Application of Areal Topography Parameters in Surface Characterization

Abstract

This review paper provides a comprehensive overview of selected 3D surface texture parameters defined by ISO 25178-2, with a focus on their metrological aspects in high-resolution measurements using atomic force microscopy (AFM). The parameters Sa, Sz, Sq, Sdq, and Sdr are analyzed in terms of their practical application, sensitivity to measurement conditions, and role in assessing surface functionality. Through a review of the literature and simulations of surface profiles with controlled geometric variations, the study demonstrates how the selected parameters respond to changes in step pitch, step width, slope, and lateral calibration errors. Experimental AFM measurements performed on a certified step height standard further illustrate the impact of calibration on the quality of measurement results. Special emphasis is placed on the importance of evaluating measurement uncertainty. The results confirm the need for rigorous instrument calibration and uncertainty assessment to ensure reliable and comparable surface characterization across different instruments and laboratories.

1. Introduction and Terminological Framework

Modern advancements in surface nanotechnology and scanning probe microscopy (SPM) have opened new avenues for detailed surface characterization at micro- and nanoscales. Areal topography parameters (3D surface parameters) provide a more comprehensive and spatially resolved description of surface geometry compared to conventional two-dimensional (2D) profile parameters, which often fail to capture the complexity of functional surface features [1,2]. This enhanced capability is essential for linking surface characteristics with critical functional properties, such as adhesion, friction, wettability, and optical performance [3].

Surface topography can be quantified through a wide array of parameters, both filtered and unfiltered, using 2D profilometry or advanced 3D measurement techniques. Traditional 2D roughness parameters (R) and waviness parameters (W) rely on filtering the measured profile according to standardized cut-off values. However, areal topography parameters—commonly referred to as 3D roughness parameters—are often left unfiltered due to the lack of established guidelines regarding appropriate cut-off values in areal measurements [4]. This has led to inconsistent terminology and interpretation across various studies [5]. The lack of standardized terminology sometimes leads to misinterpretation of results. Several surface topography parameters describe the surface condition.

In surface nanotechnology, areal topography parameters serve as a powerful tool for characterizing a wide variety of surfaces produced through techniques such as laser surface modification, thin film deposition, and micro- or nano-fabrication processes [6,7]. Surface integrity, which encompasses both microstructural and topographical features, is critical in applications where wear resistance, adhesion, and corrosion resistance are of primary importance, as surface defects can act as initiation sites for failure [8]. In this context, areal topography data can significantly influence manufacturing outcomes and, ultimately, determine the quality of the final product [9,10].

Quantitative assessment of surfaces using areal topography parameters is essential for understanding the interactions between manufacturing processes and the resulting surface properties [11,12]. Areal topography parameters are particularly vital for characterizing the surface condition of materials in additive manufacturing (AM) and machining processes [13]. The use of areal topography parameters to enhance the reliability of AM components has been demonstrated in [13]. Muniappan et al. reported that the surface roughness and material removal rates of nanoparticle-reinforced powder metallurgy alloys were significantly affected by variations in machining parameters such as depth of cut and feed rate, demonstrating the utility of areal topographic analysis in optimizing manufacturing processes [14]. Surface topography also plays a significant role in powder metallurgy [15,16]. In [17], the effects of laser beam micromachining on the surface characteristics of compacted titanium samples, produced using hydride-dehydride (HDH) titanium powder compaction, are investigated, with a specific focus on applications in dental implantology.

Surface topography characterization typically relies on areal parameters, which provide a comprehensive representation of surface features, including both roughness and waviness. In [18], the authors investigate how to accurately and efficiently measure the irregular surfaces of metallic components produced via additive manufacturing, employing an advanced optical method known as coherence scanning interferometry (CSI). The integration of high-resolution techniques such as atomic force microscopy (AFM), a prominent SPM method, allows for nanoscale mapping of surface topography, providing detailed insight into surface roughness, texture, and local defects [19]. Atomic force microscopy (AFM), one of the most widely used scanning probe microscopy techniques, enables nanometer-resolution surface characterization and has become a standard method for acquiring areal topography data. Even when optical or stylus-based profilometry is used, AFM serves as a critical reference technique for verifying and complementing roughness measurements, especially when surface features approach the nanoscale.

The use of areal topography parameters is particularly significant in the characterization of metals fabricated through additive manufacturing techniques such as selective laser melting (SLM) and laser powder bed fusion (LPBF) [20]. These processes often generate complex surface textures whose behavior cannot be adequately described using 2D metrics alone. Recent studies indicate that specific surface textures can be quantitatively characterized, revealing features closely related to process parameters such as laser inclination and energy input [21]. Furthermore, surface topography plays a crucial role in defining fundamental mechanical properties of components—such as friction and wear performance—which, in turn, have a direct impact on the operational efficiency and longevity of machine parts in industrial settings [22,23].

The integration of surface characterization into broader materials processes reflects a multidimensional approach, enhancing our understanding of how surface attributes influence macroscopic performance. The cleaning, coating, and surface treatment of metallic components are essential in industries aiming to extend component lifespan and performance [24]. As surface metrology continues to evolve, the combination of areal topography parameters with scanning probe techniques remains central to the advancement of nanostructured materials and devices [25].

2. Areal Topography Parameters from Definition to Application

A surface condition is characterized by numerous areal topography parameters, which can be classified into several categories: height, spatial, hybrid, functional, and miscellaneous [26]. Most of these parameters are defined in the ISO 25178-2:2021 standard Geometrical Product Specifications (GPS)—Surface Texture: Areal—Part 2: Terms, Definitions, and Surface Texture Parameters [26].

Although surface texture is inherently multidimensional and complex, industry and practice still rely heavily on a limited set of parameters namely Ra or Sa and Rz or Sz due to their simplicity, standardization, and ease of interpretation. Most technical drawings, specifications, and standards require only Ra or Sa, which reinforces their widespread use in quality control and production.

The most commonly used roughness parameters Ra in 2D and Sa in 3D represent the arithmetical mean deviation of the surface profile or area. Mathematically, they quantify the average absolute deviation of height values from the mean plane, but do not capture the spatial distribution, shape, or frequency of surface features. As a result, entirely different surface profiles—for example, one with smooth, widely spaced undulations and another with sharp, densely packed peaks and valleys—can exhibit the same Ra or Sa value, despite having vastly different functional or morphological properties.

Illustrative examples of surfaces with identical Sa values but distinct topographies are shown in Figure 1. Figure 1 shows three different surface profiles (sinusoidal, randomly rough, and sharply peaked) that all share the same Sa value (approximately 18 nm). In all images, color indicates surface height: brighter colors represent higher regions, while darker tones correspond to deeper regions.

It is well known that two surfaces can share the same Sa value while differing significantly in structure, function, or manufacturing quality. However, it is less often emphasized that the magnitude and reliability of the Sa parameter as well as other surface parameters do not depend solely on their mathematical definitions, but critically on the measurement conditions, instrument calibration, and data resolution. This is particularly evident in high-precision instruments such as atomic force microscopes (AFMs), where even minimal calibration errors in the x- or y-axis can distort dimensions, alter feature spacing, and lead to incorrect parameter estimation.

To provide a broader basis for surface characterization,

Table 1. Three-dimensional surface texture parameters—definitions and applications.

Symbol	Parameter Name	Category	Description	Formula	Typical Application
Sa	Average Roughness	Height	Arithmetic mean of absolute deviations from the mean plane.	$Sa=\sqrt{{\displaystyle \frac{1}{A}}\underset{A}{\iint }\left|\mathrm{z}\left(x,y\right)\right|dxdy}$	Machined surfaces, general surface assessment
Sq	Root Mean Square Roughness	Height	Root mean square of surface deviations; sensitive to outliers.	$Sq=\sqrt{{\displaystyle \frac{1}{A}}\underset{A}{\iint }{z}^2\left(x,y\right)dxdy}$	Optical surfaces, additive manufacturing
Ssk	Skewness	Height	Indicates asymmetry of surface profile; distinguishes pits vs. peaks.	$Ssk={\displaystyle \frac{1}{{Sq}^3}}\left[{\displaystyle \frac{1}{A}}\underset{A}{\iint }{z}^3\left(x,y\right)dxdy\right]$	Wear monitoring, honing, porosity analysis
Sku	Kurtosis	Height	Measures sharpness of peaks and valleys.	$Sku={\displaystyle \frac{1}{{Sq}^4}}\left[{\displaystyle \frac{1}{A}}\underset{A}{\iint }{z}^4\left(x,y\right)dxdy\right]$	Defect detection, surface quality control
Sp	Maximum Peak Height	Height	Height of the highest peak.	Sp = max(z(x,y))	Coatings, surface anomalies
Sv	Maximum Valley Depth	Height	Depth of the deepest valley.	Sv = min(z(x,y))	Lubrication systems, sealing
Sz	Maximum Height of Surface	Height	Total vertical range of the surface.	$Sz=Sp+Sv$	Surface anomalies, roughness extremes
Sal	Autocorrelation Length	Spatial	Measures texture feature spacing.	$Sal=\underset{t_x,t_y\in R}{\mathrm{m}\mathrm{i}\mathrm{n}}\sqrt{t_x^2+t_y^2}$	Tribology, friction, wear, optics
Str	Texture Aspect Ratio	Spatial	Describes surface texture isotropy.	$Str={\displaystyle \frac{\underset{t_x,t_y\in R}{\mathrm{m}\mathrm{i}\mathrm{n}}\sqrt{t_x^2+t_y^2}}{\underset{t_x,ty\in Q}{\mathrm{m}\mathrm{a}\mathrm{x}}\sqrt{t_x^2+t{y}^2}}}$	Tool wear, directional texture analysis
Sdq	Root Mean Square Surface Slope	Hybrid	Average slope of the surface texture.	$Sdq=\sqrt{{\displaystyle \frac{1}{A}}\underset{A}{\iint }\left[{\left({\displaystyle \frac{\partial z\left(x,y\right)}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial z\left(x,y\right)}{\partial y}}\right)}^2\right]dxdy}$	Sealing, appearance, coatings
Sdr	Developed Interfacial Area Ratio	Hybrid	Increased surface area due to roughness.	$Sdr={\displaystyle \frac{A_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}-A_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}}}{A_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}}}}$	Adhesion, coatings, complexity
Sds	Summit Density	Hybrid	Number of summits per unit area.	Defined as count of summits per area	Contact mechanics, material interactions
Ssc	Mean Summit Curvature	Hybrid	Average curvature of surface summits.	Statistical curvature average	Mechanical contact analysis
Sdc(mr)	Inverse Areal Material Ratio	Functional	Height corresponding to a specific material ratio.	Defined by material ratio function	Bearing surface estimation
Sk	Core Roughness Depth	Functional	Height of core material zone.	Defined from material ratio curve	Wear resistance, plateau honing
Smr(c)	Areal Material Ratio	Functional	Material ratio at a given height.	From Abbott-Firestone curve	Cylinder bore, sealing analysis
Spk	Reduced Peak Height	Functional	Height of peaks above core zone.	From material ratio curve	Wear and tribological analysis
Svk	Reduced Valley Depth	Functional	Depth of valleys below core zone.	From material ratio curve	Lubricant retention, surface wear
Sxp(p,q)	Peak Extreme Height	Functional	Height of extreme peaks within defined percentiles.	Height between q% and p%	Critical peak identification
Vm(mr)	Material Volume	Functional	Volume of material above a height level.	Integral of material ratio above mr	Bearing surface, wear analysis
Vmc(p,q)	Core Material Volume	Functional	Volume of core material between p and q levels.	Difference in volumes at p% and q%	Load-bearing capacity
Vmp(p)	Peak Material Volume	Functional	Volume of material in peak zone above p%.	Volume from p% to top	Surface coating, sealing
Vv(mr)	Void Volume	Functional	Volume of voids below a height level.	Integral of void volume below mr	Fluid retention, surface roughness
Vvc(p,q)	Core Void Volume	Functional	Void volume between p and q material ratios.	Vvc = Vv(p) − Vv(q)	Residual volume after wear
Vvv(p)	Dale Void Volume	Functional	Void volume in deepest part of surface.	Volume from p% to valley base	Debris entrapment, lubrication
Std	Texture Direction	Miscellaneous	Angular direction of surface texture.	Angle from y-axis	Sealing, directional surface analysis

presents a summary of the most relevant 3D areal parameters as defined in ISO 25178-2, along with their mathematical definitions and typical applications.

The following section provides a literature review of the application of surface texture parameters in industry and research.

The Sa and Sq parameters provide an overall measure of surface texture. However, they do not differentiate between peaks and valleys, nor do they account for the spatial distribution of surface features. Consequently, surfaces with significantly different topographic features can exhibit identical Sa and Sq values. Figure 2 illustrates two distinctly different surfaces with identical Sa and Sq values, highlighting the limitations and insensitivity of these parameters when used in isolation. Nevertheless, once the surface type or functional context is known, Sa and Sq can still serve as useful indicators of significant deviations in surface characteristics.

Sq is commonly applied in the characterization of optical surfaces [27], metal components produced by additive manufacturing [18], dental implant bonding cements [28] and adhesion of dental restorative composites [29]. Sa is typically used for assessing machined surfaces [30].

Figure 2. Different surfaces with the same parameter values [31]: Sa = 16.03 nm and Sq = 25.4 nm.

Figure 2. Different surfaces with the same parameter values [31]: Sa = 16.03 nm and Sq = 25.4 nm.

For normally distributed surfaces, Ssk = 0 and Sku = 3. The Ssk parameter is particularly useful in monitoring various types of wear, honing processes, and surface porosity [32]. The Sku parameter is commonly used to detect surface defects caused by extreme peaks or pits [31], and for identifying irregularities in critical surface components [33]. Since both Ssk and Sku involve higher-order statistical moments, a sufficient number of measurements must be conducted to obtain statistically reliable values, and appropriate filtering must be applied to remove spurious peaks or pits.

As Sp, Sv, and Sz are based on individual surface points, their values may lack repeatability. Therefore, when using these parameters, it is essential to apply appropriate cut-off settings to minimize the influence of anomalous peaks or pits. The Sz parameter is typically used in evaluating coatings and sealing surfaces and in identifying surface anomalies [34]. It is particularly relevant for applications involving fluid retention, such as lubrication or surface coating systems, where the depth profile affects functional performance [35].

The Str parameter is valuable in identifying directional features of surfaces, especially those produced by multi-step manufacturing processes. It is also used to detect subtle texture orientations on otherwise isotropic surfaces and to monitor machine tool vibrations [36].

Sal, on the other hand, quantifies the distance over which a surface texture remains statistically correlated. It helps determine the appropriate spacing between measurement points to accurately characterize surface texture. Sal finds application in the analysis of tribological behavior (e.g., friction and wear) and in studies of surface interaction with electromagnetic radiation [37,38].

Sdq is particularly useful for differentiating surfaces that share the same average roughness (Sa). It finds application in the analysis of sealing systems, surfaces where cosmetic appearance is important, and in contexts involving fluid interaction with surfaces (e.g., coatings) [39]. This parameter is influenced by both the amplitude and the spacing of the surface texture. For a given Sa value, a wider spacing between surface features generally results in a lower Sdq, compared to a surface with the same Sa but smaller spacing (Figure 3).

Sdr can distinguish surfaces that exhibit similar amplitude values (e.g., Sa). Typically, as surface complexity increases, Sdr rises regardless of whether Sa changes. This parameter is particularly relevant for applications involving surface coatings, adhesion, and fluid interactions [41]. Sdr is also influenced by texture amplitude and spacing. Thus, a surface with high Sa but widely spaced features may exhibit a lower Sdr compared to a surface with a lower Sa and finely spaced features. In [42], it is demonstrated that the parameters Svk and Spk provide valuable insight into the wear resistance of materials under various operating conditions.

Smr(c) is used to assess the amount of residual bearing surface after a certain depth of material has been removed from the surface [43]. A typical application is in specifying engine cylinder bore surfaces before running-in. The bore can be ground to form a plateau-like surface, with fine, pointed structures placed to aid the running-in process of sliding piston rings. The bore can be ground to form a plateau-like surface, with fine, pointed structures placed to aid the running-in process of sliding piston rings [44]. Figure 4 illustrates the material ratio on the observed surface.

Different volume parameters are crucial for understanding how much material can be worn away at a given depth of the bearing curve (e.g., Vmp(p)) and how much material remains to support the load after upper surface levels are worn down (e.g., Vmc(p,q)). These parameters are applied in sealing, where Vmp(p) provides insight into the amount of material available for sealing purposes [45]. Vv(mr), Vvv(p), and Vvc(p,q) quantify the volume of cavities within the surface, with height variations determined by the selected material ratio values. These parameters are useful for determining how much fluid would fill the surface between the selected material ratio heights [46]. Void volume parameters are critical for understanding fluid flow, coating applications, and debris entrapment [47]. Core void volume (Vvc) can be used to assess how much space remains available in the surface core after the surface has undergone a running-in process, leading to reduced peak heights [43]. Meanwhile, Vvv(p) can be indicative of the residual volume remaining after significant surface wear [48].

The Std parameter measures the angular direction of the dominant surface texture (Figure 5). It is defined relative to the y-axis, where a surface aligned along the y-axis has an Std value of 0°. Since no dominant surface texture direction is observed in the right image of Figure 5, the Std parameter is undefined. Currently, the Std parameter is not prescribed by ISO 25178-2 [26].

Std is useful for determining the direction of the surface texture relative to the known orientation of the measurement instrument [44]. Its application is particularly relevant in sealing, where subtle changes in the direction of surface texture can lead to adverse conditions [50]. Additionally, Std is used to detect the presence of a preliminary surface treatment process (e.g., turning) that may be removed during subsequent finishing operations (e.g., grinding) [51].

Among surface topography parameters, Sdq stands out as a hybrid descriptor that provides additional insight into the microstructural characteristics of a surface. Unlike spatially oriented parameters such as Sa and Sz, Sdq quantifies the average slope of surface features, thereby capturing the uniformity and smoothness of the coating distribution at the nanoscale. Higher Sdq values indicate abrupt height variations and pronounced roughness, whereas lower values reflect a homogeneous and smooth surface.

In the study [39], Sdq was employed as an indicator of coating distribution quality under various application and drying methods of epoxy primers. It was found that only the chemical composition of the coating, specifically, the presence of zinc powder, had a statistically significant influence on Sdq values (p < 0.001), while variations in drying techniques and application methods showed no significant effect. These findings suggest that Sdq may serve as a sensitive indicator of microporosity and surface inhomogeneity, particularly in coatings with a high content of metallic pigments. In this context, incorporating Sdq into surface topography analysis represents an important step toward more accurate characterization of functional coatings in engineering applications.

In [52], areal topography parameters of cold-forged austenitic stainless steel AISI 304 were analyzed to evaluate their influence on corrosion resistance. Surface characterization was performed using atomic force microscopy (AFM), measuring 3D roughness parameters including Sa, Sz, Sv, Sp, Sku, Ssk, and Sq. Statistical analysis (ANOVA) confirmed that surface roughness is significantly affected by the surface preparation method, whereas the degree of plastic deformation had no significant influence. These findings highlight the importance of areal topography parameters, in assessing corrosion behavior, emphasizing their practical relevance in industrial settings where ideal surface finishing cannot always be achieved.

Although ISO 25178-2 provides standardized definitions of surface parameters, it assumes ideal measurement conditions. In real-world applications, measurement systems must be calibrated, adjusted, and verified in accordance with the principles outlined in ISO 25178-700 [53] to ensure accurate and traceable results. For atomic force microscopes (AFMs), which operate at the nanometer scale, such a metrological approach is not optional; it is essential.

The aim of this review is to provide a standards-aligned and metrologically grounded overview of selected areal surface topography parameters, with particular emphasis on their behavior in atomic force microscopy (AFM) measurements and the role of uncertainty estimation an aspect often underrepresented in similar reviews.

The following chapter presents simulation and experimental results analyzing how lateral inaccuracies, geometric variations, and measurement noise affect surface parameters such as Sa, Sq, Sz, Sdq, and Sdr. The findings confirm theoretical assumptions and highlight the importance of rigorous calibration and the application of ISO 25178-700 guidelines in precise surface metrology.

An essential part of such analysis is the evaluation of measurement uncertainty in accordance with the principles outlined in the Guide to the Expression of Uncertainty in Measurement (GUM). This study applies a Monte Carlo-based approach, as described in JCGM 101, which enables a realistic estimation of uncertainty under complex models and nonlinear influences. Such quantitative evaluation is crucial for the reliable interpretation of surface parameters, for the comparison of results between different laboratories, and for assessing compliance with specification limits.

3. Assessment of the Quality of Measurement Results Obtained via an Atomic Force Microscope

The quality of measurement results obtained using an atomic force microscope (AFM) depends on several interrelated factors, including system calibration, environmental stability, and the choice of scanning parameters. An accurate interpretation of surface characteristics requires not only high-resolution data but also confidence in the traceability and repeatability of results. Therefore, the assessment of measurement quality implies the evaluation of measurement uncertainty.

Measurement uncertainty is defined as a non-negative parameter that characterizes the dispersion of quantity values attributed to a measurand [54]. According to the Guide to the Expression of Uncertainty in Measurement [55], measurement uncertainty is a parameter associated with the result of a measurement, characterizing the dispersion of values that could reasonably be attributed to the measurand. Uncertainty implies doubt, and in its broadest sense, measurement uncertainty reflects doubt about the validity of a measurement result.

According to Measurement Systems Analysis [56], measurement uncertainty is an estimate of the range within which the true value is believed to lie. A clear understanding and proper estimation of measurement uncertainty is essential for ensuring the reliability and comparability of AFM-based results, particularly in high-precision or interlaboratory applications. Although surface topography is a well-established field in dimensional metrology, there is still limited work in the literature that explicitly recognizes the importance of measurement uncertainty estimation as a quality indicator of measurement results. In many studies, results are reported without a corresponding uncertainty evaluation, which limits their reliability and comparability, especially in high-precision applications.

A significant contribution in this context is the doctoral dissertation of one of the authors, titled Estimation of Measurement Uncertainty in the Field of Atomic Force Microscopy in Dimensional Nanometrology [57], which presents research findings focused on uncertainty estimation in AFM-based surface measurements. The dissertation includes the development of mathematical models for estimating measurement uncertainty, calculations based on both Monte Carlo and Bayesian methods, and results from interlaboratory comparisons. These findings illustrate how various sources of uncertainty can significantly influence the reliability and variability of AFM measurement results.

For uncertainty evaluation, the following influential input quantities were analyzed: measured value (h_x), scan rate (r), scanning time (t), scan length (l), nominal step height (d), thermal expansion coefficient (α), temperature difference (Δϑ), repeatability (δr), reproducibility (δR), and probe-related uncertainty (δprobe). The evaluation model is given by expression (1).

h = h_x + r · t · l + d · α · Δϑ + δr+ δR + δprobe

(1)

The probability density function for the step height was obtained using 100,000 simulations and is shown in Figure 6.

The expanded measurement uncertainty at a confidence level of p = 95% is U = 2.9 nm, with the corresponding interval limits Y_0.025 = 96.0 nm and Y_0.975 = 101.8 nm (red lines in Figure 6). This result was compared with the certified value in the standard’s calibration certificate. Results are shown in

Table 2. Comparison of step height measurements with certificate value [57].

	Certificate of Calibration	Lab A
$\overline{h}$/nm	97.6	98.9
u/nm	0.7	1.45

. Our results are labeled as Lab A in the table.

The comparison of results is performed by calculating the compatibility factor En, Equation (2).

$$En={\displaystyle \frac{\left|{\overline{h}}_i-{\overline{h}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right|}{2·\sqrt{u_i^2+u_{\mathrm{r}\mathrm{e}\mathrm{f}}^2}}}\le 1$$

(2)

where ${\overline{h}}_i$—step height obtained by an individual laboratory, ${\overline{h}}_{ref}$—step height according to NIST, u_i—the estimated standard measurement uncertainty of an individual laboratory, and u_ref—standard measurement uncertainty according to NIST.

The calculated En factor is 0.4, which is significantly below the acceptance threshold of 1. This indicates good agreement between the measurement result and the reference value, thereby confirming both the accuracy of the measurement and the consistency of the estimated measurement uncertainty. In order to ensure the reproducibility of measurement results, an interlaboratory comparison was conducted. An interlaboratory comparison included four laboratories and evaluated step height h, as well as areal parameters Sa and Sz. The results are shown in

Table 3. Results of interlaboratory comparative measurements for step height h.

	Lab A	Lab B	Lab C	Lab D
$\overline{h}$/nm	98.9	101.3	100.2	100.8
u(h)/nm	1.45	2.55	1.75	2.80
En(h)	0.40	0.70	0.69	0.55
Sa/nm	15.60	29.68	25.45	16.76
En(Sa)	3.32	4.14	1.89	2.71
Sz/nm	116.9	122.8	117.8	122.1
En(Sz)	0.98	0.95	0.69	0.73

.

Interlaboratory comparisons showed good agreement in step height measurements (h) and the Sz parameter, while the Sa values differed significantly. While good agreement was observed for h and Sz, significant discrepancies were noted in Sa, especially in Lab B and Lab C, due to uncalibrated lateral components. The obtained results not only demonstrate the metrological properties of the selected parameters in AFM surface characterization, but also define a specific research direction focused on ensuring measurement reliability. To explore this further, the next section presents a simulation-based analysis of the influence of surface profile geometry on selected areal parameters using a Monte Carlo approach.

Influence of Surface Profile Geometry on the Parameters in AFM Measurements: A Monte Carlo Approach

The motivation for this analysis stems from the results of an interlaboratory comparison, which revealed significant discrepancies in the measured values of the Sa parameter between laboratories. The primary suspected cause was the uncalibrated or improperly calibrated lateral component of the measurement system. This raised the question of which surface parameters are particularly sensitive to such lateral deviations and which remain stable regardless of them. To address this issue, five areal surface texture parameters defined in ISO 25178-2, Sa, Sz, Sq, Sdq, and Sdr were selected for detailed analysis. These parameters collectively provide a representative and functionally relevant description of surface topography: Sa, Sz, and Sq represent amplitude characteristics, while Sdq and Sdr reflect slope and surface complexity, respectively.

Sa (arithmetical mean height) and Sz (maximum height) are the most commonly used parameters in industrial practice and technical standards, and are often the only parameters specified in technical drawings. However, the inconsistency in Sa across laboratories, unlike the relatively stable Sz, suggested that some parameters may be more susceptible to errors in lateral calibration. To better understand this sensitivity and to support more reliable surface characterization, the analysis was expanded to include Sq (root mean square height), Sdq (root mean square gradient), and Sdr (developed interfacial area ratio).

The parameter Sdq was selected because it captures the average surface slope, which is critical for understanding functional properties such as friction and adhesion. Its sensitivity to micro-slopes, rotation, and feature density makes it particularly useful for analyzing surface functionality. Finally, Sdr was included as a measure of the increase in actual surface area compared to its projected area, effectively quantifying surface complexity. This is especially relevant in applications dependent on contact area, such as hydrophobicity, capillary effects, and surface engineering.

All parameters were computed from simulated surface data based on a predefined topographical profile (e.g., rectangular). Surface points (x_i, y_i) were randomly sampled over a defined area, and height values z_i were assigned according to the surface geometry.

In the simulations, parameters were evaluated under various conditions, including added lateral noise in x and y, variability in step width and step pitch, and the inclusion of surface slope or rotational misalignment

Table 4. Simulation parameters.

Parameter	Value/Range
Step pitch	(2.98–3.12) µm
Step width	(0.25–0.35) µm
Surface rotation	up to 10°
Surface slope	linear increase in the y-direction
Lateral noise	±1 to ±100 nm
Step height	fixed at 97.6 nm
Sampling area	random sampling over a 20 µm × 20 µm area
Vertical noise (z-axis)	±3 nm (to mimic measurement uncertainty)
Base profile	rectangular, 3.0 µm pitch, 0.3 µm step width

.

The results of simulated surface parameters are presented in

Table 5. Simulated surface parameter values for varying step geometry.

Step Pitch (µm)	Step Width (µm)	Simulated Sa (nm)	Simulated Sq (nm)	Simulated Sz (nm)	Simulated Sdq (–)	Simulated Sdr (%)
2.98	0.25	15.65	27.67	100.44	0.22	2.12
3.02	0.28	17.18	28.96	100.39	0.21	2.11
3.08	0.30	18.41	29.98	100.40	0.20	1.82
3.12	0.35	21.92	32.71	100.49	0.20	1.82
2.98	0.35	21.06	32.09	100.99	0.24	2.09
3.12	0.25	15.68	27.67	101.35	0.22	2.14

.

• The simulations clearly showed the following:

The simulations demonstrate that parameters Sa and Sq are sensitive to the distribution and density of structures (e.g., step pitch and step width). Even small geometric variations can lead to several nanometers of difference in computed values, despite the total step height remaining constant.

Parameter Sz remains consistent across all simulations, as it is solely based on the extreme values (z_max − z_min). For this reason, it is stable even under variations in slope, rotation, or noise, making it useful for detecting depth/deformation, though in-sufficient for describing overall texture.

Although both Sdq (root mean square gradient) and Sdr (developed interfacial area ratio) are classified as hybrid parameters combining amplitude and spatial characteristics of surface topography, the results obtained in this simulation do not confirm the expected trends. The literature consistently reports that increases in slope, surface rotation, or reductions in step pitch lead to higher values of both parameters, reflecting greater surface complexity and microstructural articulation. However, in this study, the small-scale variations applied to the simulated rectangular profiles did not produce any clear or systematic increase in either Sdq or Sdr. The observed fluctuations appear to be random rather than systematic. For more conclusive insights, it would be necessary to analyze more pronounced geometric or instrumental influences.

In the simulations, an idealized rectangular profile was used without surface imperfections or noise, resulting in the Sz value matching the nominal step height (97.6 nm). However, in the case of the real reference standard, microstructural variations, local protrusions, indentations, or measurement noise are always present to some extent. These factors can cause the measured Sz value to exceed the nominal step height, even when the step height itself is accurately measured.

To validate the simulation model, the simulated parameter values were compared with those obtained from actual measurements performed on the standard. This comparison was made possible by simulating a reference standard with a rectangular profile using the certified values of step height and step width. The results are presented in

Table 6. AFM measurement results for reference step height standard.

	Sa (nm)	Sq (nm)	Sz (nm)	Sdq	Sdr (%)
	15.60	28.72	117.02	0.17	1.76
Measurement uncertainty U; k = 2; p = 95%	4.5	4.5	7	0.4	0.6

.

The simulated and experimental results show good agreement for all the parameters, confirming the validity of the model. The deviation of 7 nm between the simulated and measured values of the Sz parameter is attributed to surface noise and microstructural irregularities, which are inevitably present in real physical standards but are absent from the idealized simulation model. These findings provide a good basis for interpreting the behavior of individual parameters and evaluating their robustness in both simulated and real measurement conditions. To support more detailed empirical validation, future research should aim to develop a standardized intercomparison protocol.

4. Conclusions

This study extends the current understanding of surface topography evaluation by analyzing the behavior of five selected 3D areal surface texture parameters—Sa, Sz, Sq, Sdq, and Sdr—within the context of atomic force microscopy (AFM) measurements. The parameters were assessed through both simulation and experimental validation using a certified step height standard, with a particular focus on the influence of controlled geometric variations and instrumental factors.

Simulations using idealized rectangular profiles demonstrated that even small deviations in step pitch, width, slope, or lateral positioning can significantly affect certain parameters. The results confirmed the sensitivity of Sa and Sq to lateral components, as well as the robustness of Sz as an extreme value parameter, and highlighted the need for further investigation into Sdq and Sdr under more complex topographies. The experimental AFM measurements aligned well with simulated values, providing empirical validation of the model. The 7 nm deviation in the Sz parameter value was attributed to unavoidable surface noise and microstructural irregularities present in physical standards.

These findings emphasize the importance of rigorous calibration and comprehensive uncertainty evaluation for the accurate application of surface parameters. Without proper traceability, the use of certain amplitude or hybrid parameters, particularly Sa and Sq, may lead to inconsistent or misleading conclusions. The use of Monte Carlo simulation, in accordance with the GUM and JCGM 101 guidelines, proved effective in quantifying uncertainty and supporting metrological reliability.

Looking ahead, future work should focus on the development of integrated tools for automated uncertainty evaluation within AFM software (https://afm.oxinst.com/, accessed on 30 May 2022) environments, particularly using Bayesian and Monte Carlo approaches. Further research could also include systematic comparisons between AFM results and those obtained using other high-resolution measurement techniques, such as optical profilometry, confocal microscopy, or interferometry, in order to validate parameter values across a broader range of instruments. In this regard, it is essential to further develop calibration methods for these instruments and ensure the reliable estimation of associated measurement uncertainties.