ORIGINAL ARTICLES

Peridynamic Simulations of Nanoindentation Tests to DetermineElastic Modulus of Polymer Thin Films

Emrah Celik1 & Erkan Oterkus2 & Ibrahim Guven3

Received: 25 October 2018 /Accepted: 8 January 2019 /Published online: 12 March 2019# Springer Nature Switzerland AG 2019

AbstractThis study combines atomic force microscope (AFM) nanoindentation tests and peridynamic (PD) simulations to extract theelastic moduli of polystyrene (PS) films with varying thicknesses. AFM nanoindentation tests are applied to relatively hard PSthin films deposited on soft polymer (polydimethylsiloxane (PDMS)) substrates. Linear force versus deformation response wasobserved in nanoindentation experiments and numerical simulations since the soft PDMS substrate under the stiff PS filmsallowed bending of thin PS films instead of penetration of AFM tip towards the PS films. The elastic moduli of PS thin films arefound to be increasing with increasing film thickness. The validity of both the simulation and experimental results is establishedby 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 ofsurfaces such as corrosion resistance, wettability, adhesion,biocompatibility, morphology, and conductivity. Hence,these films are employed in many applications in nanotech-nology, automotive, biomedicine, and energy conversion.The properties and the performance of these films are cor-related to the film geometry, film chemical composition,and surface properties. Among these, mechanical proper-ties are particularly important which determine reliable useof these materials systems under different types of loadingconditions.

Several mechanical characterization techniques have beenintroduced in the literature for mechanical characterization ofthin films such as wrinkling method [1, 2], resonance method

[3, 4], and nanoindentation [5–10]. Among all, nanoindenta-tion is the most commonly used technique for mechanicalcharacterization of thin films due to the ease of the experi-ment. Atomic force microscope (AFM) can be utilized fornanoindentation of soft thin films [8–10] and nanoindenter ispreferred for the characterization of hard thin films [5–7].However, as the thin film thickness becomes very small, theresolution of the equipment for the obtainable indentationdepth becomes critical, and it determines the minimum thick-ness of the film that can be tested. Also, the effect of thesubstrate underneath the thin film becomes significant.Therefore, accurate extraction of mechanical properties ofthe film depends on the nature of the substrate modeling.The development of a new methodology becomes essentialfor mechanical characterization of ultra-thin films.

Although nanoindentation of ultra-thin films may posechallenges because of their thickness limitation, these filmscan be tested under bending loads if deposited on soft sub-strates without any limitation on the thickness. Bending testsof thin films can be performed by using an AFM which ispreferred over the traditional nanoindenters because it permitsthe application of extremely small indentation forces throughthe tip of the AFM cantilever, thus leading to truly nanometerscale deformations. However, the extraction of the mechanicalproperties requires accurate characterization of the soft sub-strate underneath the film and modeling of the bendingexperiment.

* Emrah Celike.celik@miami.edu

1 Mechanical and Aerospace Engineering Department, University ofMiami, Coral Gables, FL, USA

2 Department of Naval Architecture, Ocean and Marine Engineering,University of Strathclyde, Glasgow, UK

3 Department of Mechanical and Nuclear Engineering, The VirginiaCommonwealth University, Richmond, VA, USA

Journal of Peridynamics and Nonlocal Modeling (2019) 1:36–44https://doi.org/10.1007/s42102-019-0005-4

Finite element analysis coupled with nanoindentation ex-periments have been used to evaluate/extract material proper-ties of the thin films previously [11–14]. For ultra-thin filmshaving nano-scale thickness, the commonly accepted simula-tion technique in the literature is Molecular DynamicsSimulations (MDS). In MDS, each individual atom ismodeled, and a suitable interatomic potential is used in orderto define the interaction between the atoms of the nanostruc-ture. However, modeling each atom proves to be extremelydemanding from a computational standpoint [15]. An alterna-tive to MDS is peridynamics which can be considered as thecontinuum version of the MDS. Therefore, this study utilizesthe peridynamic theory introduced by Silling [16].Peridynamics is a version of the non-local theory which wasfirst introduced by Eringen and Edelen [17] and Kroner [18].Peridynamics has been applied for the solution of many dif-ferent problems and material systems including crackbranching [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 compu-tational approach for determining the elastic modulus of rela-tively stiff ultra-thin films deposited on soft substrates. An in-house AFM is used for the mechanical tests and Peridynamictheory for the numerical simulations. The next two sectionsdescribe the experimental setup and a description of the PDtheory. Subsequently, the experimental data and numerical

simulation results are compared in order to extract mechanicalproperties of the thin films.

2 Experimental Details

2.1 Specimen Preparation

Two different types of samples are prepared: (i) polydimeth-ylsiloxane (PDMS) in the form of bulk sample, and (ii) thin-film polystyrene (PS) deposited on the bulk PDMS. The spec-imen configurations of bulk PDMS and thin PS film areshown in Fig. 1A. The thickness of the bulk PDMS is approx-imately 5 mm while the PS film thickness is varied between700 nm and 10 μm.

The PDMS samples are prepared using a commerciallyavailable silicone encapsulant kit (Sylgard 184). Resin is firstmixed with the curing agent (10:1 mass ratio) manually with aglass stirring rod for approximately 15 min and then the mix-ture is poured over a pre-cleaned surface of a silicon wafer.PDMS is then degassed by placing it in a vacuum chamberand subjecting it to pressures below ~ 1000 mTorr for 3 to5 min. The curing is completed at atmospheric pressure at50–60 °C for 4 to 5 h. After the curing is completed, PDMSsample is peeled off from the wafer surface. The wafer-contacted side of the PDMS is used in the nanoindentationexperiments due to the low roughness of the surface.

Fig. 1 a Schemeatic of fabricatedspecimens for nanoindentation. bScanning electronic microscopeimage of PS film on PDMSsubstrate

Fig. 2 AFM surface profiles of aPDMS substrate and b PS filmedge on PDMS susbtrate

J Peridyn Nonlocal Model (2019) 1:36–44 37

The thin-film PS samples are prepared by spin coatingof PS on bulk PDMS substrate. Solid polystyrene particlesare first dissolved in toluene. The solutions with differentPS concentrations (2 to 10 wt%) are prepared in order toachieve different PS film thicknesses. PS solutions are thenspin coated on PDMS substrate at a spin velocity of 3000revolutions per minute. The thickness of the PS film is

measured to be between 700 nm and 10 μm using AFM.The SEM micrograph of the thickest film on PDMS sur-face is shown in Fig. 1B. Roughness and thickness of PSand PDMS surfaces are characterized by AFM imaging(Fig. 2). Figure 2A shows the smooth surface of wafer-contacted side of the PDMS specimen where the roughnessis measured to be less than 50 nm using the AFM.

a bFig. 3 A SEM image of the AFMcantilever used innanoindentation process. BSchematic of force andindentation calculations from theraw AFM data

Fig. 4 a Damaged andundamaged representation of aperidynamic bond. b Force-stretch relation for a peridynamicmaterial interaction

Fig. 5 Interaction of a materialpoint with its neighboring pointsin Peridynamic model

38 J Peridyn Nonlocal Model (2019) 1:36–44

Figure 2B shows the thickness measurement using AFMwhere the step height between the PDMS (dark, lowerside) and PS (higher, pink side) surfaces represents thethickness of PS films.

2.2 AFM Nanoindentation Tests

Both the bulk PDMS and PS thin film specimens are indentedwith the tip of a vendor-calibrated Mikromasch ultra-sharpcantilevers as shown in Fig. 3A. Raw piezo displacement ver-sus cantilever deflection data are converted into force versusindentation response using standard methodology as de-scribed earlier [27]. Briefly, cantilever stiffness is multipliedby the cantilever deflection to determine the applied force onthe surface. Total indentation on the sample is equal to thedifference between the rigid sample and actual (PS/PDMS)sample deflection response as denoted in Fig. 3B. The repeat-ability of the experimental tests is established by performingthe indentation test three consecutive times on each type ofsample.

Simulations of the AFM indentation experiments areperformed by employing the peridynamic theory insearch of a material constant that yields the best fit tothe measured force-deformation relation. While treatingthe tip of the AFM as a rigid indenter, this inverse ap-proach permits the extraction of the elastic moduli of PSpolymer films.

2.3 Peridynamic Formulation

In this study, bond-based peridynamics is utilized forcomputational analysis. According to bond-basedperidynamics, material points are interacting with eachother in a non-local manner. The interaction (bond) forcesbetween material points are assumed as pairwise, equal,and opposite to each other. It is also assumed that there isno interaction between material points if the distance be-tween them is greater than a specific distance (horizon).During the solution process, stretch of each bond is mon-itored and if the stretch value exceeds a critical stretchvalue, then the bond is considered to be broken as shownin Fig. 4A. Peridynamic equations are always valid re-gardless of the broken bonds. The damage parameter foreach material point can be simply defined as the ratio ofthe number of broken bonds and the number of bonds inthe undeformed configuration associated with that materi-al point. Therefore, it is straightforward to represent dam-age as opposed to classical continuum mechanics–basedapproaches. Moreover, in peridynamics, it is possible tospecify different properties to bonds representing inter-faces between different materials with respect to theirbulk material properties.

In bond-based peridynamics, the equation of motion can beexpressed as

ρ x; tð Þ ∂2u x; tð Þ∂t2

¼ ∫∙

Hx

f u x0; t

� �−u x; tð Þ; x0

−x; t� �

dVx0

þ b x; tð Þ ð1Þ

where ρ and u represent the density and the displacement ofthe material point, t denotes time, f is the bond force between

Fig. 6 Procedure to determine the reaction force on the indenter [29]

J Peridyn Nonlocal Model (2019) 1:36–44 39

material points x and x′ (see Fig. 5), and Hx is the horizon ofthe material point x. For an isotropic material, the bond forcecan be defined as (see Fig. 4B)

f u x0; t

� �−u x; tð Þ; x0

−x; t� �

¼ξþ ηξþ ηj j csð Þ; if s < s0

0 ; if s≥s0

8<: ð2Þ

in which ξ is the bond vector, i.e., x′ − x, η represents therelative 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 interms of Young’s modulus of the material and the horizon size,δ, as [28].

c ¼ 12E

πδ4: ð3Þ

2.4 Peridynamic Contact Analysis

In this study, the indenter is assumed as a rigid structure.Based on the peridynamic contact analysis approach for arigid indenter presented in [29], at each time step of thesimulation process, unphysical penetration of the indenterinside the target material should be prevented. If suchsituation occurs, material points inside the indenter aremoved to closest locations outside of the indenter surface(refer to Fig. 6). As a result of this relocation, the

modified velocity of the material point i at t + Δt, �VtþΔti ,

can be calculated as

�VtþΔti ¼ �utþΔt

i −uti� �

Δtð4Þ

where utþΔti is the modified displacement of the material point

i at t + Δt, uti is the displacement of the material point i at t,and Δt is the time increment. As a result of the contactbetween indenter and target material, a reaction force occurs.The contribution of the material point i to the total reactionforce at time t + Δt, FtþΔt

i , can be defined as

FtþΔti ¼ −1� ρi

�VtþΔti −VtþΔt

i

� �Δt

V i ð5Þ

where VtþΔti is the velocity of the material point i at time

t + Δt before relocating the material point i, ρi and Vi rep-resent the density and volume of the material point i, re-spectively. Finally, the total reaction force on the indenterat time t + Δt, Ft + Δt, can be obtained by summing up thecontributions of all material points inside the indenter as,

FtþΔt ¼ ∑N

i¼1FtþΔti λtþΔt

i ð6Þ

where

λtþΔti ¼ 1; if the point i is inside the indenter

0; if the point i is outside the indenter

�ð7Þ

and N is the total number of material points in the targetmaterial.

Fig. 7 Experimental force versusdeformation results

40 J Peridyn Nonlocal Model (2019) 1:36–44

3 Results

Force versus deformation measurements for bulk PDMS andthin PS films deposited on PDMS are shown in Fig. 7. The PSfilm thickness is varied as 700 nm, 1100 nm, 3100 nm, and10,000 nm. The representative force versus deformation datais obtained through averaging of the repeated tests with theusage of error bars. As observed in this figure, force changesnonlinearly as a function of sample deformation during theindentation of bulk PDMS suggesting a significant changein contact area between the AFM tip and the soft PDMS ma-terial. However, testing of all four PS thin films deposited onPDMS results in linear force-deformation variations. This is aclear indication that the deformationmode of hard PS thin filmis different than that of soft bulk PDMS and that the deforma-tion of hard PS thin film is dominated by film bending insteadof indentation. Figure 7 also indicates that as the thin filmthickness decreases, the amount of force to deform the sampledecreases due to the decrease in bending rigidity.

In order to validate the numerical simulation accuracy,PD analysis is first performed to model the measuredforce-deformation response of the bulk PDMS specimen.In this analysis, the following simulation parameters areused: total of 500,000 material points, grid spacing ofd = 200 nm and horizon radius of δ = 603 nm. The indentervelocity is v0 = 20 m/s. While searching for the elasticmodulus, a simple optimization algorithm is used to mini-mize the difference between the measured and computedforce-indentation depth relations. The elastic modulusresulting the best correlation was extracted as the correctelastic modulus for the material. Figure 8 shows the mea-sured force-indentation depth and its PD simulation usingthe extracted value of elastic modulus. The correlation be-tween the measurement and simulation is remarkable. In

this analysis, PDMS elastic modulus is extracted as10 MPa which compares well with the previously pub-lished studies for PDMS [30].

Next, the PD simulation of indentation experiment involv-ing thin-film PS deposited on PDMS is performed. For PDMSmaterial modulus, the extracted value from the previous sim-ulation is used. In addition, the same simulation parameters ofthe PDMS analysis are used for the PD analysis for consisten-cy in results. However, indenter velocity is reduced to v0 = 10m/s due to smaller thickness of PS films compared to the bulkPDMS. The comparison of force-deformation response ob-tained from the PD simulation against the measurements forthe PS film with a thickness of 1100 nm is shown in Fig. 9,which exhibits a very good agreement. PD simulations

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 100 200 300

]Nn[

ecrof

indenta�on [nm]

PD Theory

Experiment

Fig. 9 Comparison of the force vs. deformation results betweenexperimental and PD theory results on thin-film PS deposited on PDMSsample

Fig. 8 Comparison of forceversus deformation resultsbetween experimental and PDtheory results on bulk PDMSsample

J Peridyn Nonlocal Model (2019) 1:36–44 41

confirm the linear material response of deposited PS filmsmeasured in the tests.

The PD simulation of the deformed shape of thin PS filmdeposited on PDMS substrate is shown in Fig. 10. It is clearthat the deformation pattern, unlike the PDMS case, is a com-bination of local indentation and global bending of the PSlayer. Indentation pattern at the top surface of the PS filmunder the indenter is visible in the inlet of Fig. 10B. The tipindentation in the vicinity of the indenter is present and thebending of the PS thin film and deformation of the bottomsurface of the thin film is clearly visible. The deformed shapeof the PS thin film deposited on soft PDMS substrate in con-junction with the linear force-deformation response obtainedby experiments and PD simulations clearly indicates the bend-ing behavior of the relatively hard PS films.

The elastic moduli of the PS films were extracted in com-parison with AFM experiments and PD simulations. The var-iation of elastic modulus as a function of film thickness isgiven in Fig. 11. The figure denotes that the elastic modulusshows an increasing trend as the thickness of the film in-creases. This variation is consistent with the measurementsof Overney et al. [31] and observations of Teichroeb andForrest [32] arguing that the thin-film polystyrene is lessglassy than the bulk samples. Elastic modulus linearly in-creased as the film thickness increased. We expect that, asthe thickness increases, the thickness effect will vanish andthe elastic modulus of PS specimens will approach the bulkmodulus of PS which is reported as 3 GPa [33]. In literature,similar behavior of thin film modulus increases as a functionof film thickness and its saturation to the bulk material mod-ulus were commonly observed. The saturation thickness atwhich the film modulus equalized to that of the bulk valuewas found to be lower than 200 nm [34, 35] in some studies.However, similar to our study, continuous increase of elastic

0

400

800

1200

0 4000 8000 12000

Thickness (nm)

)a

pM

(s

ulu

do

Mcit

sal

E

Experiment

Linear Fit to Experimental Data

Fig. 11 Variation of elasticmodulus as a function of samplethickness

Fig. 10 Deformed configuration obtained by simulation using PD theoryon thin-film PS deposited on PDMS sample (a full view, b close-up view)

42 J Peridyn Nonlocal Model (2019) 1:36–44

modulus as the thickness exceeded 500 nm was observed inother studies as well [36, 37]. The change of elastic modulusin some of these studies was shown to be linearly increasing[37], bi-linear [36], and nonlinear [34, 35] as a function of filmthickness.

4 Conclusions

A new technique for extracting elastic moduli of ultra-thinfilms based on a combined experimental and computationalmethod is demonstrated. The stiff/hard material is depositedon a soft substrate with known material properties. Combinedbending/indentation deformation of the material system al-lows accurate AFM measurements and effective extractionof material properties via peridynamic theory. Unlike the con-ventional indentation experiments, testing of stiff films depos-ited on soft substrates is not limited by sample thickness andthat mechanical properties of ultra-thin films can be reliablycharacterized.

The force-deformation measurements of the bulk PDMSand PS thin-film specimens deposited on bulk PDMS sub-strates are compared against the PD simulations to extractmechanical properties of bulk PDMS and PS thin films. Theextracted values for PDMS and thin-film PS are consistentwith the previous research from the literature. Both exper-iments and simulations clearly indicate that deformationtype is indentation for soft PDMS substrates but it occursvia bending on relatively hard PS films deposited on softPDMS substrates.

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