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 turbulence
Lecce, June 7-9, 2012

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

Introduction to cluster-cluster aggregation (CCA)

Many particles of one
material dispersed in
another.
Transport: diffusive,
advective, ballistic...
Particles stick together on
contact.
Applications: surface and colloid physics, atmospheric
science, biology, cloud physics, astrophysics...


Mean-ﬁeld model: Smoluchowski’s kinetic equation
Cluster size distribution, Nm (t), satisﬁes the kinetic equation :
Smoluchowski equation :

m
∂Nm (t) 1
= dm1 dm2 K (m1 , m2 )Nm1 Nm2 δ(m − m1 − m2 )
∂t 2 0
M−m
− dm1 dm2 K (m, m1 )Nm Nm1 δ(m2 − m − m1 )
0
M
− Nm dm1 K (m, m1 )Nm1
M−m
J
+ δ(m − m0 )
m0

Source of monomers
Removal 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 )
µ ν
K (m1 , m2 ) ∼ m1 m2 m1 m2 .

Clearly β = µ + ν. Model kernel:

1 µ ν ν µ
K (m1 , m2 ) = (m1 m2 + m1 m2 )
K (m1 , m2 ) = 1.
2

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

J (1 − (ν − µ)2 ) cos((ν − µ) π/2) − β+3
Nm = m 2 . (1)
2π
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’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 m
interacts “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 differential
sedimentation:
1 1 2 2 2
K (m1 , m2 ) = 3 3
m1 + m2 m1 − m2 .
3 3


An aggregation-fragmentation problem from planetary
science: Saturn’s rings
Brilliantov, Bodrova and Krapivsky: in preparation (2012)

small particles of ice, ranging in size from micrometres to
metres. Collisions results in aggregation at low energy and
fragmentation at high energy.
dynamic equilibrium: clumping vs collisional fragmentation
with 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).
m
∂Nm (t) 1
= dm1 K (m − m1 , m1 )Nm−m1 Nm1
∂t 2 0
∞
− (1 + λ) Nm dm1 K (m, m1 )Nm1
0
∞ ∞
∂N1 (t)
= −N1 dm1 K (1, m1 )Nm1 + λN1 dm1 m1 K (1, m1 )Nm1
∂t 0 0
∞
1
+ dm1 dm2 (m1 + m2 ) K (m1 , m2 )Nm1 Nm2
2 0

λ is a relative fragmentation rate.


An exact solution

Collision kernel is worked out to be
1 1
−1 −1
K (m1 , m2 ) = (m1 + m2 )2
3 3
m1 + m2 . (2)

Brilliantov et al. argue that this can be replaced with simpler
kernel of the same degree of homogeneity:
β
K (m1 , m2 ) = (m1 m2 ) 2 with β = 1 .
6 (3)

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

λ2 β+3
Nm ∼ A exp(− m) m− 2 .
4
Kolmogorov cascade with effective source and sink provided by
fragmentation. 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 by
collisions are removed from the system. Monomers are
supplied to the system at a fixed rate, J.
Rate equations are the same except for a simplified
equation for monomer density:

m
∂Nm (t) 1
= dm1 K (m − m1 , m1 )Nm−m1 Nm1
∂t 2 0
∞
− (1 + λ) Nm dm1 K (m, m1 )Nm1 + J δm,1
0
β
Exact solution for K (m1 , m2 ) = (m1 m2 ) 2 :

J λ2 β+3
Nm ∼ exp(− m) m− 2 .
2π 4
Analogous behaviour to Brilliantov’s.

What about the non-local case?

We could obtain an asymptotic solution for the more general
µ ν ν µ
kernel K (m1 , m2 ) = 1 (m1 m2 + m1 m2 ):
2

2
λ 1+ν−µ β+3
Nm ∼ A exp − m m− 2
2

but solution fails as ν − µ → 1 (probably A → 0?) : cascade
becomes non-local.
We can at least see what the nonlocal stationary state looks
like using an alternative semi-analytic approach.


µ ν ν µ
Solution for K (m1 , m2 ) = 1 (m1 m2 + m1 m2 )
2

M M
Mµ = m µ Nm Mν = mν Nm
0 0

Masses are discrete so can solve exactly for Nm iteratively:
m−1
˜ m1 =1 K (m1 , m − m1 )Nm1 Nm−m1
Nm (Mµ , Mν ) = )
(1 + λ)(mν Mµ + mµ Mν

starting from
2J
N1 = ).
(1 + λ)(Mµ + Mν
Solution given by

M 2 M 2
µ˜ ν˜
(Mµ , Mν ) = argmin Mµ − m Nm + Mν − m Nm
(Mµ ,Mν ) 0 0


What does the nonlocal stationary state look like?

Effective cut-off, M, generated by
the fragmentation as before.
Decays exponentially for large
cluster sizes.
Does not look like a simple power
law for small/intermediate masses.
We have arguments for the
functional form ...
ν = 3/4, µ = −3/4, M = 104 ,

λ = 5 × 10−4 .


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 and
adding monomers at rate J do not seem to ever reach the
expected stationary state.
Numerics suggest the long-time kinetics are oscillatory!

Instability of the nonlocal stationary state

Our analysis computes the
stationary state directly but makes
no statement about its stability.
Because we can compute the exact
stationary state, we can perform a
linear stability analysis.
This suggests that if the cascade is
nonlocal then for M large enough,
ν = 3/4, µ = −3/4, M = 104 . the stationary state is unstable.


Bifurcation diagram

A rudimentary numerical bifurcation
traces the origin of the oscillatory
behaviour to a Hopf bifurcation of
the stationary state as M is
increased.
Structure of the instability is fairly
complicated however.
Stability diagram for the kernel None of this happens for local
K (m1 , m2 ) = 1
2
m1
m2
ν
+
m2
m1
ν
cascades underlining the necessity
for different values of ν and cut-off, M. to think about the locality of cascade
[Disclaimer: results are for hard cut-off!] dynamics in such problems.


Summary: the story so far

The transfer of mass through the space of cluster sizes in
cluster-aggregation with a source of small clusters and a
sink of large clusters is analogous to the transfer of energy
through scales in Richardson’s view of turbulence.
The intuitive idea that fragmentation can act as an effective
source and sink in an isolated system is supported by
theoretical 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 cascades
are unstable. Large time behaviour of the cascade is
oscillatory.


Non-local approximation to Smoluchowski Eqn

Write the Smoluchowski equation as:
m/2
∂Nm (t)
= dm1 [K (m − m1 , m1 )Nm−m1 − K (m, m1 )Nm ] Nm1
∂t 0
M
J
− Nm dm1 K (m, m1 )Nm1 + δ(m − m0 )
m/2 m0

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

∂Nm (t) 1 ∂
= − mν Mµ+1 + mµ Mν+1 Nm
∂t 2 ∂m
1 J
− [(mν Mµ + mµ Mν )] Nm + δ(m − m0 )
2 m0

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


Self-consistent solution of the nonlocal SE
Stationary solution of nonlocal kinetic
equation (Horvai et al 2008):

α −γ
Nm = C exp m m−ν
γ

where C is a constant of integration, γ =
ν − µ − 1 and α = Mν /Mµ+1 .
ν = 3/4, µ = −3/4, M = 104 .

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):
2 J γ log(M) m−γ −ν
Nm = M m .
M
Note Nm → 0 as Nm → ∞!

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