Nonequilibrium statistical mechanics ofcluster-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 ColloquiumUniversity of Warwick

20 May 2011

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Aggregation phenomena : motivation

Many particles of onematerial dispersed inanother.Transport is diffusive oradvective.Particles stick together oncontact.

Applications: surface physics, colloids, atmospheric science,earth sciences, polymers, cloud physics.

This talk:Today we will focus on simple theoretical models of thestatistical dynamics of such systems.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Simplest model of clustering: coalescing randomwalks

Cartoon of dynamics in1-D with k →∞.

Particles perform random walks on alattice.Multiple occupancy of lattice sites isallowed.Particles on the same site mergewith probability rate k : A + A→ A.A source of particles may or may notbe present.No equilibrium - lack of detailedbalance.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Mean field description

Equation for the average density, N(x , t), of particles:

∂tN = D ∆ N − 12 k N(2) + J

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) ∝ N2:Spatially homogeneous case, N(x , t) = N(t).

Mean field rate equation:

dNdt

= − 12 k N2 + J N(0) = N0

J = 0 : N(t) =2 N0

2 + k N0 t∼ 1

k tas t →∞

J 6= 0 : N(t) ∼√

2 Jk

as t →∞

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

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 typicallyhave a bigger collision cross-section).Micro-physics of different applications is encoded inK (m1,m2) - the collision kernel - which is often ahomogeneous function:

K (am1,am2) = aλ K (m1,m2)

Given the kernel, objective is to determine the cluster sizedistribution, Nm(t), which describes the average number ofclusters of size m as a function of time.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

The Smoluchowski equation

Assume the cloud is statistically homogeneous and well-mixedso that there are no spatial correlations.Cluster size distribution, Nm(t), satisfies the kinetic equation :

Smoluchowski equation :

∂Nm(t)∂t

=

∫ ∞0

dm1dm2K (m1,m2)Nm1Nm2δ(m −m1 −m2)

− 2∫ ∞

0dm1dm2K (m,m1)NmNm1δ(m2 −m −m1)

+ J δ(m −m0)

Notation: In many applications kernel is homogeneous:K (am1,am2) = aλ K (m1,m2)

K (m1,m2) ∼ mµ1 mν

2 m1�m2.

Clearly λ = µ+ ν.http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Self-similar Solutions of the Smoluchowski equation

In many applications kernelis a homogeneous function:

K (am1,am2) = aλ K (m1,m2)

Resulting cluster sizedistributions often exhibitself-similarity.

Self-similar solutions have the form

Nm(t) ∼ s(t)−2 F (z) z =m

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

Stationary solutions of the Smoluchowski equationwith a source of monomers

Add monomers at rate, J.Suppose particles havingm > M are removed.Stationary state is obtainedfor large t .Stationary state is abalance between injectionand removal. Constantmass flux in range [m0,M]

With some work exact stationary solution can be found:

Nm(t) ∼ CK√

J m−λ+3

2 as t →∞.

Require mass flux to be local: |µ− ν| < 1.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Violation of mass conservation: the gelation transition

Microscopic dynamics conserve mass: Am1 + Am2 → Am1+m2 .

M1(t) for K (m1,m2) = (m1m2)3/4.

Smoluchowski equation formallyconserves the total mass,M1(t) =

∫∞0 m N(m, t) dm.

However for λ > 1:

M1(t) <∫ ∞

0m N(m,0) dm t > t∗.

(Lushnikov [1977], Ziff [1980])Mean field theory violates massconservation!!!

Best studied by introducing cut-off, M, and studying limitM →∞. (Laurencot [2004])What is the physical interpretation?

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Physical interpretation of the gelation transition

As λ increases, the aggregation rate, K (m1,m2), increasesmore rapidly as a function of m. If λ > 1 the absorbtion ofsmall clusters by large ones becomes a runaway process.Clusters of arbitrarily large size (gel) are generated in afinite time, t∗, known as the gelation time.Loss of mass to the gel component corresponds to a finitemass flux as m→∞.Finite time singularities generally pose a problem forphysics: for gelling systems Smoluchowski equationusually only describes intermediate asymptotics of Nm(t).In qualitative agreement with experiments in crosslinkedpolymer aggregation (Lushnikov et al. [1990]).

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Instantaneous gelation

Asymptotic behaviour of the kernel controls the aggregation ofsmall clusters and large:

K (m1,m2) ∼ mµ1 mν

2 m1�m2.

µ+ ν = λ so that gelation always occurs if ν is big enough.

Instantaneous GelationIf 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 theintermediate asymptotics of any physical problem.Mathematically pathological?

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Droplet coagulation by gravitational settling: a puzzle

The process of gravitationalsettling is important in theevolution of the droplet sizedistribution in clouds andthe onset of precipitation.Droplets are in the Stokesregime→ larger dropletsfall faster merging withslower droplets below them.

Some elementary calculations give the collision kernel

K (m1,m2) ∝ (m131 + m

132 )2

∣∣∣∣m 231 −m

232

∣∣∣∣ν = 4/3 suggesting instantaneous gelation but model seemsreasonable in practice. How is this possible?

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Instantaneous gelation in the presence of a cut-off

M(t) for K (m1,m2) = m321 + m3/2

2 .

With cut-off, M, regularizedgelation time, t∗M , is clearlyidentifiable.t∗M decreases as M increases.Van Dongen & Ernst recovered inlimit M →∞.

Decrease of t∗M as M is very slow. Numerics and heuristicssuggest:

t∗M ∼1√

log M.

This suggests such models are physically reasonable.Consistent with related results of Krapivsky and Ben-Naimand Krapivsky [2003] on exchange-driven growth.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

"Instantaneous" gelation with a source of monomers

A stationary state is reached in the regularised systems if asource of monomers is present (Horvai et al [2007]).

Stationary state (theory vs numerics)

for ν = 3/2.

Stationary state has theasymptotic form for M � 1:

Nm =

√J log Mν−1

MMm1−ν

m−ν .

Stretched exponential for smallm, power law for large m.Stationary particle density:

N =

√J(

M −MM1−ν)

M√

log Mν−1∼

√J

log Mν−1 as M →∞.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Approach to Stationary State is non-trivial

Total density vs time forK (m1,m2) = m1+ε

1 + m1+ε2 .

“Q-factor" for ν = 0.2.

Numerics indicate that theapproach to stationary state isnon-trivial.Collective oscillations of the totaldensity of clusters.Numerical measurements of theQ-factor of these oscillationssuggests that they are long-livedtransients. Last longer as Mincreases.Heuristic explanation in terms of“reset” mechanism.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

The addition model: a completely nonlocal limit

The extreme case of nonlocal aggregation is the Additionmodel (Brilliantov & Krapivsky, 1992).Clusters can only grow by absorption of monomers:

N1 = −2 γ N21 −

M∑m=2

κ(m)N1Nm + J,

Nm = κ(m − 1)N1Nm−1 − κ(m)N1Nm 2 ≤ m ≤ M,

with κ(m) ∼ mν . Instantaneously gelling for ν > 1.Stationary state:

Nm =√γ

√J

M + 1m−ν .

Large mass behaviour agrees with Smoluchowski model ifwe choose effective monomer reaction rate γ = M−1.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Oscillatory approach to stationary state in the additionmodel

Addition model with effective monomer reaction rate capturesthe 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:

N1(τ) =

√J

γ (M + 1)(1 + Aeζτ ),

where Re ζ < 0. Asymptotics are tractable for ν = 1 + ε.Find: |Re ζ| = O(ε), Im ζ = O(1). This implies long-livedoscillatory transient.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

Summary and conclusions

Aggregation phenomena exhibit a rich variety ofnon-equilibrium statistical dynamics.If the aggregation rate of large clusters increases quicklyenough as a function of cluster size, clusters of arbitrarilylarge size can be generated in finite time (gelation).Kernels with ν > 1 which, mathematically speaking,undergo complete instantaneous gelation still make senseas physical models provided a cut-off is included since theapproach to the singularity is logarithmically slow as thecut-off is removed.Approach to stationary state for regularised system with asource of monomers is interesting when ν > 1: surprisinglylong-lived oscillatory transients.Some analytical insight can be obtained from the additionmodel with renormalised monomer reaction rate.

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech

References

R.C. Ball, C. Connaughton, T.H.M. Stein and O. Zaboronski, "Instantaneous Gelation in Smoluchowski’sCoagulation Equation Revisited", preprint arXiv:1012.4431v1 [cond-mat.stat-mech], 2010

C. Connaughton, R. Rajesh, and O. Zaboronski "On the Non-equilibrium Phase Transition inEvaporation-Deposition Models", J. Stat. Mech.-Theor. E., P09016, 2010

C. Connaughton and J. Harris, "Scaling properties of one-dimensional cluster-cluster aggregation with Levydiffusion", J. Stat. Mech.-Theor. E., P05003, 2010

C. Connaughton and P.L. Krapivsky "Aggregation-fragmentation processes and decaying three-waveturbulence ", Phys. Rev. E 81, 035303(R), 2010

C. Connaughton, R. Rajesh and O. Zaboronski, "Constant Flux Relation for diffusion limited cluster–clusteraggregation", 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 TurbulentCascade : 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 ClusterAggregation", Phys. Rev. Lett. 94, 194503 (2005)

C. Connaughton R. Rajesh and O.V. Zaboronski, "Stationary Kolmogorov solutions of the Smoluchowskiaggregation equation with a source term", Phys. Rev E 69 (6): 061114, 2004

http://www.slideshare.net/connaughtonc arXiv:1012.4431 cond-mat.stat-mech