Managing uncertain branching ratios in chemical models - MUCM

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Managing uncertain branching ratios in chemical models - MUCM

Managing uncertain branching ratios

in chemical models

P. Pernot, S. Plessis and N. Carrasco

Labo. de Chimie Physique, CNRS/Univ Paris-Sud, Orsay

UCM2012, Jul. 2-4

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 1 / 29


About this speaker

The Physical Chemistry Lab. in Orsay (CNRS/Univ. Paris-Sud)

50 researchers in fields covering elementary processes to complex

systems, from gas phase reactions to biophysics and planetary

atmospheres

Theory and Simulation group

12 researchers in theoretical chemistry and molecular simulation

My tream

2 researchers

Bayesian Data Analysis of Spectrokinetic Data (since 1980)

Uncertainty Management in Chemical Networks (since 2005)

Calibration/Prediction in semi-empirical models (since 2009)

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 2 / 29


Detailed chemistry modeling

V. De La Haye et al. (2008) Icarus 197:110-136

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 3 / 29


Chemical kinetics measurements (1)

Partial reaction rates (k1, k2)

A + B


−→ P1

−→ P2

dcP1 (t)

dt

= k1 ∗ cA(t) ∗ cB(t)

dcP2 (t)

dt

= k2 ∗ cA(t) ∗ cB(t)

dcA/B(t) = −(k1 + k2) ∗ cA(t) ∗ cB(t)

dt

(k1, k2) can be used directly in modeling

and treated as independent uncertain variables

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 4 / 29


Chemical kinetics measurements (2)

Global reaction rate (k) and branching ratios (b1, b2)

A + B


−→ P1

−→ P2

dcA/B(t) = −k ∗ cA(t) ∗ cB(t)

dt

cP1 (t)

k1

=

cP2 (t) k2

= b1

b2

1 = b1 + b2

(k1 = k ∗ b1, k2 = k ∗ b2) are used in modeling

they are NOT independent uncertain variables

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 5 / 29


Detailed chemistry modeling in a nominal world

From the chemistry lab. to the modeler

1 Experimentalist

1 acquires reaction rate data:

1 rate constant k

2 branching ratios {b1, ..., bn}

2 Database manager

1 collects and processes data to fit “one-line-per-reaction” scheme

3 Modeler

1 A + B −→ Pi; ki (with ki = k ∗ bi)

1 builds kinetic scheme from database

2 performs simulation

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 6 / 29


This is where uncertainty comes in...

Inputs Outputs

−→

Model

−→

N. Carrasco et al. (2006) Planet. Space Sci. 55:141-157

E. Hébrard et al. (2009) J. Phys. Chem. A 113:11227-11237

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 7 / 29


This is where uncertainty comes in...

Inputs Outputs

−→

Model

−→

N. Carrasco et al. (2006) Planet. Space Sci. 55:141-157

E. Hébrard et al. (2009) J. Phys. Chem. A 113:11227-11237

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 7 / 29


Yet another example in astrochemistry

V. Wakelam et al. (2012) Astroph. J. Supp. Ser. 199:21

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 8 / 29


Information Loss in Chemical Modeling

From the chemistry lab. to the modeler

1 Experimentalist

1 acquires reaction rate data:

1 rate constant k

2 branching ratios {b1, ..., bn}

2 does not publish covariances

2 Database manager

1 collects and processes data to fit “one-line-per-reaction” scheme

2 estimates uncertainty factor F (ki) for each “line”

3 Modeler

1 A + B −→ Pi; ki, Fi

1 builds kinetic scheme from database

2 performs MCUP/SA based on F (ki)

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 9 / 29


Information Loss: the consequences

Let us consider a simple reaction scheme

Conditions

I1 + M1 −→ P1 ; k11

I1 + M1 −→ P2 ; k12

I1 + M2 −→ P3 ; k2

k1 = k2 = c t = 1; cM1 = cM2 ≫ cI1(0), cI2(0)

b11 = 0.33 ± 0.12, b12 = 0.67 ± 0.12

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 10 / 29


Information Loss: the consequences

Sum-rule-correlated partial rates {k11, k12} = k1 ∗ {b11, b12} Σ=1

{b11, b12} ∼ Dirichlet (45 × {0.33, 0.67})

dcP (t) 1(2)

= k1 ∗ b11(12) ∗ cI1(t) ∗ cM1

dt

dcP3(t)

dt = k2 ∗ cI1(t) ∗ cM2

dcI1(t)

dt = −k1 ∗ (b11 + b12)

∗cI1(t) ∗ cM1 − k2 ∗ cI1(t) ∗ cM2

1

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 11 / 29


Information Loss: the consequences

Uncorrelated partial rates k11 = k1 ∗ b11, k12 = k1 ∗ b12

k11 ∼ Beta(45 ∗ 0.33, 45 ∗ 0.67)

k12 ∼ Beta(45 ∗ 0.67, 45 ∗ 0.33)

dcP (t) 1(2)

= k11(12) ∗ cI1(t) ∗ cM1

dt

dcP3(t)

dt = k2 ∗ cI1(t) ∗ cM2

dcI1(t)

dt = −(k11 + k12) ∗ cI1(t) ∗ cM1 − k2 ∗ cI1(t) ∗ cM2

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 12 / 29


Information Loss: the consequences

Sum-rule-correlated partial rates {k11, k12} = k1 ∗ {b11, b12} Σ=1

No uncertainty leaks !!!

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 13 / 29


Information Loss: the consequences

Uncorrelated partial rates k11 = k1 ∗ b11, k12 = k1 ∗ b12

Sideways/lateral uncertainty leaks !!!

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 14 / 29


Information Loss: the consequences

Neglecting the covariance of branching ratios of a reaction leads to:

1 spurious uncertainty in non-directly related outputs

2 underestimation of the uncertainty in the directly related outputs

How to trust such a scrambled uncertainty budget ???

Which impact on key reactions identification ???

Pb: this is what (almost) everybody does !!!

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 15 / 29


Example of a scrambled uncertainty budget

Chemical model of ions in Titan ionosphere

(alt=1200 km; 100+ species; 700+ reactions)

Rank correl. of outputs Rank correl. of uncertainties

Carrasco & Pernot (2007) J. Phys. Chem. A 111:3507

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 16 / 29


The geometry of positive sum-to-one variables

for a reaction with n products channels, one has

bi ≥ 0 (i = 1, n) and n

i=1 bi = 1;

these conditions define a subspace (hypertetrahedron) of R n ,

the n-simplex

n = 2 n = 3 n = 4

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 17 / 29


Sampling in the simplex

A toolbox of knowledge-adapted Dirichlet-based distributions

Preferred values and uncertainties (bi = µi ± ui)

{b1, b2, b3} ∼ Dirg (µ1, µ2, µ3 ; u1, u2, u3)

Carrasco and Pernot (2007) J. Phys. Chem. A 111:3507-3512

Plessis et al. (2010) J. Chem. Phys. 133:134110

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 18 / 29


Sampling in the simplex

A toolbox of knowledge-adapted Dirichlet-based distributions

No preference: total uncertainty

{b1, b2, b3} ∼ Diun (3)

Carrasco and Pernot (2007) J. Phys. Chem. A 111:3507-3512

Plessis et al. (2010) J. Chem. Phys. 133:134110

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 18 / 29


Sampling in the simplex

A toolbox of knowledge-adapted Dirichlet-based distributions

Ordering rule b1 ≥ b2 ≥ b3

{b1, b2, b3} ∼ Dior (3)

Carrasco and Pernot (2007) J. Phys. Chem. A 111:3507-3512

Plessis et al. (2010) J. Chem. Phys. 133:134110

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 18 / 29


Sampling in the simplex

A toolbox of knowledge-adapted Dirichlet-based distributions

Preferred intervals

{b1, b2, b3} ∼ Diut

b min

i

, bmax

3

i i=1

Carrasco and Pernot (2007) J. Phys. Chem. A 111:3507-3512

Plessis et al. (2010) J. Chem. Phys. 133:134110

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 18 / 29


Sampling in the simplex

A toolbox of knowledge-adapted Dirichlet-based distributions

Partial knowledge or heterogeneous data

{b1, b2, b3} ∼ Dirg (µ1 ⊗ Diun (2) , µ2 ; u1, u2)

Carrasco and Pernot (2007) J. Phys. Chem. A 111:3507-3512

Plessis et al. (2010) J. Chem. Phys. 133:134110

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 18 / 29


On the interest of nesting (not for birds)

Combining experimental data into a probabilistic tree

and

one experiment measured the probabilities of

{M1, M2} (B1) and M3 (B3 = 1 − B1);

another experiment measured the probabilities of

M1 (B11) and M2 (B12 = 1 − B11).

I + + e −


⎪⎨

⎪⎩

B1±∆B1

−−−−−→

B3±∆B3

−−−−−→ M3

B11±∆B11

−−−−−−−→ M1 (b1)

µ1 = B1 × B11

µ2 = B1 × B12

µ3 = B3

B12±∆B12

−−−−−−−→ M2 (b2)

(b3)

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 19 / 29


On the importance of nesting

To obtain a one-line-per-reaction description,

one has to “flatten” the tree

with

ui

µi

=

I + + e −

∆B1

B1


⎪⎨

⎪⎩

2

µ1±u1

−−−−→ M1 (b1)

µ2±u2

−−−−→ M2 (b2)

µ3±u3

−−−−→ M3 (b3)

+

∆B1i

B1i

2

; i = 1, 2

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 20 / 29


On the importance of nesting

Ex.: B1 = 0.6 ± 0.1, B3 = 0.40 ± 0.05, B11 ∈ [0, 1], and B12 ∈ [0, 1]

Flat representation Nested representation

{b1, b2, b3} ∼ Dirg (0.30, 0.30, 0.40 ; 0.18, 0.18, 0.05) {b1, b2, b3} ∼ Dirg (0.6 ⊗ Diun(2), 0.4 ; 0.1, 0.05)

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 21 / 29


What experimentalists tell us about branching ratios

CH2CHCNH + + e − [Vigren et al. (2009) ApJ 695:317-324]

Products Probability Exoergic channels

C3NHx + y H + wH2 0.50±0.04 6

C2Hx + CNHy + zH + wH2 0.49±0.04 7

C3Hx + NHy

C2NHx + CHy + zH

0.01±0.01 9

x + y + z + 2w = 4 22

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 22 / 29


What experimentalists tell us about branching ratios

CH2CHCNH + + e − [Vigren et al. (2009) ApJ 695:317-324]

Products Probability Exoergic channels

C3NHx + y H + wH2 0.50±0.04 6

C2Hx + CNHy + zH + wH2 0.49±0.04 7

C3Hx + NHy

C2NHx + CHy + zH

0.01±0.01 9

x + y + z + 2w = 4 22

What am I supposed to do with that ?

or

How hundreds of data have been left into oblivion by modelers

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 22 / 29


Experimentalists take modelers by the hand

The measurement results are translated into

the “one line per reaction” format for databases (UMIST,osu...)

Reaction k(300) / cm 3 s −1 Probability

H4C3N + + e − −→ CH2CHCN + H 8.90 × 10 −7 0.50

H4C3N + + e − −→ C2H2 + HCN + H 4.45 × 10 −7 0.25

H4C3N + + e − −→ C2H3 + HNC 4.45 × 10 −7 0.25

what happened to the 19 other channels?

Vigren et al. (2009) ApJ 695:317-324

why equipartition scenario amongst channels 2 and 3?

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 23 / 29


Example of combined data

The case of HCCCNH +

Geppert et al. [(2004) ApJ 613:1302-1309] measurements for

DCCCND + :

B1 = 0.52 ± 0.05 for {DC 3 N + D, C 3 N + D 2 }

B2 = 0.48 ± 0.05 for {DCN + C 2 D, CN + C 2 D 2 }

isotope effects expected to be small: we can transpose information to

HCCCNH +

Osamura et al. [(1999) ApJ 519:697-704]

interconversion barriers low enough for isomerisation of HC 3 N

formation of HC 3 N more likely than HC 2 NC

HCN / HNC

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 24 / 29


Example of combined data

The case of HCCCNH +

Geppert et al. [(2004) ApJ 613:1302-1309] measurements for

DCCCND + :

B1 = 0.52 ± 0.05 for {DC 3 N + D, C 3 N + D 2 }

B2 = 0.48 ± 0.05 for {DCN + C 2 D, CN + C 2 D 2 }

isotope effects expected to be small: we can transpose information to

HCCCNH +

Osamura et al. [(1999) ApJ 519:697-704]

interconversion barriers low enough for isomerisation of HC 3 N

formation of HC 3 N more likely than HC 2 NC

HCN / HNC

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 24 / 29


Example of combined data

The case of HCCCNH +

Geppert et al. [(2004) ApJ 613:1302-1309] measurements for

DCCCND + :

B1 = 0.52 ± 0.05 for {DC 3 N + D, C 3 N + D 2 }

B2 = 0.48 ± 0.05 for {DCN + C 2 D, CN + C 2 D 2 }

isotope effects expected to be small: we can transpose information to

HCCCNH +

Osamura et al. [(1999) ApJ 519:697-704]

interconversion barriers low enough for isomerisation of HC 3 N

formation of HC 3 N more likely than HC 2 NC

HCN / HNC

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 24 / 29


Example of combined data

The case of HCCCNH +

Geppert et al. [(2004) ApJ 613:1302-1309] measurements for

DCCCND + :

B1 = 0.52 ± 0.05 for {DC 3 N + D, C 3 N + D 2 }

B2 = 0.48 ± 0.05 for {DCN + C 2 D, CN + C 2 D 2 }

isotope effects expected to be small: we can transpose information to

HCCCNH +

Osamura et al. [(1999) ApJ 519:697-704]

interconversion barriers low enough for isomerisation of HC 3 N

formation of HC 3 N more likely than HC 2 NC

HCN / HNC

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 24 / 29


Example

A resulting tree is

8

HCCCNH + + e − ><

−→

>:

B1=0.52±0.05

−−−−−−−−−→

B2=0.48±0.05

−−−−−−−−−→

8

><

>:

B11∈[0,1]

−−−−−→

8

><

(

B1111∈[0,1]

B111∈[0,1] −−−−−−−→ HC3N + H

−−−−−−→

B1112∈[0,1]

B112≤B111

−−−−−−→

−−−−−−−→ C3NH + H

(

B1121∈[0,1]

−−−−−−−→ C2NCH + H

>:

>: B12∈[0,1]

−−−−−→ C3N + H2

8 (

B211∈[0,1]

>< B21∈[0,1] −−−−−−→ HCN + C2H

−−−−−→

B212∈[0,1]

−−−−−−→ HNC + C2H

B22∈[0,1]

−−−−−→ CN + C2H2

B1122∈[0,1]

−−−−−−−→ HC2NC + H

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 25 / 29


Titan Ionospheric Chemistry

Two RD models/scenarii have been built and compared

the old “H-loss” scheme

58 ions

63 partial reactions

48 neutral products

a “full scheme” implementing all available data

58 ions

448 partial reactions

62 neutral products

Plessis et al., J. Chem. Phys. (2010); Icarus (2012)

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 26 / 29


DR in Titan’s ionosphere: H-loss vs. full model

Ratio of“full model”over“H-loss”production rates of neutral species

Plessis et al., JCP (2010)

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 27 / 29


Conclusions

In many chemical modeling situations,

we have to handle very large uncertainties and

to adopt stochastic modeling (MCUP)

Explicit enforcement of conservation equations is a necessity

for reliable Uncertainty Propagation and Sensitivity Analysis

Probabilistic tree representation of branching ratios is a powerful

method to deal with partial measurements and to implement

empirical rules

In combination with Monte Carlo methods, this enables

a very promising advance in kinetic modeling of chemical plasmas

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 28 / 29


Thank you !

The workforce

S. Plessis - PhD thesis (LCP, Orsay)

N. Carrasco (LATMOS, Versailles-Saint-Quentin)

Helpful discussions with specialists of DR process

B. Mitchell (Rennes),

K. Béroff and M. Chabot (Orsay)

Fundings

CNRS, CNES, EuroPlaNet, ISSI...

P. Pernot et al. (LCP@Orsay) Managing uncertain branching ratios UCM2012, Jul. 2-4 29 / 29

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