap model Now look at atom-atom contributions to total energy partition total density overlap into atom-atom contributions, i.e. S ρ ρ = a e ( ) b r ( r) 3 ρ d r e Now refit E sr to E fit defined below to give set of coefficients K E sr E fit = K a A, b B S ρ
Refine distributed overlap model Some K are unphysical, so use K 0 as constraint and minimise = i w i i E E i fit 1 i sr w + λ ( K K ) 0 Use refined set of K to fit energies to separation and orientation dependent Born- Mayer term G exp i 2 [ α ( ρ ( Ω ))] R 2
Overall Potential Piece together short-range and long-range terms to give overall potential Refine against total energies of 128 benzene dimers using multivariate optimisation with tight harmonic constraints. Final fit has weighted r.m.s. residual error of only 0.49 kj/mol
Overall anisotropic potential Improvement over isotropic potentials Provides starting point for general PAH potential
Generalising to larger PAH Use a further 111 SAPT(DFT) dimer energies relating to larger PAH molecules: naphthalene (C 10 H 8 ), anthracene (C 14 H 10 ) pyrene (C 16 H 10 ) Start with BAP parameters, using tight harmonic constraints. Overall weighted r.m.s. residual error was found to be only 0.73 kj/mol
PAH anisotropic potential (PAHAP) Much improved agreement for non-stacked dimer configurations over W99 potential Similar high accuracy for stacked configurations
rk ure Fut wo Use new potential to simulate clusters of PAH molecules Coarse-grain new potential to look at much larger systems Study pathways on potential energy surface PAH rearrangement soot morphology Interaction with gas-phase species
Acknowledgements