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).