Simulations of Formation of Molecular Clusters

(ACDC course)

Practical matters

• Lectures+exercises Wednesday and Thursday at 2-4pm• mostly working on simulations

• coming to the classes is voluntary, but probably useful

• Grading: exercises (30%) + project (70%)

Exercises

• 3 exercise sets, deadlines 14, 21 and 28 Sep at 4 pm

• You can work on the exercises in class and finish them at home.

• Everyone hands in their answers separately, but you can work on theexercises together and/or ask me for help during classes.

Course project

• done individually

• we will discuss the topics next week

• plan and perform simulations, analyze the results and write a report

• oral presentation about preliminary results (graded as pass/fail)

• deadline for the report?

Simulations of Formation of Molecular Clusters

• Molecular cluster: two or more molecules attached to each other, butsmaller than a macroscopic particle/drop

• Formation≈growth: the process where molecules collide and sometimes stick together, but may also unstick and fly off

• Simulations: solving the time evolution numerically from differentialequations

Applications

• Atmospheric science: Which compounds form aerosol particles? How efficiently?

• Instrumentation: Condensation particle counters (CPC) and Particlesize magnifiers (PSM) are based on condensing vapor molecules on a small particle and make it grow to a detectable size.

• Industrial processes

• …

How to simulate cluster formation and growth?

• Molecular dynamics (MD): each individual molecule is treatedseparately, and its trajectory is computed based on equations of motion and interactions with other molecules (6 variables per molecule)

• Cluster population simulations: the concentration is solved as a function of size and time; individual molecules and clusters are notconsidered (one variable per cluster size)

• Monodisperse growth models: the growth of a single particle is studied based on the surrounding vapor concentration and thecomposition and size of the particle (one variable per compound)

Molecular dynamics (MD)

• most fundamental way to study cluster formation

• computationally extremely demanding:• 105-106 molecules with a very simple interaction potential

• 2 molecules for 1 picosecond with quantum chemical interactions

• useful for studying fundamentals of cluster formation

• usually not applicable for studying realistic situations

Monodisperse growth models

• most simple approach

• several compounds can participate in the growth and the particlescan grow to large sizes

• monodisperse growth: if particles have the same size and composition at time t0, they will grow at the same rate and alwayshave the same size• What does this mean on a microscopic level?

• Is it realistic?

Cluster population simulations

• several options for treating cluster sizes: • moving/fixed size bins

• modes

• molecular resolution = each cluster size is treated separately

• good compromise between a physically reasonable description and a not too overwhelming computational burden

→used in this course

• Air consists mostly of nitrogen and oxygen molecules when they collide, they just bounce off

• There are also some stickier molecules like sulfuric acid when they collide, they can stick together and form a cluster

the clusters can grow bigger through more collisions

or they can evaporate and lose molecules

Particle formation from molecules

Solving cluster concentrations

• For each cluster type, go through all processes where it can be formed or lost

• Find the rate constant for each process (i.e. how often the collision or evaporation happens)

• Write out equations for the time derivatives of the concentrations

• Integrate them to get concentrations as a function of time

Example system: HSO4– + H2SO4 + NH3

= number of clusters / volume

From rate constants to cluster concentrations

d Cid t

= Si (sources)

+

j<i

βj,i−j Cj Ci−j (formation by collisions)

+

j

γi+j,i Ci+j (formation by evaporations)

–

j

βi,j Ci Cj (loss by collisions)

–

j<i

γi,j Ci (loss by evaporations)

– LiCi (loss to external sink, e.g. walls)

Ci = concentration (number/volume) of clusters of type i

Next step:

Ci(t) =Ci(0) + 0

t d Cid t′

d t′Example system: HSO4

– + H2SO4 + NH3

Cluster properties

Oona Kupiainen-Määttä (University of Helsinki)

Cluster properties

Oona Kupiainen-Määttä (University of Helsinki)

Cluster properties

Oona Kupiainen-Määttä (University of Helsinki)

Cluster properties

Oona Kupiainen-Määttä (University of Helsinki)

Collision and evaporation rates

Neutral-neutral collision rates from kinetic gas theory:

βi+j = (8πkT)1/2(

1mi

+1mj

)1/2 (ri + rj

)2

Ion-neutral collision rates from classical electrodynamicsEvaporation rates γ from equilibrium conditions, assumingbarrierless reactions:

i + jβ−−−⇀↽−−−γ

k : Ci = e−Gi−GjkB T Cj = e

Gk −GikB T Ck

Ckγ = CiCjβ

⇒ γ = P0kB T βe

Gk −Gi−GjkB T

Sulfuric acid dimer:β ≈ 10−10 cm3/sγ ≈ 104 1/s

Oona Kupiainen-Määttä (University of Helsinki)

ACDC: Atmospheric Cluster Dynamics Code

Perl scriptInput:

list of clusters and theirGibbs free energiesdensities and molar massesof pure compounds,simulation parameters etc.

Output:Matlab function for thebirth-death equationsMatlab function for runningthe simulation

Matlab simulationInput:

initial concentrations,sources, constantconcentrationssimulation time

Output:concentrations as a functionof time, solved with ode15sformation ratesfluxes between clusters

Oona Kupiainen-Määttä (University of Helsinki)

Running ACDC (making the equations)

From Windows or Linux command line:perl acdc_2015_08_08.pl <options>

From Linux command line:./acdc_2015_08_08.pl <options>

From Matlab:system(’perl acdc_2015_08_08.pl <options>’)

Example:

perl acdc_2015_08_08.pl --cluster_set_file

cluster_set.txt --no_evap --temperature 298.15

Oona Kupiainen-Määttä (University of Helsinki)

Running the simulation in Matlab

Examples:

[C,T] = driver_acdc(Tmax, C0);

[C,T] = driver_acdc(Tmax, ’Sources_in’, filename);

Oona Kupiainen-Määttä (University of Helsinki)