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)

The Experimental System Hg 100mm Top-view Hg Ag, Au Ag, Au 1000A-0.1mm Glass Side-view Optical microscope + DIC, Video, CCD Camera thin metallic films – silver or gold. we used various thicknesses - 0.1mm, 500-6000A Small (100 micro-m) Mercury droplets spreading

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

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

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  

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

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

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

50mm 180 110 80 30 00

Silver 4200 A t=5, 15, 25 sec

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

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

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

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

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

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. +- 0.4 – the scatter of the exponents obtained from 10 experiments. 3000A 0.96 1.00 Universality Class ?? a+a/b=2

Average “row” distance – 10000 A א' 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 – 10000 A

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

Non Monotonic Width Growth 6 11 16 21 7 12 17 22 18 8 13 23 9 14 19 small 24 15 10 large 20 Non Monotonic Width Growth

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

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.

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)

Interface Fluctuations Gold - 1500 A Width Fluctuations Crossover Length from Interface Fluctuations Roughness exponent Lc = 8 microns

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.

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

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

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, 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).