Uncertainty studies of topographical measurements on steel surface corrosion by 3D scanning electron microscopy

Abstract

Pitting corrosion is a damage mechanism quite serious and dangerous in both carbon steel boiler tubes for power plants which are vital to most industries and stainless steels for orthopedic human implants whose demand, due to the increase of life expectation and rate of traffic accidents, has sharply increased. Reliable methods to characterize this kind of damage are becoming increasingly necessary, when trying to evaluate the advance of damage and to establish the best procedures for component inspection in order to determine remaining lives and failure mitigation.

A study about the uncertainties on the topographies of corrosion pits from 3D SEM images, obtained at low magnifications (where errors are greater) and different stage tilt angles were carried out using an in-house software previously developed. Additionally, measurements of pit depths on biomaterial surfaces, subjected to two different surface treatments on stainless steels, were carried out. The different depth distributions observed were in agreement with electrochemical measurements.

Introduction

The scanning electron microscope (SEM) is largely used for materials characterization due to its great versatility, large depth of field and high lateral resolution. It is well known that the changes in gray levels on SEM images are not related to changes in local height but in the slope. The image brightness can also be affected by the enhanced emission from edges and ridges, effects of surface contaminations such as local oxidation, local variations of composition, detector position, electric and magnetic properties, among others. Thus, the parameters to characterize a surface, by only using a single image, are closer to the image texture than the surface roughness. Nevertheless, these parameters can be used to quantify surface differences of samples subjected to different processes.

In modern applications the exact position of studied objects in space or topographic information about the specimen is required; therefore, the coordinates in all three dimensions are necessary.

The 3D reconstruction methods of surface topography, necessary to obtain roughness parameters, can be divided into two principal categories:

• Stereoscopy, in which a SEM image stereo pair is used. The stereo pair can be obtained by deflecting the electron beam, but it is generally implemented by tilting the specimen stage (Lane, 1969, Lane, 1972, Stampfl et al., 1996, Davies and Randle, 2001, Huang et al., 2004, Bonetto et al., 2006, Ponz et al., 2006, Jahnisch and Fatikow, 2007, Marinello et al., 2008, Ostadi et al., 2009, Ostadi et al., 2010, Malboubi et al., 2009, Fatikow et al., 2009, Azevedo and Marques, 2010, Chen et al., 2010). Both, observation of 3D images and measurement of 3D height data are possible. Particularly, by overlapping the stereo pair images, which are one in red and the other in blue or cyan, for example, it is possible to build an anaglyph image which produces a depth effect when glasses with one red lens and the other blue or cyan are used. Practically in all scientific areas, the anaglyph images were usually used as a complementary investigation technique, allowing a more comprehensive study about morphology of the samples searched with SEM (see Hortolà, 2009 for a recent example).

  The principal problem of the stereometric method is that this cannot be applied to very smooth surfaces lacking distinguishable details.

• Shape from shading method which was used first to obtain a surface height image from just a single bidimensional image of an object light-illuminated (Ikeuchi and Horn, 1981) and then, implemented for SEM images, in different versions including one or several detectors (Walker et al., 2005, Pintus et al., 2005, Pintus et al., 2008, Drzazga et al., 2006, Paluszynski and Slówko, 2008, Wzorek et al., 2009, Wzorek et al., 2010, Vynnyk et al., 2010).

The principal disadvantage of the shape from shading method (based on Lambert's angular distribution of the secondary and backscattered electrons), is that the angular distribution is far from Lambert's law in the real cases, where the samples have different local orientations concerning the incident electron beam, requiring in many cases, several detectors to obtain images from different orientations. In this work, the first method will be used.

When two images are obtained under different perspectives like in the stereo pair, surface features of different heights differ in their lateral displacement (parallax or disparity) and relative heights (z coordinate) can be calculated for each image pixel using the corresponding disparity value.

In a previous work (Ponz et al., 2006), the EZEImage program was developed to obtain height maps from SEM images. In this software, the Sun (2002) method to find the disparity map, which uses fast cross correlation and two-stage dynamic programming, was implemented. It works on epipolar rectified stereo images so the matching points lie on the same image scanlines of the stereo pair. This means that the tilt axis on the image must be exactly vertical (y axis) and the image center must be the eucentric point.

The equation to find the height values z(i,j) corresponding to each pixel (i,j), measured with respect to a plane that contains the tilt axis and forms an (90-ϕ₁) angle with the optical axis is the following (Ponz et al., 2006):

$$z(i,j)=\frac{(x_1/M)((x_i-\Delta x)\sin {\varphi }_2/WM+\cos {\varphi }_2)-((x_1-\Delta x)/M(x_1\sin {\varphi }_1/WM+\cos {\varphi }_1))}{[(1+x_1(x_1-\Delta x)/{(WM)}^2)\sin ({\varphi }_2-{\varphi }_1)+\Delta x\cos ({\varphi }_2-{\varphi }_1)/WM]}$$

where W and M are the working distance and the magnification, respectively and they are equal for both images during eucentric tilting, ϕ₁ and ϕ₂ are the tilt angles corresponding to the left and right images, respectively, x₁ is the pixel position (i,j) whose height value needs to be known on the left image, Δx is the disparity and x₁-Δx is the pixel position of the same point on the right image (measured in the epipolar and by taking the image center as coordinate origin). The x₁ and Δx parameters are measured in the same units as W. The z(i,j) expression in Ponz et al. paper (2006) is wrong because the denominator should not be squared.

Eq. (1) is the Lane, 1969, Lane, 1972 general equation adapted for eucentrically tilted stereo pairs. When the specimen is tilted ±Δϕ around a normal axis to the beam, it can be written as:

$$z(i,j)=\frac{2(x_1/M)(x_1-\Delta x)\sin \Delta \varphi /MW+\Delta x\cos \Delta \varphi /M}{[(1+(x_1/M)(x_1-\Delta x)/M{W}^2)\sin 2\Delta \varphi +\Delta x\cos 2\Delta \varphi /MW]}$$

Besides the independent variables W, Δϕ, x₁ and Δx, there is a number of additional variables that influence the quality of the reconstruction such as: sample tilt eucentricity, magnification, software algorithm robustness, sharpness of the stereo pair images, among others (Marinello et al., 2008). Bariani et al. (2005) presented a theoretical model regarding the uncertainty calculation of the vertical elevation of a single point, which depends mainly on the tilt angle accuracy and the magnification calibration. They showed that the experimental deviations from the nominal height values confirmed the trend predicted by their model, where the following expression for the variance of the vertical elevation measurement was used:

$$u_z^2={\left(\frac{\partial z}{\partial \Delta x}\right)}^2{u}^2(\Delta x)+{\left(\frac{\partial z}{\partial \Delta \varphi }\right)}^2{u}^2(\Delta \varphi )$$

With the variance on the parallax, u²(Δx) following the expression:

$$u^2(\Delta x)=\frac{s^4}{12}+\Delta x^2\frac{u^2(s)}{s^2}+\Delta x\frac{s^2}{\sqrt{3}}\frac{u(s)}{s}$$

where s is the single pixel dimension and u (s) is its uncertainty. Bariani et al. (2005) calibrated the tilt angle in their microscope by means of a laser interferometer system and they obtained 10 arcseconds as tilt angle residual error in their measured samples. Also, they calibrated the magnification and found a pixel size relative error of 1.9% and 0.81% in the 100× and 400× magnification cases, respectively.

Marinello et al. (2008) searched the critical factors in 3D stereo microscopy using a galena crystal and commercial software. They found that it is possible to obtain deviations from reference height values (a 22.95 μm galena step) within 5% of the total step height, in ideal conditions, i.e., with pixel size and stage calibrated, magnification in the interval 1000–3000× and no deviations from the eucentricity condition, while a 30% error could be expected out of these optimum conditions.

In a previous paper (Ponz et al., 2006) it was stated that due to the fact that the disparity is an integer number on a digital image, the subpixel resolution implemented in the EZEImage program (which allows to increase the precision in the height difference values, reaching an equal or smaller value than lateral resolution depending on the Δϕ angle), is valid only in the epipolar axis and on the “plateaus” with quasi constant disparity values.

Therefore, and taking into account that a smoothing method on the disparity values between two plateaus has not been implemented in this software, the estimated maximum error of the Δx disparity will be 1 pixel in microns, i.e., u_max(Δx) = 122/M [μm] for the microscope used here.

In this paper, a study about uncertainties in the corrosion pit topography in metallic samples, using EZEImage program and a Philips SEM 505 microscope, was carried out. Carbon steel samples of a boiler tube from a power thermoelectric generator were used to search these uncertainties under different experimental conditions. Reliable methods to characterize corrosion pits are becoming increasingly necessary in any type of corroded samples (API, 1998, API, 2000a, API, 2000b), and particularly, in stainless steel samples for orthopedic implants, since this damage mechanism is very common and dangerous in these biomaterial devices (Pohler, 1986, Choules et al., 2009, Wang et al., 2006). Therefore, the corrosion pit depth distributions on samples under different surface treatments to be used as implants were analyzed. The results obtained are compared with electrochemical test.