December  2018, 11(6): 1359-1376. doi: 10.3934/krm.2018053

On the convergence to critical scaling profiles in submonolayer deposition models

1. 

Departamento de Ciȇncias e Tecnologia, Universidade Aberta, Lisboa, Portugal

2. 

Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

3. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

* Corresponding author: F.P. da Costa

Received  July 2017 Published  June 2018

Fund Project: Partially funded by FCT/Portugal through project RD0447/CAMGSD/2015.

In this work we study the rate of convergence to similarity profiles in a mean field model for the deposition of a submonolayer of atoms in a crystal facet, when there is a critical minimal size $ n≥ 2$ for the stability of the formed clusters. The work complements recently published related results by the same authors in which the rate of convergence was studied outside of a critical direction $ x = τ$ in the cluster size $ x$ vs. time $ \tau $ plane. In this paper we consider a different similarity variable, $ ξ : = (x-\tau )/\sqrt \tau $, corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $ Φ_{2,n}(ξ)$ when $ x, \tau \to +∞$ with $ ξ$ fixed, as well as the rate at which the limit is approached.

Citation: Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic and Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053
References:
[1]

J. M. BallJ. Carr and O. Penrose, The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions, Comm. Math. Phys., 104 (1986), 657-692.  doi: 10.1007/BF01211070.

[2]

J. CañizoA. Einav and B. Lods, Trend to equilibrium for the Becker-Döring equation: an analogue of Cercignani's conjecture, Analysis and PDE, 10 (2017), 1663-1708.  doi: 10.2140/apde.2017.10.1663.

[3]

J. Cañizo and B. Lods, Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations, J. Diff. Equa., 225 (2013), 905-950.  doi: 10.1016/j.jde.2013.04.031.

[4]

J. CañizoS. Mischler and C. Mouhot, Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients, SIAM J. Math. Anal., 41 (2010), 2283-2314.  doi: 10.1137/08074091X.

[5]

F. P. da Costa, Mathematical Aspects of Coagulation-Fragmentation Equations, in Mathematics of Energy and Climate Change (eds. J. P. Bourguignon, R. Jeltsch, A. Pinto and M. Viana), CIM Series in Mathematical Sciences, vol. 2, Springer, Cham, (2015), 83–162.

[6]

F. P. da CostaJ. T. Pinto and R. Sasportes, Rates of convergence to scaling profiles in a submonolayer deposition model and the preservation of memory of the initial condition, SIAM J. Math. Anal., 48 (2016), 1109-1127.  doi: 10.1137/15M1035033.

[7]

F. P. da CostaH. van Roessel and J. A. D. Wattis, Long-time behaviour and self-similarity in a coagulation equation with input of monomers, Markov Processes Relat. Fields, 12 (2006), 367-398. 

[8]

O. CostinM. GrinfeldK. P. O'Neill and H. Park, Long-time behaviour of point islands under fixed rate deposition, Commun. Inf. Syst., 13 (2013), 183-200.  doi: 10.4310/CIS.2013.v13.n2.a3.

[9]

M. EinaxW. Dieterich and Ph. Maass, Colloquim: Cluster growth on surfaces: Densities, size distributions, and morphologies, Rev. Modern Phys., 85 (2013), 921-939. 

[10]

J. W. Evans and M. C. Bartelt, Nucleation, growth, and kinetic roughening of metal (100) homoepitaxial thin films, Langmuir, 12 (1996), 217-229.  doi: 10.1021/la940698s.

[11]

P.-E. Jabin and B. Niethammer, On the rate of convergence to equilibrium in the Becker-Döring equations, J. Diff. Equa., 191 (2003), 518-543.  doi: 10.1016/S0022-0396(03)00021-4.

[12]

Ph. Laurençot and S. Mischler, From the Becker-Döring to the Lifshitz-Slyozov-Wagner equations, J. Stat. Phys., 106 (2002), 957-991.  doi: 10.1023/A:1014081619064.

[13]

Ph. Laurençot and S. Mischler, Liapunov functional for Smoluchowski's coagulation equation and convergence to self-similarity, Monatsh. Math., 146 (2005), 127-142.  doi: 10.1007/s00605-005-0308-1.

[14]

G. Menon and R. L. Pego, Approach to self-similarity in Smoluchowski's coagulation equations, Commun. Pure Appl. Math., 57 (2004), 1197-1232.  doi: 10.1002/cpa.3048.

[15]

P. A. Mulheran, Theory of cluster growth on surfaces, in Metallic Nanoparticles, vol. 5 of Handbook of Metal Physics (ed. J. A. Blackman), (series editor: P. Misra), Elsevier, Amsterdam, (2009), 73–111.

[16]

R. W. Murray and R. L. Pego, Algebraic decay to equilibrium for the Becker-Döring equations, SIAM J. Math. Anal., 48 (2016), 2819-2842.  doi: 10.1137/15M1038578.

[17]

B. Niethammer, On the evolution of large clusters in the Becker-Döring model, J. Nonlinear Sci., 13 (2003), 115-155.  doi: 10.1007/s00332-002-0535-8.

[18]

R. Srinivasan, Rates of convergence for Smoluchowski's coagulation equations, SIAM J. Math. Anal., 43 (2011), 1835-1854.  doi: 10.1137/090759707.

[19]

J. A. D. Wattis, An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach, Physica D, 222 (2006), 1-20.  doi: 10.1016/j.physd.2006.07.024.

show all references

References:
[1]

J. M. BallJ. Carr and O. Penrose, The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions, Comm. Math. Phys., 104 (1986), 657-692.  doi: 10.1007/BF01211070.

[2]

J. CañizoA. Einav and B. Lods, Trend to equilibrium for the Becker-Döring equation: an analogue of Cercignani's conjecture, Analysis and PDE, 10 (2017), 1663-1708.  doi: 10.2140/apde.2017.10.1663.

[3]

J. Cañizo and B. Lods, Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations, J. Diff. Equa., 225 (2013), 905-950.  doi: 10.1016/j.jde.2013.04.031.

[4]

J. CañizoS. Mischler and C. Mouhot, Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients, SIAM J. Math. Anal., 41 (2010), 2283-2314.  doi: 10.1137/08074091X.

[5]

F. P. da Costa, Mathematical Aspects of Coagulation-Fragmentation Equations, in Mathematics of Energy and Climate Change (eds. J. P. Bourguignon, R. Jeltsch, A. Pinto and M. Viana), CIM Series in Mathematical Sciences, vol. 2, Springer, Cham, (2015), 83–162.

[6]

F. P. da CostaJ. T. Pinto and R. Sasportes, Rates of convergence to scaling profiles in a submonolayer deposition model and the preservation of memory of the initial condition, SIAM J. Math. Anal., 48 (2016), 1109-1127.  doi: 10.1137/15M1035033.

[7]

F. P. da CostaH. van Roessel and J. A. D. Wattis, Long-time behaviour and self-similarity in a coagulation equation with input of monomers, Markov Processes Relat. Fields, 12 (2006), 367-398. 

[8]

O. CostinM. GrinfeldK. P. O'Neill and H. Park, Long-time behaviour of point islands under fixed rate deposition, Commun. Inf. Syst., 13 (2013), 183-200.  doi: 10.4310/CIS.2013.v13.n2.a3.

[9]

M. EinaxW. Dieterich and Ph. Maass, Colloquim: Cluster growth on surfaces: Densities, size distributions, and morphologies, Rev. Modern Phys., 85 (2013), 921-939. 

[10]

J. W. Evans and M. C. Bartelt, Nucleation, growth, and kinetic roughening of metal (100) homoepitaxial thin films, Langmuir, 12 (1996), 217-229.  doi: 10.1021/la940698s.

[11]

P.-E. Jabin and B. Niethammer, On the rate of convergence to equilibrium in the Becker-Döring equations, J. Diff. Equa., 191 (2003), 518-543.  doi: 10.1016/S0022-0396(03)00021-4.

[12]

Ph. Laurençot and S. Mischler, From the Becker-Döring to the Lifshitz-Slyozov-Wagner equations, J. Stat. Phys., 106 (2002), 957-991.  doi: 10.1023/A:1014081619064.

[13]

Ph. Laurençot and S. Mischler, Liapunov functional for Smoluchowski's coagulation equation and convergence to self-similarity, Monatsh. Math., 146 (2005), 127-142.  doi: 10.1007/s00605-005-0308-1.

[14]

G. Menon and R. L. Pego, Approach to self-similarity in Smoluchowski's coagulation equations, Commun. Pure Appl. Math., 57 (2004), 1197-1232.  doi: 10.1002/cpa.3048.

[15]

P. A. Mulheran, Theory of cluster growth on surfaces, in Metallic Nanoparticles, vol. 5 of Handbook of Metal Physics (ed. J. A. Blackman), (series editor: P. Misra), Elsevier, Amsterdam, (2009), 73–111.

[16]

R. W. Murray and R. L. Pego, Algebraic decay to equilibrium for the Becker-Döring equations, SIAM J. Math. Anal., 48 (2016), 2819-2842.  doi: 10.1137/15M1038578.

[17]

B. Niethammer, On the evolution of large clusters in the Becker-Döring model, J. Nonlinear Sci., 13 (2003), 115-155.  doi: 10.1007/s00332-002-0535-8.

[18]

R. Srinivasan, Rates of convergence for Smoluchowski's coagulation equations, SIAM J. Math. Anal., 43 (2011), 1835-1854.  doi: 10.1137/090759707.

[19]

J. A. D. Wattis, An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach, Physica D, 222 (2006), 1-20.  doi: 10.1016/j.physd.2006.07.024.

Figure 1.  Lines with $\xi = \text{constant}$ (full), and with $\eta = \text{constant}$ (dashed) in the $(j,\tau)$ plane. The values used for these parameters are the following, in counterclockwise direction: $\xi = 5.0, 2.0, 1.0, 0.5, 0.0, -0.3, -0.5, -0.7, -0.9,$ and $\eta = 1/4,$ $1/3,$ $1/2,$ $1,$ $2,$ $3,$ $4$.
Figure 2.  Graph of the similarity profile $\Phi_{2,n}(\xi)$ for different values of $n$.
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