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).

Non ideal particles and aggregates in turbulenceLecce, June 7-9, 2012

http://www.slideshare.net/connaughtonc arXiv:1205.4445

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...


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

K (m1,m2) = 1.

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 (c.f. Kraichnan 1967)

Dim. analysis gives exponent (β + 3)/2 but not amplitude.Amplitude vanishes when |ν − µ| = 1 so Hayakawa’ssolution 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?

This is a physically relevant question. Eg differentialsedimentation:

K (m1,m2) =

(m

131 + m

132

)2 ∣∣∣∣m 231 −m

232

∣∣∣∣ .http://www.slideshare.net/connaughtonc arXiv:1205.4445

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 tometres. Collisions results in aggregation at low energy andfragmentation at high energy.dynamic equilibrium: clumping vs collisional fragmentationwith fragmentation acting as effective source and sink?


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 arXiv:1205.4445

What about the non-local case?

We could obtain an asymptotic solution for the more generalkernel 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.We can at least see what the nonlocal stationary state lookslike using an alternative semi-analytic approach.


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

(1 + λ)(mνMµ + mµMν)

starting from

N1 =2 J

(1 + λ)(Mµ +Mν).

Solution given by

(Mµ,Mν) = argmin(Mµ,Mν)

(Mµ −

M∑0

mµNm

)2

+

(Mν −

M∑0

mνNm

)2


What does the nonlocal stationary state look like?

ν = 3/4, µ = −3/4, M = 104,

λ = 5 × 10−4.

Effective cut-off, M, generated bythe fragmentation as before.Decays exponentially for largecluster sizes.Does not look like a simple powerlaw for small/intermediate masses.We have arguments for thefunctional form ...


Dynamics in the nonlocal regime

Our iterative procedure computes the stationary state directly.What about the dynamics?

ν = 3/4, µ = −3/4, M = 104, λ = 5 × 10−4.

Dynamical simulations starting from empty system andadding monomers at rate J do not seem to ever reach theexpected stationary state.Numerics suggest the long-time kinetics are oscillatory!


Instability of the nonlocal stationary state

ν = 3/4, µ = −3/4, M = 104.

Our analysis computes thestationary state directly but makesno statement about its stability.Because we can compute the exactstationary state, we can perform alinear stability analysis.This suggests that if the cascade isnonlocal then for M large enough,the stationary state is unstable.


Bifurcation diagram

Stability diagram for the kernel

K (m1,m2) = 12

(( m1m2

)ν+

( m2m1

)ν)for different values of ν and cut-off, M.

[Disclaimer: results are for hard cut-off!]

A rudimentary numerical bifurcationtraces the origin of the oscillatorybehaviour to a Hopf bifurcation ofthe stationary state as M isincreased.Structure of the instability is fairlycomplicated however.None of this happens for localcascades underlining the necessityto think about the locality of cascadedynamics in such problems.


Summary: the story so far

The transfer of mass through the space of cluster sizes incluster-aggregation with a source of small clusters and asink of large clusters is analogous to the transfer of energythrough scales in Richardson’s view of turbulence.The intuitive idea that fragmentation can act as an effectivesource and sink in an isolated system is supported bytheoretical analysis at the mean-field level.Locality of the cascade is an important consideration:several interesting physical examples are non-local.There is strong evidence that nonlocal stationary cascadesare unstable. Large time behaviour of the cascade isoscillatory.


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 SE

ν = 3/4, µ = −3/4, M = 104.

Stationary 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 arXiv:1205.4445

cluster aggregation with complete collisional fragmentation

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