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Cluster aggregation with complete collisional fragmentation

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Cluster aggregation with complete collisional fragmentation

1. 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). Non ideal particles and aggregates in turbulence Lecce, June 7-9, 2012 http://www.slideshare.net/connaughtonc arXiv:1205.4445
2. 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... http://www.slideshare.net/connaughtonc arXiv:1205.4445
3. 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. http://www.slideshare.net/connaughtonc arXiv:1205.4445
4. 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 ) = (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. http://www.slideshare.net/connaughtonc arXiv:1205.4445
5. 5. 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 http://www.slideshare.net/connaughtonc arXiv:1205.4445
6. 6. 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 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? http://www.slideshare.net/connaughtonc arXiv:1205.4445
7. 7. 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. http://www.slideshare.net/connaughtonc arXiv:1205.4445
8. 8. 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? http://www.slideshare.net/connaughtonc arXiv:1205.4445
9. 9. 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. http://www.slideshare.net/connaughtonc arXiv:1205.4445
10. 10. 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. http://www.slideshare.net/connaughtonc arXiv:1205.4445
11. 11. µ ν ν µSolution 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 http://www.slideshare.net/connaughtonc arXiv:1205.4445
12. 12. 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 . http://www.slideshare.net/connaughtonc arXiv:1205.4445
13. 13. 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! http://www.slideshare.net/connaughtonc arXiv:1205.4445
14. 14. 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. http://www.slideshare.net/connaughtonc arXiv:1205.4445
15. 15. 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. http://www.slideshare.net/connaughtonc arXiv:1205.4445
16. 16. 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. http://www.slideshare.net/connaughtonc arXiv:1205.4445
17. 17. 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 . http://www.slideshare.net/connaughtonc arXiv:1205.4445
18. 18. 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 → ∞! http://www.slideshare.net/connaughtonc arXiv:1205.4445