Interface Dynamics and Far-From-Equilibrium Phase Transitions in Multilayer Epitaxial Growth and Erosion on Crystal Surfaces: Continuum Theory Insights

Golubović, Leonardo; Levandovsky, Artem; Moldovan, Dorel

doi:10.4208/eajam.040411.030611a

Cited by 42 publications

(26 citation statements)

References 43 publications

(349 reference statements)

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“…Phase-field models take two distinct values (for instance, +1 and −1) in each of the phases, with a smooth change between both values in the zone around the interface, which is then diffused with a finite width. The phase-field method was first used to describe the microstructure evolution [7,16] and phase transition [21,25,30,36], but it has been recently extended to many other physical phenomena, such as solid-solid transitions, growth of cancerous tumors, phase separation of block copolymers, dewetting and rupture of thin liquid films, and infiltration of water into a porous medium.…”

Section: Introductionmentioning

confidence: 99%

Long Time Numerical Simulations for Phase-Field Problems Using $p$-Adaptive Spectral Deferred Correction Methods

Feng¹,

Tang²,

Yang³

2015

SIAM J. Sci. Comput.

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A high-order and energy stable scheme is developed to simulate phase-field models by combining the semi-implicit spectral deferred correction (SDC) method and the energy stable convex splitting technique. The convex splitting scheme we use here is a linear unconditionally stable method but is only of first-order accuracy, so the SDC method can be used to iteratively improve the rate of convergence. However, it is found that the accuracy improvement may affect the overall energy stability which is intrinsic to the phase-field models. To compromise the accuracy and stability, a local p-adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level. The proposed strategy is found very useful for producing accurate numerical solutions at small time (dynamics) as well as long time (steady state) with reasonably large time stepsizes. Numerical experiments are carried out to demonstrate the high effectiveness of the proposed numerical strategy.

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Section: Introductionmentioning

confidence: 99%

Long Time Numerical Simulations for Phase-Field Problems Using $p$-Adaptive Spectral Deferred Correction Methods

Feng¹,

Tang²,

Yang³

2015

SIAM J. Sci. Comput.

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“…Here we report a simple explanation of the exponent n = 1/3 in the case of faceting, proposed by Golubović et al in Ref. [14]. Two hypotheses are used in such derivation, the first being the existence of one single length scale, the wavelength of the periodic pattern.…”

Section: High Symmetry Orientationsmentioning

confidence: 78%

Coarsening dynamics at unstable crystal surfaces

Politi

2015

Comptes Rendus. Physique

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In this paper we focus on crystal surfaces led out of equilibrium by a growth or erosion process. As a consequence of that the surface may undergo morphological instabilities and develop a distinct structure: ondulations, mounds or pyramids, bunches of steps, ripples. The typical size of the emergent pattern may be fixed or it may increase in time through a coarsening process which in turn may last forever or it may be interrupted at some relevant length scale. We study dynamics in three different cases, stressing the main physical ingredients and the main features of coarsening: a kinetic instability, an energetic instability, and an athermal instability. I. PREAMBLEThis paper appears in a special issue of Comptes Rendus Physique devoted to coarsening dynamics. My contribution focuses on coarsening processes occurring at the surface of a crystal as a result of an instability. The meaning of coarsening in this context is clearly exemplified by the images (a1-a6) in Fig. 1. Here we provide Scanning Tunneling Microscope (STM) images of a growing Cu/Cu(100) surface at different times (the number of deposited monolayers (ML) is proportional to time and it is indicated on the bottom right of each image). As you can see, the crystal surface does not growth flat and it develops a clear mound structure whose typical size increases in time. Images (c1-c4) in the same figure refer to a similar growth process of Pt/Pt(111). In this case coarsening stops after a while.There are so many unstable growth processes at a crystal surface that it is not possible to cover all them in this short review. For this reason I will treat a personal shortlist of phenomena. Also, I will not try to report all relevant bibliography, because I am more interested to develop general considerations and to highlight some aspects.I have chosen three main topics with different features: i) The kinetic instability of a growing high symmetry surface. It takes place because the growth process is too fast and the surface cannot relax to its equilibrium shape (a flat surface in this case). ii) The energetic instability of a surface in heteroepitaxy. Here, dynamics is driven by the minimization of the system free energy. iii) The nonthermal instability of a surface under ion beam erosion. In this case surface morphology changes under the action of an incident beam of energetic ions.The first topic on the kinetic instability, which is introduced in Sec. II and discussed in Sec. III, is the most extended one because it has long been studied in specific relation to coarsening. In fact, experimental studies of this phenomenon show all possible scenarios. Here above we have already mentioned perpetual coarsening and interrupted coarsening. The former, illustrated in Fig. 1(a1-a6), means that the size of the emergent pattern increases in time forever. The latter, shown in Fig. 1(c1-c4), means that coarsening stops when the size reaches some special length scale, not depending on boundary conditions. Finally, it is also possible that the size of the pattern keep...

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“…We think that the square case would require some more analysis to gain more insights on the role of defects, as well as on the roles of the specific form of the current, of the initial conditions and of the simulation time. For the sake of completeness we also make our reader aware that the same considerations just expressed for the square case might be extended to the less studied rectangular case, since a different coarsening exponent can be found in the literature (n = 2/7) [31]. Again, also for this symmetry, coarsening is bound to the presence of dislocations and the specific form of the current might be relevant (e.g., its derivability from a potential).…”

Section: A the Contextmentioning

confidence: 98%

“…The field of crystal growth processes by a vapour phase has been very active in the past twenty-five years, involving experiments, simulations, and analytic studies. General references of special interest for the present article are three review papers [6,30,31] and one book [32].…”

Section: A the Contextmentioning

confidence: 99%

Universality classes for unstable crystal growth

2014

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Universality has been a key concept for the classification of equilibrium critical phenomena, allowing associations among different physical processes and models. When dealing with nonequilibrium problems, however, the distinction in universality classes is not as clear and few are the examples, such as phase separation and kinetic roughening, for which universality has allowed to classify results in a general spirit. Here we focus on an out-of-equilibrium case, unstable crystal growth, lying in between phase ordering and pattern formation. We consider a well-established 2+1-dimensional family of continuum nonlinear equations for the local height h(x,t) of a crystal surface having the general form ∂_{t}h(x,t)=-∇·[j(∇h)+∇(∇^{2}h)]: j(∇h) is an arbitrary function, which is linear for small ∇h, and whose structure expresses instabilities which lead to the formation of pyramidlike structures of planar size L and height H. Our task is the choice and calculation of the quantities that can operate as critical exponents, together with the discussion of what is relevant or not to the definition of our universality class. These aims are achieved by means of a perturbative, multiscale analysis of our model, leading to phase diffusion equations whose diffusion coefficients encapsulate all relevant information on dynamics. We identify two critical exponents: (i) the coarsening exponent, n, controlling the increase in time of the typical size of the pattern, L∼t^{n}; (ii) the exponent β, controlling the increase in time of the typical slope of the pattern, M∼t^{β}, where M≈H/L. Our study reveals that there are only two different universality classes, according to the presence (n=1/3, β=0) or the absence (n=1/4, β>0) of faceting. The symmetry of the pattern, as well as the symmetry of the surface mass current j(∇h) and its precise functional form, is irrelevant. Our analysis seems to support the idea that also space dimensionality is irrelevant.

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Interface Dynamics and Far-From-Equilibrium Phase Transitions in Multilayer Epitaxial Growth and Erosion on Crystal Surfaces: Continuum Theory Insights

Cited by 42 publications

References 43 publications

Long Time Numerical Simulations for Phase-Field Problems Using $p$-Adaptive Spectral Deferred Correction Methods

Long Time Numerical Simulations for Phase-Field Problems Using $p$-Adaptive Spectral Deferred Correction Methods

Coarsening dynamics at unstable crystal surfaces

Universality classes for unstable crystal growth

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