An Enthalpy Model for Dendrite Growth Vaughan Voller

University of Minnesota

Growing Numerical CrystalsObjective: Simulate the growth (solidification) of crystals from asolid seed placed in an under-cooled liquid melt

Some Physical Examplessnow-flakes-ice crystals

Germs

Dendrite grains in material systems

(IACS), EPFL science.nasa.gov Voorhees,Northwestern

l

TLe

~0.5 m ~5 mmComputational grid size

Process REV

~ 1 m

In terms of the process

Sub grid scale

Simulation can be achieved using modest models and computer power

Growth of solid seed in a liquid melt Initial dimensionless undercooling T = -0.8 Resulting crystal has an 8 fold symmetry

Solved in ¼Domain withA 200x200 grid

Box boundariesare insulated

Since thethermal boundarylayer is thin Boundariesdo not affect growth until seed approaches edges

By changing conditionscan generate any number of realistic shapes in modest times

PC CPU ~5mins

BUT—WHY do we get these shapes ?—WHAT is Physical Bases for Model ?Solution is with a FIXED grid -HOW does the Numerical solution work ?

IS the solution “correct” ?

Complete

garbage

First recognize that there are two under-coolings

The bulk liquid is under-cooled, i.e., in a liquid statebelow the equilibrium liquidus temperature TM

The temperature of the solid-liquid interface is undercooled

nM

L0Merfaceinterfaceint v

LT)CC(mTTT

Conc. of Solute, m < 0 slopeof liquidus Gibbs Thomson

curvature, surface tension

Kinetic vn normal interfacevelocity

MinitialBulk TTT

Can describe process with the Sharp Interface Model—for pure materialWith dimensionless numbers

c/LTTT m

* 2

** tt,xx

Assumed constant properties

Insulated domain Initiated with small solid seeds

2s TtT

l2l Tt

T

nvn

Capillary length~10-9 for metal

On interface

nls vTT nn

)4cos151(dT o

Angle between normal and x-axis

The heat flows from warm solid intothe undercooeld liquid drives solidification

Preferred growth direction and interplay betweencurvature and liquid temperature gradient determine Growth rate and shape.

)2

ntanh(121)t,y,x(f

Smear out interface

n

1f

0f

Use smooth interpolation forLiquid fraction f across interface

f=1f=0

A Diffusive interface model: Usually implies Phase Field—Here we mean ENTHALPY (see Tacke 1988, Dutta 2006)

,TtH 2

fTH

1f,1H1f0),f(T0f,H

TThe Enthalpy -sum of sensible and latent heats-change continuously and smoothly throughout domain—Hence can write Single Field Eq. For Heat transport

undercooling

Can be solved on a Fixed grid

Growth of Equiaxed CrystalIn under-cooled melt

A microstructure model

Phase change temperaturedepends on interfacecurvature, speed and concentration

Sub-grid modelsAccount for Crystal anisotropyand “smoothing” of interface jumps

)4cos151()( Four fold symmetry

Sub grid constitutive

TtH 2

fTH fHT If f= 0 or f = 1 If 0 < f < 1

2/32y

2x

yy2xxyyxxx

2y

)ff(fffff2ff

)ff(tany

x1curvature

Local direction

)(dT o

Capillary length 10-9 m in Al alloys

ENTHALPY

Numerical Solution Very Simple—Calculations can be done on regular PC

seed

Typical gridSize 200x200¼ geometry

At end of time step if solidificationCompletes in cell iForce solidification in ALL fullyliquid neighboring cells.

Physical domain ~ 2-10 microns

Initially insulated cavity contains liquid metal with bulk undercoolingT0 < 0. Solidification induced by placing solid seed at center.

Some Results

Problem: range of cells with 0 < f < 1 restricted to width of one cell Accuracy in curvature calc?

Remedial scheme: smear out f value, e.g.,

NEPmod

P f)1(ff

Remedial Scheme: Use nine volume stencil to calculate derivatives

2SWSSEEPWNWNNE

2

2

x)21()ff2f()ff2f()ff2f(

xf

Tricks and Devices

Produces nice answers BUT are they correct

4do (blue)3.25do (black)

2.5do (red)

Dendrite shape with 3 grid sizes shows reasonable independence

= 0.05, T0 = -0.65

Dimensionless time = 6000

= 0.25, = 0.75

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 1000 2000 3000 4000 5000 6000

Dim. Time

Dim

. Tip

Vel

.

Solvabillity (kim et al)

Long term tip dynamics approaches theory

BUT results begin to deteriorate if grid is made smaller !!

0

2

4

6

8

10

12

14

16

0 50 100 150 200 250

,t2s

0T)(erfce 02

Verify solution coupling by Comparing with one-d solidification of an under-cooled melt

T0 = -.5

Compare with Analytical Similarity Solution Carslaw and Jaeger

Temperatureat dimensionless time t =250

Front Movement

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 20 40 60 80 100 120 140 160

)),t(sx(,)(erfct2

xerfcTTT 00

Verification 1 Looks Right!!

k = 0 (pure), = 0.05, T0 = -0.65, x = 3.333d0

Enthalpy Calculation

Dimensionless time = 0 (1000) 60002

odtk

k = 0 (pure), = 0.05, T0 = -0.55, x = d0

Level Set Kim, Goldenfeld and Dantzig

Dimensionless time = 37,600

75.0,do5.2x Red my calculation for these parameters With grid size

The Solid color is solved with a 45 deg twist on the anisotropy and then twisted back—the white line is with the normal anisotropy

,75.0)0(,95.0)45(

,25.0

0

0

Note: Different “smear” parameters are usedin 00 and 450 case

0

50

100

150

200

250

300

350

0 2000 4000 6000

Dim. Time

Tip

Pos.

45 deg. twist in anisotropy

Tip position with time

Dimensionless time = 6000

= 0.05, T0 = -0.65

Not perfect: In 450 case the tip velocity at time 6000 (slope of line) is below the theoretical limit.

Low Grid Anisotropy

m

For binary system need to consider – solute transport, discontinuous diffusivities and solutes

k)k1(fCV

fDkD)f1(DfCC)f1(C

ls

ls

Use smoothly interpolatedVariables across the interface

isC

ilC

)CD(tC

sss

)CD(tC

lli

nlllss v)k1(CCDCD nn

Sharp Interface Model

il

is kCC

LT)CC(mTT M

L0Merfaceint

Single Domain Eq.

)VD(tC

)V1(MC)4cos151(dT

0

o

Comparison with one-d Analytical Solution

Constant Ti, Ci

k = 0.1, Mc = 0.1, T0 = -.5, Le = 1.0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 10 20 30 40x

C

T/T0k = 0.1MC0 = 0.1T0 = -0.5

Le = 1.0

Concentration and Temperatureat dimensionless time t =100

0

2

4

6

8

10

12

0 100 200 300

timeps

oitio

n

Front Movement

Effect of Lewis Number: small Le interface concentration close to C0

0Le

10Le

1Le

2.Le

k = 0.15, Mc = 0.1, T0 = -.65= 0.05, x =3.333d0

l

le DL

All predictions attime =6000

0

0.5

1

1.5

2

2.5

3

3.5

4

0 50 100 150 200 250 300 350 400 450 500

distance

Conc

entra

tion

k = 0.15, Mc = 0.1, T0 = -.55, Le = 20.0

= 0.02, x = 2.5d0

Concentration field at time = 30,000

,1,25.0

Profile along dashed line

Concentration

= 0.05, T0 = -0.65 time = 6000

= 0.25, = 0.75, x =4d0

FAST-CPUThis

On This

In 60 seconds !

Conclusion –Score card for Dendritic Growth Enthalpy Method (extension of original work by Tacke)

Ease of Coding ExcellentCPU Excellent (runs

shown here took between 1 and 2 hours on a regular PC)

Convergence to known analytical sol.

Excellent

Convergence to known operating state

Very Good (if grid is not too fine and remedial parameters well chozen

Grid Anisotropy Good (see comment above)

Alloy Promising

0

2

4

6

8

10

12

14

16

0 50 100 150 200 250

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.090.1

0 1000 2000 3000 4000 5000 6000

Dim. Time

Dim

. Tip

Vel

.

Playing Around

A Problem with Noise Multiple Grains-multiple orientations

Grains in A Flow Field

Thses calculations were performed by Andrew Kao, University of Greenwich, LondonUnder supervision of Prof Koulis Pericleous and Dr. Georgi Djambazov.

Can this work be related to other physical cases ?

Extensions:Grain Growth

Couple with porosity formation ?

100 mseconds

100 km

1000’s of years

e.g., shoreline in sedimentary basin

NCED’s purpose:to catalyze development of an integrated, predictive science of the processes shaping the surface of the Earth, in order to transform management of ecosystems, resources, and land use

The surface is the environment!

Research fields• Geomorphology• Hydrology• Sedimentary geology• Ecology• Civil engineering• Environmental economics• Biogeochemistry

Who we are: 19 Principal Investigators at 9 institutions across

the U.S.

Lead institution: University of Minnesota

Fans Toes Shoreline

Two Sedimentary Moving Boundary Problems of Interest

Moving Boundaries in Sediment Transport

1km

Examples of Sediment FansMoving Boundary

How does sediment-basement interfaceevolve

Badwater Deathvalley

sediment

h(x,t)

x = u(t)

0q

bed-rock

ocean

x

shoreline

x = s(t)

land surface

A Sedimentary Ocean Basin

An Ocean Basin

Melting vs. Shoreline movement

pressurizedwater reservoir

to water supply

solenoidvalve

stainless steelcone

to gravel recycling

transport surface

gravel basement

rubber membrane

experimental deposit

Experimental validation of shoreline boundary model

~3m

Base level

Measured and Numerical results ( calculated from 1st principles)

1-D finite difference deforming grid vs. experiment

xxt+Shoreline balance

Is there a connection

Grain Growth in Metal SolidificationFrom W.J. Boettinger

m

10km

“growth” of sediment delta into oceanGanges-Brahmaputra Delta