Persistence in Reactive-Wetting Interfaces
Haim TaitelbaumDepartment of Physics
Bar-Ilan University, Ramat-Gan, Israel
With:
Avraham Be’er, Inbal Hecht
Tal Ya’akov, Ya’el Efraim, Meital Har’el
Yossi Lereah (TAU), Aviad Frydman
Hg-Ag (Au) Reactive-Wetting in Room Temperature
Side-view Top-view
Hg
Ag, Au Glass
Ag, Au 1000A-0.1mm
Hg 100mm
Optical microscope + DIC
CCD Camera
50mm
Silver 4200A t = 5, 10, 15 sec
Bulk Spreading
Kinetic Roughening
t = 5 sec
t = 10 sec t = 15 sec
Sn spreading on 1mm Au-coated 12mm Cu (Singler et al 2008)
Kinetic Roughening in High Temperature
Dynamics of Classical Wetting
t1 t2
Side view
q(t)
R(t)
H(r=0,t)
Tanner’s Law
1/10t
3/10t
12
( ) ( )R t tq
Reactive wetting ?
Droplet placement
Spreading & reaction
Touching the glass
Halo – fast flow
Final stage
50mm
Side-View ??
Dynamical 3-D Shape Reconstruction
Side View from a Top View !
Be’er & Lereah, JOM 2002
50mm
180
110
80
30
00
0
5
10
15
20
25
30
-300 -200 -100 0 100 200 300
R (mm)
H (
mm
)
t=5, 15, 25 sec
Silver 4200 A
0
5
10
15
20
25
30
35
40
0 5 10 15 20
q(deg)
time (sec)
t
0
5
10
15
20
25
30
35
40
0 5 10 15 20
0
5
10
15
20
25
30
35
40
45
Bulk spreading
R(mm)
R ~ t
Silver 4200 A
Halo
step
contact
angledroplet
radius
Final stage – Top-view
Ag3Hg4
Ag4Hg3
SEM
Kinetic Roughening of the Reaction Band
Screen height - about 100mm
Ag thickness = 0.1 mm (foil)
Hg initial radius - about 150 mm.
Bulk height from surface - about 1 mm.
Initial time here is 15 sec
Total time of experiment = 5 min
Complex Interface Patterns
mm20
mm2
mm2
mm2
Hg mm150
Ag 4000 Å
Glass
AAg 4000
Glass
mHg m150
Top View
W L t h x t h x t2 2 2( , ) ( , ) ( , )
In isotropic systems 2
- growth
– roughness
x
h
Hg
Ag 2 1t t
0
0),(tt
tt
L
ttLW
00 Lt
W
Family-Vicsek scaling
222 ),(),(),( txhtxhtLW
L0
L1L2
L1
02.046.0
Silver 2000 A
Crossover behavior
47.076.0
“”
Scaling Exponents
Correlation Length
Summary of Results
+/=2 ?
1.000.963000A
0.760.881500A
Gold
0.600.770.1mm
0.460.662000A
Silver
ThicknessMaterial
03.076.0 02.046.0
04.082.0 02.060.0
03.085.0 03.076.0
04.086.0 04.000.1
Universality Class ??
NOISE –
Substrate Roughness
2mm
8mm
7mm
14mm
Correlation
length
What really happens
in the growth regime ???
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5
log(t)
log
(w)
40 pixels
60 pixels
80 pixels
100 pixels
120 pixels
140 pixels
150 pixels
saturation
“”
Yael Efraim 2009
The persistence measure
♠ Persistence probability P(t) is
the probability that a certain
flactuative variable never
crosses a chosen reference
level within time interval t.
Universal scaling form:
Illustration
-40
-30
-20
-10
0
10
20
0 5 10 15 20 25 30
time
A
q ttP )( Derrida et al. PRL (1994)
Persistence in Reactive-Wetting Interface
h
x x
t0
t1
t2
t0
t1
t2
Intervalt
Intervalt
Flactuative variable- “location” of points on interface
Reference level- height at time t0
Persistence – the algorithm
Single point – distance from <h> vs. t
Divide t axis to intervals in length t’ where t’=1,2…Tmax sec
Count “persistent” intervals for each t.
"trajectory" of single point
-86
-66
-46
-26
-6
14
34
54
0 5 10 15 20 25 30
t (sec)
h*
(pix
) ttallforsametheremains
txhttxhsign
thatprobistttxP
'0
)],()',([
.),,(
00
00
Average over t0 & x:
)(),,( 00 tptttxP
C. Dasgupta et. al.
Survival
Constant reference level - h*=0
ttallforsametheremains
ttxhsignthat
probistttxS
'0
)]',([
.),,(
0
00
)(),,( 00 tStttxS P(t) - Power Law decay
S(t) - Exponential decay
Persistence Exponent Results
Ising model (PRL 1996) (1d, 2d, 3d)
Potts model (Physica 1996) (2d- large q)
Random walk (PRE 1997)
Linear Langevin equations (PRE 1997)
KPZ equation (Europhys. Lett. 1999)
26.0,22.0,37.0q
86.0q
q 1s
18.1q 64.1q
5.0q
Experimental results
2d Liquid crystal (PRE 1997)
Fluctuating monatomic steps on crystal (PRL 2002)
Combustion fronts in paper (PRL 2003)
2d Soap froth (PRL 1997)
03.019.0 q
03.077.0 qt
03.066.0 q
02.088.0 q
Relation q=1- is valid even though system is non-linear
Results for Reactive-Wetting
y = 0.0659x + 1.0003
y = -0.5148x - 0.131
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-0.5 0 0.5 1 1.5
log (t [sec])L=150 pix
log (w [pix])
log (P)
First time regime:
Random walk
05.055.0 q
consttW )(
Persistence in growth regime new calculation !
Persistence – growth regime
q 1sy = 0.6679x + 1.3568
R2 = 0.9091
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-0.5 0 0.5 1 1.5
log tgrowth
log
w
y = -0.3085x - 0.1669
R2 = 0.9574-0.8
-0.6
-0.4
-0.2
0
0 0.5 1 1.5
log tgrowthlo
g p
,
Relation is valid:
07.068.0 05.037.0 q
08.004.1 qq 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-0.5 0 0.5 1 1.5 2
Persistence – saturation regime
Random walk again:
y = -0.4829x - 0.2152
R2 = 0.9617
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
log tsat
log
p
01.047.0 q
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-0.5 0 0.5 1 1.5 2
Global interpretation
-1.5
-1
-0.5
0
0.5
1
1.5
2
-0.35 -0.15 0.05 0.25 0.45 0.65 0.85 1.05 1.25
log (t [sec])L=150 pix
log (w [pix])
log (P)
Noise
Non-linear
growth
Surface
tension
evidence: q~0.5
Persistence Summary
Three kinetic regimes in experiment:
Transient regime - random walk mechanism
Growth regime -
Saturation regime - random walk again
Growth:
Persistence:
1 q
070680 ..β
05.037.0 q
Validity of “Linear” relation in a non-linear system
Summary
• Reactive-Wetting - Room Temperature
Side view from Top View
Bulk Spreading – q(t), R(t)
• Interface Kinetic Roughening
Spatio - Roughness Exponent , Lateral Correlation Length
Temporal - Growth Exponent, The Persistence Measure
References
A. Be’er, Y. Lereah, H. Taitelbaum, Physica A 285, 156 (2000)
A. Be’er, Y. Lereah, I. Hecht, H. Taitelbaum, Physica A 302, 297 (2001)
A. Be’er, Y. Lereah, A. Frydman, H. Taitelbaum, Physica A 314, 325 (2002)
A. Be’er, Y. Lereah, J. of Microscopy, 208, 148 (2002).
I. Hecht, H. Taitelbaum, Phys. Rev. E 70, 046307 (2004).
A. Be’er, I. Hecht, H. Taitelbaum, Phys. Rev. E 72, 031606 (2005).
I. Hecht, A. Be’er, H. Taitelbaum, FNL, 5, L319 (2005).
A. Be’er, Y. Lereah, A. Frydman, H. Taitelbaum, Phys. Rev. E 75, 051601 (2007).
A. Be’er, Y. Lereah, H. Taitelbaum, Mater. Sci. Eng. A 495, 102 (2008).
Y. Efraim, H. Taitelbaum, Cent. Eur. J. Phys. 7, 503 (2009).
L. Lin, B.T. Murray, S. Su, Y. Sun,
Y. Efraim, H. Taitelbaum and T.J. Singler, JPCM (2009).
