1. 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 Forum
Warwick Centre for Complexity Science
11 January 2012
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2. 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...
This talk:
Mean-field theory of aggregation with input of monomers (part
1) and collision-induced fragmentation (part 2).
http://www.slideshare.net/connaughtonc Aggregation with Collisional Fragmentation

3. 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.
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4. 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 ) = (m m + m1 m2 )
2 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.
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5. The importance of locality
J (1 − (ν − µ)2 ) cos((ν − µ) π/2) − β+3
Nm = m 2 .
2π
Dimensional argument gives exponent (β + 3)/2 but not
amplitude.
Note that amplitude vanishes when |ν − µ| = 1
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?
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6. 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 ) =
m ν Mµ + m µ Mν
starting from
2J
N1 = .
Mµ + Mν
Solution given by
M 2 M 2
µ ν
(Mµ , Mν ) = argmin Mµ − m Nm + Mν − m Nm
(Mµ ,Mν ) 0 0
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7. 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 .
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8. 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 .
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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9. Instability of the solution and persistent oscillatory
kinetics
This analysis computes the
stationary state directly and makes
no statement about its stability.
Dynamical numerical simulations
suggest that for M large enough,
nonlocal stationary state is unstable.
Long-time behaviour of Smoluchowski equation with source
and sink seems to be oscillatory for |ν − µ| > 1!
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10. 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
metre.
dynamic equilibrium: clumping vs collisional fragmentation.
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11. 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.
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12. 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?
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13. 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.
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14. What about non-locality?
µ ν ν µ
Asymptotic solution for 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.
Earlier iterative method can be adapted to calculate the
stationary state in the case ν − µ > 1.
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15. Partial solution of the nonlocal kinetic equation
Nonlocal kinetic equation with fragmentation
∂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
λ
− [(mν Mµ + mµ Mν )] Nm
2
Effective cut-off, M.
Intermediate masses:
m0 m M: same as before.
Large masses: m M:
Mµ+1
Nm ∼ exp −λ m m−ν
Mµ
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16. Instability of the nonlocal stationary state
Dynamical simulations starting from the exact nonlocal
stationary state again exhibit instability in the presence of
fragmentation.
Oscillatory kinetics are not a result of the "hard" cut-off
used previously and are not transient.
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17. Conclusions and ongoing work
Stationary nonlocal solutions of the Smoluchowski
equation have been presented in the case of aggregation
with a source and sink.
They are dynamically unstable and vanish as the cut-off is
removed.
A set of models with collisional fragmentation were studied
and it was found that collisional fragmentation can act as
an effective source and sink for monomers to produce
solutions which carry an effective mass flux.
Locality remains a important property for the models with
fragmentation
The original Brilliantov kernel may behave differently to the
simplified product version.
http://www.slideshare.net/connaughtonc Aggregation with Collisional Fragmentation