COMPUTATIONAL MATERIALS ENGINEERING Episode 11 ppsx

The time stepδtcan be set according to thestability condition in equation 7.20.phase-Case 1: Energy Equation—Dirichlet and Neumann Boundary Conditions In this first application, we consi

15 potentialEnergy = A*w(argPhi[i][j]);

16 drivingForce = B*f(argPhi[i][j]);

17

18 // Summation of terms for the update of the phi matrix

19 newPhasefield = argPhi[i][j] + delta_t * ( 2*gamma*

LISTING 7-3 Description of Input Data for a Phase-Field Simulation

7 # boundary_type_* defines the type of the boundary condition.

8 # For each boundary a special type can be defined.

9 # Possible values:

10 # 0 = Dirichlet

11 # 1 = Neumann

12 # 2 = Periodic

13 # If the Dirichlet boundary condition is chosen, values

14 # at the boundaries need to be set by rand_wert_*.

33 # value_interiorbody initializes the filled region

34 # with a defined value The remaining region is set

35 # to value_exteriorbody.

36

37 value_interiorbody=1

40 # Parameters of the phase-field equation:

41 # epsilon is the thickness of the diffuse interface.

To illustrate the evolution of the phase-field in time and space, the C++ program may produce

a MatLab file “data file.m” as output file This file contains a number of successiveφmatrices

at preselected time steps Applying further MatLab script files “show ij.m,” “show xy.m,” or

“show 3d.m” allows the graphical illustration of the phase field either as 2D colored images

or as a 3D surface plot The Listings 7-4 and 7-5 now give examples of the MatLab codes

LISTING 7-4 Structure of a MatLab Output File

1 %%%%%

2 % data_file.m - MatLab output file with the matrices

4 %%%%%

7 % Loop over n frames illustrating the temporal evolution

8 % of the phase field

For the subsequent case study, the following parameters occurring in equation (7.19) aretaken:

τ = 1.0,  = 1.0, γ = 1.0andN x= 100,N y= 10,N t= 1000

By choosing appropriate values forAandBgiven subsequently, the certain terms in the field equation are switched on and off, respectively The time stepδtcan be set according to thestability condition in equation (7.20).

phase-Case 1: Energy Equation—Dirichlet and Neumann Boundary Conditions

In this first application, we consider the case in which the phase-field equation reduces to a heattransport equation of the type∂ t u (x, t) = D  u(x, t)by setting the control coefficientsA= 0and B = 0 The influence of different boundary conditions is displayed in Figure 7-12 Thediagrams show the profile of the phase-field variableφ (x, t)starting fromφ (x, t) = 0at time

t= 0everywhere in the domain

Case 2: Phase-Field Equation

To investigate the phase-field equation, we switch on the potential entropy contributionw ,φ (φ)

and the bulk driving forcef ,φ (φ)by setting the coefficientsA= 1andB = 1 The gray scalesindicateφ= 1in white,φ= 0in black, and the diffuse interface region in varying colors

1 Di ffuse Interface Thickness: A planar solid–liquid front is placed in the center of the

domain atN x/2with a sharp interface profile, with zero driving forcem = 0and withNeumann boundary conditions on each side The effect of different values of the smalllength scale parameter: = 1and  = 10responsible for the thickness of the diffuseinterface is shown in Figure 7-13

FIGURE 7-12 Left diagram (pure Dirichlet condition): The left (west) boundary is set to φW = 1

and the right (east) boundary to φE= 0 A linear profile is established; Right diagram (combination

of the Dirichlet and the Neumann boundary condition): The left (west) boundary is set constant to

φW = 1and the right (east) boundary is isolated (Neumann) The complete domain “warms up” approaching the constant value φ = 1

2 Driving Force: As a next configuration, the three simulations shown in Figure 7-14(b)–(d)

were performed with= 1with different values of the driving force and with the initialconfiguration of Figure 7-14(a) Form = 0, the initial planar front remains stable, for

m = −1the solid phase (light color) grows, whereas form= 1the solid phase shrinks

Bild-Nr.: 10/10

8 4

X

5

10 20 25 30 35 40 45 Bild-Nr.:15/15

X

5

10 20 25 30 35 40 45 Bild-Nr.:15/15

X

5

10 15 20 25 30 35 40 Bild-Nr.:15/15

X

5

10 15 20 25 30 35 40 Bild-Nr.:15/15

X

5

10 15 20 25 30 35 40 Bild-Nr.:15/15

X

(a)

(b)

FIGURE 7-15 Two-dimensional phase-field simulation with (a) a solid nucleus in the center of the

domain and Neumann boundary conditions and with (b) a solid nucleus at the bottom of the domain and periodic boundary conditions.

Case 3: 2D Phase-Field Simulations of a Growing Nucleus

For the simulation in Figure 7-15, a solid nucleus is set in a 2D domain ofN x × Ny = 50 × 50

grid points, withA= 1,B= 1and with a driving forcem = −2.5 Due to periodic boundary

conditions in Figure 7-15(b) the particle grows across the lower boundary and appears at thetop boundary

7.4 Model for Multiple Components and Phases

7.4.1 Model Formulation

Based on the phase-field model for pure substances, a more general approach can be derived

in a thermodynamically consistent way allowing for an arbitrary number of phases (or grains)and components [GNS04] The model that will be described in the following sections uses

N-order parameters to describe either different phases in alloy systems or different orientationalvariants in polycrystals The formulation can be defined solely via the bulk free energies ofthe individual phases, the surface energy densities (surface entropy densities, respectively) ofthe interfaces, the diffusion and mobility coefficients Thus, the full set of phase-field evolutionequations is defined by quantities which can be measured Since the bulk free energies determinethe phase diagrams (see, e.g., Chalmers [Cha77], Haasen [Haa94]), the phase-field model can

be used to describe phase transitions, in principal, for arbitrary phase diagrams The field model for a general class of multicomponent and multiphase (or polycrystalline) alloysystems is formulated consisting of K components andN different phases (or grains) in adomainΩ ⊂ IR3 The domain Ωis separated in phase regionsΩ1, ,ΩN occupied by the

phase-N phases as schematically illustrated in the left image of Figure 7-16 The middle and rightimages show examples of an Al–Si grain structure with grains of different crystallographicorientations and of a real multiphase structure with primary dendrites and an interdendriticeutectic substructure

The concentrations of the components are represented by a vectorc(x, t) = (c1(x, t), ,

c K (x, t)) Similarly, the phase fractions are described by a vector-valued-order parameter

(x, t) = (φ1(x, t), , φ N (x, t)) The variable φ α (x, t)denotes the local fraction of phaseα.The phase-field model is based on an entropy functional of the form

FIGURE 7-16 (a) Schematic drawing of a domain separation by four different phase regions,

(b) polycrystalline grain structure, and (c) multiphase solidification microstructure with dendrites and an interdendritic eutectic structure.

We assume that the bulk entropy densitysdepends on the internal energy densitye, the trationsc, and the phase-field variable
We require that the concentrations of the componentsand of the phase-field variables fulfill the constraints

7.4.2 Entropy Density Contributions

It will be convenient to use the free energy as a thermodynamical potential We therefore tulate the Gibbs relation

We note that given the free energy densities of the pure phases, we obtain the total free energy

f as a suitable interpolation of the free energiesf αof the individual phases in the system Byinserting the free energyf into the phase-field method enables to model systems with a verygeneral class of phase diagrams In the way it is formulated, the model can describe systemswith concave entropiess α (e, c)in the pure phases This corresponds to free energiesf α (T, c)

which are convex incand concave inT In the case wheref (T, c)is not convex in the variablec,the free energy needs to contain gradients of the concentrations (as in the Cahn–Hilliard model).Choosing the liquid phase to be the last componentφ N of the phase-field vector
, an idealsolution formulation of the bulk free energy density reads

h (φ α)

+ K

i = 0andL α

i,i = 1, , K, α = 1, , N − 1, being the latent heat per unit volume

of the phase transition from phaseαto the liquid phase and of pure componenti Furthermore,

T α

i ,i = 1, , K, α = 1, , N −1is the melting temperature of theith component in phaseα,

T M is a reference temperature.c v, the specific heat andv m, the molar volume are assumed to

be constant,R gis the gas constant With a suitable choice of the functionh(
)as introduced inequations (7.4) – (7.6) satisfyingh(0) = 0andh(1) = 1, for example,h (φ α ) = φ αorh (φ α) =

φ2

α (3 − 2φ α), the free energy densityfis an interpolation of the individual free energy densities

f α We can calculate

T α

i h (φ α)



− K i=1

ij depending on the parameter ν For M= 0, theRedlich–Kister–Muggianu ansatz takes the form of a regular solution model In most appli-cations, in particular to metallic systems,M takes a maximum value of two A ternary term

∼ c i c j c kcan be added to describe the excess free enthalpy.

The thermodynamics of the interfaces gives additional contributions to the entropy given by

a Ginzburg–Landau functional of the form

a (
, ∇
) =

α<β

γ αβ (a αβ (q αβ))2|q αβ |2 (7.27)

whereγ αβrepresents the surface entropy density of theα–βinterface andq αβ = (φ α ∇φ β −

φ β ∇φα) is a generalized gradient vector oriented in the direction of the normal to anα–β

interface The formulation using the generalized gradient vectorsq αβallows to distinguish thephysics of each phase (or grain) boundary by providing enough degrees of freedom Anisotropy

of the surface entropy density is modeled by the factor (a αβ (q αβ))2 depending on the entation of the interface Isotropic phase boundaries are realized by a αβ (q αβ) = 1 Weaklyanisotropic crystals with an underlying cubic symmetry can be modeled in 3D by a straightfor-ward extension of the expression in equation (7.10):

αβ , k = 1, , n αβare then αβcorners of the Wulff shape of theα–βtransition leading

to flat crystal faces with sharp edges These evolve in the direction of the cusps In principle,equation (7.29) allows to model arbitrary crystal shapes withn αβcorners For a comparison, we

display in Figure 7-17(a) and (b) the smooth and faceted formulation of the functiona αβ (q αβ)for a cubic crystal symmetry In Figure 7-17(c) the simulation results of two crystals with45◦rotated orientation growing from adjacent nuclei are shown Each grain develops its minimumenergy surfaces in contact with the melt and at their interface

The interfacial entropy density contribution w(
) is a nonconvex function with N

global minima corresponding to theN phases in the system As an extension of the standarddouble well potential in equation (7.9) and Figure 7-4, one may take the standard multiwellpotential

0.5 0 -0.5 -1

FIGURE 7-17 Three-dimensional surface plot of, (a) a smooth and, (b) a faceted cubic anisotropy,

(c) contour plots of two adjacent growing, 45◦ misoriented cubic crystals applying the smooth anisotropy formulation in equation (7.28) with δ= 0.2.

We refer to refs [GNS99a] and [GNS99b] for a further discussion of the properties of thesurface termswst, w˜st, wob, andw˜ob We assume for simplicity thataandwand, hence, theinterfacial contributions to the entropy, do not depend on(T, c).

7.4.3 Evolution Equations

The energy and mass balance equations can be derived from the energy fluxJ0and from the

fluxes of the componentsJ1, , J Kby

0.6 0.8 0.80.60.4 0.2 0

x

y

0

0 (b)

FIGURE 7-18 Plot of the multiwell potential wst (
)for N = 3 and equal surface entropy sities γαβ

den-0.6 0.4 0.2 0

-0.2 -0.4 -0.6

0.4 0.2 0

0.60.80

in such a way that the second law of thermodynamics is fulfilled in an appropriate local version

In order to derive the expressions for the fluxesJ0, , J K, we use the generalized namic potentials

K

i=1

L ij = 0, j = 0, , K

which implies K

i=1 J i= 0, and, hence,∂ t( K i=1 c i ) = ∇ · ( K i=1 J i ) = 0.We further assumethatLis symmetric (Onsager relations) In addition,Lis assumed to be positive semidefinite;that is,

whereδ ijdenotes the Kronecker delta andL α

i are the latent heats of fusion The formulation

in equations (7.35)–(7.37) takes bulk diffusion effects including interdiffusion coefficients intoaccount The dependence of the mass and heat diffusion coefficients on
can be realized by,for example, linear expansions To also consider enhanced diffusion in the interfacial region ofphase or grain boundaries, additional terms proportional toφ α φ βwith interfacial diffusion coef-ficientsD αβ

i (T, c, q αβ)need to be added Altogether, we suggest for mass and heat diffusion

For the nonconserved phase-field variablesφ1, , φ N, we assume that the evolution is such

that the system locally tends to maximize entropy conserving concentration and energy at thesame time Therefore, we postulate

τ ∂ t φ α = ∇ · a ,∇φ α (
, ∇
) − a ,φ α (
, ∇
)

− 1

 w ,φ α (
) − f ,φ α (T, c,
)

where, as already introduced in equation (7.8), we denote witha ,φ α,w ,φ α,f ,φ α, anda ,∇φ αthederivative with respect to the variables corresponding toφ αand∇φα, respectively For mate-rial systems with anisotropic kinetics, the kinetic coefficientτ may depend on the generalizedgradient vectorsq αβin a similar way as the gradient energiesa (
, ∇
)in equation (7.27) Thequantityτ = τ(
, ∇
)in equation (7.40) models an anisotropic kinetic coefficient of the form

αβ.ζ αβdetermines the strength of the kinetic anisotropy similar

toδ αβin equation (7.28) for the surface energy anisotropy Systems with isotropic kinetics arerealized by settingζ αβ = 0 The parameterλis an appropriate Lagrange multiplier such thatthe constraint N

α=1 φ α= 1in equation (7.22) is satisfied; that is,



= v m c v R˜v

With the dimensionless mobility coefficients [equations (7.35)–(7.37)] ˜L00, ˜ L 0j , ˜ L i0 , ˜ L ij of

the form

L00

JKsm



m2s

By making the choiceγ0= c vandτ = τ0

c v, the phase-field equation retains its original form(equation 7.40) with dimensionless quantities instead of the dimensional ones

Finally, if we consider the Gibbs–Thomson equation describing the motion of a sharp face with the curvatureκ

m2K



κ

1m

+([f] β α − i ¯µ i [c i]β α)

T

J

we have = 4.0 · 10 −8m

7.4.5 Finite Difference Discretization and Staggered Grid

The complete set of evolution equations of the model [equations (7.32)–(7.34)] can be treated

in the following divergence form:

All terms including field variables and their spatial derivatives are arguments of the valued function  In case of the phase-field equation [equation (7.34)] the right-hand side(rhs) contains additional source terms, which in general are small by a value compared to thedivergence term, so that the general character of equation (7.45) is kept preserved.

vector-Concerning the time discretization, an explicit forward Euler scheme can be applied with atime update according to

u l n+1 = u l n + ∆t · (rhs) n

(time steps are superscripted) This explicit scheme requires a control of the temporal step width

∆tfor each individual equation For the case of a 3D simulation and an identical grid step width

∆xin each space dimension, the criterion for stability suggests a step width of

∆t ≤ min



∆x26kmax,

∆x26Dmax,

∆x2

6γ αβ

τ0



wherekmaxandDmaxare the maximum values for all heat and mass diffusion coefficients,γ αβ

is the maximum surface entropy coefficient among all appearing phase boundaries, andτ0is thekinetic coefficient

Since the right-hand sides of the conserved order parameter equations [equations (7.32)–(7.33)] as well as the first term in equation (7.34) consist of a divergence term, a two-stepalgorithm can be applied: First, the vector flux quantities are calculated using right-sided finitedifferences and they are stored in a memory buffer for multiple access This buffer holds theflux values of three adjacent 2D layers during the layerwise calculation, shifted through the 3Dgrid along thez-direction In a second step, the divergence is evaluated using left-sided differ-ences This results in an extremely memory saving numerical scheme without any redundantcalculations, since for each field variable only a single 3D array must be stored in memory Thespatial discretization of the phase-field and of the balance equations may be treated differently

as described in the following paragraph

Balance Equations

In the nonlinear energy and mass diffusion equations, the physical diffusion coefficients areincorporated in the Onsager coefficientsL ij, which may depend on
,c, andT The discretiza-tion on a regular grid with spatial indices(i, j, k)for the following simplified energy equation(mass diffusion cross terms have been omitted for clarity)

com-respective intermediate grid positions, in the center boundary of two adjacent grid cells [seeFigure 7-20(b)] For example, thexcomponent of the energy flux reads: