Stability of travelling wave solutions for coupled surface and grain boundary motion

Beck, Margaret; Pan, Zhenguo; Wetton, Brian

doi:10.1016/j.physd.2010.05.008

Cited by 7 publications

(3 citation statements)

References 24 publications

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“…While our study of the steady states is so far not yet exhaustive, we have established a parametric framework which allows for their complete characterization. Moreover, many dynamic questions lie ahead which are of concern to materials scientists, such as the role of the size and shape of grains which border on or lie in close proximity to holes [36], and many analytic stability questions can be readily formulated [21,1]; such as, stability of the steady states with respect to non-axi-symmetric perturbations.…”

Section: Discussionmentioning

confidence: 99%

Coupled Surface Diffusion and Mean Curvature Motion: Axisymmetric Steady States with Two Grains and a Hole

Golubkov¹,

Novick-Cohen²,

Vaknin³

2021

Preprint

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The evolution of two grains, which lie on a substrate and are in contact with each other, can be roughly described by a simple model in which the exterior surfaces of the grains evolve by surface diffusion and the grain boundary, namely the contact surface between the adjacent grains, evolves by motion by mean curvature. For simplicity we consider an axi-symmetric two grain system, which is contained within an inert bounding semi-infinite cylinder with unit radius and which is bounded below by a planar substrate. The two grain system is assumed to have a hole along the axis of symmetry, where the substrate is exposed. Boundary conditions are imposed reflecting the considerations of W.W. Mullins, 1958. The resultant dynamic problem conserves the total volume of the two grain system and dissipates the total energy, where the total energy is defined as sum of the areas of the various participating surfaces, weighted according to their respective surface free energies.We focus here on the steady states of this system. We demonstrate that at steady state, the exterior surfaces have constant and equal mean curvatures, and the grain boundary has zero mean curvature. Taking into account the geometry and the boundary conditions, it then follows that the exterior surfaces are nodoids and the grain boundary surface is a catenoid. The physical parameters in the model can be expressed via the angles β and θc, which depend on the surface free energies, where, in the meridian cross-section, β ∈ (π/2, π) is the angle between the grain boundary and each of the exterior surfaces, and θc ∈ (0, π], the contact angle, is the angle between the inner grain and the substrate. Typically if a steady state solution exists for given values of (β, θc), then there exists a continuum of such solutions with varying volumes and energies. In particular, we prove that there exists a continuum of solutions with θc = π for any β ∈ (π/2, π). While many open questions remain, with regard to both the steady states and the full dynamic problem, our study already provides insight into the possible steady states and their structure. In particular, the relative volumes and heights of the inner and outer grain can be seen to be roughly in accordance with experimental predictions, for realistic values of the physical parameters.

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Section: Discussionmentioning

confidence: 99%

Coupled Surface Diffusion and Mean Curvature Motion: Axisymmetric Steady States with Two Grains and a Hole

Golubkov¹,

Novick-Cohen²,

Vaknin³

2021

Preprint

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“…The answer will depend on the character of perturbations, but some results indicate stability in relatively general settings; see, e.g. the analysis of Y-shaped GRTs (named 'equal-order grain boundary model') in [22]. Assuming stability, the simplest approach to testing would be to start a simulation using an analytically obtained translating boundary network and check whether the simulated structure does move by translation.…”

Section: Discussionmentioning

confidence: 99%

On 2D interface networks that are translation-invariant under curvature-driven migration

Morawiec¹

2018

Modelling Simul. Mater. Sci. Eng.

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The issue of constructing interface networks translating under curvature-driven migration is addressed. The translating networks are the main accessory for testing capillarity driven migration mode in grain growth simulations. They also arise in experimental studies of boundary mobility and junction drag. The networks are constructed of pieces of simple translating curves—lines and grim reapers (GRs)—meeting at triple junctions. The primary method of building the networks relies on a certain property of GRs which reduces the construction to simple algebraic operations. At the outset, networks of interfaces having different tensions are considered. More detailed discussion concerns a restricted model in which the migration speed does not depend on the tensions. In this second case, the networks have noteworthy properties: some cell dimensions are directly related to angles at triple junctions, there are simple expressions for area sweeping rates, and there is a convenient and elegant alternative way of constructing these particular networks.

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“…There are also studies of triple junction between free surfaces and GBs that include surface effects. 3,4 However, twodimensional models are used as a simplication of the problem. Conventional studies of GB or interfaces can only infer the processes involved in the experiments; in addition, since the interface is buried, direct observation of friction experiments are rare.…”

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confidence: 99%

Direct observation of liquid-like behavior of a single Au grain boundary

et al. 2013

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Behavior of matter at the nanoscale differs from that of the bulk due to confinement and surface effects. Here, we report a direct observation of liquid-like behavior of a single grain boundary formed by cold-welding Au nanoparticles, 40 nm in size, by mechanical manipulation in-situ TEM. The grain boundary rotates almost freely due to the free surfaces and can rotate about 90 degrees. The grain boundary sustains more stress than the bulk, confirming a strong bonding between the nanoparticles. Moreover, this technique allows the measurement of the surface diffusion coefficient from experimental observations, which we compute for the Au nanoparticles. This methodology can be used for any metal, oxide, semiconductor or combination of them.

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Stability of travelling wave solutions for coupled surface and grain boundary motion

Cited by 7 publications

References 24 publications

Coupled Surface Diffusion and Mean Curvature Motion: Axisymmetric Steady States with Two Grains and a Hole

Coupled Surface Diffusion and Mean Curvature Motion: Axisymmetric Steady States with Two Grains and a Hole

On 2D interface networks that are translation-invariant under curvature-driven migration

Direct observation of liquid-like behavior of a single Au grain boundary

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