American Institute of Mathematical Sciences

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January  2022, 15(1): 161-177. doi: 10.3934/dcdss.2021014

Energetics and coarsening analysis of a simplified non-linear surface growth model

Hamza Khalfi 1,, , Amal Aarab 2, and  Nour Eddine Alaa 2,

1. 

ENSA, LIPIM laboratory, University Sultan Moulay Slimane, Khouribga, Morocco
2. 

FST, LAMAI Laboratory, University Cadi Ayyad, Marrakesh, Morocco

* Corresponding author: Hamza Khalfi

Received  August 2020 Revised  December 2020 Published  January 2022 Early access  January 2021

We study a simplified multidimensional version of the phenomenological surface growth continuum model derived in [6]. The considered model is a partial differential equation for the surface height profile
$ u $
 which possesses the following free energy functional:
$ E(u) = \int_{\Omega} \left[ \frac{1}{2} \ln\left(1+\left|\nabla u \right|^2\right) - \left|\nabla u \right| \arctan\left(\left|\nabla u \right|\right) + \frac{1}{2} \left|\Delta u \right|^2 \right] {\rm d}x, $
where
$ \Omega $
 is the domain of a fixed support on which the growth is carried out. The term
$ \left|\Delta u \right|^2 $
 designates the standard surface diffusion in contrast to the second order term which phenomenologically describes the growth instability. The energy above is mainly carried out in regions of higher surface slope
$ \left( \left|\nabla u \right| \right) $
. Hence minimizing such energy corresponds to reducing surface defects during the growth process from a given initial surface configuration. Our analysis is concerned with the energetic and coarsening behaviours of the equilibrium solution. The existence of global energy minimizers and a scaling argument are used to construct a sequence of equilibrium solutions with different wavelength. We apply our minimum energy estimates to derive bounds in terms of the linear system size
$ \left| \Omega \right| $
 for the characteristic interface width and average slope. We also derive a stable numerical scheme based on the convex-concave decomposition of the energy functional and study its properties while accommodating these results with 1d and 2d numerical simulations.
Keywords: Epitaxial growth, coarsening, interface width, energy minimization, D. C. Algorithm, convergence to equilibrium.
Mathematics Subject Classification: Primary: 65M70, 35A15; Secondary: 65M12, 74K35.
Citation: Hamza Khalfi, Amal Aarab, Nour Eddine Alaa. Energetics and coarsening analysis of a simplified non-linear surface growth model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 161-177. doi: 10.3934/dcdss.2021014
References:
[1]

W. Chen, S. Conde, C. Wang, X. Wang and S. M. Wise, A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 52 (2012), 546-562.  doi: 10.1007/s10915-011-9559-2.

[2]

W. Chen, C. Wang, X. Wang and S. M. Wise, A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 59 (2014), 574-601.  doi: 10.1007/s10915-013-9774-0.

[3]

L. Chen, J. Zhao and X. Yang, Regularized linear schemes for the molecular beam epitaxy model with slope selection, Appl. Numer. Math., 128 (2018), 139-156.  doi: 10.1016/j.apnum.2018.02.004.

[4]

K. Cheng, Z. Qiao and C. Wang, A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability, J. Sci. Comput., 81 (2019), 154–185. arXiv: 1903.03296. doi: 10.1007/s10915-019-01008-y.

[5]

Q. Cheng, J. Shen and X. Yang, Highly efficient and accurate numerical schemes for the epitaxial thin film growth models by using the SAV approach, J. Sci. Comput., 78 (2019), 1467-1487.  doi: 10.1007/s10915-018-0832-5.

[6]

Z. Csahók, C. Misbah and A. Valance, A class of nonlinear front evolution equations derived from geometry and conservation, Phys. D, 128 (1999), 87-100.  doi: 10.1016/S0167-2789(98)00320-0.

[7]

L. Golubović, Interfacial coarsening in epitaxial growth models without slope selection, Phys. Rev. Lett., 78 (1997), 90. doi: 10.1103/PhysRevLett.78.90.

[8]

M. Grasselli, G. Mola and A. Yagi, On the longtime behavior of solutions to a model for epitaxial growth, Osaka J. Math., 48 (2011), 987-1004. 

[9]

M. Guedda and H. Trojette, Coarsening in an interfacial equation without slope selection revisited: Analytical results, Phys. Lett. A, 374 (2010), 4308-4311.  doi: 10.1016/j.physleta.2010.08.052.

[10]

L. Ju, X. Li, Z. Qiao and H. Zhang, Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection, Math. Comp., 87 (2018), 1859-1885.  doi: 10.1090/mcom/3262.

[11]

H. Khalfi, N. E. Alaa and M. Guedda, Coarsening properties of a nonlinear front evolution equation, J. Adv. Math. Stud., 11 (2018), 295-305. 

[12]

H. Khalfi, N. E. Alaa and M. Guedda, Period steady-state identification for a nonlinear front evolution equation using genetic algorithms, International Journal of Bio-Inspired Computation, 12 (2018), 196-202.  doi: 10.1504/IJBIC.2018.094647.

[13]

H. Khalfi, M. Pierre, N. E. Alaa and M. Guedda, Convergence to equilibrium of a DC algorithm for an epitaxial growth model., Int. J. Numer. Anal. Model., 16 (2019), 98-411. 

[14]

H. G. Lee, J. Shin and J.-Y. Lee, A second-order operator splitting fourier spectral method for models of epitaxial thin film growth, J. Sci. Comput., 71 (2017), 1303-1318.  doi: 10.1007/s10915-016-0340-4.

[15]

W. Li, W. Chen, C. Wang, Y. Yan and R. He, A second order energy stable linear scheme for a thin film model without slope selection, J. Sci. Comput., 76 (2018), 1905-1937.  doi: 10.1007/s10915-018-0693-y.

[16]

B. Li and J.-G. Liu, Thin film epitaxy with or without slope selection, European J. Appl. Math., 14 (2003), 713-743.  doi: 10.1017/S095679250300528X.

[17]

B. Li and J.-G. Liu, Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, J. Nonlinear Sci., 14 (2004), 429-451.  doi: 10.1007/s00332-004-0634-9.

[18]

C. Misbah, O. Pierre-Louis and Y. Saito, Crystal surfaces in and out of equilibrium: A modern view, Reviews of Modern Physics, 82 (2010), 981. doi: 10.1103/RevModPhys.82.981.

[19]

O. Pierre-Louis, C. Misbah, Y. Saito, J. Krug and P. Politi, New nonlinear evolution equation for steps during molecular beam epitaxy on vicinal surfaces, Physical Review Letters, 80 (1998), 4221. doi: 10.1103/PhysRevLett.80.4221.

[20]

P. Politi and J. Villain, Ehrlich-Schwoebel instability in molecular-beam epitaxy: A minimal model, Physical Review B, 54 (1996), 5114. doi: 10.1103/PhysRevB.54.5114.

[21]

P. Politi and C. Misbah, Nonlinear dynamics in one dimension: A criterion for coarsening and its temporal law, Phys. Rev. E, 73 (2006), 036133, 15 pp. doi: 10.1103/PhysRevE.73.036133.

[22]

Z. Qiao, Z.-Z. Sun and Z. Zhang, Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection, Math. Comp., 84 (2015), 653-674.  doi: 10.1090/S0025-5718-2014-02874-3.

[23]

M. Rost, P. Šmilauer and J. Krug, Unstable epitaxy on vicinal surfaces, Surface Science, 369 (1996), 393-402.  doi: 10.1016/S0039-6028(96)00905-3.

[24]

R. Schwoebel, Step motion on crystal surfaces. II, Journal of Applied Physics, 40 (1969), 614-618.  doi: 10.1063/1.1657442.

show all references

References:
[1]

W. Chen, S. Conde, C. Wang, X. Wang and S. M. Wise, A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 52 (2012), 546-562.  doi: 10.1007/s10915-011-9559-2.

[2]

W. Chen, C. Wang, X. Wang and S. M. Wise, A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 59 (2014), 574-601.  doi: 10.1007/s10915-013-9774-0.

[3]

L. Chen, J. Zhao and X. Yang, Regularized linear schemes for the molecular beam epitaxy model with slope selection, Appl. Numer. Math., 128 (2018), 139-156.  doi: 10.1016/j.apnum.2018.02.004.

[4]

K. Cheng, Z. Qiao and C. Wang, A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability, J. Sci. Comput., 81 (2019), 154–185. arXiv: 1903.03296. doi: 10.1007/s10915-019-01008-y.

[5]

Q. Cheng, J. Shen and X. Yang, Highly efficient and accurate numerical schemes for the epitaxial thin film growth models by using the SAV approach, J. Sci. Comput., 78 (2019), 1467-1487.  doi: 10.1007/s10915-018-0832-5.

[6]

Z. Csahók, C. Misbah and A. Valance, A class of nonlinear front evolution equations derived from geometry and conservation, Phys. D, 128 (1999), 87-100.  doi: 10.1016/S0167-2789(98)00320-0.

[7]

L. Golubović, Interfacial coarsening in epitaxial growth models without slope selection, Phys. Rev. Lett., 78 (1997), 90. doi: 10.1103/PhysRevLett.78.90.

[8]

M. Grasselli, G. Mola and A. Yagi, On the longtime behavior of solutions to a model for epitaxial growth, Osaka J. Math., 48 (2011), 987-1004. 

[9]

M. Guedda and H. Trojette, Coarsening in an interfacial equation without slope selection revisited: Analytical results, Phys. Lett. A, 374 (2010), 4308-4311.  doi: 10.1016/j.physleta.2010.08.052.

[10]

L. Ju, X. Li, Z. Qiao and H. Zhang, Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection, Math. Comp., 87 (2018), 1859-1885.  doi: 10.1090/mcom/3262.

[11]

H. Khalfi, N. E. Alaa and M. Guedda, Coarsening properties of a nonlinear front evolution equation, J. Adv. Math. Stud., 11 (2018), 295-305. 

[12]

H. Khalfi, N. E. Alaa and M. Guedda, Period steady-state identification for a nonlinear front evolution equation using genetic algorithms, International Journal of Bio-Inspired Computation, 12 (2018), 196-202.  doi: 10.1504/IJBIC.2018.094647.

[13]

H. Khalfi, M. Pierre, N. E. Alaa and M. Guedda, Convergence to equilibrium of a DC algorithm for an epitaxial growth model., Int. J. Numer. Anal. Model., 16 (2019), 98-411. 

[14]

H. G. Lee, J. Shin and J.-Y. Lee, A second-order operator splitting fourier spectral method for models of epitaxial thin film growth, J. Sci. Comput., 71 (2017), 1303-1318.  doi: 10.1007/s10915-016-0340-4.

[15]

W. Li, W. Chen, C. Wang, Y. Yan and R. He, A second order energy stable linear scheme for a thin film model without slope selection, J. Sci. Comput., 76 (2018), 1905-1937.  doi: 10.1007/s10915-018-0693-y.

[16]

B. Li and J.-G. Liu, Thin film epitaxy with or without slope selection, European J. Appl. Math., 14 (2003), 713-743.  doi: 10.1017/S095679250300528X.

[17]

B. Li and J.-G. Liu, Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, J. Nonlinear Sci., 14 (2004), 429-451.  doi: 10.1007/s00332-004-0634-9.

[18]

C. Misbah, O. Pierre-Louis and Y. Saito, Crystal surfaces in and out of equilibrium: A modern view, Reviews of Modern Physics, 82 (2010), 981. doi: 10.1103/RevModPhys.82.981.

[19]

O. Pierre-Louis, C. Misbah, Y. Saito, J. Krug and P. Politi, New nonlinear evolution equation for steps during molecular beam epitaxy on vicinal surfaces, Physical Review Letters, 80 (1998), 4221. doi: 10.1103/PhysRevLett.80.4221.

[20]

P. Politi and J. Villain, Ehrlich-Schwoebel instability in molecular-beam epitaxy: A minimal model, Physical Review B, 54 (1996), 5114. doi: 10.1103/PhysRevB.54.5114.

[21]

P. Politi and C. Misbah, Nonlinear dynamics in one dimension: A criterion for coarsening and its temporal law, Phys. Rev. E, 73 (2006), 036133, 15 pp. doi: 10.1103/PhysRevE.73.036133.

[22]

Z. Qiao, Z.-Z. Sun and Z. Zhang, Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection, Math. Comp., 84 (2015), 653-674.  doi: 10.1090/S0025-5718-2014-02874-3.

[23]

M. Rost, P. Šmilauer and J. Krug, Unstable epitaxy on vicinal surfaces, Surface Science, 369 (1996), 393-402.  doi: 10.1016/S0039-6028(96)00905-3.

[24]

R. Schwoebel, Step motion on crystal surfaces. II, Journal of Applied Physics, 40 (1969), 614-618.  doi: 10.1063/1.1657442.

Figure 1.  Thin film surface height in a co-moving frame ($ O{x_1}{x_2}z $)
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Figure 2.  Evolution dynamics of the front equation (10) starting from the initial condition (40) where $ \varepsilon^2 = 0.25 $, $ \lambda = 10^{-2} $, $ M_k = 2^8 $ and $ \gamma = 1 $
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Figure 3.  Evolution of the energy (6) in one dimension starting from the initial configuration (40) where $ \varepsilon = 0.25 $, $ \lambda = 10^{-2} $, $ M_k = 2^8 $ and $ \gamma = 1 $
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Figure 4.  Evolution of the energy (6) in two dimensions starting from the initial configuration (41) where $ \varepsilon = 0.1 $, $ \lambda = 10^{-2} $, $ M_k = 2^8 $ and $ \gamma = 0.2 $
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Figure 5.  Evolution of the energy (6) in two dimensions starting from the initial configuration (41) where $ \varepsilon = 0.01 $, $ \lambda = 10^{-2} $, $ M_k = 2^8 $ and $ \gamma = 0.2 $
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Figure 6.  Evolution of the interfacial width or roughness $ W_u(t) $ starting from the initial configuration (41) where $ \varepsilon = 0.1 $, $ \lambda = 10^{-2} $, $ M_k = 2^8 $ and $ \gamma = 0.5 $
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Figure 7.  Evolution of the $ L^2 $-error $ \| u^{n+1} - u^{n} \|_{L^2} $ (6) starting from the initial configuration (41) where $ \varepsilon = 0.01 $, $ \lambda = 10^{-2} $, $ M_k = 2^8 $ and $ \gamma = 0.2 $
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Figure 8.  Evolution dynamics of the front equation (10) starting from the initial condition (41) where $ \varepsilon = 0.1 $, $ \lambda = 10^{-2} $, $ M_k = 2^8 $ and $ \gamma = 0.2 $
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Figure 9.  Evolution dynamics of the front equation (10) starting from the initial condition (41) where $ \varepsilon = 0.01 $, $ \lambda = 10^{-2} $, $ M_k = 2^8 $ and $ \gamma = 0.2 $
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