Decoupling subsurface inhomogeneities: a 3D finite element approach for contact nanomechanical measurements^*

In low-load AFM measurements, the solid material can be purely deformed in an elastic fashion. Thus, the elastic behavior of the tip-sample system can be studied using the classical Hertzian theory. The theory of Hertz is subject to a number of assumptions [67, 68]; however, nowadays this theory is still widely used to capture the elastic contact phenomena in (nano)indentation experimental and numerical works. In this study, the analytical solution of the Hertzian theory is used to validate FE numerical models with comparable contact radii. For a spherical tip, the normal force, P, is [67, 68]

$$\begin{eqnarray}&&P=\displaystyle \frac{4{R}^{\ast 0.5}{\delta }^{1.5}{E}^{\ast }}{3},\end{eqnarray} \tag{ 1 }$$

where R^* is the composite radius, δ is the indentation displacement, and E^* is the indentation modulus, also known as the composite elastic modulus. The composite radius is: ${R}^{\ast }={\left[(1/{R}_{1}+1/{R}_{2})\right]}^{-1}$, being equivalent to the radius of the probe's tip for a flat sample surface of the sample; and the E^* modulus is defined for isotropic materials as: $1/{E}^{\ast }\,=(1-{{\nu }_{1}}^{2})/{E}_{1}+(1-{{\nu }_{2}}^{2})/{E}_{2}$ where the subscripts 1 and 2 correspond to the cantilever and the sample, respectively, and ν is the Poisson's ratio. Contact radius a and δ are in order

$$\begin{eqnarray}&&a=\sqrt[3]{\displaystyle \frac{3{{PR}}^{\ast }}{4{E}^{\ast }}}\,\,\,\,\,\,\,\,\,\,{\rm{and}}\,\,\,\,\,\,\,\,\,\,\delta =\displaystyle \frac{{a}^{2}}{{R}^{\ast }}.\end{eqnarray} \tag{ 2 }$$

Upon spherical indentation, the contact stiffness, k, is typically related to the composite elastic modulus via Hertzian contact mechanics as

$$\begin{eqnarray}&&k=\displaystyle \frac{{\rm{d}}P}{{\rm{d}}\delta }=\sqrt[3]{6{R}^{\ast }P{{E}^{\ast }}^{2}}.\end{eqnarray} \tag{ 3 }$$

When modeling linear-elastic indentation in inhomogeneous materials, E^* can be extracted via several different Hertzian relations. For an indentation-based modulus, ${E}_{\delta }^{\ast }$, this is

$$\begin{eqnarray}&&{E}_{\delta }^{\ast }=\sqrt{\displaystyle \frac{9}{16}\displaystyle \frac{{P}^{2}}{{R}^{\ast }{\delta }^{3}}}.\end{eqnarray} \tag{ 4 }$$

In inhomogeneous materials, the surface and subsurface mechanical behavior can also depend on the proximity of any material inhomogeneity to the contact area. In such non-Hertzian contact mechanics, the unknown contact radius can be studied based on analytical and approximate formulations from classical elasticity theory originally applied to layered materials of any thickness D (i.e. $0\leqslant D\leqslant \infty$) [69, 70] having a high-stiffness mismatch and later applied to continuum mechanics of composites with embedded inhomogeneities [71]. Considering the Hertzian contact radius of the coating material, a^c, and that of the substrate or inhomogeneity material, a^s, are known, the affected contact radius of the inhomogeneous material, ${a}_{{\rm{a}}}^{{\rm{i}},{\rm{l}}}$, is

$$\begin{eqnarray}&&{a}_{{\rm{a}}}^{{\rm{i}},{\rm{l}}}={a}^{{\rm{s}}}+({a}^{{\rm{c}}}-{a}^{{\rm{s}}})\left\{1-\exp \left[(-{\pi }^{(1/4)}{\left(\displaystyle \frac{D\sqrt{\alpha }}{{a}^{0}}\right)}^{\pi /4}\right]\right\},\end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{c}\begin{array}{l}{a}^{0}={a}^{{\rm{s}}}+({a}^{{\rm{c}}}-{a}^{{\rm{s}}})\left\{1-\exp \left[(-{\pi }^{(1/4)}{\left(\displaystyle \frac{2D\sqrt{\alpha }}{{a}^{{\rm{c}}}+{a}^{{\rm{s}}}}\right)}^{\pi /4}\right]\right\},\\ {\rm{and}}\,\,\,\,\alpha \equiv \displaystyle \frac{{a}^{{\rm{c}}}}{{a}^{{\rm{s}}}}.\end{array}\end{array}\end{array}\end{eqnarray} \tag{ 6 }$$

AFM can be used for qualitative and quantitative measurements of the material properties of polymer-reinforced nanocomposites via contact interactions. Contact measurements can generate material stiffness information using a variety of AFM methods such as CR-AFM. In CR-AFM, the AFM probe is in contact with a sample at a constant force. The AFM cantilever is excited by a small-amplitude oscillation to detect the resonance frequencies, or eigenvalues, of the cantilever-sample system. The continuous nature of the cantilever allows the output of contact resonance frequency to be converted into contact stiffness data using theoretical formulations such as point-mass and beam mechanics models. Any shift in the contact frequency can be interpreted as a change in the probed-material mechanical properties. For a perfectly linear-elastic probed material, the continuous Euler–Bernoulli (E–B) beam mechanics can be modified to relate the cantilever beam resonant frequency to the stiffness of the sample, k^s, without damping [72, 73]. The contact interaction using CR-AFM behaves similarly to a spring and can be represented as a homogeneous and uniform cantilever of length L rigidly clamped at one end and a spring coupled at the free end of length L₁, where L₁ < L, and has a stiffness k^s equivalent to the probed material stiffness.

In the E–B model, the wave equation is used for pure bending of the beam:

$$\begin{eqnarray}&&{E}^{{\rm{p}}}I\displaystyle \frac{{\partial }^{4}z}{\partial {x}^{4}}+{\rho }^{{\rm{p}}}A\displaystyle \frac{{\partial }^{2}z}{\partial {x}^{2}}=0,\end{eqnarray} \tag{ 7 }$$

where E^p, I, ρ^p, and A are the elastic modulus, the moment of inertia, the density, and the cross-sectioned area of the cantilever probe, in that order. For each eigenfunction z_n, there is a corresponding wave number κ_n associated with an eigenfrequency f_n. The boundary conditions to solve equation (7) neglect moment and shear forces. In probe-sample contact interaction, the contact resonance frequency ${f}_{n}^{{\rm{R}}}$ emerges and the normalized contact stiffness of the sample material, ${\bar{k}}^{{\rm{s}}}$, can be derived (see [62, 72, 73] for details)¹ .

Cuboidal (AgNP^c) and near-spherical (AgNP^s) APS = 78 nm silver NPs (nanoComposix, Inc., San Diego, California) were used as received. The AgNP^c were suspended 0.1 mass% in ethanol and the AgNP^s were suspended 0.5 mass% in ultrapure water. Figures 1(a)–(e) show the sample-process schematic for embedded cuboidal particles; samples with embedded spherical particles were prepared in a similar fashion. AFM topography maps in figures 1(a), (c), and (e) also show the material surface during that sample step. Particles were drop cast onto toluene and ultraviolet-ozone cleaned silicon substrates, dried, then rinsed aggressively to remove loosely bound particles and residual surfactant (figure 1(a)). The AFM topography in figure 1(a) confirms the planar orientation of the deposited cubes relative to the Si substrate. Substrates were then imaged with intermittent contact AFM to confirm regions of well-dispersed, isolated single particles (≈10 NPs per 5 μm × 5 μm scan area). Polystyrene sheet stock (PS, # ST313120, Goodfellow, Coraopolis, PA) was cut into ≈1 mm³ pieces and dissolved in toluene (Sigma Aldrich, St. Louis, MO) to produce a 5 mass% solution. Approximately 1 ml of the solution was drop cast onto the particle-coated substrates, producing a  5 μm thick film with particles located at the bottom surface of the film (figure 1(b)). The films were peeled from the substrates with tweezers (figure 1(c)). The topography in figure 1(c) shows that the cubes are only faintly evident, with few-nm height variation from the surrounding polymer, and levelness from figure 1(a) intact. The films were then flipped 180°, resulting in a polymer substrate with particles whose top-surfaces were located at the surface of the thick film supported by a silicon substrate (figure 1(d)). To bury the particles a controlled depth, additional nanometer-thin PS films were fabricated by spin coating. The above PS solution was diluted to concentration c (mass%) and cast in a two-step process (at velocity v₁ for time t₁, then v₂ for time t₂) with the parameters in table 1 to produce thickness D = 17 nm, 40 nm, and 60 nm homogeneous films.

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Table 1.  Polymer spin casting details of sample fabrication.

PS section	D (nm)	c ($\mathrm{mass} \%$)	v1 (rad s−1)	t1 (s)	v2 (rad s−1)	t2 (s)
	17	0.4	104.72	30	523.60	30
Top film	40	1.0	314.16	30	523.60	30
	60	1.0	107.72	30	523.60	30

The thin PS films were scored with a razor blade, then shallowly immersed in ultrapure water to float the films onto the water surface. Subsequently, the resultant sandwich structures (figure 1(e)) were annealed at 120 °C to create a homogeneous matrix with well-controlled particle depth. The AFM topography in figure 1(e) shows that the NPs buried at D = 60 nm are no longer visible due to the presence of the cover layer.

CR-AFM [74, 75] was performed with a Cypher AFM (Oxford Instruments, Santa Barbara, CA). Measurements used a 2.67 N m⁻¹ cantilever (FMR-PPP, Nanosensors, Neuchatel, CH) with a free resonance frequency of 72.48 kHz. (CR)-AFM images were acquired with dual AC resonance tracking [76] while the cantilever was actuated photothermally with a 405 nm laser. Images were obtained with an applied load of 30 nN. Typical CR frequencies for the 1st eigenmode were 254.60–319.77 kHz. Scan size was 1024 × 1024 resolution and 5 nm pixel (1024 pixels × 1024 pixels and each pixel is a square 5 nm on a side). Maps of the CR frequency were converted to contact stiffness with the E–B model in equation (7).

Table 2 gives the mechanical properties used in the FE nanoindentation model. The FE approach is accomplished using ANSYS Mechanical software (Canonsburg, Pennsylvania). The structural model uses a frictionless surface-to-surface contact pair that allows a rigid-to-flexible contact interaction [77]. In an approach similar to that of the AFM experiments, a perfectly flat contact surface is assumed, thus, R* = 60 nm [59]. The bottom of the PS body in figures 1(c) and (d) is fully constrained and a quasistatic contact interaction occurs between the rigid body and the elastic body upon a normal load of P = 30 nN.

Table 2.  Material properties of the FE probe-nanocomposite system.

Body	E (GPa)	ν
Rigid sphere	$2\times {10}^{8}$	0.3
PS matrix	4	0.3
AgNP	83	0.3

In all cases studied, surface contact elements are 100% 1.25 nm hexahedral elements with aspect ratio of unity. Fine subsurface elements in the contact vicinity are also hexahedral and biased from the surface (bias factor of two with increasing vertical aspect ratio along y and an invariant aspect ratio along plane xz). Figures 3(a)–(d) show 3D cut mesh mid-sections of the FE models with D = 17 nm embedded NPs depicting the two different morphologies and various orientations with respect to the normal contact in the xz plane. In order to deconvolve this contact problem and relate the numerical solution to AFM measurements, this work studies the affected contact stiffness, k^FE, of the polymer with a subsurface AgNP at a given depth D based on the contribution of: (1) particle geometry: nanosphere, AgNP^s (figure 3(a)) versus flat-surface nanocube, ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ (figure 3(b)) of equal specific surface area; (2) cuboidal particle orientation: nanocube's flat surface is parallel to the plane of contact (xz), face-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ (figure 3(b)); a nanocube is rotated 45° with respect to the z axis, edge-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{2}}$ (figure 3(c)); and a nanocube is rotated 45° with respect to the y- and z axes, vertex-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{3}}$ (figure 3(d)); (3) lateral distance to the probe's axis of symmetry with respect to the particle centerline, to simulate the AFM probe translational movement {x, z}; and (4) particle subsurface depth D along the y axis.

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This section validates the numerical models for the spherical nanoindentation on the PS-AgNP dilute composite based on two contact mechanics problems: (a) the Hertzian classical theory [67, 68] for the homogeneous PS matrix, which yields an exact solution that can be used to optimize the model surface and subsurface mesh and computational time; and (b) the affected contact-radius empirical approach formulated in [69–71] for the inhomogeneous PS-AgNP material. All models are based on the schematic of the bulk material dimensions depicted in figure 1 and are adjusted as necessary for the studied methods.

4.2.1. Homogeneous model

For the uniform PS matrix, this approach uses the particle-matrix mesh for the subsurface ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ that is closer to the contact surface (D = 17 nm) and centered along the axis of symmetry of the spherical indenter's tip (see figure 3(b)) to approximate the maximum error possible in the FE models. Numerical results are denoted with the 'n' subscript and superscript 'PS' for the homogeneous PS matrix model and are compared to the Hertzian-based analytical solution using equations (2)–(4). Table 3 provides the validation results in terms of: a^PS, ${E}_{\delta }^{* \mathrm{PS}}$, and ${k}^{* \mathrm{PS}}$ with their respective relative errors, ${e}_{{a}^{\mathrm{PS}}}$, ${e}_{{E}_{\delta }^{* \mathrm{PS}}}$, and ${e}_{{k}^{* \mathrm{PS}}}$. These results demonstrate that the numerical solution for the homogenous-material problem has high accuracy with respect to the analytical solution; showing an adequate stiffness error% ${e}_{{k}^{* \mathrm{PS}}}=-0.67$.

Table 3.  Hertzian-based numerical results for homogeneous PS matrix.

P (nN)	${a}_{{\rm{n}}}^{\mathrm{PS}}$ (nm)	${E}_{\delta ,{\rm{n}}}^{\ast {\rm{PS}}}$ (GPa)	${k}_{{\rm{n}}}^{* \mathrm{PS}}$ (N m−1)	${e}_{a}^{\mathrm{PS}}$ (%)	${e}_{{E}_{\delta }^{\mathrm{PS}}}$ (%)	${e}_{{k}^{* \mathrm{PS}}}$ (%)
30	6.82	4.44	59.71	−1.21	−0.90	−0.67

4.2.2. Inhomogeneous model

Following the affected contact-radius approach in equations (5) and (6), this model has a PS nano-film of thickness D that is perfectly bonded to an Ag bulk substrate. This validation technique considers that the mechanical mismatch behavior of the material interface between the top polymeric matrix and the embedded nanocube's surface oriented parallel to the free surface resembles that of a PS film to an Ag substrate, where the particle centerline is aligned to the probe's axis of symmetry (figure 3(b)). The numerical contact radius for this coated material is denoted as ${a}_{{\rm{n}}}^{{\rm{i}},{\rm{l}}}$. Second, for the current study, the ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ model (see figure 3(b)) can be thought of as a local film-substrate material, considering ${a}^{\mathrm{PS}}\ll \tfrac{1}{2}$ APS and denoting the resulting numerical contact radius as ${a}_{{\rm{n}}}^{{\rm{i}},{\rm{p}}}$. Using equation (2), the Hertzian contact radius for the PS and silver Ag are a^PS = 6.75 nm and a^Ag = 2.23 nm.

Table 4 shows that contact-radius numerical solutions for the inhomogeneous material model are accurate with respect to the theoretical approach for both the film-coated material and the embedded particle models at the P and D parameter values studied. The mechanical response is consistent between the PS-Ag coated material and the PS-${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ composite material; indicating that the local response at the centerline of the embedded cuboidal particle with the flat-surface orientation is equal to that of a film-based material. Numerical approximations are more accurate in cases wherein the material interface is further away from the free surface, i.e. D ≥ 40 nm. Errors in contact-radius for the inhomogeneous materials, ${e}_{{a}^{{}^{{\rm{i}}}}}$, are within the range: $-3.15\leqslant {e}_{{a}^{{}^{{\rm{i}}}}}\leqslant -1.85$.

Table 4.  Numerical results for PS-${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ composite material.

P (nN)	D (nm)	${a}_{{\rm{a}}}^{{\rm{i}},{\rm{l}}}$ (nm)	${a}_{{\rm{n}}}^{{\rm{i}},{\rm{l}}}$ (nm)	${a}_{{\rm{n}}}^{{\rm{i}},{\rm{p}}}$ (nm)	${e}_{a}^{{\rm{i}},{\rm{l}}}$ (%)	${e}_{a}^{{\rm{i}},{\rm{p}}}$ (%)
	17	6.67	6.88	6.88	−3.15	−3.15
30	40	6.74	6.87	6.87	−1.85	−1.85
	60	6.74	6.87	6.87	−1.85	−1.85

Figures 4(a)–(b) show the k^FE profiles for the six models studied at D = 17 nm and the four models at D = 40 nm and D = 60 nm. These stiffness profiles correspond to indents along the x axis from the center of the particle at l = 0 nm to ±60 nm away from it, where z = 0 nm. In the case of the 17 nm nanocube particles with asymmetrical orientation with respect to the plane of contact xz: edge-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{2}}$ and vertex-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{3}}$ models, a second profile along the z axis is included (figure 4(a)), where x = 0. Figures 4(c)–(f) complement figure 4(a) to illustrate 3D maps of the local stiffness of the PS-AgNP composite with subsurface ${{\rm{AgNP}}}^{{\rm{s}}}$, ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$, ${\mathrm{AgNP}}^{{{\rm{c}}}_{2}}$, and ${\mathrm{AgNP}}^{{{\rm{c}}}_{3}}$ particles, in that order (D = 17 nm). These stiffness maps have a consistent ${e}_{{k}^{* \mathrm{PS}}}\approx -0.67 \%$ with respect to its analytical value (see table 3) and depict how the geometry and orientation of the NPs can significantly contribute to the inhomogeneity of k^FE distribution across the free surface by capturing the symmetry of the subsurface particles in relation to the plane of contact xz. Results in figure 4(a) show a considerable geometry effect of subsurface ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ and ${\mathrm{AgNP}}^{{{\rm{c}}}_{2,x}}$ particles on ${k}^{\mathrm{FE}}$, depicting a plateau-like stiffness shape in the embedding surface above them. This k^FE shape is induced by the flat top surface of ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ (figure 4(d))and the nanocube edge of ${\mathrm{AgNP}}^{{{\rm{c}}}_{2}}$ (figure 4(e)); resembling the local response of geometrically uniform layered materials. At 17 nm, a peak-like ${k}^{\mathrm{FE}}$ shape occurs when embedding AgNP^s (figure 4(c)), edge-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{2,x}}$ (figure 4(e)), and vertex-high particle ${\mathrm{AgNP}}^{{{\rm{c}}}_{3,x}}$ and ${\mathrm{AgNP}}^{{{\rm{c}}}_{3,z}}$ (figure 4(f)), resembling the geometry of the embedded sphere while exposing the orientation of the cuboidal ${\mathrm{AgNP}}^{{{\rm{c}}}_{2,x}}$, ${\mathrm{AgNP}}^{{{\rm{c}}}_{3,x}}$, and ${\mathrm{AgNP}}^{{{\rm{c}}}_{3,z}}$. The stiffness contact is similar along the x and z axes for the vertex-high particle ${\mathrm{AgNP}}^{{{\rm{c}}}_{3}}$ since this orientation has at least trigonal symmetry with respect to the indenter. The different stiffness profiles in figure 4(a) are the result of stress-strain response of the composite system that encompasses not only the matrix but also the stresses that are transferred to the AgNPs and in the case of asymmetrical orientation with respect to the contact interaction plane, induced stress gradients. Figures 5(a)–(f) show the contour plots along the axis of symmetry of subsurface normal stress, σ_y, for 17 nm embedded AgNP models at l = 0 nm. Subsurface stresses are uniformly distributed in both the PS matrix and the embedded ${{\rm{AgNP}}}^{{\rm{s}}}$ (figure 5(a)) and face-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ (figure 5(b)). However, the non-symmetrical orientation of edge-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{2}}$ (figures 5(c)–(d)) and vertex-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{3}}$ (figures 5(e)–(f)) contribute to the non-uniform σ_y field distribution. The mechanical behavior inside the non-axisymmetric inhomogeneity is also intrinsically non-uniform in figures 5(c)–(f). In all cases, except when the nanocube face is parallel to the surface (face-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$), these subsurface stress images also show the edge of the particle, because these edges are not perpendicular to the normal stress. In the case of face-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$, since the normal stress is continuous at an interface, this matrix-particle edge is not seen in figure 5(b).

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In the peak-like ${k}^{\mathrm{FE}}$-group, the ${{\rm{AgNP}}}^{{\rm{s}}}$ impart a higher stiffness to the surrounding matrix when −40 ≤ l ≤ 40. The stiffness profile shows a dramatic change in terms of shape and magnitude when $| l| \gt 0$ nm when embedding vertex-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{3,x}}$. This nanocube orientation results in a very high inhomogeneous stiffness distribution along the principal axes {x, z}. Relating to the degree of stiffness contribution at D = 17 nm, k^FE at l = 0 increases 23.46%, 18.53%, 17.13%, and 19.63% with respect to ${k}_{{\rm{n}}}^{* \mathrm{PS}}$ for embedded ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$, ${\mathrm{AgNP}}^{{{\rm{c}}}_{2}}$, ${\mathrm{AgNP}}^{{{\rm{c}}}_{3}}$, and ${{\rm{AgNP}}}^{{\rm{s}}}$ (see figure 3); respectively. Embedded face-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ noticeably yields the higher stiffness at all depths studied, as shown in figures 4(a)–(b), since more of the NP material; is close to the surface versus the other orientations. To relate to different nanoreinforcement systems, three additional FE models were run at l = 0 nm using the 17 nm embedded nanocube model orientation that yields the less stiffness contrast. This is the case of the ${\mathrm{AgNP}}^{{{\rm{c}}}_{3}}$ model for the actual PS:AgNP modulus ratio of 1:20 (figure 4(a)). Decreasing the modulus ratio between the PS matrix and a vertex-high embedded nanocube, ${{\rm{NP}}}^{{{\rm{c}}}_{3}}$, so that the PS:${{\rm{NP}}}^{{{\rm{c}}}_{\text{3}}}$ modulus ratio is: (a) 1:2; (b) 1:5; and (c) 1:10 results in a stiffness increase compared to the homogeneous matrix of 3.77%, 4.96%, and 10.70%, respectively (1:20 ratio results in 17.13% stiffness increase). Relative to AFM measurement uncertainties, the 1:10 particle would still be seen, but the 1:5 and 1:2 stiffness particles would probably not be seen by the AFM.

Figure 4(b) depicts an abrupt decrease in ${k}^{\mathrm{FE}}$ when particles are embedded at deeper subsurfaces (D ≥ 40 nm), particularly in cases where $| l| \geqslant 40$ nm. Figure 4(b) shows that at D = 40 nm, the effect of particle geometry of the nanocube with respect to that of the nanosphere can be distinguished as the trace of the cube is noticeably flatter than that of the sphere, whereas at D = 60 nm the particle geometry effect is negligible since the shapes of the two curves are very similar. At l = 0 nm, the increment in k^FE in relation to ${k}_{{\rm{n}}}^{* \mathrm{PS}}$ is: 5.39% and 2.75% when embedding ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$, ${{\rm{AgNP}}}^{{\rm{s}}}$ at D = 40 nm and 3.53%, and 1.67% when embedding ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$, ${{\rm{AgNP}}}^{{\rm{s}}}$ at D = 60 nm. These ${k}^{\mathrm{FE}}$ increments in the polymeric surface can be considered small. In this regard, the ISO 14577-1 standards for instrumented (nano) indentation tests on metallic materials [78], which is generally used for most materials, indicate that the material interface in coated substrates can significantly affect the (nano) indentation measurements when such interface is within $D\leqslant 6a$ of the indented film. Based on the Hertzian contact-radius in equation (2) for the current contact problem, this relation yields D = 40.5 nm for the PS homogeneous material, which agrees well with data in figure 4(b). This indicates that the contact stiffness at the surface cannot distinguish particle shape and/or orientation below D ≥ 40 nm. The present numerical solutions demonstrate that both the geometry and orientation of subsurface particles can influence the surface nanomechanics in the degree of heterogeneity of the local composite stiffness of the embedding matrix whereas the proximity of AgNPs to the free surface quantitatively delimits k^FE. The mechanical response of this contact interaction depends on the subsurface material interfaces and how different 3D phase morphologies and orientations to acting loads result in a different heterogeneous mechanical behavior.

Figures 6(a)–(h) show the CR-AFM imaging results of the six PS-AgNP composite samples studied. Figures 6(a)–(c) correspond to the 17 nm embedded ${\mathrm{AgNP}}^{{\rm{c}}}$ sample and illustrate the data extraction process carried out in the study for the AFM measurement interpretation. The AFM contact resonance frequency ${f}_{n}^{{\rm{R}}}$ results in figure 6(a) are converted into absolute stiffness ${k}_{{\rm{a}}}^{\mathrm{AFM}}$ data using equation (7), as shown in figure 6(b). The AFM imaging results in figures 6(a)–(b) reveal a high variation of the sample background frequency and stiffness, respectively. These findings suggest that a change in the nanomechanics between the cantilever tip and the sample surface occurred during scanning. The significance of this varying background stiffness is directly attributed to the in situ change of the probe's geometry and surface contamination, which have critical roles in the instrument resolution. To provide accurate AFM interpretation, a uniformly distributed PS background stiffness exhibiting sharp particle exposure is needed. This can be attained using a flattening method that consists of an offset value that sets to zero the medium absolute stiffness value per scanned line across the map area (figure 6(c)). This results in relative stiffness ${k}_{{\rm{r}}}^{\mathrm{AFM}}$ data, as shown in figure 6(c). Figures 6(c)–(h) show the relative-stiffness imaging of the PS samples with embedded nanocubes and the embedded nanosphere samples, respectively, at D ≈ 17 nm, D ≈ 40 nm, and D ≈ 60 nm, in that order. Subsurface NPs are clearly detected through the AFM scans in cases wherein D ≈ 17 nm (figures 6(c) and (f)) with relatively low noise measurement effect. This noise, ϕ, can be interpreted in terms of $\phi ={k}_{{\rm{r}},\min }^{\mathrm{AFM}}/{k}_{{\rm{r}},\max }^{\mathrm{AFM}}$, where ${k}_{{\rm{r}},\min }^{\mathrm{AFM}}$ corresponds to the PS/background stiffness and ${k}_{{\rm{r}},\max }^{\mathrm{AFM}}$ is the maximum stiffness measured in the embedding surface. Such noise was on the order of 0.22 and 0.25 for the 17 nm embedded ${\mathrm{AgNP}}^{{\rm{s}}}$- and ${\mathrm{AgNP}}^{{\rm{c}}}$ samples, respectively. NPs are detected to some extent when embedded at D ≥ 40 nm underneath the contact interaction (figures 6(d), (e), (g), and (h)). In these cases, measuring any stiffness changes due to subsurface interfaces were affected by a higher level of AFM noise in the range of 0.38 ≤ ϕ ≤ 0.60. In addition, the nanostructure mechanics depend on the proximity of the NPs to the free surface. Contrary to FE numerical solution, there is not a distinct difference in stiffness shape pattern between nanocube- and nanosphere samples at D ≈ 40 nm due to the noise-affected AFM measurements.

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To quantitatively relate to numerical and experimental results, the material stiffness is normalized by the maximum stiffness value per case studied, so that: $\bar{k}={k}^{\mathrm{FE}}/{k}_{\max }^{\mathrm{FE}}$ and $\bar{k}={k}_{{\rm{r}}}^{\mathrm{AFM}}/{k}_{{\rm{r}},\max }^{\mathrm{AFM}}$ for FE- and AFM stiffness results, respectively. Figures 7(a)–(b) report the normalized stiffness, $\bar{k}$, AFM profiles of three (I–III) individual 17 nm embedded ${\mathrm{AgNP}}^{{\rm{s}}}$ and ${\mathrm{AgNP}}^{{\rm{c}}}$ sample particles, respectively (refer to figures 6(c) and (f)). The FE solution of the 17 nm embedded ${{\rm{AgNP}}}^{{\rm{s}}}$ and ${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ models are also shown in figures 7(a) and (b), respectively. Figures 7(a) and (b) show that the FE solutions predict a higher stiffness along l with respect to AFM profiles. This can be the result of the mutual effect of: (i) FE ideal particle geometry, size, and orientation, (ii) no noise effect in numerical simulations, (iii) in situ changes to probe tip radius as well as an absolute difference between the modeled 60 nm probe radius and the actual tip radius, (iv) uncertainties in the matrix, and/or (v) stiffer PS matrix used in FE models compared to the actual PS used in the samples. Regarding the latter, the PS elastic modulus can range between ≈2 and 4 GPa, yielding a Hertzian stiffness range of 37 N m⁻¹ $\leqslant \,{k}^{* \mathrm{PS}}\leqslant 60\,{\rm{N}}$ m⁻¹ for the current contact problem studied; FE studies are based on a 4 GPa matrix modulus. Figure 7(a) shows a fairly consistent peak shape response for ${{\rm{AgNP}}}^{{\rm{s}}}$ samples II and III. T he I-${{\rm{AgNP}}}^{{\rm{s}}}$ sample's k^AFM profile indicates this response can correspond to a non-spherical particle, since such a profile resembles that of the numerical edge-high ${\mathrm{AgNP}}^{{{\rm{c}}}_{2,x}}$ model in figure 4(a). Figure 7(b) depicts a plateau-like stiffness shape for the three experimental I–III-${{\rm{AgNP}}}^{{\rm{c}}}$ profiles. These profiles are qualitatively close to the numerical FE-${\mathrm{AgNP}}^{{{\rm{c}}}_{1}}$ result in figure 7(b), particularly in the case of I-${{\rm{AgNP}}}^{{\rm{c}}}$. This study's combined numerical- and experimental results show that despite the stiffness measurement limitations related to noise and probe tip changes, the CR-AFM technique can comprehensibly characterize and distinguish particle geometry using material stiffness variations when embedding AgNPs at D ≈ 17 nm from the matrix free surface. The CR-AFM results indicate that the PS-AgNP material interphase, which is of great relevance in understanding the nanodynamics of nanoparticulate composites [27, 28], did not limit the probe measurement capability to differentiate the effect of particle geometry on the local stiffness.

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Figures 8(a.1)–(a.3) and (b.1)–(b.3) illustrate 3D ${\bar{k}}^{\mathrm{AFM}}$ maps for D = 17 nm embedded particles corresponding to the dashed boxes in figures 6(c) and (f). The local stiffness maps for individual 'nearly' spherical ${\mathrm{AgNP}}^{{\rm{s}}}$ particles in figures 8(a.1)–(a.3) reveal a comparatively more pronounced peak similar to that anticipated in the 3D numerical ${\mathrm{AgNP}}^{{\rm{s}}}$ (figure 4(c)) and ${\mathrm{AgNP}}^{{{\rm{c}}}_{3}}$ maps (figure 4(f)). This indicates that the geometry of this particular ${\mathrm{AgNP}}^{{\rm{s}}}$ could have protuberant surfaces at the nanoscale. Figures 8(b.1)–(b.3) illustrate a relative plateau-like stiffness shape induced by the subsurface nanocube. Based on 3D FE predictions in figures 4(d)–(f) and considering that the 3D profile maps in figures 8(b.2)–(b.3) are relatively symmetrical, this CR-AFM 3D map captures particle orientation, suggesting that the flat surface of the particle is slightly rotated with respect to the plane of contact (xz) but not as much as in the edge-high and vertex-high cases.

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