# Peridynamic Simulations of Nanoindentation Tests to Determine Elastic Modulus of Polymer Thin Films

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## Abstract

This study combines atomic force microscope (AFM) nanoindentation tests and peridynamic (PD) simulations to extract the elastic moduli of polystyrene (PS) films with varying thicknesses. AFM nanoindentation tests are applied to relatively hard PS thin films deposited on soft polymer (polydimethylsiloxane (PDMS)) substrates. Linear force versus deformation response was observed in nanoindentation experiments and numerical simulations since the soft PDMS substrate under the stiff PS films allowed bending of thin PS films instead of penetration of AFM tip towards the PS films. The elastic moduli of PS thin films are found to be increasing with increasing film thickness. The validity of both the simulation and experimental results is established by comparison against those previously published in the literature.

## Keywords

Atomic force microscopy (AFM) Nanoindentation Peridynamic theory Elastic moduli## 1 Introduction

Polymer thin films can significantly alter the properties of surfaces such as corrosion resistance, wettability, adhesion, biocompatibility, morphology, and conductivity. Hence, these films are employed in many applications in nanotechnology, automotive, biomedicine, and energy conversion. The properties and the performance of these films are correlated to the film geometry, film chemical composition, and surface properties. Among these, mechanical properties are particularly important which determine reliable use of these materials systems under different types of loading conditions.

Several mechanical characterization techniques have been introduced in the literature for mechanical characterization of thin films such as wrinkling method [1, 2], resonance method [3, 4], and nanoindentation [5, 6, 7, 8, 9, 10]. Among all, nanoindentation is the most commonly used technique for mechanical characterization of thin films due to the ease of the experiment. Atomic force microscope (AFM) can be utilized for nanoindentation of soft thin films [8, 9, 10] and nanoindenter is preferred for the characterization of hard thin films [5, 6, 7]. However, as the thin film thickness becomes very small, the resolution of the equipment for the obtainable indentation depth becomes critical, and it determines the minimum thickness of the film that can be tested. Also, the effect of the substrate underneath the thin film becomes significant. Therefore, accurate extraction of mechanical properties of the film depends on the nature of the substrate modeling. The development of a new methodology becomes essential for mechanical characterization of ultra-thin films.

Although nanoindentation of ultra-thin films may pose challenges because of their thickness limitation, these films can be tested under bending loads if deposited on soft substrates without any limitation on the thickness. Bending tests of thin films can be performed by using an AFM which is preferred over the traditional nanoindenters because it permits the application of extremely small indentation forces through the tip of the AFM cantilever, thus leading to truly nanometer scale deformations. However, the extraction of the mechanical properties requires accurate characterization of the soft substrate underneath the film and modeling of the bending experiment.

Finite element analysis coupled with nanoindentation experiments have been used to evaluate/extract material properties of the thin films previously [11, 12, 13, 14]. For ultra-thin films having nano-scale thickness, the commonly accepted simulation technique in the literature is Molecular Dynamics Simulations (MDS). In MDS, each individual atom is modeled, and a suitable interatomic potential is used in order to define the interaction between the atoms of the nanostructure. However, modeling each atom proves to be extremely demanding from a computational standpoint [15]. An alternative to MDS is peridynamics which can be considered as the continuum version of the MDS. Therefore, this study utilizes the peridynamic theory introduced by Silling [16]. Peridynamics is a version of the non-local theory which was first introduced by Eringen and Edelen [17] and Kroner [18]. Peridynamics has been applied for the solution of many different problems and material systems including crack branching [19], plasticity [20], viscoelasticity [21], viscoplasticity [22], composite materials [23, 24], nanowires [25], and bounded and unbounded domains [26].

This study presents a combined experimental and computational approach for determining the elastic modulus of relatively stiff ultra-thin films deposited on soft substrates. An in-house AFM is used for the mechanical tests and Peridynamic theory for the numerical simulations. The next two sections describe the experimental setup and a description of the PD theory. Subsequently, the experimental data and numerical simulation results are compared in order to extract mechanical properties of the thin films.

## 2 Experimental Details

### 2.1 Specimen Preparation

The PDMS samples are prepared using a commercially available silicone encapsulant kit (Sylgard 184). Resin is first mixed with the curing agent (10:1 mass ratio) manually with a glass stirring rod for approximately 15 min and then the mixture is poured over a pre-cleaned surface of a silicon wafer. PDMS is then degassed by placing it in a vacuum chamber and subjecting it to pressures below ~ 1000 mTorr for 3 to 5 min. The curing is completed at atmospheric pressure at 50–60 °C for 4 to 5 h. After the curing is completed, PDMS sample is peeled off from the wafer surface. The wafer-contacted side of the PDMS is used in the nanoindentation experiments due to the low roughness of the surface.

### 2.2 AFM Nanoindentation Tests

Simulations of the AFM indentation experiments are performed by employing the peridynamic theory in search of a material constant that yields the best fit to the measured force-deformation relation. While treating the tip of the AFM as a rigid indenter, this inverse approach permits the extraction of the elastic moduli of PS polymer films.

### 2.3 Peridynamic Formulation

*ρ*and

**u**represent the density and the displacement of the material point,

*t*denotes time,

**f**is the bond force between material points

**x**

*a*nd

**x**

^{′}(see Fig. 5), and

*H*

_{x}is the horizon of the material point

**x**. For an isotropic material, the bond force can be defined as (see Fig. 4B)

*ξ*is the bond vector, i.e.,

**x**

^{′}

**− x**,

*η*represents the relative displacement between two material points, i.e.,

**u**(

**x**

^{′},

*t*) −

**u**(

**x**,

*t*), and stretch can be defined as

*s*= (|

*ξ*

**+**

*η*| − |

*ξ*| )/ ∣

*ξ*∣. Moreover, the bond constant,

*c*can be expressed in terms of Young’s modulus of the material and the horizon size, δ, as [28].

### 2.4 Peridynamic Contact Analysis

*i*at

*t*+

*∆t*

**,**\( {\overline{V}}_i^{t+\Delta t} \)

**,**can be calculated as

*i*at

*t*+

*∆t*, \( {\mathbf{u}}_i^t \) is the displacement of the material point

*i*at

*t*, and

*∆t*is the time increment. As a result of the contact between indenter and target material, a reaction force occurs. The contribution of the material point

*i*to the total reaction force at time

*t*+

*∆t*, \( {F}_i^{t+\Delta t} \), can be defined as

*i*at time

*t*+

*∆t*before relocating the material point

*i*,

*ρ*

_{i}and

*V*

_{i}represent the density and volume of the material point

*i*, respectively. Finally, the total reaction force on the indenter at time

*t*+

*∆t*,

*F*

^{t + ∆t}, can be obtained by summing up the contributions of all material points inside the indenter as,

*N*is the total number of material points in the target material.

## 3 Results

*d*= 200 nm and horizon radius of

*δ*= 603 nm. The indenter velocity is

*v*

_{0}= 20 m/s. While searching for the elastic modulus, a simple optimization algorithm is used to minimize the difference between the measured and computed force-indentation depth relations. The elastic modulus resulting the best correlation was extracted as the correct elastic modulus for the material. Figure 8 shows the measured force-indentation depth and its PD simulation using the extracted value of elastic modulus. The correlation between the measurement and simulation is remarkable. In this analysis, PDMS elastic modulus is extracted as 10 MPa which compares well with the previously published studies for PDMS [30].

*v*

_{0}= 10 m/s due to smaller thickness of PS films compared to the bulk PDMS. The comparison of force-deformation response obtained from the PD simulation against the measurements for the PS film with a thickness of 1100 nm is shown in Fig. 9, which exhibits a very good agreement. PD simulations confirm the linear material response of deposited PS films measured in the tests.

## 4 Conclusions

A new technique for extracting elastic moduli of ultra-thin films based on a combined experimental and computational method is demonstrated. The stiff/hard material is deposited on a soft substrate with known material properties. Combined bending/indentation deformation of the material system allows accurate AFM measurements and effective extraction of material properties via peridynamic theory. Unlike the conventional indentation experiments, testing of stiff films deposited on soft substrates is not limited by sample thickness and that mechanical properties of ultra-thin films can be reliably characterized.

The force-deformation measurements of the bulk PDMS and PS thin-film specimens deposited on bulk PDMS substrates are compared against the PD simulations to extract mechanical properties of bulk PDMS and PS thin films. The extracted values for PDMS and thin-film PS are consistent with the previous research from the literature. Both experiments and simulations clearly indicate that deformation type is indentation for soft PDMS substrates but it occurs via bending on relatively hard PS films deposited on soft PDMS substrates.

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