1. Nonequilibrium statistical mechanics of
cluster-cluster aggregation
Colm Connaughton
Mathematics Institute and Centre for Complexity Science,
University of Warwick, UK
Collaborators: R. Ball (Warwick), P. Krapivsky (Boston), R. Rajesh (Chennai), T.
Stein (Reading), O. Zaboronski (Warwick).
Mathematics Colloquium
University of Warwick
20 May 2011
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

2. Aggregation phenomena : motivation
Many particles of one
material dispersed in
another.
Transport is diffusive or
advective.
Particles stick together on
contact.
Applications: surface physics, colloids, atmospheric science,
earth sciences, polymers, cloud physics.
This talk:
Today we will focus on simple theoretical models of the
statistical dynamics of such systems.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

3. Simplest model of clustering: coalescing random
walks
Particles perform random walks on a
lattice.
Multiple occupancy of lattice sites is
allowed.
Particles on the same site merge
with probability rate k : A + A → A.
A source of particles may or may not
be present.
Cartoon of dynamics in No equilibrium - lack of detailed
1-D with k → ∞. balance.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

4. Mean field description
Equation for the average density, N(x, t), of particles:
∂t N = D ∆ N − 1 k N (2) + J
2
k - reaction rate, D - diffusion rate, J - injection rate, N (2) -
density of same-site pairs.
Mean field assumption:
No correlations between particles: N (2) ∝ N 2 :
Spatially homogeneous case, N(x, t) = N(t).
Mean field rate equation:
dN
= − 2 k N2 + J
1
N(0) = N0
dt
2 N0 1
J = 0 : N(t) = ∼ as t → ∞
2 + k N0 t kt
2J
J = 0 : N(t) ∼ as t → ∞
k
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

5. A more sophisticated model of clustering:
size-dependent coalescence
A better model would track the sizes distribution of the clusters:
Am1 + Am2 → Am1 +m2 .
Probability rate of particles sticking should be a function,
K (m1 , m2 ), of the particle sizes (bigger particles typically
have a bigger collision cross-section).
Micro-physics of different applications is encoded in
K (m1 , m2 ) - the collision kernel - which is often a
homogeneous function:
K (am1 , am2 ) = aλ K (m1 , m2 )
Given the kernel, objective is to determine the cluster size
distribution, Nm (t), which describes the average number of
clusters of size m as a function of time.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

6. The Smoluchowski equation
Assume the cloud is statistically homogeneous and well-mixed
so that there are no spatial correlations.
Cluster size distribution, Nm (t), satisfies the kinetic equation :
Smoluchowski equation :
∞
∂Nm (t)
= dm1 dm2 K (m1 , m2 )Nm1 Nm2 δ(m − m1 − m2 )
∂t 0
∞
− 2 dm1 dm2 K (m, m1 )Nm Nm1 δ(m2 − m − m1 )
0
+ J δ(m − m0 )
Notation: In many applications kernel is homogeneous:
K (am1 , am2 ) = aλ K (m1 , m2 )
µ ν
K (m1 , m2 ) ∼ m1 m2 m1 m2 .
Clearly λ = µ + ν.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

7. Self-similar Solutions of the Smoluchowski equation
In many applications kernel
is a homogeneous function:
K (am1 , am2 ) = aλ K (m1 , m2 )
Resulting cluster size
distributions often exhibit
self-similarity.
Self-similar solutions have the form
m
Nm (t) ∼ s(t)−2 F (z) z=
s(t)
where s(t) is the typical cluster size. The scaling function, F (z),
determines the shape of the cluster size distribution.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

8. Stationary solutions of the Smoluchowski equation
with a source of monomers
Add monomers at rate, J.
Suppose particles having
m > M are removed.
Stationary state is obtained
for large t.
Stationary state is a
balance between injection
and removal. Constant
mass flux in range [m0 , M]
With some work exact stationary solution can be found:
√ λ+3
Nm (t) ∼ CK J m− 2 as t → ∞.
Require mass flux to be local: |µ − ν| < 1.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

9. Violation of mass conservation: the gelation transition
Microscopic dynamics conserve mass: Am1 + Am2 → Am1 +m2 .
Smoluchowski equation formally
conserves the total mass,
∞
M1 (t) = 0 m N(m, t) dm.
However for λ > 1:
∞
M1 (t) < m N(m, 0) dm t > t ∗ .
0
(Lushnikov [1977], Ziff [1980])
M1 (t) for K (m1 , m2 ) = (m1 m2 ) 3/4
. Mean field theory violates mass
conservation!!!
Best studied by introducing cut-off, M, and studying limit
M → ∞. (Laurencot [2004])
What is the physical interpretation?
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

10. Physical interpretation of the gelation transition
As λ increases, the aggregation rate, K (m1 , m2 ), increases
more rapidly as a function of m. If λ > 1 the absorbtion of
small clusters by large ones becomes a runaway process.
Clusters of arbitrarily large size (gel) are generated in a
finite time, t∗ , known as the gelation time.
Loss of mass to the gel component corresponds to a finite
mass flux as m → ∞.
Finite time singularities generally pose a problem for
physics: for gelling systems Smoluchowski equation
usually only describes intermediate asymptotics of Nm (t).
In qualitative agreement with experiments in crosslinked
polymer aggregation (Lushnikov et al. [1990]).
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

11. Instantaneous gelation
Asymptotic behaviour of the kernel controls the aggregation of
small clusters and large:
µ ν
K (m1 , m2 ) ∼ m1 m2 m1 m2 .
µ + ν = λ so that gelation always occurs if ν is big enough.
Instantaneous Gelation
If ν > 1 then t ∗ = 0. (Van Dongen & Ernst [1987])
Worse: gelation is complete: M1 (t) = 0 for t > 0.
Instantaneously gelling kernels cannot describe even the
intermediate asymptotics of any physical problem.
Mathematically pathological?
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

12. Droplet coagulation by gravitational settling: a puzzle
The process of gravitational
settling is important in the
evolution of the droplet size
distribution in clouds and
the onset of precipitation.
Droplets are in the Stokes
regime → larger droplets
fall faster merging with
slower droplets below them.
Some elementary calculations give the collision kernel
1 1 2 2
K (m1 , m2 ) ∝ (m1 + m2 )2 m1 − m2
3 3 3 3
ν = 4/3 suggesting instantaneous gelation but model seems
reasonable in practice. How is this possible?
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

13. Instantaneous gelation in the presence of a cut-off
With cut-off, M, regularized
∗
gelation time, tM , is clearly
identifiable.
∗
tM decreases as M increases.
Van Dongen & Ernst recovered in
limit M → ∞.
3
2 3/2
M(t) for K (m1 , m2 ) = m1 + m2 .
Decrease of ∗
tM as M is very slow. Numerics and heuristics
suggest:
∗ 1
tM ∼ .
log M
This suggests such models are physically reasonable.
Consistent with related results of Krapivsky and Ben-Naim
and Krapivsky [2003] on exchange-driven growth.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

14. "Instantaneous" gelation with a source of monomers
A stationary state is reached in the regularised systems if a
source of monomers is present (Horvai et al [2007]).
Stationary state has the
asymptotic form for M 1:
J log M ν−1 m1−ν −ν
Nm = M m .
M
Stretched exponential for small
m, power law for large m.
Stationary state (theory vs numerics)
Stationary particle density:
for ν = 3/2.
√ 1−ν
J M − MM J
N= ∼ as M → ∞.
M log M ν−1 log M ν−1
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

15. Approach to Stationary State is non-trivial
Numerics indicate that the
approach to stationary state is
non-trivial.
Collective oscillations of the total
density of clusters.
Total density vs
1+
time for
1+
Numerical measurements of the
K (m1 , m2 ) = m1 + m2 .
Q-factor of these oscillations
suggests that they are long-lived
transients. Last longer as M
increases.
Heuristic explanation in terms of
“reset” mechanism.
“Q-factor" for ν = 0.2.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

16. The addition model: a completely nonlocal limit
The extreme case of nonlocal aggregation is the Addition
model (Brilliantov & Krapivsky, 1992).
Clusters can only grow by absorption of monomers:
M
˙ 2
N1 = −2 γ N1 − κ(m)N1 Nm + J,
m=2
˙
Nm = κ(m − 1)N1 Nm−1 − κ(m)N1 Nm 2 ≤ m ≤ M,
with κ(m) ∼ mν . Instantaneously gelling for ν > 1.
Stationary state:
√ J
Nm = γ m−ν .
M +1
Large mass behaviour agrees with Smoluchowski model if
we choose effective monomer reaction rate γ = M −1 .
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

17. Oscillatory approach to stationary state in the addition
model
Addition model with effective monomer reaction rate captures
the essential features of the full problem when ν > 1.
It can be (almost) linearised by a change of variables:
dτ (t) = N1 (t)dt,
τ (0) = 0.
Look for solution of the resulting monomer equation of the form:
J
N1 (τ ) = (1 + Aeζτ ),
γ (M + 1)
where Re ζ < 0. Asymptotics are tractable for ν = 1 + .
Find: |Re ζ| = O( ), Im ζ = O(1). This implies long-lived
oscillatory transient.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

18. Summary and conclusions
Aggregation phenomena exhibit a rich variety of
non-equilibrium statistical dynamics.
If the aggregation rate of large clusters increases quickly
enough as a function of cluster size, clusters of arbitrarily
large size can be generated in finite time (gelation).
Kernels with ν > 1 which, mathematically speaking,
undergo complete instantaneous gelation still make sense
as physical models provided a cut-off is included since the
approach to the singularity is logarithmically slow as the
cut-off is removed.
Approach to stationary state for regularised system with a
source of monomers is interesting when ν > 1: surprisingly
long-lived oscillatory transients.
Some analytical insight can be obtained from the addition
model with renormalised monomer reaction rate.
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

19. References
R.C. Ball, C. Connaughton, T.H.M. Stein and O. Zaboronski, "Instantaneous Gelation in Smoluchowski’s
Coagulation Equation Revisited", preprint arXiv:1012.4431v1 [cond-mat.stat-mech], 2010
C. Connaughton, R. Rajesh, and O. Zaboronski "On the Non-equilibrium Phase Transition in
Evaporation-Deposition Models", J. Stat. Mech.-Theor. E., P09016, 2010
C. Connaughton and J. Harris, "Scaling properties of one-dimensional cluster-cluster aggregation with Levy
diffusion", J. Stat. Mech.-Theor. E., P05003, 2010
C. Connaughton and P.L. Krapivsky "Aggregation-fragmentation processes and decaying three-wave
turbulence ", Phys. Rev. E 81, 035303(R), 2010
C. Connaughton, R. Rajesh and O. Zaboronski, "Constant Flux Relation for diffusion limited cluster–cluster
aggregation", Phys. Rev E 78, 041403, 2008
C. Connaughton,R. Rajesh and O. Zaboronski , "Constant Flux Relation for Driven Dissipative Systems",
Phys. Rev. Lett. 98, 080601 (2007)
C. Connaughton,R. Rajesh and O. Zaboronski , "Cluster-Cluster Aggregation as an Analogue of a Turbulent
Cascade : Kolmogorov Phenomenology, Scaling Laws and the Breakdown of self-similarity", Physica D 222,
1-2 97-115 (2006)
C. Connaughton R. Rajesh and O.V. Zaboronski, "Breakdown of Kolmogorov Scaling in Models of Cluster
Aggregation", Phys. Rev. Lett. 94, 194503 (2005)
C. Connaughton R. Rajesh and O.V. Zaboronski, "Stationary Kolmogorov solutions of the Smoluchowski
aggregation equation with a source term", Phys. Rev E 69 (6): 061114, 2004
http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech