Dynamic boundary conditions in the interface modeling of binary alloys. (English) Zbl 1431.35195
Summary: We study the initial boundary value problem with dynamic boundary conditions to the Penrose-Fife equations with a ‘memory effect’ for the order parameter and temperature time evolutions. The dynamic boundary conditions describe the process of production and degradation of surface crystallite near the walls, which confine the disordered binary alloy at a nearly melt temperature during the fast cooling process. The solid-liquid periodic distributions, which were obtained in 1D case, represent asymptotically periodic piecewise constant spatial-temporal impulses in a long timedynamics. It is confirmed that, depending on parameter values, the total number of discontinuity points of such periodic impulses can be finite or infinite. We refer to such wave solution types as relaxation or pre-turbulent, respectively. These results are compared with experimental data.
MSC:
| 35Q70 | PDEs in connection with mechanics of particles and systems of particles |
| 35B10 | Periodic solutions to PDEs |
| 80A22 | Stefan problems, phase changes, etc. |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 80A19 | Diffusive and convective heat and mass transfer, heat flow |
| 35C07 | Traveling wave solutions |
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
References:
| [1] | R, Surface segregation at the binary alloy Cu Au (100) studied by low-energy ion scattering, Surface Science, 643, 138-141 (2016) |
| [2] | K, Surface-directed spinodal decomposition versus wetting phenomena: Computer simulations, Faraday Discuss, 112, 103-117 (1999) |
| [3] | A, Non-Fickian diffusion and tau approximation for numerical turbulence, Phys. Fluids, 16, 1020-1028 (2004) · Zbl 1186.76074 |
| [4] | H, Surface composition analysis by low-energy ion scattering, Surface Science Reports, 62, 63-109 (2007) |
| [5] | T, Order-disorder and segregation behavior at the Cu3Au(001) surface, Phys. Rev. Lett., 51, 43-46 (1983) |
| [6] | G, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. An., 92, 205-245 (1986) · Zbl 0608.35080 |
| [7] | C, On thermodynamically consistent schemes for phase field equations, Open Syst. Inf. Dyn., 5, 99-123 (1998) · Zbl 0907.60090 |
| [8] | C, Surface density oscillations in disordered binary alloys: X-ray reflectivity study of Cu3Au(001), Appl. Phys. A, 64, 383-390 (1997) |
| [9] | J, Modeling of convection, temperature distribution and dendritic growth in glass-fluxed nickel melts, Journal Crystall Growth, 471, 66-72 (2017) |
| [10] | M, Surface alloys, surface rumpling and surface stress, Surface Science, 572, 309-317 (2004) |
| [11] | I, Spatial-temporal oscillations of order parameter in confined diblock copolymer mixtures, International Journal of Computational Materials, Science and Engineering, 2, 135006 (2013) |
| [12] | I, Long-time oscillating properties of confined disordered binary alloys, Journal of Advanced Research in Applied Mathematics, 7, 1-16 (2015) |
| [13] | I, Surface-directed multi-dimensional wave structures in confined binary mixture, International Journal of Computational Materials Science and Engineering, 4, 1550023 (2015) |
| [14] | I, Impulse Spatial-Temporal Domains in Semiconductor Laser with Feedback, Journal of Applied Mathematics and Physics, 4, 1714-1730 (2016) |
| [15] | I, Stationary Solutions for the Cahn-Hilliard equation coupled with Neumann boundary conditiona, Bulletin of the South Ural State University. Ser., Mathematical Modelling, Programming Computer Software (Bulletin SUSU MMCS), 9, 60-74 (2016) · Zbl 1349.35218 |
| [16] | N, Evolution of the Structure Factor in a Hyperbolic Model of Spinodal Decomposition, European Physical Journal Special Topics, 179, 165-175 (2009) |
| [17] | T, Period Three Implies Chaos, The American Mathematical Monthly, 82, 985-992 (1975) · Zbl 0351.92021 |
| [18] | Y. L. Maistrenko, E. Y. Romanenko, About qualitative behaviour of solutions of quasi-linear differential-difference equations, The research of differential-difference equations, Kiev, Institute Mathematics, 1980. |
| [19] | J, On the dynamical theory of gases, Philos. T. R. Soc. A, 157, 49-88 (1867) |
| [20] | H, Low-energy ion scattering at surfaces, Surface Science Reports, 17, 213-303 (1993) |
| [21] | H. O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals, New Frontiers of Science, Springer-Verlag, New York, 2004. · Zbl 1036.37001 |
| [22] | O, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43, 44-62 (1990) · Zbl 0709.76001 |
| [23] | J. du Plessis, Surface Segregation, Series Solid State Phenomena, volume 11, Sci.-Tech. Pub., Vaduz, 1990. |
| [24] | M, The interplay of surface segregation and atomic order in alloys, Surface Science Reports, 38, 127-194 (2000) |
| [25] | S, Surface effects on spinodal decomposition in binary mixtures and the interplay with wetting phenomena, Phys. Rev. E, 49, 5359-5377 (1994) |
| [26] | E, Ideal turbulence: attractors of deterministic systems may lie in the space of random fields, Int. J. Bifurcat. Chaos, 2, 31-36 (1992) · Zbl 0899.76238 |
| [27] | E, From one-dimensional to infinite-dimensional dynamical systems: Ideal turbulence, Ukrainian Math. J., 48, 1817-1842 (1996) · Zbl 0936.37021 |
| [28] | E, Self-stochasticity in deterministic boundary value problems, Nonlinear Boundary Value Problems, Institute of Applied Mathematiccs and Mechanics of the NAS of Ukraine, 9, 174-184 (1999) · Zbl 0973.35058 |
| [29] | A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko, Difference equations and their applications, Ser. Mathematics and Its Applications, 250, Klüwer Academic, Dordrecht, The Netherlands, 1993. · Zbl 0806.39001 |
| [30] | A, Universal Phenomena in Solution Bifurcations of Some Boundary Value Problems, J. Nonlinear Math. Phy., 1, 147-157 (1994) · Zbl 0955.37501 |
| [31] | E. Taglauer, Low-energy ion scattering and Rutherford backscattering, Surface Analysis - The Principal Techniques, 2nd ed., eds. by J.C. Vickerman and I.S. Gilmore, JohnWiley & Sons, Ltd., 269-331, 2009. |
| [32] | J, Oscillatory segregation at a metal alloy surface: Relation to ordered bulk phases, Phys. Rev. B, 42, 10965-10968 (1990) |
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