Thesis supervisor: Gabriella Vadászné Bognár
Location of studies (in Hungarian): Searching for analytic and numerical solutions to models relevant to describe surface growth phenomena Abbreviation of location of studies: GET
Description of the research topic:
One of the great challenges of physics and materials science is to understand the growth and surface morphology of the interfaces, both in nature and in technological applications. A recently developed, highly active field of research in statistical physics is dealing with the understanding of surface growth processes [1-7]. Researchers are challenged to explore the relationship between the structure and properties of nanostructured materials, and to develop nanoscale structures in a conscious and planned way. The industrial application of coating processes allows thin layers with prescribed properties to be formed on a solid substrate [2]. Different layer-forming techniques are used to produce films, such as ion-beam sputtering (IBS), molecular beam epitaxia (MBE), chemical vacuum deposition (CVD) or physical vacuum deposition (PVD).
To understand the physical mechanisms of various surface growth mechanisms are essential in engineering, e.g. for ion implantation or for manufacturing various thin layers in optics or in microelectronics. These processes are highly non-linear in origin and take place in time and space, therefore, the natural mathematical description tools are partial differential equations (PDEs). There is no existing general mathematical theory of non-linear PDEs, however there are some trial functions like self-similar solutions, traveling waves which give us physically relevant reasonable solutions helping to get a deeper insight into the internal properties of such phenomena.
In the last years we investigated the basic surface growth PDEs, the Kardar-Paris-Zhang equation and described dozen solutions for numerous external noises [8, 9]. The PhD candidate should engage such researches. The two-component KPZ equation and other higher order equations like the Kuramoto-Sivashinsky (KS) equation with analytic and numerical methods have to be investigated.
For additional background information see papers at: http://www.kfki.hu/~barnai
1. Krug, J. Origins of scale invariance in growth processes, Taylor & Francis: London, 2008.
2. Cross, M. C.; Hohenberg, P.C. Pattern Formation Outside of Equilibrium. Reviews of Modern Physics 1993, 65, 851.
3. Barabasi A.L.; Stanley H.E. Fractal concepts in surface growth, Cambridge University Press, 1995.
4. Kardar M.; Parisi G.; Zhang Y.C. Dynamical scaling of growing interfaces. Phys. Rev. Lett. 1986, 56, 889.
5. Meakin P.; Coniglio A.; Stanley H.E. Witten T.A. Scaling properties for the surfaces of fractal and nonfractal objects: An infinite hierarchy of critical exponents. Phys. Rev. 1986, A 34, 3325.
6. Barabasi A.L. Roughening of growing surfaces: Kinetic models and continuum theories. Comp. Mater. Sci. 1996, 6, 127
7. Raible M.; Linz S.J.; Hanggi P. Amorphous thin film growth simulation methods for stochastic deposition equations. Acta Physica Polonica 2002, 33, 1049-1061.
8. Barna, I.F. ; Bognár, G.; Guedda, M.; Hriczó, K.; Mátyás, L. Analytic self-similar solutions of the Kardar-Parisi-Zhang interface growing equation with various noise term, Mathematical Modelling and Analysis 2020, 25, 241-256.
9. Barna, I.F. ; Bognár, G.; Guedda, M.; Hriczó, K.; Mátyás, L. Analytic traveling-wave solutions of the Kardar-Parisi-Zhang interface growing equation with different kind of noise terms, https://arxiv.org/abs/1908.09615
10. Bognár, G. Roughening in nonlinear surface growth model, Appl. Sciences 2020, 10:4, Paper 1422.
Expectations:
Solid knowledge of basic mechanics and the theory of ordinary differential equations
Good communication skills in English
Basic knowledge of the Latex, Maple and/or Mathematica and Office software
Number of students who can be accepted: 1
Deadline for application: 2021-01-31
2024. IV. 17. ODT ülés Az ODT következő ülésére 2024. június 14-én, pénteken 10.00 órakor kerül sor a Semmelweis Egyetem Szenátusi termében (Bp. Üllői út 26. I. emelet).
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