Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Diffusion coefficient of linear chainsIs active surface a useful concept?

Lorenzo Isella Yannis Drossinos

European CommissionJoint Research Centre

Ispra (VA), Italy

European Aerosol Conference 2009, Karlsruhe

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Objectives

Physical systemFractal aggregate (e.g., soot particle) composed of k(spherical) monomersBrownian motion in a quiescent fluidContinuum regime

MotivationMonomers in an aggregate are shieldedLangevin simulation with unshielded monomers generateideal clusters

Aimto determine an algorithm for the calculation of the frictioncoefficient of a monomer of a k -monomer aggregatesto use it in Langevin simulations of aggregate formation interms of monomer properties

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Friction coefficient of a k -aggregate

Aggregate equation of motion

mkdvk

dt= −fkvk + F(t)

Stokes drag force arises from collisions with carrier-gasmolecules

Aggregate friction coefficient fk = k(

m1βk

)βk average friction coefficient per monomer (unit mass)

Random force models fluctuating force resulting fromthermal motion of carrier gas molecules

Stokes drag and friction coefficient similar origin:Fluctuation Dissipation Theorem relates them

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Shielding factor

Average monomer shielding factor ηk in a k -aggregate

fkkf1

=km1βk

km1β1=

βk

β1≡ ηk

Stokes-Einstein diffusion coefficient

Dk =kBT

km1βk= D1

1kηk

Mobility radius Rk defined through Dk ≡ kBT/(6πµRk )

Rk

R1= kηk

Ideal clusters: ηk = 1

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Gas molecule-monomer interactions

Active surface

Description of (carrier gas) molecule-monomer interactionsin term of Active Surface (or Fuchs surface)

Fraction of geometrical surface area directly accessible(exposed) to gas molecules

Active surface determines condensational growth,adsorption kinetics

Surface area active in mass and momentum transfer

Experimentally measurable: attachment rate of diffusingions (diffusion charger) or radioactively labelled atoms(epiphaniometer)

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Gas molecule-monomer interactions

Experimental observation

Figure: From A. Keller, M. Fierz, K. Siegmann, H.C. Siegmann, A.Filippov, “Surface science with nanosized particles in a carrier gas”, J.Vac. Sci. Technol. A 19(1), 1-8 (2001).

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Gas molecule-monomer interactions

Scaling law (1)

Mass transfer coefficient Kk (∝ attachment probability)times agglomerate mobility bk is independent of k for avariety of aggregate sizes and shapes

Kk × bk =Kk

fk= constant

Argument:Attachment probability ∝ active surface, the surface areaaccessible to diffusing moleculesFriction coefficient (inversely proportional to particlemobility), ∝ active surface

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Gas molecule-monomer interactions

Scaling law (2)

Attachment probability is the gas-molecule monomercollision probability (stricking coefficient of unity)Molecular collision rate Kk with an aggregate consisting ofk monomers

Kk =

∫S

J · s dS

J (steady-state) diffusive flux towards the aggregate, s unitvector perpendicular to SFrom the experimental scaling law

Kk

kK1=

fkkf1

=βk

β1= ηk

The calculation of the average monomer shielding factor(and the ratios of the friction coefficients) reduces tocalculating relative molecular diffusive fluxes

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Gas molecule-monomer interactions

Diffusive flux

Molecular diffusive flux

J = −Dg∇ρ

Gas density from steady-state diffusion equation(continuum regime)

Dg∇2ρ(r) = 0

Boundary conditions

ρ → ρ∞ for |r| → ∞, and ρsur = 0 for r = S

Sticking probability unity, no multiple scattering events:absorbing boundary conditions at the aggregate surface

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Linear chains: Effect of anisotropy

Linear chains: Numerical method (1)

Use the scaling law to determine the diffusion coefficient oflinear chains

The approach mimics closely the experimental procedure

Steady-state diffusion equation, with appropriate boundaryconditions, solved with the finite-element software ComsolMultiphysics in cylindrical co-ordinates

Collision rate obtained by numerical integration of thediffusive flux over the aggregate geometrical surface

Linear chains of up to k = 64

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Linear chains: Effect of anisotropy

Linear chains: Anisotropic friction coefficients

Linear chains are anisotropic

Anisotropic friction coefficients: β‖k , β⊥k

Random orientations (Brownian motion)

βk =3β

‖kβ⊥k

β⊥k + 2β‖k

Does the diffusive flux to a monomer have differentperpendicular and parallel components (anisotropicfluxes)?

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Linear chains: Effect of anisotropy

Diffusive flux to a (spherical) monomer (3)

If the monomer is considered a rotation solid, the rotationaxis breaks rotational symmetry

(a) (b)

Rotation (symmetry)axis

Rotation (symmetry)axis

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Linear chains: Effect of anisotropy

Diffusive flux to a (spherical) monomer (2)

Perpendicular collision rate: diffusive flux perpendicular tothe rotation axis

K⊥1 =

∫S

J · s⊥dS =

∫S

J⊥dS = π2DgR1ρ∞

Parallel collision rate: molecular flux parallel to thesymmetry axis

K ‖1 =

∫S

J · s‖dS =

∫S

J‖dS = 2πDgR1ρ∞

Explicit calculation confirms K⊥1 6= K ‖

1Anisotropic shielding factors

η⊥k =β⊥kβ1

=K⊥

k

kK⊥1

, η‖k =

β‖k

β1=

K ‖k

kK ‖1

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Numerical results

Concentration field: dimer and 8-mer

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Numerical results

Parallel (axial) and perpendicular (radial) diffusive flux: dimer

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Numerical results

Parallel (axial) and perpendicular (radial) diffusive flux: 8-mer

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Numerical results

Monomer shielding factor in a linear chain (1)

Comparison with previous calculations (linear chains,continuum regime)

Analytical solutions of the velocity field for steady-stateviscous flow (Stokes)

Happel and Brenner (1991): dimerFilippov (2000): arbitrary aggregates of k spheres

Extrapolated experimental data: Dahneke (1982)Creeping flow coupled to Darcy flow within the porousaggregate

Vainshtein, Shapiro, and Gutfinger (2004)Garcia-Ybarra, Castillo, and Rosner (2006)

and many others . . .

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Numerical results

Monomer shielding factor in a linear chain (2)

Filippov Happel Brenner Dahneke Collision Rate

β2/β1 0.692 0.694β3/β1 0.569 0.574β4/β1 0.507 0.507β5/β1 0.461 0.463β8/β1 0.390 0.389

β‖2/β1 0.645 0.639 0.633

β‖8/β1 0.324 0.313

β⊥2 /β1 0.716 0.719 0.725β⊥8 /β1 0.435 0.434 0.428

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Numerical results

Anisotropic friction coefficient

0 10 20 30 40 50 60

0.2

0.3

0.4

0.5

0.6

n

ββ n||ββ 1

Diffusion simulationsFitVainshtein et al.

0 10 20 30 40 50 60

0.3

0.4

0.5

0.6

0.7

n

ββ n⊥⊥ββ 1

Diffusion simulationsFitVainshtein et al.

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Numerical results

Isotropic friction coefficient, Mobility radius

0 10 20 30 40 50 60

0.3

0.4

0.5

0.6

0.7

n

ββ nββ 1

Diffusion simulations

ββn from ββn|| and ββn

⊥⊥

0 10 20 30 40 50 602

46

810

1214

n

r nr 1

βk obtained from β‖k and β⊥k for random aggregate

orientationsIdeal clusters: mobility radius Rk/R1 = k

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Shielded Langevin equations

Monomer Langevin equations of motion (1)

3d equations of motion for the i-th monomer in ak -monomer linear chain

m1r i = Fi − β1im1r i + Wi(t)

Intra-chain isotropic friction coefficient β1i

β1i

β1=

K1i

K1≡ η1i ; ηk =

1k

∑i=1,k

η1i

Steady-state collision rate K1i on the i-monomer

Fluctuation Dissipation Theorem

〈W ji (t)W

j ′

i ′ (t′)〉 = Γiδii ′δjj ′δ(t − t ′)

Γi = 2β1im1kBT = 2η1iβ1m1kBT

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Shielded Langevin equations

Langevin Dynamics: Diffusion coefficient of a linear chain (1)

Mean-square displacement of chains: k = 5, 8 monomers

limt→∞

〈δR2CM(t)〉 = 6Dk t

0 20 40 60 80 100

ββ1t

⟨⟨δδr C

M2

⟩⟩d 12

025

5075

100

125

n=8n=5Linear fit for n=8Linear fit for n=5

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Shielded Langevin equations

Langevin simulations: Diffusion coefficient of a linear chain (2)

Ratios of diffusion coefficients

Collision Rate Langevin simulations

D5/D1 0.432 0.428

D8/D1 0.321 0.319

Equivalent descriptionsAggregate diffusion in terms of an average monomershielding factor, Fluctuation Dissipation Theorem applies tothe whole aggregateIndividual monomer shielding factor, Fluctuation DissipationTheorem applies to each monomer in the aggregate

Motivation Dynamics of linear chains Langevin simulations Revisited Conclusions

Conclusions

Importance of the shielding factor of a monomer in anaggregate

Active surface may be a useful concept

Diffusion and friction coefficients may be obtained from thecalculation of the molecular collision rate to an aggregate

Calculated coefficients in reasonable agreement withprevious theoretical calculations

Approach is based on mass transfer only, momentumtransfer is treated approximately

Not clear whether this approach may be coupled tosimulations of aggregate formation by Langevin dynamics