Nonlinear Dynamics (2006) 44: 307–313 DOI: 10.1007/s11071-006-2011-8

c Springer 2006 

Quantitative Methods for the Characterization of Complex Surface Structures ALEJANDRO MORA∗ and MARIA HAASE Institute for Computer Applications (ICA II), University of Stuttgart, Stuttgart, Pfaffenwaldring 27, D-70569, Stuttgart (Vaihingen), Germany; ∗ Author for correspondence (e-mail: [email protected]; fax: +49-711-6853758) (Received: 13 August 2004; accepted: 30 April 2005)

Abstract. Attempting to model the processes resulting in complex pattern formation and small-scale roughness of surfaces and to compare with experimental measurements calls for numerical methods which allow a quantitative characterization being as complete as possible. New methods incorporating wavelets and stochastic approaches based on the theory of Markov processes allow a stepwise characterization of increasing completeness and unambiguousness. In this paper we demonstrate the underlying numerical approaches taking electropolished and laser-jet etched surfaces for demonstration. Key words: electropolished surfaces, Markov analysis, scaling, wavelet transform

1. Introduction The development of new technologies like high-precision microstructuring of metals calls for numerical methods which allow a characterization of the topography and a stochastic description of the surface roughness [1]. The aim is not only to classify different surface structures but also to ensure that model descriptions do not contradict the measured data. Since the 1980s surface roughness has predominantly been analyzed on the basis of self- or multi-affinity and has thus been described within the framework of fractal dimensions or multifractal spectra. However, many surfaces although sharing the same multifractal properties, and thus having the same two-point correlations, might still differ in their N -point correlations [2]. Here, we present a number of numerical approaches which allow a description of increasing completeness. These include in addition to classical spectral methods more recently developed wavelet approaches which allow to determine characteristic length scales and provide robust methods for the calculation of multifractal spectra. In order to incorporate the complete stochastic information in the description a new method based on the theory of Markov processes is applied [2, 3]. The paper is organized as follows. In Section 2, we analyze the characteristic wavelengths and scaling regions for two exemplary surface profiles, an electropolished brass sheet and a laser-jet etched steel surface. In contrast to the classical Fourier techniques, the continuous wavelet transform (CWT) allows a space-scale resolution of the surface profile. After a brief review of the wavelet transform modulus maxima (WTMM) method [4, 5], the multifractal spectra for the electropolished surface are estimated in Section 3. As can be seen from the power spectra, two processes are interacting leading to different multifractal behaviour on different scales. Section 4 deals with the evolution of the probability density functions (pdf) of surface height increments for varying scales and gives a short introduction into the recently developed stochastic approach, which is used for a complete stochastic characterization of the profiles [2, 3]. Finally, Section 5 presents our conclusions and perspectives of further investigations.

308 A. Mora and M. Haase 2. Characteristic Length Scales and Scaling Properties Figure 1a shows a laser-profilometer scan of a surface which is obtained by electropolishing a brass sheet in an electrolyte solution containing methanol (Figure 1a) [6, 7], while Figure 1b displays a typical scanning electron microscope (SEM) image of a kerf in a steel sheet structured by a recently developed laser-jet assisted wet etching process [8]. Electropolishing was performed in the transpassive region with vertically arranged electrodes, the workpiece acting as an anode. The surface structure is a result of two competing processes, namely the dissolution of metal leading to a falling film of spent electrolyte containing dissoluted metal and the hydrolysis of water, where oxygen is formed at the anode causing gas bubbles to rise forming an unwanted pattern of so-called gas lines [6]. Characteristic length scales can easily be obtained by classical Fourier methods. For the spectrum transversal to the gas lines (Figure 2a), two regions with different decay can be seen, indicating the interplay of two different processes. We estimate the bandwidth of scales introduced by the gaslines from the ratio the power spectral density of ensemble averages transversal and parallel to the gas lines (Figure 2b). In Figure 3a the average profile in the bottom of a kerf obtained by laser-jet etching is shown together with the power spectral density displaying a sharp peak for a wavelength of 40 μm (Figure 3b). In order

Figure 1. (a) Laser scan of the height profile of a brass surface electropolished in methanol-electrolyte, (b) SEM micrograph of a kerf obtained by laser-jet etching.

Figure 2. (a) Power spectra of ensemble averages transversal and parallel to gas lines for electropolished brass sheet. (b) Ratio of power spectral densities measures the anisotropy of the surface.

Quantitative Methods for Characterization of Complex Surface Structures 309

Figure 3. (a) Average profile in the bottom of the kerf obtained by laser-induced etching. (b) Power spectral density E(k). (c) Wavelet transform (Morlet wavelet, ω0 = 25).

to resolve the profile in space and scale, we apply a continuous wavelet transform (CWT) to the profile 1 W f (a, b) = a



+∞ −∞

  x −b ¯ f (x)ψ d x, a

(1)

where a, b ∈ R, a > 0. The CWT decomposes the function f (x) ∈ L 2 (R) hierarchically in terms of elementary components ψ((x − b)/a) which are obtained from a single mother wavelet ψ(x) by ¯ dilations and translations. ψ(x) denotes the complex conjugate of ψ(x), a the scale and b the shift parameter. A unique reconstruction of the function f (x) is ensured if ψ(x) ∈ L 1 (R) has zero mean. 2 2 Figure 3c shows the CWT using a Morlet-type progressive wavelet: ψ(x) = ddx 2 (e−x /2 eiω0 x ). At a scale correponding to λ = 40 μm, a dominant band can be seen displaying a strong variation of the wavelet coefficients of the ripple structure.

3. Multifractal Analysis Using Wavelet Techniques The power spectral density E(k) gives only limited information about the mono- or multifractal properties of the surface roughness. It only allows us to estimate a global H¨older exponent h via the relation E(k) ∼ k −1−2h . Local fluctuations in the degree of roughness call for a location-dependent H¨older exponent h(x). In turbulence, the standard way to extract the multiscaling properties of a function f (x) is

310 A. Mora and M. Haase to study the scaling behaviour of the structure functions Sq (r ) ≤ δ fr q ∼ r ζq of order q of the increments δ fr = f (x + r/2) − f (x − r/2). Multifractal behaviour leads to a nonlinear scaling exponent ζq . The spectrum D(h) of H¨older exponents is obtained by Legendre transforming the exponents ζq leading to D(h) = minq (qh − ζq + 1) [9]. A severe drawback of this method is that one has only access to H¨older exponents 0 < h < 1, i.e. singularities in the derivatives of the function cannot be identified. In addition, negative moments q < 0 lead to divergencies. These limitations can be circumvented using the wavelet framework [4]. Choosing derivatives of the 2 Gaussian function as wavelets in Equation (1), ψ0 (x) = e−x /2 , ψn (x) = ddx ψn−1 (x) with n ∈ N, n ≥ 1, and assuming a cusp singularity with H¨older exponent h(x0 )  (n, n + 1) at x0 , the CWT scales like |Wψ f (a, x0 )| ∼ a h(x0 ) ,

a → 0+ ,

(2)

provided the analyzing wavelet chosen has n ψ > h(x0 ) vanishing moments. In contrast, if one chooses a wavelet with n ψ < h(x0 ), the CWT scales with an exponent n ψ . It can be shown, that this scaling behaviour is also valid along the maxima lines of the modulus of the CWT, which point to the singularities [4]. The so-called wavelet transform modulus maxima (WTMM) method [4, 5] is a generalization of the classical multifractal formalism [9–11] and allows a robust estimation of the full spectrum of singularities. A partition function Z (q, a) is defined Z (q, a) =



 bi ∈max lines

q

sup |Wψ f (a , bi )|

a ≤a

(3)

containing the qth moments of the contributions of |Wψ f | along the maximal lines, where the supremum in Equation (3) is related to a Hausdorff-like covering with scale-adapted wavelets removing divergencies due to negative order moments [4]. From the power-law behaviour of the partition function (cf. Equation 3), Z (q, a) ∼ a τ (q) , a → 0+ , the whole spectrum of H¨older exponents D(h) is obtained by Legendre transforming the scaling exponents τ (q): D(h) = minq (qh −τ (q)). Figure 4 shows the spectra of H¨older exponents for the electropolished surface. An ensemble of five profiles transversal to the direction of the gas lines containing 12 800 maxima lines is used for the calculation of the partition function Z (q, a). According to the observation of two different scaling regions in the transversal power spectrum (Figure 2a), two regions with different power law behaviour occur, which lead to different distributions of the corresponding H¨older exponents.

Figure 4. Spectra of H¨older exponents D(h) for different scaling regions.

Quantitative Methods for Characterization of Complex Surface Structures 311

Figure 5. Probability density functions for electropolished surface.

4. Stochastic Approach Based on the Theory of Markov Processes From a stochastic point of view, the multifractal characterization is still incomplete, since only 2-point correlations are involved in this formulation. The height increment zr (x) = z(x + r/2) − z(x − r/2) of the surface profile z(x) can be considered as a stochastic variable in the length scale r [2]. Figure 5 displays the evolution of pdfs of zr as r is varied. The distributions are normalized to their respective standard deviations σr and shifted in vertical direction for clarity. For small scales the shapes of the curves deviate strongly from Gaussian distributions indicating pronounced intermittency effects. In a series of papers a new approach for the stochastic analysis has been proposed which allows us to extract the explicit form of the underlying stochastic process directly from experimentally measured data without making any assumptions, provided the process is Markovian [2, 3]. The aim is to describe the evolution of the conditional probability density functions as r is varied, where the conditional pdf p(z 1 , r1 | z 2 , r2 ) describes the probability for finding the increment z 1 on scale r1 provided that the increment z 2 is given on scale r2 . A stochastic process is Markovian, if the conditional probability densities fulfill the relations p(z 1 , r1 | z 2 , r2 ; · · · ; z n , rn ) = p(z 1 , r1 | z 2 , r2 ),

(4)

where r1 < r2 < · · · < rn . In this case, the conditional pdf satisfies a master equation. Expanding the distribution function into a Taylor series, the evolution equation can be written as [3]  ∞   ∂ ∂ k −r p(zr , r | z 0 , r0 ) = − Dk (zr , r ) p(zr , r | z 0 , r0 ), ∂r ∂zr k=1

(5)

where the so-called Kramers–Moyal coefficients Dk (zr , r ) = lim Mk (zr , r, r ) r →0

can be directly estimated from experimental data:  −∞ r Mk (zr , r, r ) = (˜z − zr )k p(˜z , r − r | zr , r ) d z˜ . k!r ∞

(6)

(7)

Figure 6 shows a test of Markov properties of the electropolished surface data. In Figure 6a, the contour plots of p(z 1 , r1 | z 2 , r2 ; z 3 , r3 ) (black lines) as well as p(z 1 , r1 | z 2 , r2 ) (grey lines) are shown

312 A. Mora and M. Haase

Figure 6. (a) Contour lines of conditional pdfs p(z 1 , r1 | z 2 , r2 ) (grey lines) and p(z 1 , r1 | z 2 , r2 ; z 3 = 0, r3 ) (black lines) for r1 = 10 μm, r2 = 108 μm, r3 = 216 μm. (b) and (c) Cut through the conditional pdfs for z 2 = ±σ/2.

in units of the standard deviation σ of the z-data. The good correspondence over several orders of magnitude is corroborated by two cuts for z 2 = ±σ/2 displayed in Figure 6b and c indicating the validity of the necessary condition Equation (4). However, choosing different scale increments, for example r1 = 52 μm, r2 = 60 μm, r3 = 68 μm, the two sets of contour lines strongly deviate from each other. The minimal increment rMarkov , for which Markovian properties hold, is the so-called Markov length. In the Markovian range, drift and diffusion coefficients D1 , D2 can be estimated directly from the measured data without making any assumption for the underlying process. If D4 is small as compared to D1 and D2 , the evolution of conditional probabilities can be described by a Fokker–Planck equation.

5. Conclusions We presented various numerical techniques including wavelet analysis and stochastic methods for a characterization of complex surface structures. The multifractal scaling behaviour is contained in the singularity spectra D(h) which are estimated using the WTMM method. If Markov properties can be verified, a complete stochastic description of the surface is given by a Fokker–Planck equation (for D4 negligible), which describes the evolution of conditional pdfs over scales. The corresponding Langevin equation would open the possibility for a direct synthetization of surface profiles [12]. In addition, dynamical and measurement noise and even their magnitude can be extracted from experimental data [13].

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