On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions

AbstractWe show short time existence for the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For the existence analysis we first write the geometric problem over a fixed reference surface and then use for the resulting analytic problem an approach in a parabolic Hölder setting.

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Thermodynamics and Kinetics of 1D Structural Elements and Stability of Nanocrystalline Materials

Diffusion Foundations ◽

10.4028/www.scientific.net/df.5.173 ◽

2015 ◽

Vol 5 ◽

pp. 173-195

Author(s):

Günter Gottstein ◽

Lazar S. Shvindlerman

Keyword(s):

Grain Boundary ◽

Grain Growth ◽

Triple Junction ◽

Nanocrystalline Materials ◽

Triple Line ◽

Second Phase ◽

Structural Elements ◽

Triple Junctions ◽

Line Energy ◽

The Impact

Grain boundary triple junctions are the structural elements of a polycrystal. Recently it was recognized that they can strongly impact the microstructural evolution, and therefore there engender new opportunities to control and to design the grain microstructure of fine-grained and nanocrystalline materials due to their effect on recovery, recrystallization and grain growth. The measurement of triple junction energy and mobility is thus of great importance. The line energy of a triple junction constructs an additional driving force of grain growth. Taking the triple line energy into account, a modified form of the Zener force and the Gibbs-Thomson relation can be derived to reveal the influence of the triple line energy on second phase particles and the change of the equilibrium concentration of vacancies in the vicinity of voids at a grain boundary. The impact of triple junctions on the sintering of nanopowders is discussed. The role of “grain boundary - free surface” triple lines in the adhesive contact formation between spherical nanoparticles is considered. It is shown that there is a critical value of the triple line energy above which the nanoparticles do not stick together. Based on this result, a new nanoparticle agglomeration mechanism is proposed, which accounts for the formation of large agglomerates of crystallographically aligned nanoparticles during the nanopowder processing.

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Nonlinear stability results for the modified Mullins–Sekerka and the surface diffusion flow

Journal of Differential Geometry ◽

10.4310/jdg/1567216953 ◽

2019 ◽

Vol 113 (1) ◽

pp. 1-53 ◽

Cited By ~ 2

Author(s):

E. Acerbi ◽

N. Fusco ◽

V. Julin ◽

M. Morini

Keyword(s):

Surface Diffusion ◽

Nonlinear Stability ◽

Diffusion Flow ◽

Surface Diffusion Flow ◽

Stability Results

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Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow

Mathematics in Engineering ◽

10.3934/mine.2022054 ◽

2022 ◽

Vol 4 (6) ◽

pp. 1-104

Author(s):

Serena Della Corte ◽

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Antonia Diana ◽

Carlo Mantegazza ◽

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...

Keyword(s):

Surface Diffusion ◽

State Of The Art ◽

Gradient Flow ◽

Necessary And Sufficient Condition ◽

Diffusion Flow ◽

Critical Set ◽

Surface Diffusion Flow ◽

Area Functional ◽

Existence And Stability ◽

Necessary And Sufficient

<abstract><p>In this survey we present the state of the art about the asymptotic behavior and stability of the <italic>modified Mullins</italic>–<italic>Sekerka flow</italic> and the <italic>surface diffusion flow</italic> of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the <italic>strict stability</italic> property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $ W^{2, p} $–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth <italic>strictly stable critical</italic> set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.</p></abstract>

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