Molecular Insights into the Interfacial Adhesion and Chain Adsorption of Silicone Polymers via Nanoindentation

Abstract

Silicone-based polymers, particularly polydimethylsiloxane (PDMS), are esteemed for their exceptional thermal stability, hydrophobicity, and biocompatibility. This study leverages atomistically informed coarse-grained molecular dynamics (CG-MD) simulations to explore the interfacial adhesive characteristics of PDMS films subjected to nanoindentation, with a focus on the influences of interfacial interaction strength between nanoindenter and polymer chains, temperature, and cross-link density, interpreted through the classic Johnson–Kendall–Roberts (JKR) model. Our findings reveal that increasing the interfacial interaction strength significantly enhances adhesion, necessitating a greater energy for separation. Notably, beyond a certain threshold, the adhesion exhibits a plateau, as quantified by the apparent critical energy release rate, G _c. This saturation in G _c can be attributed to chain adsorption on the indenter tip. Such an interfacial adsorption phenomenon becomes more pronounced at elevated temperatures along with a concomitant decrease in G _c, due to enhanced chain mobility. Additionally, increasing cross-link density of the PDMS network reduces chain adsorption during indentation, thereby resulting in a higher apparent G _c. Our simulation results, confirmed by the experimental Atomic Force Microscopy (AFM) measurements, offer valuable insights into interfacial behavior of silicone-based polymers, highlighting the intricate interplay among interaction strength, temperature, and cross-link density in quantifying adhesive properties of PDMS films.

Introduction

Silicone-based polymers and elastomers are extensively utilized in scientific and industrial fields due to their exceptional thermal stability, flexibility, hydrophobicity, low surface energy, and biocompatibility. − Polydimethylsiloxane (PDMS) in particular stands out for its remarkable chemical stability, corrosion resistance, high optical transmittance, oleophilicity, and stretchability. − The straightforward and cost-effective preparation processing of PDMS further enables its use in large-scale applications. For example, PDMS-based coatings can reduce ice adhesion in low-temperature environments, enable the selective separation of oil or organic solvents from water, , enhance material performance of infrastructure, and protect transportation systems. − By adjusting the cross-link density and formulation of PDMS, the thermomechanical responses and interfacial properties can be tailored for diverse applications across various environmental conditions. , Understanding the surface properties of PDMS thin films, including interfacial toughness, surface energy, adhesion, and interactions with substrates, is essential for most of these applications. Adhesion plays a critical role as the force required to separate two different contacting surfaces impacts the stability, surface structure, and chemical properties of polymer films, influencing the overall functional performance. , Understanding the adhesive properties of PDMS will provide insights into optimizing its performance, ensuring its effective application in the industrial and scientific fields.

Nanoindentation is a powerful technique for evaluating the physical properties of material surfaces at the nanoscale. It enables high-precision measurements of elastic, viscoelastic, and plastic deformation behavior, making it particularly suitable for thin film materials to determine fracture toughness, adhesion, and other mechanical properties. ,− Atomic Force Microscopy (AFM) is commonly used for nanoindentation and nanoscratch experiments. − Numerous studies have been published highlighting factors that influence adhesion force measurements using AFM, − such as tip degradation, which alters tip–sample interactions over time. − In general, AFM-based experiments are recognized as complex and very sensitive to variations in tip properties, surface characteristics, and environmental conditions. , That said, AFM measurement still represents one of the most commonly used methods to measure the tiny interaction forces between an object (i.e., the AFM tip) and a substrate. Using molecular dynamics (MD) simulations, interaction forces can be translated into measurements of material stiffness and, during separation, a critical energy release rate (G _c, an interfacial property related to adhesion). , There is currently only limited information regarding the dynamics of polymer chains during AFM measurements, which could impact interpretation of the measurement data. To address these limitations and gain deeper insights into microscopic deformation mechanisms, molecular-level modeling is essential.

MD simulation provides detailed insights into the deformation behavior at the atomic and molecular levels, addressing the limitations of experimental methods and continuum models such as the finite element method (FEM). − MD simulations reveal nanoscale mechanisms that are often challenging to observe experimentally, enabling systematic analysis of the mechanical responses under varying conditions. Hu et al. investigated the nanoindentation of pure polyethylene (PE) and porous polyhedral oligomeric silsesquioxane (POSS–PE)using MD simulations with different diamond indenter shapes, showing that the flat indenter produces higher Young’s modulus due to stress concentration at the edge. Similarly, Peng and Zeng observed a pronounced size effect in polyethylene nanoindentation, where hardness increased sharply with contact depth, peaking at approximately 1.5 nm before gradually decreasing. Furthermore, Fang et al. reported that rising temperatures reduce Young’s modulus, hardness, and elastic recovery in single-crystal copper during nanoindentation. These studies demonstrate the effectiveness of MD simulations in uncovering molecular-level mechanisms, providing valuable insights into material deformation and mechanical properties. ,

Nanoindentation simulations based on MD often rely on all-atom models, which are limited by their time and spatial scales. Chemistry-specific coarse-grained molecular dynamics (CG-MD) addresses these limitations to a large extent by mapping clusters of atoms onto CG beads, which reduces the system’s degrees of freedom while retaining essential molecular characteristics. , Ikeshima et al. combined AFM with CG-MD simulations to study polycarbonate (PC) deformation, identifying chain bending as a critical factor during the initial yielding stage, with results closely aligning with experiments. Wang et al. demonstrated that in poly­(3-alkylthiophene) (P3AT) films, higher molecular weights improved toughness, while longer side chains reduced Young’s modulus. These studies collectively demonstrate the capability of CG-MD simulations to replicate experimental nanoindentation phenomena. However, most nanoindentation analyses still rely on the Hertzian contact theory and thus neglect adhesion effectan omission that in soft, elastomeric materials can dramatically enlarge the true contact area and skew the force–displacement response. , Since adhesive forces depend on material properties and environmental conditions, their inclusion is essential for accurately describing elastomer contact mechanics. Adhesive contact frameworks address this by introducing the Tabor parameter, a single dimensionless measure of the relative roles of surface energy, elastic modulus, probe radius, and interaction distance. , When the Tabor parameter is less than approximately 0.1, adhesion is restricted to the contact perimeter and the Derjaguin–Muller–Toporov (DMT) model is applicable. In contrast, when the Tabor parameter exceeds roughly 5, adhesion dominates across the entire contact area, and the Johnson-Kendall-Roberts (JKR) model should be used.

This study aims to investigate the interfacial adhesive behavior of PDMS films under nanoindentation using CG-MD simulations, combined with the JKR theory, to systematically examine the effects of temperature, cross-link density, and indenter–film interactions on interfacial adhesion. Our simulations reveal that polymer chains flow onto the indenter upon contact, significantly impacting the interfacial adhesion. We validate our simulation findingsspecifically the flow of polymer chains onto the AFM tipusing a model silicone elastomer. The experiment measures the critical energy release rate between the elastomer and the AFM tip. The study highlights the challenge of interpreting adhesive force measurement without additional information (as many others have reported over the years). ,,,− Simulation results indicate that the tip becomes covered with polymer after its first interaction with the surface. Based on this insight, the interaction can be modeled effectively, allowing for an estimation of G _c. These simulations provide valuable insights into the deformation mechanisms and adhesive behavior of PDMS films, offering guidance for interpreting experiments, advancing materials research, and improving device design.

Models and Methods

Overview of the Coarse-Grained Model System

All MD simulation processes are performed using the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The simulation systems and results are visualized with OVITO. An object-oriented Python toolkit, MDAnalysis, is used to analyze molecular dynamics trajectories. Following the energy renormalization (ER) approach, , temperature-dependent scaling factors α_p(T) and β_p(T) are introduced to the Lennard-Jones (LJ) parameters ε and σ for CG modeling of polymer (eq ). In this work, we applied this approach to construct a CG model of PDMS. This approach ensures that the CG model can accurately reproduce the density and dynamics of the all-atom (AA) model across a wide temperature range and for different polymer architectures. The introduction of temperature dependence compensates for the loss of configurational entropy and many-body effects inherent to the coarse-graining process, thereby greatly improving the transferability of the CG model to other thermodynamic states and chemical structures. The nonbonded potential energy is represented as a GROMACS style 12–6 Lennard-Jones (LJ) potential:

$$U_{\mathrm{L}\mathrm{J}}(r_{\mathrm{L}\mathrm{J}},T)=4\epsilon (T)\{{\left[\frac{\sigma (T)}{r_{\mathrm{L}\mathrm{J}}}\right]}^{12}-{\left[\frac{\sigma (T)}{r_{\mathrm{L}\mathrm{J}}}\right]}^6\}+S_{\mathrm{L}\mathrm{J}}(r_{\mathrm{L}\mathrm{J}})$$ 1

where σ­(T) controls the effective van der Waals radius of a bead in the CG model and marks the radial distance at which the potential function crosses the zero-energy line, ε­(T) is the depth of the potential well in energy units related to the strength of the cohesive interactions of the material. In the CG simulations, the energy switching function S _LJ is used, which smoothly reduces the energy to zero between an inner cutoff value R _i = 12 Å and an outer cutoff value R _o = 15 Å. The cross-interaction terms take the arithmetic mean for ${\sigma }_{ij}=\frac{1}{2}({\sigma }_{ii}+{\sigma }_{jj})$ and the geometric mean for ${\epsilon }_{ij}=\sqrt{{\epsilon }_{ii}\times {\epsilon }_{jj}}$ , where i and j denote different particle types for polymer and indenter. All bonded and nonbonded parameters are given in Tables S1 and S2.

For nanoindentation simulations, we constructed a thin film system consisting of 1000 CG polymer chains, each composed of 30 repeating units, with each unit represented by a single CG bead. The system dimensions are 150 × 150 × 320 Å, and the thin film thickness is about 160 Å at 300 K. An indenter with a diamond lattice structure is placed above the film, with a radius of 25 Å and a height of 85 Å, as shown in Figure b. For the cross-linked thin film system, we added 500 tetrafunctional CG cross-linkers, each linker consisting of five beads, to the linear thin film system. The total number of polymer CG beads is 36,680 for the noncross-linked thin film and 39,180 for the cross-linked thin film with cross-linkers. To simplify the model, the same potential is applied to all CG beads in the system. The cross-linked network is established through the formation of bonds between reactive sites, , and the cross-linking process is shown in Figure c. The axis of the indenter is positioned at the center of the x–y plane and aligned parallel to the z-plane. The initial distance between the tip of the indenter and the upper surface of the film is approximately 25 Å, as shown in Figure d. The nonbonded interaction parameters of the indenter are set to 0.821 kcal/mol and 3.46 Å to mimic the diamond tip. To modulate the interaction strength between the indenter and polymer, a scaling factor, α_int, is introduced. Specifically, the interfacial interaction strength is defined as ε_int = α_int × ε _ij , where ε_int represents the interfacial interaction strength parameter for the indenter–polymer interface.

1.

(a) Coarse-grained (CG) mapping scheme for PDMS, showing the transition from the all-atomistic (AA) model (left) to the CG model (right). (b) PDMS thin film (red), positioned between two vacuum layers, with an indenter (yellow) placed above. (c) Illustration of the PDMS cross-linking process, showing linear CG chains (red) and tetra-functional CG cross-linkers (blue). (d) Snapshot of the structure of the nanoindentation model.

CG-MD Simulation Details

The CG-MD simulation consists of two stages: equilibration and indentation. Periodic boundary conditions are applied in the x and y directions, while a fixed boundary condition is imposed in the z direction. The expansion space in the z direction is adjusted to be sufficiently large to simulate a free surface, ensuring consistency with the experimental conditions. To equilibrate the thin film system, we minimized the energy using the conjugate gradient algorithm, followed by annealing in the NVT ensemble (Nose–Hoover) at 800 K for 2 ns. Subsequently, the thin film is annealed from 800 K to the target temperature, allowing it to relax gradually with a cooling rate of 0.1 K/ps. The system is further relaxed at the target temperature for additional 2 ns. During the indentation stage, the indenter is fixed as a rigid body and initially placed 25 Å above the thin film. To represent a rigid substrate, a 20 Å-thick bottom layer of the polymer film CG beads is held fixed along z. In most MD simulations of nanoindentation, the indentation speed typically ranges from 0.01 to 1 Å/ps. ,, In this study, the speed of the indenter during the loading/unloading process is set to 0.2 Å/ps (i.e., 20 m/s), with a total penetration depth of approximately 25 Å. To improve the statistical reliability, 20 independent models are calculated for each variable.

Property Calculation

To analyze the adhesion behavior, the energy release rate G is defined as the derivative of the strain energy with respect to the contact area A between two surfaces. Upon unloading, adhesive forces may cause the indenter to stick to the surface, often producing an inflection point at complete separation. According to the JKR theory, the force on the indenter during the nanoindentation process is given by the following equation: ,,

$$F=\frac{4E^{*}{r}^3}{3R}-\sqrt{8\pi E^{*}G{r}^3}$$ 2

where the first term represents the Hertzian model, E* is the reduced elastic modulus, R is the indenter radius, and r is the radius of the contact patch. The critical energy release rate G _c, defined at $\frac{\mathrm{d}G}{\mathrm{d}A}=0$ , can be calculated from the maximum pull-off force (F _c), obtained as the absolute minimum of the force–indentation depth curve, together with the radius of the indenter: ,,

$$F_{\mathrm{c}}=\frac{3}{2}\pi R{G}_{\mathrm{c}}$$ 3

The Debye–Waller factor (DWF), ⟨u ²⟩, can be used to characterize the dynamic heterogeneity and local stiffness of the thin films. In this work, ⟨u ²⟩ is determined as the value of the mean square displacement (MSD) ⟨r ²(t)⟩ curve at t ≈ 4 ps (i.e., a segmental caging timescale) and MSD is calculated as

$$⟨r^2(t)⟩=⟨\frac{1}{N}\sum _{a=1}^N{[r_a(t)-r_a(0)]}^2⟩$$ 4

Here, r _a (t) is the center-of-mass position of the bead a at time t, and ⟨···⟩ represents the ensemble average of all N polymer chain beads in the system.

To characterize the local atomic stress distribution under nanoindentation, the von Mises stress is calculated for each bead considering the Voronoi volume. The formula is as follows:

$${\sigma }_{\mathrm{v}}=\sqrt{0.5[{({\sigma }_{xx}-{\sigma }_{yy})}^2+{({\sigma }_{yy}-{\sigma }_{zz})}^2+{({\sigma }_{xx}-{\sigma }_{zz})}^2]+3({\sigma }_{xy}^2+{\sigma }_{yz}^2+{\sigma }_{xz}^2)}$$ 5

where σ _ij , with i and j denoting the combinations of the x, y, and z directions, represents the 6-element symmetric stress tensor components. The atomic virial stress tensor can be calculated as ,

$${\sigma }_{ij}^a=-\frac{1}{V_a}[m^a{v}_i^a{v}_j^a+\frac{1}{2}\sum _{b\ne a}{r}_i^{ab}{f}_j^{ab}]$$ 6

where V _a is the Voronoi volume assigned to bead a, m ^a and v _i are the mass and the i-th component of the velocity of bead a, respectively. r _i is the i-th component of the displacement vector from bead a to bead b, and f _j is the j-th component of the force exerted by bead b to bead a. The summation is over all beads interacting with bead a.

Material Synthesis

The silicone/SiO₂ nanocomposite is prepared with a 2 wt % silicon dioxide (silica) concentration. The matrix used in this study is a silicone elastomer which is synthesized in house following the procedure outlined in our earlier work. Briefly, silicone chains denoted as DMS-V22 (divinyl-terminated polymethylsiloxane, Mw = 9400 g mol^–1) polymer are mixed with the HMS-151 (15%–18% methylhydrosiloxane–dimethylsiloxane copolymer, trimethylsiloxane terminated, Mw = 1900–2000 g mol^–1) cross-linker in the presence of the platinum catalyst SIP6830.3 (3% Pt catalyst in vinyl-terminated PDMS, Mw = 475 g mol^–1) and moderator, SIT7900 (1,3,5,7-tetravinyl-1,3,5,7 tetramethylcyclotetrasiloxane, Mw = 345 g mol^–1, molecular formula: C₁₂H₂₄O₄Si₄). All components are supplied as received from Gelest Inc. (PA, USA). In this case, a ratio between the polymer and cross-linker of 1:1.3 is used. Samples are mixed with a speed-mixer and cured at room temperature for 1 week prior to use. Figure a shows the basic chemical scheme. The SiO₂ nanoparticles with 10–20 nm average radius are used as received from Sigma-Aldrich.

6.

AFM measurement of model silicone elastomers. (a) The chemical scheme for the silicone elastomer used in this work. (b) Topographical images of the tip check sample with tip used (i) before starting, (ii) after run 1, (iii) after run 2, (iv) after completing the adhesion characterization of the PDMS elastomer. (c) Representative AFM force–displacement curve from a typical measurement with a newer tip showing a JKR fit to the pull-off portion of the experimental curve. Note that the spring constant of the cantilever has been accounted for. (d) Pull-off force for several scans of the same PDMS substrate. [e­(i)] SEM images of a new, unused AFM tip and (ii) same tip after scanning the PDMS sample. (f) Calculated adhesion energy release rate G _c based on tip radius obtained from the JKR fit. The standard deviation is within the size of the marker.

Experimental Measurement

An Oxford Instruments, Asylum Research, Jupiter XR AFM instrument is used to scan and characterize the adhesion properties of surfaces at the nanometer scale. Fast Force Mapping (FFM) mode is used to measure the adhesive properties of the sample. Individual force–displacement curves are collected for each pixel of an image. A scan size of 2 μm, a set point (normal force/target point) of 20 nN, and a distance of 400 nm are used. Different speeds of the cantilever, indenting into the substrate, in the z direction (z-rate) of 6.2, 10, 20, 50, and 100 Hz are used (corresponding to speeds of 5 × 10^–6, 8 × 10^–6, 1.6 × 10^–5, 4 × 10^–5, and 8 × 10^–5 m/s, respectively). The experiment is carried out in a typical laboratory environment (23 °C and 50% relative humidity). High-density diamond-like carbon tips with a spherical shape are utilized (supplied from Nanotools company, Germany). The cantilever has a nominal length of 450 μm, width of 50 μm, thickness of 2 μm with a spring constant of 0.2 N/m, and resonance frequency of 13 kHz. The tip radius is 20 ± 5 nm. Additionally, the spring constant of each cantilever is determined with the common thermal noise method.

Results and Discussion

Effects of Temperature and Interfacial Interaction

To perform nanoindentation simulations at large length and time scales, we use an atomistically informed PDMS CG model, which was developed with an energy-renormalization (ER) approach. The nanoindentation model is depicted in Figure d. The nanoindentation simulation process consists of both loading and unloading phases during which we measure the force applied to the indenter as a function of the indentation depth. This measurement, conducted without altering the interaction between the indenter and the polymer, results in the nanoindentation curve shown in Figure a at 300 K. We introduce the coefficient α_int, which controls the interfacial interaction strength defined as ε_int = α_int × ε _ij , where ε_int denotes the interfacial interaction strength between the indenter and polymer and ε _ij denotes the cohesive interaction strength between polymer chains. For special cases of α_int, when α_int = 0, there is no interaction between the indenter and the polymer. When α_int = 1, the indenter-polymer interaction strength remains comparable with polymer-polymer interaction strength. For α_int >1, the indenter-polymer interaction becomes stronger than the polymer-polymer interaction, indicating stronger adhesion between the indenter and polymer.

2.

(a) Force–depth curve from nanoindentation of a linear PDMS CG model at 300 K with an initial interaction strength (α_int = 1) between the polymer and the indenter. (i–vi) Snapshots of the model at different indentation depths. (b–g) Force–depth curves under varying temperatures and interaction strengths, with force values scaled by α_int.

To elucidate the indentation behavior at different depths, we identify six critical points on the curve to capture configurations of the system. At point (i), the indenter is separated from the film, resulting in zero applied force. As the indenter approaches the film surface, the force reaches a minimum (negative value) at point (ii), indicating the onset of interaction between the indenter and the film. With increasing indentation depth, polymer chains start to adsorb onto the indenter surface. At point (iii), the adhesive force from the film and the repulsive force due to film deformation reach equilibrium, resulting in a zero net force. As the indentation depth increases further, the repulsive force dominates, and the force rises to its maximum at point (iv). During the unloading phase, the force on the indenter decreases as the indentation depth decreases. However, the adhesive force maintains contact between the indenter and film, resulting in a negative force. As the indentation depth further decreases, the force reaches its maximum negative value at point (v), when the adhesive force reaches its peak. Finally, at point (vi), the indenter lifts off the thin-film surface. A residual interaction force, resulting from the adsorption of a small number of polymer chains onto the indenter surface, stabilizes at a low negative value close to zero.

To study how indenter–film interactions affect adhesion, we vary the coefficient α_int, which controls the interaction strength. A comparative analysis of Figure b–g reveals a pronounced evolution in curve characteristics as α_int increases, ranging from 0.2 to 10. With α_int set to 0.2, the adhesive force generated by the interaction is minimal (Figure b). During loading, both the occurrence and magnitude of negative adhesive force are limited, with negligible variation across temperatures. However, as α_int increases, the negative force appears earlier and spans a broader range. Specifically, in Figure f, with α_int = 5, a negative force emerges as early as 10 Å above the film surface and persists until the indenter penetrates 15 Å into the film. This indicates that stronger interfacial interactions substantially enhance adhesion and extend its range of influence. Another notable observation is that when α_int exceeds 2, the loading curves transition into a concave shape, which results from the increased adhesion between the indenter and polymer as the penetration depth increases. The adhesive force dominates the process, extending the range of negative force until equilibrium with the repulsive force is achieved at greater depths. Although temperature modulates the overall force–depth response, the qualitative trends are preserved at fixed α_int value. This consistency underscores the predominance of interaction strength over temperature in determining the adhesive behavior of silicone polymers.

To provide a clearer and more rigorous comparison of the effects of temperature and α_int on the nanoindentation behavior, we examine the maximum force (F _max) exerted when the indenter reaches its deepest penetration during the loading phase, the maximum detachment (or pull-off) force (F _c), taken as the absolute value of force magnitude during the unloading phase, and the critical energy release rate (G _c) as calculated by eq . As depicted in Figure a–c, despite variations in α_int, the trend with respect to temperature remains consistent, exhibiting a decreasing pattern as the temperature increases. This observation indicates that the polymer chains become more mobile at elevated temperatures, facilitating easier deformation and flow. Consequently, less force is required to press the indenter to the same depth in this case.

3.

(a–c) Variation of F _max, F _c, and G _c with temperature at different interaction strengths (α_int). (d–f) Dependence of F _max, F _c, and G _c on interaction strength (α_int) under various temperatures. (g–i) ΔE _pe, ΔE _nonbonded, and ΔE _bonded as a function of α_int at different temperatures.

In Figure d–f, at fixed temperature, we observe that the trend in F _max is inversely related to α_int. This indicates that stronger interactions result in greater adsorption of polymer chains onto the indenter, generating an additional pulling force upon indentation. This additional force aids the indenter in penetrating the film, thereby reducing the actual external force required and consequently decreasing F _max as the interaction strength increases. Conversely, F _c and G _c initially increase with α_int. However, once α_int exceeds a certain threshold, the detachment force stabilizes, because the surface of the indenter is almost completely covered with the polymer. At this stage, the F _c mainly reflects the force needed to detach the adsorbed chains from the film. Despite this stabilization, the observed trends with temperature variations persist, indicating that the interaction strength has a more substantial impact on nanoindentation behavior compared to temperature. These results are consistent with the trends observed in Figure , reinforcing the observations that both temperature and interaction strength significantly influence nanoindentation behavior.

To clarify the energy variations during the indentation process, we further analyze the total potential energy difference (ΔE _pe), the nonbonded energy difference (ΔE _nonbonded), and the bonded energy difference (ΔE _bonded) at the point of maximum indentation depth, relevant to the initial pre-indentation stage. The results are shown in Figure g–i. These energy differences are governed by the equation: ΔE _pe = ΔE _nonbonded + ΔE _bonded, where the nonbonded energy represents the nonbonded interaction energy between polymer beads, and the bonded energy encompasses bond, angular, and dihedral energies. As α_int increases, the total potential energy difference decreases. This phenomenon can be attributed to the enhancement of nonbonded interactions, which bring the polymer chains and the indenter surface closer together. As a result, the polymer chains become less conformationally flexible, forming stiffer structures that require more energy to maintain these conformations. Although the bonded energy difference increases with higher α_int values, it remains relatively small compared to the dominant nonbonded energy difference contribution under strong interaction conditions. At higher temperatures, thermal fluctuations and increased conformational entropy reduce the energy required to maintain the conformation of polymer chains in the system, leading to smaller bond energy differences. However, the decrease in the total potential energy becomes less significant at higher temperatures because the enhanced chain mobility inhibits the system from transitioning to a lower energy state. Therefore, the interplay among temperature, interaction strength, and energy difference reveals a complex equilibrium in the energetics of the system during indentation.

Interfacial Chain Adsorption

To understand polymer adsorption on the indenter surface under various conditions, we analyze the adsorption behavior during the unloading phase when the indenter is lifted up to 25 Å above the film surface. The reference surface is defined at the lowest point of the indenter, and the polymer below this surface is rendered with increased transparency to enhance clarity (Figure a). At fixed temperature, increasing the interaction strength leads to a significant rise in the number of adsorbed repeating units, eventually resulting in complete coverage of the indenter surface. As temperature rises, the mobility and flexibility of the polymer chain increase, causing polymer chains to transition from an elongated state to a coiled state and leaving the indenter surface almost completely covered. At 350 K, the adsorbed repeating units on the indenter surface further attract free repeating units from the surrounding film. The statistical analysis of adsorbed chain segments (i.e., number of repeating units) above the lowest point of the indenter tip (Figure b) indicates negligible adsorption at lower temperatures (e.g., below 250 K) and weaker interactions (e.g., α_int <1). However, the number of adsorbed repeating units rises with increasing temperature and interaction strength. A comparative analysis indicates that although both temperature and interaction strength influence repeating unit adsorption, they do so through distinct mechanisms. Interaction strength directly determines the adsorption behavior between polymer chains and the indenter surface, whereas temperature primarily affects polymer mobility and conformation.

4.

(a) Adsorbed polymer chain segments on the indenter under different interfacial interaction strength α_int and temperatures. The two dashed lines indicate the reference surface at z = 0, defined by the indenter’s lowest point. (b) Adsorbed repeating unit count as a function of α_int at various temperatures. (c) Spatial distribution of local stiffness k _B T/⟨u ²⟩ as a function of film depth at different temperatures. (d) Distribution of local von Mises stresses σ_v at varying temperatures for different α_int values, where the indenter reaches the maximum depth. The gray regions indicate areas beyond the cutoff range (15 Å) of interfacial interactions between the indenter and chains. Corresponding stress histograms are shown in Figure S3.

To understand the stress heterogeneity, we calculate the local von Mises stress (σ_v) of polymer chains adjacent to the indenter at maximum indentation depth, as shown in Figure d. The analysis considers a spherical region with a radius of 50 Å from the center of the indenter. Elevated σ_v values occur at stronger interfacial interactions, primarily due to increased interfacial adhesion between the indenter and thin film under such conditions. Enhanced adhesion hinders the motion of the indenter during loading, resulting in a concentrated stress distribution within the contact region between the indenter and thin film. To further analyze the effect of temperature, we refer to DWF. Previous studies have shown that the DWF ⟨u ²⟩ can be used to predict the dynamical heterogeneity of glassy states in confined systems. , In this work, ⟨u ²⟩ represents MSD at t = 4 ps, capturing the harmonic vibrations of chain segments within a cage formed by surrounding particles. It follows the relation ⟨u ²⟩ ∼ k _B T/k, where k _B is the Boltzmann constant, T is the temperature, and k is the local elastic constant. Therefore, the local molecular stiffness of the polymer can be approximated as k _B T/⟨u ²⟩. Figure c presents the distribution of k _B T/⟨u ²⟩ along the z-axis of the film at different temperatures. The results clearly show that k _B T/⟨u ²⟩ increases both at the surface and within the film as the temperature decreases. Accordingly, at lower temperatures a greater force is required to displace chains near the indenter to achieve the same penetration depth. Reduced chain mobility impedes deformation, leading to a higher σ_v within the contact zone. Based on the data presented in Figures and , we observe that as the temperature increases, polymer chains exhibit enhanced mobility, allowing them to better conform to the indenter (qualitative increase in the contact area as shown in Figure a). However, F _c and G _c decrease with increasing temperature (Figure a–c), primarily due to the softening of the material and the enhanced mobility of polymer chains, which exhibit greater conformational flexibility and are thus more easily deformed under external forces. Together, these results reveal the complex interplay between polymer interfacial adhesion and mechanical response under elevated temperature conditions.

Effect of Cross-Linking on Interfacial Adhesion

We next investigate the impact of cross-link density on the interfacial adhesion behavior between the indenter and PDMS film. The cross-link density (c) of the system in our simulations is defined as the ratio of the number of new bonds formed to the maximum number of bonds that could be formed. As c increases, the F _max exhibited during indentation rises correspondingly (Figure b). The increase in c reduces the mobility of polymer chains, thereby limiting their movement and decreasing the number of polymer repeating units available for adsorption onto the indenter surface. Consequently, the depth at which the adhesion-induced pulling force counteracts the indentation force shifts from approximately 7 Å at c = 0% to about 5 Å at c = 80%, quantitatively reflecting the weakening of the "pull-in" effect in highly cross-linked films. Figure d shows a pronounced decrease in the number of repeating units adsorbed on the indenter surface as c increases during the detachment of the indenter from the film. Furthermore, the increase in c enhances the overall molecular stiffness of the film as measured by k _B T/⟨u ²⟩ (Figure f). This increased molecular stiffness further impedes polymer mobility, resulting in stress concentration when the indenter approaches its deepest penetration point, leading to an elevated σ_v value. The reduced mobility of polymer chains near the interface leads to an upward trend of F _c and G _c with increasing c (Figure b). Some of these observed trends are supported by previous studies. − For example, Ogawa et al. conducted AFM nanoindentation experiments and found that epoxy adhesive interfaces with lower cross-link density exhibited reduced stiffness and increased molecular mobility, making them more susceptible to deformation under applied stress. Furthermore, their study demonstrated that lower cross-link density leads to a decrease in indentation load and yield strength. These findings further support the impact of cross-link density on interfacial adhesion mechanics, indicating that an increase in cross-link density not only reduces surface adsorption but also enhances stress concentration at the interface, ultimately elevating F _c and G _c.

5.

(a) Nanoindentation force–depth curves for PDMS thin films with varying cross-link densities c (%) at 300 K with α_int = 1. (b) Curves of F _max, F _c, and G _c as a function of cross-link density c (%). (c) Number of adsorbed repeating units as a function of cross-link density c (%). (d) Visualization of the adsorbed polymer repeating unit on the indenter at c = 0%, 40%, and 80% at 300 K with α_int = 1. (e) Distribution of von Mises stresses σ_v at c = 0%, 40%, and 80% of the film when the indenter reaches the maximum depth. The gray regions indicate areas beyond the cutoff range (15 Å) of interfacial interactions between the indenter and chains. Corresponding stress histograms are shown in Figure S4. (f) Spatial distribution of local stiffness k _B T/⟨u ²⟩ for different cross-link densities c (%) as a function of film depth.

AFM Measurement and Analysis

In this section, we demonstrate how experimental AFM tips quickly become covered in the polymer, in agreement with the predictions of the simulations. The goal of a typical AFM adhesion measurement is to collect a force–displacement curve and then to use a mechanics model to extract material properties such as the modulus or the energy release rate from an experiment. Figure d shows the maximum force (pull off force) as a function of the velocity of the tip for three different locations on a sample. The results show the mean value calculated from 5 measurements, and the error bars represent the standard deviation. We also scanned the tip-check sample before and after each location is scanned (Figure b). Of note, each scan shows different peak force values, which would imply each location has different material properties. More precisely, a two-sample t-test gives a value of 41.3 when comparing the first set with the third set, clearly beyond the 0.05 significance level. The t-test indicates that these measurements cannot be considered the same. Of course, several parameters can influence adhesion measurement, including surface roughness, − , testing geometry, − ,,,,− and experimental protocol (e.g., contact time, force, and area). , To ensure reproducible and reliable results, these parameters must be carefully controlled in AFM testing. Since all the experiments in this work are carried out in the same laboratory using systematic conditions, it is unlikely that roughness or protocol is the issue.

Figure b illustrates four topographical images of the tip-check sample scanned by the tip over the course of the experiment. Specifically, Figure b­(i) shows the tip-check sample scanned with the tip before starting the adhesion experiment; Figures b­(ii) and b­(iii) are collected after the first and second scans, respectively, and b­(iv) is collected at the end of the experiment. It is obvious that the level of clarity in the topography images of the tip check sample is reduced by continuing the use of the tip during adhesion characterization. We additionally used SEM imaging to examine the tip before and after the experiment. Figure e shows that the as-received tip has a radius of ∼20 nm, but the tip becomes significantly larger after the experiment having “picked-up” significant material from the soft elastomeric sample, which has been illustrated in Figure d. There is some precedent for tips changing during AFM scanning, and it is well known that enlarged tips correlate with increased adhesion forces. For example, Skulason et al. have shown that changes in the radius of the tip due to wear, which is not uncommon during scanning in AFM, have linear correlation with the pull-off force (force of adhesion) in agreement with the JKR model. It is also known that changes in humidity can affect adhesion measurements, ,, but these occur more often with hydrophilic substrates. In our work, each consecutive scan is conducted at the same fixed humidity level, so this should not affect the results. Experiments with larger probes clearly reveal that fluid is exuded under pressure.

Acknowledging material accumulation on the tip clearly alters the interpretation of the peak forces measured by AFM or simulation and highlights two unavoidable conclusions. First, beyond the initial contact with the substrate, the interaction is not tip-to-silicone but silicone-to-silicone, further complicated by chain entanglement and other physical interactions during indentation and pull-off. Second, the tip geometry shifts from a sharp 20 nm radius ridged indenter to a much larger radius and a softer indenter. Both ideas agree with observations made through experiments and with the simulations.

Taking into account the increased tip radius by using a fit to eq to determine the contact radius (modulus is assumed to be its bulk value), the average G _c was found to be 0.16 ± 0.05 N/m, with little or no speed dependence. This value is higher than the values reported by Petroli et al. (colloidal probe AFM) and Vaenkatesan et al. (PDMS lens) but lower than the values reported by Choi (PDMS lens). We assume that sample differences make up some of the reported variation but also note that the larger probes would experience less change in radius due to polymer build-up and thus are likely more accurate. In short, our radius-corrected measurement yields values similar enough to those published to validate our basic premise: material build-up must be accounted for in analysis of AFM measurements of elastomers that contain some portion of uncross-linked chains or with uncross-linked polymer melts.

Moreover, the significance of the probe material and its interaction with the polymer surface has been repeatedly demonstrated in both our experiments and previous studies. For instance, in our experiments using diamond-like tips, we observed clear pull-in phenomena. Lin and Kim used steel microsphere probes to measure PDMS films on silicon substrates and found that the tip material significantly affects adhesion behavior. Wang et al. observed pronounced pull-in and pull-off events with diamond tips, further demonstrating that differences in surface energy and chemical properties of the tip material can alter adsorption between the tip and polymer chains, thereby affecting the overall interfacial adhesion. Additionally, Bellido-Aguilar et al. employed glass and ice lenses as probes in JKR indentation tests to measure adhesion between PDMS elastomers and glass or ice and similarly found that the measured adhesion energy strongly depends on the probe material. Collectively, these experimental observations qualitatively support the simulation predictions presented in this study.

In fact, the clear material accumulation observed on the AFM tip during indentation experiments directly validates the chain adsorption behavior predicted by the simulations. Although there are differences between the simulation and experimental setupssuch as the indenter model and interfacial parametersboth approaches consistently reveal the key phenomenon of chain adsorption and material buildup, thus further strengthening the correlation between experimental and simulation results.

One possible confounding factor in the experiments is that the portion of free chains in the elastomer is not well controlled. In addition, chain scission under load cannot be ruled out as an additional contributor to adsorption on the AFM tip. Each sample created using the steps outlined in our materials section has the same sol:gel ratio (here about 5% sol) as measured by weight; however, the sol content contains uncross-linked chains, moderator, and cross-linker. Furthermore, partially cross-linked material is also present. A more detailed study could isolate this content by extracting an elastomer and then reswelling it with a controlled molecular weight PDMS fluid. Differing cross-link densities could also broaden understanding of the specific interactions between a PDMS-covered tip and an elastomer sample. Regardless of the exact makeup of the material coating the AFM tip, our main point is clear: material coats an AFM tip during measurement of silicone elastomers, and this buildup of material affects the measured adhesive forces in a similar way to what is found in simulations.

Conclusions

In this study, we employ CG-MD simulations, together with validating experiments, to investigate the interfacial adhesion of PDMS. Specifically, we examine how indenter tip-polymer interfacial interaction strength, temperature, cross-link density influence the interfacial adhesion behavior of polymer chains. Our simulations reveal that, as the interfacial interaction strength increases, polymer chains at the film surface increasingly adsorb and wrap around the indenter, causing both the detachment force and the critical energy release rate to increase until they plateau. Higher temperatures enhance the polymer mobility, facilitating chain rearrangement and migration at the interface. This results in a greater contact area and a higher probability of polymer adsorption onto the indenter. In contrast, increased cross-link density restricts polymer chain motion, reducing adsorption but simultaneously increasing resistance to detachment due to limited chain mobility. Experimental characterization based on AFM and SEM analyses confirms polymer chains transfer onto the probe, further highlighting the role of polymer adsorption in adhesion measurement. By integrating computational and experimental approaches, our study systematically elucidates the pivotal roles of interaction strength, temperature, and cross-link density in governing the contact mechanics during nanoindentation. This comprehensive analysis provides valuable insights into how these parameters modulate the adhesion behavior and mechanical response of polymer films under nanoindentation.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.macromol.5c01840.

• JKR contact mechanics model; dynamic mechanical analysis; bonded potentials of PDMS CG models; nonbonded potentials for PDMS CG models; ε­(T) and σ­(T) vs temperature; stress histograms vs temperature and α_int; and stress histograms vs cross-link density at 300 K (PDF)

The authors declare no competing financial interest.