GROWING NUMERICAL CRYSTALS

Vaughan Voller, and Man Liang, University of Minnesota

We revisit an analytical solution in Carslaw and Jaeger for the solidification of an an under-cooled melt in a cylindrical geometry.

We show that when the one-d axi-symmetry is exploited a fixed grid enthalpy model Produces excellent results.

BUT—when a 2-D Cartesian solution is sort—”exotic” numerical crystals grow.

This IS NOTA numerical crystal

The Carslaw and Jaeger Solution for a cylindrical sold seed in an under-cooled melt

Consider a LIQUID melt infinite in extentAt temperature T< 0BELOW Freezing Temp

At time t = 0 a solid seed attemperature T = 0 is placed in the center

This sets up a temp gradient that favorsthe growth of the solid

Rrr

Tr

rrt

T

,1

,

,0 0

t

R

r

T

TrTT

R

Similarity Solution

x

de

zE1

x

exE

dx

d x1

Exponential integral

Also develop similarity solutions forplanar and spherical case

Rr

E

trETT

RrT

C

,)(

)4/(0,0

21

210

0

tR C2

0)( 022

1

2

TeE CCC

Assume radius grows as

Then

With c Found from

Enthalpy Solution in Cylindrical Cordiantes

Assume an arbitrary thin diffuse interface whereliquid fraction

01 f

fTH Define

Throughout Domain a single governing Eq

0,1

rr

Tr

rrt

H

0lim0

r

Tr 0TrT

Numerical Solution Very Straight-forward

)(1

112

ii

noutii

ninn

ii

newi TTrTTr

rr

tHH

10 ifIf ]1],0,(max[min newnewi Hf

newi

newi

newi fHT

Initially

99.,0

15.0,1,5.0

11

0

fT

HfTT iii

seed

Set

Transition: When

1 and 0 1 newi

newi ff 99.0 1

newif

R(t)

0

2

4

6

8

10

12

14

0 20 40 60

time

soli

d-fr

ont R

(t)

Enthalpy

Analytical

Excellent agreement with analytical when predicting growth R(t)

Rrr

Cr

rrLet

C

,11 2

2

dt

dRCk

r

C

rCCRC

fix

R

fix

)1(

,1,

Similarity and enthalpy solutions can be extended to account for a binary alloyand a spherical seed

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3 4 5 6

position

conc

entr

atio

n

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

3 4 5 6

position

tem

pera

ture

0

0.5

1

1.5

2

2.5

0 20 40 60

position

conc

entr

atio

n

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 20 40 60

position

tem

pera

ture

Concentration and Temperature Profiles for spherical seed at time

Time 20, Le= 50 Time 250, Le= 2

)()(2 PEPWPSPNi

newi TTTTTTTT

tHH

10 ifIf ]1],0,(max[min newnewi Hf

newi

newi

newi fHT

A 2-D Cartesian application of enthalpy model

Start with a single solid cell

When cell finishes freezing “infect” -- seed liquid cells in mane compass directions

Initial-Seed Infection

This choice will grow a fairly nice four-fold symmetry dendritic crystal is a stableconfiguration

(1) where is the anisotropy

(2) Why is growth stable (no surface tension of kinetic surface under-cooling)

Pleasing at first!!! But not physically reasonable

1. The initial seed, grid geometry and infection routine introduce artificial anisotropy

2. The grid size enforces a stable configuration—largest microstructure has to be at grid size

Initial-Seed Infection

Demonstration of artificial anisotropy induced by seed and infection routine

Similarity

Solution

Similarity

Solution

tR C2

Serious codes impose anisotropy-and include surface tension and kinetic effects

But the choice of seed shape and grid can (will) cause artificial anisotropic effects

With the Cartesian grid Hard to avoid non-cylindrical perturbations Which will always locate in a region favorable for growthIf imposed anisotropy is weak this feature will swampPhysical effect and lead to a Numerical Crystal

Numerical Crystal

Can The similarity solution be used to test the intrinsic grid anisotropy in numerical crystal growth simulators ?

Conclusions:

1. Growth of a cylindrical solid seed in an undercooled binary alloy melt can (in the absence of imposed anisotropy, surface tension and kinetic effects) be resolved with a similarity solution and axisymmetric enthalpy code

0

2

4

6

8

10

12

14

0 20 40 60

timeso

lid-

fron

t R(t

)

Enthalpy

Analytical

2. On Cartesian structured grid however the enthalpy method breaksdown and—due to artificial grid anisotropy grows NUMERICAL crystals stabilized by grid cell size.

3. Similarity solutionstringent test of abilityOf a given method to suppress grid anisotropy

What about unstructured meshes ? We get a sea-weed pattern

Can we use this as a CA solver for channels in a delta