Growing Numerical Crystals Vaughan Voller, SAFL Feb 1 2006.

Growing Numerical Crystals Vaughan Voller, SAFL Feb 1 2006

Growing Numerical Crystals

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

Some Physical Examples

snow-flakes-ice crystalsGerms

Dendrite grains in metal systems

(IACS), EPFL science.nasa.gov

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

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 ModelHOW does the Numerical solution work, IS the solution “correct”

Complete

garbage

A Physical Basis

What do I mean when I say that the bulk liquid is undercooled

Bulk Undercooling

We think of a solid changing to a liquid at a single equilibrium temperature Tm (e.g., in ice-water Tm = 273.2 K)

Consider the liquid volume following the cooling curve below

Time t

T

Tm

As volume tempDrops below Tm wewould expect it to change tosolid

But due to the thermodynamicsof the phase change it is more Than likely that the liquid will become undercooledi.e. remain in the liquid state until a temperture Tu below Tm

is reached

Bulk Undercooling

We think of a solid changing to a liquid at a single equilibrium temperature Tm (e.g., in ice-water Tm = 273.2 K)

As volume tempDrops below Tm wewould expect it to change tosolid

Time t

T

Tm

Tu

But due to the thermodynamicsof the phase change it is more Than likely that the liquid will become undercooledi.e. remain in the liquid state until a temperture Tu below Tm

is reached

Consider the liquid volume following the cooling curve below

STHG u

T

Tm

Tu

The undercooling behavior can be quantifiedBy considering the thermodynamic state of a system givenby the the Gibbs free energy (free enthalpy) G

Enthalpy (totalheat) larger inliquid due tolatent heat

Entropy-measureof chaos-largerin liquid (less atomicstructure)

LuLL STHG

SuSS STHG

LuLL STHG

mu TT

SuSS STHG

A plot of the molar Gibbs free energies of the liquid and solid states looks like

LIQUID SOLID

A substance will alwaystend to find a state that minimizes G

Liquid here is meta-stable –given “encouragement”will transform to solid state.

Tm

Mola

r Fr

ee E

nerg

y

GL

GS

T

Mola

r Fr

ee E

nerg

y

GL

GS

T

TTm

Liquid at undercooled temperature is meta-stable –-given “encouragement” will transform to solid state.

TTT mu

Tu

The process can be facilitated by the introduction of a small solid particle (nuclide) So called —heterogeneous nucleation— resulting in the growth of a solid with a crystalline morphology

Note columnar crystals form when we have another type of undercooling-Constitutional under-cooling

For a given undercooling Not just any particle will be able to nucleate the solidification

can be undercooled below the equilibrium melting temperature

In order to work (grow solid) the solid seed particle needs to be able to establish an equilibrium with the under-cooled bulk liquid.

To see how this works we need to understand how a solid-liquid interface

A solid changes to a liquid at a singleequilibrium temperature Tm (in ice Tm = 273.2 K)

Interface undercooling

Conditions can exist, however, where the temperature at the solid-liquidinterface, Ti, is below the equilibrium melting temperature

ckmi TTTTT

Curvature-surface energyGibbs -Thomson

Kinetic

solute

ckmi TTTTT

Solute Undercooling

Tm

C –solute concentration

liquidus

The distribution Of impurities (solutes)Will change the meltingtemperature




ckmi TTTTT


Kinetic

Solute-presence of impurities lower meltTemp.

ckmi TTTTT

Kinetic undercooling

Attachment of atoms to the solidCould be “sluggish”, if front advance is rapidPosition of front will lag Tm isotherm




ckmi TTTTT


Kinetic sluggish attachment of atoms


Mola

r Fr

ee E

nerg

y

GL

GST

TTmTu

ckmi TTTTT

Gibbs-Thomson-Surface Energy Undercooling

Consider Bulk free energies in liquid and solid

m

mo

TTL

Then consider spherical particle in liquid

The particle will have an additional free energy due surface energy induced pressure difference between liquid and solid

momoex pG

curvature

r/1mo

Equating

LT

T m

Surface energy J/m2

ckmi TTTTT

Gibbs-Thomson-Surface Energy Undercooling

LT

T m

1

2

m1 TT

m2 TT

LT

T m




ckmi TTTTT


Kinetic sluggish attachment of atoms


LT

T m

1

2

m1 TT

m2 TT

Explained Undercooling BUT why does solid growAND how is Morphology Controlled

How does solid growT

Initial StateSmall seedAt bulk undercoling

Early Stage Liquid around seed is encouraged to change stateIncrease in temperature in surrounding region due to latent heat release

Later Stage Temperature gradient removesLatent heat and heats bulk

Solid Liquid

What Controls Morphology

Preferred growth direction in crystal structure Manifest in anisotropic surface tension, e.g.,

)4cos151()(

Four fold symmetry

0.25

0.25 1

45

B45

90

B904590

TTTTTT

since

LT)(

TT mmi

TB

90T

45T

Will drive sold. FasterIn preferred direction

90

45

4590

TB

90

45

4590

Due to the “pinching”In preferred dir. GrowthLooks unstable

TB

But increase in will reduceT90—can reach a Balance between Temp. grad. and

Steady tip vel;“Operating point”

Can describe process with the Sharp Interface Model

With dimensionless numbers

c/LTT

T m*

2

** tt,

xx

Assumed constant properties

Insulated domain Initiated with small solid seed

s2s T

tT

l2l T

tT

n

vn

Capillary length~10-9 for metal

On interface

nls vTT nn

)4cos151(dT o

Angle between normal and x-axis

Solutions Based on Sharp Interface Model

s2s T

tT

l2l T

tT

H. S. Udaykumar, R. Mittal, Wei Shyy

Interface Tracking-reconstruction

Juric Tryggvason

Zhao and Heinrichnls vTT nn

)4cos151(dT o

Kim, Goldenfeld, DantzigAnd Chen, Merriman, Osher,andSmereka

Level Set function distance

)1(T)21)(1(d

tLc1 20

tT

tT 2

)

2n

tanh(121

Applies throughout domain

obtain an evolution equation for that satisfies curvature and kinetic undercooling

Can be done byMinimize a free energy functionalORDirect geometric modeling (Beckermann)

Diffusive interface model 1: Phase Field

Smear out interface

n

1

0

Liquid fraction

Advantage can be calculatedON A FIXED UNIFORM GRID

TtH 2

)

2n

tanh(121

f

, where enthalpy Also Applies throughout domain

Diffusive interface model 2: Enthalpy (extension of Tacke)

Smear out interface

n

1f

0f

Liquid fraction

fTH

f=1

f=0

1f,1H

1f0),f(T

0f,H

T

)4cos151(dT o

2/32y

2x

yy2

xxyyxxx2y

)ff(

fffff2ff

A non-linear system for evolution of f

If we only consider curvature undercooling

with undercooling

curvature

and orientation

)ff

(tany

x1

In a time step

Solve for H (explicit time integration on FIXED grid)

If 0 < f < 1 then

Calculate curvature and orientation from current nodal f field

Calculate interface undercooling Update f from enthalpy as

Check that calculated liquid fraction is in [0,1]

Update

Iterate until

At end of time step—in cells that have just become all solidintroduce very small solid seed in ALL neighboring cells.Required to advance the solidification

Numerical Solution

)fTH(ff i

fHT

tol)fTH( i

Very Simple—Calculations can be done on regular PC

Typical gridSize 200x00¼ geometry

Seed calc With smallSolid part.

Produces nice answers BUT are they correct

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

0

2

4

6

8

10

12

14

16

0 50 100 150 200 250

,t2s

0T)(erfce 0

2

Verification 2

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

xerfc

TTT 00

Verification 3

Compare calculated dimensionless tip velocity withSteady state operating state calculated from the microscopic solvability theory

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

time

velocity

5do

3.333do

2.5do

x

dvv ogridtip

Verification 4

Check for grid anisotropy

))4cos(151(dT oi

Solve with

4 fold symmetry twisted 45o

Then Twist solution back

Dimensionless time = 6000

0d5.2x

Verification 5: Curvature Calculation

2/32y

2x

yy2

xxyyxxx2y

)ff(

fffff2ff

Operates over narrow band

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

time

velocity

5do

3.333do

2.5do

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

Ease of Coding Excellent

CPU Very Good (runs shown here took between 5 and 20 minutes on a regular PC)

Convergence to known analytical sol.

Excellent

Convergence to known operating state

Good (further study with finer grid indicates problem

Grid Anisotropy Reasonable (further study with finer grid required)

Curvature OK-But I have doubts--grid dependence ?

0

2

4

6

8

10

12

14

16

0 50 100 150 200 250

Extensions

Alloy—Solve for concentration

H. S. Udaykumar, R. Mittal, Wei Shyy

Interface Tracking-reconstruction

Juric Tryggvason

Zhao and Heinrich Fotis

10km

Does any of this this have any relevanceto shorelines Matt

Growing Numerical Crystals Vaughan Voller, SAFL Feb 1 2006.

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