Multi-phase systems

LetM ∈Nbe the number of possible phases. The domain Ω is now decomposed into subdomains Ω1(t), . . . ,ΩM(t), t ∈ I, which are called phases (and, more precisely, correspond to grains in applications; see the discussion in the Introduction). The phases are not necessarily connected but it is assumed that each one consists of an finite number of connected subdomains. The phase boundaries

Γαβ(t) := Ωα(t)∩Ωβ(t), 1≤α, β≤M, α6=β,

are supposed to be piecewise smoothly evolving points, curves or hypersurfaces, depending on the dimension (cf. Definition C.1 in Appendix C). The unit normal on Γαβ pointing into phase β is denoted byναβ. The external boundary of phase Ωα is denoted by

Γα,ext:= Ωα(t)∩∂Ω.

If d≥ 2 the intersections of the curves or hypersurfaces are defined by (for pairwise different α, β, δ∈ {1, . . . , M})

Tαβδ(t) := Ωα(t)∩Ωβ(t)∩Ωδ(t).

Besides the phase boundaries can hit the external boundary. The sets of these points are denoted by

Tαβ,ext(t) := Ωα(t)∩Ωβ(t)∩∂Ω.

Ifd= 2 thenTαβδis a set of triple junctions, i.e., piecewise smoothly evolving points. Ifd= 3 triple lines can appear which are piecewise smoothly evolving curves. The triple lines can intersect and form quadruple junctions. Then the following sets are well-defined for pairwise differentα, β, δ, ζ∈ {1, . . . , M}:

Qαβδζ(t) := Ωα(t)∩Ωβ(t)∩Ωδ(t)∩Ωζ(t).

Besides the triple lines can hit the external boundary. The sets of these points are denoted by Qαβδ,ext(t) := Ωα(t)∩Ωβ(t)∩Ωδ(t)∩∂Ω.

1.2 Remark During evolution, it may happen that one of the connected subdomains of a phase or even a whole phase vanishes, namely if the adjoining phase boundaries coalesce. It is also possible that a piece of a phase boundary vanishes so that one of the setsTαβδ includes a quadruple point or line. The latter configuration is not in mechanical equilibrium and will instantaneously split up forming new phase boundaries.

It is supposed that such singularities only occur at finitely many timest∈Iduring the evolution.

This is why only piecewise smooth evolution is assumed. The following evolution equations are stated for times at which no singularity occurs.

In each phase Ωα, α∈ {1, . . . , M}, the smooth fields as in the previous Subsection 1.1.1 are present. They are denoted by c^α_i, e^α, µ^α_i, T^α and s^α (here, α is always an index, no exponent).

Additionally, surface fields on the phase boundaries Γαβare taken into account. The surface tension σαβ(ναβ) and a capillarity coefficientγαβ(ναβ) can depend on the orientation of the interface given byναβ. Bothσαβ andγαβ are one-homogeneously extended toR^d\{0}, i.e.,

σαβ(lν) =lσαβ(ν), γαβ(lν) =lγαβ(ν) ∀l >0.

Then the gradient∇γαβ(ν) is well-defined wheneverν 6= 0. Furthermore there is a mobility coeffi-cientmαβ(ναβ) that can also depend on the orientation of the interface. It is zero-homogeneously extended toR^d\{0}, i.e.,

mαβ(lν) =mαβ(ν) ∀l >0.

Besides it is assumed that for allα6=β σαβ(ναβ) =σαβ(−ναβ) =σβα(νβα)

and analogously forγαβandmαβso that the anisotropic surface fields are even and do not depend on the order of the indices. This assumption is not really necessary but shortens the following presentation and analysis.

The surface tensionsσαβ and the mobilitiesmαβ are physical quantities that may be measured in experiments. Given some reference temperature Tref, the capillarity coefficients are related to the surface tensions by setting

γαβ(ναβ) := σαβ(ναβ) Tref

. (1.9)

Based on ideas of [WSW⁺93] (see the Remark 1.3 below) the entropy is defined by S(t) =

XM

α=1

Z

Ωα(t)

s^α(e^α,ˆc^α)dL^d− XM

α<β, α,β=1

Z

Γαβ(t)

γαβ(ναβ) dH^d−1 (1.10)

1.3 Remark Surface tensions usually decrease if temperature is increased. Similarly there can be a dependence on the concentrations of the adjacent phases Ωαand Ωβor on the chemical potential.

In [Gur93] the case of a pure material in two dimensions is considered. Temperature dependent surface fields for free energy, entropy and internal energy are defined and analysed yielding analogous relations as valid for the bulk fields. In particular, there is a contribution to the internal energy by the present surfaces which must be taken into account in the energy balance and which leads to additional terms in the jump condition for the energy (1.13c). These terms are often supposed to be small and are neglected (cf. [Dav01], Section 2.2.1). But in the following Gibbs-Thomson condition (1.14) the γ-term is necessary to generate capillarity effects leading to structures as in Fig. 1 and 2.

If the surface tension is linear in the temperature, i.e., σ=_T^γ

refT, then following [Gur93] there is indeed no surface contribution to the internal energy, and the surface entropy, given by−∂Tσ, is independent of the temperature as defined in (1.10). This yields the desired capillarity term in (1.14) without changing (1.13c). The following chapters deal with phase field models, and in that context such a definition of the entropy is motivated in [WSW⁺93]. The analysis of a more general dependence ofσonT and also onµis left for future research.

The evolution must be defined in such a way that energy and mass are conserved and that local entropy production is non-negative. In every phase α balance equations hold for the conserved quantities, i.e.

∂te^α=−∇ ·J₀^α, ∂tc^α_i =−∇ ·J_i^α, 1≤i≤N, (1.11) and the coefficients of the fluxes which are defined as in the previous Subsection 1.1.1 can depend on the phase:

J₀^α = L^α₀₀∇ 1 T^α−

XN

j=1

L^α_0j∇µ^α_j

T^α, (1.12a)

J_i^α = L^α_i0∇ 1 T^α−

XN

j=1

L^α_ij∇µ^α_j

T^α, 1≤i≤N. (1.12b)

These equations are coupled to conditions on the moving phase boundaries Γαβ. To ensure con-servation of e and the ci the potentials _T¹ and ⁻_T^µj, 1 ≤ j ≤ N, (or, equivalently, temperature

1.1. IRREVERSIBLE THERMODYNAMICS

and generalised chemical potential difference) are continuous and jump conditions (or Rankine-Hugoniot-conditions) have to be satisfied (cf., for example, [Smo94]):

T^α = T^β, (1.13a)

µ^α_i = µ^β_i ∀i, (1.13b)

[e]^β_α vαβ = [J0]^β_α·ναβ, (1.13c)

[ci]^β_α vαβ = [Ji]^β_α·ναβ ∀i. (1.13d)

Here,h^αstands for the limit of the fieldhfrom the adjacent phaseαand [·]^β_αdenotes the jump of the quantity in brackets across Γαβ, e.g., [e]^β_α=e^β−e^α. The quantityvαβ is the normal velocity towardsναβ.

The evolution of the phase boundaries is coupled to the thermodynamic fields by the Gibbs-Thomson condition. To ensure that entropy is maximised during evolution a gradient flow of the entropy is considered to describe the phase boundary motion. Computing the variation of the entropy (1.10) under the constraint that energy and mass are conserved (see the next subsection) yields the following condition on Γαβ:

mαβ(ναβ)vαβ=−∇Γ· ∇γαβ(ναβ) + 1 T

hf(T,ˆc)−µ(T,ˆc)·ˆciβ

α. (1.14)

The fieldf^α is the (Helmholtz) free energy density of phase α. By ∇Γ· the surface divergence is denoted. In the case of an isotropic surface entropy, i.e.,γαβ(ν) =γ_αβ|ν| with some constantγ_αβ independent of the direction, there is the identity−∇Γ· ∇γαβ(ν) =γ_αβκαβ whereκαβ is the mean curvature (see Section 1.3). In thermodynamic equilibrium the right hand side of (1.14) vanishes.

To obtain a well-posed problem for the evolution of the Γαβ(t) initial boundaries Γ⁰_αβare given.

Besides if d= 2,3 certain angle conditions in points where a phase boundary of Γαβ hits ∂Ω or another phase boundary are satisfied. As mass density is constant and there is not transport (except diffusion) mechanical equilibrium is ensured. The angle conditions are due to local force balance or, equivalently, local minimisation of the surface energy (cf. [GN00], Section 2). The surface tensions are demanded to fulfil the constraint

σαβ+σβδ > σαδ for pairwise differentα, β, δ

uniformly in their arguments. Otherwise undesired wetting effects could appear (cf. [Haa94], Section 3.4, for a discussion and references).

On a phase boundary belonging to Γαβ there is the vector field

ξαβ(ναβ) :=∇σαβ(ναβ) =σαβ(ναβ)ναβ+∇^Γσαβ(ναβ) (1.15) where∇^Γ is the surface gradient. The identity∇=∇^Γ+ναβ· ∇was used as well as the fact that σαβ is one-homogeneously extended implying

∇σαβ(ναβ)·ναβ=σαβ(ναβ). (1.16)

The idea of using those ξ-vectors originally stems from [CH74] where also the relation to the capillary forces acting on the phase boundary is established. For a short outline, [WM97] is a suitable reference.

In the three-dimensional case Tαβδ consists of triple lines that can be oriented so that, to each point x on the triple line, a unit tangent vector ταβδ(x) can be assigned. If the whole space is cut with the plane orthogonal to ταβδ(x) through x then the picture in Fig. 1.1 is obtained.

Observe that this plane is spanned by the vectors ναβ(x) and ταβ(x). The force with that Γαβ

acts on x is given byξαβ(ναβ(x))×ταβδ(x), ×: R³×R³ →R³ being the vector product. Since (ταβ(x), ναβ(x), ταβδ(x)) is an oriented orthonormal system of R³ it follows that (evaluation at x which is omitted here)

ξαβ(ναβ) = (∇σαβ(ναβ)·ταβ)ταβ+ (∇σαβ(ναβ)·ναβ)ναβ+ (∇σαβ(ναβ)·ταβδ)ταβδ,

whence for the force there results the identity

ξαβ(ναβ)×ταβδ = (∇σαβ(ναβ)·ταβ)(ταβ×ταβδ) + (∇σαβ(ναβ)·ναβ)(ναβ×ταβδ)

= (∇σαβ(ναβ)·ταβ)(−ναβ) +σαβ(ναβ)ταβ. (1.17a) Mechanical equilibrium means that the sum of the capillary forces acting onxis zero, i.e., setting A:={(α, β),(β, δ),(δ, α)}:

0 = X

(i,j)∈A

ξij(νij(x))×ταβδ(x). (1.17b)

The set Γαβ(t)∩∂Ω consists of lines to that a unit tangent vector ταβ,ext(x) can be assigned to every pointx∈Γαβ(t)∩∂Ω similarly asταβδ(x) as before. The force acting onxis given by

ξαβ(ναβ(x))×ταβ,ext(x). (1.17c)

Force balance inximplies that this force is not tangential to∂Ω. Since it is already orthogonal to ταβ,ext(x) by definition this is true if and only if

ξαβ(ναβ(x))·νext(x) = 0 (1.17d)

because then ξαβ(ναβ(x)) is tangential to ∂Ω implying that the force ξαβ(ναβ(x))×ταβ,ext(x) is normal to∂Ω.

The two-dimensional case can be handled by extending identically the situation into the third dimension such that one getsταβδ= (0,0,1). The conditions (1.17b) and (1.17d) hold true also in this case. Observe that then∇σαβ(ναβ)·ταβ=∇Γσαβ(ναβ).

All the identities that are derived for the σαβ hold also true for the γαβ by the relation (1.9).

A full list of the equations governing the evolution is given in Section (1.2).

1.4 Remark The principle of local thermodynamic equilibrium implies that the entropy locally is maximised, hence its variation should vanish. This yields a Gibbs-Thomson condition (1.14) with mαβ≡0. But it turned out that a mobility coefficient is necessary to describe certain phenomena (cf. the introduction of the kinetic coefficient in [Dav01] in Section 2.1.4; in Section 5 also its anisotropy is motivated). But there may be situations where the kinetic term can be neglected, cf., for example, [JH66], Section III.

Im Dokument Derivation and Analysis of a Phase Field Model for Alloy Solidification (Seite 15-18)