Cross-Permeability of the Semisolid Region in Directional ... · 2/11/2015  · Ni-based super alloys.15,16 Although numerous phase-field studies on direc-tional dendritic solidification

Cross-Permeability of the Semisolid Region in DirectionalSolidification: A Combined Phase-Field and Lattice-BoltzmannSimulation Approach

B. BOTTGER,1,3 C. HABERSTROH,2 and N. GIESSELMANN2

1.—Access e.V, 52072 Aachen, Germany. 2.—Institut fur Industrieofenbau und Warmetechnik,RWTH, 52072 Aachen, Germany. 3.—e-mail: [email protected]

Based on the results of microstructure simulations, fluid flow through thesemisolid region during directional solidification of the technical Ni-base alloy718 has been studied. Three-dimensional microstructures at different posi-tions in the semisolid region were obtained by using a multicomponent mul-tiphase-field model that was online coupled to a commercial thermodynamicdatabase. For the range of five different primary dendrite distances k1 be-tween 50 lm and 250 lm, the flow velocity and the permeability perpendic-ular to the dendrite growth direction was evaluated by using a proprietaryLattice-Boltzmann model. The commercial CFD software ANSYS FLUENTwas alternatively applied for reference. Consistent values of the average flowvelocity along the dendrites were obtained for both methods. From the resultsof the fluid flow simulations, the cross-permeability was evaluated as afunction of temperature and fraction liquid for each of the five different pri-mary dendrite distances k1. The obtained permeability values can beapproximated by a single analytical function of the fraction liquid and k1 andare discussed and compared with known relations from the literature.

INTRODUCTION

In simulation of solidification processes, fluid flowphenomena play a crucial role. Numerous types ofdefects arise during solidification as the formationof dendritic structures in the mushy zone causesspecific fluid flow patterns. Although most of thesedefects are located at the micro- or meso-scale, theycan be linked to the macro-scale by the melt per-meability of the mushy zone. As stated by severalauthors, the specific permeability is a determiningparameter in fluid flow phenomena like freckles, aswell as in defects like porosity and hot crackingamong others.1–4

Improvements of simulation techniques and thecombination of multiple tools working on differentlength scales through Integrated ComputationalMaterials Engineering (ICME)5 open up new pos-sibilities of investigations on the impact of perme-ability on casting defects. Phase-field models havebecome very popular for the simulation of themicrostructure evolution during solidification pro-cesses of alloys. To describe their thermodynamic

properties, idealized descriptions of the phase dia-grams (ideal solution approximation6 and linearphase diagrams7) have been used for binary andpseudo-binary alloys. But this approximation is notsuitable for use in multicomponent multiphasesystems. Instead, using Gibbs energy descriptionsassessed from experimental data via the Calphadapproach,8 together with software tools for Gibbsenergy minimization,9 seems to be most promising.

MICRESS�10 has been developed by Access11 atAachen Technical University (RWTH). It is based ona phase-field concept for multiphase systems thathas been consequently extended to multicomponentsystems12–14 by direct coupling to thermodynamicdatabases via the TQ Fortran interface to Thermo-Calc.9 Since then the software has been developedfurther and applied to different alloy systems and toNi-based super alloys.15,16

Although numerous phase-field studies on direc-tional dendritic solidification have been published,the three-dimensional (3D) simulation of represen-tative parts of the mushy zone for technical multi-component alloys, which are needed for a reliable

JOM, Vol. 68, No. 1, 2016

DOI: 10.1007/s11837-015-1690-3� 2015 The Minerals, Metals & Materials Society

(Published online November 2, 2015) 27

evaluation of the liquid permeability, is still achallenge. The present paper reports phase fieldsimulations carried out with MICRESS� thatallowed subsequent investigations concerning thepermeability during the solidification process. Theresults of the 3D dendrite simulations were sub-jected to further calculations by using the Lattice-Boltzmann method and the commercial CFD soft-ware ANSYS FLUENT. This allowed for assessmentof the relation between temperature, fraction liquidand cross permeability for isothermal fluid flow,which plays a crucial role for freckle formation intechnical remelting processes (e.g., of alloy 718).2

The obtained results are compared with empiricalrelations provided by Poirier17 and Madison.18

PHASE-FIELD SIMULATION

The multiphase-field theory describes the evolu-tion of multiple phase-field parameters /að~x; tÞ intime and space. The phase-field parameters reflectthe spatial distribution of different grains of differ-ent orientation and/or of several phases with dif-ferent thermodynamic properties. At the interfaces,the phase-field variables change continuously overan interface thickness g that can be defined as beinglarge compared with the atomic interface thicknessbut small compared with the microstructure lengthscale. The time evolution of the phases is calculatedby a set of phase-field equations deduced by mini-mization of the free energy functional:13

_/a ¼X

b

Mab ~nð Þ r�ab ~nð Þ Kab þpg

ffiffiffiffiffiffiffiffiffiffiffi/a/b

qDGab ~c;Tð Þ

� �

ð1Þ

Kab ¼ /br2/a � /ar2/b þp2

g2/a � /b

� �: ð2Þ

In Eq. 1, Mab is the mobility of the interface as afunction of the interface orientation, described bythe normal vector ~n. r*ab is the anisotropic surfacestiffness, and Kab is related to the local curvature ofthe interface. The interface, on the one hand, isdriven by the curvature contribution r*ab Kab and,on the other hand, by the thermodynamic drivingforce DGab. The thermodynamic driving force, whichis a function of temperature T and local composition~c ¼ c1; c2; . . . ; ck

� �, couples the phase-field equations

to the multiphase diffusion equations for the kalloying elements:

_~c ¼ rXN

a¼1

/a~Dar~ca with~cdefined by~c ¼

XN

a¼1

/a~ca ð3Þ

and ~Da being the multicomponent diffusion coeffi-

cient matrix for phase a. ~Da is calculated online fromdatabases for the given concentration andtemperature.

These equations are implemented in the softwarepackage MICRESS�10 being used for the simula-tions throughout this paper. Direct coupling tothermodynamic and mobility databases is accom-plished via the TQ-interface of Thermo-Calc Soft-ware.9 The thermodynamic driving force DGab andthe solute partitioning are calculated separately byusing the quasi-equilibrium approach,13 and theyare introduced into the equation for the multiplephase-fields Eq. 1. This allows the software packageto be highly flexible with respect to thermodynamicdata of a variety of alloy systems and not to berestricted by the number of elements or phasesbeing considered. Various extrapolation schemes14

have been implemented to minimize the thermody-namic data handling, especially for complex alloysystems.

Simulation Conditions

For simulation of 3D dendritic microstructures,the smallest representative domain was chosen bytaking advantage of the fourfolded mirror symmetryof the dendrites (Fig. 1). Correspondingly, symmet-ric boundary conditions (Neumann condition withsymmetry plane through the centers of the bound-ary FD grid cells) have been used on the four sidesof the domain for the phase field and the concen-tration fields. For the top concentration boundary, afixed condition was used to ensure that the far-fieldconcentration remains constant (and equal to theaverage alloy composition). At the bottom, an iso-lation (Neumann) boundary condition was used.

Simulation started with a temperature of 1733 Kat the bottom of the domain. A constant tempera-ture gradient of G = 40 K/cm and cooling rate ofV = 0.40 K/s were applied in all cases, ensuring aconstant secondary arm spacing,19 the average ofwhich has been determined as 31.2 lm. TiN parti-cles were nucleated at the beginning according to aseed-density model by using an arbitrary density-radius distribution20,21 and were given time to growand ripen before the primary dendrite appeared. Toreach stationary growth conditions as quickly aspossible, nucleation of the initial seed of fcc phasewas performed exactly at the stationary tip tem-perature of 1618.8 K at the lower left corner of thesimulation domain. Due to the high composition ofalloying elements in alloy 718, stationary growth isachieved almost immediately, and the tip tempera-ture remained at this value until the dendritereached the top boundary.

To reduce the required calculation time, spatialresolution was reduced to a minimum and a gridspacing of Dx = 1.0 lm was chosen. The total heightof the domain was 1000 grid cells or 1 mm, and thewidth varied from 25 to 150 cells, according to aprimary dendrite spacing of 50 lm to 300 lm. Theinterface mobility value for the liquid/fcc boundarywas calibrated such that the tip temperature stayedconstant during the period of free dendrite tip

Bottger, Haberstroh, and Giesselmann28

growth, as will be discussed. This value of0.058 cm4 J�1 s�1 was applied for all simulations.For other interfaces (TiN-liquid, TiN-fcc), themobility values were estimated. Respective numer-ical parameters are given in Table I.

In the phase-field simulations, the elements Ni,Cr, Fe, Nb, Mo, Ti, Al, C, N, and B and the phasesLIQUID, FCC_A1, and FCC_A1#2 (Ti(N,C)) weretaken into account. Thermodynamic data wereobtained directly by coupling to the thermodynamicdatabase TTNI7. Diffusion coefficients for the solidphases were taken from the mobility databaseMOBNI1;9 for the liquid phase, they were estimatedto 1 9 10�5 cm2 s�1 for all elements.

Calibration of Interface Kinetics

It is well known that phase-field models sufferfrom numerical artifacts if spatial resolution is notsufficiently high.22 If the interface thickness is notmuch smaller than the diffusion length of all ele-ments, the interface kinetics deviate from the sharpinterface solution due to artificial solute trapping.As an additional problem, the interface even may

get instable if the driving force varies too much overthe length of the diffuse interface. For the case ofsolidification, there have been attempts to correctfor these artifacts by introducing an anti-trappingcurrent to the diffusion equations and by applying asuitable correction to the interface mobility (the so-called thin interface limit).23 Those approaches areoften referred to as quantitative phase-field models.But even when rigorous thin interface correctionsare considered, the interface thickness has to be onthe order of the capillary length, and the requiredgrid would still be too fine for 3D application of thephase-field method to multicomponent and multi-phase technical alloys.

A typical approach is to compensate the artificialtrapping effect by the choice of suitable interfacemobility values. This has been done recently bycalibration against a representative high-resolutionreference simulation.21 In the present case, how-ever, the high computational effort makes such acalibration impossible, particularly as no substan-tial reduction of the domain size is possible withoutaffecting the selection of the correct stationary tiptemperature. Therefore, in this work, another

Fig. 1. Growth of an alloy 718 dendrite at a primary spacing k1 = 250 lm.

Table I. Numerical parameters

InterfaceInterface

energy (J cm22)Static anisotropy

coefficientInterface mobility

(cm4 J21 s21)Kinetic anisotropy

coefficient

Liquid/fcc 2.0 9 10�5 0.3 0.058 0.3Lquid/TiN 1.0 9 10�4 – 1.0 9 10�8 –fcc/TiN 1.0 9 10�4 – 1.0 9 10�11 –

Cross-Permeability of the Semisolid Region in Directional Solidification:A Combined Phase-Field and Lattice-Boltzmann Simulation Approach

29

approach has been used to obtain quantitativeinterface kinetics: A numerical approach based onKGT theory24 was used to estimate the stationarytip temperature for the given temperature gradientand cooling rate. A pseudo-binary phase diagramdescription in the form

c�s ¼X

i

cis; c�l ¼ c�s þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXicil � cis� �2

qð4Þ

m�s ¼

Pi cil � cis� �

misffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

i cil � cis� �2

q ; m�l ¼

Pi cil � cis� �

milffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

i cil � cis� �2

q ð5Þ

was used for the KGT model, leading to a value of9.6 K for the dendrite tip undercooling. By system-atic variation of the interface mobility of the liquid-fcc interface, a calibrated value of 0.058 cm4 J�1 s�1

was obtained that guaranteed a correct tip temper-ature despite the relatively coarse grid resolution.

Furthermore, an averaging of the driving forceDG along the normal vector of the interface wasperformed to reduce ‘‘artificial solute trapping’’ andto stabilize the interface profile, and a small inter-face thickness (g = 3Dx = 3 lm) was used that fur-ther helps reducing trapping artifacts. Artifactsoriginating from the small number of interface gridpoints were minimized by using a correctionscheme for numerical discretization errors.25

Results of 3D Dendrite Simulations

Figure 1 shows the shape of the growing dendriteas an iso-surface with / ¼ 0:5 at three differenttimes (t = 3, 9, and 37 s). Small spherical TiN par-ticles can be observed in front of the dendrite tip,which have formed at a much higher temperature.

The dendrite emerges and overgrows tertiary armsthat compete with the primary trunk. After hittingthe top boundary, dendrite coarsening is the mainphenomenon that can be observed. Due to theimmense calculation time, only in the case of thesmaller simulation domains (lower primary spac-ing), an overall fraction of solid above 0.5 could bereached.

It is well known that for given G/V ratio, k1 ishistory dependent and may adopt different valueswithin a certain stability range.26 A comparison ofthe simulated growth morphologies of alloy 718dendrites for different values of the primary den-drite spacing k1 is shown in Fig. 2. For the smallestspacing of 50 lm, only small side branches can form,leading to a cellular-dendritic growth mode. With anincreasing value of k1, the side branches evolvemore and more and eventually show tertiarybranching.

Furthermore, these tertiary branches that growin the direction of the temperature gradient com-pete with the primary dendrite. When a primarydistance of k1 = 300 lm is reached, those tertiaryarms cannot be blocked any more. This indicatesthat the upper limit of the stability range has beenpassed. Hence, this simulation set has been dis-carded from further evaluation.

SIMULATION OF PERMEABILITY

In the process of solidification, the mushy zone isa domain of solid phases with emerging dendriticstructures and an interconnected void phase filledwith liquid. It has been proven to be consistent tomodel the mushy zone as a porous medium showinga spatially varying permeability.27

Fig. 2. Comparison of growth morphology for six different values of the primary dendrite spacing.

Bottger, Haberstroh, and Giesselmann30

As the criteria of a large area of porous mediumand negligible boundary effects are fulfilled, theinterdendritic flow follows Darcy’s law.28 Therefore,the pressure gradient rp is proportional to the flowvelocity vector �v:

rp ¼ lK

�v: ð6Þ

Here, l is the (dynamic) viscosity and K is the per-meability. The dendritic structure determined byphase-field simulations follows the assumption of asymmetric array of dendrites. Therefore, a periodicboundary condition can be used to calculate numeri-cally the fluid flow in a specific symmetric volume.

The calculations are carried out by using theLattice-Boltzmann method29 or alternatively theCFD software ANSYS FLUENT. As the velocitiesand characteristic lengths are very small, in theorder of 10�8 m/s and 10�5 m, respectively, a lami-nar, steady flow can be assumed in both models.

ANSYS FLUENT solves continuity Eq. 7 andmomentum Eq. 8 equations on a finite volume grid:

@q@t

þr q�vð Þ ¼ 0 ð7Þ

��s ¼ lD r�vþr�vT� �

� 2

3r �v��eð Þ

� �: ð8Þ

Here, q is the melt density, ��s the stress tensor, lD thedynamic viscosity, and ��e the unity tensor. Solvingthese equations numerically has been proven valid inmany cases, especially considering a laminar, steadyfluid flow. Analogous to the Lattice-Boltzmannmodel, a periodic flow perpendicular to the primarydendrites is being assumed. Due to the high geo-metric complexity of the dendritic structure, a stairstep grid is generated including only cells above aspecific fraction liquid. This specific fraction liquidhas been chosen in a manner in which the averagefraction liquid is being conserved.

The Lattice-Boltzmann method, however, is foun-ded on particle interaction on a microscopic and amesoscopic level. It can be derived from the lattice gasautomata (LGA) on the approach of fluid or gas as abig number of randomly moving particles that allowsthe use of Navier–Stokes equations on a macroscopicscale.30 A transformation of the particles into a sin-gle-particle distribution function results in the latticeBoltzmann equation (LBE). Several different LBE

models denoted as DdQq exist, where d refers to thedimension and q to the amount of velocity direc-tions.31 The following study uses the LBE modelD3Q19, a 3D space with 19 orthogonal velocity vec-tors shown in Fig. 3. Between the several existingcubic models, it is considered to combine both a highcomputational reliability and efficiency, therebypromising consistent results.30

For athermal fluids, the equilibrium distributionfunction introduced by He and Luo can be appliedfor D3Q19:29

f eqi ¼ qwi 1 þ 3

c2ei � uþ 9

2c4ei � uð Þ2� 3

2c2u � uð Þ

� �:

ð9Þ

The function depends on the weighting factor wi,the discrete velocity ei, and the lattice speed c.Hereby is lattice speed, the quotient of the latticeconstant and the time step: c ¼ dx=dt. The weightingfactor and the set of discrete velocities vary with therespective model; in the case of D3Q19, they aredefined as follows. The lattice structures with thediscrete velocity vectors ei are shown in Fig. 3.30

wi ¼1=3 i = 0 rest particle1=18 i = 1,2, . . . 6; group I1=36 i = 7,8; . . .18; group II

:

8<

: ð11Þ

This LBG model does not require a scattered matrixbut uses a quasi-linear LBE variation. This impliesa change from a multi-to a single-time relaxation

ei ¼ð0; 0; 0Þ i = 0 rest particleð�1; 0; 0Þc; ð0; �1; 0Þc; ð0; 0; �1Þc i = 1,2, . . . 6; group Ið�1; �1; 0Þc; ð0; �1; �1Þc; ð�1; 0; �1Þc i = 7,8; . . . 18; group II

8<

: ð10Þ

Fig. 3. Lattice structure of the D3Q19 (Ref. 29).

Cross-Permeability of the Semisolid Region in Directional Solidification:A Combined Phase-Field and Lattice-Boltzmann Simulation Approach

31

that gives a fixed time scale for all modes. Theindividual particle motion is neglected; the particlesinstead follow an ensemble average.27,32 As therequirements of mass and momentum conservationare no longer guaranteed through sum rules of theformerly scattered matrix, local equilibrium has tobe imposed by the definition of the local density q(12) and momentum q�u (13) as particle velocitymoments of the distribution function fi.

27 Eq. 12enables us to step back from a discrete microscopicvelocity to a continuous macroscopic velocity, suchas the fluid motion.33

q ¼XN

i¼0

fi ð12Þ

q�u ¼XN

i¼0

fiei: ð13Þ

The discretization in space and time allows the fit-ting of the LBEs in the collision model of Bhatna-gar-Gross-Krook (BGK), resulting in the latticeBhatnagar-Gross-Krook (LBGK) approximation forthe contribution function fi:

fi xþ ei; tþ 1ð Þ � fi x; tð Þ ¼ � 1

sfi x; tð Þ � f eqi x; tð Þ� �

:

ð14Þ

Hereby, fieq is the equilibrium value and the char-

acteristic time s ¼ k=dt, with k the relaxation time.The LBGK model has become widely used due to itseasy implementation and numerical effectiveness.In Eq. 14, two steps are implemented, equatingstreaming with collision. While in the streamingstep the distribution function fi moves to the nextlattice place, the collision step relaxes fi toward fi

eq.Both the step of streaming and the collision of thefluid particles have to be separately approached tosolve the LBGK Eq. 14.33 Although the collisionstep is entirely local, the streaming step is consis-tent. The simulation of solid boundaries follows thebounce back condition where particles are reverselybounced back from the wall points. The bounce backboundary condition is integrated in the collisionstep.

RESULTS

The original results from the MICRESS� simu-lations comprise precise fraction liquid data for allgrid cells. For simulation of fluid flow, the fractionliquid information had to be reduced to zero or onein each cell. This was achieved by using a thresholdvalue for the fraction liquid close to 0.5 below whichthe cell was assumed to be completely solid andotherwise completely liquid. This threshold valuewas chosen such that the total fraction liquid in thewhole simulation domain was conserved. Duringthis procedure, all TiN particles that were not in

contact with the fcc dendrite were converted intoliquid phase because they moved with the melt flowand such did not contribute to the pressure drop.

Figure 4 illustrates the smallest representativedomain for simulation of fluid flow across the mushyzone when a symmetric structure of the dendritenetwork is assumed. Accordingly, the computa-tional domain of the phase-field simulations had tobe doubled by mirroring in the x-direction.

With the described procedures, according to thevalue of k1 in the phase-field simulation, cubicdomains with a height of 1000 cells and base areasbetween 50 9 25 and 250 9 125 cells with a cell sizeof 1 lm3 were obtained that comprise a sharpinterface between solid and liquid. These weredirectly used as representative domains for CFDcalculation of the permeability of the mushy zone fordifferent temperature and primary spacing.

The presented equations and considerations leadto the velocities in x-, y-, and z-direction. As thegeneral stream direction is assumed normal to theprimary dendrite arms (Fig. 5), the velocity in the x-direction is the decisive factor. The permeabilitycoefficient represents a characteristic value thatdoes not require additional information of theinternal structure of the porous media; only thevelocity’s x-component on the periodic faces of thecomputational domain is crucial. The velocitiesincrease near the dendritic solidified structure asthe flow cross section becomes smaller.

To check the accuracy of the results, some of theCFD calculations were done by using both ANSYSFLUENT and the Lattice-Boltzmann method. Fur-thermore, the grid resolution was doubled (‘‘fine’’resolution, Dx = 0.5 lm) to reduce discretizationerrors. A very high conformity of the results provedevidence of correctness and applicability of bothapproaches. Figure 6 shows the average flow veloc-ities obtained with the different methods andresolutions.

Therefore, for the further simulation, the averagevelocities (v) at the left border of the mesh gridobtained by using the Lattice-Boltzmann method and

Fig. 4. Symmetry of the dendrite array assumed for fluid flowsimulation.

Bottger, Haberstroh, and Giesselmann32

the coarse grid were identified as representative forthe velocity trend. With a viscosity of l¼ 0:009 Pa � s

and a pressure gradient of rp ¼ 200Pa=m, the per-meability is presented in Eq. 15:

kx ¼l � vmw

rpm2� �

: ð15Þ

Due to the dendritic structure including the sec-ondary arms, the fraction liquid presents a strongfluctuation along the z-coordinate. Therefore, the

liquid fractions as well as the velocity are averagedover the distance of two secondary dendrite armspacings (k2).

Calculation of fluid flow was done for the fivephase-field simulations for k1 = 50, 100, 150, 200,and 250 lm which are shown in Fig. 2. For eachcase, the results at 4–12 different intermediateoutput times were selected to span the temperaturerange between the dendrite tip and the lowesttemperature reached in each phase-field simulation(Fig. 7). Note that for large values of k1, only infor-mation for higher temperatures could be obtaineddue to the high computation times of the corre-sponding phase-field simulations.

To allow for a comparison of the findings toexisting empirical relations, the simulated perme-ability values also have been plotted as a function ofthe liquid fraction (Fig. 8). Although ford2 = 100 lm, d3 = 150 lm, d4 = 200 lm, andd5 = 250 lm, the differences are only small, for thelowest value of k1 (d1 = 50 lm), considerably highervalues of the permeability are obtained. This can beexplained with the mostly cellular morphologyobtained under these conditions (Fig. 2).

The permeability as a function of fraction solidcan be approximated by the following relation,which has been obtained empirically (Fig. 8):

kx ¼10�10:3 � fL � d0:6�ð0:9�fLÞ

i 0:3 � fL � tv

10�5:8 � fL1�fL

0:7 � 4

� 10�4 þ di

10

tv � fL � 1

8<

: :

ð16Þ

Fig. 6. Comparison between left: ANSYS FLUENT and the Lattice-Boltzmann method, right: normal and double grid resolutions for a simulationtime of 20 s and k1 = 150 lm.

Fig. 5. Representation of the 3D flow field around the dendrite arms.

Cross-Permeability of the Semisolid Region in Directional Solidification:A Combined Phase-Field and Lattice-Boltzmann Simulation Approach

33

The transition between the steep curve at highfraction liquid values and the continuous descent inthe lower fraction liquid region can be related to thetransition between the free growth of the dendritetip and a restricted growth once the dendrite sidebranches interact with the neighboring dendrite.Not only the temperature for this transition (as canbe clearly seen in Fig. 2) but also the correspondingfraction liquid value tv is found to vary with theprimary dendrite arm spacing:

tv ¼ 3194 � d2i � 7:481 � di þ 0:888: ð17Þ

DISCUSSION

It is interesting to compare the results of thecurrent study with well-established empirical rela-tions that are frequently used for mesoscopic ormacroscopic simulation of fluid flow in semisolidregions (e.g., for predicting freckles, microporosity,

or hot cracking in different types of casting).Figure 9 puts our results in contrast with theequations presented by Madison et al.18 and Poirieret al.17 For all three cases, the curves are shown fork1 = 100, 150, 200, and 250 lm (d2–d5), plotted as afunction of the fraction liquid fL.

Although there is very good agreement for thecross-permeability curves for intermediate values ofk1 (150–200 lm) and the fraction liquid range below0.65, there are strong deviations otherwise. Moststriking is the steep decay of the cross-permeabilitycurve in the current results occurring already athigh values of fL (>0.85), whereas it was predictedby Poirier for the range 0.65< fL< 0.75.17 Thedecay is attributed to the transition between free

0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-13

10-12

10-11

10-10

10-9

10-8

10-7

Fraction Liquid

Per

mea

bilit

y [m

2 ]

0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8

10-11

Fraction Liquid

Per

mea

bilit

y [m

2 ]

Fig. 8. Averaged permeability for different primary spacings k1 as a function of liquid fraction. Approximation curves according to Eq. 16 havebeen included.

Fig. 7. Averaged permeability for different primary spacings k1 as afunction of temperature.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

Fraction Liquid

Per

mea

bilit

y

Current Study

Porier (1987)

Current StudyMadison et al (2012)Poirier (1987)

Fig. 9. Permeability curves comparing Madison et al.,18 Poirier,17

and the current study (inside each series, the top curve is for d5(k1 = 250 lm), followed by d4 (k1 = 200 lm), d3 (k1 = 150 lm), andd2 (k1 = 100 lm)).

Bottger, Haberstroh, and Giesselmann34

growth in the dendrite tip region and restrictedgrowth below and (at least under these conditions)is wrongly predicted by the empirical relation ofPoirier by up to a factor of 100 for high fractionliquid values. For prediction of fluid-flow phenom-ena in the upper part of the mushy zone (e.g., theinitiation of freckles during electro-slag remelting(ESR)),2 this can have significant consequences.

The second strong difference is the much smallerdependency of the cross-permeability on k1 in ourstudy. This can be easily explained by the fact thatin our phase-field simulations, the primary dendritedistance k1 was modified ‘‘artificially’’ withoutchanging the local thermal parameters (tempera-ture gradient and cooling rate). To the contrary, theformulae by Poirier17 and Madison18 assume thechange of k1 to be attributed to a change of the localthermal condition (i.e., they assume that the den-drites have selected their primary distance accord-ing to the local conditions). Thus, no similarityshould be expected with respect to the k1-depen-dency of the cross-permeability because the maineffect, which is the change of k2 due to an alteredlocal solidification time, has not been included inour study. Direct comparisons of the datasets shouldonly be made for average (stable) values of the pri-mary distance (k1 = 150–200 lm).

Although for application a representation as afunction of temperature (Fig. 7) may be most prac-tical, plotting permeability in terms of the interfa-cial surface area (ISA) has been proposed as a morecomprehensive description.34 Whether the ISA,which can be easily evaluated from simulatedmicrostructures, provides a suitable metric for pre-diction of permeability will be the subject of futureresearch.

CONCLUSION AND OUTLOOK

The coupling between the 3D phase-field simula-tion of multicomponent alloy microstructures byusing MICRESS� and fluid-flow simulations carriedout with ANSYS FLUENT or the Lattice-Boltzmannmethod presents an important contribution to thestill developing and improving field of computa-tional simulation of casting processes in an ICMEframework. Besides the methodological aspects, thestudy also gives new knowledge and insight into thecross-permeability of the interdendritic zone andpoints out the opportunities offered by simulationapproaches. By direct simulation of the dendritemorphology and its change downward through themushy zone and by a systematic variation of theprimary dendrite distance in the phase-field simu-lations, the dependency of the effective cross-per-meability on the fraction liquid and the effect of theprimary dendrite distance could be quantitativelyevaluated for the technical Ni-base alloy 718 andcompared with empirical relations from theliterature.

The study not only presents a methodology to linkdifferent software tools and exchange data betweenthem but also proposes a way to aggregate detailedresults on the microscale and transform them bysystematic variation and averaging into empiricallaws that can be applied on the macroscale. Thepath is now open, and more dimensions can beincluded by variation of other parameters likecooling rate, temperature gradient, or alloy compo-sition to obtain more comprehensive description ofthe permeability and other properties of the mushyzone.

ACKNOWLEDGEMENT

The author B. Bottger acknowledges funding bythe German Research Foundation (DFG). Parts ofthis work were conducted in the context of theCollaborative Research Centre SFB1120 ‘‘PrecisionMelt Engineering’’ at RWTH Aachen University.

REFERENCES

1. D. Ma, J. Ziehm, W. Wang, and A. Buhrig-Polaczek, IOPConf. Ser. Mater. Sci. Eng. 27, 012034 (2012).

2. B. Bottger, G.J. Schmitz, F.-J. Wahlers, J. Klower, J. Tewes,and B. Gehrmann, High Temp. High Press 42, 115 (2013).

3. B. Goyeau, T. Benihaddadene, D. Gobin, and M. Quintard,Metall. Mater. Trans. B 30, 1107 (1999).

4. R. Nadella, D. Eskin, and L. Katgerman, Mater. Sci. Tech-nol. 23, 1327 (2007).

5. U. Prahl and G. Schmitz, Integrative Computational Mate-rials Engineering: Concepts and Applications of a ModularSimulation Platform (Weinheim: Wiley, 2012).

6. B. Nestler and A.A. Wheeler, Physica D 138, 114 (2000).7. J. Tiaden, B. Nestler, H.J. Diepers, and I. Steinbach, Phys-

ica D 115, 73 (1998).8. N. Saunders and A. Miodownik, CALPHAD Calculation of

Phase Diagrams: A Comprehensive Guide (New York: Else-vier, 1998).

9. Themo-Calc Software, http://www.thermocalc.se.10. http://www.micress.de. Accessed 29 Oct 2015.11. www.access-technology.de. Accessed 29 Oct 2015.12. B. Bottger, U. Grafe, D. Ma, and S.G. Fries, Mater. Sci.

Technol. 16, 1425 (2000).13. J. Eiken, B. Bottger, and I. Steinbach, Phys. Rev. E 73,

066122 (2006).14. B. Bottger, J. Eiken, and M. Apel, Comput. Mater. Sci. 108,

283 (2015).15. J. Rosler, M. Gotting, D. Del Genovese, B. Bottger, R. Kopp,

M. Wolske, F. Schubert, H.J. Penkalla, T. Seliga, A. Thoma,A. Scholz, and C. Berger, Adv. Eng. Mater. 5, 469 (2003).

16. B. Bottger, M. Apel, B. Laux, and S. Piegert, IOP Conf. Ser.Mater. Sci. Eng. 84, 012031 (2015).

17. D.R. Poirier, Metall. Trans. B 18, 245 (1987).18. J. Madison, J.E. Spowart, D.J. Rownhorst, L.K. Aagesen, K.

Thornton, and T.M. Pollock, Metall. Mater. Trans. A 43, 369(2012).

19. W. Kurz and D.J. Fisher, Fundamentals of Solidification(Switzerland: Trans Tech Publications, 1989), p. 85.

20. B. Bottger, J. Eiken, and I. Steinbach, Acta Mater. 54, 2704(2006).

21. B. Bottger, M. Apel, B. Santillana, and D.G. Eskin, Metall.Mater. Trans. A 44, 3765 (2013).

22. A. Karma and W.J. Rappel, Phys. Rev. E 57, 4323 (1997).23. S.G. Kim, Acta Mater. 55, 4391 (2007).24. W. Kurz, B. Giovanola, and R. Trivedi, Acta Metall. 34, 823

(1986).25. J. Eiken, IOP Conf. Ser. Mater. Sci. Eng. 33, 012105 (2012).26. D. Ma, Metall. Mater. Trans. B 35, 735 (2004).

Cross-Permeability of the Semisolid Region in Directional Solidification:A Combined Phase-Field and Lattice-Boltzmann Simulation Approach

35

27. S. Chen and G.D. Doolen, Ann. Rev. Fluid Mech. 30, 329(1998).

28. K. Vafai, Handbook of Porous Media (Boca Raton, FL:Taylor & Francis Group, 2005).

29. X. He and L. Luo, Phys. Rev. E 56, 6811 (1997).30. M. Renwei, W. Shyy, and Y. Dazhi, ICASE Report No. 2002–

2017.31. Y.H. Qian, D. d’Humieres, and P. Lallemand, Europhys.

Lett. 17, 479 (1992).

32. S. Succi, The Lattice Boltzmann Equation for FluidDynamics and Beyond (Numerical Mathematics and Scien-tific Computation) (New York: Oxford University Press,2001).

33. M.C. Sukop and D.T. Thorne Jr, Lattice Boltzmann Model-ing—An Introduction for Geoscientists and Engineers (Ber-lin: Springer, 2006).

34. J. Madison, J. Spowart, D. Rowenhorst, L.K. Aagesen, K.Thornton, and T.M. Pollock, Acta Mater. 58, 2864 (2010).

Bottger, Haberstroh, and Giesselmann36