1. Interface Roughening Dynamics of Spreading Droplets
Haim Taitelbaum (Bar-Ilan University)
With:
Avraham Be’er, Inbal Hecht, Ya’el Hoss
Aviad Frydman
Yossi Lereah (Tel-Aviv U)

2. The Experimental System
Hg 100mm
Top-view Hg
Ag, Au
Ag, Au A-0.1mm
Glass
Side-view
Optical microscope + DIC, Video, CCD Camera
thin metallic films – silver or gold.
we used various thicknesses mm, A
Small (100 micro-m) Mercury droplets spreading

3. Optical microscope + DIC
+ CCD camera
Data analysis
Image analysis
Frame Graber
0.04 sec resolution

4. Why does the Mercury Spread ???
1. Rough Substrate
2. Reaction Hg-Ag

5. Silver 4200A t = 5, 10, 15 sec Bulk Spreading Kinetic Roughening
50mm
t = 5 sec
Bulk Spreading
Kinetic Roughening
t = 10 sec
t = 15 sec

6. Bulk Spreading - Reactive Wetting
Side view
q(t)
R(t)
H(r=0,t)
Tanner’s Law (Non-Reactive)

7. Dynamical 3-D Shape Reconstruction Side View from a Top View !
Reactive Wetting ??
Dynamical 3-D Shape Reconstruction Side View from a Top View !
Be’er & Lereah, JOM 2002
50mm

8. Differential Interference Contrast (DIC)
1. Steps of 1/20 of l
Slope = Color
Calibration !
Ocular Lens
4. Analyzer
3. l Plate
Light Source
1. Polarizer
2. Rochon Prism
Objective Lens

9. 50mm
180
110 80 30 00

10. Silver 4200 A
t=5, 15, 25 sec

11. q(deg) time (sec) Silver 4200 A R(mm) t R ~ t Bulk spreading
Reaction band t
Bulk spreading
R(mm)
R ~ t

12. Kinetic Roughening of 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

13. Chemical Reaction Optical microscope SEM Studies Ag3Hg4 Ag4Hg3 Ag
2 intermediate phases – products of the chemical reaction between the silver and the Mercury.
We can see that the chemical reaction plays a crucial role in the spreading process.
The surface tension
SEM Studies

14. x h Hg Ag
Top View x h W
b - growth
a – roughness
In isotropic systems

15. Silver 2000 A Crossover behavior Log W NEW! Log W log t
1.2
The roughness exponent a 1
Slope=0.66
0.8
0.6
log W
Log W
0.4
0.2
-0.2
0.5 1
1.5 2
2.5
-0.4
log L
0.3
The growth exponent b
0.2
NEW!
0.1
Slope=0.46
Log W
log W
0.2
0.4
0.6
0.8 1
1.2
-0.1
-0.2
-0.3
-0.4
log t
1.2
End of Propagation
The roughness exponent a 1
0.8
0.6
Slope1=0.76
Slope2=0.47
Log W
log W
0.4
Crossover behavior
0.2 EW
KPZ
-0.2
0.5 1
1.5 2
2.5
-0.4
-0.6
log L

16. Exponents Results a+a/b=2 Material a b Silver 2000A 0.66 0.46 0.1mm
NOISE –
Substrate Roughness
Material
Thickness a b
Silver
2000A
0.66
0.46
0.1mm
0.77
0.60
Gold
1500A
0.88
0.76
We investigated systems with several materials and several thiknesses. The scaling relation a+a/b=2, which is characteristic for isotropic systems, is obeyed in all cases.
We give here the values of the exponents a and b for several cases.
These values, in particular (__) belong to a NEW universality class and have not been measured in experiments before. .
Average over 10 experiments – the scatter of the exponents obtained from 10 experiments.
3000A
0.96
1.00
Universality Class ??
a+a/b=2

17. Average “row” distance – 10000 Aא'
Gold – 1500 A
Silver – 2000 A
Average pin height – 100 A
Average pin width – 500 A
Average pin height – 200 A
Average pin width – 1000 A
Silver foil 0.1 mm
Average pin height – 250 A
Average “row” distance – A

18. Single Interface Growth Fluctuations
Silver foil 0.1 mm
Width
but if we look closer on the graphs, we can see that while the graph of alpha is a straight line, the graph of beta is non-monotonic and the width fluctuates around the average slope. This kind of fluctuations appear in all the other systems, for all thickness, for silver and for gold.
we can also see the crossover in the slope of alpha, which indicates the typical correlation length of the system. So, if we want to know the typical correlation length of our system, we should wait long enough (time longer then t0) and then plot log W vs. log L.
time

19. Non Monotonic Width Growth6 11 16 21 7 12 17 22 18 8 13 23 9 14 19
small 24 15 10
large 20
Non Monotonic Width Growth

20. Qualitative Description of Non-Linear Interface Growth
3. Non- Linear Growth
4. Surface Tension
2. noise
1. Initially flat interface

21. Quantitative Analysis of Fluctuations
We want to analyze the fluctuations quantitatively; therefore we define the size of the fluctuations of a given interface as the average deviation from the straight line (which has the slope of beta). We take the square of this deviation because of the positive and negative values.

22. Fluctuations and the Correlation Length
if we plot the size of the fluctuations for different “window” sizes we can see a clear crossover - there is a length, beyond which the fluctuations don’t increase anymore. It means that this length has the characteristics of the entire interface – namely it is the correlation length!!
If we compare it with the graph of alpha, from which the correlation length is usually calculated, we can see that these two methods give the same correlation length.
But the new method that we’ve just seen gives the correlation length in much earlier times than the regular method.
Correlation Length
from
Interface Fluctuations
Correlation Length
from
Roughness Exponent (a)

23. Interface Fluctuations
Gold A
Width Fluctuations
Crossover Length
from
Interface Fluctuations
Roughness exponent
Lc = 8 microns

24. Interface Fluctuations
Water Imbibition
in Paper
Family et al (1992)
Width Fluctuations
Crossover Length
from
Interface Fluctuations
Roughness exponent
Lc = 9 mm
We used the data of Family et al, who made an experiment of water spreading on paper. This system has no chemical reaction, and has the length scale of centimeters.
We can see that very different systems, with different length scales, have the same nature of behavior of the interface width.

25. Temporal Survival Probability (Also Spatial Survival Prob.)
Persistence of Growth Fluctuations h
Work in Progress….
BLUE
RED
Temporal Survival Probability
Exponential Decay ? …
(Also Spatial Survival Prob.) x t
height fluctuation field x

26. Summary Bulk Spreading Side view from Top View
Reactive Wetting – q(t), R(t)
Interface Roughening Dynamics
Growth Exponents
Growth Fluctuations
Correlation Length, Persistence

27. 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 and Y. Lereah, J. of Microscopy, 208, 148 (2002).
I. Hecht, H. Taitelbaum, Phys. Rev. E 70, (2004).
A. Be’er, I. Hecht, H. Taitelbaum, Phys. Rev. E, 72, (2005).
I. Hecht, A. Be’er, H. Taitelbaum, FNL, 5, L319 (2005).