Viscoelastic Characterization and Modeling of Polymer ... · PDF fileViscoelastic Characterization and Modeling of Polymer Transducers for Biological Applications ... and simulations

JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 18, NO. 5, OCTOBER 2009 1087

Viscoelastic Characterization and Modeling ofPolymer Transducers for Biological Applications

I-Kuan Lin, Member, IEEE, Member, ASME, Kuang-Shun Ou, Yen-Ming Liao, Yan Liu,Kuo-Shen Chen, and Xin Zhang, Member, IEEE

Abstract—Polydimethylsiloxane (PDMS) is an important poly-meric material widely used in bio-MEMS devices such as micro-pillar arrays for cellular mechanical force measurements. Theaccuracy of such a measurement relies on choosing an appropriatematerial constitutive model for converting the measured struc-tural deformations into corresponding reaction forces. However,although PDMS is a well-known viscoelastic material, many re-searchers in the past have treated it as a linear elastic material,which could result in errors of cellular traction force interpre-tation. In this paper, the mechanical properties of PDMS werecharacterized by using uniaxial compression, dynamic mechani-cal analysis, and nanoindentation tests, as well as finite elementanalysis (FEA). A generalized Maxwell model with the use of twoexponential terms was used to emulate the mechanical behavior ofPDMS at room temperature. After we found the viscoelastic con-stitutive law of PDMS, we used it to develop a more accurate modelfor converting deflection data to cellular traction forces. Moreover,in situ cellular traction force evolutions of cardiac myocytes weredemonstrated by using this new conversion model. The resultspresented by this paper are believed to be useful for biologists whoare interpreting similar physiological processes. [2008-0318]

Index Terms—Cellular traction force, finite-element analysis(FEA), polydimethylsiloxane (PDMS), transducer, viscoelasticity.

I. INTRODUCTION

THE MECHANICAL interaction between cells and theirneighboring extracellular matrix is believed to be of fun-

damental importance in various physiological processes suchas cell division, cell growth, apoptosis, and migration [1]–[4].The mechanical traction forces generated by living cells aretypically on the order of piconewtons to micronewtons andcannot be measured by using conventional force transducers[5], [6]. In recent efforts to measure the cellular traction forcesof living cells, researchers have developed a technique that usespolymer micropillar arrays as extracellular matrices that cells

Manuscript received December 24, 2008; revised May 6, 2009. Firstpublished September 11, 2009; current version published September 30, 2009.This work was supported in part by the National Science Foundation underGrants CMMI-0826191, CMMI-0239163 and CMMI-0700688 and in part bythe National Science Council of Taiwan under Grant NSC96-2628-E-006-006-MY3. Subject Editor S. M. Spearing.

I-K. Lin and X. Zhang are with the Department of Mechanical Engi-neering, Boston University, Boston, MA 02215 USA (e-mail: [email protected];[email protected]).

K.-S. Ou, Y.-M. Liao, and K.-S. Chen are with the Department ofMechanical Engineering, National Cheng Kung University, Tainan 70101,Taiwan (e-mail: [email protected]; [email protected];[email protected]).

Y. Liu is with Entegris, Inc., Burlington, MA 01803 USA (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JMEMS.2009.2029166

Fig. 1. (a) SEM image of a cell that was cultured on a PDMS micropillar arrayand (b) the schematic view of pillar bending during cell movement.

can grow on [7], [8], as shown in Fig. 1. Instead of measuringthe cell reaction forces directly, this approach measures thedeflection first and converts the deflections into reaction forcesusing appropriate mechanics. As a result, the micropillar arraymust exhibit high sensitivity in cellular traction force detectionas well as good biological compatibility. More specifically,these micropillars must have: 1) low stiffness allowing for themeasurement of mechanical forces of cells in the piconewtonsto micronewtons level; 2) they must be able to operate inbiological environments, such as cultivated liquid or normalsaline; and 3) they must have good biological compatibility, sothat cells can survive on the micropillar array for a long periodof time. To satisfy these requirements, the micropillars mustbe fabricated with nontoxic soft polymers [9]–[11]. The highlycompliant biocompatible polydimethylsiloxane (PDMS) meetsall the required material characteristics for cellular tractionforce measurements and is thus chosen for the fabrication ofmicropillar arrays. Once fabricated, cells are cultivated onto themicropillar array, and well-established microscopy techniquesare used to measure the deflection of each individual micropillaras a function of time [12].

The cellular interactions with the substrate have resultedin the deflection of the micropillars, which can be converted

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1088 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 18, NO. 5, OCTOBER 2009

into localized forces induced by the cells. Traditionally, theconversion from micropillar deflection to force requires knowl-edge of the bending stiffness of the micropillars, which iscalculated based on the elementary beam theory with theYoung’s modulus of PDMS assumed to be a constant (∼1 MPa)[8], [9]. However, applying this theory has some significantdisadvantages. Using the elementary beam theory for cellulartraction force calculations can cause significant errors becausethe PDMS micropillars fabricated by soft lithography are rel-atively short beams [13] that are located on a soft substrate.Thus, they do not satisfy the slender beam (aspect ratio > 10)and small deflection assumptions associated with the beamtheory. Therefore, shear deflection must be taken into ac-count [14], [15]. Moreover, PDMS is a viscoelastic material,and its Young’s modulus changes with the loading rate andthe elapsed time [16], [17]. Neglecting the loading rate- andtime-dependent characteristics of PDMS, as seen in previousresearch, can result in errors of cellular traction force mea-surements and erroneous interpretations of the measured data,which can lead to a misunderstanding of the cellular responsemechanisms. Hence, it is of vital importance to perform anaccurate material characterization on PDMS materials and todevelop a more appropriate model for converting micropillardeflections to cellular traction forces.

In order to address the aforementioned need, as mentionedin our previous work [18], we proposed a rational analysis/characterization flow to perform the viscoelastic characteriza-tion and finite element analysis (FEA) of PDMS micropillarsto develop a more accurate deflection to force conversion forcellular force measurements. Following the same scope, thedetailed design, experiments, and simulations of the PDMS ma-terial characterization and modeling were further discussed inthis paper; a more in-depth and physically accurate conversionmodel for polymer transducer applications was consequentlyintroduced, and an application for calculating the cellular forceof cardiomyocytes was also presented.

Our contribution includes integrating the results from recentresearch and sophisticated mechanics to form a comprehensivemethodology for the viscoelastic characterization, modeling,and analysis of cellular traction forces exerted on PDMS mi-cropillars. A research flow chart of this paper is shown inFig. 2. Section II describes the three kinds of PDMS specimenfabrications for macroscale PDMS viscoelastic characteriza-tion, nanoindentation stress-relaxation tests, and the bendingcharacterization of PDMS micropillars. Section III covers howthe mechanical characterization of the viscoelastic PDMS wasprobed macroscopically using conventional punch tests anddynamic mechanical analysis (DMA). The results were imple-mented into a viscoelastic FEA model developed in tandem, andthen validated on a microscale with a nanoindentation stress-relaxation test, which is explained in Section VI. In Section V,the validated viscoelastic FEA model was applied to analyzethe micropillar bending behavior and was compared with ex-perimental results. The validated viscoelastic FEA model wasultimately utilized to determine the mechanical forces of livingcells from the measured deflection of PDMS micropillar arrays;covered in Section IV. This paper entails a balance betweenexperimental characterization of the viscoelastic material and

Fig. 2. Overall research structure and process.

Fig. 3. PDMS specimens in form of (I.) bulk, (II.) film, and (III.) micropillarswere fabricated for punch, nanoindentation, and micropillar bending charac-terization, respectively. The schematic fabrication flow of PDMS specimens.(a) Prepare different molds. (b) Pour mixed PDMS with curing agent intomolds. (c) Remove air bubble in PDMS by using vacuum system and curingat 65 ◦C for 90 min. (d) Remove PDMS specimens from molds.

the development of a viscoelastic FEA model for cellulartraction force measurements.

II. SPECIMEN PREPARATION AND FABRICATION

PDMS is a silicone-based organic polymer that can be ther-mally cured through a crosslinking reaction, and its glass tran-sition temperature is about −123 ◦C [15], [19]. Because PDMScannot be patterned by photolithography, the PDMS speci-mens were fabricated by soft lithography techniques which canduplicate inverted structures from a rigid mold [20]. In thispaper, three fabricated forms of PDMS material: bulk, films,and micropillar arrays, are described using the processes shownin Fig. 3. The bulk and film specimens were then used for punchand DMA tests described in Section III, and nanoindentationcharacterization in Section VI, respectively. The micropillararrays were characterized using specifically designed nano-indentation techniques to perform bending force experiments


LIN et al.: VISCOELASTIC CHARACTERIZATION AND MODELING OF POLYMER TRANSDUCERS 1089

on individual micropillars, shown in Section V. In order tofabricate three types of PDMS specimens, three different moldscorresponding to three types of PDMS specimens were used:1) a 30 × 30 × 3-mm3 metallic mold for the bulk specimen;2) a 25 × 75-mm2 glass slide for the film specimen; and 3) apatterned 40-μm-thick SU-8-50 mold (MicroChem SU-8-50)with 20 × 20-μm2 holes, as shown in Fig. 3(a). In the SU-8-50mode of fabrication, the SU-8-50 molds were fabricated on a〈100〉 silicon wafer which was cleaned with 40% HF to strip offthe native oxide. The 40-μm SU-8-50 photoresist was coatedon the wafer at 3000 r/min followed by a 65 ◦C prebake anda 95 ◦C soft bake in an oven for 5 and 15 min, respectively.Subsequently, the photoresist-coated wafer was exposed usingmask aligner (Electronic Visions Model EV620) for 18 s with alight intensity of 10 mW/cm2 and then postbaked at 95 ◦C for5 min. The unexposed SU-8-50 was removed from the waferin SU-8 developer and then the patterned SU-8-50 was bakedat 150 ◦C in the oven. Subsequently, the SU-8-50 molds wereformed with deep holes and ready for the following PDMSmolding. These PDMS specimens were obtained by mixingPDMS prepolymer (Sylgard 184, Dow Corning) and a curingagent in a ratio of 10 : 1. The mixed PDMS prepolymer waspoured into different molds [see Fig. 3(b)] and placed in avacuum chamber for 5 min to remove residual air bubbles. ThePDMS specimens were then cured at 65 ◦C for 90 min, asshown in Fig. 3(c). Finally, the specimens were removed fromtheir molds, completing the fabrication process, as shown inFig. 3(d).

III. MACROSCALE PDMS VISCOELASTIC

CHARACTERIZATION

A. Experiment

The mechanical behavior of the macroscale PDMS wascharacterized by standard techniques, namely, punch and DMAtests. The goal of these characterizations was to establish a con-stitutive law for PDMS. Although these testing methods are lessinnovative, the standard procedures will generate more credibledata, which are important for the subsequent constitutive lawdevelopment.

The punch test was conducted to assess the role of loading ef-fects in determining the deformation in bulk PDMS specimens.More specifically, the punch test measures the stiffness of bulkPDMS in various situations such as a quasi-static compressionor stress-relaxation test. A self-developed punch test system,shown in Fig. 4(a), was used to measure the load-displacementrelationship. The system operated with a 6-mm-diameter flatpunch tip at room temperature. During a typical experiment, thebulk PDMS specimen was placed on the Z-axis stage and thestepper motor moved the specimen up, approaching the punchtip. Once the bulk PDMS was in contact with the punch tip,a laser displacement sensor (05MGV 80, Wenglor) measuredthe displacement of the Z-stage with a 5-μm resolution whilea load cell (ELFS-B3-100N, Entran) measured the force with312.8 mV/100 N sensitivity. The force–displacement data werecontinuously monitored and recorded with a data acquisitionsystem for analysis.

Fig. 4. Macroscale PDMS characterization systems. (a) Punch test system.(b) DMA system.

On the other hand, in order to characterize the frequency-dependent material properties of PDMS, a self-developed DMAsystem was used. As shown in Fig. 4(b), during a typicalDMA experiment, a bulk PDMS specimen was placed on theZ-stage and brought in contact with a 6-mm-diameter flatpunch indenter tip, which was connected to a piezoelectricstack (PA100/12, Piezosystem Jena) with a maximum load andmaximum actuated distance of 270 N and 100 μm, respectively.The piezoelectric stack was then actuated by a sinusoidal signal.As the specimen was dynamically loaded, a capacitance sensor(ASP-10-CTA, MTI Instruments) and amplifier (Accumeasure9000, MTI Instruments), with an accuracy and measurementrange of 0.254–254 μm, and a load cell (208B01, PCB) withsensitivity of 122.1 mV/N measured the displacement andforce, respectively.

Both the punch and DMA tests were performed several timesto obtain meaningful statistical results. Two typical experimen-tal data sets for the punch test are shown in Fig. 5. In Fig. 5(a),the loading-deformation characteristics of the specimen areshown and the data can be further used to obtain the Young’smodulus. On the other hand, a typical stress-relaxation testingcurve is shown in Fig. 5(b) and a strong viscoelastic phenom-enon is evident. Additionally, the viscoelasticity of PDMS wasalso characterized using the DMA shown in Fig. 4(b), and thecorresponding storage and loss moduli with respect to loadingfrequency is shown in Fig. 5(c). In parallel, finite elementmodels were also established. In conjunction with the test datashown above, these models were used to extract the materialproperties of PDMS and are addressed below.

B. FEA Model of Macroscale PDMS Testing

The material characterization described above can onlyprovide information on extensive properties of viscoelastic



Fig. 5. Experimental and simulation results of (a) a quasi-static load-deflection relation, (b) a time-domain stress-relaxation test using punch testingsystem, from which the time relaxation constant of PDMS was obtained, and(c) the PDMS storage and loss moduli as a function of loading frequency fromstandard frequency-domain tests using DMA system.

materials, namely, the reaction loads and the applieddisplacements. The information can only be treated as theresponses of a structure. In order to find the constitutive law,viscoelastic FEA must be conducted. By using the controlledinputs applied during the experiment, the analysis aims to fitthe experimental response data by tuning the parameters of theadapted constitutive laws in the associated viscoelastic FEA

Fig. 6. FEA of the macroscale punch test. (a) Typical finite element mesh.(b) Schematic view of punch test. (c) Convergence study of punch testsimulation.

model. In this paper, the viscoelastic FEA model with the samegeometry as the specimens was constructed. A Prony seriesfor generalized Maxwell model with two exponential termswas selected as the constitutive law because of its generalityand relative ease of fitting. The general-purpose finite elementpackage ABAQUS v.6.4 was used to perform the work.

As shown in Fig. 6(a), the model consists of 9261 nodesand 3000 CAX8 axisymmetric elements [21] for a 3-mm-thickPDMS. The rigid punch was represented by a surface constrainmotion of the material in contact with the punch tip. Theboundary conditions are also shown in Fig. 6(b). From theconvergence study shown in Fig. 6(c), the percent error ofthe FEA results from different element numbers as comparedwith 30 × 100 elements can be express as

Error(%) =FEAa×b − FEA30×100

FEA30×100× 100%. (1)

When the mesh is more than 2000 elements (i.e., 20 × 100),the error of FEA results between different mesh is smaller than



0.15%. Hence, the FEA model with 3000 elements (i.e., 30 ×100) in the punch test simulation provides a converged result.By utilizing the equation from the punch test, developed fromcontact mechanics analysis, the plane strain Young’s modulusE∗ is obtained by [18], [22]

E∗ =E

1 − ν2=

P

hD(2)

where P , h, and D are the load, PDMS bulk thickness, andpunch diameter, respectively. By applying (2) to the loadingportion of the load-deflection curve in Fig. 5(a), it is estimatedthat the Young’s modulus of the bulk PDMS is 1.45 MPa. Thelinear elastic FEA result with a Young’s modulus of 1.45 MPawas also shown in Fig. 5(a). This result would be used for thesubsequent simple linear analysis.

Next, the viscoelastic properties of PDMS were modeled.The shear relaxation function G(t) is given in (3), with gR(t)as a Prony series shown in (4) [21], [23]. This equation isapplicable for any linear viscoelastic solids as long as anappropriate number of terms is used. In this paper, two terms areused. From the experimental data shown in Fig. 5(b), the shearrelaxation modulus (G(t)) can be obtained. The viscoelasticfunctions appear appropriate in the time domain over a veryshort period of time; however, the complex modulus cannot beobtained. Therefore, DMA needs to be conducted to derive thecomplex modulus [17], [18], [24]. Thus, the Young’s relaxationmodulus (E(t)) can be determined from (5) under the conditionof a constant Poisson’s ratio

G(t) = gR(t)G0 (3)

gR(t) = 1 −N∑

i=1

gi[1 − e−tτi ] (4)

G(t) =E(t)

2(1 + ν). (5)

In (4), gi and τi are material parameters. Following thisprocedure, the constitutive law of PDMS was obtained andcould be employed in the viscoelastic FEA model. The testdata and the viscoelastic FEA results, found by using a five-parameter linear solid model, are all shown in Fig. 5(b). Byapplying a punch displacement of 392.5 μm and recording thetotal reaction force, the simulation data can be obtained. Thefitting between the experimental data and the associated simu-lation can be achieved by varying the parameters incorporatedin the model. The results indicate that the simulation resultsessentially agree with the test data. By using such a Prony seriesapproach, the viscoelastic behavior at room temperature of thePDMS can be modeled as

G(t) = 0.455(1 −

[0.08(1 − e−t/0.165)

+ 0.03(1 − e−t/5)])

(MPa) (6)

where G(t) is the shear modulus. Notice that (6) is only validwithin the time frame considered. Since the Poisson’s ratio ofpolymers can be assumed to be approximately 0.5 [19], theE(t) is therefore three times that of G(t), as shown in (5). The

Fig. 7. Comparison between the experimental and simulated stiffness,graphed in the frequency domain.

obtained viscoelasticity constitutive law would then serve as thestandard material model for converting micropillar deflectioninto cellular traction force, to be done later. However, beforethis model can be applicable for microscale applications, it mustbe further validated by a nanoindentation test, which will beaddressed in Section VI.

The complex modulus of PDMS was also characterized.In viscoelastic FEA model [using the frequency-domain vis-coelasticity model presented in (6)], the DMA tip was modeledas a rigid cylindrical body that indents the bulk PDMS atdifferent loading frequencies and the PDMS was modeled aseight-noded viscoelastic elements. By using the viscoelasticFEA model and applying a controlled sinusoidal displacementwith different frequencies, the frequency response of the PDMSstructure’s stiffness can be determined. From the stiffness inthe frequency domain, shown in Fig. 7, it can be seen that thesimulation results essentially agree with the experimental data.

IV. MICROSCALE PDMS NANOINDENTATION

CHARACTERIZATION

A. Testing System and Methodology

The constitutive law of PDMS obtained using the macroscalestandard testing procedure was further examined by a nanoin-dentation relaxation test, shown schematically in Fig. 8(a).By such a test, it is possible to validate and modify theobtained constitutive relations on a microscale level becauseboth the deformation and reaction force were obtainable andcontrollable. The PDMS specimen used in the nanoindentationstress-relaxation characterization has dimensions of 25 × 75 ×1 mm3. The stress-relaxation experiments were performed us-ing a TriboIndenter (Hystrion, Inc.) [25]. In order to reduce thesnap-to-contact effect, the system is operated under a displace-ment feedback control mode using a conical diamond tip with aradius of 20 μm [26]. The PDMS specimen was mounted on asteel stub and placed on the magnetic stage of the TriboIndenter.The TriboIndenter moved the tip in contact with the specimenand then withdrew the tip to a distance sufficient to ensure thatthere was not any force or contact before the stress-relaxation



Fig. 8. (a) Schematic view of nanoindentation on PDMS film and (b) experi-mental and simulation results from the nanoindentation tests.

test was performed. The displacement loading function wasapplied to the transducer of the nanoindenter in three stages:1) loading to 1000 nm at a constant rate; 2) holding the settingdisplacement for 10 s; and 3) unloading at a constant rate, whichis the same as the loading segment. Displacement loading ratesof 100 nm/s were used in this paper. In Fig. 8(b), it is shownthat the experimental results exhibit strong scattering due to theimperfection of the specimen surface [27], [28]. Nevertheless,the general tendency is clear, i.e., a decrease in reaction force isobserved, which indicates that a possible stress relaxation alsooccurs. However, nanoindentation is a triaxial process, and thedetailed material behavior model cannot be easily synthesizedby the experimental data. As a result, a corresponding finiteelement simulation should also be performed.

B. Viscoelastic FEA Model for NanoindentationStress Relaxation

As stated in previous sections, an associated finite elementmodel should be constructed to simulate a response of nanoin-dentation stress relaxation on PDMS film. By using the materialmodel obtained from the macroscale testing and the testing con-ditions described above, the correlation between the simulationresult and the test data can be used to validate or modify the

Fig. 9. (a) Finite-element mesh of nanoindentation on PDMS film with 20-µmconical tip and (b) an enlarged view of spherical end.

material models. As shown in Fig. 9, the specimen was modeledby 5714 axisymmetric CAX8 elements using 14536 nodes andthe conical indenter tip is modeled by R2X2 rigid elements[21]. The mesh density is determined based on the conclusionof our previous study [29], i.e., at least 20 to 30 elements shouldbe in contact during indentation for viscoelastic materials. Theconstitutive relation obtained from the previous section [i.e.,(6)] was incorporated into the finite element. The displacementof the indenter was linearly increased between t = 0 to 10 swith a ramp rate of 100 μm/s until the loading reached itsmaximum. This maximum displacement was held constant toperform the stress-relaxation simulation.

Fig. 8(b) shows the comparison between the viscoelasticFEA prediction and the experimental results. It can be seenthat the tendencies of both sets of data essentially agree witheach other, further validating the accuracy of the viscoelasticFEA model of the PDMS. Notice that, quantitatively, there arestill certain discrepancies between model and test data, but thiscan be further improved by using a more sophisticated model.Nevertheless, in this paper, we have chosen to concentrate onthe qualitative characterization and have left detail modelingfor future work.

V. MICROSCALE PDMS MICROPILLAR

BENDING CHARACTERIZATION

After the validation of the viscoelastic FEA model withthe constitutive law of PDMS by the nanoindentation testingdescribed in Section VI, the deformation behavior of PDMS mi-cropillars subjected to bending forces were further studied forfuture biological applications. For validation purposes, in thispaper, the forces exerted on a micropillar array were measured



Fig. 10. 20 × 20 × 40-µm3 micropillars were used to characterize the bend-ing behavior of PDMS microstructures.

Fig. 11. Experimental and simulation results of PDMS micropillarbending test.

and compared with the viscoelastic FEA results before utilizingthe model for cellular traction force measurements.

A. Testing System and Methodology

A nanoindenter with a Berkovich tip was used to performthe bending test to characterize the mechanical behavior of20 × 20 × 40-μm3 micropillars (Fig. 10). A typical bendingdisplacement, loading/unloading rate, and stress-relaxation pe-riod were 4 μm, 1.33 μm/s, and 30 s, respectively; experimentaldata are shown in Fig. 11. Due to possible faulty control of thetesting environment, the test data exhibit strong scattering inthe holding period. Nevertheless, the tendency is clear, i.e., thereaction force initially increases with the displacement. Duringthe constant displacement phase, the reaction force decreaseswith time—indicating a strong stress-relaxation behavior.

Notice that the deformation of the experimental data actuallycontains at least three different deformations: First, the bendingof micropillars (X1); second, the deformation of soft substrate(X2); and finally, penetration under indentation (X3), as shownin Fig. 12. It is important to distinguish each individual’s contri-

Fig. 12. Three major deformation behaviors during micropillar bending,including (a) micropillar bending, X1, (b) substrate deformation, X2, and(c) penetration under indentation, X3.

bution. As a result, the corresponding FEA, in conjunction withthe experimental work, is conducted and presented below.

B. Viscoelastic FEA Model for PDMS Micropillar Bending

Again, the corresponding finite element models were con-structed, and the aforementioned experimental data were usedfor comparison and validation. Notice that, in actuality, thenanoindentation micropillar bending process involves the threemajor deformation behaviors, including micropillar bending(X1), substrate deformation (X2), and penetration under inden-tation (X3), addressed above in Section V-A. The experimentaldeformation data are actually the summation of these threecontributions. However, it is too complicated to conduct a faith-ful 3-D contact analysis. Nevertheless, because the force actedupon the specimen is the same, the entire process can be treatedas three springs connected in series and the total deformation isthe lumped sum of these three major deformation behaviors. Asa result, two simplified finite element models were adapted tosimulate the micropillar bending behavior.

As shown in Fig. 13(a), the model is a simple beam modelwith a relatively concentration load and it aims to find theforce. Consistent with the experimental setup, the concentratedload was not applied directly at the cantilever end, as shown inFig. 13(a). This model uses a 2-D planar strain approximationof 6000 quadratic elements (18 654 nodes) [21]. Another modelis actually the same as the model used in Section VI but withBerkovich tip, as shown in Fig. 13(b). The constitutive lawobtained using macroscale standard test procedures is againused in this model.

During the micropillar bending test, the soft substrate de-formation and nanoindenter tip penetration may affect theaccuracy of micropillar stiffness as we mentioned before.Fig. 13(c) shows the total load-deflection behavior of the mi-cropillar subjected to the bending force. Most of the defor-mation comes from the micropillar bending (54.3%), and thecontributions from substrate deformation and nanoindenter tippenetration are approximately 11.5% and 34.2% of the total



Fig. 13. (a) Finite element mesh of PDMS micropillar, (b) the finite elementmesh of nanoindentation on PDMS film with Berkovich tip, and (c) thesimulation result of a PDMS micropillar load-deflection behavior subjected toa bending force.

deformation, respectively. The tip penetration could be reducedusing blunt tips such as 20-μm conical tip. Fig. 11 showsthe comparison between the micropillar bending test data andthe associated viscoelastic FEA modeling. Due to the exper-imental data scattering in the holding period, the comparison

Fig. 14. Typical force–displacement relationship of a PDMS micropillarshown by: 1) the elementary beam theory; 2) the elastic FEA model for shortbeam; and 3) the viscoelastic FEA model with different loading rate.

cannot be performed quantitatively. Nevertheless, qualitatively,the simulation results essentially agree with those performedexperimentally. Further improvement of the scattering wouldbe important for future work. Upon the validation in Fig. 11,the bending behavior of micropillars associated with cellularforce measurement experiment only includes pure bending andsubstrate deformation, and can be derived from the FEA byremoving the tip penetration, as shown in the Fig. 14.

One important issue to be addressed here is the accuracyof the different models used to extract the applied loads.Traditionally, the cellular traction forces are usually obtainedby converting micropillar deflection data using elementarybeam theory. However, as shown in Fig. 14, the beam theorytends to overestimate the reaction force by 200%, potentiallyresulting in considerable error [8], [9]. In addition, note thatthe viscoelastic FEA and the elastic short beam FEA stronglyagree with each other for the slow loading rate used. However,at higher loading rates, viscoelastic effects will cause theviscoelastic FEA to diverge from the elastic FEA result [18],[30]. The stiffness of the PDMS micropillars increases with theloading rate due to the inherent viscoelasticity considered in ourmodel. When increasing the loading rate by 50 times, the stiff-ness is increased by 8.1% (Fig. 14). This loading rate-dependentresponse, nonexistent in the elastic FEA, is important in cellulartraction force analysis where high loading rates exist. Forexample, the mechanical forces generated by muscle cells haveworking frequencies that fall in the viscoelastic regime of thePDMS microcantilevers. For many biological measurements,the loading rate is slow and the difference between theviscoelastic FEA model and elastic FEA model may not besignificant. However, some cells contract or beat quickly (e.g.,cardiac myocytes) [9], [31], and therefore cellular tractionforce measurement of these cells can be improved by usingour viscoelastic model. In addition, PDMS force transducersmay be used in other fields, which can experience loading ratesgreater than cardiac myocytes. In such applications, without



Fig. 15. Dimensionless form of force–displacement relationships for a PDMSmicropillar shown by: 1) the elementary beam theory; 2) the elastic FEA modelfor short beams; and 3) the viscoelastic FEA model with different loading rates.

understanding the viscoelastic nature of PDMS, the forcetransducers may yield significant errors. We believe that theresults of this paper are not restricted to cellular traction forcemeasurement but benefit other applications as well.

C. Dimensionless Form of Force–Displacement Relationshipfor a PDMS Micropillar

A more general plot based on the simulation results usingdimensionless form is shown in Fig. 15. The dimensionlessdisplacement (X∗) and loading (P ∗) can be expressed as

X∗ = X/L (7)

P ∗ = P/P0 (8)

where L is the micropillar length and P0 is the cellular tractionforce calculated by the elementary beam theory. Notice thatthere are two thresholds of the viscoelastic FEA model shownin Fig. 15. When we applied an extremely fast loading rate(200 L/s) as a step loading function on a PDMS micropil-lar, we found that there was an upper threshold where theforce–displacement relationship was the greatest. When weapplied an extremely slow loading rate (0.005 L/s) as a quasi-static loading function on a PDMS micropillar, we found alower threshold where the force–displacement relationship wasthe smallest.

The percent error as a function of dimensionless displace-ment (X/L) and loading rate is shown in Fig. 15. We graph andcompare: 1) the elastic FEA with the elementary beam theory;2) the viscoelastic FEA with elementary beam theory; and3) the viscoelastic FEA with the elastic FEA.

1) Elastic FEA Versus the Elementary Beam Theory: PDMSmicropillars are relatively short beams which do not satisfythe slender beam and small deflection assumptions associatedwith the elementary beam theory. Xiang and LaVan [15] haveinvestigated the problem using the elastic FEA. From Fig. 15,the percent error as a function of dimensionless displacement

between the elastic FEA (short beam) and the elementary beamtheory can be expressed as

percent error =

∣∣∣∣∣

(PP∗

)Elastic FEA

− (PP∗

)Beam Theory(

PP∗

)Elastic FEA

∣∣∣∣∣×100%. (9)

As shown in Fig. 16(a), the percent error is about 43% whena micropillar has 0.05 dimensionless displacement; such errorincreases with the dimensionless displacement. The elastic FEAassumes that PDMS is an elastic material, and the stiffness ofthe PDMS micropillars does not change with the loading rate;hence, there is only one curve shown in Fig. 16(a).

2) Viscoelastic FEA Versus the Elementary Beam Theory:Moreover, PDMS is a viscoelastic material, and its Young’smodulus changes with loading rates and the elapsed time. Ne-glecting the time- and loading rate-dependent nature of PDMScould result in errors in data interpretation. Again, in Fig. 15,the percent error as a function of dimensionless displacementbetween the elastic FEA and the elementary beam theory canbe expressed as

percent error=

∣∣∣∣∣

(PP∗

)Viscoelastic FEA

−(PP∗

)Beam Theory(

PP∗

)Viscoelastic FEA

∣∣∣∣∣×100%. (10)

As shown in Fig. 16(b), the percent error between the vis-coelastic FEA and the elementary beam theory increases withdisplacement. Although this relationship is similar to the oneshown in Fig. 16(a), our viscoelastic FEA shows the loadingrate dependent characteristic of PDMS, which is nonexistentin the elastic FEA. As shown in Fig. 16(b), we have plottedmultiple error curves in different loading rates. For example,when the micropillar has 0.05 dimensionless displacement[Fig. 16(b)], the percent error between the viscoelastic FEA andthe elementary beam theory is in the region of 28% to 44%,depending on the loading rate.

3) Viscoelastic FEA Versus the Elastic FEA: AlthoughXiang and LaVan [15] have studied the short beam problem,they have assumed the PDMS is an elastic material withoutloading rate dependent characteristic. Inspired by their work,we further compared the viscoelastic and the elastic FEA results[Fig. 16(c)]. As shown before (in Fig. 15), the percent error asa function of dimensionless displacement between the elasticFEA and the elementary beam theory can be expressed as

percent error =

∣∣∣∣∣(

PP∗

)Viscoelastic FEA

− (PP∗

)elastic FEA(

PP∗

)Viscoelastic FEA

∣∣∣∣∣×100%. (11)

When we applied an extremely slow loading rate (0.005 L/s)as a quasi-static loading function on the PDMS micropillars, theresult was the same as the elastic FEA because the latter usesthe Young’s modulus from the quasi-static punch tests. Whenwe compared the elastic FEA with the viscoelastic FEA, whichunderwent a step loading function, the error increased up to11%. The percent error between the viscoelastic FEA and the



Fig. 16. Percent error as a function of dimensionless displacement (a) be-tween the elastic FEA and the elementary beam theory, (b) between the vis-coleastic FEA and the elementary beam theory, and (c) between the viscoelasticFEA and the elastic FEA.

elastic FEA is in the region of 0% to 11%, depending on theloading rate. This percent error is found when the micropillarhas a dimensionless displacement of 0.05, and it increases withthe further dimensionless displacement.

Fig. 17. Displacement and corresponding traction forces of an individualpillar underlying the subject cardiac myocyte which were measured (a) beforeand (b) after the ISO perfusion.

VI. DISCUSSION

A. Contributions

In this paper, a rational flow for characterizing the materialbehavior of PDMS to achieve better cellular traction forcemeasurements is presented. By using the standard macroscalepunch tests, a preliminary general Maxwell fluid model wasobtained. By using nanoindentation stress-relaxation and bend-ing tests, in conjunction with viscoelastic FEA, this model wasproved to be suitable for predicting the behavior of PDMSmicropillars. The contribution of this paper can be itemized asthe following.

First, to the best of our knowledge, this paper is the firstto introduce the material viscoelasticity behavior in cellulartraction force measurements. The presence of viscoelasticitymakes the measurement process loading rate or time dependent.The impact of these effects on the accuracy of cellular tractionforce measurements has never been discussed.

Second, the micropillars, in general, are too short with re-spect to their width, and they cannot be accurately representedby simple cantilever beams. This paper considers both geomet-rical nonlinearity and viscoelasticity and works out a correctionchart for converting the measured deflections into appropriatebending forces. With a more realistic conversion, it is possibleto provide biologists with a more accurate understanding of cel-lular interaction, which may lead to a more precise cell responsemechanism. For example, the cellular traction force response ofa cardiac myocyte in situ was monitored from the measureddeflection during the isoproterenol (ISO) perfusion [31], asshown in Fig. 17. After about 300 s from the beginning of the



perfusion, the beating frequency of cardiac myocyte started todrop from 3.68 to 1.82 Hz. Hence, the stiffness also changeddue to the inherent viscoelasticity of the PDMS micropillar,which can be found in our conversion model. It can be seenthat the elementary beam theory significantly overestimates thecell force by approximately 63%. A huge difference such asthis may lead biologists to misunderstand the cellular responsemechanisms.

B. Future Work

Although the results showed the feasibility and importance ofthis paper, further material characterization and sophisticatedmodeling are necessary to improve the constitutive law andthe displacement–force conversion. For example, most of thebiological experiments, including cellular force measurements,are conducted at human body temperature [31], [32]; hence, thematerial characterization at 37 ◦C is desired. The nanoinden-tation could also be employed to simultaneously measure thebulk and shear relaxation functions without assuming constantPoisson’s ratio, in order to obtain more accurate materialproperties [33]. The environmental and experimental controlsshould be further improved to reduce the large scatter in exper-imental data, likely due to the PDMS nature and as-preparedsurface roughness of the specimens [27], [28]. Even so, theobserved trend is clear. With such an improvement, a more com-plicated viscoelasticity model may be incorporated to providea better correlation between model prediction and experimentalverification.

It is also important to point out that as the scale reducesfurther, there are other surface forces, such as van der Waalsforces, surface tension, and possible electrostatic forces, thatcould affect the deflection. Therefore, it could be challenging tocompletely distinguish the cell force from a global interactionforce. Additionally, since cells are very soft materials, a moreadvanced model with the consideration of adhesion, such asJohnson–Kendall–Roberts–Sperling, may be needed to modelthe contact between cells and the PDMS micropillars. Lastly,in the cellular force measurement experiment, as shown inFig. 1, the micropillars may be too close together, causing thereto be lateral contact between adjacent pillars. Thus, it couldsignificantly change the force–displacement relationship fromthe case where a single micropillar deforming with no contactwith other micropillars was considered. In the future, a morecomprehensive FEA model should be established to calculatethe force–displacement relationship for touching micropillars.Furthermore, a new device should be designed with largerspacing between pillars to avoid pillar interaction.

VII. SUMMARY AND CONCLUSION

The understanding of the PDMS micropillar bending behav-ior is critical to the acquisition of the cellular traction force. Inbasic cellular biological research, the viscoelasticity propertiesof the PDMS structures are often neglected, for example, whenusing micropillars to measure the cellular traction forces. It istypically assumed that the PDMS micropillars are elastic beamswith a constant Young’s modulus. However, this assumption

is highly questionable, due to the inherent viscoelasticty (andhyperelasticity) of PDMS. In this paper, a comprehensive vis-coelastic characterization is demonstrated, providing a morein-depth and physically accurate conversion model for forcemeasurement applications.

ACKNOWLEDGMENT

The authors would like to thank the Photonics Center, BostonUniversity, for all of the technical support throughout the courseof this paper, and members at the Laboratory for MicrosystemsTechnology, Boston University, and System Dynamics Labora-tory for Mechatronics and Microsystems, National Cheng KungUniversity, particularly B. Hansen, A. Bonhomme-Isaiah, andE. Frohlich for PDMS specimens fabrication at the PhotonicCenter, Boston University.

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I-Kuan Lin (M’08) received the B.S. degree in me-chanical and electrical engineering (double major)from Da-Yeh University, Changhua, Taiwan, in 2002and the M.S. degree in mechanical engineering fromNational Cheng Kung University (NCKU), Tainan,Taiwan, in 2004. He is currently working towardthe Ph.D. degree in the Department of MechanicalEngineering, Boston University, Boston, MA.

He was a Research Assistant with the SystemDynamics Laboratory for Mechatronics and Mi-crosystems, Department of Mechanical Engineering,

NCKU, and the Advanced Lithography Division, National Nano Device Lab-oratory, during his master’s study. From 2004 to 2006, he was a ResearchAssistant with the Institute of Physics, Academia Sinica, Taipei, where heworked on fabrication of 25-nm X-ray phase zone plates using e-beam litho-graphy and electroplating. His Ph.D. research includes mechanical analysisand material characterization of polymer MEMS and amorphous thin film,and thermomechanical behavior of multilayer microcantilever structures withnanocoating.

Mr. Lin is a member of the American Society of Mechanical Engineers(ASME).

Kuang-Shun Ou was born in Penghu, Taiwan, in1979. He received the B.S. degree in mechanicalengineering from the Chinese Culture University,Taipei, Taiwan, in 2001, and the M.S. and Ph.D.degrees in mechanical engineering from NationalCheng Kung University (NCKU), Tainan, Taiwan, in2003 and 2008, respectively.

He is currently a Postdoctoral Associate in theDepartment of Mechanical Engineering, NCKU. Hisresearch interests include solid mechanics, vibration,and MEMS reliability assessment.

Yen-Ming Liao was born in Taipei, Taiwan, in 1983.He received the B.S. degree in civil engineeringand the M.S. degree in mechanical engineering fromNational Cheng Kung University, Tainan, Taiwan, in2006 and 2008, respectively.

He is currently with the Department of MechanicalEngineering, National Cheng Kung University. Hisresearch interests include mechanical and materialcharacterization, and finite-element simulation.

Yan Liu received the Ph.D. degree in metallurgical engineering from theUniversity of Missouri, Rolla, in 2004.

Between 2005 and 2007, he conducted two postdoctoral appointments withBall State University, Muncie, IN, and Boston University, Boston, MA. SinceJanuary 2008, he has been a Materials Scientist with Entegris, Inc., Burlington,MA. His research interests include thin-film deposition and characterization,nanomaterials synthesis and devices development, microelectromechanical sys-tems, and microscopic and spectroscopic techniques.

Kuo-Shen Chen received the B.S. degree (summacum laude) in power mechanical engineering fromNational Tsing Hua University, Hsinchu, Taiwan, in1989, and the M.S. and Ph.D. degrees in mechani-cal engineering from the Massachusetts Institute ofTechnology (MIT), Cambridge, in 1995 and 1999,respectively.

From 1998 to 1999, he was a Postdoctoral As-sociate with the Active Materials and StructuresLaboratory, Department of Aeronautics and Astro-nautics, MIT, where he worked on developing the

overall system dynamics of a piezo-based microenergy harvester. Since August1999, he has been with the Department of Mechanical Engineering, NationalCheng Kung University, Tainan, Taiwan, where he is currently a Professor.His research interests are in mechanics, materials science, and mechatronics,including mechanical analysis and material characterization of MEMS, systemdynamics and control of mechatronics and MEMS, and vibration control ofsmart structures.



Xin Zhang (M’97) received the Ph.D. degree inmechanical engineering from Hong Kong Universityof Science and Technology, Hong Kong, in 1998.

From 1998 to 2001, she was a PostdoctoralResearcher and then a Research Scientist at theMassachusetts Institute of Technology, Cambridge,where she worked on the “MicroEngine” project,applying microfabrication technology to develop mi-crogas turbine engines on a chip. She is currently anAssociate Professor in the Department of Mechan-ical Engineering and Division of Materials Science

and Engineering, Boston University (BU), Boston, MA. She has been withthe BU faculty since January 2002. In 2006, she was promoted to AssociateProfessor with tenure, and in 2008, she was appointed as Associate Chair forgraduate programs in the Department of Mechanical Engineering.

Dr. Zhang was the recipient of the Boston University SPRInG Award in 2002,a National Science Foundation Faculty CAREER Award in 2003, a BostonUniversity Technology Development Award in 2004, and Boston UniversityDean’s Catalyst Award in 2009. In 2007, she became an Invitee of the NationalAcademy of Engineering (recognized as one of the top engineers in the countrybetween the ages of 30–45). In 2008, she was named the inaugural Distin-guished Faculty Fellow, a five-year appointment given to a tenured College ofEngineering faculty member who is on a clear trajectory toward an exemplaryleadership career in all dimensions of science and engineering.


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