A comprehensive study of AFM stiffness measurements on inclined surfaces: theoretical, numerical, and experimental evaluation using a Hertz approach

Abstract

Atomic Force Microscopy (AFM) is a leading nanoscale technique known for its significant advantages in the analysis of soft materials and biological samples. Traditional AFM data analysis is often based on the Hertz model, which assumes perpendicular indentation of a planar sample. However, this assumption is not always valid due to the varying geometries of soft materials, whether natural, synthetic or biological. In this study, we present a new theoretical model that incorporates correction coefficients into Hertz’s model to account for cone-like and spherical probes, and to consider local tilt at the probe-sample interface. We validate our model using finite element analysis (FEA) simulations and experimental AFM measurements on tilted polyacrylamide gels. Our results highlight the need to include local tilt at the probe-sample contact to ensure accurate AFM measurements. This represents a step forward in our understanding of the elastic properties at the surface of soft materials in the broadest sense.

Introduction

Atomic force microscopy (AFM), first introduced by Binnig et al. in 1986¹, has significantly advanced multiple materials characterization by providing an outstanding spatial resolution and force sensitivity. Its ability to non-destructively probe a wide range of materials - from rigid structures such as cellulose fibrils and stiff polymers to sensitive biological samples such as living cells - has established AFM as a central tool in both materials science and biological research^2–6.

Since the early 1990s, AFM has gained significant interest in the study of soft matter and biological samples, as evidenced by the increasing number of publications linking AFM to cells or polymers^5,7,8. Early demonstrations of AFM’s capabilities highlighted its potential to probe biological samples under near-physiological conditions, including aqueous environments^9–13. This versatility makes AFM indispensable for elucidating the intricate mechanical and structural properties of natural, synthetic and biological samples.

The modified Hertzian solution, introduced by Sneddon in 1965¹⁴, is a mathematical framework widely used in soft matter research and cell mechanics to estimate the elastic properties of materials from indentation data in AFM experiments^15,16. This approach assumes that the sample is a planar elastic medium vertically indented by a probe, which amounts to considering it as an infinite half-space. However, this assumption does not always correspond to the complex nature of the surface topographies of a number of materials, which are often non-planar.

This topographical complexity arises from various factors, such as chemical composition and manufacturing processes, as seen in hydrogels—a class of soft matter¹⁷—and cellular geometry. The shape of cells can vary significantly depending on their phenotype, function, and environmental interactions, ranging from rounded to spread-out forms^18,19. Consequently, the use of the Sneddon-Hertzian solution in AFM measurements of soft matter and biological systems can lead to inaccuracies due to the discrepancy between the assumed planar surfaces and the actual topography of these samples. This discrepancy can lead to inaccurate stiffness evaluations, highlighting the need for corrections that incorporate the angle between the probe and the surface to improve the evaluation of elastic properties at the surface of soft materials in general.

Results

Our study improves AFM-based cell mechanics measurement by integrating theoretical adjustments, simulations, and experiments. We reconsider Sneddon’s classical Equations¹⁴ to account for sample tilt angles during indentation with conical and spherical probes, and introduce corrective factors to improve measurement accuracy.

Second, our approach uses numerical simulations to accurately calculate the force required to indent a material with conical or spherical probes at various angles, providing a detailed understanding of the indentation process.

The study concluded with AFM stiffness measurements on polyacrylamide (PAA) gels. 3D-printed holders were used to provide different tilt angles to understand the effect of sample-probe positioning on stiffness assessment.

The aim of the integrated study is to improve the accuracy of AFM measurements by analyzing the impact of tilt angles. This will enhance our understanding of mechanical properties in materials and biological sciences.

Theoretical model

Indentation testing involves applying a load to the surface of a material using an indenter. This test is used to determine the local elasticity or Young’s modulus of the material. During the test, the indenter applies a compressive force to a small contact area on the surface of the material. The resulting local surface displacement is then measured. The material’s Young’s modulus can be calculated by analyzing the relationship between the applied force, the contact area and the surface displacement.

For axisymmetric indenters, Sneddon¹⁴ showed that the material surface stiffness, denoted , can be defined as the derivative of the applied force, , with respect to the surface displacement, , expressed as . This material surface stiffness is directly related to the Young’s modulus, , of the material and the contact area radius . Here, represents the contact area between the indenter (assumed to be rigid) and the material surface. A graphical illustration of these relationships is presented in Fig. 1a.

Fig. 1.

Schematic representation of a cone with half-angle of aperture that indents a horizontal half-space elastic material in (a) and a half-space elastic material tilted by an angle in (b). Schematic of a sphere with a radius that indents a horizontal half-space elastic material in (c) and a half-space elastic material tilted by an angle in (d).

This way,

1

The dimensionless Poisson’s ratio of the material is denoted as . The relationship between the applied force and the indentation depth is influenced by the contact radius , which varies depending on the shape of the indenter used. In AFM, commonly used indenter shapes include spherical, pyramidal, or conical probes. When elastic materials are indented by axisymmetric probes, the material surface at the contact line experiences a displacement. This displacement, , is a constant fraction, denoted as , of the total displacement of the indenter.

2

According to Sneddon’s findings, the value of the constant fraction varies with the indenter type: for conical indenters and for spherical ones. Therefore, the depth of the indenter at the contact line can be calculated accordingly:

3

Case of a conical indenter

To establish the relationship for a conical indenter penetrating an elastic material in half-space, we refer to expression (1) and consider the geometric aspects shown in Fig. 1a.

For conical geometries with small deformations, the contact radius, denoted as , can be calculated as follows.

4

Subsequently, by using Eq. (3), this relationship is refined:

5

This allows to derive the formula for ,

6

Here, represents the cone’s half-opening angle, as illustrated in Fig. 1a.

Assuming negligible adhesion effects between the indenter and the sample surface, the relationship between the applied force and the indentation depth can be derived by integrating Eq. (6) with Eq. (1).

7

To establish a theoretical relationship between the force and indentation depth for a conical indenter on an elastic material in a half-space inclined at angle , as depicted in Fig. 1b. As a working hypothesis, we assume that the formula in Eq. (1) can still be applied along the vertical axis , even though the axisymmetry is disrupted by the tilt. We consider that the elastic properties of the material remain unchanged, and we analyze how the contact radius is affected by the inclination angle of the plane. This requires consideration of the specific geometry and orientation of both the indenter and the inclined surface to accurately determine the contact area radius (Fig. 1b).

8

where and are the force and indentation depth along the vertical axis , respectively, and the radius of the contact area is the equivalent radius of the ellipse formed at the contact line in section plane .

Assuming the contact area is elliptical, the equivalent radius can be approximated by dividing the area of the ellipse by :

9

Assuming the contact area is elliptical, the semi-major axis and semi-minor axis can be calculated from the indentation depth , cone half-opening angle , and inclination angle . The formulas for and are and , detailed in Appendix B equation (B5) for and Appendix C equation (C3) for (see Fig. B1). Then, Eq. (9) is as follows:

10

Since , using expression (10), Eq. (8) simplifies to:

11

and represent the force and indentation measured along the vertical axis by the AFM system, respectively.

12

when , the classical Hertz equation for a conical probe is restored. To accommodate contact with an inclined surface for conical geometry, a correction factor can be introduced. This factor, dependent on the tilt angle and the half-opening angle , adjusts the contact mechanics to the specific geometry of the tilted surface:

13

In other words,

14

where is:

15

The apparent modulus of elasticity, , measured by the conical indenter test using the classical Hertz equation, estimates the material’s modulus of elasticity from the indentation data. In turn, the actual modulus of elasticity, denoted , is the intrinsic modulus characteristic of the material tested.

Case of a spherical indenter

To examine the relationship between force and indentation depth with a spherical indenter, consider the geometry shown in Fig. 1c, where a sphere of radius indents a half-space elastic material. The relationship between , the probe radius, the contact radius , and the indentation depth at the contact line is given by:

16

By applying Eq. (3) and considering for a spherical indenter, we derive:

17

This simplifies to:

18

This approximation is evaluated in Appendix A.

Neglecting adhesive forces, we integrate Eq. (18) with Eq. (1) to determine the relationship between and :

19

To establish a theoretical relationship between force and indentation depth using Eq. (1) for a spherical probe indenting an elastic material in inclined half-space, as shown in Fig. 1d. We assume that Eq. (1) applies along the axis, which is aligned with the line connecting the center of the sphere to its point of contact on the surface.

20

Assuming from Eq. (19) and relationships and , we adapt the formula for an inclined surface:

21

22

In this situation, when , the classical Hertz equation for a spherical indenter is restored, introducing a correction factor for spherical geometry:

23

Thus, Eq. (15) remains applicable for a spherical probe in this context.

Theoretical results

In our analysis, we introduced tilt correction factors for conical and spherical probes, shown in Fig. 2. These factors, and , change with tilt angle , based on Eqs. (13) and (23). Notably, our findings confirm the spherical probe formulation in Eq. (22), previously proposed by Fujii and Okajima²⁰.

Fig. 2.

Tilt adjustment coefficients - Displays for conical probes and for spherical probes, compensating for the impact of sample surface tilt on indentation measurements. The curves are shown for conical probes with half-opening angles of 20, 30, and 45 degrees.

The correction factor for a theoretically infinite height cone varies significantly with its tilt and half-opening angle (shown by dashed lines in Fig. 2). For example, if a sample tilted by 30 degrees is indented by a cone with a half-opening angle of 30 degrees, the correction factor () becomes 1.14. This results in a 14% overestimation of the force measured by the AFM compared to the force measured on a non-tilted surface, and consequently an overestimation of the apparent Young’s modulus calculated using Hertz’s model. Increasing the half-opening angle () to 45 degrees for the same tilt angle increases the overestimation to 32%, highlighting the significant influence of the half-opening angle of the cone on the Young’s modulus measurements.

The correction factor approaches infinity as the tilt angle nears . At this angle, a cone of infinite height would have an infinite contact area with the inclined surface, leading to an infinite force. This behavior is depicted by the asymptotic increase of the correction factor in Fig. 2’s dashed lines as approaches this critical value.

For a spherical probe, the correction factor depends only on the tilt angle , decreasing from 1 to 0 as varies from 0 to , as shown by the solid line in Fig. 2. At a 30-degree tilt, the correction factor is 0.70, leading to a 30% underestimation of the normal force applied to the sample surface and, consequently, of the Young’s modulus calculated using Hertz’s model. As approaches , the factor drops to 0, indicating that the probe switches from indentation to sliding along the surface, reducing the measurable force to zero under frictionless conditions.

The accuracy of modulus measurements varies considerably with the angle of tilt. At an inclination of 15 degrees, the errors are relatively small (+ 3% for conical probes with = 30 degrees and − 8% for spherical probes), but these deviations increase at larger angles, for example to + 21% and − 39% for conical and spherical probes, respectively, at = 35 degrees. This highlights the importance of the angle of inclination in determining measurement accuracy, with conical probes overestimating the apparent modulus and spherical probes underestimating it.

Model validation

Finite element analysis (FEA) validation

Numerical simulations evaluate the vertical component of the reaction forces when the AFM probe indents the material at different sample tilt angles. These simulations measure how the vertical reaction forces change with sample tilt by comparing the ratio , where is the vertical reaction force component without tilt, and is the vertical reaction force component when the sample is tilted by an angle (see Fig. 3b and d). For each tilt angle, we calculate the final ratio as the average of the ratios obtained over indentation depths ranging from 0.35 to 0.5 μm.

Fig. 3.

(a and c) Show the computational mesh for the cone (a) and sphere (c) indentations used in the simulations. These meshes, a discretized representation of the geometric models, are key to solving the finite element equations and analyzing the material behavior under indentation. (b and d) View finite element analysis results for cone (b) and sphere (d) indentations, highlighting material response and resultant force (R) at the base of the computational domain, represented by a rectangular parallelepiped. The color gradients in these plots show the von Mises stress, reflecting the stress state of the material.

Numerical correction factors for conical probes and for spherical probes are derived by averaging these ratios. These correction factors adjust for variations in both tilt angle and indentation depth, ensuring accurate indentation data in AFM experiments by compensating for the effects of tilt.

Figure 4 compares the correction factors and for the conical and spherical probes shown in 4a and 4b, respectively, as a function of the tilt angle . This analysis uses numerical simulations and theoretical approaches. The results show that the tilt angle has a significant effect on the calculated elastic moduli and this effect varies with probe shape, validating the theoretical approach.

Fig. 4.

(a) and (b) show a comparison of theoretical correction factors (dotted lines, labeled for conical probes and for spherical probes) with those derived from finite element analysis (FEA) simulations (scatter plots). These factors, labeled are averaged from ratio values over indentation depths ranging from 0.35 to 0.5 μm. Figure (A) includes a range of cone half-opening angles (). Error bars represent the standard error of the mean (SEM), highlighting the variability in the numerical data.

For the conical probe, the FEA correction factors closely match the theoretical predictions, showing an increase with plane tilt angle. It is also noted that the increase in correction factors tends to infinity as the plane tilt angle approaches 50 and 40 degrees for cones with half-opening angles of 40 and 50 degrees, respectively. This pattern is consistent with theoretical predictions suggesting an asymptotic behavior of the correction factors at tilt angles close to .

Conversely, the spherical probe shows a decrease in the ratio as the tilt angle increases, indicating a reduction in the resultant reaction forces and thus the force causing indentation. This trend is different from that of the conical probe, with both numerical and theoretical results reflecting the unique response of the spherical geometry to tilt angles.

Overall, the agreement between numerical simulations and theoretical predictions enhances the understanding of probe geometry responses under different tilt angles and highlights the importance of integrating theoretical and numerical data in AFM measurement analysis.

Experimental validation

For each measurement on the polyacrylamide (PAA) gel surface, the local tilt angle is calculated using the height coordinates of two adjacent points along the slow scan direction, as shown in Fig. 7b. The local Young’s modulus at each point is then calculated using Eq. (24) for pyramidal probes (part of the conical probe family) or Eq. (19) for spherical probes.

Fig. 7.

(a) Customized 3D-printed 14 mm coverslip holder designed specifically for handling PAA gels. The holder ensures steady angular positioning of the coverslip throughout the experimental procedure. (b) Schematic for calculating the local tilt angle at different points on the PAA gel surface. The surface of the gel is assessed using a grid. At each point on the grid (measurement point), adjacent contact points are identified. The local tilt angle at each point on the gel is calculated from the spatial coordinates of adjacent contact points. (c, d, e, and f) Scanning Electron Microscope (SEM) images of the µmash pyramidal probe and the spherical probe are presented from different viewpoints. (c) shows a bottom view of the µMash pyramidal probe, while (d) provides a side profile of the same probe. Similarly, (e) shows a bottom view of the spherical probe, while (f) shows the side profile. All images are annotated with a scale bar indicating a length of 10 μm. Additionally, (g) presents an Atomic Force Microscope (AFM) scan of the spherical probe, providing a detailed topographical representation of the spherical bead surface. This scan is complemented by a cross-sectional profile highlighting height variations and surface features.

An experimental correction factor is calculated for each tilt angle , which represents the ratio of the average modulus at a given tilt angle to the average modulus when the gel surface is planar. This ratio, , provides insight into how the slope of the gel surface affects the Young’s modulus measurement. By comparing the Young’s moduli at various tilt angles to the baseline at , variations are quantified, allowing for a better understanding of how surface tilt affects the mechanical properties of the PAA gel measured with conical (Fig. 5a) and spherical (Fig. 5b) probes.

Fig. 5.

Solid lines represent theoretical correction factors for the 22-degree half-opening angle of the pyramidal probe, classified as part of the conical probe family (a), and the spherical probe (b). These are aligned with the experimental factors, with each data point showing the average Young’s modulus ratio, labeled with the standard error of the mean (SEM). These ratios compare the modulus measured on an inclined polyacrylamide (PAA) gel surface to that measured under horizontal conditions ( = 0 degrees) and illustrate the effect of surface tilt on modulus measurements.

The experimental analysis investigated how local tilt affects stiffness measurements using two types of probes. One probe was cone-like, essentially a four-sided pyramid, while the other was spherical, consisting of a microsphere mounted on a cantilever.

The experimental data, presented in Fig. 5a and b, show correction factors for each tilt angle . These factors, represented as the ratio , compare the average Young’s modulus when the PAA gel surface is tilted to when it’s horizontal. Each data point is labeled with the standard error of the mean (SEM), highlighting discrepancies in the measured modulus. Theoretical correction factors, for the conical probe and for the spherical probe, are also plotted, showing the expected trends based on probe-specific equations.

Results from the conical probe show an overestimation of the Young’s modulus as the tilt angle increases, particularly for angles less than 29 degrees, which agrees well with theoretical predictions as shown in Fig. 5a. This demonstrates a strong agreement between experimental and theoretical data and confirms the accuracy of our model for the response of the pyramidal probe to tilt.

Conversely, the spherical probe tends to underestimate the Young’s modulus for tilt angles greater than zero. This trend is consistent with theoretical predictions up to a tilt angle of about 20 degrees, as shown in Fig. 5b, demonstrating good but limited agreement between experimental observations and theoretical models.

Discussion

In this work, we assume that the theoretical correction factor given for conical shapes is still valid for evaluating results from a pyramidal probe. The experimental increases quadratically with tilt angles up to 30 degrees but exhibits a different behavior beyond that angle. As the tilt angle increases, peaks and then declines steadily, contrary to the theoretical model. For example, at 41 degrees, is 1.06, below the predicted 1.23, indicating a 14% deviation from the expected trends, as shown in Fig. 5a.

We propose a hypothesis to explain the observed behavioral changes when the tilt angle of the pyramid probe exceeds 30 degrees, suggesting that altered contact with the gel surface reduces the calculated Young’s modulus and lowers the correction factor. The SEM images in Fig. 7d show that when the tilt angle approaches 35 degrees, the cantilever probe contacts the PAA gel surface, rendering the application of Hertz’s model inappropriate for calculating Young’s modulus due to the altered contact situation.

Experimental results from a spherical probe indicate that the Young’s modulus is underestimated for tilt angles up to 20 degrees, which is consistent with theoretical predictions and confirms a close correlation between theory and experimental data under these conditions. Beyond this angle, however, discrepancies appear. At an inclination of 34 degrees, the experimental results show that the Young’s modulus is underestimated by 24% compared to the theoretical prediction of 37%, indicating significant differences in estimates at higher angles. These results suggest that mechanical factors not accounted for by the theoretical model influence stiffness measurements at tilt angles greater than 20 degrees.

Between 20 and 34 degrees of tilt, the experimental observations show less underestimation of the Young’s modulus than the theoretical predictions would indicate. From 34 to 43 degrees, the experimental correction factor for the spherical probe stabilizes around an average value of 0.67, suggesting that the theoretical model may not fully capture the factors influencing this behavior. The SEM image (Fig. 7f) clarifies that the cantilever tip of the spherical probe only contacts the gel surface at angles greater than 56 degrees, ruling out probe geometry as a cause of the discrepancies seen with the pyramidal probes. In addition, AFM images (Fig. 7g) reveal microstructures on the polystyrene microsphere that may interact with the gel, adding complexity such as frictional forces that could explain the observed discrepancies between experimental and theoretical results. This suggests that current non-frictional theoretical models may not accurately represent these interactions, thereby affecting stiffness measurements.

To test the role of friction in probe-gel interactions, we added a friction model to the numerical simulation of a spherical probe. This addition improves the accuracy of the simulation by capturing the effects of friction at different tilt angles, as shown in Fig. 6. Friction introduces additional resistance, resulting in increased forces corresponding to an increased apparent modulus of elasticity. These effects can be seen quantitatively in the increased correction factor shown in Fig. 6.

Fig. 6.

Variation of the numerical correction factors for a spherical probe as a function of the tilt angle . There are two curves: one considering friction according to Coulomb’s law (dashed line) with a friction coefficient of 0.5, and one without friction (solid line). The x-axis represents the tilt angle of the material surface, while the y-axis represents the correction factors . This comparative plot illustrates the effect of friction on the correction factors, showing how friction modifies the indentation results at different tilt angles, thereby affecting the accuracy of the measurements.

In simulations using a Coulomb friction coefficient of 0.5, the effects of friction become noticeable at a 10-degree slope, increasing the forces compared to frictionless cases. As the tilt angle increases, these effects intensify, peaking at a 60-degree tilt, bringing the simulation results in line with experimental measurements. This suggests that friction has a substantial effect on the mechanical response, especially at higher angles.

Experimentally, the correction factor stabilizes around 0.67 for angles between 34 and 43 degrees, indicating that friction compensates for the effects of tilt, supporting the hypothesis that friction is a key factor in probe-gel interactions. The agreement between numerical simulations and experimental data underscores the role of friction in explaining discrepancies between theoretical predictions and observed results, particularly in how probe geometry affects measurement bias at different tilt angles.

The experimental results confirm theoretical and numerical analyses of the effect of local tilt angles on modulus measurements and highlight the importance of probe geometry on measurement accuracy. Conical and pyramidal probes, such as those used in our study, tend to overestimate modulus at higher tilt angles due to increased contact area and interaction force. In contrast, spherical probes, which are prone to sliding and friction, tend to underestimate it due to decreasing contact area.

Our experiments also revealed challenges due to suboptimal probe-sample contact, exacerbated by probe geometry and sample tilt. These interactions, especially with spherical probes, can significantly affect the accuracy of AFM measurements by altering the contact mechanics.

This study confirms the validity of the conical model for understanding the behavior observed with pyramidal probes and emphasizes the need for correction factors when tilt angles exceed 20 degrees. Recognition of these complexities is critical for accurate data analysis and underscores the need for a thorough understanding of the specifics and challenges of performing AFM measurements on inclined surfaces.

Conclusions

Our study demonstrates that by accounting for the local geometrical factors affecting probe-sample contact in AFM, specifically the tilt angle between the probe and sample surface, we can achieve more accurate Young’s modulus measurements. We proposed a correction to Hertz’s model, which was validated through both experimental and numerical methods. This correction introduces coefficients to account for deviations from the traditional assumption of a planar surface and perpendicular to the probe. These coefficients have been specifically formulated for the two most commonly used AFM probe types—conical and spherical—enabling more precise determination of Young’s modulus from AFM data.

In addition, our study has identified several factors that become significant at extended tilt angles of the sample surface. Recognizing and accounting for these factors is essential to ensure the accuracy and reliability of results and to gain a more complete understanding of AFM data for elastic materials.

While our approach focuses on purely elastic materials for which Hertzian mechanics is valid, extending this approach to viscoelastic or dissipative materials presents additional challenges that need to be evaluated. In addition, measuring the local tilt angle in the case of a soft dynamic system such as the cell body can be sensitive due to local shape changes, and the use of high speed AFM may be a relevant solution to be evaluated.

Our study improves AFM-based measurement techniques to more accurately determine Young’s modulus in elastic materials, thereby improving the characterization of soft materials and providing more detailed information about their mechanical properties. These results support more accurate interpretations in soft materials research and open avenues for future studies of topographically complex materials.

Materials and methods

Finite element modeling

Numerical simulations were performed using COMSOL Multiphysics^® 5.3 to model AFM indentation experiments. Two probe types were considered in the simulations: a conical probe and a spherical probe, reflecting those used in actual AFM experiments. The indented material was modeled as a linear elastic material to reflect common material behavior in AFM studies. The sample tilt was varied between 0 and 40 degrees relative to the horizontal plane to fully analyze the effect of tilt angle. The simulations included a conical probe with a half-opening angle ranging from 20 to 50 degrees, allowing the effect of different geometric configurations on indentation results to be examined. For the spherical probe, a 10 μm diameter probe was chosen to be representative of the standard sizes used in AFM experiments. The simulations were designed to deepen our understanding of indentation mechanics and to compare with theoretical and experimental results.

Material properties and geometry mesh

In the simulations, the material under test - representing the body indented by the AFM probe - was assumed to be homogeneous, isotropic, and exhibit linear elastic behavior. The material properties were set to a Young’s modulus of 1 kPa, a density of 1000 kg/m³, and a Poisson’s ratio of 0.4, typical of soft materials such as gels, polymers, or biological tissues^21–23.

The COMSOL simulations used the geometries shown in Table 1. The elastic body was modeled as a rectangular parallelepiped, with dimensions tailored for each indenter type to minimize edge effects and optimize mesh distribution. For the conical indenter (Fig. 3a), the dimensions were length − 10 μm, width − 10 μm, and height − 5 μm. For the spherical indenter (Fig. 3c), the dimensions were length − 20 μm, width − 20 μm, and height − 5 μm. These configurations ensure that the spatial extent of the elastic body is adequate to accurately simulate indentation.

Table 1.

Number of vertices and mesh elements (tetrahedra or hexahedra) in each geometry domain, as determined by the selected meshing and discretization settings in COMSOL simulations.

Geometry	Mesh type	Number of vertices	Number of tetrahedra/hexahedra	Average skewness quality
Sphere	Tetrahedron	495	2,070	0.6633
Rectangular parallelepiped (Sphere)	Tetrahedron	8,560	39,932	0.6608
Cone	Tetrahedron	2,689	11,990	0.6212
Rectangular parallelepiped (Cone)	Hexahedron	726	500	N/A

The mesh density and element distribution were customized for each simulation scenario based on the desired level of mesh refinement and geometry complexity, balance accuracy, and computational performance.

Boundary conditions

The bottom surface of the rectangular parallelepiped was fixed to prevent any displacement in all three dimensions. The vertical displacement of the probe ranges from 0.35 to 0.5 μm, indicating the depth of penetration of the probe. A contact pair was created to manage the interaction between the probe and the top surface of the parallelepiped. For the contact formulation, we used a penalty approach, a common method for its effectiveness in simulating contact interactions. The penalty factor, , is key to controlling the contact mechanics and is calculated using the formula , where is the modulus of elasticity and is the indentation depth.

Friction

COMSOL Multiphysics^® simulations were used to investigate the effects of potential sliding and friction between the spherical probe and the planar surface. This sliding can affect the contact mechanics and the correction factor . To account for this, we integrated a friction model based on Coulomb’s law, which states that the frictional force is proportional to the normal force and is determined by the friction coefficient. For these simulations, we chose a friction coefficient of 0.5 to represent the interaction between the spherical probe and the planar surface. It’s worth noting that this friction model is used only in the case of the spherical probe, since sliding is considered less important for conical probes and was therefore omitted from these simulations.

Experimental assessments

Preparation of polyacrylamide gels

Polyacrylamide (PAA) gels were prepared on 14 mm glass coverslips. The coverslips were first treated to activate the surface for gel polymerization by immersion in a 0.3% solution of 3-methacryloxypropyltrimethoxysilane (Bind-Silane, Sigma, Cat. No. 440159) in a mixture of acetic acid diluted to 3% concentration and absolute ethanol. This activation step lasted 3 min to improve gel adhesion, followed by three ethanol washes and 45 min of air drying.

The gel itself consisted of a final concentration of 5% acrylamide from a 40% stock solution and 0.225% bisacrylamide from a 2% stock solution. This was amplified with 0.5% ammonium persulfate at 10% and 0.05% tetramethylethylenediamine (Sigma), both in aqueous solution. To form the gels, 11 µl of the mixture was applied to a 14 mm diameter coverslip and covered with a 10 mm diameter coverslip pretreated with Repel-Silane ES (Merck, Cat. No. GE17-1332-01) to prevent adhesion. The assembly was left undisturbed for 5 min to allow the gel to set.

After a 30-minute polymerization period, the top coverslip was removed, and the gels were rinsed three times with phosphate-buffered saline (PBS) to remove any residual components. Gels were prepared one day prior to AFM experiments and stored in water at 4 °C to ensure integrity and hydration.

Tilted supports

The 3D-printed holders are designed to securely mount 14 mm diameter glass slides, ensuring that they remain in place during PAA gel indentation tests (see Fig. 7a). These holders allow controlled tilting of the glass slides at predetermined angles relative to the horizontal plane, allowing detailed analysis of tilt effects on indentation responses. For the experiments, we used holders with tilt angles of 0, 5, 10, 15, and 20 degrees to thoroughly investigate how tilt affects the mechanical behavior of PAA gels.

AFM probes

AFM probes were examined by SEM as shown in Fig. 7c-f. Cone-like pyramidal probes µMash CSC38/NO AL (MikroMasch^®, Sofia, Bulgaria) with a half-opening angle of 22° were used for a subset of experiments. For experiments with spherical probes, polystyrene microspheres (Polybead©, Polysciences Inc., USA) with an average diameter of 9.02 ± 0.67 μm were attached to ARROW-TL1 cantilevers (NanoWorld^® AG, Neuchâtel, Switzerland) according to the guidelines of a JPK Instruments technical note²⁴. In addition, an AFM scan of the spherical probe was performed (Fig. 7g), providing detailed insight into the surface features of the microspheres and enhancing the information from the SEM images.

AFM assessments on PAA gels

Young’s modulus of PAA gels were determined using a JPK NanoWizard Sense + AFM system (Brucker, Billerica, Massachusetts, USA) integrated with a Zeiss Axio-Observer z1 inverted microscope using a methodology adapted from Ben Bouali et al.¹⁷. The measurements used the above-mentioned AFM cantilevers and were calibrated using the thermal noise method originally described by Hutter et al. to accurately determine the nominal spring constant of the cantilevers²⁵. The measurements were performed at room temperature in an aqueous medium, focusing on the central region of the gel. The pyramidal probe, with a fine mesh size of 0.2 μm, covered a measurement area of 2.4 × 2.4 μm², while the spherical probe, with a larger mesh size of 2.0 μm, covered a larger area of 48 × 48 μm², allowing a detailed evaluation of the mechanical properties of the PAA gels.

To calculate the local tilt angle on the PAA gel surface, a systematic approach using a grid of the surface is used. For each measurement point on the grid, adjacent contact points are identified, and their x, y, and z coordinates are used to accurately determine the local tilt angle. This information is critical for characterizing surface deformations and irregularities that could affect the AFM measurement results (see Fig. 7b).

The local tilt angles are essential for understanding how surface features affect the results of AFM measurements. Bilodeau’s Eq. (24)²⁶ is applied to the spectroscopy curves from the pyramidal probe:

24

where is the applied force, is the modulus of elasticity of the gel, is the half-opening angle of the probe, is the indentation depth, and is Poisson’s ratio, approximately 0.4 for PAA gels^21,22.

For the spherical probe, a Hertz-based Eq. (19) is typically used to analyze AFM spectroscopy curves. Both pyramidal and spherical probe results are processed using an in-house developed Matlab^® program that uses these equations to optimize fits to the experimental data, allowing accurate determination of Young’s modulus for the PAA gel samples.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Author contributions

A.A.N.: Theoretical model, 3D holders, AFM experiments.B.M.: SEM images.F.S.: AFM experiments, PAA gels preparation, wrote the main manuscript text.F.R.: Theoretical model, numerical simulation, AFM experiments, AFM analysis, wrote the main manuscript text, prepared all figures.A.A.N., F.S., and F.R. reviewed the manuscript.

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-75958-1.