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Sample text
Moreover, the total energy E = E S + E L is fixed. E S and E L are contributions to the energy E from liquid and solid phase, respectively. In addition, NS n Cˆ S Cˆ L ˆS Q ˆL Q : : : : : : total number of particles in solid phase number of particles of species B in solid phase concentration of species B in solid phase: Cˆ S ≡ n/N NB −n concentration of species B in liquid phase: Cˆ L ≡ N −NS ˆ S ≡ E S /NS energy per particle in solid phase: Q ˆ S ≡ E−E S . energy per particle in liquid phase: Q N −NS Here variables Cˆ S , Cˆ L , ...
G. 47) (φ → Φ) [48, 154]. Thus for thermodynamic equilibrium configurations microscopic and macroscopic derivation yield the same ansatz for G(Φ, X, r). Furthermore the microscopic approach provides additional insight into the nature of diffuse interface models. It yields an understanding, that the finite interface thickness of a diffuse interface model and the surface energy term originate from finite correlation lengths on a microscopic scale. Moreover – assuming nearest neighbor interaction and constant coupling constant J – interface thickness and surface energy are proportional to J.
The derivation of the energy equation is more involved. 80) into V ρ d De Du + ρu − ∇ · (m · u) + ∇qE dV + Dt Dt dT V Further the identity d dT V 1 2 2 ξ Γ (∇Φ)dV = 2 E V 1 2 2 ξ Γ (∇Φ)dV = 0 . 85) which is proven in the appendix of [15], is employed, where QG = ∇ · Γ Σ DΦ Dt − DΦ 1 ∇ · (Γ Σ) − Γ ∇u : Σ ⊗ ∇Φ + Γ 2 ∇ · u . 86) Dt 2 Here ⊗ refers to the outer tensor product and : to the double contraction of the tensor product. Σ denotes the Cahn–Hoffmann capillary vector mentioned in the introductory part of this section.