3D spherical-cap fitting procedure for (truncated) sessile nano- and micro-droplets & -bubbles

TL;DR

This paper introduces a comprehensive 3D spherical-cap fitting procedure for analyzing nanostructures from AFM images, providing more accurate and consistent morphological parameters than traditional 2D methods.

Contribution

A novel 3D fitting algorithm that automatically extracts nanostructure geometry from AFM images, improving accuracy over existing 2D cross-sectional methods.

Findings

3D-SCFP yields consistent and accurate morphological parameters.

Compared to 2D fitting, 3D method uses all valid data points for higher fidelity.

The procedure is applicable to both complete and truncated spherical caps.

Abstract

In the study of nanobubbles, nanodroplets or nanolenses immobilised on a substrate, a cross-section of a spherical-cap is widely applied to extract geometrical information from atomic force microscopy (AFM) topographic images. In this paper, we have developed a comprehensive 3D spherical cap fitting procedure (3D-SCFP) to extract morphologic characteristics of complete or truncated spherical caps from AFM images. Our procedure integrates several advanced digital image analysis techniques to construct a 3D spherical cap model, from which the geometrical parameters of the nanostructures are extracted automatically by a simple algorithm. The procedure takes into account all valid data points in the construction of the 3D spherical cap model to achieve high fidelity in morphology analysis. We compare our 3D fitting procedure with the commonly used 2D cross-sectional profile fitting method to determine the contact angle of a complete spherical cap and a truncated spherical cap. The results from 3D-SCFP are consistent and accurate, while 2D fitting is unavoidably arbitrary in selection of the cross-section and has a much lower number of data points on which the fitting can be based, which in addition is biased to the top of the spherical cap. We expect that the developed 3D spherical-cap fitting procedure will find many applications in imaging analysis.

Tables

Table 1. Table 1: Comparison of the calculated contact angles through 3D-SCFP and 2D cross-sectional profile fitting method. The cross section labels correspond to those in figures  4 and 5 . Two kinds of contact angle are compared: θ f ​ s subscript 𝜃 𝑓 𝑠 \theta_{fs} , contact angle of nanodroplets or nanoocap on flat substrate, and θ m ​ c subscript 𝜃 𝑚 𝑐 \theta_{mc} , contact angle of the nanodroplet on a nanocap.

	CS-1	CS-2	CS-3	CS-4	CS-5	CS-6a,b,c	3D fit
${\theta }_{fs}$	24.63o	25.38o	25.34o				24.82o
a: ${\theta }_{fs}$ (drop)				29.08o		27.32o	26.99o
a: ${\theta }_{mc}$ (drop)				30.73o			28.87o
b: ${\theta }_{fs}$ (drop)					27.61o	28.16o	27.82o
b: ${\theta }_{mc}$ (drop)					29.42o		30.28o
c: ${\theta }_{fs}$ (cap)				7.16o	7.45o	7.81o	7.65o

Equations

$$|h(r,\phi )-h_{fit}^{3D}(r,\phi )|/H_0.$$

$$h^{\phi }(r)=\frac{1}{2\pi }{\int }_0^{2\pi }h(r,\phi )d\phi ,$$

$$I_f=\frac{l_B{I}_A+l_A{I}_B}{l_A+l_B},$$

Full text

{affiliations}

Physics of Fluids group, Department of Science and Technology, Mesa+ Institute, and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands,

Soft Matter & Interfaces Group, School of Engineering, RMIT University, Melbourne, VIC 3001, Australia,

Center for Combustion Energy & Department of Thermal Engineering, Tsinghua University, China,

Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany.

1 Introduction

Atomic force microscopic (AFM) imaging has been one of the most popular techniques in the study of surface nanobubbles, nanodroplets and many other spherical cap nanostructures [1, 2, 3, 4]. Thanks to the high spatial resolution of AFM measurements, the size of nanobubbles and nanodroplets immobilised on the substrate can be characterised accurately in all three dimensions. Geometrical features of the spherical cap structures, such as lateral extension of the footprint, height and contact angle, can be compared on a quantitative level [1, 5, 6, 7]. Although most of those parameters are directly read out from the images, the contact angle of nanobubbles and nanodroplets requires further imaging analysis and data processing. As surface nanobubbles or nanodroplets are usually spherical caps due to dominance of capillarity on such small dimension, the data analysis requires the construction of an ideal spherical cap to fit the AFM data.

Up to now, fitting a two-dimensional (2D) cross-sectional profile is the most common approach for determining the contact angle of nanobubbles and nanodroplets [7, 8, 9, 10, 11, 12, 13, 14, 15]. With an AFM off-line software, the cross sectional profile through the centre of bubbles or droplets can be conveniently read out from the image. For instance, Simonsen et al. [10] and Wang et al. [12] calculated the contact angle of nanobubble, based on the measured height and footprint size in the cross-section. Wang et al. [7] determined the droplet size by simply averaging the values calculated from several different cross-section profiles. In a slightly improved method, the cross-sectional profile is fitted with a portion of an ideal circle (i.e. an arc). The angle subtended by the arc and the flat baseline is the contact angle of the measured nanostructures. This method has been applied by Zhang et al. [8, 9, 14, 15] and Yang et al. [16] to obtain the contact angle of nanobubbles and nanodroplets on several flat substrates. However, the drawbacks from 2D fitting are the arbitrary selection of the cross-section, which leads to large variation in the contact angle measured along different directions in the image [7, 11, 17, 18].

To obtain a contact angle accurately, the contribution of all valid the data points on the nanostructure must be taken into account in the reconstruction of the spherical cap model. This requires the development of a comprehensive 3D fitting [19, 20]. Although Song et al. [20] applied a 3D spherical-cap fitting to analyse the morphology of nanobubbles, sensible exclusion of unreliable data points is an important aspect in 3D fitting which has not been considered so far. Particularly for highly curved bubbles or droplets, the data points close to the three-phase contact line should be excluded (i.e. contact angle larger than $90^{o}$), due to the tip-sample convolution [1, 21]. In the analysis of an image of very small bubbles and droplets, there may be also potential effects from the disjoining pressure [22, 23], and hence only the data points above a certain threshold are valid for the fitting [19, 20]. Furthermore, no 3D fitting procedure has been reported for the analysis of a truncated spherical cap on a substrate with physical structures. A simple example of truncated droplets is that the droplets form at the rim of a microcap [15]. The shape of those truncated droplet resembles an apple slice with a part bitten off from the flat face.

In this paper, we have developed a 3D spherical-cap fitting procedure (3D-SCFP). The procedure employs the techniques of digital image processing to recognise the features from a spherical cap. It performs well for an isolated nanodroplet sitting on a flat substrate, but also for a truncated nanodroplet on the rim of a microcap [15, 24]. In the latter case, a feature extraction method, the Circle Hough Transform, is performed to distinguish the AFM data points from the part of the underlying microcaps and from the truncated nanodroplets [25, 26, 27].

With our proposed 3D-SCFP, we will analyse polymerised nanodroplets in AFM images collected from our previous work  [24], as represented in figure 1a. The nanodroplets (indicated by blue solid arrows) to be analysed include both complete spherical caps on a flat area and truncated droplets by the underlying microstructures (indicated by red dashed arrow). 3D-SCFP will translate AFM data points to ideal spherical caps that nicely match the nanodroplet morphology, as displayed in figure 1b. For those truncated nanodroplets, the procedure recognises the data points from a part of the spherical cap and reconstruct a whole spherical cap model.

The following sections of the paper are organised as follows: Section 2 details the algorithm of 3D-SCFP, and section 3 discusses the influence of an important parameter, the threshold of height cut-off, on the 3D fitted result. After that, the paper gives two examples in section 4, showing the comparison between 3D-SCFP and 2D cross-sectional profile fitting method. The results reveal the 3D-SCFP is robust, compared to the 2D fit method. The Matlab codes of 3D-SCFP are provided in the supplementary material, with free access to the readership.

2 Detailed 3D fitting procedure

The algorithm consists of four parts: raw image preprocessing, objective detection, objective identification, and cut-off of vertical data points near the rim.

2.1 Raw image pre-processing

The AFM images pretreated by second-order flattening by AFM off-line software are used in this step. A representative image is shown in $I$ (Fig. 2a). We create a so-called gamma encoded image $I^{\prime}$ (Fig. 2b) from the raw image $I$. Gamma encoding is defined as $V_{out}=AV_{in}^{\gamma}$ with $1>\gamma>0$ [28], where $V_{out}$ and $V_{in}$ present pixel intensity of images $I^{\prime}$ and $I$, respectively, and $A$ is a constant. The operation was implemented through “imadjust”, a basic function in MATLAB. After the operation, the pixel intensity of the whole image $I^{\prime}$ is nonlinearised, resulting in a sharper contact line with the flat substrate. The obtained image is shown in figure 2b. The sharp boundary in the image increases the reliability of the feature detection in the next step [29].

2.2 Objective detection

This step determines the circular footprint of a complete nanodroplet on the flat substrate, but also the footprints of multiple truncated nanodroplets and the underlying microcaps (blue circles in Fig.2b). An edge detection technique, namely the Circle Hough Transform [25, 26, 27], is applied to the gamma encoded image $I^{\prime}$. The Circle Hough Transform is a feature extraction technique for circle detection with robustness in the presence of noise and occlusions [30]. In order to have all the data points of a complete droplet included in the circle of the footprint, we set a parameter $pa_{0}$ to adjust the radii of the fitting. As demonstrated in the zoom-in in figure 2c, the initially detected circle (blue) does not enclose all the data points of a nanodroplet. By increasing the radius, a new circle (red) is generated, now wrapping up all data points (pixels) in the image. The rim of the truncated nanodroplet on the flat substrate is part of a circle, which together with the rim of the underlying nanocap can be detected through the advanced feature extraction method, i.e., the Circle Hough Transform. As shown in figure 2b, those blue connected rings fit the boundaries of the nanocap and the truncated droplets nicely.

2.3 Objective identification

If there are only isolated nanodroplets in the AFM image, the step discussed in this subsection is not essential. For the case with nanodroplets sitting on the rim of a microcap, it is crucial to automatically identify whether the detected objective is a nanodroplet or a microcap. The identification is based on the characteristics from experiments: (i) Nanodroplets can sit above microcaps, while the reverse does not apply; (ii) Serval nanodroplets can nucleate on the rim of an identical microcap; (iii) The truncated nanodroplets are higher or lower than the microcaps, depending on their materials (details refer to Ref. [24]).

Based on those information, we perform some conditional statements to determine the nanodroplet and the microcap. Then the data points in the overlap region are attributed to the recognised nanodroplets according to characteristics (i) (see appendix A for the details). In figure 2d, the identification numbers are displayed. Behind the label number of each microcap, the label numbers of its overlapping nanodroplets are also listed in the parentheses.

2.4 Cut-off of vertical data points near the rim

As already mentioned in the Introduction, the data points near the rim are subject to the influence of tip convolution and the disjoining pressure. Hence, an appropriate cut-off of the droplet rim morphology is needed for proper profile fitting, no matter 2D or 3D fitting method is applied. Here we take those data points above a threshold height for 3D spherical cap fitting (details are discussed in section 3). The threshold is a new parameter, $h_{cutoff}$ (sketch in Fig.2f). The valid data points are shaded in different colours in figure 2d, and fitted with a spherical cap by the program. The fitting is generated by minizing the cost function, which is the summed distance from the data points to the ideal spherical surface. The system of the nonlinear equation for the optimisation is directly solved by MATLAB. As shown in figure 2e, the whole topographic AFM image (the coloured surface) is perfectly matched by the corresponding ideal spheres (the black dot lines for nanodroplets and magenta for microcap). The fitting results are saved as ideal spheres’ radii and their centre-point coordinates with respect to the flat substrate surface. Based on these parameters, the calculations of footprint lateral extension $L$, height $H$, contact angle on flat substrate $\theta_{fs}$ (Fig.2f), contact angle on the nanocap $\theta_{mc}$ and other relevant geometrical parameters are determined mathematically.

3 Evaluation of the effects from cut-off threshold

In this section, we show the effect of the threshold by analysing the droplet in Fig. 4a. The nanodroplet is analysed by 3D-SCFP with 31 different $h_{cutoff}$ from 1 nm up to 30 nm (around 10% height of the nanodroplet), while all the other fitting parameters are fixed. With an increase in the threshold $h_{cutoff}$, the fraction of data points above the threshold decreases. The obtained contact angles from 3D-SCFP with different given $h_{cutoff}$ are plotted in figure 3a, showing a strong dependence on $h_{cutoff}$ for $h_{cutoff}<8$ nm. We define the fit error $\delta_{rms}$ as root mean square of the radial distance between the data points and the ideal sphere. The plot of the error versus the threshold in figure 3b shows that inclusion of the data points close to the substrate leads to derivation of the fitting from an ideal sphere. Too small $h_{cutoff}$ causes large 3D spherical-cap fitting error, while too larger $h_{cutoff}$ leads to less contributions from valid data points. An appropriate cut-off is determined to be the point just after the turning point in the plot of figure 3b, marked by an arrow. For each group of AFM images from a certain experimental condition, the “turning points” was determined by looping the procedure.

4 3D-SCFP versus 2D fitting

We provide in this section the comparison between the results of 3D-SCFP and 2D cross-sectional profile fitting method in two cases: an isolated nanodroplet and two truncated nanodroplets.

4.1 Case 1: An isolated nanodroplet on the flat substrate

The threshold $h_{cutoff}$ for the fitting is set to be 8 nm, which is determined in the way described in section 3 (Fig. 3a). In figure 4a, the 3D fitting result (the coloured spherical surface) shows an excellent agreement with the AFM data points (the black dots). To quantify the error of 3D fitting, the residual is defined as:

[TABLE]

The height of each point $h(r,\varphi)$ is expressed as function of the cylindrical coordinates (sketch Fig. 2f). $h_{fit}^{3D}(r,\varphi)$ is the corresponding ideal 3D profile by 3D fitting. The absolute difference is normalised by the droplet height $H_{0}$. As displayed in the residual map (Fig. 4b), the residual within most region is less than 1%, which is comparable to the signal noise. The maximum deviation, around 8%, occurs at the region near the contact line. From 3D fitting result, the contact angle $\theta_{fs}$ is calculated and listed in Table 1.

We introduce an averaged cross-sectional profile $h^{\varphi}(r)$ defined as:

[TABLE]

where $h(r,\varphi)$ is the height of the data point in the cylindrical coordinates (sketch Fig. 2f). The origin of the coordinate system is the centre point of the circular footprint of the ideal sphere on the flat substate. As displayed in figure 4c, the 3D spherical-cap fitting result (the red line) and the averaged cross-sectional (the black square line) profile perfectly match with each other as expected.

In 2D fitting, the nanodroplet is assumed to be an ideal spherical cap. Part of three circles are used to fit three different cross-sectional profiles of the same nanodroplet: CS-1 (along $45^{o}$ direction), CS-2 ($0^{o}$) and CS-3 ($135^{o}$), as shown in figures 4d, e and f, respectively. The data for the fitting are selected as the points that are higher than 8 nm, which are displayed as solid dots in figures 4d-f. The fitting results show that the 2D fitting circle (the black line) and the profile data (the circular dots) in each cross section match well. The contact angles fitted from those three profiles data are listed in Table 1. $\theta_{fs}$ obtained from CS-1 is smaller than the fitted results of other two cross-sectional profiles. We display the residuals along different cross-section directions, i.e. $|h^{CS}(r)-h_{fit}^{3D}(r)|/H_{0}$, in figure 4g-i. The deviation profiles vary along different cross-section directions and can be asymmetry. Hence, different selections of the direction of the cross-section in 2D cross-sectional profile fitting give inconsistent results, as also concluded from the fitted contact angles shown in Table 1. The inconsistency of the 2D fitting results has already been noticed by other researchers [11, 17, 18]. Such difference among different cross-sections might be associated with different extent of experimental errors along different directions of the image: longer scanning time required to collect the data points across different scan lines and along the same line and hence the cross-section along the scan line in (Fig. 4g) is more accurate. Another possibility is that the droplet is never an ideal spherical cap, due to the unavoidable chemical heterogeneities on the surface [31, 32]. If so, the heterogeneity may lead to the irregularity on the base boundary and a non-circular droplet footprint. Overall, the variation of the extracted cross-sectional profile jeopardise the reliability of 2D cross-sectional profile fitting method.

4.2 Case 2: Truncated nanodroplets sitting on the rim of a nanocap

We now analyse the image with two truncated nanodroplets A and B sitting on the rim of a nanocap C (Fig. 5a). The threshold $h_{cutoff}$ for nanocap and nanodroplets is set to be 10 nm and 20 nm, respectively. 3D-SCFP automatically distinguishes the data points on nanodroplets A, B and on the nanocap C, respectively, and then fits valid data points of each them to an ideal sphere. As shown in figure 5a, the fitting result is displayed in coloured spherical surface and compared with AFM data points in black dots. A residual map of 3D fitting defined as in equation (1) is given in figure 5b ($H_{0}$ for the map is the height of nanodroplet B here). The high performance of 3D fitting is thus quantitively verified. The maximum residual is less than 10% and is located at the contact region between the nanodroplets and the nanocap. The map here also exposes the inconsistency between different cross-sections. Based on the definition in equation 2, we also build their averaged cross-sectional profiles. Three averaged cross-sectional profiles (black square) are integrated in a height-radial coordinate with the origin defined as the footprint centre of the nanocap C (Fig. 5c). The 3D fitting results are displayed as the red lines, showing perfect agreement to the averaged profiles. Here, we introduce the contact angle of the nanodroplet on the overlapped nanocap, labeled as $\theta_{mc}$. Then both $\theta_{fs}$ and $\theta_{mc}$ are calculated from the 3D fitting results and are listed in Table 1.

For this case, it is no longer convenient to apply 2D cross-sectional profile analysis any more. To obtain the correct contact angle $\theta_{mc}$, we must pick up the cross section that passes both the centre point of the nanodroplet and that of the nanoocap. However, there are unavoidable artificial errors of locating the centra positions of the nanodroplet and the nanocap. When the cutting line of the selected cross section is not in the column direction or row direction of the image pixel matrix, the profile of an selected cross section can not be precisely depicted, because the cutting line may not pass pixel centres (as sketched in Fig. 7 in appendix B). In low pixel resolution image, this problem is even more crucial. In order to reduce this influence as much as possible in this section, firstly we choose an AFM image with a relatively high resolution. The lateral dimension of a nanodroplet is depicted by more than 40 pixels (Fig. 5a). Secondly, the extraction of the cross-sectional profile in a particular direction is improved by applying a linear fitting, as described in appendix B.

In figures 5d, e and f, two cross sections (CS-4 and CS-5) that pass two centre points of the objectives and three complete cross-sectional profiles of each objective (CS-6a, b and c) are displayed. Notably, the profile of nanocap C in CS-4 is not only overlapped by nanodroplet A, but also slightly by the edge of nanodroplet B, which is not conspicuous. The overlap reduces the number of the valid data points for a 2D fit. This situation tends to be common when there are more nanodroplets sitting on a same nanocap. We only apply 2D fits to the valid segment of the extracted profiles that are higher than 10 nm for nanocap C and higher than 20 nm for nanodroplets A and B. As shown in figures 5d-f, the 2D fitting results (the black circles) have a good agreement to the corresponding profile data (solid dots). Based on the 2D fitting results, $\theta_{fs}$ and $\theta_{mc}$ of the nanodroplets and the $\theta_{fs}$ of the nanocap are calculated and are listed in Table 1. Again, the normalised deviation of $h^{CS}(r)$ from $h_{fit}^{3D}(r)$ is calculated and is displayed in figures 5g-i. It shows that the deviation increases in the profile part nearby the contact point/line. Especially for nanodroplet A in CS-4, its profile severely departs from its 3D fitting profile (Fig. 5g), leading to 2o difference for both $\theta_{fs}$ and $\theta_{mc}$ (Tab. 1). However, with the partial profile of nanodroplet B in CS-5, the difference of fitted results of the 2D method to that of the 3D-SCFP is less than 1o. We also acquired similar fitting result $\theta_{fs}$ when applying 2D fits to the complete extracted profiles CS-6a, b and c. But in general the contact angle $\theta_{mc}$ is unaccessible from the complete profiles as in general the cross sections do not go through the centres of both the nanodroplets and the nanocap. Thus again the 3D-SCFP fit demonstrates its superiority in this complex case.

5 Conclusion and Outlook

In this paper, we provide a 3D spherical-cap fitting procedure (3D-SCFP) with many advantages and capabilities. The procedure integrates powerful feature extraction method, namely the Circle Hough Transform. Through this method, the data points of the truncated nanodroplets and the overlapped microcaps can be accurately separated. Then the procedure applies 3D fits to the nanodroplet and the microstructure separately. The details of the procedure and the MATLAB program are provided. We also provide a comparison between 3D-SCFP and the often-used 2D cross-sectional profile fitting method by applying them on two AFM images: one with an isolated nanodroplet and the other complex one with two truncated nanodroplets sitting on a nanocap. The 2D fits have the following shortcomings: (i) the uncertainty caused by the arbitrary selection of the cross-sectional profile; (ii) the limited number of data points taken for the fitting and the too large statistical weight of data points close to the top of the spherical cap; and (iii) inconvenience in the truncated nanodroplet case lies in the small amount of fitted data points, the artificial error of the cross section selection and the possible distortion of the extracted profile nearby the contact line. The first two shortcomings lead the inconsistent results of 2D cross-sectional profile fitting method. When a required cross section is not along the pixel array in the truncated nanodroplet case, the difficulty of extracting an accurate cross-sectional profile makes the last shortcoming more obvious. However, all the defects above are overcome in the 3D-SCFP as it considers all the valid data points and fits them in 3D. The comparison confirms that for both cases. Therefore the 3D-SCFP is strongly preferable.

We expect our 3D spherical-cap fitting procedure to find many applications in image analysis. It may be particularly useful for case 2 in which (nano)droplets overlap with a (nano)cap (or other droplets), as then the potential of the procedure is fully explored. This case 2 is omnipresent in nature [33, 34, 35, 36] and (nano)technology [1, 15, 24]. The morphology of this case with unconventional shapes can be used as effective templates [37, 38, 39] in fabrication. In recently published work [40] it was reported that the droplet nucleation on convex shape, such as microcaps, significantly enhances the heat transfer. That paper underlines various important applications of convex shapes in water collection and heat transfer connected with phase transitions. One famous biomimetic structure is the back of a desert beetle, where water droplets form on the spherical lumps of the hydrophilic domain. Another famous biomimetic concept is the ‘Lotus Effect’ due to micro- and nanoscopic architecture under a droplet [41, 42, 43]. A mimicked lotus-leaf surface is fabricated by making structures with convex shapes on top of a surface [44]. The morphology of droplets on caps with unconventional shapes also provides effective templates [37, 38, 39] in fabrication of sophisticated microparticles. How to precisely extract morphologic characteristics of truncated spherical caps from experimental data is crucial for all these cases. The method and codes created in this paper are readily implemented for imaging analysis of similar (nano)structures, and are accessible to all users through the supplementary materials online.

{addendum}

We thank Ivan Dević for discussion and testing of 3D spherical-cap fitting procedure. We gratefully acknowledge NWO because of support through the MCEC Zwaartekracht program. D.L. in addition acknowledges the support from an ERC Advanced Grant and X.H.Z the support from Australian Research Council (FT120100473, DP140100805). H. T. thanks for the financial support from the China Scholarship Council (CSC, file No. 201406890017).

The authors declare that they have no competing financial interests.

h.tan@utwente.nl or xuehua.zhang@rmit.edu.au or d.lohse@utwente.nl.

Appendix A. Data points division

The projector of the contour of the truncated nanodroplet on the top surface of the microcap (the black solid line) is a part of an ellipse. It is within the overlapping region of two detected circles. In the “Objective detection” step of 3D-SCFP, we prepare data points inside the microcap-circle and outside the nanodroplet-circle (Data group 1) for 3D fit of the microcap, while the data points inside the nanodroplet-circle and outside the microcap-circle (Data group 2-1) for 3D fit of the nanodroplet.

In order to also consider contributions from the data points in the overlapping region, we divide the region into two parts with a straight line (the red dash-line). The line passes through two intersection points of two circles. Once the nanodroplet is recognised in “Objective identification” step, we can determine the part that all belongs to the nanodroplet. The part (Data group 2-2) is labeled with a red-line pattern as displayed in figure 6. Hence, both the data group 2-1 and the data group 2-2 are used in 3D fit of the overlapping nanodroplet in the final step of 3D-SCFP.

Appendix B. Cross-sectional profile extraction

An AFM image is a raster image. Each pixel has a relative x-y position in the image. The pixel intensity correspond to the sample height (or cantilever deflection) at that location. When we extract a cross-sectional profile along any direction except $0^{o}$, $45^{o}$ or $90^{o}$, it is common to come across the situation shown in figure. 7. The cutting line of the cross section does not pass through any pixel centres in a pixel column (row). In order to obtain the height value $I_{f}$ at position $(x_{0},y)$, we assume there existing a linear variation between the height values at positions $(x_{0},y_{i})$ and $(x_{0},y_{i+1})$. Hence, $I_{f}$ is calculated as,

[TABLE]

where $I_{A}$ and $I_{B}$ are the distances from the investigating point to the neighbour pixel centres. The AFM image used in figures 5 has a high resolution, so the linear fit is precise enough. For a low resolution AFM image, it is required to apply a high order fit which considers contributions from more neighbour pixels.

Appendix C. Matlab Codes

Matlab codes of 3D spherical-cap fitting procedure are available for any applications and future development for any purpose. A resulting publications should cite this paper.

Reference