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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