Cluster-cluster aggregation with (complete) collisional fragmentation

Cluster-cluster aggregation with (complete)collisional fragmentation

Colm Connaughton

Mathematics Institute and Centre for Complexity Science,University of Warwick, UK

Collaborators: R. Rajesh (Chennai), O. Zaboronski (Warwick).

Complexity ForumWarwick Centre for Complexity Science

11 January 2012

http://www.slideshare.net/connaughtonc Aggregation with Collisional Fragmentation

Introduction to cluster-cluster aggregation (CCA)

Many particles of onematerial dispersed inanother.Transport: diffusive,advective, ballistic...Particles stick together oncontact.

Applications: surface and colloid physics, atmosphericscience, biology, cloud physics, astrophysics...

This talk:Mean-field theory of aggregation with input of monomers (part1) and collision-induced fragmentation (part 2).


Mean-field model: Smoluchowski’s kinetic equation

Cluster size distribution, Nm(t), satisfies the kinetic equation :

Smoluchowski equation :

∂Nm(t)∂t

=12

∫ m

0dm1dm2K (m1,m2)Nm1Nm2δ(m −m1 −m2)

−∫ M−m

0dm1dm2K (m,m1)NmNm1δ(m2 −m −m1)

− Nm

∫ M

M−mdm1K (m,m1)Nm1

+J

m0δ(m −m0)

Source of monomersRemoval of clusters larger than cut-off, M.


Stationary state of CCA with a source and sink

Kernel is often homogeneous:

K (am1,am2) = aβ K (m1,m2)

K (m1,m2) ∼ mµ1 mν

2 m1�m2.

Clearly β = µ+ ν. Model kernel:

K (m1,m2) =12(mµ

1 mν2 + mν

1mµ2 )

Stationary state for t →∞, m0 � m� M (Hayakawa 1987):

Nm =

√J (1− (ν − µ)2) cos((ν − µ)π/2)

2πm−

β+32 . (1)

Describes a cascade of mass from source at m0 to sink at M.


The importance of locality

Nm =

√J (1− (ν − µ)2) cos((ν − µ)π/2)

2πm−

β+32 .

Dimensional argument gives exponent (β + 3)/2 but notamplitude.Note that amplitude vanishes when |ν − µ| = 1Hayakawa’s solution exists only for |ν − µ| < 1.If |ν − µ| < 1, cascade is local: a cluster of size m interacts“mostly" with clusters of comparable size.If |ν − µ| > 1, cascade is nonlocal: a cluster of size minteracts “mostly" with the largest clusters in the system.

Question:What replaces Eq.(1) in the nonlocal case |ν − µ| > 1?


Solution for K (m1,m2) =12(m

µ1mν

2 + mν1mµ

2)

Mµ =M∑0

mµNm Mν =M∑0

mνNm

Masses are discrete so can solve exactly for Nm iteratively:

Nm(Mµ,Mν) =

∑m−1m1=1 K (m1,m −m1)Nm1Nm−m1

mνMµ + mµMν

starting from

N1 =2 J

Mµ +Mν.

Solution given by

(Mµ,Mν) = argmin(Mµ,Mν)

(Mµ −

M∑0

mµNm

)2

+

(Mν −

M∑0

mνNm

)2


Non-local approximation to Smoluchowski Eqn

Write the Smoluchowski equation as:

∂Nm(t)∂t

=

∫ m/2

0dm1 [K (m −m1,m1)Nm−m1 − K (m,m1)Nm]Nm1

− Nm

∫ M

m/2dm1K (m,m1)Nm1 +

Jm0

δ(m −m0)

Nonlocal assumption: major contribution to first integrand isfrom the region where m1 � m. Taylor expand:

∂Nm(t)∂t

= −12∂

∂m[(

mνMµ+1 + mµMν+1)

Nm]

− 12[(mνMµ + mµMν)] Nm +

Jm0

δ(m −m0)

Obtain linear PDE for Nm but coefficients are moments of Nm.


Self-consistent solution of the nonlocal SEStationary solution of nonlocal kineticequation (Horvai et al 2008):

Nm = C exp[α

γm−γ

]m−ν

where C is a constant of integration, γ =ν − µ− 1 and α =Mν/Mµ+1.

α is obtained by solving the consistency condition

α =Mν(α)/Mµ+1(α)

C is then fixed by global mass balance (Ball et al 2011):

Nm =

√2 J γ log(M)

MMm−γ

m−ν .

Note Nm → 0 as Nm →∞!http://www.slideshare.net/connaughtonc Aggregation with Collisional Fragmentation

Instability of the solution and persistent oscillatorykinetics

This analysis computes thestationary state directly and makesno statement about its stability.Dynamical numerical simulationssuggest that for M large enough,nonlocal stationary state is unstable.

Long-time behaviour of Smoluchowski equation with sourceand sink seems to be oscillatory for |ν − µ| > 1!


An aggregation-fragmentation problem from planetaryscience: Saturn’s rings

Brilliantov, Bodrova and Krapivsky: in preparation (2012)

small particles of ice, ranging in size from micrometres tometre.dynamic equilibrium: clumping vs collisional fragmentation.


Complete fragmentation - Brilliantov’s Model

Very complex kinetics in general. Assume:Eagg = Efrag = const.All clusters have the same kinetic energy on average.Fragmentations are complete (produce only monomers).

∂Nm(t)∂t

=12

∫ m

0dm1K (m −m1,m1)Nm−m1Nm1

− (1 + λ)Nm

∫ ∞0

dm1K (m,m1)Nm1

∂N1(t)∂t

= −N1

∫ ∞0

dm1K (1,m1)Nm1 + λN1

∫ ∞0

dm1m1 K (1,m1)Nm1

+12

∫ ∞0

dm1dm2 (m1 + m2)K (m1,m2)Nm1Nm2

λ is a relative fragmentation rate.


An exact solution

Collision kernel is worked out to be

K (m1,m2) = (m131 + m

132 )

2√

m−11 + m−1

2 . (2)

Brilliantov et al. argue that this can be replaced with simplerkernel of the same degree of homogeneity:

K (m1,m2) = (m1m2)β2 with β = 1

6 . (3)

Exact asymptotics for λ� 1 and 1� m� λ−2:

Nm ∼ A exp(−λ2

4m)m−

β+32 .

Kolmogorov cascade with effective source and sink provided byfragmentation. But (3) is local (ν − µ = 0) whereas (2) is not(ν − µ = 7/6 > 1). Does this matter?


Simplified fragmentation model with source

Introduce model in which the monomers produced bycollisions are removed from the system. Monomers aresupplied to the system at a fixed rate, J.Rate equations are the same except for a simplifiedequation for monomer density:

∂Nm(t)∂t

=12

∫ m

0dm1K (m −m1,m1)Nm−m1Nm1

− (1 + λ)Nm

∫ ∞0

dm1K (m,m1)Nm1 + J δm,1

Exact solution for K (m1,m2) = (m1m2)β2 :

Nm ∼√

J2π

exp(−λ2

4m)m−

β+32 .

Analogous behaviour to Brilliantov’s.http://www.slideshare.net/connaughtonc Aggregation with Collisional Fragmentation

What about non-locality?

Asymptotic solution for K (m1,m2) =12(m

µ1 mν

2 + mν1mµ

2 ):

Nm ∼ A exp

[−(λ

2

) 21+ν−µ

m

]m−

β+32

but solution fails as ν − µ→ 1 (probably A→ 0?) : cascadebecomes non-local.Earlier iterative method can be adapted to calculate thestationary state in the case ν − µ > 1.


Partial solution of the nonlocal kinetic equation

Nonlocal kinetic equation with fragmentation

∂Nm(t)∂t

= −12∂

∂m[(

mνMµ+1 + mµMν+1)

Nm]

− 12[(mνMµ + mµMν)] Nm +

Jm0

δ(m −m0)

− λ

2[(mνMµ + mµMν)] Nm

Effective cut-off, M.Intermediate masses:m0 � m� M: same as before.Large masses: m� M:

Nm ∼ exp[−λMµ+1

Mµm]

m−ν


Instability of the nonlocal stationary state

Dynamical simulations starting from the exact nonlocalstationary state again exhibit instability in the presence offragmentation.Oscillatory kinetics are not a result of the "hard" cut-offused previously and are not transient.


Conclusions and ongoing work

Stationary nonlocal solutions of the Smoluchowskiequation have been presented in the case of aggregationwith a source and sink.They are dynamically unstable and vanish as the cut-off isremoved.A set of models with collisional fragmentation were studiedand it was found that collisional fragmentation can act asan effective source and sink for monomers to producesolutions which carry an effective mass flux.Locality remains a important property for the models withfragmentationThe original Brilliantov kernel may behave differently to thesimplified product version.


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Cluster-cluster aggregation with (complete) collisional fragmentation