A Closed Form Simulation of a Coarsening Analog System Vaughan Voller, University of Minnesota

a·nal·o·gy

Similarity in some respects between things that are otherwise dissimilar.

http://www.thefreedictionary.com/

E.g., the coarsening of a froth and grain growth in material microstructure

MORRIS COHEN—grains in a thin film

Two Uses of analogy

1. May provide physical insight into your process of interest

2. Allows for the development and testing of cross-cutting modeling technologies

N < 6 n = 6 n > 6

A Fundamental Coarsening Law: The Von-Neumann-Mullins Growth law

For an individual isolated 2-D bubbleA balance of pressure and surface-tension forces shows that

)n(Ddt

dan 6

n-number of sides, D- diffusivityan area of bubble with n -sides But in array of bubbles

topological change willcreate new n < 6 bubbles

Propose modified array version ofvon-Neumann-Mullins Growth law

)n(Ddt

ad

dt

ad n 6

Rate of change ofAverage n-sided bubble area

Rate of changeof average area

Experimental Verification of array form of von-Neumann-Mullins Growth law

)n(Ddt

ad

dt

ad n 6

-10

0

10

20

30

40

50

60

2 3 4 5 6 7 8 9

Sides

d<

an >

/dt (

pix

cels

/min

)

dt

ad)6n(76.9

dt

ad 6n

coarsening soap-froth structure formed by colloidal particles. Mejía-Rosales, et alPhysica A 276 30 (2000).

What might we want to know

1. The rate of change of the average area <a>(t) ~ 1/N(t) , N(t) number of bubbles

2. The rate of change of the area of the average n-sides bubble <an>

A simple conceptual model for soap froth coarsening

Model each bubble in 2-D domain as a pointundergoing a random walk

When two points approach within A distance “d” they combine into one point

In this way the bubbles will reduce over time

Can visualize the bubble array at a pointIn time by creating a Voronoi diagramaround the remaining “bubble points”

Similar to the colloidal aggregation model ofMoncho-Jordá, et alPhysica A 282 50 (2000)

Could develop a direct simulationbut prefer to develop a “conceptual”solution

A conceptual solution of random walk model: Basic

Let—assuming multiple realizations—the average time forthe destructive meeting of two particles to be

A

21

“diffusivity”

Domain area A

With 3-particels it is reasonable to project that--since there will possible meetings –

the average time-from multiple realizations—will be

323 C

At 61

3

With 4-particels 1224

21

4 AA Ct

With k particles )k(kCt AkAk 122

1

If this holds for any number of particles k—the mean time togo from an initial particle (bubble) count of N0 to N particles is

011

00

1 NN)k(ktt

AAN

Nk

AN

Nkk

Matches long-termbubble coarsening dynamicsderived from Dim. Anal.

Is meeting time valid if bubble count k is large

)k(kCt AkAk 122

1

A conceptual solution of random walk model: Extension

With many of bubbles (k>>1) the distance betweenwill become relatively uniform—i.e., the variance

about the mean distance will be small k

Admean

The mean meeting time ~ dmean

And the time to go from N0 to N (>>1) may be bettergiven by

0

1

N

Nk

A

kt

“velocity”

100

1000

10000

1 10 100

time(hours)

N n

um

be

r o

f b

ub

ble

s

0

1

N

Nk

A

d kt

Compare with experiments of Glazier et al Phys Rev A 1987

0NNt

AA

C

dNN

CNN tt

00

1

A simple Linear CombinationOf the time scales

100

1000

10000

1 10 100(Hours)

(Nu

mb

er

of B

ub

ble

s)

Compare with experiments of Glazier et al Phys Rev A 1987

A three parameter Fit

81261265 .,.,.

0NNt

AA

C

0

1

N

Nk

A

d kt

dNN

CNN tt

00

1

Note in long time limit the average area )t(N/Aa

0 ataa l

2. The rate of change of the area of the average n-sides bubble <an>

Start with the Array version ofvon-Neumann-Mullins Growth law

)n(Ddt

ad

dt

ad n 6

Integrate to

nfa)n(Dtan 6

anaa ln 6 D

Where

)n(a)n(f 60 Set So that

Choice of )n(f justified by noting that in long time limit—as full disorder is reached the Lewis Law

)n(aan 16 Is recovered—consistent with theoreticalResult of Rivier

100

1000

10000

1 10 100(Hours)

(Nu

mb

er

of B

ub

ble

s)s/mm.D. 258250

Value measured in experiment D= 2.742

Glazier et al Phys Rev A 1987

16

na

aaa ln

0

1

2

3

2 4 6 8 100

1

2

3

4

2 4 6 8 10

0

1

2

3

4

5

6

7

2 4 6 8 10 12 14 16

SidesN

orm

aliz

ed

Are

a

00.5

11.5

22.5

33.5

44.5

5

2 4 6 8 10

Other work

At a point in the bubble coarsening model Visualization of the bubble froth can be obtained using a Voronoi Diagram

How do the statistics of this Visualization comparewith real bubble froths

0

0.5

1

1.5

2

2.5

3 4 5 6 7 8 9 10 11

sides (n)

Mo

difi

ed

No

rma

lize

d A

rea

nfnaan 16

=0.222

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

time

Va

ria

nce

2

Variance of bubble sides

n

22 )nn)(n(

Summary

1. Based on a conceptual random walk model the mechanisms for an early and late time-scale for froth coarsening have been hypothesized.

0NNt

AA

C

0

1

N

Nk

A

d kt

A simple linear combinationprovides excellent agreement with experiments

Comparison with more cases is needed

Key features of soap froth coarsening can be recovered with simple closed form models

100

1000

10000

1 10 100(Hours)

(Nu

mb

er

of B

ub

ble

s)

dNN

CNN tt

00

1

2. From the Proposed Von Neumann-Mullins Modification

)n(Ddt

ad

dt

ad n 6

0

1

2

3

4

5

6

7

2 4 6 8 10 12 14 16

Sides

No

rma

lize

d A

rea

Best fit value consistent with independently measured D

A more general version of the Lewis law since slope depends ontime

16

na

aaa ln

coarsening dependent equation for relationship between avaerage bubble areaWith different sides

3.

Voronoi Visualization exhibits features ofCoarsening systems BUT NOT with the same coefficients

4. The Open Question—How is this related to Metals ?

MORRIS COHEN—grains in a thin film