Rubio, Dougherty, and Gollub reply

Abstract

Rubio, Dougherty, and Gollub Reply:In the preceding Comment 1 Horvath, Family, and Vicsek (HFV) report results on experiments suggested to be similar to the ones described in our paper, 2 and compare them with their own reanalysis of our data in Fig. 1 of Ref. 2. They find that the value of the roughness exponent in their experiment is /3=0.88 ±0.08, in agreement with the value obtained by reanalyzing the interfaces in our paper, /3 =0.91 ±0.08. Both of these values differ from our reported result of p =0.73 ± 0.03. They conclude that this discrepancy is particularly important "because it is expected to be relevant from the point of view of universality classes of surface growth phenomena."Because of space limitations, we did not report all of the checks we have made on our roughness calculations, although some have been published elsewhere. 3 As stated in Ref. 2, we also checked our results against direct computations of the box and divider dimensions, and obtained good agreement. Furthermore, we checked all three algorithms on self-affine Weierstrass-Mandelbrot curves for a wide range of roughness exponents. Therefore, we are confident that the results reported in our paper are correct.The discrepancy between our result and that obtained in the reanalysis of our data by HFV is probably due to the process they used to obtain the data. This process included the plotting of the interfaces with a pen of finite width, the printing of the figure, and the redigitization of the data with 740x600 resolution. The first two processes smooth the interfaces on length scales smaller than the actual thickness of the lines in the final printed figure. The final stage involves digitizing the interfaces with higher spatial resolution (in the horizontal direction) than that of the original data.In a system with scaling behavior over many decades, this process should introduce a crossover from rough interfaces at large length scales to smooth ones at smaller scales. In our system, though, where the scaling range is somewhat limited, it is not surprising that the result is simply a higher apparent roughness exponent. Indeed, their "reanalyzed" data do not exhibit as clear a scaling range as the original data. We have checked this hypothesis quantitatively by taking the original data representing the interfaces reported in Fig. 1 of Ref. 2, and processing it in a way that corresponds approximately to that used in Ref. 1. The value obtained for the resulting interfaces was p =0.84 ±0.04, distinctly higher than the correct value we reported. Variations in the details of the interface-finding algorithm might also have some effect. We conclude that a large part of the difference is simply an artifact resulting from the fact that they did not have access to the original data. Thus their reanalysis has little relevance either to our data or to their experimental value.Without additional experimental details, it is difficult to make direct comparisons between our work and the

Roughness and growth in a continuous fluid invasion model

“…Martys, Cieplak, and Robbins [3] calculated this exponent for the wetting regime (below c ) and found ␣ = 0.81± 0.05 for = 25°as a representative angle for this regime. It was claimed that this value of ␣ agrees very well with experimental data on wetting invasion [8][9][10][11], unlike most growth models (e.g., Kardar-Parisi-Zhang (KPZ) [29]) which give ␣ = 0.5. In fact, as was pointed out by Roux and Hansen [16], as well as later by Albert et al [18], there exists some scatter in the determination of ␣ from experimental results, most of them performed in Hele-Shaw cells.…”

“…In the past two decades, many papers have addressed this issue, both experimentally and theoretically . Most experiments were done using a Hele-Shaw cell [8][9][10][11]13,17], tubes network [5][6][7], or paper [12], with fluids such as water, glycerol, or ink. Several models were introduced in order to describe flow dynamics and interface characteristics under nonequilibrium conditions.…”

Self-organized pinning and interface growth in a random medium

“…There has recently been much interest in nonequilibrium growth models and their dynamic universality classes. These studies may have a number of practical applications, e.g., in chemical vapor deposition [1], electrochemical deposition [2], molecular beam epitaxy [3], growth of bacteria colonies [4], and in fluid invasion of porous media [5][6][7]. Theoretical studies have fallen into two classes.…”

“…The results of this algorithm are the well-known scalings [11] for the ensemble averaged width w of the interface, w'^ = ((/i -(ft))2> oc t^^fit^l^lL) with X = 1/2 as the roughness exponent of the saturated interface and with /5 = xl^ = 1/3 describing the transient roughening. Experimentally determined x for one-dimensional interfaces ranges from x = 0.55 ±0.06 in electrochemical deposition [2], X = 0.78 ± 0.07 in growth of bacterial colonies [4], to a X of 0.63±0.04 ( [7]), 0.73±0.03 ( [5]), and « 0.81 ( [6]) measured for fluid invasion of porous media. In all experiments the measured x are above the one predicted by the KPZ universality class.…”

Anisotropic percolation and the d-dimensional surface roughening problem

“…with (Y = 1 and p = 1. For many phenomena in d = 1 + 1 -from bacterial growth [ 111 and viscous flows [12,13] to the wetting [13-E] and burning [16] of paper-self-affine surfaces are found with anomalous exponents (Y and /3 significantly larger than the KPZ values but less than 1. Recent experimental data in d = 2 + 1 also show anomalously large values of (Y -for mountain surfaces (Y = 0.58 [17,18], for wetting of porous media (Y = 0.5 [19], and for ion beam erosion of metal surfaces (Y = 0.53 [20].…”