Toward a better modulus at shallow indentations—Enhanced tip and sample characterization for quantitative atomic force microscopy

Approximations of the geometry of indenting probes, particularly when using shallow indentations on soft materials, can lead to the erroneous reporting of mechanical data in atomic force microscopy (AFM). Scanning electron microscopy (SEM) identified a marked change in geometry toward the tip apex where the conical probe assumes a near linear flat‐punch geometry. Polydimethylsiloxane (PDMS) is a ubiquitous elastomer within the materials and biological sciences. Its elastic modulus is widely characterized but the data are dispersed and can display orders of magnitude disparity. Herein, we compare the moduli gathered from a range of analytical techniques and relate these to the molecular architecture identified with AFM. We present a simple method that considers sub‐100 nm indentations of PDMS using the Hertz and Sneddon contact mechanics models, and how this could be used to improve the output of shallow indentations on similarly soft materials, such as polymers or cells.

nominally flat surfaces are brought together the surface roughness leads to contact at discrete points along their contacting regions.
These discrete points constitute the real contact area, and it is usually only a small proportion of the expected contact area, had the surfaces been perfectly smooth (Bhushan, 1998). Several different mathematical models of contact mechanics seek to address these factors, each with their own strengths and limitations. The Hertz (Hertz, 1882), Sneddon (Sneddon, 1948), Johnson-Kendal-Roberts (JKR;Johnson et al., 1971), or Derjaguin-Muller-Toporov (DMT;Derjaguin et al., 1975) models are widely used for the assessment of soft materials. The merits and limitations of each are hotly contested in the literature, and such debates are beyond the scope of this paper. For the researcher with little experience in contact mechanics, or those who may be restricted to a software embedded model, their options may be limited. We have utilized the Hertz and Sneddon models, which are widely used in the material (Denisin & Pruitt, 2016;Domke & Radmacher, 1998;Lim & Chaudhri, 2004;Mesarovic & Fleck, 1999) and biomechanical sciences (Chopinet et al., 2013;Eaton et al., 2008;Smolyakov et al., 2016a;Solopova et al., 2016). These models do not consider the adhesive contribution, and are best suited when adhesion is negligible-such as when analysis is performed in an aqueous environment (Suriano et al., 2014). Despite this limitation, many studies of soft materials are published each year using the Hertz model.
Moreover, even when competing models show differences in elastic modulus on the same sample they are typically small, and reside within the same order of magnitude (Suriano et al., 2014). Incorrectly identifying the indenter radius can lead to differences of elastic moduli spanning many orders of magnitude within the same sample. With adequate insight into the correct geometries of an indenting probe and by minimizing the influence of adhesion-particularly if analyzing in an aqueous environment, it is possible to extract reliable, quantitative data using the Hertz model.
The shape at the end of the tip is critical when indenting to depths much smaller than the indenter radius (Cohen & Kalfoncohen, 2013). During instrumented nanoindentation (INI), the experimenter makes great attempts to fully characterize the geometry of the indenter. However, the atomic force microscopy (AFM) community do not appear to be as zealous. Often, the manufacturer designated nominal tip radii or conical half-angles are used in the contact mechanics models routinely employed. Alternatively, blind reconstruction of the tip geometry may be utilized (Bailey et al., 2014b;Cohen & Kalfon-cohen, 2013). This technique has proved useful to identify changes in tip geometry which can occur over the course of repeated scans on hard substrates, and transmission electron microscopy (TEM) can provide high resolution micrographs of these changes (Liu et al., 2010;Vahdat et al., 2013). Equally, many of the published reports using scanning electron microscopy (SEM) also relate to the wear and damage to the tip following repeated scans on a hard surface (Park et al., 2014;Xue et al., 2014). However, the resolution of these images does not always allow for detailed radii or angle measurements near the tip apex. Moreover, tip wear and changes in geometry should be negligible when scanning on soft substrates, such as those used in this study. Polydimethylsiloxane (PDMS) is an organosilicon elastomer, widely used in the life sciences. It is extensively used in microfluidic devices (Fujii, 2002;Johnston et al., 2014;Tsai et al., 2011) as it has shown biocompatibility with a diverse range of biomolecules (Bélanger et al., 2001). Its soft and elastic nature allows for reversible deformations and it can be lithographically molded with high fidelity (Jahed et al., 2017;Liu et al., 2009a). Being optically transparent to wavelengths from the infrared to the ultraviolet (Liu et al., 2009b) it displays little autofluorescence (Piruska et al., 2005), thus rendering it as a useful substrate for optical microscopy. Tensile loading or INI are widely used to characterize the elastic modulus of PDMS. In tensile loading the sample is stretched uniaxially and the stress-strain curve is analyzed to derive the modulus. INI applies a compressive, calibrated force, and the accurate determination of the contact point is then used to define the depth of indentation. However, for soft materials, the initial contact point is difficult to determine, which can lead to erroneous reporting of indentation depth and modulus (Cohen & Kalfon-cohen, 2013). It has been suggested that the modulus increases at increased depths, due to a greater contact of the polymer and the indenter (De Paoli & Volinsky, 2015) and others suggest the opposite-that the modulus decreases during increased indentations, with higher moduli at the surface. This has been attributed to a spherical indenter geometry having a large contact area at the surface (Charitidis, 2011), a dependence of the film thickness (Liu et al., 2009a), or as a result of the molecular properties of PDMSwhere it is suggested that there is a greater crosslinking density between the surface and down to 300 nm, whereby the bulk properties change after this depth (Charitidis & Koumoulos, 2012). INI typically operates within a range of hundreds of nanonewtons (nN) to low micronewtons (μN) and is routinely used to determine the mechanical properties of very stiff materials, at micrometer depths. If very small indentations, on the order of tens of nanometers (nm), on a soft substrate are required, INI may not be an ideal choice. AFM is more suited for the characterization of soft materials as it permits the application of smaller loading forces (Celik et al., 2009). Here, a calibrated displacement is applied to the probe and its deflection is typically measured optically (Meyer & Amer, 1988). This interaction between the probe and the sample generates a force-distance (F-D) curve that contains a variety of quantitative data. The F-D curve is then usually analyzed by fitting against a contact mechanical model.
Long-chain polymers typically exist in disordered random coils.
The polymeric PDMS has a molecular chain width of around 0.7 nm and the disordered coil should be around 10 nm in thickness (Granick et al., 2003;Yamada, 2003). From an imaging standpoint these structural details are yet to be elucidated with sufficient resolution. Superresolution microscopy has investigated PDMS microchannels within a fluidic device (Cheng et al., 2013) and SEM and AFM have observed the surface of native and coated PDMS (Davis et al., 2021;Liamas et al., 2021;Nourmohammadi et al., 2015;Yu et al., 2013), which showed considerable porosity. In this work we have characterized the molecular architecture of PDMS at the surface and sub-surface respectively and obtained high-resolution electron micrographs of a commonly used AFM probe. Using these structural details to infer possible contact angles between the two surfaces at a range of indentation depths, we adjusted the indenter half-angle value within the Hertzian models to constrain the calculated elastic modulus of PDMS and effectively use it to tune the AFM system.

| MATERIALS AND METHODS
2.1 | PDMS substrate preparation PDMS was prepared using SYLGARD ® 184 silicone elastomer kit (Dow Corning) containing 10% (w/w) cross-linking agent The mixture was gently stirred for 1 min with a pipette tip, degassed for $5 min under vacuum at room temperature and cured at 70 C for 2 h.

| Nanoindentation analysis
Freshly prepared and aged samples were analyzed on a TI Premier TriboIndenter ® (Hysitron) using a 5 μm Z-axis transducer. The transducer was calibrated at the start of every experiment, following the manufacturers tip to optics calibration routine. The transducer piezo was allowed a rest period prior to PDMS analysis, to minimize drift.

| Tensile analysis
Five pieces (freshly made and up to 4 weeks old) of PDMS of varying thickness, with a total area between 3 and 12 mm 2 , respectively, were trimmed into rectangles to fit into the specimen grips of a zwickiLine Z0.5 (Zwick Roell). The modulus was determined from a gradient of $14% on the stress/strain curves, using testXpert ® II software (Zwick Roell). The outputted data were exported into Microsoft Excel (2013) v.15.0.5015 for Windows and analyzed further.

| AFM multiparametric imaging of PDMS
A NanoWizard ® 3 AFM, software v. 5.0.51 was used in Quantitative Imaging (QI™) mode and all images and F-D curves were analyzed using JPK Data Processing software v.spm-5.1.13 (JPK Instruments).
All experiments were performed in liquid (brain heart infusion broth, 37 g l À1 ) (Fluka) passed through a 0.22 μm filter (Millipore) at 37 C with the same settings. QI™ setpoint was either 1 nN ( Figure S2b) or 5 nN. Z-length was set to 900 nm with an additional 50 nm added before going to the next pixel. Approach and retract times were 40 ms, which equated to 22.5 μm/s. Motion and acceleration times were 1.0 ms and sample rate was 100 kHz ( Figure S2a). The resolution was 64 Â 64 pixels over a 10 Â 10 μm scan region. MLCT silicon nitride cantilevers (Bruker Corporation) with silicon nitride tips were used throughout unless otherwise stated, with a new cantilever used for every experiment. The inverse optical lever sensitivity (InvOLS) (Cleveland et al., 2006) was performed at the start of each experiment in air and liquid on freshly cleaved mica or 1 molar potassium hydroxide-cleaned glass coverslip. The thermal noise method (Hutter & Bechhoefer, 1993) was used to determine the cantilever spring constant using the calibration routine in the JPK software with corrections applied for the 10 cantilever tilt (Hutter, 2005). The spring constant from the air calibration routine was used with the InvOLS value from the liquid measurements. All F-D curves were subject to $200 pN of hydrodynamic drag force. These forces can be problematic for force measurements on soft samples, particularly with oscillating probes. However, there remains predictability with these forces when scanning in the same liquid medium (as in this study), and the drag force dependence on tip speed exhibits linear behavior (Alcaraz et al., 2002;Berthold et al., 2017). The same loading speed was maintained in all our experiments. In principle, we could have subtracted 200 pN from each F-D curve or performed slower scans. Crucially, however, QI™ mode utilizes a "dynamic baseline adjustment" during imaging, which takes account of hydrodynamic effects, and there is no sinusoidal oscillation of the cantilever, nor feedback loop (Chopinet et al., 2013;JPK Instruments AG, 2011). These combined effects allow for controlled loading on the samples, at all preselected loading rates used in this study, leading to a consistent $200 pN drag force and accurately defined loads (i.e., 1, 5, 7, and 10 nN) across all F-D curves ( Figure S2-4). Because of the predictability, consistency of experimental conditions and the QI™ corrective factors, we could disregard the hydrodynamic contribution.

| AFM indentation method
When formulating the method that led to the adoption of the 85 cone half-angle settings a range of cantilevers was used for comparative analysis of the PDMS substrate modulus. Standard MLCT cantilevers using the triangular D, E, or F cantilever, or B500_CONTR (nanotools) were used. A range of QI™ setpoints were used (1, 5, 7, or 10 nN) to either ensure that the large spherical indenter reached the same indentation depth of smaller radii tips, or to specifically determine the modulus at increased depths ( Figure S4). Freshly made PDMS substrates were scanned at 0.5 nN and immediately after at 1 nN to establish, and subsequently nullify (not shown), if there was a loading force dependence on the reported modulus and the tip geometry at differing indentation depths. Importantly, the tips share the same dimensions across all the cantilevers on the MLCT chip, and so the angle measurements should be consistent from either cantilever.
The inbuilt Hertz/Sneddon model in the AFM software allows the user to change the indenter geometry. The Hertz fit for a spherical indenter and the Sneddon fit for a conical indenter were both used.
Following the elucidation of the tip angles, a half-angle of 85 was used for all PDMS modulus data. Indentation of elastic solids has been studied for over a century. Heinrich Hertz first pioneered the contact between elastic bodies (Hertz, 1882) where he approximated shallow indentations for a smooth elastic sphere onto a rigid flat surface, according to the equation where E s is the sample surface modulus, v s is the Poisson's ratio, r is the tip radius of curvature, and δ is the displacement of the indenter.
The model was extended to study the contact problem between two linearly isotropic solids (Boussinesq, 1885). Ian Sneddon took the approach by Boussinesq to derive the load-displacement relationship for a rigid conical indenter (Sneddon, 1948) to derive the equation where α is the half opening angle of the indenting cone. He later extended his work for other indenter geometries (Sneddon, 1965).

| AFM F-D curve batch processing
All F-D curves were manually analyzed using JPK Data Processing software v.spm-5.1.13 (JPK Instruments) using the approach portion of the curves. The inbuilt operators were loaded in a specific, necessary order, and were saved as a user "process" ( Figure S3). This process could be easily loaded for all future batch, or individual, curve processing. At least 10 F-D curves were analyzed for every sample unless otherwise stated. Further adjustment was typically required to better define the contact point, using the "subtract baseline," "contact point," and the "correct height for cantilever bending" operators.

| Elastic modulus calculations
Often, custom scripts developed with third party software, such as MATLAB, are utilized for the calculation of elastic moduli. These typically employ an equation of contact mechanics, like the Hertz model, and a variety of written code seeks to identify the contact point between probe and sample (Denisin & Pruitt, 2016b;Dhahri et al., 2013). We elected to use the manufacturer software and embedded Hertz models. Each individual F-D curve was analyzed manually. The position around the zero crossing point was zoomed into and the identification of deflection was considered only where the noise-that deviated above and below the baseline-remained consistently above the baseline. PDMS is relatively stiff (compared to cells) and this made identification of the deflection point easy ( Figures S3 and S4). Fortunately, even if this identification under-or over-estimated the exact moment of contact due to the noise, the error is negligible in many AFM applications (Dufrene et al., 2013).  Figure S8). The internally

| PDMS roughness
The top surface and three sub-surface AFM scans from Figure 4 were quantified using NanoScope Analysis. The arithmetic mean roughness, R a , the RMS roughness, R q , and the skewness, R sk and kurtosis, R ku were calculated from the whole image. All images were processed with third order plane fitting and first order flattened.  (Gaudière et al., 2012;Kingsley et al., 2019;Niu et al., 2018). Strain rates below 15% utilized in this study are within the linear elastic regime for PDMS and this behavior allows the elastic modulus to be calculated via Hooke's law where the tensile stress-strain curve is simply the reverse of the compressive stressstrain curve (Johnston et al., 2014;Niu et al., 2018). When characterized within this regime fiber network models demonstrate identical moduli for both tension and compression (van Dillen et al., 2008).
INI relies on the accurate calculation of the indenter tip geometry and elastic moduli are determined by using the Oliver and Pharr model (see Figure S1), which accounts for the changes in contact area at different locations along the unloading portion of an indentation curve (Oliver & Pharr, 1992). The model is based on a conical indenter. INI has reported widely dispersed values of elastic modulus for 10:1 (w/w) elastomer to crosslinker PDMS between $0.6 and 50 MPa (Charitidis, 2011;Lin et al., 2008;Liu et al., 2009b), with many suggesting an elastic modulus $3-4 MPa (Deuschle et al., 2008;Shen et al., 2008). In this present work, we tested a variety of PDMS sub- loading rates are low (e.g., 2 μm/s) the molecules are able to move and recover to their original conformation, whereas for faster loading rates (e.g., >500 μm/s) the molecules cannot move fast enough to follow the induced deformation, and subsequently the material behaves more like a stiff material rather than an elastic one-leading to increased elastic moduli (Kim et al., 2008). It has been shown that PDMS exhibited no time-dependent effects on loading curves when indented at speeds between 0.2 and 200 mm/min (i.e., 3-3300 μm/s) (Lim & Chaudhri, 2004

| SEM reveals changes in geometry toward the tip apex
Because the elastic moduli calculated from the spherical B500_CONTR probe and tensile loading were in close agreement we sought to identify the conical half-angle and radius of the MLCT tip in greater detail, to determine if our approximated conical half-angles were realistic. Typically, samples are coated with a metal to reduce any charging effects and to provide good contrast. However, the coating is often 10-20 nm in thickness, and often not of a uniform distribution ( Figure S5). Given that the tip diameter is only around 40 nm ( Figure 2b,c,e) this could be problematic. As we were interested in the terminal 10 nm from the tip apex, we used uncoated tips and obtained high-resolution micrographs with only moderate signs of charging ( Figure 2b-g). The manufacturer assigned a tip height between 2.5 and 8.0 μm and a side angle of 17.5 ± 2.5 . We measured a tip height of $7 μm, and the side angle was approximated as 16.3 , in close agreement with the manufacturer (Figure 2d). The angle was found to change very little when increasing the magnification to show $4.4 μm exactly from the tip apex to represent the contact geometry between the probe and PDMS that would be required to constrain the elastic modulus at $1.3 MPa, when indenting at 10, 50, and 100 nm, respectively.
Our method suggests that at 10 nm indentations the full angle had to be adjusted to nearly 180 representing an almost flat punch contact area between the probe and the sample. As the probe is indented further, the angle becomes steeper as more material begins to contact the sides of the probe, which seems logical.
Materials either sink-in or pile-up when indented. A soft metal, such as aluminum has been shown to pile-up during indentation (Van Vliet et al., 2004). Conversely, Deuschle and coworkers extensively studied 10:1 (w/w) PDMS using a combination of INI, SEM, AFM, and optical microscopy. They used a cube corner probe and found that even at 15 μm indentations a clear sink-in effect was observed, and that the shape of the impression, rather than being pyramidal, was more conical. All of their indentations recovered fully, confirming true rubber-like properties of PDMS (Deuschle et al., 2008).
3.3 | Characterizing the molecular architecture and sink-in of PDMS with AFM and optical microscopy with dense bundles of $10 nm widths (red arrows), which correlate with the expected structures (Granick et al., 2003;Yamada, 2003). A 50 Â 50 nm profile in x and y was produced to examine the topogra-  Figure S6b,c). It should be noted that indenting with the softer MLCT cantilevers did not produce a visible sink in, compared to those observed in Figure 3g,h, using our top mounted optics. It is expected that any sink-in would naturally start small at the nanoscale, as depicted in the cartoon in Figure 3f, and grow increasingly larger under a greater load. Thus, this nanoscale sink-in would be obscured by the greater size of the cantilever when viewed from above.
Deuschle and colleagues also utilized finite element analysis to infer possible contact area between their sample and indenter, which showed good agreement (albeit slightly smaller), with their experimental contact mechanics model (Deuschle et al., 2008). In this regard, the use of finite element modeling could be explored in future work as it may help to address some of the limitations of this study and potentially offer new insights into the deformation of soft materials at shallow indentations.

| Internal architecture of PDMS
The molecular organization of the surface (Figure 4b) was compared to the molecular organization deeper within the 10:1 (w/w) PDMS at nm (Figure 4c i), μm (Figure 4c ii) and mm (Figure 4c iii) depths by moving the probe along the exposed internal surface, which was created by snap-freezing in liquid nitrogen and cracking open (see Section 2 and Figure S8). Initial scans were performed on a bladesliced sample ( Figure S7), but the steep cutaway led to some difficulties during scanning. The PDMS was secured to the AFM stage with the internal surface facing upwards towards the probe. Optically, there were numerous, seemingly ordered, and slightly concentric lines evenly dispersed throughout the interior ( Figure S7 and S8). Areas between these lines were scanned with the AFM. The molecular architecture at all depths appeared to be similar to the surface, but with a greater number of dense bundles (Figures 4c, S7, and S9). The then allows for an area percentage to be assessed. All images shared a similar mean percentage of 24.2 ± 1.06 (Figure 4a). Although the internal structure does not initially appear to be as porous as the surface, the increased depth likely added to the area percentage calculations. Structurally, the apparent reduction in porosity could be due to the image contrast in the AFM scans. We binary filtered the internal structure images, and these more closely resembled the top surface ( Figure S9).

| Roughness measurements at the surface and subsurface
To further understand how the PDMS and the AFM probe interact at sub-100 nm indentations we analyzed the roughness at the surface and the subsurface from the whole image (Table 1). R a and R q both represent surface roughness and refer to variations in the height of a surface relative to a plane of reference. They are standard measures used in engineering and tribology. R a is calculated as the arithmetic average of the peaks and valleys, and R q is the root mean square (RMS) of the same measurements. A single large peak or valley would raise the R q value more than the R a . It can be seen from Figure 4c and the tabulated data in Table 1, that the subsurface is rougher and denser than the top surface of PDMS, with a mean roughness of 3.9 ± 2.5 nm. However, at sub-100 nm indentations, and most certainly at 10 nm indentations, it will be the surface architecture that bears the greatest impact on the contact geometry between an indenting probe, and thus, the reported moduli from modifications to a contact mechanics model, such as the Hertz/Sneddon model utilized in this work. The small, but numerous asperities that we postulate would be pushed aside, coupled with the bulk sink-in effect of PDMS, may explain the almost flat punch-like contact geometry on a seemingly spherical tip apex, as less material would be in contact with the probe at very small indentations.

| CONCLUSIONS
This study sought to better understand the contact geometry relationship between an indenting probe and PDMS, and how the molecular properties of the polymer may lead to contact angles that differ to the expected values. We compared the molecular architecture of PDMS at the surface and within the bulk material. Our data suggest that the porosity is similar between the surface and the interior, but that the PDMS may be slightly denser within the bulk material, with more apparent bundles. Using the classical Hertzian contact mechanics model, we identified that adjusting the half-angle geometry of the indenter tip has a marked influence on the reported moduli of the sample under investigation. Using high magnification SEM of an AFM probe we showed that the shape changes markedly toward the tip apex and the progressively steeper narrowing away from the apex show a similar trend to our approximated angles. Ultimately, we used the conical half-angle as an adjustable parameter and fixed it at a range of indentation depths (from 0 to 100 nm) to constrain the elastic modulus of PDMS at $1.3 MPa, which we calculated from tensile loading and from a spherical probe with a well-defined radius. To this end, the PDMS was used as a calibrant to infer the AFM tip geometry.
With a reasonable knowledge of the probe geometry, and the sample properties, this method may allow the investigator a simple method to improve the mechanical quantification of a variety of soft materials, and minimize the often widely dispersed data reported in the literature. for their assistance with tensile loading and INI. Heartfelt gratitude to Jamie K. Hobbs for assistance with the preparation of this manuscript.

CONFLICT OF INTEREST
There are no conflicts to declare. This sub-surface scan only, being within a few hundred nanometers of the top surface, is likely to influence sub-100 nm indentations. R a is the arithmetic average, R q the root-mean-square, R max is the maximum roughness. R sk is skewness and R ku is kurtosis.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.