Thermal Grooving by Surface Diffusion: a Review of Classical Thermo-Kinetics Approach

Abstract

P olycrystalline materials are composed of tiny perfect crystalline regions (grains) in between internal interfaces called grain boundaries. Grain boundaries and the external interfaces (i.e. free surfaces which separate the material from the environment) determine the morphology of the material at a major extent. A material may change its morphology through interface motion if a driving force exists. An important special case, which is the focus of this review, is the grain boundary grooving. Wherever a grain boundary intersects a free surface and whenever the topographic variation associated with the atomic motion is favored by total free energy dissipation, the surface grooves. Grooving can occur via several mass transport mechanisms, such as surface diffusion, bulk diffusion, and evaporation and condensation. Surface diffusion dominates for temperatures far below the melting temperature, and for grooves less than 10 micron in size [1]. If the surface evolution is driven solely by the total excess free energies associated with the interfaces the resulting force for motion is conventionally called a capillary force,

Determination of the time-dependent thermal grooving coefficient

“…More advanced models of thermal grooving by surface diffusion have recently been revisited in [2]; we can also mention here the fractional sub-diffusion modelling given by ∂ α t u = −Bu xxxx , where α ∈ (0, 1], see [1,18]. We consider the inverse problem of finding the time-dependent B(t) > 0 for t ∈ [0, T ], together with u(x, t) satisfying…”

“…Assume that there exist two solutions (B i (t), u (i) (x, t)), i = 1, 2, of the inverse problem (2.3) and (2.4). For the difference of these solutions B(t (2) (x, t), we obtain the following problem:…”

“…Using that u(x 0 , t) = 0 for t ∈ [0, T ], from the first equation in (3.12) we obtain B 1 (t)u xxxx (x 0 , t) = B(t)u (2) xxxx (x 0 , t).…”

“…Let (B i (t), u (i) (x, t)), i = 1, 2, be the solutions of problem (2.3) and (2.4) with same initial data ϕ(x) and free term f (x, t), but different measurement data E 1 (t) and E 2 (t) from the class Σ. For the difference of these solutions B(t) (2) (x, t), we obtain the problem same as (3.12) with the overdetermination condition…”

Inverse Problem for the Time-Fractional Euler-Bernoulli Beam Equation

“…In the case of 0 < α ≤ 1 the Equation (1.1) is a model of fractional sub-diffusion equation [6,8]. In particular, Equation (1.1) is a well-known model of thermal grooving by surface diffusion for α = 1, see [1,3,15]. In the case of α = 2, the Equation (1.1) is the simplest model for the transverse motion of a beam called the Euler-Bernoulli beam model.…”